SAMO Junior 2015 to 1998 Questions

Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION

2015 FIRST ROUND JUNIOR SECTION: GRADE 8 12 March 2015

Time: 60 minutes

Number of questions: 20

Instructions 1. This is a multiple choice choi ce question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. Scoring rules: 2.1. Each correct answer is worth 5 marks. 2.2. There is no penalty for an incorrect answer or any unanswered question. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments instruments are not permitted. permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. The centre page is an information and formula sheet. Please tear out the page for your own use. 7. Start when the invigilator tells you to do so. 8. Answers and solutions solutions will be available at www.samf.ac.za

Do not turn the page until you are told to do so. Draai die boekie om vir die Afrikaanse vraestel

PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9372 Email: [email protected]

Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns

Compiled by and downloaded from www.erudits.com.ng

1.

The value of 2 – (0 (0 – (1 (1 – 5)) is (A) 3

2.

(B) 1

If

12

x for x for x??

(B) 120

If

20 3

(A)

5.

(B) 5

120 

y

(E) 120 000

(C) 6

(D)

7

(E) 8

9

(B)

12

(C)

15

(D)

18

(E)

20

0

(B)

1

(C)

2

(D)

4

(E)

5

If today is Thursday, what day of the week will it be in 150 days from now? (B) Monday

(C) Tuesday

(D) Wednesday (E) Thursday

It is said that you can only fold a piece of paper in half 7 times. Harold folds a sheet of paper in half 5 times and then makes a hole in the folded paper. How many holes does the sheet of paper have after it is unfolded again? (A)

8.

(D) 12 000

then y then y equals equals

(A) Sunday

7.

(C) 1 200

The last digit of 2011 × 2013 × 2015 – 2010 2010 × 2012 × 2014 is (A)

6.

(E) – 2

is a natural number number and x and x is is a natural number, how many possible values are there

(A) 4

4.

(D) – 1

Human hair grows at a rate of about 1 centimetre per month. This is equivalent to about how many millimetres every ten years? (A) 12

3.

(C) 0

32

(B)

36

(C)

81

(D)

50

(E)

64

Alfred and four other people want to be in a group photograph. In how many different ways can they be arranged in a row with Alfred in t he middle? (A) 4

(B) 8

(C) 12

(D) 24

(E) 40


9.

If a  b means the value of ab + ab + a + b, and 5  x = x = 35, the value of x must x must be (A) 5

10.

(B) 7

(A)

22°

(E) 15

BCO is ˆ

(B)

27°

(C)

30°

(D)

45°

(E)

60°

Ten cubes are glued together as shown in the diagram and then the entire figure is painted. How many of the cubes are painted on exactly four faces?

(A) 10

12.

(D) 12

In the figure, triangle ABC is inscribed in the circle with centre O and diameter AC. If AB = AO, the size of angle

11.

(C) 9

(B) 8

(C) 6

(D) 5

(E) 4

Children were asked about their favourite juice. The results of the survey are shown in the bar graph and also in the pie chart, drawn to sc ale. The size of the angle in the shaded sector is

(A) 45°

(B) 60°

(C) 72°

(D) 75°


(E) 90°

13.

The diagram shows three intersecting straight line segments. The average of a , b , c and d is

(A) 45°

14.

(B) 55°

(C) 65°

(D) 75°

(E) 85°

The shaded area in the diagram is formed when a square and a circle overlap. The shaded area is 5 8

of the area of the circle and

1 2

the area of the

square. If the area of the circle is 80 cm2, the length of a side of the square, in cm, is

(A)

15.

70

(C)

80

90

(D)

10

(E)

A tick and a cross are to be placed in the grid of 16 blocks alongside, no more more than one in a block. No column of four blocks may contain both symbols and no row of four blocks may ma y contain both symbols. In how many ways can this be done? (A) 24

16.

(B)

(B) 36

(C) 42

(D) 108

(E) 144

ABCD is a square. M is the midpoint of BC, and of AP. The size of CBP is ˆ

(A) 22,5°

(B) 30°

(C) 36°

(D) 45°


(E) 50°

120

17.

ABCD is a rectangle and P is a point on BC. If the area of triangle ABP is one third of the area of the rectangle, then the ratio BP : PC is

(A)

18.

3:2

3:2

(C)

2:1

(D)

3:1

(E)

9:4

(B)

2:1

(C)

4:3

(D)

5:3

(E)

5:4

How many of the integers between 97 and 199 are multiples of 2 or 3? (A) 33

20.

(B)

The midpoints of the sides of a square are joined to form a new square. Inside that one a third square is formed by joining the midpoints of the second square. The ratio of the area of the triangle marked a to the area of the triangle marked b is

(A)

19.

5:2

(B) 40

If t toffees cost (A)

100rc

t

c

(C) 55

(D)

60

(E) 68

cents, the number of toffees that can be bought for r rands is

(B)

100rt c

(C)

100 r

ct

(D)

rt 100 c


(E)

100 c

rt



Time: 60 minutes


Instructions 1. This is a multiple choice choi ce question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. Scoring rules: 2.1. Each correct answer is worth 5 marks. 2.2. There is no penalty for an incorrect answer or any unanswered question. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments instruments are not permitted. permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. The centre page is an information and formula sheet. Please tear out the page for your own use. 7. Start when the invigilator tells you to do so. 8. Answers and solutions solutions will be available at www.samf.ac.za

Do not turn the page until you are told to do so. Draai die boekie om vir die Afrikaanse vraestel

PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9372 Email: [email protected]

Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


1.

The value of 2 – (0 – (0 – (1 – (1 – 5)) 5)) is (A) 3

2.

(E) – 2

(B) 120

(C) 1 200

(D) 12 000

(E) 120 000

(B) 30%

(C) 25%

(D) 20%

(E) 10%

Ten cubes are glued together as shown in the diagram and then the entire figure is painted. How many of the cubes are painted on exactly four faces?

(A) 10

5.

(D) – 1

One of the factors of 48 is chosen at random. What is the probability that the chosen factor is NOT a multiple of 4? (A) 40%

4.

(C) 0

Human hair grows at a rate of about 1 centimetre per month. This is equivalent to about how many millimetres every ten years? (A) 12

3.

(B) 1

(B) 8

(C) 6

(D) 5

(E) 4

Children were asked about their favourite juice. The results of the survey are shown in the bar graph and also in the pie chart, drawn to sc ale. The size of the angle in the shaded sector is

(A) 45°

(B) 60°

(C) 72°

(D) 75°


(E) 90°

6.

It is said that you you can only fold a piece of paper in half 7 times. Harold folds a sheet of paper in half 5 times and then makes a hole in the folded paper. How many holes does the sheet of paper have after it is unfolded again? (A)

7.

32

(B) 16

In the diagram,

(A) 240°

10.

81

(D)

50

(E)

64

(C) 24

(D) 48

(E) 80

(B) 4

(C) 6

(D) 8

(E) 9

(D)

(E) 300°

w  x  y  z equals

(B) 255°

(C) 270°

295°

A hiker walks 1 km km East, East, then 2 km North, then 3 km West, then 4 km South, then 5 km East and finally 6 km North. N orth. The hiker’s straight-line straight -line distance in km from the starting point is

(A) 4

11.

(C)

The digits from 1 to 9 are added, in order, over and over again until the total is 460. 1+2+3+4+5+6+7+8+9+1+2+3+…. The last digit that was added is (A) 2

9.

36

Mollie, Alfred and four other people want to be in a group photograph. In how many different ways can they be arranged in a row with Mollie and Alfred together in the middle? (A) 8

8.

(B)

(B) 5

(C) 6

(D)

7

(E) 8

When a is increased by 20% and b is decreased by 20% the resulting values are equal. The ratio of a to b is

(A)

1 2

(B) 1

(C)

2 3

(D)

3 4

4


(E)

3

12.

A tick and a cross are to be placed in the grid of 16 blocks alongside, no more more than one in a block. No column of four blocks may contain both symbols and no row of four blocks may contain both symbols. In how many ways can this be done? (A) 144

13.

If

a +

(B) 108

(B) 2

(C) 3

(D) 4

(E) 5

6

(B)

8

(C)

(D)

10

(E) 4

12

ABCD is a rectangle and P is a point on BC. If the area of triangle ABP is one third of the area of the rectangle, then the ratio BP : PC is

(A)

16.

(E) 24

Δ ABC is rightright-angled at A, and Δ ABP is equilateral with AB = 2 . The length of AC is

(A)

15.

(D) 36

2b = 13 and 5a – 2 2b = 5, the value of b is

(A) 1

14.

(C) 42

5:2

(B)

3:2

(C)

2:1

(D)

The number of different positive integers such that (A) 0

(B) 1

(C) 2

3:1

n n4

(D)

4 

9

(E)

9:4

is

3


(E) 4

17.

ABCD is a square. M is the midpoint of BC, and of AP. The size of CBP is ˆ

(A) 22,5°

18.

(B) 30°

(E) 50°

(B) 40

(C) 55

(D)

60

(E) 68

Two squares are adjacent to each other as shown. One has sides of length of 5 cm and the other has sides of length 7 cm. The area in cm 2 of the shaded region is

(A)

20.

(D) 45°

How many of the integers between 97 and 199 are multiples of 2 or 3? (A) 33

19.

(C) 36°

35

(B)

35,5

(C)

36

(D)

36,5

(E)

37

Two circles have the same centre O, and the radius of the smaller one is 1. The radii OA and OB are such that the area of the shaded region is equal to one quarter of the area of the sector OAB. The length of PA is

(A)

3

2 2

(B)

2

3 3

(C)

5

3

(D)

3

5


5 5

(E)

3 2

5


2015 SECOND ROUND JUNIOR SECTION: GRADE 8 & 9

13 May 2015

Time: 120 minutes


Instructions 1. The answers to all questions are integers integers from 0 to 999. Each question has only one correct answer. answer. 2. Scoring rules: 2.1. Each correct answer is worth 4 marks in Part A, 5 marks in Part B and 6 marks in Part C. 2.2. There is no penalty for an incorrect answer or any unanswered question. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments instruments are not permitted. permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. Start when the invigilator tells you to do so. 7. Answers and solutions solutions will be available at www.samf.ac.za

Do not turn the page until you are told to do so. Draai die boekie om vir die Afrikaanse vraestel.

PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9372 Email: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


HOW TO COMPLETE THE ANSWER SHEET

The answers to all questions are integers from 0 to 999. Consider the following example question:

21.

If 3x − 216 = 0, determine the value of x.

The answer is 72, so you must complete the block for question 21 on the answer sheet as follows: shade 0 in the hundreds row, 7 in the t ens row, and 2 in the units row:

Write the digits of your answer in the blank blocks on the l eft of the respective rows, as shown in the example; hundreds, tens and units from top to bottom. The three digits that you wrote down will not be marked, since it is only for your convenience — only only the shaded circles will be marked.


Part A (4 marks each) 1.

What is the value of (5 – 1) 1) + (0 – 2) 2) ?

2.

How many positive factors of 128 are not factors of 120?

3.

Marks are drawn on a thin strip of paper dividing the strip into 4 equal lengths. Marks are also drawn dividing the strip into 3 equal lengths. After the strip has b een cut at each mark, how many pieces will there be?

4.

For how many positive values of n are both

5.

When simplified, the fraction

2 0

5

2

n

and 2n two-digit integers?

equals

1 

1

2 

0

1 

5

Part B (5 marks each) 6.

In a group of 108 people, 1 in 4 of them has a pen and 1 in 3 has a pencil. What is the minimum possible number of people that have something to write with?

7.

The smallest number bigger than 2015 that is divisible by all of 2, 3, 4, 5 and 6 is

8.

If

9.

The circle centre O has radius 4 cm. If the area of

n



13

3

is a positive integer, then the remainder when

sector OPQ is

1 π 2

n

is divided by 13 is

, and if the probability that a point

chosen randomly in the circle lies in the shaded sector is

1

, then the value of n is

n

10.

The diagram shows a 6 cm by 8 cm rectangle ABCD. The diagonals intersect at point E. The midpoints of AE, BE, CE and DE are joined with straight line segments. The area of the shaded trapezium in cm 2 is

11.

When a water-tank is 30% empty it has 30 litres more in it than when it is 30% full. How many litres can the tank hold when it is full? Compiled by and downloaded from www.erudits.com.ng

12.

A shape consisting of 1000 small squares is made by continuing the arrangement shown.

If each small square has a side length of 1 cm, find the perimeter of the whole shape in cm.

13.

Points A1, A2, A3 … are constructed as follows: the length OA1 is 4, OA1A 2 = 90° ˆ

and the length A 1A2 = 1; then a right angle is constructed at A2 to find A 3, and so on as shown in the diagram. The length of OA 21 is

14.

If p, q, r and and s are different prime numbers less than 20, what is the greatest possible value of

15.

p  q

2 r  s

?

The largest integer value of

x

x

such that 4 divides exactly into

10  9  8  ...  3  2  1 is

Part C (6 marks each) 16.

Three circles and a rectangle touch each other as shown. The radius of the larger circle is 9, and the two smaller circles both have radius 4. The length of a longer side of the rectangle is


17.

How many pairs of non-negative integers  and  are solutions of

18.

Black bricks, white bricks and grey bricks are laid to form the pattern in the diagram. The three squares have the same centre, and the diagonals of the squares have lengths 3, 4 and 5 respectively. If 120 bricks are needed for the central grey area, then the number of black bricks needed is

19.

In the diagram, A, B, C, D and E must each be replaced by one of five consecutive positive positive integers, not necessarily in that order. The numbers inside the triangle add up to 29. The numbers inside the circle add up to 47. The numbers inside the square add up to 30. All five numbers add up to 75. The value of C is

20.

The large quadrilateral shown has 2 right angles, and two of the sides have lengths 7 cm and 10 cm. B and D are on the other sides so that AB = 2 cm and DC = 6 cm. The area of the shaded quadrilateral ABCD in cm 2 is


x

20



y

15



1?

SOUTH AFRICAN MATHEMATICS OLYMPIAD


2015 THIRD ROUND JUNIOR SECTION: GRADES 8 AND 9 29 July 2015

Time: 4 Hours

Number of questions: 15 TOTAL: 100

Instructions · Answer all the questions. · All working details and explanations must be shown. Answers alone will not be awarded full marks. · The neatness in your presentation of the solutions may be t aken into account. · Diagrams are not necessarily drawn to scale. · No calculator of any form may be used. · Use your time wisely and do not spend all your time on one question. · Answers and solutions will be available at: www.samf.ac.za

Do not turn the page until you are told to do so. Draai die boekie om vir die Afrikaanse vraestel. PRIVATE BAG X173, PRETORIA, P RETORIA, 0001 TEL: (012) 392-9372 392-9372 FAX: (012) 392-9312 392-9312 E-mail:[email protected] E-mail:[email protected] Organisations involved: involved: AMESA, SA Mathematical Mathematical Society, SA Akademie vir Wetenskap en Kuns

Compiled by and downloaded from www.erudits.com. www.erudits.com.ng ng

Question 1

M

The diagram shows a 5 by 5 table.

A

T

H

M

A

T

S

The top row contains the letters M, A, T, H and S. The fourth row contains the letters M, A and T as shown. The remaining squares must be b e filled with the letters M, A, T, H and S such that su ch that no row, column or diagonal contains the same letter more than once. Redraw and complete the table in your answer book. [4]

Question 2

If the average of four different positive inte gers is 8, what is the largest possible value of any one of these integers? [4] Question 3

How many zeroes will there be in the answer if the following number is divided by 111?

1110222222003333333330004444444444440000 [4] Question 4

Eskom announces that there is a 60% chance of Stage 1 load shedding for a specific week. In a specific suburb S tage 1 load shedding is from 10:00 10 :00 to 12:30 on a Monday, Wednesday, Friday and Sunday. What is the probability that in that specific suburb there t here will be load shedding at a moment in the week, without knowing the day, or time or whether or not it is day or night? [4] Question 5

If we place a 3 at both ends of a number, its value value is increased b y 3372. Find the original number. [6] 1


Question 6

a)

What is the side-length of the largest solid cube you can build, u sing at most 500 unit cubes?

b)

By gluing some of the unit u nit cubes together, you can build a cube t hat looks solid from any point on the outside, but is is hollow on the inside. What is the side-length of the largest such cube you can build, using at most 500 unit cub es? [6]

Question 7

Let A, Let A, B, B, and C be be distinct points on a straight line with AB with AB = AC = = 1.

D

Square ABDE Square ABDE and and equilateral

E G

F

triangle ACF triangle ACF are are drawn on the same side of line BC line BC . Lines EC Lines EC and BF and BF cut cut in G. B

Ù

What is the size of EGB ?

C

[6]

Question 8

A rectangular sheet of paper can be used to form a cylinder by joining two opposite sides together: OR

Should the short edges or the long edges be joined together to obtain the largest volume of the cylinder? NB: Show all your working!

[6] 2


Question 9

A stack of cubes are built such that there is 1 cube in the first la yer, 3 in the second layer, 6 in the third layer, and so on, as shown:

Stack 1

Stack 2

Stack 3

When these stacks are viewed as shown, we can see 9 edges in the first stack a nd 24 edges in the second stack. a)

How many edges are visible in the third stack?

b)

How many edges would be visible in a stack of 20 layers?

[8]

Question 10

Two players alternate placing 2 ´1 tiles, with no overlap, on a 5 ´ 5 chessboard until no p layer can place a tile. A number of 1´ 1 squares remain empty.

a)

Prove that there must be b e an odd number of empty squares remaining.

b)

Of the empty squares remaining, are more coloured black or white?

c)

What is the maximum number of remaining squares? [8]

Question 11

A new sequence is sequence is formed b y deleting numbers from the sequence 1, 2, 3, 4,

…,

500 such that the

sum of any two numbers of the new sequence is sequence is never a multiple of seven. What is the maximum length of the new sequence? [8] 3


Question 12

Find three prime factors of:

52015 + 5 2016 + 52017 + 3 12016

NB: Show all your working! [8]

Question 13

Prove that the alphanumeric below does not have a solution. Note: Different letters represent different digits.

TWENTY +TWENTY CR I CKET [8]

Question 14 Note:

The floor function êë úû is defined as the greatest integer less than or equal to x. x.

E.g. ëê3, 6úû = 3;

ëê-3, 5ûú = -4

ëê3ûú = 3 .

and

ê7ú ê x ú If ê ú + ê ú = 2 , find all all real solutions for x for x.. ë x û ë7 û [10] 4


Question 15

D ABE and DBCF are equilateral triangles

D

C

and ABCD is a square.

E

Prove that D DEF DEF is an equilateral tr iangle.

F

B

[10]

Total: 100

THE END

5


NICE TO KNOW! PLEASE NOTE: DO AT HOME – NOT NOT FOR THIS PAPER ’S PURPOSE!

See if you can figure o out ut what (if anything) went wrong. A farmer died leaving his 17 horses to his three sons. When his sons opened up his last will and testament, it read:

My eldest son must get one half of my horses; My middle son must get one third of my horses; horses; My youngest son must get one ninth of my horses.

As it is impossible to divide 17 into 2, 3 or 9, the three sons started to fight with each other. So, they decided to go to a farmer friend who they considered quite smart, to see if he could work it out for them. The farmer friend read the farmer ’s last will and testament patiently and after given it due thought, he brought one of his own horses over and added it to the 17. That increased the total to 18 horses. Now, he divided the horses horses according to their father s last will and testament. ’ ’

1 2

1 3 1 9

of 18 = 9. So he gave the eldest son 9 horses. of 18 = 6. So he gave the middle son 6 horses. of 18 = 2. So he gave the youngest son 2 horses.

Here is the horses each son got: Eldest son:

9

Middle son:

6

Youngest son:

2

TOTAL:

17.

Now this leaves one horse over, over, so the farmer ’s friend took his own horse back to his farm. Problem Solved! Or is it?

6




Time: 60 minutes


Instructions 1. This is a multiple choice question paper. Each question is followed foll owed by answers marked A, B, C, D and E. Only one of these is correct. 2. Scoring rules: 2.1. Each correct answer is worth 5 marks. 2.2. There is no penalty for an incorrect answer or any unanswered question. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments are not permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. The centre page is an information and formula sheet. Please tear out the page for your own use. 7. Start when the invigilator tells you to do so. 8. Answers and solutions will be available at www.samf.ac.za




Grade 8 First Round 2014

1.

The value of (A) 0

2.

2  0 1  4 2  0 1  4

is

(B) 1

(C) 2

(D) 4

(E) 5

x

In the grid the three numbers in each horizontal row, vertical column and diagonal add up to 15. The value of x is

5 6

(A) 0

3.

(D) 8

(E) 9

(B) 240

(C) 72

(D) 36

(E) 0

Some squares in a grid have been filled in. If the picture must be symmetrical about the diagonal line, more squares need to be shaded. When that has been done the maximum number of small squares not shaded is

(A) 3

5.

(C) 7

360  20 – 24  300 = (A) 360

4.

(B) 3

7

(B) 4

(C) 5

(D)

6

(E) 7

Two identical squares meet at a vertex as shown. The size of the angle marked x is

(A) 105°

(B) 120°

(C) 130°

(D) 150°

(E) 160°


6.

Which of the following is an odd number? (A)

2 + 0  1  4 (D)

7.

2  0 + 1

2+01+4



(E)

4

(C)

2  0 + 1 + 4

2 + 0 + 1  4

The product of three prime numbers is 42. The sum of these three numbers is (A) 12

8.

(B)

(B) 13

(C) 14

(D) 15

(E) 16

If the pattern shown is continued, the number that will appear directly below 49 is 1

10

2

3

4

5

6

7

8

9

11

12

13

14

15

16

.......

(A) 63

9.

Let

(B) 64

be defined as ab – c. The value of

(A) 17

10.

(C) 65

(B) 18

(C) 19

(D) 66

(E) 67

is

(D) 20

(E) 21

(D) 21

(E) 23

Some arrangements of blocks are shown in the diagram:

The number of blocks in Pattern 10 is

(A) 15

11.

(B) 17

(C) 19

A 10  8 grid is made up of squares each with side 1 cm. The area of the shaded region is

(A) (A) 10 cm

(B) (B) 16 cm

(C) (C) 20 cm

(D) (D) 24 cm

(E) (E) 28 cm


12.

120

Boris has been training for four weeks. Each week he records the total distance he ran that week in a bar graph. How far must he run in the fifth week for his average distance per week to be 60 km?

100 80 60 40 20 0 1

(A) 20 km

13.

1 6

(E) 100 km

(B)

1

(C)

12

1 18

(D)

1 36

(E)

1 72

(B) (B) 30 cm

(C) (C) 32 cm

(D) (D) 36 cm

(E) (E) 40 cm

(D) 30

(E) 36

A cube has all its corners cut off as shown. How many edges does the solid have now?

(A) 12

16.

(D) 80 km

A rectangle of length 10 cm has a triangle shaded in it, as shown. The length CP is 2 cm and the shaded area is 12 cm2. The area of the rectangle is

(A) (A) 28 cm

15.

(C) 60 km

3

Three normal dice are rolled. The probability that they all show the same number is (A)

14.

(B) 40 km

2

(B) 18

(C) 24

A rectangle is divided into four smaller rectangles by lines parallel to its sides. The areas of three of the small rectangles are given in the diagram, which is not drawn to scale. The area of the fourth small rectangle is

(A) 25

(B) 30

(C) 35

(D) 40

(E) 45


4

17.

The value of (A) 4

18.

 9

(B) 9

is (C) 27

(D) 36

(E) 81

(B) (B) 27 cm

(C) (C) 30 cm

(D) (D) 36 cm

(E) (E) 40 cm

A square is divided into four identical rectangles as shown. The perimeter of each of the four rectangles recta ngles is 20 units. The perimeter of the square is

(A) 28 units

20.



4

P is the midpoint of side BC of a rectangle, and Q is the midpoint of side DC. The area of rectangle ABCD is 72 cm2. The area of triangle APQ is

(A) (A) 21 cm

19.

9

4

(B) 32 units

(C) 36 units

(D) 40 units

(E) 48 units

Four points are on a line segment as shown below. If AB : BC = 1 : 3 and and BC : CD = 9 : 5 then AB : BD is

(A) 3 : 14

(B) 1 : 4

(C) 2 : 9

(D) 1 : 5

(E) 1 : 6




Time: 60 minutes


Instructions 1. This is a multiple choice question paper. Each question is followed foll owed by answers marked A, B, C, D and E. Only one of these is correct. 2. Scoring rules: 2.1. Each correct answer is worth 5 marks. 2.2. There is no penalty for an incorrect answer or any unanswered question. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments instruments are not permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. The centre page is an information and formula sheet. Please tear out the page for your own use. 7. Start when the invigilator tells you to do so. 8. Answers and solutions will be available at www.samf.ac.za




Grade 9 First Round 2014

1.

Two identical squares meet at a vertex as shown. The size of the angle marked x marked x is is

(A) 105°

2.

(B) 13

Let

(B) 3

The value of (A) 4

6.

(D) 15

60.01 0.98



(C) 6

be defined as ab – c. The value of

(A) 17

5.

(D) 150°

(C) 14

Which integer is nearest in value to (A) 0

4.

(C) 130°

(E) 160°

The product of three prime numbers is 42. The sum of these three numbers is

(A) 12

3.

(B) 120°

(B) 18

9

4



 9

(B) 9

3.96

(E) 16

? (D) 8

(E) 9

is

(C) 19

(D) 20

(E) 21

(C) 27

(D) 36

(E) 81

4

is

A 10  8 grid is made up of squares each with side 1 cm. The area of the shaded region is

(A) (A) 10 cm

(B) (B) 16 cm

(C) (C) 20 cm

(D) (D) 24 cm

(E) (E) 28 cm


7.

A rectangle has one corner placed at the centre of a quarter-circle and the opposite corner on the circumference of the ci rcle. If the rectangle has width 6 cm and length 8 cm, then the shaded area in cm 2 is

(A) 9π – 48

8.

(A) 30°

(C) 25π – 48

(D) 36π – 48

(E) 49π – 48 48

C ˆ

= 70°, then the size of the angle marked x marked x is is

(B) 36°

(C) 40°

(D) 45°

(E) 46°

If x = x = w + 5, w = y = y – 3 3 and y and y = = 2, then the value of x is (A) 0

10.

16π – 48

In ∆ ABC, AC = AB. P is a point on AC so that BP = BC. If

9.

(B)

(B) 1

(C) 2

(D) 3

Boris has been training for four weeks. Each week he records the total distance he ran that week in a bar graph. How far must he run in the fifth week for his average distance per week to be 60 km?

(E) 4

120 100 80 60 40 20 0 1

(A) 20 km

11.

(B) 40 km

(C) 60 km

(D) 80 km

2

3

(E) 100 km

Three normal dice are rolled. The probability that they all show the same number is (A)

1 6

(B)

1 12

(C)

1 18

(D)

1 36

(E)


1 72

4

12.

A rectangle is divided into four rectangles by two perpendicular lines. The perimeters of these smaller s maller rectangles are 4, 5, 6 and 7. The perimeter of the original rectangle is

(A) 11

13.

(D) 20

(E) 22

(B) 29 m

(C) 31 m

(D) 33 m

(E) 35 m

P is the midpoint of side BC of a rectangle, and Q is the midpoint of side DC. The area of rectangle ABCD is 72 cm2. The area of triangle APQ is

(A) (A) 21 cm

15.

(C) 17

A school corridor has three sections at right angles. Some dimensions are given. The length of the dashed line down the centre of the corridor is

(A) 27 m

14.

(B) 14

(B) (B) 27 cm

(C) (C) 30 cm

(D) (D) 36 cm

(E) (E) 40 cm

Four points are on a line segment as shown below. If AB : BC = 1 : 3 and BC : CD = 9 : 5 then AB : BD is

(A) 3 : 14

(B) 1 : 4

(C) 2 : 9

(D) 1 : 5

(E) 1 : 6


16.

The value of 202 – (65 (652 – 63 632) is (A) 120

17.

(C) 169

(D) 180

(E) 200

(D) 103

(E) 104

The last digit of 5n – 2 2n is 7, so n could be (A) 100

18.

(B) 144

(B) 101

(C) 102

The shape alongside, made up of seven equal squares, is divided into two parts of equal area by the line shown. The line cuts the shape at points E and F. If the sum of the lengths AE and CF is 56 cm, the area of each individual square is

(A) (A) 144 144 cm

19.

(C) (C) 189 189 cm

(D) (D) 225 225 cm

(E) (E) 256 256 cm

A two-digit number is chosen and its digits are added together. When this original number is increased by 5, the new sum of digits is double what it was originally. The number of two-digit numbers for which this is possible is (A) 6

20.

(B) (B) 169 169 cm

(B) 5

(C) 4

(D) 3

(E) 2

The diagram shows a rectangle PQRS with a semicircle on PQ as diameter. The length of PQ is 2 units. The area that is inside the rectangle but outside the semicircle is equal to the area that is inside the semicircle but outside the rectangle. The length of PS is (A)

2 3

(B)



3

(C)



4

(D)

2 5

(E)




4


2014 SECOND ROUND JUNIOR SECTION: GRADE 8 & 9 13 May 2014

Time: 120 minutes


Instructions 1. The answers to all questions are integers from integers from 0 to 999. Each question has only one correct answer. 2. Scoring rules: 2.1. Each correct answer is worth 4 marks in Part A, Part A, 5 marks in Part B and 6 marks in Part C. 2.2. There is no penalty no penalty for for an incorrect answer or any unanswered question. and geometry 3. You must use an HB pencil. HB pencil. Rough work paper, work paper, a ruler and an eraser are permitted. are permitted. Calculators and geometry instruments are not permitted. permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your Indicate your answers on the sheet provided. sheet provided. 6. Start when the invigilator tells you tells you to do so. 7. Answers and solutions will be available at www.samf.ac.za

Do not turn not turn the page the page until you are told to do so. Draai die boekie om vir die vir die Afrikaans Afrikaansee vraestel.



HOW TO COMPLETE THE ANSWER SHEET The answers to all questions are integers from 0 to 999. Consider the following example question: 21 21..

If 3x − 3x − 216 = 0, determine the value of x.

The answer is 72, so you must complete the block the block for question 21 on the answer sheet as follows: shade 0 in the hundreds row, 7 in the tens row, and 2 in the units row:

Write the digits of your answer in the blank the blank blocks blocks on the left of the respective rows, as shown in the example; hundreds, tens and units from top to bottom. to bottom. The three digits that you wrote down will not not be be marked, since it is only for your convenience — only the shaded circles will will be be marked.


Part A (4 marks each)

201  4  ( 2  0  1)4 ?

1.

What is the value of

2.

O is the centre of the circle through A, B and C. CB is parallel to OA and What is the size of

3.

ˆ B= 70°. OA

ˆ B in degrees? CO

ˆ C and AB ˆ C are right angles. What is the length of AB? AD

A

4 B 1 D

4.

What is the smallest natural number K so that 315  K is a perfect square?

5.

How many times in a 24-hour day do the hands on a 12-hour clock point in exactly the same direction?

7

C

Part B (5 marks each)

6.

In the diagram, SAMF is a parallelogram. Point R cuts SA in the ratio 1:3 while point P cuts FM in the ratio 3:1. The length of RP is 8. If the perimeter of SAMF is 52, find the perimeter of RAMP. S

R

A

8

F

7.

P

M

What is the smallest prime number that divides exactly into 312 + 513 + 714 + 1115 ?


8.

Divide 77 into three parts so that one of the parts is one-and-a-half times each of the other two. What is the value of the largest part?

9.

8 pieces of string are lying on the floor and I pick up two of the ends at random. 1 If the probability that I am holding the two ends of the same piece of string is , what k is the value of k ?

10.

Identical rectangles of length 4 and breadth 1 are arranged on a plane as shown. Find the total total perimeter of the shape.

11.

In the figure shown, ABCD and CDEF are rectangles, AB = 9 and BC = 6. 6. Determine the area of the shaded region in square units.

12.

In the sequence of numbers 1, 4, 3, … each term after the first two is calculated as the term preceding it minus the term preceding that. So, for example, the third term is the second term minus the first term, i.e. 3 = 4 – 1. Find the sum of the first 2014 terms of the sequence.

13.

How many positive two-digit numbers become bigger when their digits are reversed?

14.

A circle starts with a radius of 5 cm and its circumference is increasing at a rate of π cm each minute. How many minutes will it take for the area to become four times as big as at the start?

15.

How many two-digit numbers N have the property that the sum of N and the number formed by reversing the digits of N is a perfect square?


Part C (6 marks each)

16.

A game requires an L-shaped piece to be placed on a 33 grid and then a round piece to be placed in any one of the remaining squares. The L-shaped piece can be picked up and turned over or rotated rotated but must exactly cover 4 squares. In how many different ways can the pieces be placed?

17.

In the grid alongside, the sum of each row and of each column is the same.

What must be the value of x – y?

18.

The square SAMF has an area of 169. It contains two overlapping squares. The smaller of these squares has an area that is one quarter of the larger of them, and the area of their overlap is 4. What is the area of the shaded region?

19.

Four people put their hats on the table as they they arrive. When When they leave, each person picks up one hat. It so happens that no-one has picked up his own hat. In how many ways can this have happened?

20.

The numbers 1 to 2014 are arranged in columns as shown: a

b

c

d

1

2 8 10 16 …

3 7 11 15 …

4 6 12 14 …

9

e

5 13

How many multiples of 3 are in column e?




2014 THIRD ROUND JUNIOR SECTION: GRADES 8 AND 9 12 September 2014

Time: 4 Hours

Number of questions: 15 TOTAL: 100

Instructions Answer all the questions. · All working details and explanations must be shown. Answers alone will not be · awarded full marks. This paper consists consists of 15 questions for a total of 100 marks marks as indicated i ndicated.. · The neatness in your presentation of the solutions may be t aken into account. · Diagrams are not necessarily drawn to scale. · · No calculator of any form may be used. Use your time wisely and do not spend all your time on one question. · Answers and solutions will be available at: www.samf.ac.za ·

Do not turn the page until you are told to do so. Draai die boekie om vir die Afrikaanse vraestel. PRIVATE BAG X173, PRETORIA, P RETORIA, 0001 TEL: (012) 392-9323 392-9323 FAX: (012) 392-9312 392-9312 E-mail:[email protected] E-mail:[email protected] Organisations involved : AMESA, SA Mathematical Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

What is the largest 5-digit number that is a multiple of 9, but which has no two digits the same? [4]

Question 2

Find the area of the shaded region. The grid is made up of 1 cm ´1 cm squares. Vertices of the shape lie on the corners of squares. 1 cm

[4]

1 cm

Question 3

5 factorial, written as 5! is defined as 5! = 5´ 4´ 3 ´ 2 ´1 ; also 3! = 3 ´ 2 ´ 1 . In general: n! = n ´ (n - 1) ´ (n - 2) ´ ... ´3 ´ 2 ´ 1. If 10 ! = ( a !) ´ (b!) !) ´ (c !) !) an and a ¹ b ¹ c ¹ 0, fi find a , b and c. [6] Question 4 3

On the six-by-six s ix-by-six (6 ´ 6) grid we need to place

0

One (3 ´ 1) rectangle

2

Two ( 2 ´ 1) rectangles and

2

Three (1 ´ 1) squares, so that none of the

2 1

above figures touch each other, 4

not even diagonally. The rectangles may be placed vertically

1

2

0

1

2

One

rectangle

Two

rectangles

Three

squares

or horizontally within the grid. The numbers at the bottom and on the right of the grid show how many squares in the corresponding rows and columns are occupied by the figures. One square, which is part of a rectangle or one of the square figures is indicated on the grid. Show on the grid where all six figures are located by colouring in the squares. [6] 1


Question 5

The sum of four positive integers is 396. If 5 is added to the first number, 5 is subtracted from the second, the third is multiplied by 5 and the fourth is divided by 5, we get four equal positive integers. Find the four original integers. [5]

Question 6

You are given a set of scales. The black and white rods on which the pans are hanging are all of exactly the same length and the weight of the rods and pans can be ignored. Assign the weights of 3 kg; 4 kg; 5 kg; 6 kg and 7 kg to the scales A, B, C, D and E such that the scales balance.

A

D

E

B

C [5]

2


Question 7

Bangkok is East of Jo hannesburg and and New York is West of Johannesburg.

New York

Bangkok

Johannesburg

Flight time from Johannesburg to Bangkok Bangkok is 11 1 1 hours and 40 minutes and the time difference is 7 hours. Flight time Johannesburg to New York is 15 hours and the time difference is 8 hours. Flight time New York to Bangkok is 17 hours. If I depart from Johannesburg at 06h00 (SA time) and fly to New York, stop over there for 24 hours and then fly to Bangkok, what will the local Bangkok time be when I arrive there? [6]

3


Question 8

The diagram shows two squares A and B inside a bigger square.

A

Find the ratio of the area of A to the area of B.

B [6]

Question 9

The number N is the product of two primes. The sum of the positive divisors of N that are less than N is 2014. Find the value of N. [6]

Question 10

Jack and Gill play p lay a game on a grid of white squares. They take turns in colo uring in any size squares (i.e. 1´ 1 or 2 ´ 2 or or 3 ´ 3 etc. ) which are still white. The last player who is able to colour in a square, wins. (a)

If Gill starts starts the game on a 5 by 3 board, can she definitely win? If so, what is her strategy to force a win?

(b)

If Gill starts, can she always win on a 20 by 14 grid? If so, describe her str ategy. ategy.

(c)

Show that on an odd-by-odd grid of squares, if Gill starts, she can always win. Describe the strategy she will use to force a win.

[10]

4


Question 11 th

For a positive integer n, the n triangular number is T ( n) = (a)

n(n + 1) 2

.

There is one positive integer, k , such that: T ( 4) + T (k ) = T (10 )

Determine k . (b)

Determine the smallest value of the integer b, where b > 2014, such that integer x.. T (b + 1) - T (b ) = T ( x ) for some positive integer x [8]

Question 12

S pider In a rectangular room (a cuboid) with dimensions

2,5 m

Fly

11 m by 2,5 m by 2,5 m, a spider is located in the middle of one 2,5 m by 2,5 m wall 0,5 metres below

2,5 m

the ceiling.

11 m

A fly is in the middle of the opposite wall, 0,5 metres above the floor. If the fly remains stationary, what is the shortest total distance the spider must crawl along the walls, ceiling, and/or floor in order to capture the fly? Note: The answer is NOT 13,5 m

[8]

P

Question 13

Ten points, P, Q, R,

Q

Y , Y, are equally spaced around

⋯

X

R

W

S

a circle of radius one unit. What is the difference in the lengths of the lines PQ and PS? V

T U

5


[8]

Question 14

In the sum

a b

+

c d

each letter represents a distinct digit picked from 1 to 9.

Find the biggest sum less than 1. [8] Question 15

The table cloth in the picture consists of squares with differently coloured circles at their vertices. In black indicated, you can see a 3-by-3 square containing 8 yellow circles. How many yellow circles are there o n a n-by-n -by-n square?

[10]

Total: 100

THE END 6



�� SOUTH AFRICAN MATHEMATICS FOUNDATION

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1.

If January 1st 1985 was a Tuesday, how many Tuesdays were there in 1985? (A) 50

2.

(E) 54

(C) 2

(D) 3

(E) 4

(B) 2

(C) 6

(D)

7

(E) 9

(B) 25

(C) 40

(D)

99

(E) 100

1 4

(B)

1

(C)

3

n+2 2n + 1

gets closer and closer to

1

(D) 1

2

(E) 1,5

A bathroom floor is covered by square tiles: the floor is 5 tiles wide and 8 tiles long. If one of the floor tiles is chosen at random, what is the probability that it is at the edge of the floor? (A)

7.

(B) 1

As n gets larger and larger the value of

(A)

6.

53

(2 + 4 + 6 + ... + 50) – (1 + 3 + 5 + ... + 49) = (A) 23

5.

(D)

The three digit number 7d 2 is divisible by 3 and by 11. The digit d must must be (A) 1

4.

(C) 52

When 30012 is written as a normal number (in the decimal system), the number of times the digit 0 appears is (A) 0

3.

(B) 51

19 40

(B)

1

(C)

2

21

(D)

40

A circle is divided into four sectors. Angle A is

2 3

11 20

(E)

23 40

the angle C

while angle D is twice angle B. Angles B and C are supplementary. The size of angle C is

(A) 100°

(B) 110°

(C) 120°

(D) 135°

(E) 145°


8.

ABCD is a rectangle, and BP = 2 units with AD = 7 units. ˆ C = CA ˆ D .The length of AP is PA

(A)

9.

15

1 50

(E) 5

(C) 8

(D) 5

(E) 7

(B) 1

(B)

(C) 1,5

(D) 2

(E) 2,1

(C)

(D)

(E)

=

1 100

99 100

49 50

1 25

(B) 47

(C) 48

(D) 49

(E) 50

A set of 12 numbers has average 18, but the smallest and largest have average 28. What is the average of the others? (A) 14

14.

18

A petrol tank weighs 34 kg when empty and 58 kg when full. Its Its weight in kg when it is two-thirds full is (A) 46

13.

(B) 2

(1 − 13 )(1 − 14 )(1 − 15 )...(1 − 1010 ) (A)

12.

(D)

The diagram shows two concentric circles. If the circumference of one exceeds the circumference of the other by 6 cm, then its radius exceeds the other radius by approximately (in cm)

(A) 0,5

11.

(C) 4

The decimal form of 3÷7 is the recurring decimal 0.428571428571........ 0.428571428571 ........ The digit in the 2013th decimal place is (A) 4

10.

(B) 3

(B) 15

(C) 16

(D) 17

(E) 18

Four teams play in a knock-out tournament (which means that two pairs compete, and the two winners then play each other). Team A beat Team D, and Team B beat Team A. Who beat Team C? (A) B only

(B) A only

(C) D only

(D) B and A

(E) B and D


15.

In the sequence 5; 11; 17; ...... how many terms are smaller than 1000? (A) 163

16.

(B) 90°

(C) 80°

(D) 70°

(E) 60°

(B) 5m ≥ f ≥ f

(C)

m ≤ 5 f

(D)

5m ≤ f ≤ f

(E) m + f + f ≥ 5

(B) 16

(C) 18

(D) 19

(E) 20

How many factors does the product 11 × 13 × 17 × 19 have? (A) 8

20.

(E) 175

The positive integers are written in a long sequence 12345678910111213........ 12345678910111 213........ When the sequence contains 100 digits, how many of those are 1s? (A) 14

19.

(D) 172

It has been observed that in a herd of gazelle there is always at least one male for every 5 females. If m is the number of males and f and f the number of females, which is true? (A) m ≥ 5 f

18.

(C) 169

M is the midpoint of AB and is joined to the third vertex of ∆ ABC, with MC = AM = MB. The value of x + x + y y is

(A) 100°

17.

(B) 166

(B) 10

(C) 12

(D) 16

(E) 24

A certain value increases by a fixed amount each year. If the increase during the first year was 10%, then the percentage increase during the third year was (A) 7.5

(B) 8

(C) 8.33

(D) 9

(E) 9.5

The SA Mathematics Olympiad Training Programme is a free distance-learning problem solving course course for high school school learners, learners, presented by by the SAMF. All you have to do to participate is to complete an application form online at http://www.samf.ac.za/SAMO_Training or or phone 012 392 9362 for an application form.




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1.

10,1 + 1,01 = (A) 11,01

2.

(B) 51

(C) 52

(D)

53

(E) 54

(B) 2

(C) 6

(D)

7

(E) 9

(B) 2

(C) 8

(D)

5

(E) 7

(B) 25

(C) 40

(D)

99

(E) 100

A bathroom floor is covered by square tiles: the floor is 5 tiles wide and 8 tiles long. If one of the floor tiles is chosen at random, what is the probability that it is at the edge of the floor? (A)

19 40

7.

(E) 10,01

(2 + 4 + 6 + ... + 50) – (1 + 3 + 5 + ... + 49) = (A) 23

6.

1,1001

The decimal form of 3÷7 is the recurring decimal 0,428571428571........ 0,428571428571 ........ The digit in th the 2013 decimal place is (A) 4

5.

(D)

The three digit number 7d 2 is divisible by 3 and by 11. The digit d must must be (A) 1

4.

(C) 11,11

If 1st January 1985 was a Tuesday, how many Tuesdays were there in 1985? (A) 50

3.

(B) 10,11

(B)

(C)

1 2

21

(D)

11

40

(E)

20

A circle is divided into four regions by radii. Angle A is

2 3

23 40

the angle

C while angle D is twice angle B. Angles B and C are supplementary. Angle C is

(A) 100°

8.

(B) 110°

(C) 120°

(D) 135°

(E) 145°

(D)

(E) 60°

ABCD is a parallelogram. BP = DP = BC. The size of x of x is is

(A) 52°

(B) 54°

(C) 56°

58°


9.

2

(A) 2

10.

(B) 3

(B) 41

(A)

(E) 6

(D) 39

2

 A + a  π  (B)  2 

1 4

(E) 38

a  π  A −  (C) 2 

(B)

1

(C)

3

n+2 2n + 1

π( − a)

2

 A  π − a 2 

(D)

2

(E)

gets closer and closer to

1

(D) 1

2

(E) 2

The sequence 123456789123456789123 123456789 123456789123... ... is continued until the sum of all the digits used is 460. The last digit in the sequence is (A) 3

14.

(C) 40

As n gets larger and larger the value of

(A)

13.

(D) 5

Two semicircles are placed in a rectangle of length A length A.. The shortest distance between the semicircles is a. The total area of the semicircles (shaded) is

2

12.

(C) 4

P is a point on side AB of the right-angled triangle ABC. The distances of P from the vertices of the triangle are as shown. The length of BC is

(A) 42

11.

2

How many zeros are there in the result of 5675 – 4325 ?

(B) 4

(C) 5

(D)

6

(E) 7

P and M are points on side AC of the right-angled triangle ABC. AB = BM, and BP is perpendicular to AC. Which statement is not necessarily true? A ˆ P = PB ˆM AB 1. 2. 3. 4. 5. (A) 1

ˆ = BM Â A ˆ P = MB ˆC AB

P M

ˆ = PB ˆM C APˆ B = BPˆ M (B) 2

C

B

(C) 3

(D) 4

(E) 5


 π 

−a  2 

2

15.

I have five books, one of each colour red, yellow, green, blue, white. In how many ways can I place them in a row? (A) 5

16.

(B) 10

(B) 3

2 −1

(D) 5

(E) 6

2 +1

(B)

(C)

2 2−2

3 −1

(D)

(E)

5−2

(B) 15

(C) 16

(D) 17

(E) 18

Five identical rectangles are placed to form a new rectangle. The width of the new rectangle is 15 cm. The area of the big rectangle (in cm2) is

(A) 270

20.

(C) 4

A set of 12 numbers has an average of 18, but the smallest and largest have an average of 28. What is the average of the others? (A) 14

19.

(E) 120

A rectangular sheet of paper with sides 2 and 1 has been folded as shown, so that one corner meets the th e opposite long edge. The length d is is

(A)

18.

(D) 60

The number in each box is the product of the numbers in the two boxes below it. In this case the value of xy of xy is is

(A) 2

17.

(C) 30

(B) 300

(C) 330

(D) 360

(E) 450

The long sides of a rectangle are divided into four equal parts and the short sid es are divided into three th ree equal parts. One point on a short side is joined to all the others. The fraction of the original rectangle that is shaded is

(A)

3 4

(B)

1 2

(C)

7 12

(D)

6 11

(E)


13 24



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Part A (4 marks each) 1.

3 What is the value of (1 + 74 ) ÷ (1 − 14 )?

2.

Oleg thinks of a number, doubles it and adds 1. Ravi starts with the same number, but subtracts 2 and then multiplies by 3. If Oleg and Ravi have the same final result, what was the original number?

3.

Twelve identical wooden cubes are packed on one level in a rectangular tray. What is the minimum perimeter that tray can have if the cubes have a side length of 2 units?

4.

The proper factors of a number are those factors of it that are not 1 or the number itself. What is the sum of the two largest proper factors of 2013?

5.

Bottles are packed in boxes of either 6 or 12. The number of small boxes must be at least half the number of big boxes. If 240 bottles are to be boxed, what is the minimum number of boxes needed?

Part B (5 marks each) 6.

What is the value of 1032 + 1012 – 1002 – 1022 ?

7.

If ab = ab = 2, bc = bc = 12 and ac= ac= 6 with a, b and c all natural numbers, what is the value of a + b + c?

8.

If x If x and and y y are are whole numbers between x + x + y y have? have?

9.

If 2A + 3B = 8 and 3A + 2B = 12, what is the value of 4A + 4B ?

10.

Jane has three shirts, four skirts and some belts. Every combination of shirt, skirt and belt is an outfit, and Jane knows she can wear at least 50 different outfits. What is the minimum number of belts she must have?

11.

ABC is a right-angled triangle, and points P,Q R are chosen on the sides so that AP = AQ, CP = CR. What is the value of x of x ? ?

12.

How many numbers in the sequence 6; 66; 666; 6666; ... are perfect squares?

39 and

224 224 , then how many different values can


13.

Which integer is closest in value to

21000 + 21008 1001

2

+

1001

2

?

14.

The figure shows five straight lines all passing through the s ame point, with some segments joined to form five triangles. What is the value of a + b + c + ... + j + j ? ?

15.

When two identical rectangles overlap as shown, the area of the overlap is exactly half the total area covered by the two rectangles. If AD has a length 2 cm, what is the value of AC2 in cm2?

Part C (6 marks each) 16.

In the figure the square with side length 8 has two vertices on the circle, and one side touching the circle. What is the length of the radius of the circle?

17.

How many multiples of 4 less than 1000 (excluding 4 itself) do not contain any of the digits 6, 7, 8, 9 or 0?

18.

The natural numbers are written in seven columns:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 …

If the number of columns is changed to m, then 114 appears in the same column as 70, and 208 is one column to the right of 152. What is the value of m? m?

19.

Equal arcs with centres at the ends of a diameter of a circle intersect each other on that same circle. If the length of the radius of the whole circle is 4 m, what is the area of the shaded region in m 2?

20.

Some positive integers have cubes whose last two digits are 88. What is the sum of the two smallest such integers? Compiled by and downloaded from www.erudits.com.ng

HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD

Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION THIRD ROUND 2013 JUNIOR SECTION: GRADES 8 AND 9 9 SEPTEMBER 2013 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 TOTAL: 100 Instructions  Answer all the questions.  All working details and explanations must be shown. Answers alone will not be awarded full marks. This paper consists consists of 15 questions for a total of 100 100 marks as indicat ind icated. ed.  The neatness in your presentation of the solutions may be taken into account.  Diagrams are not necessarily drawn to scale.   No calculator of any form may be used. used.  Use your time wisely and do not spend all your time on one question.  Answers and solutions will be available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 392-9312 E-mail: E-mail:[email protected] Organizations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

How many digits does the number 2013 have? [4]

Question 2

A palindromic number is a number that reads the same when its digits are reversed, e.g. 1623261.

What is the largest palindromic 8-digit number which is exactly divisible by 45? [6]

Question 3

a)

68o

Calculate the value of P + Q. 90o

Q P 54o

45o

b)

Prove that P + Q = a + b + c + d . b a Q P c

d

1


[6]

Question 4

This is a real image of a Zimbabwean bank note:

The thickness of one of these Zimbabwean bank notes is 0,1 mm. If we stack t wenty trillion (short scale – scale – see see below) such notes on top of each other, how high would the pile be? a) To the roof of your classroom? b) As high as the Telkom tower in Pretoria? c) As high as Table Mountain? d) As high as Mount Everest? e) As high as a 747 Jet flies from f rom King Shaka International to Oliver Thambo International airport? f) Higher than to the moon?

Explain your answer.

[4]

Nice to know: There are different number naming systems which sometimes create confusion. Value in Scientific notation 10 6 10 1012 1015 10 etc.

Value in numerals

Short Scale Name

1 000 000 1 000 000 000 1 000 000 000 000 1 000 000 000 000 000 1 000 000 000 000 000 000 etc.

Logic

million 1 000×1 0001 1 000×1 000 billion trillion 1 000×1 0003 quadrillion 1 000×1 0004 quintillion illion 1 000×1 000×1 000 To get from one named order of magnitude to the next: multiply by 1 000

Long Scale Name

million thousand million or milliard billion thousand billion or billiard trillion llion To get from one named order of magnitude to the next: multiply by 1 000 000

Logic

1 000 0001 1 000 0002 1 000 000

The word milliard , or its translation, is found in many European languages and is used in those languages for 10 9. However, it is unknown in American English, which uses billion, billion, and not used in British English, which preferred to use thousand million. The root mil in in "million" refers to the Latin word for "thousand" (milia). The word million derives million derives from the Old French milion from milion from the earlier Old Italian milione, milione, an intensification of mille, mille, a thousand. 2


Question 5

The Fibonacci sequence is given by 1; 1; 2; 3; 5; 8; 13; 21; …, where the next number is generated by summing the previous two. Fibonacci numbers were made famous by the rabbit problem, because it explained rabbit breeding. It is less well-known that one can also use Fibonacci numbers to convert miles to kilometers. To do so, one must realise that every positive integer can be uniquely expressed as the sum of different, non-consecutive Fibonacci numbers. To convert integer miles into kilometres, miles are expressed as the unique sum of non-consecutive Fibonacci numbers, then each Fibonacci number is changed to the next Fibonacci number. The new sum approximately gives the kilometres. For example, 50 miles = 34 + 13 + 3 miles, where each number on the right hand side is a Fibonacci number. Using the conversion above, the right-hand side becomes 55 + 21 + 5 km = 81 km. Now use this method to convert convert 120 miles into kilometres. Show your working. working. [6] [You might want to check your answer by using the conversion on the cartoon.]

Question 6

All boxes in a 3  3 table are occupied by zeroes. Suppose that we can choose any 2  2 sub-table and increase all the numbers in it by 1. Example:

Possible next two moves:

0

0

0

1

1

0

1

1

0

0

0

0

1

1

0

1

2

1

0

0

0

0

0

0

0

1

1

Prove that we cannot obtain the table below using these operations. 4

9

5

10

18

12

6

13

7 [6] 3


Question 7

a) Find all integers a and b such that (a  b 15 ) 2

 31  8

15

b) Find the value of 31  8 15



31 8 15 [6]

Question 8

The game of TacTic is a board game for two people. i) During a turn, a player selects a row or column and removes at least one stone, or any number of adjacent stones from that row or column. ii) The player, who removes the last stone(s) from the board, wins.

It is your turn.

a) Describe all possible possible first moves moves to guarantee a win for this configuration.

b) Find a winning move move for this configuration and explain why it works.

[8]

4


Question 9

In a certain city all the skyscrapers are arranged in a square s quare grid and each has either one, two, three or four storeys. Jonathan is surveying 4  4 sections of the city and noting how many skyscrapers he can see from a certain position positi on in a certain direction. For example, Jonathan came across t he following section of the city (viewed from the top):

Each number represents the number of storeys in

1

3

4

2

4

2

1

3

2

1

3

4

3

4

2

1

2

2

1

3

3

1

3

4

2

2

one with 3 storeys and one with 4 storeys.

1

4

2

1

3

2

The building with 1 storey is hidden behind

3

2

1

3

4

1

the building with 2 storeys.

2

3

4

2

1

3

2

1

3

2

the building located there.

He jots down the following data: E.g. Viewing from the circled position Jonathan can see three buildings, one with 2 storeys, store ys,

Suppose Jonathan jots down the following data for a different city: 4

Jonathan knows that each row and each column in the grid contains exactly one of each of the numbers 1; 2; 3; 4 and 5 (storeys of the buildings). Complete the grid by filling in the missing numbers (storeys).

1

2

2

3

2

3

2

2

2

3

1

4

5

1

2

3

2

2

1 [6]

5


Question 10

Molly and Fred had an argument about the next term in the sequence: 2; 4; 6;… Molly says it is 8. Fred says it is – is – 4, 4, which he got by using the formula: Tn  2n  k (n 1)(n  2)(n  3) ,

where k = = 2.

a) What is the fifth term in Fred’s sequence? b) Find a formula that defines the sequence: 2 ; 4 ; 6 ; 38 ; …? [6] Question 11

a) The number 15 can be written as 7 + 8 or 4 + 5 + 6 or 1 + 2 + 3 + 4 + 5 (the sum of 2 consecutive integers, 3 consecutive integers and 5 consecutive integers). Find a positive integer that can be expressed as the sum of 3 consecutive integers, 5 consecutive integers and 7 consecutive integers. b) Prove that if an integer n can be written as the sum of p consecutive integers where p is odd, then n is divisible by p. p. [8] Question 12

Four regular dodecagons (12-sided shapes) are placed as shown. If each side has length 1, what is the area of the black region?

You will be given four Canadian 1c coins (dodecagons) which you may use. You may keep the coins as a souvenir.

[6] 6


Question 13

Let A be any set of 19 numbers chosen from the arithmetic progression 1; 4; 7; 10; 13; 16: … ; 100. Prove that there will always be two numbers in A whose sum is 104. [8]

Question 14

a) The numbers 1 through 12 must be filled into the circles such that the sum along each line is 32. Do it!

b) Prove that it is impossible to arrange the numbers 1 through 12 such that each sum is 31.

c) The numbers 1 through 12 are arranged in the circles such that each sum is S. What are the possible values of S? [10]

7


Question 15

If

 ABC has

sides of length a, b and b and c,

then we define the semi-perimeter, s semi-perimeter, s,, by: s 

a bc

2

A

c b

B C

The area of a) If

 ABC

a

is given by Heron’s Formula:

 ABC has

K 

s( s  a)(s  b)(s  c )

sides of lengths with with the consecutive consecutive integers 13, 13, 14, and 15, show that the

area is twice the perimeter. b) Find all sets of 3 consecutive even integers even integers such that the triangle tr iangle with sides of these lengths has an area exactly equal to its perimeter. [10]

Total: 100

THE END

8




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1.

Which of the following is the largest? (A) 2,010

2.

(E) 2,101

(B)

203,032

(C) 201,32

(D) 203,212

(E) 202,312

31

(C) 15

(D) 13

(E) 25

(B)

(B)

1

(C) 2

(D) 3

(E) 6

(B)

30º

(C) 34º

(D) 38º

(E) 40º

The value of x is is

(A) 28º

6.

(D) 2,011

The number of positive even factors of 18 is (A) 0

5.

(C) 2,100

23 + 32 is equal to (A) 17

4.

2,001

2,012 + 201,2 = (A) 203,32

3.

(B)

A shop promoting soap bars encourages you to Buy

Three Get Another One Free!

If you want 23 bars, what is the least number you have to pay for? (A) 17

7.

(C) 19

(D) 20

(E) 21

(B)

11

(C) 12

(D) 13

(E) 14

(D) 6

(E) 5

The number 41 x 356 356 is divisible by 9. The digit x is is (A) 9

9.

18

If the square roots of the natural numbers from 1 to 200 are written down, how many of them are whole numbers? (A) 10

8.

(B)

(B)

8

(C) 7

South America and Africa are drifting apart at 30 cm per century. Approximately how many millimetres is that per week? (A) 60

(B)

30

(C) 6

(D) 0,6

(E) 0,06


10.

Two primes add up to 36. Their product is not (A) 323

11.

If

(D) 203

(E) 161

(B)

4

(C) 3

(D) 2

(E) 1

(D) 15

(E) 17

* means 2n + 1, then the value of (3*)* is (B)

11

(C) 13

A racing cyclist circles the cycling track every 2 minutes and 40 seconds. How many full laps will he complete in four hours at the same rate? (A) 80

14.

(C) 155

n

(A) 9

13.

299

John says a number out loud; Jane doubles it but Rebecca multiplies it by 5 and then subtracts 6. Both girls get the same result. The number John mentioned was (A) 5

12.

(B)

(B)

85

(C) 90

(D) 95

(E) 100

My friend and I both both bought bought the same same thing thing at a shop. shop. We both both paid paid using R5 and R2 coins only, but each of us paid with a different number of each of the coins, and each of us used R5-coins as well as R2-coins. The least possible price of the item we bought is (A) R 11

(B)

R 13

(C) R 15

(D) R 17

(E) R 23

15.

If M and N are natural numbers, numbers, and if only only one one of of the following following sentences is true, which is it? (A) M is odd (B) N2 is even (C) M – N is odd (D) N is odd (E) M, N have no common factors other than 1

16.

The diagram diagram shows part part of a regular regular polygon, polygon, and the size size of one of the interior interior angles. The number of sides of the polygon is

(A) 8

17.

(B)

9

(C) 10

(D) 12

(E) 15

Anne is now now three times as old old as she was three three years years before before she was half as old as she is now. Anne’s age now is (A) 9

(B)

12

(C) 15

(D) 16

(E) 18


18.

M is the midpoint of the side DC of rectangle ABCD. The fraction of the rectangle that is shaded is is

(A)

19.

2 5

1 2

(C)

1 4

(D)

3 5

(E)

1 3

How many three-digit odd numbers become bigger when their digits are reversed? (A) 120

20.

(B)

(B)

145

(C) 200

(D) 260

(E) 360

The perfect squares are written as a sequence of digits 149162536….. 149162536….. The 85th digit in this sequence is (A) 4

(B)

5

(C) 6

(D) 7

(E) 8

The SA Mathematics Olympiad Training Programme is a free distance-learning problem solving course for high school learners, presented by the SAMF. All you have to do to participate is to t o complete an application form online at http://www.samf.ac.za/SAMO_Training or or phone 012 392 9362 for an application form.




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1.

1 2

+

(A)

2.

1 3

=

1 5

6

(C)

3 5

(D)

4

(B) 30º

(C) 34º

(D) 38º

(B) 32 mm

(C) 64 mm

910

(C) 920

P3

(D) 930

(E) 940

(D) 203

(E) 161

P4

(B)

299

(C) 155

South America and Africa are drifting apart at 30 cm per century. Approximately Approximately how many millimetres is that per week? (A) 60

(B)

30

(C) 6

(D) 0,6

(E) 0,06

A racing cyclist circles the cycling track every 2 minutes and 40 seconds. How many full laps will he complete in four hours at the same rate? (A) 75

8.

(B)

P2

Two primes add up to 36. Their product is not (A) 323

7.

(E) 160 mm

Betty makes patterns of dots as shown. If she continues like this, the number of dots in P30 will be

(A) 900

6.

(E) 40º

(D) 128 mm

P1

5.

(E) 1

5

A sheet of paper has been folded 4 times, and now has thickness 2,5 mm. m m. If it were possible to fold it 10 times its thickness would become (A) 25 mm

4.

5

The value of x is is

(A) 28º

3.

(B)

(B)

80

(C) 85

(D) 90

(E) 95

My cellphone costs me a monthly subscription plus a charge per minute of talking. If I talk for 12 minutes the total cost is R25, and if I talk for 15 minutes the total cost is R 28. The total cost if I talk for 25 minutes will be (A) R 34

(B)

R 38

(C) R 42

(D) R 46


(E) R 50

9.

If

* means 2n + 1, then the value of (3*)* is

n

(A) 9

10.

(B)

11

(C) 13

O

OQ is parallel to SR and PS = PQ. ˆ S is In terms of x , the size of R Q

(D) 15

(E) 17

P

Q

x

R

S (A)

11.

x

3

5 9

2

(D) 45º – x

(E) 45º + x

(B)

14

(C) 16

(D) 18

(E) 20

(B)

1 3

(C)

2 3

(D)

7

(E)

9

4 9

Anne is now three three times times as as old old as as she was three three years years before she was half as old old as she is now. Anne’s age now is (A) 9

14.

x

ABCD is a square of side 3 units. Points that divide its sides in the ratio 2:1 are joined to form a new, shaded, s haded, square. The proportion of the original square which is shaded is

(A)

13.

(C)

In order order to achieve a total of 400 the the number of terms terms in the sum 1 + 3 + 5 + 7 + .... must be (A) 12

12.

(B) 90º – x

(B)

12

(C) 15

(D) 16

(E) 18

The shaded quarter-circle has area 9. The perimeter of the ̟ shaded region is

(A)

3 ̟

(B)

3( + 4) ̟

(C) 6 ̟

(D) 6 + 4 ̟

(E) 6 + 12 ̟


15.

How many three-digit odd numbers become bigger when their digits are reversed? (A) 120

(B)

145

(C) 200

(D) 260

(E) 360

16.

If M and N are natural numbers, and if exactly one of the following sentences is true, which is it? (A) M is odd (B) N2 is even (C) M – N is odd (D) N is odd (E) M, N have no common factor other than 1

17.

M is the midpoint of the side DC of rectangle ABCD. The fraction of the rectangle that is shaded is

(A)

18.

2 5

(B)

1 4

(C)

1 2

(D)

3

(E)

5

1 3

Shona has already scored a practical mark of 82%, and will also write a test. If mark is obtained from the practical and the other

1 4

3 4

of her final

from the test, then if she wants a final

mark of exactly 80% the score she needs to get in the test is (A) 70 %

19.

(B) 72 %

(C) 74%

(D) 76 %

The remainder when 12 + 32 + 52 + 72 + …. + 10132 is divided by 8 is (A) 0

(B)

1

(C) 2

(D) 3

(E) 4

A 20.

(E) 78 %

The rectangle ABCD is folded about the line CP so that D falls on AB in the position marked D’. BC = 6 cm and CD = 10 cm. The distance DP is (in cm)

D'

P

6

D

(A)

10 3

(B) 3

(C)

8 3

B

(D)

13


C

10

(E)

10


2012 SECOND ROUND JUNIOR SECTION 23 May 2012

Time: 120 minutes


Instructions 1. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. Scoring rules: 2.1. Each correct answer is worth 4 marks in part A, 5 marks in part B and 6 marks in part C. C. 2.2. For each incorrect answer one mark mark will be deducted. There is no penalty for unanswered unanswered questions. 3. You must use an HB pencil. Rough work paper, a ruler and an eraser are permitted. Calculators and geometry instruments are not permitted. 4. Figures are not necessarily drawn to scale. 5. Indicate your answers on the sheet provided. 6. The centre page is an information and formula sheet. Please tear out the page for your own use. 7. Start when the invigilator tells you to do so. 8. Answers and solutions will be available at www.samf.ac.za




Part A: (Each correct answer is worth 4 marks) 1.

Arthur takes 8 subjects at school, and for each one receives a code code from 1 to 7. If the sum of his codes is 50, then the least number of 7s he must have received is (A)

2.

6

25

(B)

13

(B)

4

(D) 3

(E) 2

30

(C)

35

(D) 40

(E) 45

10

(C)

9

(D) 5

(E) 4

The vertex P of a regular hexagon is joined to the other vertices. The size of

(A)

5.

(C)

The five-digit number 24 X 8Y is is divisible by 4 and by 5 and by 9. The sum of the digits X and and Y is is (A)

4.

5

A man gives away half of his money to his friend, and after that 10% of what he has left to charity. The percentage of his original amount that he keeps is (A)

3.

(B)

RPS (marked ˆ

20°

(B)

x in in the diagram) is

30°

(C)

40°

(D) 45°

(E) 50°

(D)

(E) 99

The value of 1 – (2 (2 – (3 (3 – (…… (100) )… – (…… – (100) )… ) is (A) – 50

(B)

50

(C) – 100

100

Part B: (Each correct answer is worth 5 marks) 6.

The natural numbers are written in seven columns:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 … A square is drawn around a block block of four numbers, and the sum of those four numbers is 312. The number at top left of the square is (A)

7.

If

67

x  1 x

(A)

8.



(B)

y

and

y

69

1

(C)

 x

72

(D)

74

(E) 76

then the value of x – y is is – y

y

2

(B) – 2

(C)

3

(D) – 3

(E) 1

A positive integer N has exactly three different prime factors (1 is not a prime) and is not divisible by any square. How many different factors does N have? (A)

4

(B)

5

(C)

6

(D) 7

(E) 8


9.

a , b , c , d , e are are five positive numbers, with bd < < cd < < ab < < bc < < ae . The smallest of the five numbers is

(A)

10.

(B)

4

(C)

5

(D) 6

(E) 9

1

(B)

3

2

(C)

3

3 4

(D)

4 5

5

(E)

6

12 13 min

(B)

13 13 min

(C)

14 min

(D)

14 13 min

(E)

15 13 min

20

(B)

24

(C)

28

(D) 34

(E) 36

John spent R 19.00 at the the tuck shop: he bought 2 chocolate bars and 3 packets of chips. The amount he spent on chips was R3 greater than the amount he spent on chocolate. Jane wants to buy 3 chocolate bars and 3 packets of chips: how much will that cost her? (A)

15.

3

100 cards are placed placed face down in a line. line. Alice turns every second second one over, starting with the second in line. Brenda then turns over every third card that is still face down (so starts with w ith the fifth card in the line). After this, how many cards remain face down? (A)

14.

(E) impossible to tell

Dimitri can travel from P to Q in 10 minutes at constant speed; Olga can make the same journey in 20 minutes, also at constant speed. Boris goes from P to Q at the average aver age of Dimitri’s and Olga’s speeds: how long will it take him? (A)

13.

(D) d

c

Everyone in my class has toffees or chocolates, but half of them have both. Twice as many people have toffees only as have chocolates only. The proportion of people with chocolates who also have toffees is (A)

12.

(C)

b

A circular logo is made up of three circles with the same centre and radii in the ratio 1:2:3. A point is chosen randomly inside the logo. How many times more likely is the point to be in the outer ring than in the shaded centre?

(A)

11.

(B)

a

R 24.40

(B)

R 23.60

(C)

R 23.00

(D) R 22.60

(E) R 21.40

Peter travels along the circle and Quentin along the square; both travel at the same speed. If they both start at A, moving clockwise, then where is Quentin when Peter reaches A again for the first time?

(A)

at B

(B)

somewhere on AB

somewhere (C) on BC

somewhere (D) on CD


somewhere (E) on DA

Part C: (Each correct answer is worth 6 marks)

16.

The 81st term of the sequence 1; 2; 2; 3; 3; 3; 4; 4; 4; 4; ... is (A)

17.

(C)

13

(D) 14

(E) 15

40

(B)

45

(C)

48

(D) 60

(E) 120

1 2 2

1

(B)

3

(C)

2  1

(D)

3 

2

(E)

1 3

A three-digit three-digit number X has has its its digits reversed to become Y. Y. The The sum sum of of X and Y is 1535. The sum of the digits of X is (A)

20.

12

Squares APQS and ABCD both have sides of length 1 unit. P lies on the diagonal AC. The shaded area a rea that is common to both squares is (in square units)

(A)

19.

(B)

Among the pupils at a school, the ratio of seniors seniors to juniors juniors is exactly exactly 5 : 3. Among the juniors the ratio of boys to girls is exactly 3 : 2, and among the seniors it is exactly 2 : 3. What is the minimum number of pupils in the whole school? (A)

18.

11

11

(B)

12

(C)

13

(D) 15

Points P and Q are chosen on sides of the square ABCD so that the lines lines AP and AQ divide the the square into three regions of equal area. The sides of the square have length 1. The ratio of the lengths AQ:QC is

(A)

5

(B)

13

(C)

4

(E) 16

A

B

P

(D)

D

Q

15

(E) 3


C


Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION FOUNDATION THIRD ROUND 2012 JUNIOR SECTION: GRADES 8 AND 9 6 SEPTEMBER 2012 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 TOTAL: 100 Instructions • Answer all the questions. • All working details and explanations must be shown. Answers alone will not be awarded full marks. • This paper consists of 15 questions for a total of 100 marks as indicated. • For Question 9 and Question 13 you need a number of rings that will be provided. • There is an Answer Sheet at the end of the paper on which you need to do Question 1 and which you need to hand in. • The neatness in your presentation of the solutions may be taken into account. • Diagrams are not necessarily drawn to scale. • No calculator of any form may be used. • Use your time wisely and do not spend all your time on one question. • Answers and solutions will be available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 392-9312 E-mail:[email protected] E-mail:[email protected] Organizations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

(This question must be done on the answer sheet)

The figure is made up of squares. Draw a straight line through P to divide the shape into two equal parts and explain why your line in fact works.

P [4]

Question 2

This string of beads was made according to a certain pattern. How many beads are hidden in the box?

[4]

Question 3

In the right-angled triangle ABC, BC = 12 and AC = 5. DC is perpendicular to AB. Find x.

B

12

(This answer will of course get you no marks)

D x

C

5

A [6]

1 Compiled by and downloaded from www.erudits.com.ng

Question 4

Calculate each of the following and thus state which of them are integers. (a)

((1 ÷ 2) ÷ 3) ÷ 4

(b)

(1 ÷ 2) ÷ (3 ÷ 4)

(c)

1 ÷ ((2 ÷ 3) ÷ 4)

(d)

(1 ÷ (2 ÷ 3)) ÷ 4

(e)

1 ÷ (2 ÷ (3 ÷ 4)) [6]

Question 5

C

When you travel from A to B, you can either travel along the big semi-circle (i.e. via C) or you can travel along all the smaller semi-circles.

A

B

Which is the shorter route and why? [6] Question 6

The Luhn algorithm is used in computer systems to check the validity of a credit card number as follows: • • • • •

Starting from the right, colour each second digit in the number red. Count the number number of red red digits that are are 5 or greater. greater. Let this number number be R. Add up all the digits in the t he number, and add each red digit again. Finally add add the number R. If the final sum is divisible by 10, then the credit card number is valid.

For example:

4532 0972 5042 9185

R=4

Sum = (4+5+3+2+0+9+7+2+5+0+4+2+9+1+8 (4+5+3+2+0+9+7 +2+5+0+4+2+9+1+8+5) +5) + (4+3+0+7+5+4+9+8) (4+3+0+7+5+4+9 +8) + 4 = 110 Since 110 is divisible by 10 we have a valid credit card number. Find the missing digit in the following valid credit card numbers. (a) 4565 3328 9132 662 X

(2)

(b) 5232 7198 3402 Y781

(4) [6]


Question 7

Fred cycles up a hill at a constant speed speed of 5 km/h (Yes it’s very steep). He cycles down the hill at a constant speed of 45 km/h. (a)

What is his average speed?

(4)

(b)

If he is able to increase his speed on the downhill, what is the maximum average speed he will be able to t o achieve? (2) [6]

Question 8

The picture shows shows a gift that you can can buy in curio shops. shops. It is a calendar which tells you the date and consists of two loose cubes which can be moved and rotated in any way. There must always be two numbers on display and in this case the date is 16 February. (Don’t worry about the month which is displayed below the cubes) What numbers must be on the six faces of each of the cubes so that all the necessary days of any month can be displayed? [6] Question 9

A certain type of ring has an outer diameter of 58 mm and an inner diameter of 40 mm and a thickness of 1 mm. If one stacks enough rings on top of each other, it is possible to stand another ring vertically on top of the pile in such a way wa y that the ring doesn’t touch the ground. What is the minimum number of rings you need to stack on top of each other so that the vertical ring just doesn’t touch the ground? (You have been given some rings (washers) (washers) to help you with this question. They do not have the same dimensions as

in the question but you can build the model with them if you need to)

[8]


Question 10

Did you know that if you form a four digit number using any four non-zero digits on the corners of any rectangle on a calculator the number will always be divisible by 11!! In the example in the picture we have 7128. Look :

7128 11

=

648 which is an integer.

5236 also works:

5236 11

=

476 which is an integer.

It even works if you rotate your calculator!! (i.e. if you use numbers like 2365 or 4697) (a)

Prove it.

(b)

Prove that it even works if you rotate the calculator 90 degrees clockwise. [8]

Question 11 N students are playing an elimination game with a paintball gun. The one student that doesn’t get shot is the winner! The method of elimination works as follows: All students stand in a circle. A paintball gun is given to student 1, and he has to shoot the student to his left (who is then eliminated). He then passes the paintball gun to the next “alive” “ alive” student on his left, who, in turn will shoot the student on his left, and pass p ass the paintball gun to the next “alive” student on his left, and so on. For example, let N = 5. Call them students 1, 2, 3, 4 and 5. •

1 shoots 2, and passes the paintball gun to 3.

•

3 shoots 4, and passes the paintball gun to 5.

•

5 shoots 1, and passes the paintball gun to 3 (2 is already “dead”)

•

3 shoots 5, and 3 is the last one left and so he is the winner!

Who will be left with the paintball gun (not shot) and therefore the winner for N = 100? [8] Question 12 Put the positive integers from 5 to 9 into the blocks (with no repeats) such that the resulting product will be the greatest.

× [8]


Question 13

3

The age-old game of NIM is a game for two players. Three piles of rings lie on the table with

5 4

any number of rings in each pile. The players take it in turns to take any number of rings from any pile and the last player to take a ring loses. There is a winning strategy for this game. If the piles have 3, 4 and 5 rings in them respectively, explain the winning strategy and whether you should go first or second.

[8] Question 14 (a)

A goat is tethered to the corner of a shed which consists of a square and an equilateral triangle. The square has side length 2 m and the rope is 5 m long.

Top view

What is the maximum area that the goat can graze outside of its shed? Give your answer in terms of π. (5)

(b)

This time the goat is tethered to the corner of a rectangular shed 4m by 5m, but with two ropes of length 4 2 m and 4 m as shown. What is the area the goat can graze? Again, give your answer in terms of π.

(5)

5m 4m 4m Top view

4 2m

[10]


Question 15 2012 is a special number. Look at its property in terms of powers of two:

20 – 12 = 8 = 2

3

and

20 + 12 = 32 = 2

5

Which is the next year with this property? [8]

Total: 100

THE END Please turn over for the Answer Sheet to answer Question 1


ANSWER SHEET This Answer Sheet needs to be handed in! Question 1 Practice Grid

P

Final answer

P

Explanation: ________________________ _______________________________________ _____________________________ _____________________ _______ _____________________________ ___________________________________________ _____________________________ ____________________________ _____________ _____________________________ ___________________________________________ _____________________________ ____________________________ _____________ _____________________________ ___________________________________________ _____________________________ ____________________________ _____________ _____________________________ ___________________________________________ _____________________________ ____________________________ _____________ _____________________________ ___________________________________________ _____________________________ ____________________________ _____________

Name:

School:

Grade:




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1.

2 + 3 × 10 = (A) 15

2.

(B)

4.

5.

(B)

(B) 4 years

The value of 4 14

−

(A) 1.00

(B)

(C) 0.056

(D) 0.0056

(E) 0.00056

(C) 5 years

(D) 6 years

(E) 7 years

(C) 1.25

(D) 1.50

(E) 1.75

3.25 is 1.15

(B) 4

(C) 5

(D) 8

(E) 9

Which one of the following numbers is a multiple of 7?

If

a

(B)

b

c

(B)

2020

means

13

(C) 2030

a +b÷c,

(D) 2040

7

then the value of

(C) 15

(D) 18

(E) 2050

4

2

is

(E) 22

Which one of the following fractions is nearest to 1? (A)

9.

(E) 50

The square of an integer never has a last digit equal to

(A) 9

8.

0.56

(A) 3 years

(A) 2010

7.

(D) 45

A child is 1 500 days old. How old will he become on his next birthday?

(A) 1

6.

(C) 42

The value of 0.014 × 0.4 is (A) 5.6

3.

32

9 10

(B)

14 13

(C)

19 20

(D)

121 120

(E)

211 212

2011 – 201.1 is (A) 180.99

(B)

1809.9

(C) 1908.9

(D) 180


(E) 190.9

x

10.

The size of the angle marked x is is

140°

(A) 50°

11.

(B) 60°

(D) 80°

(E) 90°

A nurse gives 3 patients their medicines at different intervals. Cherry has to take her medication every 3 hours. Sandy has to take his medication every 4 hours. Nishi has to take her medication every 6 hours. All three were given their t heir medication at 06:00. When will all three next take their medication at the same time? (A) 09:00

12.

(C) 70°

300°

(B)

12:00

(C) 15:00

(D) 18:00

(E) 21:00

The graph shows the heights of four girls. 175 150 m 125 c n i 100 t h g 75 i e H

50 25

Names of Girls

The names are missing from the graph. Debbie is the tallest. Amy is the shortest. Dawn is taller than Sarah. How tall is Sarah? (A) 50 cm

(B)

75 cm

(C) 100 cm

(D)

125 125 cm

A 13.

Q

B

P 2 cm

C

The fraction of rectangle ABCD that is shaded is

D

(A)

14.

(E) 150 cm

1 8

(B)

1 6

(C)

1 5

4 cm

(D)

1 4

(E)

1 3

Assume that 5 miles is 8 kilometres. Then a speed of 120 km per hour expressed expressed in miles per hour is (A) 60

(B) 75

(C) 90

(D) 105


(E) 192

15.

A pattern of numbers is arranged in columns as shown. In which column does the number 163 lie?

(A) A

16.

(B) B

(B)

C

D

E

1 28 31

4 25 34

7 22 etc

10 19

13 16

(C) C

(D) D

(E) E

20

(C) 22

(D) 24

(E) 26

The sum of the numbers in the series 1 – 2 + 3 – 4 + 5 – ….. + 2011 is (A) 1002

18.

B

The 11 numbers in a list list have an average of 18. When the number 42 is added to the list, the new average of all twelve numbers is (A) 18

17.

A

(B)

1004

(C) 1006

(D) 1008

(E) 1010

Which one of the following is not divisible by 5?

(A) 2312 – 2112

(B)

2132 – 2122

(C)

2132 + 2312

(D)

2132 + 2112

(E)

2132 + 2122

19.

Stage 1

Stage 2

Stage 3

A crystal grows as illustrated. How many polygons are there in Stage 100? (A) 290

20.

(B)

298

(C) 299

(D) 301

(E) 305

A soccer ball is made up of 12 pentagons (5-sided figures) and 20 hexagons (6sided figures) which are stitched together along their edges to form seams. How many seams does the soccer ball have? (A) 30

(B)

60

(C) 90

(D) 120


(E) 150



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1.

2 + 3 × 10 = (A) 15

2.

9 10

0.56

(C) 0.056

(D) 0.0056

(E) 0.00056

(B)

14 13

(C)

19 20

(D)

121 120

(E)

211 212

(B)

2020

(C) 2030

(D) 2040

(E) 2050

(B) 75

(C) 90

(D) 105

(E) 192

(B)

13

(C) 14

(D) 15

(E) 16

The 11 numbers in a list have an average of 18. When the number 42 is added to the list, the new average of all twelve numbers is (A) 18

8.

(B)

A cricket ball and a soccer ball roll in a straight line along the ground and cover the same distance. The cricket ball rotates 27 times while the soccer ball rotates only 9 times. If the radius of the cricket ball is 4 cm, the radius of the soccer ball (in cm) is (A) 12

7.

(E) 50

Assume that 5 miles is 8 kilometres. Then a speed of 120 km per hour expressed in miles per hour is (A) 60

6.

(D) 45

Which one of the following numbers is a multiple of 7? (A) 2010

5.

(C) 42

Which one of the following fractions is nearest to 1 ? (A)

4.

32

The value of 0.014 × 0.4 is (A) 5.6

3.

(B)

(B)

20

(C) 22

The angle marked y is is equal to

(D) 24

y

x + 10° (A) 3 x – 40°

(B) 2 x – 30°

(E) 26

(C) x – 20°

(D) x – 50°


2 x – 40° (E) x + + 20°

9.

The Olympics are held every four years, and the Holympics every six years. They were held in the same year in 1968. How many times will they be held in the same year between the years 2000 and 2200? (A) 11

10.

(B)

(B)

(C) 12

(D) 16

(E) 20

(B)

1004

(C) 1006

(D) 1008

(E) 1010

(B)

2132 – 2122

(C)

2132 + 2312

(B) B

(C) C

2132 + 2112

(D)

(E)

2132 + 2122

A

B

C

D

E

1 28 31

4 25 34

7 22 etc.

10 19

13 16

(D) D

(E) E

If the number x is is increased by 50% and the number 2 x is is decreased by 30%, then the difference between the first new number and the second new number is (A) 0

15.

8

A pattern of numbers is arranged in columns as shown. In which column does the number 163 lie?

(A) A

14.

(E) 17

Which one of the following is not divisible by 5?

(A) 2312 – 2112

13.

(D) 15

The sum of the numbers in the series 1 – 2 + 3 – 4 + 5 – ….. + 2011 is (A) 1002

12.

(C) 14

The longer side of a rectangle has a length of 63 cm and the diagonals both have a length of 65 cm. The width of the rectangle (in cm) is (A) 4

11.

13

(B) 0.05 x

(C) 0.1 x

(D) 0.15 x

(E) 0.2 x

A number from 1 to 99 (including 1 and 99) is chosen at random. The probability that exactly one of its digits is 3 is (A)

1 9

(B)

2 9

(C)

1 11

(D)

2 11


(E)

3 11

B 16.

In the diagram, AB = AC, BC = BD and DE = AD.

E

ˆ is equal to ˆ = 30°, then BAC If EDB

30°

(A) 24°

17.

(B) 32°

(B)

(E) 60°

15

(C) 18

(D) 24

(E) 30

(B) 29 : 16

(C) 30 : 15

(D) 31 : 14

(E) 32 : 13

(D)

(E)

Three equal arcs of circles are drawn centred on the vertices of an equilateral triangle; they touch but do not intersect. intersect. The sides of the triangle triangle are of leng length th 2 unit units. s. The The shade shaded d area area is

(A)

20.

(D) 48°

A 45-litre tank is filled with wine. Nine litres are removed and replaced with water. Then ten litres of the mixture are removed and replaced by water. What is the ratio of wine to water in the final mixture? (A) 28 : 17

19.

(C) 40°

A man is is now now twice as old as his son. Fifteen years ago he was three times as old as his son was then. How old is the son now? (A) 12

18.

C

D

A

3 2

− π

(B)

3 2

−

π

2

(C)

3−

π

2

2 3 − π

2 3 − 2π

A soccer ball is made up of 12 pentagons (5-sided figures) and 20 hexagons (6sided figures) which are stitched together along their edges to form seams. How many seams does the soccer ball have? (A) 30

(B)

60

(C) 90

(D) 120


(E) 150

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Part A: (Each correct answer is worth 4 marks)

1.

Which fraction is closest to 2? (A)

2.

29 19

49

(D)

39

59 49

(E)

69 59

(B)

5

(C) 8

(D) 10

(E) 20

(B)

344

(C) 364

(D) 384

(E) 404

When 2011 is calculated, the number of digits in the result is (A) 11

5.

29

(C)

The diagram shows part of some scaffolding which consists of many equal rods joined together by connectors. The diagram shows a structure of length 2 units. The number of rods required to make a structure of length 40 units is

(A) 324

4.

39

When Johan added the factors of 40 he left one out by mistake and thus arrived at a total of 70. The factor he left out was (A) 4

3.

(B)

(B)

12

(C) 13

(D) 14

(E) 15

Rebecca has some money. She gives 10% of it to her sister, and then 20% of what is left to her brother. The percentage of the money that she kept for herself hers elf is (A) 80

(B)

72

(C) 66

(D) 50

(E) 30

Part B: (Each correct answer is worth 5 marks) 6.

The formula for converting ºF into ºC is C =

5 9

( F − 32) . The temperature in ºC

which is twice as big when converted to ºF is (A) 80º

7.

(B)

100º

(C) 120º

(D) 140º

(E) 160º

Boris is 4 km West of a fixed point O, travelling towards O at a steady speed of 10 km per hour. Otto is 8 km East of O, travelling away from O at a steady speed of 6 km per hour. When Boris reaches O, the distance of Otto from O will be (A) 5,6 km

(B) 9,6 km

(C) 9,8 km

(D) 10,4 km


(E) 23 km

8.

The diagram shows a triangular area that has been covered using identical identical square tiles of size 1 unit. If the triangle has sides of length 3 units and 6 units, and if each square tile can be cut at most once, then the minimum number of tiles needed to cover this triangle is (A) 7

9.

The value of 1 12 (A) 5

(B)

×

1 13

(B)

8

×

(C) 9

1 14

×

10

...

×

3 6 (D) 10

(E) 11

(D) 20

(E) 25

1 191 is

(C) 15

A 10.

The area of ∆ADP is

1 5

B

of the area of the rectangle.

The proportion of the rectangle that is shaded is

D (A)

11.

3 10

(B)

2 5

(C)

4 15

(D)

1

C

P (E)

3

2 7

Four different natural numbers all leave a remainder of 6 when divided by 7. If no two of them have a common factor, what is the least possible value of their sum? (A) 87

(B)

101

(C) 115

(D) 125

(E) 145

12.

A leap year has 366 consecutive days. What is the probability that a leap year has 53 Sundays in it? 53 1 2 2 4 (A) (B) (C) (D) (E) 366 366 7 53 7

13.

Avril has cards: whenever an even number is on one side of the card, there must be an odd number on the other other side. Six of her cards lie on a table, and four of those show an odd number. What is the maximum number of odd numbers we might see if we were able to look at both sides of each of those six cards? (A) 4

(B)

5

(C) 8

(D) 10

(E) 12 2

14.

3

A square piece of paper has squares cut off its corners. The sides of the removed squares have lengths 1 cm, 2 cm, 3 cm and 6 cm. Because of this the area of the piece of paper has been halved. The perimeter of the remaining re maining piece of paper is 1

(A) 36 cm

(B) 40 cm

(C) 44 cm

(D) 48 cm


6

(E) 52 cm

15.

Among the pupils at a school, school, the ratio of boys to girls is 2 : 3. If 5 more boys joined the school that that ratio would become 7 : 10. The number of girls in the school is (A) 50

(B)

75

(C) 100

(D) 125

(E) 150

A


16.

When two circles of radius 2 cm each pass through the centre of the other, the distance in cm between the points of intersection A and B is B

(A)

17.

5

(C)

3 +1

(D)

(E)

2 3

2 5

(B)

2–4 ̟

(C)

̟

–2

(D)

π

2

−

1

(E) 4π – 2

A three-digit number is written down; then the same digits are reversed to give a new three-digit number. The smaller of these numbers n umbers is subtracted from the larger. Which of the following might be the result? (A) 729

19.

3+ 2

The figure shows a square with semicircles drawn on its sides. The length of a side of the square is 2 units. The area of the four-leaf shaded region is

(A) 4π – 8

18.

(B)

(B)

189

(C) 198

(D) 459

(E) 759

A rectangular sheet of paper has a square removed from one end. The remaining rectangle has a square removed from one end. The rectangle now remaining has perimeter 83 that of the original rectangle. The ratio of the shorter side of the original rectangle to its longer side is (A) 3 : 5

20.

(B)

1:3

(C) 3 : 8

(D) 1 : 2

(E) 5 : 8

Three-digit numbers are formed that use only different odd digits (so not 551, for instance). The sum of all the possible numbers is (A)

22 200

(B)

33 300

(C)

44 400

(D)

55 500


(E)

66 600


Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION THIRD ROUND 2011 JUNIOR SECTION: GRADES 8 AND 9 7 SEPTEMBER 2011 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 TOTAL: 100 Instructions  Answer all the questions.  All working details and explanations must be shown. Answers alone will not be awarded full marks. This paper consists consists of 15 questions for a total of 100 100 marks as indicat ind icated. ed.  For Question 4 and Question 12 you need two Pula and two rulers that will be provided.   There is a Working Sheet at the end of the paper to help you to answer Question 4 and 12 which you don’t need to hand in unless you have done work on it that needs to be marked.  The neatness in your presentation of the solutions may be taken into a ccount.  Diagrams are not necessarily drawn to scale. No calculator of any form may be used. used.  Use your time wisely and do not spend all your time on one question.   Answers and solutions are available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 320-1950 E-mail: E-mail:[email protected] Organizations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

One millionth of a second is called a micro second. Roughly how long is a micro century? [4] Question 2

N is is a positive integer such that N 2 – 200 200 is a perfect square. square. How many possible values are there for N ? Explain how you got to your answer. [4] Question 3

Prove that there is only one right angled triangle whose sides are consecutive integers. [6] Question 4

(You may use the Working Sheet to help you to ans wer this question)

A Botswana 1 Pula coin is not circular, but nevertheless has a constant diameter. Two Pula coins and and 2 rulers are given to you. Place the Pula coins on the space provided with the rulers besides them (see worksheet). Move the rulers back and f orth so that the Pula coins roll between them. Wow – Wow – no no bumps?!! If this constant diameter of the coin is Q, what is the perimeter? [6] Question 5

The diagram shows a rectangular piece of paper with a circular hole cut through it. (a) Explain how to cut the shaded area exactly in half with a single straight cut. (b) Is this always possible for any location of any circle that fits into the rectangle? Explain. [6] Question 6

Potatoes are made up of 99% water and of 1% solid "potato matter." Vladimir bought 100 kg of potatoes and left them outside in the sun for a while. When he returned, he discovered that the potatoes had dehydrated and were now now only made up of 98% water. water. How much did the potatoes now weigh? [6]


Question 7

ABCDE × 4 = EDCBA. Find A, B, C, D, and E where each is a unique integer from 1 to 9. [6] Question 8 Some years ago the Old Mutual Math 24 game game was very popular in South Africa. You are given a card with 4 numbers on it and you have to get the number 24 by using the 4 numbers. All the numbers have to be used – used – but but each only once. You can use any of the operations: addition, subtraction, multiplication or division. (a) Make 24 from 2; 6; 8; 4.

6 4

2

8 (b) Make 24 from 8; 8; 3; 3.

8 3

3

8 [6] Question 9

A car travels downhill at 72 km/h. On level ground it travels at 63 km/h, km/h, and uphill at only 56 km/h. The car takes 4 hours hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes. Find the distance between the two towns. [8]

Question 10

These were scenes from the first democratic election in South Africa. A number of men and women are standing in single file in a row to enter the tent to cast their vote. Saskia arrives a little late, and wants to join the queue. Prove that Saskia can always join a queue of any length in such a way that the number of men in front front of her is the same as the number number of women women behind behind her. her.


[8]

Question 11

A 5-by-5 square consists of 25 1-by-1 small squares. (a) Is it possible to tile this square with the non-overlapping shown in the figure?

- shapes

(b) If the shaded square is removed, is it possible to cover the rest of the square using 8 of the - shapes shown above? (If it is possible, draw a solution. s olution. If it is not possible, prove it)

(c) If one corner square is removed, prove that it is not possible to cover the rest of the squares by eight 3-by-1 rectangles as shown in the figure.

[8] Question 12

(You may use the Working Sheet to help you to answer this question)

A row of blocks is provided in your worksheet. Place the 2 Pula coins you have been given on any two squares on the grid. For example:

The Moving coins game is an interesting game in which players pla yers take it in turns to move any to to the right. right. You are not permitted permitted to move back, one of th e two coins any number number of blocks or to jump over another coin. The first player who cannot move loses. Play the game a couple of times to make sure you understand it. Can Player 1 (i.e. the player who makes the first move) always force a win? Explain your answer.

[8]

Question 13

On a 26 question test, 5 points are deducted for each incorrect answer, 2 points are scored for each unanswered question and 8 points are scored for each corre ct answer. Susan writes the test and obtains a final score of 0 (zero). How many questions did she correctly answer if: (a) Susan answered all the questions? (b) Susan did not answer all the questions?


[8]

Question 14

Fig. 1

Fig. 2

12

12

11

11

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

Funnels of Death

Legend has it that an ancient king used these funnels to determine by lot which of his captives should die. They contained contained white sugar-pills and black poison-pills. Each had a spring-release at the end to let one pill fall out at a time. Fig. 1 shows shows a black pill just about to fall. Each captive’s fate depended upon the second pill pill which fell into his hand. He had to replace the first the first pill in the funnel, no matter what its colour, and swallow the second one.

Referring to the picture and description, answer the following questions:

(a)

Fig. 1 shows 12 pills ready to be drawn by 9 captives. Captive #1 draws the black pill first, replaces it at the top and eats Pill #2. Then captive #2 draws Pill #3, replaces it at the top and eats Pill #4. Which one of the 9 captives will be the first to die?

(b)

Fig. 2 shows no poison-pills. Again supposing there are 9 captives, where would you put the 3 black pills so that none would would be drawn as a second choice? Show your solution by drawing the funnel in your answer book and blackening in three of the pills. [8]

Question 15

Find the largest positive integer which for all positi ve integers, n, is a factor of

(

n n

1)2 (n  2)3 (n  3)4

[8]

Total: 100

THE END Please turn over for the Working Sheet to help you to answer Questions 4 and 12


WORKING SHEET Use this working sheet to help you to answer Questions 4 and 12 Question 4

RULER 1

RULER 2

Name:

School:

Grade:

Q u e s t i o n 1 2



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PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


1.

4 × 5 + 3 × 6 is equal to (A) 18

2.

When

19 11

(A) 1

4.

(E) 360

(B) 2

(C) 3

(D) 5

(E) 7

is written as a recurring decimal, how many different digits appear? (B) 2

(C) 3

(D) 4

(E) 5

(B) 2

(C) 4

(D) 5

(E) 6

(B)

15

(C) 20

(D) 25

(E) 30

(D) 4

(E) 5

The sum of the digits of the product 5104 × 452 is (A) 1

7.

(D) 192

The smallest number which must be added to 2010 to arrive at a perfect square is (A) 10

6.

(C) 80

The number of two-digit prime numbers that can be written using only digits from the list 2; 3; 5; 7 is (A) 1

5.

38

The units digit of the product 29 × 37 × 21 × 55 × 43 × 39 is (A) 1

3.

(B)

(B) 2

(C) 3

How many different arrangements of the letters A C T O R are there if the O and the R must be next to each other? (A) 4

(B)

8

(C) 16

(D) 48


(E) 256

8.

Did you know …

2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24 ?

This means that the value of

(A)

9.

109

(B)

111

65

11!

+

9!

is

(C)

25 26

138

(D)

137

(E)

150 151

A jug was 60% full of water. After 20% of that water is removed the jug contains 192 ml. The maximum amount of water (in ml) that the jug could contain is (A) 275

10.

56

11! − 9 !

(B)

300

(C) 325

(D) 350

Five equal squares each with side 2 cm are used to make the figure alongside. M is the midpoint of PQ.

(E) 400

P M Q

The area of the shaded region r egion 2 (in cm ) is

(A) 14

11.

(B)

12

(C) 10

(D) 8

(E) 6

In the given diagram, the number in any box is equal to the sum of the numbers in the two boxes immediately below it.

A

The value of A is

21 10

(A) 88

12.

If

a b

= 6 ;

(A) 3

(B)

b c

=

1 4

67

(C) 21

(D) 97

15 (E) 12

and a + c = = 30 then the value of b is

(B)

18

(C) 12

(D) 60


(E) 10

x cm

13.

A square of side length 5 cm is removed from each corner of a square piece of cardboard of side length x cm. cm. The sides are then turned up to make an open box. If the volume of the box is 605 cm3, the length of a side (in cm) of the original piece of cardboard is

(A) 5

14.

(B)

605

(C) 16

5 cm

(D) 10

(E) 21

A car takes 2 hours to travel from Apetown to Beeville. If its speed is reduced by 30 km/h, it will take 3,2 hours. The distance from Apetown to Beeville (in km) is (A) 30

(B)

90

(C) 120

(D) 140

(E) 160

A 15.

In the figure, AB = 8 cm, AC = 10 cm, BD = 3 cm, CD = 7 cm and BC = 2 x + + 1 cm. If x is is an integer, the sum of the possible values of x is is

8

10

2 x + 1

B 3

C 7

D (A) 10

16.

(B)

9

(C) 8

(D) 7

(E) 6

Identical matchsticks are used to make the different figures shown. 4 matchsticks make figure 1, 10 matchsticks make figure 2, 18 matchsticks make figure 3, 28 matchsticks would make figure 4, and so on.

(1)

(2)

(3)

The number of matchsticks required to make figure 20 is is (A) 420

(B)

440

(C) 460

(D) 480


(E) 500

17.

In a grade of 100 learners, 40% of the boys scored A symbols and 50% of the girls scored A symbols. Four more boys than girls scored A symbols. The number of boys in the grade is (A) 80

(B)

70

(C) 60

(D) 50

(E) 40

L 8 cm

18.

ABCD and BJKL are two identical squares with sides of length 8 cm. c m. M is the midpoint of AD and also of JK.

B

A

K

The area of the hexagon BCDMKL (in cm2) is

M

8 cm

J C

D

(A) 64

19.

80

(C) 96

(D) 112

(E) 128

The sum of eleven consecutive even numbers is p. The largest of the numbers, in terms of p, is (A)

20.

(B)

p 5

+5

(B)

p 11

+5

(C)

p 5

+ 10

(D)

p 11

+ 10

(E)

p 6

+ 10

Anne says Barbara is lying. Barbara says Catherine is lying. Catherine says Barbara is lying. Diane says Anne is lying. How many girls are lying? (A) 0

(B) 1

(C) 2

(D) 3

(E) 4

The Mathematical Talent Search is a free correspondence based problem solving course for high school learners, presented by the SAMF. All you have to do to participate is to complete an application form form and to solve four questions. questions. The application form and questions are available on www.samf.ac.za/MathTalentSearch




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1.

The number 2010 has the property that the number formed by its first two digits is twice the number formed by its last two t wo digits. The number of four-digit numbers with this property is (A) 9

(B) 10

(C) 40

(D) 45

(E) 50 A

2.

ABCDE is a regular pentagon. The size of the angle marked x is

x E

B

D

(A) 18°

3.

(D) 36°

(E) 42°

(B) 50

(C) 53

(D) 55

(E) 60

Today John has X CDs. Tomorrow he will will give Jane four of them, and and then she will have twice as many as he will have. The number of CDs that Jane has today is (A) 2 X – 4

5.

(C) 33°

A rectangular tank has dimensions 2 m by 2 m by 4 m. Water fills the tank at a rate of 5 litres lit res per second. The number of minutes required to fill the t he tank is nearest to (A) 3

4.

(B) 27°

X – 4) (B) 2( X

(C) 2 X + 4

X – 6) (D) 2( X

(E) 2 X + + 8

(D) 0

– b (E) – b

If a + b = c – d and a + c = b – d , then a + d is is (A) b

(B) c

(C) – c

Part B: (Each correct answer is worth 5 marks)

6.

Charlie is now twice as old as Arthur, and four years older than Bernard. In six years’ time the sum of all their ages will be 69. Bernard’s age now is (A) 11

(B) 13

(C) 18

(D) 22


(E) 26

C

7.

Some of the natural numbers less than 1000 leave a remainder of 1 when divided by 21. The number of these that also leave a remainder of 1 when divided by 35 is (A) 8

8.

(B) 17 : 21

(C) 15 : 19

(D) 5 : 3

(E) 5 : 4

(B) 5

(C) 6

(D) 7

(E) 8

(B) 30

(C) 48

(D) 36

(E) 32

(D) 375

(E) 325

The value of 1 – 4 + 9 – 16 + 25 + … + 625 is (A) 725

12.

(E) 12

12 x y If 4 × 6 = 48 then the value of x + y is

(A) 24

11.

(D) 11

John cycles 10 km/h faster than Dave, and takes one third of the time that Dave takes. They both cover the same distance. Dave’s speed in km/h is (A) 4

10.

(C) 10

Among the pupils at a school, the ratio of seniors to juniors is 5 : 3. Among the juniors the ratio of boys boys to girls is 3 : 2, and among among the seniors it is 2 : 3. The ratio of boys to girls in the whole school is (A) 19 : 21

9.

(B) 9

(B) 575

(C) 425

Lebo is asked to find a four-digit number such that the first digit leaves a remainder of 1 when divided by 2, the second digit leaves a remainder of 2 when divided by 3, the third digit leaves a remainder r emainder of 3 when divided by 4 and the last digit leaves a remainder of 4 when divided by 5. The number of dif ferent four-digit numbers he can find with this property is (A) 48

(B) 60

(C) 80

(D) 120


(E) 180

B

A

13.

The midpoint M of a square ABCD is joined to a point P on the side BC. If the square has sides of length 1 unit, and the area of ∆ BMP is 15 that

M P

of the square, then the length BP is

C

D

(A)

3

(B)

5

4

5

(C)

5

7

(D)

9

(E)

9

7 10

Q x

A

14.

ABCD is a square of side 3 units. PQ is a line through O, the centre of the square, meeting AB at a distance x units from B. The shaded area in terms of x is

(A)

15.

x + 9

4

(B)

2 x + 3 2

(C)

3 4

( x + 3)

(D)

9 4

(5 − x )

3

O

D

C

P

(E)

9 16

(4 − x )

The figure shows a square with a circle just fitting into it and another another circle fitting into the corner. The sides of the square are of length 2 units. The radius of the smaller circle is

(A)

2 −1 2

(B)

2 −1 2 +1

(C)

2 −1

(D)

1 2


(E)

B

2 1+ 2 2


16.

3

At the beginning of the day I have 1 000 cm of a mixture that t hat is 90% water and 10% oil. Evaporation during the day means that we lose 50% of the water and none of the oil. The percentage of water in the remaining mixture is approximately (A) 82%

17.

(C) 50%

(D) 45%

(E) 35%

The figure shows an equilateral triangle with a square fitting inside it: all four vertices of the square lie on sides of the triangle. The value of a b

b

a

is

(A)

18.

(B) 78%

3 2

( 3 − 1)

3 +1

(B)

2

(C)

1 4

(4 + 3 )

2

(D)

3

(E)

Four circles of equal size lie in a circle so that each touches the larger circle and also passes through its centre. The arrangement is symmetrical, with the centres of the smaller circles all on the vertices of a square. The larger circle has radius 2 units. The difference in area between the two shaded parts is

(A)

1 8

(B)

π

12

(C)

π

18

(D)

π

24


(E) 0

3

19.

8 cards all show different numbers; four of those numbers are even, the others are odd. If two of the cards are chosen at random, the probability that the sum of their numbers is even is (A)

20.

3 8

(B)

3 7

(C)

1 2

(D)

5 8

Adam, Bob and Chris play different sports. Four statements are true: (1) If Bob plays soccer, Adam plays cricket. (2) If Bob plays rugby, Adam plays soccer. (3) If Adam plays soccer or cricket, Chris does not play rugby. (4) If Chris does not play rugby, Bob does not play rugby. Which one of the following statements must be correct? (A) (B) (C) (D) (E)

Bob plays soccer Bob plays rugby Adam plays rugby Chris plays cricket Adam plays soccer


(E)

5 7


Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION THIRD ROUND 2010 JUNIOR SECTION: GRADES 8 AND 9 8 SEPTEMBER 2010 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 TOTAL: 100 Instructions • Answer all the questions. • All working details and explanations must be shown. Answers alone will not be awarded full marks. • This paper consists of 15 questions for a total of 100 marks as i ndicated. • For Question 9 you need a strip of paper that will be in your answer book. • The neatness in your presentation of the solutions may be taken into account. • Diagrams are not necessarily drawn to scale. • No calculator of any form may be used. • Use your time wisely and do not spend all your time on one question. • Answers and solutions are available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 320-1950 E-mail:[email protected] E-mail:[email protected] Organizations involved: involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

One of the following five numbers is the average of the other four. Which one is it? 26

;

30

;

37

;

28

;

29 [4]

Question 2

2 Place a single digit in each empty square in the diagram so that each row, each column and each jigsaw piece contains all numbers

4

from 1 to 5.

1 5

(Fill in the numbers on the answer sheet)

[4]

Question 3

Find the size of the smaller angle a ngle between the hands of a clock cl ock when the time is 19:30. [6] Question 4

A

Nine 2 x 2 squares of paper, each labelled

B

C

D

with a different letter, are placed on a table

F E

after each other, resulting in the 4 by 4 square shown.

G

I H

In what order were the 9 squares of paper placed from first to last?

[6] Question 5

(a)

Find 28 consecutive integers that add up to 294.

(b)

Prove that it is not not possible possible to find 3 consecutive integers that that add up to 50. [6]

1 Compiled by and downloaded from www.erudits.com. www.erudits.com.ng ng

Question 6

22

(a)

Is the statement

(b)

The figure shows a circle drawn into a square.

π=

7

true or false? Why?

If the perimeter of the shaded region is 25 cm, estimate the area of the circle taking

π

t o be

22 7

.

[6] Question 7

A

ABC point D is on AC, In ∆ABC

D

AB = AD, ∧

and

∧

ABC − ACB = 30 . 

∧

Find Find CBD CBD . C

B [6]

Question 8

Find the smallest positive integer K which, when successively divided by 6 ; 5 ; 4 ; 3 ; 2, leaves remainders remaind ers of 5 ; 4 ; 3 ; 2 ; 1 respectively. [6] Question 9

You have been given 2 strips of paper in your answer book. Tie a knot into one of the strips of paper and pull tight (The second strip is just if you don’t get it right the fi rst time) (a)

What shape do you get? Cool hey!!

(b)

You will see that in some places your shape is 2 layers, 3 layers or 4 layers thick. t hick. Describe the shape that is 4 layers thick. [8]


Question 10

Consider the following pattern. Row 1:

1 +

2

=

3

5 +

6

=

7 +

+ 10 + 11 + 12

=

13 + 14 + 15

Row 2:

4 +

Row 3:

9

8

Row … … … Row n:

(a)

Give a formula, in terms of n, for the last term on the right-hand side of row n.

(b)

Give a formula, in terms of n, for the last term on the left-hand side of row n.

(c)

Find a formula for the sum of either side of the equation of row n. [8]

Question 11

n

Loop 1

Loop 2

Does

end in an even number?

n

No

Sum of the digits of n.

Yes

Does

n

end in a 0?

No n

Yes

2

Stop

(a)

If

n =

49 997 how many times does it go through each loop and what is the final output? output?

(b)

Some positive even numbers go through both loops at least once. Describe these numbers.

(c)

Some positive odd numbers less than 150 go through both loops at least once. Describe these numbers. [8]


Question 12

Before his last Maths test, Bongani’s average for Maths was 33%. In his last test he scored 40% which increased his average to 34%. What must he score in his next test to increase his average to 35%? [8]

Question 13

Fred puts 11 plastic bags inside another plastic bag. Each of the 11 bags is either empty or itself contains another 11 bags. All together 6 bags contain other bags. Of all the bags, how many remain empty? [8] Question 14

On a far-away planet in a far-away galaxy the people have 13 fingers and so they count a little differently – they have three extra letters: x, y, z. Our numbers:

1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14

Their numbers: 1 ; 2 ; 3 ; x ; 4 ; 5 ; y ; 6 ; 7 ;

8 ;

z

;

9 ; 10 ; 11

Our 20 is represented represented by 1 y and our 100 by y7 and so on. What will the square of 1 x be written like on that far-away planet? [8] Question 15

Mr Mahlanyana had 9 children and 31 grandchildren. In his last will and testament he l eft an amount of money to each each grandchild. Each girl was was to get R7 more than each boy. All 31 grandchildren were alive when Mr Mahlanyana died a nd their legacies totaled R470. Of this amount R74 went to Mrs Zweni’s children (she (she was Mr Mahlanyana’s Mahlanyana’s eldest daughter). How many daughters did Mrs Zweni have? [8]

Total: 100

THE END Please turn over for the answer sheet for questions 2 4 Compiled by and downloaded from www.erudits.com. www.erudits.com.ng ng

ANSWER SHEET Please hand in together with your answer booklet.

Name:

School:

Question 2

2

4 1 5


Grade:



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1. Which of the following numbers is the smallest? (A) 0,2009 (B) 0,209 (C) 0,029 (D) 0,02009

(E) 0,0209

2. Themba Themba and James James sell cakes cakes at school school socials. socials. At the first social they sold 50 cakes, cakes, and at the second social they sold 58 cakes. The percentage increase in their sales was: (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 3. The value of 3 ÷ 38 is (A) 3 (B) 6

(C) 7

(D) 8

(E ) 9

4. A group of children children see a herd of cattle cattle in the veld. They count count the total number of legs and the total number of ears of the cattle. The difference between these two numbers is 92. The number of animals in the herd is: (A) 23 (B) 46 (C) 92 (D) 184 (E) Impossible to tell 5. The desks in a classroom are lined up in straight rows. Vusani’s desk is in the third row from the front and the fourth row from the back of the classroo classroom. m. His desk desk is also also the fourth fourth from the left and the sixth from the right. The total number of desks in the classroom is: (A) 24 (B) 70 (C) 40 (D) 72 (E) 54 6. Pieter and Jacob share a packet of sweets in the ratio 7 : 5. Pieter gets 14 sweets more than Jacob. The number of sweets that was in the packet is: (A) 84 (B) 24 (C) 56 (D) 49 (E) 26

1


7. In the given diagram, the two shapes which are reflections of each other in the line y = x are:

(A) (A) P and Q (B) P and S (C) R and and S (D) (D) Q and R

(E) (E) P and R

8. The perimeter of the figure ABCDEF is:

(A) 75

(B) 80

(C) 85

(D) 90

(E) 95

9. Points P , Q, R, and S are marked on the sides of square ABCD so that each side is divided in the ratio 2 : 1, and therefore PQRS is is a square. The ratio of the area of PQRS to the area of ABCD PQRS to ABCD is:

(A)

√

6

9

(B)

4 9

(C)

√

5

9

(D)

5 9

2


(E)

2 3

10. A fraction that lies between (A)

19 56

(B)

4

and

7

32

5

(C)

56

is:

8

33 56

(D)

35 56

11. Liesl has three types of toys: teddybears, cars and jets. All her toys except 21 are jets. All her toys except 23 are teddybears. All her toys except 26 are cars. The number of jets she has is: (A) 14 (B) 13 (C) 12 (D) 11

(E)

37 56

(E) 10

12. 1287a45b is an 8-digit number, where a and b are not zero. The number is divisible by 18. The maximum possible difference between a and b is: (A) 4 (B) 5 (C) 6 (D) 7 (E ) 8 13. The value of h in the figure is:

(A)

24 5

(B)

5 3

(C)

√

61

(D)

18 5

(E)

√

10

3

14. A beadworker is threading beads onto a straight wire; he has four green beads and two red beads and will use them all. The number of different arrangements he can make is: (A) 13 (B) 14 (C) 15 (D) 16 (E) 17

3


15. A small wheel of radius 2 cm rolls around the circumference of a larger wheel of radius 5 cm without slipping. The small wheel has a black spot on it at the point marked P. If the small wheel starts rolling from the position shown in the diagram, how many times will the point P touch the circumference of the larger wheel before P comes back to its original position (and the small circle comes back to its original position)?

(A) 2

(B) 3

(C) 4

(D) 5

(E ) 6

16. ABD is a triangle with points C and E on its sides so that C B = AB = AC = C E = E D.

The size of ∠D is:

(A) 15◦

(B) 20◦

(C) 25◦

(D) 30◦

(E) 35◦

17. A field is in the shape of a rhombus; the length of any one side is 80 m. A path path of consta constant nt width width 2 m goes all all round round the field. field. The area area of the 2 path, in m , is:

(A) 320 + π (B) 320 + 2π (C) 640 + 2π (D) 640 + 4π

(E) 4(80 + π)

4


18. Part of Pascal’s Triangle is shown: 1

Row 1 1 1 Row 2 1 2 1 Row 3 1 3 3 1 Row 4 1 4 6 4 1 Row 5 If this pattern is continued, in which row does the 2009th number number appear? (A) Row 60 (B) Row 61 (C) Row 62 (D) Row 63 (E) Row 64 19. The sum of the digits of (111 111)2 is: (A) 25 (B) 36 (C) 49

(D) 64

(E) 81

20. The number of positive integers n for which 3n − 6 is divisible by n − 1 is: (A) 0 (B) 1 (C) 2 (D) 3 (E ) 4

5




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1.

Andrew thinks of a natural number, reduces it by 3, squares the result and then adds 1. If he ends up with 10, the number he thought of was: (A) 6

2.

4:5

(D) 3

(E) 2

(B)

5:4

(C)

9 : 20

(D)

20 : 9

(E)

4:9

The smallest positive integer which must be added to 2009 in order to get a perfect square is: (A) 7

4.

(C) 4

If the ratio x : y is 3 : 4 and the ratio y : z is 3 : 5, then the ratio x : z is: (A)

3.

(B) 5

(B) 9

(C) 16

(D) 25

(E) 41

(D) 16

(E) 17

ABC is a three-digit number such that

+

ABC ABC ABC CCC

The sum of the digits A, B and C is: (A) 13

5.

(B) 14

(C) 15

If ab = a + b then we say that b is the ‘sumprod partner’ of a. The sumprod partner of 5 is: (A)

6 5

(B)

5 6

(C)

4 5

(D)

5 4


(E) 4

Part B: (Each correct answer is worth 5 marks)

6.

A rectangle has an area which is numerically equal to its perimeter, where both the length and the breadth are integers. The rectangle is not a square. The length of the shortest side of the rectangle is: (A) 8

7.

(B) 6

(C) 5

(D) 3

(E) 1

The integers greater than 1 are written in a pattern as shown: Row 1: Row 2: Row 3: Row 4:

2 3 4 5 6 7 8 9 10 11 etc th What is the last number in the 20 row? (A) 210

(B) 211

(C) 212

(D) 213

(E) 214

B 8.

This map shows a grid of one-way streets. How many different routes are there from A to t o B? (A) 10

9.

(C) 7

(D) 5

(E) 3

Alan leaves Cape Town at 9 a.m. travelling at a constant speed of 20 km/h. Half an hour later Beatrice sets out from the same place along the same road, but at a constant speed of 18 km/h. At 1 p.m. Alan is ahead of Beatrice by: (A) 8 km

10.

(B) 8

A

Did you know …

(B) 17 km

(C) 63 km

(D) 80 km

(E) 143 km

2! = 2 × 1 3! = 3 × 2 × 1 4! = 4 × 3 × 2 × 1 ?

Therefore 100! – 98! is equal to (A)

9889(98!)

(B)

9899(98!) (C)

9900(98!) 9900(98!)

(D) 9901(98!) (E)


9902(98!) 9902(98!)

11.

The figure shows a rectangle made up of three squares. If the side of each square is 2 units, and ATUC is a diagonal of the rectangle, the length of TU is:

P

B

10

4

(B)

3

10

2

(C)

3

10

(D)

3

Q

10

3

10

3

4

C

B

12.

DFGH is a small rectangle inside a larger square ABCD, with GF = 2GH and G is on the diagonal AC. The ratio of the areas of ∆ CGH and square ABCD ABCD is:

(A)

13.

1 : 18

(B)

The last two digits of 7 (A) 56

14.

2009

(C)

1:8

(D)

G

H

A

F

2:9

(E)

D

1 : 12

are:

(B) 01

(C) 49

(D) 43

(E) 07

Which one of the following is prime? 2

2

(A) 99 – 97 (B)

15.

1: 9

D

S

(E)

2

C U

T A

(A)

R

2

2

99 – 98 (C)

2

2

99 + 98 (D)

2

2

98 + 96 (E)

2

2

99 + 97

The digits of 20098 can be arranged in any order. For each arrangement, the ‘score’ is the sum of the positive differences between successive digits. The maximum score that can be achieved is: (A) 24

(B) 27

(C) 28

(D) 32


(E) 34


16.

In a certain game a player can score either 12 or 13 points. points. Vusani plays this game game more than once and adds all his scores to get a total score. The number of total scores less than 100 which are possible is: (A) 15

17.

18.

(C) 31

(D) 37

(E) 39

Quarter circles are drawn centred on opposite vertices of a square, just touching each other. One of the arcs goes through two vertices of the square. If the square has sides of length 1 unit, then the shaded area is:

(A)

1−

(D)

1−

π

2 π

2

(2

−

(

2

2 −

)

(B)

)

(E)

1

1−

π

2

2π 2

( −

2

−

2

)

(C)

1− π 2 − 2

1

4

The diagram shows a square, with one diagonal drawn. A second line is drawn which passes through the centre of the square and meets two sides at a distance x from a corner. The square has sides of length 4 cm. The letters a and b represent the areas of the regions, and b = 2a. The value of x is:

(A) 8/3

19.

(B) 25

(B) 7/3

(C) 4/3

(D) 5/3

b

x

a

(E) 2/3

Some three-digit numbers have a special property. For each of these numbers, when its digits are reversed the new number is bigger than the original by 297. How many different t hreedigit numbers have this property? (A) 24

(B) 36

(C) 54

(D) 60

(E) 72


20.

A pattern goes as follows: 2; 5; 8; 11; 14; 17; 20; 23; 26; 29; …. The numbers in the pattern are added in groups of four as shown. The difference between the sum of the numbers in the sixth group and the sum of the numbers in the first fi rst group is: (A) 266

(B) 260

(C) 254

(D) 248


(E) 240


Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION THIRD ROUND 2009 JUNIOR SECTION: GRADES 8 AND 9 8 SEPTEMBER 2009 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 Instructions • Answer all the questions. All working details and explanations must be shown. Answers alone will • not be awarded full marks. • This paper consists consists of 15 questions for a total of 100 marks as indica indicated. ted. • Questions 4, 5 and 11 should be done on the Answer Sheet provided (Please remember to write your Name and School on the answer sheet) The neatness in your presentation of the solutions may be taken take n into • account. • Diagrams are not necessarily drawn to scale. No calculator of any form may be used. • Use your time wisely and do not spend all your time on one question. • • Answers and solutions are available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 320-1950 E-mail:[email protected] E-mail: [email protected] Organizations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1 A

B

C

D

Grasp the two loose ends of each rope firmly in your mind. Then imagine yourself pulling them until you have a straight piece of rope – either with a knot or without one. Which of these four ropes will give you a knot? [4]

Question 2

A structure is built with identical cubes. The top view, the front view and the side view are shown below. What is the least number of cubes required to build this structure?

Top view

Front view

Side view [4]

Question 3

The digits of a two-digit number AB are reversed to give the number BA. These two numbers are added. For what values of A and B will the sum be a square number? [6]


Example

Question 4

Much like crossword puzzles, there are also

3

19 6

Cross Number puzzles in which numbers from 1

3

1

2

to 9 need to be filled fill ed into the blocks in such a way

14

2

9

3

8

1

3

3

2

1

that each vertical column and horizontal row adds

12

up to the number shown above that column or to

4

the left of that row. A number may not be 14

repeated in any row or column.

26

7

An example is given.

12

16

Now complete this Cross Number puzzle. (Fill in the numbers on the answer sheet)

27

USE

13

ANSWERSHEET 3

Question 5

[6]

Correct Example

Connect adjacent dots with vertical or

1

horizontal lines so that a single closed loop is formed with no crossings or branches. Each number indicates how many lines surround it,

2 3 3 2 1

Incorrect Examples

while empty cells may be surrounded by any number of lines.

1

One correct and three incorrect i ncorrect examples are

2 2 2 1

given.

Use the Answer Sheet to complete this question.

1 2 3 0 2

3 2 0 1

3

3

1

3 3 3

2

(Two closed loops)

2 3 3 2 1

(Not a closed loop)

1

1

3 2 2 0

2 3 3 3 1

(Has branches) 2 Compiled by and downloaded from www.erudits.com.ng

[6]

Question 6

Nine squares are arranged to form a rectangle as shown in the diagram. Square P has an area of 1.

a)

Find the area of square Q.

R b)

Prove that the area of square R is 324 times that of square P.

Q

P [6] Question 7 2

In the figure the small rectangles are identical and each has an area of 8 cm . C and D are points on the line l ine segment AB as shown. If CD =

2 3

2

AB , find the shaded area in cm .

A

G

C

E D B

F

[6]


Question 8

On a distant planet, railway tracks are built using one solid railway bar. A railway is built between two towns 20 km apart on a big flat section of the planet. Unfortunately the bar was made one metre too long and the constructor decided to lift it in the middle to try to t o make the ends fit.

Approximately how high does he have to lift it in the middle? Is it 1 cm, 10 cm, 1 m, 10 m, 100 m or 1 km?

a)

Guess one of the above, without doing any calculations.

(1)

b)

Calculate the answer and comment on how it compares with your guess.

(5) [6]

Question 9

Two candles of the same height are lit at the same time. The first candle is completely burnt up in 3 hours while the second candle is completely burnt up in 4 hours. At what point i n time is the height of the second candle equal to twice that of the first fi rst candle? [8]

Question 10

Nick and John play the following game. They put 100 pebbles on the table. During any move, a player takes at least one and not more than eight pebbles. Nick makes the first move, then John makes his move, then Nick makes a move again and so on. The player who takes the last pebble is the winner of the game.

a)

What strategy can you offer Nick to win the game?

b)

Can you offer John, as the second player, such a strategy? (Give reasons for your answers) [8]


Question 11

Example The numbers 1 to 9 must be placed in the

2

squares on the grid. The numbers in the circles

1

1

are the positive differences between the numbers in adjacent squares.

1

S is the sum of the t he numbers in the circles.

7

An example is given.

8

3

1

6 8

9

1 4

2 1

7

4 5 1

1

6

S = 34 a)

Show how to arrange the numbers 1 to 9 in the squares so that S is a maximum.

b)

Explain clearly why no other arrangement could give a larger total than yours.

Use the diagram on the Answer Sheet to complete the above question

[8]

Question 12

Place algebraic operations

+

;

−

;

÷

or

×

between the numbers 1 to 9 in that order so that the

total equals 100. You may also freely use brackets before or after any of the digits in the expression and numbers may be placed together, such as 123 and 67 (see example). Two examples are given below: i)

123 + 45 − 67 + 8 − 9 = 100

ii)

1 + [( 2 + 3) × 4 × 5] − [(6 − 7) × (8 − 9)] = 100

Four solutions will be awarded 2 marks each. Any other solution will get a bonus of 1 mark each to a maximum of 3 bonus marks. [8] Question 13

(a)

Find three different positive integers, the sum of any two of which is a perfect square.

(3)

(b)

Find a general formula that will help generate other such triplets.

(5) [8]


Question 14

An age-old problem states the following: A camel sits next to a pile of 3 600 bananas at the edge of a desert. He has to get as many bananas as possible, across this desert which is 1 000 km wide. He can only carry a maximum of 1 200 bananas at any one time. To survive he has to eat one banana for each kilometer he travels. What is the maximum number of bananas that he can get to the other side of the desert?

Desert

1 000 km

[8]

Question 15 28

Prove that 3

b)

Prove that 22009 + 5 2010 is not a prime number.

+

7

51

a)

is not a prim primee num number. ber.

(2) (6) [8]

Total: 100

THE END

Please turn over for the answer sheet for questions 4, 5 and 11


ANSWER SHEET Please hand in together with your answer booklet.

Name:

School:

14

Question 4

Grade:

26

7 12

16

27 13 3 Question 5

1 3

2

0

2

3 3

3

2

3

0

2

1 3

3

1 3

2

2

0

Question 11


HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION. Sponsored by HARMONY O!D MININ.

FIRST ROUND 2008 JUNIOR SECTION: SECTION: GRADES 8 AND 9 18 MARCH 2008 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Ins!"#$%ns: ". Do not not ope open n thi thiss boo boo#$ #$et et %nt %nti$ i$ to$ to$d d to to do do so so by by the the in&i in&igi gi$a $ato tor. r. '. This This is a (%$tip (%$tip$e $e )hoi)e )hoi)e *%estio *%estion n paper. paper. Ea)h *%estio *%estion n is +o$$o, +o$$o,ed ed by ans,ers (ar#ed A- - C- D and E. On$y one o+ these is )orre)t. /. Ea)h )orre)t ans,er is ,orth 0 (ar#s. There is no pena$ty pena$ty +or an in)orre)t in)orre)t or an %nans,ered *%estion. 1. Yo% (%st %se an H pen)i$. Ro%gh paper- a r%$er and an eraser are per(itted. C&'#"'&%!s &n( )*%+*!, $ns!"+*ns &!* n% -*!+$*(. 0. Diag Diagra ra(s (s are are not ne)e ne)ess ssar ari$ i$y y dra, dra,n n to s)a$ s)a$e. e. 2. The The )en )entr tree pag pagee is is an an in+ in+or or(a (ati tion on and and +or +or(% (%$a $a shee sheet. t. 3$ea 3$ease se tear tear it o%t o%t +or +or yo%r %se. 4. Indi Indi)a )ate te yo%r o%r ans, ans,er erss on the the shee sheett pro& ro&ided ided.. 5. Start tart ,hen ,hen the the in& in&ig igi$ i$at ato or te$ te$$s $s yo% to do so. so. Yo% ha&e 26 (in%tes to )o(p$ete the *%estion paper. 7. Ans, Ans,er erss and and so$% so$%ti tion onss ,i$$ ,i$$ be a&ai a&ai$a $ab$ b$ee at ,,,.sa(+.a).8a

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOE/IE OM IR DIE AFRI/AANSE A FRI/AANSE RAESTEL 3RI9ATE A :"4/- 3RETORIA- 666" TE!; <6"'= /7'>7/'/ /7'>7/'/ E>(ai$; e$$ie?sa(+.a).8a Organisations in&o$&ed; AMESA- SA Mathe(ati)a$ So)iety- SA A#ade(ie &ir @etens#ap en %ns


".

The &a$%e o+ 6-1 B 1 is A. 6-"

'.

. 6-'

C. 6-1

D. 6-5

E. "-'

D. "6'6

E. "666

i&e i&e an an est esti( i(at atee o+ o+ the the +o$$ +o$$o, o,in ing; g; '665  "4"6 /1'" A. "656

/.

. "626

Find Find the the &a$% &a$%ee o+ /  R'66 R'665 5  4  R'66 R'665. 5. A. R'665-66 . '66-56

1.

E. R'66-65

. 1/1 47'

C. 1/1 55' D. 1/1 452 E. 1/1 555

. '6

C. '"

D. ''

E. '/

Ho, (any triang$es o+ di++erent si8es are there in the +o$$o,ing +ig%re 

A. 4 4.

D.'56-66

Ho, Ho, (any (any n%( n%(be bers rs bet bet,ee ,een n "66 "66 and and 066 066 are are di&i di&isi sib$ b$ee by both both 2 and 7  A. "7

2.

C. '6-65

I+ '7"5  "/ '45 G /5 410 '61 and '7"5  "/ "'7 G /5 /"6 1''then +ind the &a$%e o+ ' 7"5  "17. A. 1/1 45'

0.

C. "616

. "2

C. '6

D. ''

E. '1

Fi&e Fi&e prot protra) ra)tors tors )ost )ost the the sa(e sa(e as se&en se&en set set s*%a s*%ares. res. The )ost o+ +i&e +i&e protra)tors and a set s*%are is e*%a$ to R2'-66. @hat ,o%$d one pay +or '5 s%)h set s*%ares A. R"74

. R'64

C. R'"4

D. R''4


E. R'/4

5.

A gro% gro%p p o+ gir gir$s $s shar sharee "06 "06 app$ app$es es e*% e*%a$ a$$y $y and and "60 "60 do% do%gh ghn%t n%tss e*%a$$y. The $argest n%(ber o+ gir$s in this gro%p is... A. 2

7.

. 7

C. "'

D. "0

E. "5

Ho, (any nat%ra$ ra$ n%(bers n are there s%)h that n' $ies bet,een "6" and /66  A. /

. 1

C. 0

D. 2

E. 4

"6. "6. Cons Consid ider er the the +o$ +o$$o $o,i ,ing ng n%( n%(be bers; rs;>> "2 "/ "6 5. T,o n%(bers are se$e)ted +ro( the abo&e set and added. The re(aining t,o n%(bers are se$e)ted +ro( the abo&e set and added. These possib$e s%(s are s%btra)ted. Ho, (any di++erent positi&e ans,ers are possib$e A. 4

. 2

C. 0

D. 1

E. /

"". "". Cons Consid ider er the the +o$ +o$$o $o,i ,ing ng 1 n%( n%(be bers; rs; a; b; c; d "

b is greater than a by "

b is $ess than c by

2

0

and

is greater than a by d is I+ a +b + c + d G G A. "

7

-

" "0

" '

.

then +ind the &a$%e o+ a.

. '

C. /

D. 1

E. 0

"'. Ti)#ey and Sipens Sipensee )o$$e)te )o$$e)ted d (oney (oney +or +or )harity )harity in the the ratio ratio o+ 0;' 0;' respe)ti&e$y. I+ Sipense )o$$e)ted R266 $ess than Ti)#ey then ,hat ,as the tota$ a(o%nt )o$$e)ted by Ti)#ey and Sipense  A. R"606

. R""'6

C. R"166

D. R"406

E. R"5'6

"/. The digit digit 0 is ,ritten ,ritten bet,een bet,een the digits digits o+ a t,o>digit t,o>digit n%(ber to +or( a three>digit n%(ber. This n%(ber is 1"6 (ore than the origina$ t,o>digit n%(ber. I+ the s%( o+ the digits o+ the three>digit n%(ber is "'- then ,hat is the di++eren)e bet,een the digits o+ the t,o>digit n%(ber  A. 6

. "

C. /

D. 0


E. 4

"1. Find Find the the &a$% &a$%ee o+ /77 J /74 /74  /70 J /7/ /7/  ..  5/ 5/ J 5". A. "01

. "02

C. "05

D. "26

E. "2'

"0. @hat @hat is is the the s(a$$e s(a$$est st positi& positi&ee inte integer ger n s%)h that the +o$$o,ing prod%)t ends in si 8eros "2  /1  40  '"  "/  n A. "'0

. 2'0

C. "'06

D. "6 666

E. " 666 666

"2. Deter(i Deter(ine ne the &a$%e &a$%e o+ <" −

A.

" 06

.

" 16

" 0

=<" −

" 2

=<" −

C.

" 4

=....<" −

" '6

" 166

D.

=.

" "6

E.

" 0

"4. The di(en di(ensio sions ns o+ a re)tan re)tang%$ g%$ar ar +$oor +$oor are are 4"6)( 4"6)( by 1/6)(. 1/6)(. @hat @hat is is the (ai(%( n%(ber o+ ,ho$e ti$es ,ith di(ensions '0)( by '6)( ,hi)h ,i$$ +it onto this +$oor ,itho%t )hanging the dire)tion o+ the ti$es  A. 055 "5. Consider;

. 076

C. 070

D. 072

E. 075

" / 0 4 7 "" "/ "0 "4 "7 '" '/ '0 '4 '7 /" ..

@hat is the (idd$e n%(ber o+ the 2" st ro, A. 4"'"

. 4''"

C. 4/'"

D. 41'"

E. 40'"

"7. KL20'G KL1 ''0 G 1''0G"/ and KL220'G "7- ,here K L x x' gi&es the s%( o+ the digits o+ x' < x x )onsists o+ a )ertain n%(ber o+ 2s +o$$o,ed by a 0=. Find x i+ KL x'G 17. A. 22 220 C. 2 222 220 E. 222 222 220

. 222 220 D. 22 222 220


'6. '6. The The net net o+ a )%b )%bee is is gi& gi&en en;;

So(e o+ the )%bes be$o, )an be obtained by +o$ding the abo&e net
@hi)h one o+ the +o$$o,ing state(ents is tr%e A. . C. D. E.

On$y On$y )%be )%be

THE SOUTH AFRICAN MATHEMATICS OLYMPIAD _________________________________________________________ Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION __________________________ ____________ ___________________________ __________________________ ___________________________ _________________ ___

SECOND ROUND 2008 JUNIOR SECTION: SECTION: GRADES 8 AND 9 22 MAY 2008 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions 1) Do not open this booklet until told to do so by the invigilator. 2) This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3) Scoring rules: a) Each correct answer is worth 4 marks in part A, 5 marks in part B and 6 marks in part C. b) For each incorrect answer one mark will be deducted. There is no penalty for unanswered questions. 4) You must use an HB pencil. Rough paper, a ruler and an eraser are permitted. Calculators and geometry instruments are not no t permitted. 5) Diagrams are not necessarily drawn to scale. 6) The centre page is an information and formula sheet. Please tear it out for your use. 7) Indicate your answers on the sheet provided. 8) Start when the invigilator tells you to do so. You have 120 minutes to complete the question paper. 9) Answers and solutions will be available at www.samf.ac.za in June. ____________________________________ _____________________________________________________ ______________________________ _______________________ __________ DO NOT TURN THE PAGE UNTIL YOU ARE A RE TOLD TO DO SO DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE A FRIKAANSE VRAESTEL ____________________________________ _____________________________________________________ ______________________________ _______________________ __________ PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Mathematical Society, SA Akademie vir Wetenskap en Kuns


Part Part A: Four marks marks each. each.

1. How many of the following numbers are divisible by 48? 1008; (A) 0

2 008 ;

3008;

(B) 1

4008;

(C) 2

500 8 (D) 3

2. How many pairs of positive integers (m, n) satisfy mn = 16? (A) 0 (B) 1 (C) 2 (D) 3 3. Evaluate: 3 (A)

−

5 3

(B)

−

2 11

− 3 −2 (C)

(E) 4

(E) 4

2 3

11 6

(D)

15 7

(E)

28 11

4. Sipho Sipho buys buys shares shares in the stock stock mark market et for for R500. R500. Over Over the next next three three years, the value of the shares increases by 40% and Sipho sells his shares at that price. If he pays a 7% transaction fee on both his purchase price and sale price, then how much profit does Sipho make? (A) R116 (B) R130 (C) R151 (D) R172 (E) R193 5. In the given triangle, AB = B D = DC D C and A BD = C BD . Find the size of B AC . 





A

D

B

(A) 36◦

(B) 45◦

C

(C) 60◦

(D) 72◦

2


(E) 75◦

Part Part B: Five Five marks marks each. each.

6. A three digit number has (2x +1) as its hundreds digit, (x 1) as its tens digit and x as its units digit. The three digit number, in terms of x, is (A) 4x (B) 2x3 x2 x (C) 211x + 90 (D) 211 + 90x 2 (E) 211x + 90x

−

− −

7. If a + b = 23 and a2 b2 = 23, then what is the value of a? (A) 12 (B) 13 (C) 14 (D) 15

−

(E) 16

8. What is the smallest value of n n such that the product n! = 1 2 3 n ends in at least 10 zeroes? (A) 30 (B) 35 (C) 40 (D) 45 (E) 50

× × ×···×

9. You are given a rectangular piece of perspex measuring 50cm 32cm, which is then cut into several pieces and rearranged into a square. What is the length of the side of the square? (A) 182 (B) 40 (C) 41 (D) 120 (E) 1600

×

√

10. John drives from Johannesburg to Cape Town at an average speed of 90 kilometers per hour, and he drives back at an average speed of 110 kilometers per hour. What is John’s average speed for the whole journey (in km/h)? (A) (A) 98 (B) 99 (C) 100 100 (D) (D) 101 (E) (E) Impos possible to dete etermi rmine. 11. If a is smaller than b, c is smaller than d and b is smaller than d, then which number is the smallest? (A) a (B) b (C) c (D) d (E) Impossible Impossible to determine. determine. 12. ABCD is a parallelogram, F is the midpoint of AD and E is is a point on = 1 : 3. BC such that BE : E C = of  BEF Calculate the value of area of area . parallelogr parallelogram am ABCD F

A

B

(A)

1 8

(B)

C

E 1 4

D

(C)

1 3

(D)

3 7

(E)

1 2

13. There are five sticks measuring 1cm, 2cm, 3cm, 4cm and 5cm. How many different triangles can one form using three sticks at a time? (A) 2 (B) 3 (C) 5 (D) 7 (E) 9

3


14. The radius r of the small small circle circle is 1cm. Determi Determine ne the radius radius R of the large (quarter) circle.

R

(A)

√

3 2 2

(B) 1 +

r

√ 2

(C)

√

5 2

(D) 2 2

(E) 3

15. The set of odd numbers are arranged as follows: 1 3 5 7 9 11 13 15 17 .. ... ... . What is the middle number of the 20th row? (A) 759 (B) 761 (C) 763

(D) 765

4


( E ) 767

Part Part C: Six marks marks each. each.

16. In the sequence of numbers 1, 2, 3, . . . , 2008, it is possible to choose two (different) numbers whose sum is divisible by 11. A new sequence is formed by excluding some numbers from the original sequence in such a way that it is impossible to choose two numbers from the new sequence whose sum is divisible by 11. What is the maximum number of numbers in this sequence? (A) 910 (B) 911 (C) 915 (D) 916 (E) 1097 17. Two intersecting straight lines divide the 2-dimensional plane into 4 parts, and three straight lines (intersecting in different points) divide the plane into into 7 parts. parts. How How many many lines will divide divide the plane plane into into 172 parts? parts? (As(Assume that no two lines are parallel and that no three lines pass through the same point.) (A) 11 (B) 16 (C) 17 (D) 18 (E) 19 18. In a certain suburb, the power supply is interrupted during peak hours on average once every 7 days, and the power supply is interrupted during off-peak hours on average once every 17 days. On some days, the power supply supply is interr interrupte upted d both during peak hours and off-peak off-peak hours. Peak Peak hours are between 6am and 9am and again between 5pm and 9pm. What is the probability that the suburb’s power supply is interrupted on a given day? 1 11 1 23 24 (A) 119 (B) 288 (C) 12 (D) 119 (E) 119 19. Jeremy can build a wall in 16 hours if he works alone. Mpume can build the same wall in 12 hours if she works alone. If they work together they can build the wall in 8 hours, but because they sometimes get in each other’s way, they build 16 bricks less per hour than they would if they did not get in each other’s way. How many bricks are there in the wall? (A) 867 (B) 687 (C) 876 (D) 678 ( E ) 768 20. A rectangu rectangular lar box with with integr integral al dimensi dimensions ons (i.e. its side-len side-lengths gths are integers) has a volume of 288 cubic units and a surface area of 288 square units. What is the sum of the side-lengths of the box (in units)? (A) 16 (B) 21 (C) 22 (D) 23 (E) 31

5


Formula and information sheet

1. (a) The natual natual num numbers bers are 1, 2, 3, 4, 5, 5, . . . (b) The whole numbers (counting numbers) are 0, 1, 2, 3, 4, 5, . . . (c) The integers are . . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, . . .

− − − −

2. In the fraction ab , a is called the numerator and b the denominator. 3. (a) Exponen Exponentia tiall notation: notation: 5

2 3

×2×2×2×2= 2 ×3×3×3×3×3= 3 a × a × a × a × · · · × a = a 6

n

(n factors of a) (a is the base and n is the index (exponent)) (b) Factorial notation: 1

×2×3×4 1×2×3×···×n

= 4! = n!

4. The area of a (a) tr triangle is: .. .. .. .. .. .. .. .. .. .. .. .. . 12 (base height) = 12 (b h); (b) rectangle is is: .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . length breadth = lb; (c) sq s quare is is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . side side = s2 ; ( d ) r h o m b u s i s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 (product of diagonals);

×

×

×

×

·

×

(e) trap esium is is:. .. .. .. .. .. .. .. .. .. . 12 (sum of parallel sides) height; (f ) ci circle is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . πr 2 (r = radius).

×

×

5. The surface area of a (a) rectangular prism is:..................2 lb + 2bh + 2 hl (h = height); (b) sphere is is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 πr 2 ; (c) (c) righ rightt prism prism is: is: . . . . . . . . . (per (perim imit iter er of base base h) + (2 area of base).

×

×

6. The perimiter of a (a) rectangle is: ..................... 2 length + 2 breadth = 2l + 2 b; (b) sq square is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 s; (c) circle (its circumference ) is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 πr .

×

×


7. The volume of a (a) cu cub e is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s s s = s 3 ; (b) rec rectangular pr prism is: .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. l b h; ( c ) c y l i n d e r i s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . πr 2 h; (d) (d) righ rightt pri prism is: is: .. . . . . . . . . . . . . . (a (area of of bas base) (perpendicular height) height) = (area of cross-section) (perpendicular height). height).

× × × ×

×

×

8. The sum of the interior angles of a polygon is equal to 180◦ where n = number of sides. 9. Distance = speed Speed = distance Time = distance

× time ÷ time ÷ speed

× (n − 2),

(d = s t) (s = dt ) (t = ds )

×

10. Pythagoras: if ABC is a right angled triangle, then a2 = b 2 + c2 .

 

B

c

A

11. Conversions: 1 cm3 = 1 m 1000 m = 1 km

a

b

1000 cm3 = 1  1000 g = 1 kg

C

100 cm = 1 m



Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION THIRD ROUND 2008 JUNIOR SECTION: GRADES 8 AND 9 4 SEPTEMBER 2008 TIME: 4 HOURS NUMBER OF QUESTIONS: 15 Instructions • Answer all the questions. All working details and explanations must be shown. Answers alone will • not be awarded full marks. • This paper consists consists of 15 questions for a total of 100 marks as indica indicated. ted. • Questions 2, 4 and 5 should be done on the Answer Sheet provided – see last page. (Please remember to write your Name and School on the answer sheet) The neatness in your presentation of the solutions may be taken take n into • account. Diagrams are not necessarily drawn to scale. • • No calculator of any form may be used. Answers and solutions are available at: www.samf.ac.za •

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 320-1950 E-mail:[email protected] E-mail: [email protected] Organizations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


Question 1

A teacher has to buy exactly 106 sweets. The sweets are sold in packs of 5 which cost R6 per pack, or packs of 7 which cost R7 per pack. What is the lowest cost at which the teacher can buy the sweets? (4) Question 2

Without rotating the small squares on the right, arrange them into t he pattern shown in the diagram on the left, so that the t he number next to each large triangle equals the sum of the four numbers in that triangle. 32 11

26

1

7

28

5

4 3

30

2

13

17

7

6

8

5

15

9 3 (4)

Question 3

The same pile of blocks is seen from 2 different sides. It is made from only two types of blocks (A and B). How many of each type are there i n the pile? Back

Block A

Front Front Block B

Back

(6)


Question 4

0 Draw a star (*) in i n 10 empty squares in

2

the diagram alongside so that each 2

numbered square accurately indicates i ndicates 1

how many immediately adjacent squares (horizontally, vertically or

4

diagonally) contain a star.

1

4 2

2

1

An example is given below: 0

1

1

1

2

*

1

*

2

(6)

Question 5

In the faraway land of Mathopillis, along the south coastal road, lies a string of eight beautiful little villages. They are, in the order one would pass through them when travelling from West to East, Alpha, Beta, Circa, Dode, Epsilon, Flora, Gamma and Hexa. The chart below indicates the distances in km between some of the t he villages. (For example, the distance between Alpha and Dode is 28km.) Find the distance between each village and the next one and complete the t he chart on the answer sheet.

Alpha Beta Circa 28

Dode 27

43

Epsilon 25

Flora 22

38

Gamma 24

Hexa (6)


Question 6

On a wooden rod, there are markings for three different scales. The first set of markings divides the rod into 10 equal parts; the second set of markings divides the rod into 12 equal parts; the third set of markings divides the rod into 15 equal parts. If one cuts the rod at each marking, how many pieces of wood does one get? (6) Question 7

A fish tank is 100 cm long, 60 cm wide and 40 cm high. If it is i s tilted, as shown, resting on the 60 cm edge, then the water reaches the midpoint of the base. If it is i s then put down so that the base is horizontal again, what is the depth of the water?

40 cm 100 cm

60 cm

(6) Question 8

A cuboctohedron is a polyhedron that can be formed by slicing a cube at the midpoints of all its edges. In the diagram, one of the vertices has been sliced off. Find the surface area of the cuboctohedron formed from a cube having a side of length 4 cm. 4 cm (6) Question 9

Find the length of the hypotenuse of a right-angled triangle in terms of its area A and its perimeter P. (8)


Question 10

Find the smallest two digit number that satisfies the foll owing conditions:

o

The number is not an odd number.

o

It has exactly four factors, including itself and 1.

o

If you reverse the digits, a prime number is formed.

o

The sum of the digits is a two digit prime number.

o

One of the digits is a square number. (8)

Question 11

Find the smallest integer that is divisible by all integers from 2 to 13 except for one pair of two consecutive integers in this range. (8)

Question 12

Consider the following sequence in which th

and t n is the n term:

t1 =

(a)

Find

t 2007

.

(b)

Find

t 2008

.

1; t2

=

2 and t

n

=

t 1

is the first term, t 2 is the second term

 n−3  t  n −1 

n−

2

, where n > 2 .

(8)

Question 13

Observe that 39 = 3 × 9 + 3 + 9 . (a)

Find all other two-digit numbers which are equal to the product of their digits plus the sum of their digits.

(b)

Prove that there are no three-digit numbers which are equal to the product of their digits plus the sum of their digits. (8)


Question 14

(a)

Calculate:

4 × 3 × 2 ×1 + 1 .

(b)

Using the above, determine

(c)

Find and and prove prove the general general formula for for the square root of the product product of of four consecutive

51× 50 × 49 × 48 +1 without actually multiplying out 51× 50 × 49 × 48 .

integers plus 1. (8) Question 15

In a killer Sudoku, just like in a conventional Sudoku, the aim is to fill each row, each column and each 3-by-3 block with all the numbers from 1 to 9. Your clues in a Killer Sudoku are the caged numbers that represent the sum of the numbers within that cage. Duplicate numbers cannot exist within a cage. 11

Keep in mind that

could be any of the following combinations:

11

9

2

2

9

8

3

3

8

7

4

4

7

6

5

5

6

11 16

4

7

16

A 4

5

19

9

8

B

15

5

11

13

C 18

D

16

11

6

E 20

23

11 5

12

17 11

15

15

6

19

12

6 11

7

8

6

16

23

10

13

11

11

In the Sudoku above find the numbers represented by A, B, C, D and E. (8)

Total: 100

THE END

Please turn over for the answer sheet for questions 2, 4 and 5 5 Compiled by and downloaded from www.erudits.com.ng

ANSWER SHEET FOR QUESTIONS 2, 4 AND 5

Name:

Please hand in together with Answer booklet.

School:

Grade: Question 2

32

28

26

30

Question 4

0 2 2 1 4 1

4 2

2

1

Question 5

Alpha Beta Circa 28

Dode 27

43

Epsilon 25

Flora 22

38

Gamma 24

Hexa


THE HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION. Sponsored by HARMONY GOLD MINING.

FIRST ROUND 2007 JUNIOR SECTION: GRADES 8 AND AND 9 20 MARCH 2007 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth 5 marks. There is no penalty for an incorrect or an unanswered question. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. The centre page is an information informat ion and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 60 minutes to complete the question paper. 9. Answers and solutions will be available at www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X173, PRETORIA, 0001. TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


1.

1−

1 2

A. 2.

× 2 is

1 4

4

E. 1

6

D. 2

E. 1

B. 10,31

C. 9,51

D. 9,84

E. 9,78

B.

1 4

C.

3 8

D.

2 11

E. 11%

B. 330

C. 345

D. 360

E. 375

B. 20

C. 30

D. 50

E. 60

A lady has 17 buttons in a bag. She has 8 green ones, 5 blue ones and 4 red ones. What is the minimum number of buttons she must take out of her bag, without replacing them, in order to ensure that she has one of each colour? A. 4

8.

C. 3

3

Vusi invests R2500 at the beginning of the year. He is promised that his investment will grow by 5% of the original investment per year. After how many years will his investment have doubled? A. 10

7.

B. 5

6

3 −2

D.

During a 3-day festival, the number of visitors trebled each day. If the festival ended on day 3 with 3105 visitors on that day, then how many visitors attended on day 1? A. 315

6.

2

C. 0

Which of the following numbers is the smallest? A. 0,125

5.

1

Green missed the 100 metre sprint record by 0,03 seconds. What is the record, in seconds, if Green’s Green’s time was 9,81 seconds? A. 10,00

4.

B.

Write down the units digit of A. 7

3.

equal to:

B. 8

C. 12

D. 14

E. 16

( 6 y − 10 ) ( 3 y + 4 ) and ( 6 y + 2 ) ( 3 y − 4 ) are both even numbers on a number line with one even number between them. The larger of the two numbers is: A. 98

B. 100

C. 102

D. 104

E. 106


9.

Dan bought a box of apples. While unpacking he found that 12 apples, which is the same as 15% of all bought apples, had gone gone bad. How many apples would represent 35% of all bought apples? A. 5

10.

B. 12

C. 28

D. 33

E. 62

ABCD is a rectangle with mid-point of DC at E, as shown. What fraction of the area of the rectangle does the area of ∆ BEG (the shaded region) represent if GE  BC? A

B G

C D E

A.

11.

1 5

1 8

C.

1

D.

4

2 7

E.

2 9

Semi-circles Semi-circle s with diameter of 28cm are drawn on AD and BC, as shown. If AB=DC=77 cm, then the perimeter of the figure is: A

B

D

C

A. 154 12.

B.

B. 154+14 π

C. 154+28 π

D. 182 π

E. 154+56 π

Joy found that two numbers added up to 20. Five times the one number is ten more than four times the other number. What is the product of the two numbers? A. 84

B. 91

C. 96

D. 99.

E. 100


13.

If p=5q and 10q = 3t, then t =…. A.

14.

1 3

p

2 3

p

C. p

D.

3 2

p

E. 3p

The total number of triangles in the diagram below is:

A. 54 15.

B.

B. 60

C. 66

D. 72

E. 78

Linda works a 40-hour week (standard). The overtime wage rate is 1

1 2

times the standard wage rate. In the first week of the month she worked 10 hours overtime and received total wages of R2 750. What amount does she earn in a week if she does not work overtime? A. R1 000

16.

B. R1 500

C. R2 000

D. R2 500

E. R3 000

A group of girls shared 54 red beads, 90 green beads and 108 blue beads such that each girl gets an equal number of beads of each colour. What is the maximum number of girls in this group? A. 6

B. 12

C. 15

D. 18

E. 21


17.

.

B

.

A

In the above Cartesian plane, 1 cm represents 1 unit on both axes, which are not shown. If B’s co-ordinates are (9;4), then which of the following coordinates would represent point A? A. (3;-1) 18.

D. (-9;4)

E. (-1;6)

B. 4

C. 3

D. 2

E. 1

In a sports club 32 do not play soccer, 50 do not play volleyball and 40 play volleyball. How many play soccer? A. 8

20.

C. (5;9)

Sipho makes 33 straw hats in 4 days. Pretty makes 33 straw hats in 6 days. Abongile makes 33 straw hats in 3 days. If all 3 work work together and at their usual rate, how many days will it take to make 99 straw hats? A. 5

19.

B. (9;7)

B. 32

C. 40

D. 50

E. 58

A large solid cube is built of identical smaller cubes such that more than half of the small cubes are not visible from outside. What is the smallest number of small cubes that are used to build the large cube? A. 1728

B. 1331

C. 1000

D. 729

E. 125


HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION. Sponsored by HARMONY GOLD MINING.

SECOND ROUND 2007 JUNIOR SECTION: SECTION: GRADES 8 AND 9 15 MAY 2007 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: 3.1 Each correct answer is worth 4 marks in Part A, 5 marks in part B and 6 marks in part C. 3.2 For each incorrect answer one mark will be deducted. There is no penalty for unanswered questions. 4. You must use an HB pencil. Rough paper, a ruler and an eraser are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily necessaril y drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 120 minutes to complete the question paper. 9. Answers and solutions will be available at www.samf.ac.za/samo/

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 392-9323 E-mail:[email protected] E-mail:[email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


PART A: 4 MARKS EACH

1.

A two-digit number is divisible divisibl e by 18; 30 and 45. The smallest such number is: A) 60

B) 90

C) 120

D) 150

E) 180

th

2.

If the 29 day of a month falls on a Thursday, what is the date of the second Saturday in that month? A) 9

B) 10

C) 11

D) 12

E) 13

3. A

10

B

135° 5

D

C

y

The length of y in trapezium ABCD is given by: A) 15

4.

C) 25

D) 30

E) 35

Sipho is three times as old as Pam, Pam's age can be found by adding the digits of Sipho’s age. The sum of their ages is (A) 48

5.

B) 20

(B) 56

(C) 32

(D) 36

(E) 60

An angle x is 30° more than half of its complement. What is the supplement of x? A) 70°

B) 80°

C) 135°

D) 160°

E) 130°

PART B: 5 MARKS EACH

6.

x, y and z are three integers such that

(A) 3

7.

(B) 54

(C) 32

xy

=2

and

(D) 4

3

xyz

=6.

The value of z is

(E) 60

If the sum of a two-digit number is subtracted from the number, then the answer is 45. How many two digit numbers have this property? (A) 10

(B) 15

(C) 3

(D) 4

(E) 18


8.

A collection of 25 coins, whose total value is R2.75, is composed of 5c, 10c, and 20c coins. If the 5c coins were 10c coins, the 10c 10c coins were 20c coins, coins, and the 20c coins were 5c coins, the total would be R3.75. How many 20c coins are there in the collection? (A) 3

9.

(E) 7

B) 70

C) 75

D) 80

E) 85

(B) 75

(C) 85

(D) 168

(E) 196

Maggie rolls an ordinary six-sided die repeatedly, keeping track of each number as she rolls, and stopping as soon as any number is rolled for the fifth time. If Maggie stops after her ninth roll, and the sum of these numbers she has rolled is 20, then how many combinations of numbers could she have rolled? (A) 10

12.

(D) 6

Steve was at at a party and knew that three guests were born on the same day of the week and in the same month of the year. year. He also knew that all all the guests were born in the first six months of the year. year. What is the least number of people (including Steve) that could have been at the party? (A) 70

11.

(C) 5

Study the pattern below and find the value of n if n is a natural number: 2 2 2 6 + 8 = 10 2 2 2 8 + 15 = 17 2 2 2 10 + 24 = 26 2 2 2 12 + 35 = 37 . . . 2 2 2 18 + n = x A) 65

10.

(B) 4

(B) 12

(C) 14

(D) 15

(E) 16

D) 260

E) 270

Observe the following computations: 2+4=6 2 + 4 + 6 = 12 2 + 4 + 6 + 8 = 20 2 + 4 + 6 + 8 + 10 = 30 The value of: 2 + 4 + 6 + 8 + …….+ 30 is given by: A) 230

B) 240

C) 250


13.

The lengths of three consecutive sides of a quadrilateral are equal. If the angles included between these sides have measures of 60 ° and 100 ° , then what is the measure of the largest angle of the quadrilateral? (A) 130°

14.

(B) 145°

(C) 155°

(D) 160°

(E) 165 °

A squared rectangle is a rectangle whose interior can be divided into two or more squares. An example of a squared rectangle is shown. The number written inside a square is the length of a side of that square. Determine the area of the squared rectangle shown.

5

15 (A) 1024

15.

(B) 1056

(C) 2005

(D) 2120

(E) 2475

A giant watermelon weighs 50 kg of which 98% is water. After lying in the sun some of the water evaporates so that the water now makes up 96% of the weight. The new weight of the watermelon in kg is: (A) 30

(B) 44

(C) 32

(D) 22

(E) 25

PART C: 6 MARKS EACH

16.

P is the remainder when each of the numbers 478, 392, and 263 is divided by N , where N is is an integer greater than 1. The value of N – P is:

(A) 5

17.

Did you know

(B) 28

27

=

(C) 33

(D) 38

9×3 = 9 × 3

=3

(E) 43

3 ?

If the numbers below are arranged from smallest to largest, then the number in the middle is (A) 3 + 3 + 10

(B) 5 + 12

(D) 3 + 27

(E)

(C)

78

48 + 3


18.

80 switches in a row operate 80 lights. The ‘on’ and ‘off’ switches operate automatically, in the following sequence: All lights are switched ‘ on’ • • Every second light is switched ‘off’ • Every third light is switched either ‘ on’ or ‘off’ • Every fourth light is switched either ‘ on’ or ‘off’ th And so the pattern goes on until every 80 light is switched either ‘ on’ or ‘off’ How many lights will be ‘off’ after the last operation? A) 60

B) 64

C) 72

D) 78

E) 80

19. B

A

C

D

The above diagram from A to D represents a flat passageway at an international airport. Each of the eight sections is 80m long. The shaded sections represent a conveyor system, on which a person stands if she/he wants to be moved from, for example B to C (80 metres). When the conveyor system is operational it takes Jessie 8 minutes to completely cover the passageway. passagewa y. When the conveyor system is not working it takes Jessie, by walking at the same average speed, 10 minutes to cover the distance. Approximately Approximate ly how many minutes will it take Jessie to move from A to D if the conveyor system is working and in addition, she walks on the conveyor system maintaining the same constant speed? A) 4.5

20.

B) 5

C) 5.5

D) 6

E) 7

Consider the following addition pattern:

1 + 1 1 + 11 1 + 111 1 + ……………… . . + 1 1 1 1…………..1 1 row 63 _________________________ …………………………………(Answer) rd

In the 63 row there are 63 '1s'. The sum of the digits in the answer is: A) 1953

B) 2016

C) 136

D) 261

E) 276


       Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION in collaboration with the SUID-AFRIKAANSE AKADEMIE VIR WETENSKAP EN KUNS, AMESA and SAMS. Sponsored by HARMONY GOLD MINING.

Third Round 2007 Junior Section: Grades 8 and 9 Date: 6 September 2007 Instructions • Answer all the questions. • All working details and explanations must be shown. Answers alone will not be awarded full marks. • This paper consists of 15 questions for a total of 100 marks as indicated. • The neatness in your presentation of the solutions may be taken into account. • The time allocated is 4 hours. • No calculator of any form may be used. • Answers and solutions are available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, X173, PRETORIA, 0001 TEL: (012) 392-9323 392-9323 FAX: (012) 320-1950 E-mail:[email protected] E-mail: [email protected]


Question 1 th

Find the 2007 letter in the sequence: ABCDEFGFEDCBAABCDEFGFEDCBAABCD………..

(4)

A

Question 2

Given that AB = AF, find the relationship between a, b and c.

B F c

b

a

E

D

C

(4) Question 3

The figure shows four circles with the same centre and radii of 1, 2, 3 and 4 respectively.

Ring 1 Ring 2

(a)

Calculate the area of ring 3 in terms of

(b)

If there are 2007 rings, find the

.



Ring 3

area of the outside ring in terms of . (6) Question 4

Water drips at a constant rate into a container, as shown in the diagram. (a) Draw a graph of the height of the water in the container against time, until the container is full. (b) Describe your graph in words.

r e t a w f o t h g i e H

(6)

Time 1 Compiled by and downloaded from www.erudits.com. www.erudits.com.ng ng

Question 5

The sum of the factors of 120 is 360. Find the sum of the reciprocals of the factors of 120. (6) Question 6

In the right angled triangle below, for which positive integer value(s) of x is the hypotenuse y an integer? y

12

x (6) Question 7

The following question was given to a class for homework: 8 x

"In the diagram, diagram, find the the length of x "

Saskia did it as follows:

15

p

8

q x

p 2

+

x2

=

64

(1)

q2

+

x2

=

225

(2)

82 + 15 152 ∴ p +

15 (2) − (1 (1)

q

2

−

( p + q) 2

=

q = 17

p

2

(3)

= 161

(q − p)( q + p ) = 16 1 61 ∴q −

(3) + (4) :

161 17

(4)

450

2q = ∴q =

From (2) :

p=

17 450

34 450 2 225 − ( ) = x 2 34 120 ∴ x = 17

Anil did it in a far quicker and elegant way. Find Anil's solution to the problem. (6)


Question 8

In the diagram, the log A has radius R. A hole of radius r is drilled through the centre of log A at right angles to the axis. Another log B of radius r passes through the hole. Find the length S in terms of R and r .

Log A

Length S in in terms of R and r

Log B

2 r

S

2 R (6) Question 9

Two rugby teams (thirty players) have been selected to play for their school. Five of the players speak Sotho, Afrikaans and English. Nine of them speak only Sotho and English. Twenty speak Afrikaans, of which twelve also speak Sotho. Eighteen speak English. No one speaks only Sotho. How many players speak only English?

(8)

Question 10

Molly is in Grade 3 and in a test on fractions she wrote the following: 1 1 2 +

2 (a)

=

3

5

If Molly used the same method, what would her answer to the following sum be? 2 5 +

=

3 (b)

4

Prove that there are no integer values for a and b such that 1 1 2 +

a

(c)

=

b

a+b

There are however integer values for a, b, c and d such such that a c a+c +

b

=

d

b + d

make the Find a set of values for a, b, c and d (a ≠ b ≠ c ≠ d ) that equation above true.


(8)

Question 11

(a)

Find the value of 1 + 2(1 + 2(1 + 2(1 + 2)))

(b)

Find the value of 1 + 2(1 + 2(1 2(1 + …… ) … ))))

{

(8)

2007 brackets Question 12

In the game “Move out” , two players take it in turns to move any one of the three counters on the board any number of squares to the right, beginning from their ‘start’ blocks, until all three counters are in the ‘end’ blocks. The last player to move a counter loses. Start 1

2

3

4

5

6

A

End 7

Start 1

2

3

4

5

6

7

B

1

2

3

4

5

6

7

C

Start

(a) (b)

8

9

10

End 8

9

8

9

10

End 10

If it was your your turn to play in the the game above, which counter counter would you move move and to which position, to guarantee you a win. Explain your answer. (8)

Question 13

Observe the following patterns: (a)

23 × 28 = 20 × 31 + 3 × 8 = 620 + 24 = 644 51× 59 = 50 × 60 + 1× 9 = 3000 + 9 = 3009 Calculate 65 × 69 in the same way.

(b)

Prove that the pattern holds for any two two 2-digit 2-digit numbers numbers that have the same tens-digit. tens-digit. (8)

Question 14

In a computer game, you have to score the largest possible number of points. You score 9 points each time you find a jewel and 5 points each time you find a sword. There is no limit to the number of points you can score. Of course it is impossible to score 6 or 11 points. (a) What is the largest number number of points points impossible impossible to score? (b) Prove that this is in fact the largest number. number. (8) Question 15

(a) Is the following divisible by 3? 3

3

3

3

3

3

2007 – 2006 + 2005 – 2004 ……… – 2 + 1 (b)

Prove your answer. (8) Total: 100 THE END



FIRST ROUND 2006 JUNIOR SECTION: GRADES 8 AND 9 23 MARCH 2006 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth 5 marks. There is no penalty for an incorrect or an unanswered question. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 60 minutes to complete the question paper. 9. Answers and solutions will be available at www.samf.ac.za

DO NOT TURN THE PAGE P AGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X173, PRETORIA, 0001. TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


1.

6 ×111 − 3 ×111 is equal to:

(A) 222

2.

(B) 333

3

(A)

1 3

The number 1 2

3 10

(C)

; 0,313 0,313; 0,303 0,303

3 10

(D) 31%

(E) 0, 303

1 2

×

1 2

÷

1 3

(C) R525

(D) R600

(E) R675

is equal to:

(C) 2

(D)

1 4

(E)

A three-digit number is divisible by 8, 12 and 30. possible number is: (B) 120

(C) 240

(D) 360

3 4

The smallest

(E) 480

The hundreds digit of the product 7777 × 9999 is: (A) 2

7.

(B) R450

(B) 1

(A) 108

6.

(E) 666

“Diamond Stores” gives its customers four points for every R75 spent. Sipho earned 36 points. How much did Sipho spend at Diamond Stores?

(A)

5.

; 31%; 31%;

(B) 0,313

(A) R375

4.

(D) 555

If the fractions are arranged from lowest to highest then the middle fraction is: 1

3.

(C) 444

(B) 3

(C) 4

(D) 5

(E) 6

Each child of the Robertson family has at least three sisters and at least one brother. The minimum number of children in this family is: (A)

4

(B)

5

(C)

6

(D)

7

(E)


8

8.

The number of whole numbers that lie between 42 and 43 is: (A) 45

9.

(B) 46

(C) 47

(D) 48

(E) 49

A watch keeps exact time, but it has only an hour hand. When the 2

hour hand is

5

of the distance between the 4 and the 5, the correct

time is: (A) 04:10

(B) 04:20

(C) 04:22

(D) 04:24

(E) 04:26

_____________________________ _________________ _________________________ ____________________________ _____________________ ______ 10.

A protest march goes through town from the mall (M) to the community centre (CC).

M

E

S CC

If the march can only travel east or south, then the number of different possible routes is: (A) 6 (B) 10 (C) 4 (D) 8 (E) 9 _____________________________ _________________ _________________________ ____________________________ _____________________ ______ 11.

If the number 12 2 × 4 × 3 is written in the form number, then n is: (A) 12

12.

(B) 24

(C) 36

n

(D) 48

3

, where

n is

a natural

(E) 60

A bag contains six white beads, eight black beads and two green beads. A lady draws beads out of the bag without looking at them and without putting them back. What is the least number of beads that she must take out of the bag to ensure that she has taken out three beads of the same colour? (A) 3

(B) 5

(C) 7

(D) 9

(E) 11


13.

The graph below represents the motion of a car. The graph shows us that the car is: Distance

Time (A) accelerating (C) travelling north-east (E) travelling at a constant speed

(B) standing still (D) travelling uphill

_____________________________ _________________ _________________________ ____________________________ _____________________ ______ 14.

The product of two consecutive whole numbers is p . The square of the larger number minus the smaller number is: (A)

15.

p

2

(B)

p − 1

(C)

p

2

(D)

p + 1

(E)

2 p + 1

A vendor has an equal arm balance and four weights that she uses to weigh her fruit. The weights are are 1kg, 2k, 4kg and 8kg. If the weights are only placed on one end of the balance and the fruit is placed on the other end, how many different weight combinations can she use?

(A) 15

(B) 13

(C) 11

(D) 9

(E) 7


16.

In the given regular octagon, the size of angle x in degrees, is:

x

(A) 22 ½

17.

(B) 45

(D) 90

(E) 112 ½

Find the value of: 1 1× 2

(A)

18.

(C) 67 ½

0

−

(B)

1 2×3 1 49

−

1 3×4

−

1 4 ×5 1

(C)

51

−

−

!

(D)

1 49 × 50 1 40

1

(E)

50

Consider the following triangular arrangement of numbers. 1 2 4 7 11 •

3 5

8 12

•

6 9

13 •

10 14

•

15 •

•

The middle number of the 51st row is: (A) 1352

19.

(B) 1301

(C) 1251

(D) 1275

(E) 1326

42 equal sized matchsticks are used to make the figure below. The figure is a parallelogram, parallelogram, which includes the longer diagonal. In how many different ways can you make matching figures using all 42 matchsticks?

(A) 4

(B) 6

(C) 10

(D) 12

(E) 14


20.

The area of the shaded triangle, written as a fraction of the regular hexagon is:

(A)

1 6

(B)

1 5

(C)

1 4

(D)

1 3

(E)


1 2


SECOND ROUND 2006 JUNIOR SECTION: GRADES 8 AND 9 17 MAY 2006 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these these is correct. 3. Scoring rules: 3.1 Each correct answer is worth 4 marks in Part A, 5 marks in part B and 6 marks in part C. 3.2 For each incorrect answer one mark will be deducted. There is no penalty for unanswered questions. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 120 minutes to complete the question paper. 9. Answers and solutions will be available at www.samf.ac.za/samo/

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


PART A: 4 MARKS EACH

1.

15% of R560 – 15% of R500 is: (A)

R13

(B)

R12

(C)

R11

(D)

R10

(E)

R9

2. Y V

U

W

X

A piece of paper is is cut out and labeled as shown in the the diagram. It is folded along the dotted dotted lines to make an open open box. If the box is placed on a table so that the top of the box is open, then the label at a t the bottom of the box is: (A)

3.

U

(B)

V

(C)

W

(D)

X

(E)

Y

If the numbers 3 9 ; 5 ; 1; 2; 3 are arranged in in order order of magnitude, then the middle number is : 3

(A)

9

(B) (B)

5

(C)

1

(D)

2

(E)

3

4. Oudsthoorn

Uniondale x A T

George

The map shows roads joining Uniondale, George and Oudtshoorn via the T-junction at T. At point A there is a sign which shows that A is 34 km from T, 60 km from George, George, and 68 km from Oudtshoorn Oudtshoorn via T. The distance, in kilometres, via T, from Oudtshoorn to George is: (A)

148

(B)

122

(C)

60

(D)

78

(E)


52

5.

R

Q

P

A motor has a sequence of 3 wheels that drive a windmill. Wheel P has radius ra dius 36 cm, wheel Q has radius 12 cm and wheel wheel R has radius 6 cm. The wheels all touch each other and rotate without slipping. If wheel P turns 360 ° in a clockwise direction, then wheel R will turn: (A) (B) (C) (D) (E)

6 x 360 ° in an anti-clockwise a nti-clockwise direction 3 x 360 ° in an anti-clockwise a nti-clockwise direction 2 x 360 ° in an anti-clockwise a nti-clockwise direction 3 x 360 ° in a clockwise direction 6 x 360 ° in a clockwise direction

PART B: 5 MARKS EACH

6.

A popular puzzle game is called Harmony. In this game, you are given a 4 x 4 grid which is further divided into four bordered 2 x 2 squares. You are given some letters in the grid. You have to fill in the letters G, O, L and D in each row, column c olumn and 2 x 2 square such that no letter appears more than once in each row, each column and each 2 x 2 square. L

G

D

D

O

X

L

The letter marked X is: (A)

G

(B)

O

(C)

L

(D)

D

(E)


H

7.

8.

On earth there are about 10 000 000 000 000 000 ants and 6 000 000 00 0 000 humans. The ratio of humans to ants is approximately equal to: (A)

60 000 to 1

(B)

1 666 667 to 1

(D)

1 to 1 666 667

(E)

1 to 60 000 000

1 to 6000

It takes a car 11 minutes to travel a distance of 15 kilometers. If the car travels at an average speed of x km/h, then: (A) (D)

9.

(C)

50 ≤ x < 60 80 ≤ x < 90

× y

Let x ∗ y =

+ y

(B) (E)

60 ≤ x < 70 90 ≤ x < 100

, for example, 4 ∗ 3 =

4×3 4+3

=1

(C)

70 ≤ x < 80

remainder 5 .

If 5 ∗ x = 2 remainder 5 , then x is: (A)

10.

9

(B)

8

(C)

5

(D)

12

(E)

7

The arrangement below is called Pascal’s Triangle 1 1 1

1 2

1

3

1

4

1 3

6

1 4

1

…………………………………………………………………………….. The sum of the numbers in the first row is 1. The sum of the numbers in the first 2 rows is 3. The sum of the numbers in the first 3 rows is 7, etc. If this triangle arrangement is continued then the sum of the numbers in the first 15 rows is:

(A) 214

−

1

(B) 2 15 +1

(C) 215 − 1

(D) 216 +1

(E) 214 +1


11.

The number n is a perfect square. What is the next perfect square bigger than n ?

(A)

n

2

(D)

12.

(B)

+1 n

2

+ 2n + 1

If the fraction then ,

3 7

=

3 7

13.

7

(E)

n + 1 n

2

(C)

+2

n

2

+n

n +1

is written as an infinite decimal fraction,

0, a1 a 2 a3 ,... where

The digit in the

(A)

2

a2006 position

(B)

1

a1 , a 2 , a 3,.... are

digits.

is:

(C)

4

(D)

5

(E)

2

The six-digit number 4m61n2 is divisible by both 11 and 4. The number of different combinations of m and n that satisfy the above condition is: (A)

14.

4

(B)

6

(C)

8

(D)

10

(E)

12

Three different digits are used to make all possible three-digit numbers. Of the three digits, digits, one is 4 and one is three three more than another. If the sum of all such three-digit numbers is 2886, then the three digits are: (A)

1; 2; 4 (B)

4; 5; 7 (C)

3; 4; 6 (D)

2; 4; 5 (E)


4; 6; 9

15.

In the diagram, lengths are shown. The area of the shaded region is:

2

(A)

4

3

(B)

2

2

(C)

2

2

x

(D)

2

3

(E)

5 x 2 2

PART C: 6 MARKS EACH

16.

The four-digit integers from 1994 to 2006 are written consecutively and the number, N=1994199519……..20052006 is formed. If 3k is the highest power of 3, by which N is divisible, then k is equal to:

(A)

17.

0

(B)

1

(C)

2

(D)

3

(E)

4

132x is the notation to show that we are working in base x. The number 30 (in base 10), can be expanded as 1 x 42 + 3 x 4 + 2 x 1. Therefore Therefore 30 can be written written as 1324. If it is true that 14y x 14y = 232y , then y is:

(A)

5

(B)

6

(C)

7

(D)

8

(E)


9

18.

The sum of the lengths of edges of a rectangular prism is 68 cm. If the lengths of the sides are whole numbers and the area of the base is 18 cm2, then the possible volumes volumes of the prism are:

19.

(A)

54 and 72

(B)

(D)

108 and 144 (E)

108 and 72

(C)

27 and 144

216 and 288

15 one centimetre cubes with all blue faces, 16 one centimetre cubes with all yellow faces, and 33 one centimetre cubes with all black faces are glued together to form one large cube. What is the least number of one centimetre squares on the surface of the larger cube that are black?

(A)

22

20.

(B)

24

(C)

.

.

.

.

.

.

.

.

.

26

(D)

32

(E)

34

Nine points lie in a plane, as shown above. If any 3 points are joined joined to form a triangle, then the number of all possible triangles that can be drawn are:

(A)

72

(B)

84

(C)

64

(D)

78

(E)


76

HARMONY SOUTH SOUTH AFRICAN MATHEMATICS OLYMPIAD Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION in collaboration with the SUID-AFRIKAANSE AKADEMIE VIR WETENSKAP EN KUNS, AMESA and SAMS. Sponsored by HARMONY HARMONY GOLD MINING.

Third Round 2006 Junior Section: Grades 8 and 9 Date: 7 September 2006 Instructions • Answer all the questions. • All working details and explanations must be shown. Answers alone will not be awarded full marks. • This paper consists of 15 questions for a total of 100 marks as indicated. • The neatness in your presentation of the solutions may be taken into account. • The time allocated is 4 hours. • No calculator of any form may be used. • Answers and solutions are available at: www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X11, ARCADIA, 0007 TEL: (012) 392-9323 FAX: (012) 320-1950 E-mail:[email protected]


1.

Write the number 1 000 000 as the product of two positive integers neither of which has any zeros in it.

(4)

2.

Find the product of 968 880 726 456 484 032 and 125.

(4)

3.

(a)

Find five positive integers, not necessarily necessarily different, whose sum is equal to their product.

(b)

4.

Show Show that the problem in (a) has at least three solutions.

Find all integer solutions of the equation:

1

+

x

1

=

y

1 2

(6)

.

(6)

∧

5.

In the figure below, AD = DC; ED = BD and BDC = 40 . B

A

C

E o

40

∧

Find the size of ABE . D 6.

(6)

Evaluate the following sum: 2 3 1 2 1 2 3  1 2 3 4   1 + +  + +  + + +  + + + +⋯ +  2 3 3  4 4 4  5 5 5 5  100 10 100  100 10

1

+

+⋯ +

99  . 100  (6)

7.

Write the numbers a = 0,16 ;

b = 3 0, 063;

c = 5 0, 01025 and d

=

(0, 2) 2

in increasing order.

8.

(6)

An equilateral triangle of side 2 is rolled along the inside of a square of side 6 until the vertex A returns to its original position, as shown.

6

Determine the length of the path

A

the point A has travelled. Leave your answer in terms of π.

(6) 2

2

6 1 Compiled by and downloaded from www.erudits.com.ng

9.

The letters M, A, T, H, S denote denote positive real numbers such that M × A = 12 , T × H = 30 , A × H = 24 , A × T = 20 and H × S = 42 . Find the value of M × A ×T × H × S .

(6) 10.

A string of 2006 digits begins with the digit 6. Any number formed by two consecutive digits is divisible by 17 or 23. Write down the last five digits.

11.

(6)

Show that it is possible to fill in a 5 by 5 table using numbers such that the sum of all of them is positive, but the sum of any four numbers forming a 2 by 2 block is negative.

(6) 12.

The figure alongside has 3 oblique lines and 5 horizontal lines and contains 15 triangles of different shapes and sizes. How many triangles (of any size) does a similar figure with s oblique lines and h horizontal lines have?

Copy this table into your answer book and fill in as many numbers as you need to find the number of triangles with s oblique lines and h horizontal lines.

Number of oblique

Number of horizontal

Number of triangles

lines

lines

3

5

15

s

h

? (8)


13.

PINE , In the alphanumeric puzzle OAK + OAK + … + OAK = PINE

equal letters correspond to equal digits and different letters correspond to different digits, and O does not stand for 0. Find the maximum possible number of OAK’s that will satisfy the equation and explain why the number you have found is in fact a maximum.

14.

(8)

Pizza

Andy and Bongi order pizza. The pizza is divided into four pieces with two straight perpendicular cuts that do not pass through the centre. Andy can choose

P

Q

S

R

either P and R or Q and S. Which pieces should Andy choose in order to get more pizza than Bongi? Justify your answer.

(8)

15.

(a)

Find the sum of the digits of the product

99999 × 66666.

(b)

Find an expression in n and/or k for the sum of the digits of

(2)

the product below and prove that it holds for all n and/or k:

} 999 ... 9

×

n 9's

(c)

} kkk ... k n k 's

(where n, k ∈

ℕ and

0 < k < < 10)

(8)

Find an expression for the sum of the digits of

} 999 ... 9 n 9's

×

} kkk ... k m k 's

(where n, m, k ∈ ℕ and 0 < k < < 10)

(4) TOTAL: 100

THE END



FIRST ROUND 2005 JUNIOR SECTION: GRADES 8 AND 9 15 MARCH 2005 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth 5 marks. There is no penalty for an incorrect or an unanswered question. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 60 minutes to complete the question paper. 9. Answers and solutions will be available at http://science.up.ac.za/samo/

DO NOT TURN THE PAGE P AGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X173, X173, PRETORIA, 0001 TEL: (012) 392-9323 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


1.

Calculate: 2 − 2 × 2 + 2 . A)

2.

0

B)

1

C)

2

D)

4

E)

6

In 2004, 16 June falls on a Wednesday. Wednesday. On what day of the week will 16 June fall in 2010? A) D)

Monday Thursday

B) E)

Tuesday Friday

C)

Wednesday

3. In a magic square the sum of the numbers in each row, in each diagonal and in each column are equal. In this magic square the value of x is: 9

A)

4.

8

C)

11

D)

12

E)

13

39

B)

40

C)

42

D)

45

E)

48

E)

45

The sum of the digits of the following product is: 999 × 555 A)

6.

B)

x

If half of a number is 30, then three-quarters of that number is: A)

5.

7

14 10 6

9

B)

18

C)

27

D)

36

In the given figure the area of the shaded region, in cm2 , is 6 cm

12 cm

10 cm

A)

40

B)

44

C)

50

D)

54


E)

60

7. Three positive integers have have a sum of 28. product that these integers can have is: A)

720

B)

756

C)

792

D)

The greatest greatest possible

810

8. Jack was trying to tessellate regular pentagons. following figure.

E)

852

He managed managed the

a

The size of angle angle ‘a ’ is: A)

36°

B)

30°

C)

24°

D)

18°

E)

12°

E)

21

9. In what follows, ! and " are different numbers. When 503 is divided by ! the remainder is 20. When 503 is divided by " the remainder is 20. When 493 is divided by ! x " the remainder is : A)

10.

3

B)

7

C)

10

D)

20

If the area of the shaded region of the regular hexagon in the diagram below is 36 cm2, the area of the whole hexagon in cm2 is:

A)

90

B)

108

C)

117

D)

126


E)

144

11.

The distance on a map between between Harmony town and Sterling is 24 cm. The actual distance between these two towns is 360km. 360km. The scale of the map is: A) D)

12.

1 : 150 000 1 : 150

C)

1 : 15 000

35

B)

50

C)

60

D)

70

E)

90

In a certain certain code S A T ; S E T and T E N are written as 1 5 4 ; 3 1 4; 3 2 1 respectively. respectively. The codes are are not necessary necessary in the order of the letters. Using the same code in the correct order, order, how would the word S E N T be represented? A)

14.

B) E)

In the given figure, the length of of AC, AC, in cm, is:

A)

13.

1 : 1 500 000 1 : 1 500

3421

B)

3451

C)

1234

D)

4123

E)

4321

The following following information is given given for a box with integer valued dimensions: Area of face A is 24 cm2 Area of face B is 40 cm2 The volume of the prism is 240 cm3. Find the area of face C in cm2.

A)

40

B)

48

C)

60

D)

72


E)

80

15.

In the adjacent figure six squares squares make make up the rectangle rectangle ABCD. ABCD. The perimeters of all six squares were added to give 72 cm. The area of ABCD, in cm2, is: A

B

D A)

16.

54

B)

C

48

C)

36

D)

30)

E)

24

In the following addition addition problem, each of the letters a ; b and c stands for a digit.

+

The value of A)

20

a a c

b c

c b a

4

a + b + c is:

B)

19

C)

18

D)

17

E)

16

17. x

y

tw o digit number. x is greater than y by 3. x and y are the digits of a two When this two digit number is divided by the sum of its digits the quotient is 7 with a remainder of 3. The sum of the digits of this t his two digit number is: A)

1

B)

3

C)

5

D)

7


E)

9

18.

The number of al possible triangles in the given figure is:

A)

19.

34

B)

30

C)

26

D)

24

E)

20

If you continue the given number number pattern, in what row and in what position in that row will the number 320 be? 1 2 4 7

3 5 8

6 9

-----------------------------------------------------

10

row 1 row 2 row 3 row 4

The answers are given in the order order of row ; position. A) D)

20.

18 ; 15 24 ; 20

B) E)

20 ; 20 25 ; 20

C)

27 ; 25

A lady, her brother, her son and her daughter (all related by birth) played volleyball. The worst player’s twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are of the same age. Who cannot be the worst player(s)? A) C) E)

brother only son and daughter only lady only

B) D)

daughter only lady and daughter only


Formula and Information Sheet

1.1

The natural numbers are 1; 2; 3; 4; 5; …

1.2

The whole numbers (counting numbers) are 0; 1; 2; 3; 4; 5; …

1.3

The integers are

2.

3.1

In the fraction

a b

,

…; –4; –3; –2; –1; 0; 1; 2; 3; 4; 5; …

a is

called the numerator and

b

the denominator.

Exponential notation: 2 × 2 × 2 × 2 × 2 = 25 3 × 3 × 3 × 3 × 3 × 3 = 36 a×a×a×a×

...

×a = a

n

( n fa factors of a )


3.2

Factorial notation: 1 × 2 × 3 × 4 = 4! 1× 2 × 3 × . . .

4 4.1 4.2

× n = n!

Area of a triangle is: rectangle is: square is:

1

× (base × height)

=

1

2 2 length × width = lw length × breadth = lb side × side = s2

(b.h)

4.3 4.4

rhombus is:

4.5

trapezium is:

4.6

circle is:

1 2 1 2

× (product × (sum 2

π r

of diagonals)

of parallel side ides) × height

(r = = radius)


5

Surface area of a:

5.1

rectangular prism is:

5.2

sphere is:

6

Perimeter of a:

6.1

rectangle is:

2lb + 2lh + 2bh ( h = height ) 4π r 2

2 × length 2l + 2b

+ 2 × breadt h

or 2l + 2 w

6.2

square is:

7.

Circumference of a circle is:

8.

Volume of a:

8.1

cube is:

8.2

rectangular prism is:

8.3 9.1

9.2

( w = width )

4s 2π r

s × s × s = s

cylinder is:

3

l ×b×h π r

2

h

Volume of a right prism is: area of cross-section × perpendicular height or area of base × perpendicular height Surface area of a right prism is: (perimeter (perimeter of base × h) + (2 × area of base)

10.

Sum of the interior angles of a polygon is: 180! ( n − 2)

11.

Distance

=

speed × time

(d = s × t )

Speed

=

distance ÷ time

(s =

Time =

distance ÷ speed

12

t

d

(t =

s

d = s × t

d

)

s =

t

)

t =

B

Pythagoras:

a

c

A 13.

d

[ n = number of of si s ides]

Conversions: 1 cm3 = 1 m" 1000 m = 1 km

b

; ;

If ∆ABC is a right-angled triangle, then a 2 = b 2 + c 2

C

1000 cm3 = 1 " 1000 g = 1 kg

;

100 cm = 1 m


d t d s

THE HARMONY SOUTH AFRICAN MATHEMATICS MATHEMATICS OLYMPIAD Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION. FOUND ATION. Sponsored by HARMONY GOLD MINING.

SECOND ROUND 2005 JUNIOR SECTION: GRADES 8 AND 9 10 MAY 2005 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, A, B, C, D and E. Only one of these is correct. 3. Scoring rules: 3.1 Each correct answer is worth worth 4 marks in Part Part A, 5 marks in part part B and 6 marks in part C. 3.2 For each incorrect answer one mark will be deducted. There is no penalty for unanswered questions. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answers on the sheet provided. 8. Start when the invigilator tells you to do so. You have 120 minutes to complete the question paper. 9. Answers and solutions will be available in June 2005 at www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG BAG X173, PRETORIA, PRETORIA, 0001 TEL: (012) 392-9323 E-mail: [email protected] Organisations involved: AMESA, SA Mathematical Society, SA Akademie vir Wetenskap en Kuns


PART A: four marks each 1.

Calculate: 1 −

A)

2.

1 4

1 2

×

1 2

B)

1

3

C)

2

4

A)

8

11

B)

12

7

x

10

C)

13

D)

14

E)

15

44%

B)

40%

C)

36%

D)

32%

E)

28%

The sum of any two-digit number and the number formed by interchanging the digits of that number is always a multiple of: A)

5.

8

3

Terry received an initial discount of 20% on the marked price of a used car. The salesman, who was very sympathetic, sympathetic, then gave Terry a further discount of 10% on the already discounted price. According to Terry's calculations, this was the same as a single discount of: A)

4.

E)

In the magic square below, the sum of the numbers in each of the rows, in each of the diagonals and in each of the columns is equal. Find x. 9 8

3.

1

D)

7

B)

9

C)

11

The sum of two numbers is – 8. numbers is 7. The numbers are: A) (–1 and –7) D) (1 and –7)

B) E)

D)

13

E)

15

The product of the same two

(1 and 7) (1 and –9)

C)

(–1 and 7)

PART B: five marks each 6.

Evaluate: 4,32 2

A)

7

B)

8

−

3,32 2

C)

9

+

1,36

D)

10


E)

11

7.

Calculate: 1−

A)

8.

1 2004

B)

1

1−

3

1

1−

4

3

1−

5

1 2005

1

C)

2004

1

2

D)

2005

2005

3

E)

2005

A rocket is travelling towards Mars at 60 000 km/h. If Mars is wil l the rocket take to get there? 1, 2 × 108 km away, how long will A) B) C) D) E)

Just less than 1 month. Just less than 2 months. Just less than 3 months. Just less than 4 months. Just less than 5 months.

∧

9.

In the figure B H C is equal to:

∧

A)

3A

∧

B)

360

0

−

∧

∧

C)

A

180

0

−

A

D)

2A

∧

E)

10.

A

Given

−

6 < x < 10,

−

2 < y < −

1 2

and a <

x y

<

b

Find: a × b A)

0

B)

– 50

C)

– 60

D)

– 120


E)

– 240

11.

Veli looks at a calendar for the year 20 xy He notices that April 20 xy has exactly four Mondays Mondays and four Fridays. 28 April April 20 xy would then fall on a: .

A) D) 12.

Saturday Friday

B) E)

Sunday Wednesday

A)

402

B)

400

C)

(100 cubes) 202

D)

200

E)

50

Consider a rhombus. If the sizes of any two angles that are not opposite to each other differ by 40 0, what is the size of one of the smaller angles?

A) 14.

Monday

The surface area of a cube is 6 cm2. If 100 cubes are put put together end to end as shown, what is the surface area of the resulting shape (in cm2)? ---

13.

C)

700

B)

800

C)

900

D)

1000

E)

1100

The average amount a waitress earns in tips is worked out by dividing the total amount earned over a certain period by the number of days worked. worked. Marianne earned earned an average of R78 R78 over a 20 day period. What is the total amount she must earn in tips over the next five days if she wishes to achieve an average of R90 per day for the 25 day period? A)

R600

B)

R690

C)

R780

D)

R510


E)

R90

15.

A

B

A and and B are the digits of a two-digit number. A > B; A and B differ by x. When this two-digit two-digit number number is divided by the sum of its digits, the quotient is 7 and the remainder is x. What is the value of x? A)

1

B)

3

C)

5

D)

2

E)

4

PART C: six marks each 16.

A three-digit number has (2 x − 1) as its units digits and x as its hundreds digit. digit. This number number is represented by 112 x + 29 . Find, in terms of x the tens digit. ,

A) 17.

x + 1

B)

x + 2

C)

x + 3

D)

x + 4

E)

x + 5

In the following multiplication, each * stands for a digit. These digits are not necessarily different. **5 1* * 2**5 13*00 ***00 **77* The second three-digit number is: A)

18.

133

B)

189

C)

181

D)

145

E)

147

Sipho purchased purchased six identical Mandela gold coins from a nonreputable dealer. He discovered discovered that that there were three counterfeit counterfeit (false) coins, which were lighter in mass than the real coins. coi ns. Using only the coins that he has bought and a scale balance, what is the least number of weighings that Sipho must do, to identify at least one counterfeit coin?

A)

1

B)

2

C)

3

D)

4


E)

5

19.

Eric finds the sum of the digits for every possible eight-digit number. Which sum does he find fi nd occurs most often? A) D)

20.

Both 27 and 28 Both 36 and 37

B) E)

41 39

C)

32

A quadrant quadrant of a circle, with centre centre 0 and radius radius 4 cm is drawn.

Two semi-circles are drawn as shown. What is the area of the shaded region? A) E)

B) 4π 4π − 8

4π − 2

C)

4π − 4

D)


4π − 6

TH THE HA RMONY SOUTH A FRIC RICA N MA THEM THEMA A TIC TICS OLYM LY MPIA PI A D Organised by the SOUTH AFRICAN MATHEMATICS FOUNDATION in collaboration with, AMESA, SAMS and the SUID-AFRIKAANSE AKADEMIE VIR WETENSKAP EN KUNS Sponsored by HARMONY GOLD MINING

Third Round 2005 Junior Section: Grades 8 and 9 Date: 12 September 2004 Instructions • Answer all the questions. • All working details and explanations must be shown. Answers alone will not be awarded full marks. • This paper consists of 15 questions for a total of 100 marks: Questions 1 to 10 are worth 6 marks each. Questions 11 to 15 are worth 8 marks each. • The neatness in your presentation of the solutions will be taken into account. • The time allocated is 3 hours. • No calculator of any any form may be used. used. • Answers and solutions are available at: ! !

www.samf.ac.za

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. PRIVATE BAG X173, PRETORIA, 0001 TEL: (012) 392-9323 FAX: (012) 392 9312 E-mail:[email protected]

Compiled by and downloaded from www.erudits.com www.erudits.com.ng .ng

1.

The area of one side of a rectangular box is 126 cm2. The area of another side of the rectangular box is 153 cm 2. The area of the top of the rectangular box is 238 cm 2.

What is the volume of the box? (6 Marks) 2.

In a Hockey tournament, each team played each other team once. The final league table was:

WINS

DRAWS

LOSSES

POINTS

BULLS

1

2

0

4

CHEETAHS

1

1

1

3

LIONS

1

1

1

3

SHARKS

1

0

2

2

If Sharks beat Cheetahs, then which of the statements are true?

(i)

Lions defeated Cheetahs, but lost to Sharks.

(ii)

Bulls won against either Cheetahs or Lions.

(iii)

In matches against Sharks, Cheetahs were more more successful than Lions.

(iv)

In matches against Lions, Lions, Bulls were more successful than Cheetahs.

(v)

Lions were undefeated, except against Cheetahs. (6 Marks)

3.

A sequence sequence has first term 12, after which every term is the sum of the squares squares of the digits of the preceding term. Thus the second term is 12 22

+

52

=

+

22

=

5 , the third term 52

=

25 , the fourth term

29 , and so on.

Find the 2005 th term of the sequence. (6 Marks)


4.

A 4 by 4 “Antimagic square” is an arrangement of the numbers

4

5

7

6

13

3

some order. The diagram shows an incomplete antimagic square.

11

12

9

Complete the square.

10

14

1 to 16 in a square grid such that the totals of each of the four rows and columns and the main diagonals are 10 consecutive numbers in

(6 Marks) 5.

Each letter in the addition addition sum shown below stands for a different digit, with S standing standing for 3.

SO + MANY SUMS

What is the value of Y × O ? (6 Marks) 6.

A circle of radius 2 cm with centre O , contains three smaller circles as shown in the diagram; two of them touch the outer circle, and touch each other at O, and the third touches each of the other circles.

O

Determine the radius of the third circle, in centimeters.

(6 Marks) 7.

Find the smallest natural number which when multiplied by 123 yields a product that ends in 2005. (6 Marks)

8.

Let A = 200420042004 200420042004

×

2005200520052005

and B = 200520052005 200520052005

×

20042004

Find the value of

A B

in simplest form. (6 Marks)


9.

Alfred, Brigitte, Carodene, Dolly, Dolly, and Effie play a game in which each is either a dog or a mouse. A dog’s statement is always false while a mouse’s statement is always true. Alfred says that Brigitte is a mouse. Carodene says that Dolly is a dog. Effie says that Alfred is not a dog. Brigitte says that Carodene is not a mouse. Dolly says that Effie and Alfred are different kinds of animals. How many dogs are there? (6 Marks)

positive integers to different letters and then multiplied their values 10. We have assigned different positive together to make the values of words. For exa examp mple le,, if C = 4; A = 8 and and T = 12, 12, then then CAT CAT = 4 × 8 ×12 = 384 384 .

Given that

HILL = 15 PHOTO = 8470 HILLS = 195 HIPHOP = 3300

Find the value of PITSTOP. (6 Marks)

number pattern using only the odd natural numbers: 11. Consider the following number 1 3 7 13

(Row 1) 5

(Row 2)

9 15

11 17

(Row 3) 19

(Row 4)

and so on.

Numbers

A

B C

are taken from the pattern, where A and B are two

adjacent numbers in a certain row, and C is the number in the next row, directly below, and between, A and B. If A + B + C = 2093, 2093, find the value of C. (8 Marks)


A ∧

∧

∧

∧

∧

∧

∧

12. In the diagram, A B C D E F G +

+

+

+

+

+

=

p × 90 o .

B

Find p. C G D

F E

(8 Marks)

13. Paul and James go out for a cycle and are 16 km from home when Paul runs into a tree

damaging his bicycle beyond repair. They decide to return home and that Paul will start on foot and James will start on his bicycle. After some time, James will leave his bicycle beside the road and continue on foot, so that when Paul reaches the bic ycle he can mount it and cycle the rest of the distance. Paul walks at 4 km per hour and cycles at 10 km per hour, while James walks at 5 km per hour and cycles at 12 km per hour. For what length of time should James ride the bicycle, if they are both to arrive home at the same time? (8 Marks) 14. The diagram shows a large rectangle whose whose perimeter is 300 cm. It is divided up

as shown into a number of identical rectangles, e ach of perimeter 58 cm. Each side of these rectangles is a whole number of centimeters. Show that there are exactly two ways of splitting up the rectangle a s described above.

(8 Marks)

15. A complete sentence of 20 words is concealed was was

now now

old old

was was

now

as

S ue Sue

S al Sal

half half

as

is

a

o old ld

is is

when

as

S ue Sue

S al Sal

third thir d

as

here. If you start with Sal and then follow the jumps jumps of of a chess chess knight knight from word to word word along the correct route, a simple problem will be revea revealed. led. Sue is in in her her teens, teens, so how old is Sal?

(8 Marks) Total: 100

THE END 4 Compiled by and downloaded from www.erudits.com.ng

THE HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SUID-AFRIKAANSE AKADEMIE VIR WETENSKAP EN KUNS in collaboration with HARMONY GOLD MINING, AM ESA and SAMS

FIRST ROUND 2004 JUNIOR SECTION: GRADES 8 AND 9 18 MARCH 2004 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions : 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules : Each correct answer is worth 5 marks. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily necessaril y drawn to scale. 6. The centre page is an information and formula sheet. Please tear it out for your use. 7. Indicate your answer on the sheet provided. 8. Start when the invigilator invigilat or tells you to. You have 60 minutes to complete the question paper. 9. Answers and solutions are available at http://science.up.ac.za/samo/

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO D O SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X11, ARCADIA, 0007 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail:[email protected] E-mail:[email protected]


1.

Which one of the following numbers is the smallest? A) 0,068

2.

4.

B) 0,96

Find the value of

1008 ÷

= 24

A) 32

B) 42

D) 0,087

E) 0,2443

C) 1,9

D) 9

E) 0,9

C) 52

D) 62

E) 72

if

In the diagram, four equal circles fit perfectly inside a square; their centres are the vertices of the smaller square. The area are a of the smaller square is 4. The area of the larger square is A) 4

5.

C) 0,2

0, 9 ( 0, 4 + 0, 6 ) equals A) 0,09

3.

B) 0,07

B) 8

C) 12

D) 16

E) 20

Jackie said that 12% of the oranges were not sold. s old. Pam said that is the same as 360 3 60 oranges! How many oranges were sold? A) 2 400

6.

B) 2 640

C) 3 000

D) 3 600

E) 4 320

If b = 3a and c = 2b, then a + b + c is equal to A) 6a

B) 8a

C) 10a

D) 12a

E) 14a


7.

A regular six point star is formed by extending the sides of a regular hexagon. If the perimeter of the star is 96 cm then the perimeter of the hexagon (in cm) is A) 30

8.

The last digit of 2 A) 0

9.

B) 36

2004

C) 42

D) 48

E) 54

C) 2

D) 3

E) 4

− 2 is

B) 1

The game Pyramaths works as follows: 2 adjoining blocks' sum is equal to the block above the 2 adjoining blocks, e.g. a + b = c . 25

Row 1

c a

Row 2

b

4

Row 3

If the sum of the numbers in row 3 is 17, then the value of a is A) 2

10.

B) 3

C) 4

D) 5

E) 7

In a mathematics class, the learners voted to have a new operation on numbers called “super op” and used the symbol symbol # for the operation. 1 1 They defined it as: a ( # ) b = + + ab a b 1 The value of ( # ) 6 is 3 A) 5

1 6

B) 18

C) 2

1 2

D) 2

E) 8


1 3

11.

Vishnu has displayed his technology project as a mobile and hung it from the classroom ceiling. It is perfectly perfectl y balanced (figure 1). Ceiling 20 cm

10 cm

Figure 1 10 g

60 g 20 g

Sipho wants to display his project in the same way (figure 2). Ceiling 25 cm

x cm

Figure 2 25 g

16 g 15 g

What must the length ( x) of the wire be for his mobile to be perfectly balanced? [Ignore the mass of the wire] A) 5

12.

B) 10

C) 15

D) 20

E) 25

In the diagram, the numbers occupying opposite sectors are related in the same way. The relationship of y in terms of x is y A) y = 3 x – 1

11

5

2

4

B) y = 2 x + 1 C) y = 2 x + 2 D) y = x – 1

x

E) y = 2 x – 1


13.

Kelly took 60 minutes to cycle 25 kilometres after which she increased her average speed by 5 kilometres per hour. How long will it take her to cover the next 25 kilometres if she maintains the new average speed? A) 35 min

14.

B) 40 min

C) 45 min

The area of the shaded triangle P 1 2 is 4 cm . Angles Angles PQR and QAB QAB 2 o are right angles (90 ). QR = 4 and AB = 3.

A

The size of angle ABQ is

Q A) 15

!

15.

D) 50 min E) 55 min

B) 30

!

C) 45

!

(Figure NOT drawn to scale)

3

B

R

4 D) 60

E) 75

!

!

Two 3-digit numbers are multiplied. multipli ed. A star (*) represents any digit. **5

1 3-digit number

1**

2nd 3-digit number

st

2**5 13*00 ***00 **77* The second 3-digit number is A) 140

B) 189

C) 180

D) 155

E) 147


16.

Below is a list of numbers with their corresponding codes. Determine the three digit number w. Number

Code

589

524

724

386

1346

9761

w

485

A) 945

17.

B) 543

C) 425

D) 623

E) 925

A rectangular rectangular right prism has the dimensions x cm by x cm by h cm. 2 The surface area of the prism is 14 x 2 cm x

Find h in terms of x.

h x

A) 3 x

18.

B)

x

2

C) 4 x

D) 2 x

E)

x

Zama used six digits, 3; 7; 4; 6; 2 and and 5 to make two-digit numbers e.g. 37; 44 etc. If 7 cannot be used as the ten’s digit and 3 cannot be used as the units’ digit, then the sum of all possible two digit numbers is A) 1 000

B) 1 040

C) 1 080

D) 1 120

E) 1 160


19.

Four one-centimetre squares are joined as in the figure alongside. One or two one-centimetre squares may be added to this figure.

Example:

The sides of the added blocks must fit against the sides of a block in the original figure.

A A

A

B

B

Line of symmetry

What is the maximum number of ways this can be done to create different figures which are symmetrical about a line of symmetry? A) 8

20.

B) 9

C) 10

D) 11

E) 12

D

ABCD is a rectangle. N is the midpoint of AB. F is the midpoint of DA. DA produced meets CN produced at M.

C

F A

The area of ∆FNM , as a fraction of the area of rectangle ABCD is

A)

1 8

B)

3 8

C)

5 8

N

M

D)

1 4

E)

THE END


1 2

B

THE HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD organised orga nised by the SUID-AFRIKAANSE SUID-AFRIKAANSE AKADEMIE AKADEMIE VIR WETENSKA WETENSKAP P EN KUNS in collaboration with HARMONY GOLD MINING, AMESA and SAMS

SECOND ROUND 2004 JUNIOR SECTION: GRADES 8 AND 9 13 MAY 2004 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions : 1. 2. 3.

4.

5. 6. 7. 8. 9.

Do not open this booklet until told to do so by the invigilator. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. Scoring rules : For each correct answer in Part A: 4 marks in Part B: 5 marks in Part C: 6 marks For each wrong answer: –1 mark For no answer: 0 marks You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments i nstruments are not permitted. Diagrams are not necessarily drawn to scale. The centre page is an information information and formula sheet. Please tear it out for your use. Indicate your answers on the sheet provided. Start when the invigilator tells you to do so. You have 120 minutes to complete the question paper. Answers and solutions will be available in June at http://science.up.ac.za/samo/

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG BAG X11, ARCADIA, 0007 0007 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail:[email protected]


PART A

1.

If

6 5

= 1, 2

A) 1, 2

2.

B) 0,12

0,06 is 0,5

C) 0,012

D) 0,0012

E) 0,00012

If x ! y is defined to be the remainder when x is divided by y (for example 8 ! 5 = 3 ), then the value of 13 ! (11 ! 3) is A) 0

3.

, then the value of

B) 1

C) 2

D) 3

E) 4

If 10 x ⋅ 10 y ⋅ 10 z = 10 6 , then the average of x, y and z is A) 1

B)

5 3

C) 2

D)

7 3

E) 3

4.

The length of the broken line, in metres, down the middle of a road is A) 67

5.

B) 67,5

C) 68

D) 69

E) 70

In an isosceles triangle ABC , AB = 2BC. If the perimeter perimete r of triangle ABC is 300 mm , then the length of AC in millimetres is. A) 40

B) 60

C) 80

D) 100


E) 120

PART B 6.

Half of 22004 is A) 2

7.

1002

B) 2

2002

2003

C) 2

D) 1

2004

1002

E) 1

You are given four fractions 5 ; a ; b ; c 12

Two fractions a and b are equally spaced between a + b =

A)

8.

7 12

5 and c . 12

If

4 , then find the value of c . 3

B)

2 3

C)

3 4

D)

5 6

E)

11 12

What is the sum of the digits of the following product? 999 999 × 666 666 A) 54

9.

B) 63

C) 72

D) 81

E) 90

A lady travels by car at a uniform speed, from A to B and then from C to D . Determine the average travelling speed of the vehicle from from A to D in km/h.

A) 92

B) 96

C) 100

D) 104


E) 120

10.

The maximum number of integer values that could be obtained from 100 where n is a natural number, number, is 2n − − 1

A) 9

11.

B) 7

C) 5

D) 3

E) 1

In the Harmony South African Mathematics Olympiad the scoring rules are as follows:For each correct answer in Part A: 4 marks, in Part Part B: B: 5 marks, marks, in Part Part C: C: 6 marks. marks. For each wrong answer: -1 mark. For no answer: 0 marks. There are five questions in Part A, ten questions in Part B and five questions in Part C. Jessie answered every question on the paper. She had four Part A questions correct and and seven Part B questions correct. How many Part C questions were correct if she scored 63% for the Olympiad? A) 1

12.

B) 2

C) 3

D) 4

E) 5

In the diagram, AB : BC = 1 : 3 and BC : CD = 5 : 8. The ratio AC : CD in the sketch is

A) 3 : 4

B) 3 : 5

C) 5 : 6

D) 4 : 5


E) 2 : 3

13.

Twenty 1 centimetre cubes all have white sides. Forty four 1 centimetre cubes all have have blue sides. These 64 cubes are are glued together together to form one large cube. What is the minimum surface area that could be white? A) 20

14.

B) 16

C) 14

D) 12

E) 8

Four matchsticks are used to construct the first figure, 10 matchsticks for the second figure, 18 matchsticks for the third figure and so on.

th

How many matchsticks are needed to construct the 30 figure? A) 900

15.

B) 99 990

C) 10 1080

D) 2 700

E) 3 000

After careful observation, the value and location of one number of every triangle is derived. Determine the missing missing number number at the apex of triangle D.

A) 9

B) 8

C) 7

D) 6


E) 5

PART C 16.

The product of the HCF and LCM of two numbers is 384. If one number is 8 more than the other number, then the sum of the two numbers is A) 48

17.

B) 40

C) 36

D) 24

E) 18

o ! In the given figure ∆ ABC , A = 20 . DE , DC, EF and FC are joined such that AD = DE = EF = FC = BC.

! AC D is The size of ACD

A) 10

18.

o

B) 20

o

C) 30

o

D) 40 40

o

E) 60

o

The value of 100 2

2

2

2

2

9 8 + 96 9 6 − 94 94 + " + 8 − 6 − 98

2

+

42

−2

2

is A) 5 200

B) 5 100

C) 5 000

D) 4 900


E) 4 800

19.

20.

E BA is an equilateral triangle. ABCD is a square of In the diagram, ∆ EBA sides 6. E is is the centre of the circle which passes through points A and shaded region is B. The area of the shaded

A) 9π − 27

B)

6π − 27

D) 6π − 3 27

E)

4π − 3 27

C)

9π − 3 27

The diagram is a “non-traditional” “non-traditional” magic square that totals 105. This total can be obtained obtained by adding adding the 4 numbers numbers along a diagonal. diagonal. There are other sets of 4 numbers numbers giving giving the same total. The maximum number of other combinations that give a total of 105 is 12 16 18 13 A) 16

B) 18

19 23 25 20

28 32 34 29

C) 20

35 39 41 36 D) 22 22

THE END


E) 24

The South African Mathematics Olympiad Third Round 2004 Junior Section: Grades 8 and 9 Question 1 P

PQRS is a rectangle. Determine which, if any, of the shaded rectangles A and B has the larger area.

Q

B

A R

S

(6 Marks) Question 2

B

The diagram shows a long room. An ant wants to walk from A to B. It can walk along the walls and ceiling of the room. What is the shortest distance it could walk?

3m 5m

A

15m


Fred added up all the positive integers from 1 to p on his calculator and obtained a total of 2004. By mistake, however, he had entered one number twice. Find the correct total that Fred should sh ould have obtained and the number that he added twice. (6 Marks) Question 4

All six faces of a cube of side length n are painted. The cube is then t hen cut up into n3 cubes of the same size. Among these small cubes, there are some s ome that do not have paint on any of their faces, some have one, two or three faces painted. For what value of n is the number of small cubes that do not have any paint on them equal to the number of small cubes that have exactly one painted face? (6 Marks) Question 5

In a class of 30 learners, learn ers, five take Maths, Geography and Biology. Nine take Maths and Biology. Twenty learners take Geography, of whom twelve also take Mat hs. Eighteen take Biology. Every learner takes at least one subject, and no one takes only Maths. How many learners take only Biology? (6 Marks)

2


Question 6

In the addition sum below, different letters stand for different digits. (For example, if T is 3, then S cannot be 3.) Find what each letter stands for.

WHO I S T H I S+ I D I O T (6 Marks) Question 7 B

ABC is an isosceles right-angled triangle with AC = AB = 2. A circular arc of radius 2 with centre C meets the hypotenuse at D, and a circular arc of radius 2 with centre B meets the hypotenuse at E. Find the area of the shaded region ADE in terms of π.

D

E

A

C


Two cars race around a circular track at constant speeds starting at the same point. If they travel in opposite directions, then they meet e very 30 seconds. If they travel in the same direction, then they meet every two minutes. If the track is 1800m long, what is the speed of each car? (6 Marks) Question 9

Sakhile, Danny, Nelly and Pravin are manager, coach, captain and goalkeeper for a social hockey team, but not necessarily in that order. 1) Sakhile is the brother of the manager. 2) The coach and the manager are not related. 3) Nelly is an only child. 4) The captain is older than Danny, but is in the same class. 5) Pravin is in the same class as Nelly. 6) The goalkeeper is not in the same class as Pravin or Danny. 7) Danny is Nelly's cousin. 8) Pravin was disappointed not to have got the position of coach. Determine which person has which position. (6 Marks) Question 10

All possible four-digit numbers are formed using the digits 1, 2 and 4, for example 2441 and 1112. How many of of these numbers are divisible by three? (6 Marks) 3


Question 11

C

(a)

Show how to place the numbers 1 to 8 at the vertices of a cube so that the sum of the four numbers at the corners of each face is the same.

(b)

Explain why the two numbers at A and B and the two numbers at C and D must always have the same sum.

A

D

B


We write out all the integers from 1 to 30, and cross out some of these so that in the remaining list no number is equal to twice any other. Find the maximum number of integers that can appear in i n this remaining list and justify why this is the maximum. (8 Marks) Question 13

Here is an example of five f ive consecutive positive integers whose sum is 1000: 198 + 199 + 200 + 201 + 202 = 1000. Find the largest number of consecutive positive integers whose sum is exactly 1000 and justify why you think this must be the largest number. (8 Marks) Question 14

(a)

Is it possible to cover a 7 by 7 square with non-overlapping 3 by 1 rectangles? Explain your answer.

(b)

Is it possible to cover a 10 by 10 square with non-overlapping T-shaped pieces of the kind shown on the right? Explain.


X, Y and Z are three different differ ent digits from 1 to 9 forming the number XYZ. For example, the digits 2, 6 and 4 would form the number 264. Find the smallest value of

ΧΥΖ Χ + Υ + Ζ

and explain why the value you have found is

in fact the smallest. (8 Marks)

4


THE HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SUID-AFRIKAANSE AKADEMIE VIR WETENSKAP EN KUNS in collaboration with HARMONY GOLD MINING, AMESA and SAMS

FIRST ROUND 2003 JUNIOR SECTION: GRADES 8 AND 9 18 MARCH 2003 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions : 1. Do not open this this bookl booklet et until until told to do so by the invigi invigilat lator. or. 2. This This is is a multi multiple ple choice choice questi question on paper paper.. Each Each questi question on is is foll followed owed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scorin ring rules : Each correct answer is worth 5 marks. There is no penalty for an incorrect incorrect answer or an unanswered unanswered question. 4. You You must use an HB pen pencil cil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagr Diagram amss are are not nece necess ssari arily ly drawn drawn to scale scale.. 6. The centre centre page page is is an an info informat rmation ion and formul formulaa sheet sheet.. Ple Please ase tear tear it it out out for your use. 7. Indi Indica cate te your your answe answerr on the the shee sheett provid provided ed.. 8. Star Startt when when the the invi invigi gila lato torr tells tells you to. to. You have 60 minutes to complete the question paper. 9. Answe Answers rs and and solut solutio ions ns are are avail availabl ablee at http://science.up.ac.za/samo/ at http://science.up.ac.za/samo/

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DRAAI DIE BOEKIE OM VIR DIE DIE AFRIKAANSE AFRIKAANSE VRAESTEL VRAESTEL PRIVATE PRIVATE BAG X11, X11, ARCADIA, ARCADIA, 0007 TEL: (012) (012) 328-5082 FAX (012) (012) 328-5091 E-mail:[email protected]


1.

The value of 1,1 × 10 is A) 1

2.

4

3

÷

B) 11

D) 11,1

E)

110

C) 4

D) 16

E)

32

2

2 is equal to

A) 2

3.

C) 1,1

B) 3

decimal number system system (base 10) ten ten Did Did you know? know? In the decimal different different digits, 0 to 9, are used to write write all the numbers. In the binary number number system (base 2) two different different digits digits are used, i.e. 0 and 1. Which one of the following numbers is not a valid number in the octal number system (base 8)? A) 128

4.

C) 126

D) 125

E)

124

At a recent Athletics Athletics Championship Championship Sipho Sipho broke the old record of 11,01 seconds for the 100 metre race by 0,04 seconds. The new record for the race in seconds is A) 10,01

5.

B) 127

B) 10, 04 04

C) 10 1 0, 61 61

What What is the units units digit digit of 4 A)

0

B) 2

2003

D) 11 11, 05 05

E) 10, 97 97

D) 6

E)

?

C) 4

8


6.

The following net is given:

Which one of the following cubes cannot be folded from this net? A)

7.

B)

C)

D)

E)

Susan’s Susan’s average average mark mark for the the four terms terms (numb (numbered ered 1 to 4 on the horizontal axis) of 2002 was 70%. The mark for each term is out of 50 and and carr carrie iess the the same same weig weight ht when when cal calcula culati ting ng the the aver averag age. e. Use the the graph graph to dete determ rmin inee what what her her perc percen entag tagee was was for the the firs firstt term term of 2002 2002..

k r a m e g a r e v A

50 45 40 35 30 25 20 15 10 5 0

A) B) C) D) E)

?

1

2

3 Terms

4

equ equal to or more ore than than 70% 70% equ equal to or less ess than than 20% 20% betw betwee een n 20% 20% and and 30% 30% betw betwee een n 40% 40% and and 50% 50% betwe etweeen 50% 50% and and 70% 70%


If 20 x − 25 is expressed in the form a (4 x + b) , then the value of a + b is

8.

A)

20

−

B)

10

−

C) 0

D) 10

E)

20

Did you know? In a magic square the sum of the numbers in every horizontal row, vertical column and diagonal are all the same.

9.

In this magic square, the row total is equal to 15.

6

x

5

1

The value of x x is A) 1 10.

B) 2

C) 3

D) 5

E) 8

How many many numbers numbers between 800 and 1000 1000 are divisible divisible by both 7 and 8? A) 2

B) 3

C) 4

D) 5

E)

6

∧

11.

Isoscel Isosceles es triangl triangles es have have been drawn drawn between between AB and AC with with BAC = 9o . B

C

o

9

A

What is the size of the largest angle in the shaded triangle? A) 72

o

o

B) 81

C) 9 0

o

D) 126

o

E) 144


o

12.

1−

A)

13.

1 2

1−

1 3

1 2003

1−

1 4

B)

1−

1 5

2 2003

.

C)

.

.

202 2003

1−

1 2003

D)

is equal to 24 2003

E)

2002 2003

Nomava visits visits a shop with one 20c coin and ten 50c coins. The shopkeeper can offer change but has only two 20c coins and nine R1 coins. She buys one item and receives the correct change. Which one of the following is a possible price for this item? A) R4,60

14. 14.

B) R5 R5,40

C) R1,40

D) R4,40

E) R5 R5,10

Did Did you you kn know ow? ?

i) A regul egular ar poly polyg gon with with five ive

ii) ii) A regul egular ar poly polyg gon with with six six

sides (pentagon) has five diagonals,

sides (hexagon (hexagon)) has nine diagonals, diagonals,

indi indica cate ted d by the the dott dotted ed lin lines. es.

indi indica cate ted d by the the dott dotted ed lin lines. es.

The number of diagonals that can be drawn in a regular polygon with twenty sides (icosagon), is A)

190

B)

180

C)

170

D)

380

E)


19

15. 15.

Did Did you you kn know ow? ?

Formul rmulaa for spe speed: ed:

aver verage spe speed =

total distance total time

The distance from Sipho’s home to Harmony Gold mine is 120 km. Sipho travels at 60 km/h from his home to the mine. He is in a hurry to get back to his home in the afternoon, and travels at 120 km/h. What is his average speed for the journey there and back in km/h? A) 110

16.

B) 100

C) 95

D) 90

E) 8 0

A store sold 213 213 bicycles bicycles during the year year 2002. For the the first few months months they sold 20 bicycles per month, then for some months they sold 16 bicycles per month and in the remaining month(s) they sold 25 bicycles per month. For how many months did they sell only 16 bicycles per month? A)

17. 17.

5

B)

6

C)

7

D)

8

E)

9

Did Did you you kn know ow? ? 3 1 litre is the same as 1000 cm . 2

The area of the cross-section cross-section of a pipe is 250 cm . Water flows through the pipe at a rate of 3 litres per second. 2

250 cm

The speed at which the water flows through the pipe in cm/s is A) 15

B) 1,2

C) 8

D) 6

E) 12


18.

The areas areas of the faces faces of the rectangul rectangular ar box are A, B and C. B

C

A If the volume of the box is V, then A × B × C is equal to

A)

19.

V

B)

2V

In the formula formula M

=

2

10n 1 + 2n

C)

b

l

V

2

D)

V

E)

h

3

V

, n is any positive integer.

n increases (gets bigger and bigger), M will If n

A) B) C) D) E)

20.

decrease incre crease stay the the sam same first first incre increase ase and and then then decre decrease ase firs firstt decr decrea ease se and and then then incr increa ease se

A meal made made with four four eggs and 60 g cheese contains contains 560 calories. calories. Another meal made with six eggs and 20 g cheese also contains 560 calories. How many calories does one (1) egg contain? A)

60

B)

70

C)

80

D)

90

E)

THE END


100

THE HARMONY SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SUID-AFRIKAA SUID-AFRIKAANSE NSE AKADEMIE AKADEMIE VIR WETENSKAP WETENSKAP EN KUNS in collaboration with HARMONY GOLD MINING, AMESA and SAMS

SECOND ROUND 2003 JUNIOR SECTION: GRADES 8 AND 9 20 MAY 2003 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions :

1. 2. 3.

4.

Do not open open this this bbook ooklet let until until told told to to do do so so by by the invigi invigilat lator. or. This This is is a multip multiple le choice choice questi question on paper. paper. Each Each ques questio tionn is follow followed ed by answers marked A, B, C, D and E. Only one of these is correct. Sco Scoring rules : For For each each corr correc ectt answ answer er in Part Part A: 4 marks arks in Part Part B: 5 marks rks in Part Part C: 6 marks rks For each wrong answer: –1 mark For no answer: 0 marks Youu must Yo ust use use an HB penc pencil il.. Rough paper, a ruler and a rubber are permitted.

Calculators Calculators and geometry geometry instruments instruments are not permitted.

5. 6. 7. 8. 9.

Diagra Diagrams ms are not necessa necessaril rilyy drawn drawn to scale. scale. The centre centre page is an information information and formula formula sheet. sheet. Please tear it out for your use. Indica Indicate te your your answer answerss on the sheet sheet provid provided. ed. Start Start when when the invigi invigilat lator or tells tells you you to do so. You have 120 minutes to complete the question paper. Answ An swer erss and solu soluti tion onss are are avai availa labl blee at http://science.up.a http://science.up.ac.za/samo/ c.za/samo/ DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO.

DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE PRIVATE BAG X11, ARCADI ARCADIA, A, 0007 TEL: (012) 328-5082 328-5082 FAX (012) 328-5091 328-5091 E-mail:[email protected]


PART A: 1.

Two consecutive natural numbers add up to 2003. The smaller of these two numbers is A) 10 1001

2.

(Each correct answer is worth 4 marks)

B) 1002

C) 1003

D) 1004

E) 1000

I recently returned from a trip. Today is Friday. Friday. I returned four days before the day after tomorrow. On which day did I return? A) Mond Monday ay B) Tuesd Tuesday ay C) Wedn Wednes esda dayy D) Thur Thursd sday ay E) Frid Friday ay

3.

The pie chart shows the breakdown of the 500 runs scored by one of the South African batsmen over the last three years.

2003

2002

225 runs

20% 2001

The percentage of runs in 2003 is A) 45 % 4.

B) 20 % C) 15 %

D) 65 65 % E) 35 %

Consider the following pattern: 1st row: 2nd row: 3rd row: 4th row:

1 1 1 .

3 3 .

5 .

.

The difference between the sums of the numbers in the 9 th and 10th rows is A) 17 5.

B) 18

C) 19

D) 21

E)

Which one of the following is an odd number? A) B) C) D) E)

20012 + 3 2002 2 + 10 20032 + 7 20042 + 1 20052 + 9


22 22


PART B: 6.

The measurement of Â is . . .

A

100o

30o

B A) 30 7.

°

If

a

and

A) 12 9.

B) 40

°

C) 50

°

D) 60

°

E)

70 70

°

A supermarket supermarket always always prices prices its goods at ‘so many many Rands and ninetynine cents’. If a shopper who has bought different items has to pay R41,71, how many items did she buy? A) 41

8.

C

B) 39

C) 30

b are integers, and a ⊗ b =

B) 4

C) 6

D) 19 b

2

a

−

b a

E)

29 29

, then 3 ⊗ 6 is equal to

D) 8

E) 10

The regions regions marked marked A are equal in area, area, and the regions regions marked marked B are equal in area. 6

2 B

area A The ratio of is area B

A 0

A)

1 5

B)

1 4

B

C)

A

2

1 3

D)

8

1 2

E)


1 1

10

10.

2

If x + y = 4 , y + z = 7 and x + z = 5 the value of ( x + y + z ) is A) 36

11.

∆ABC

B) 64

C) 100

D) 144

E)

256

has D on BC such that BD = 2 and DC = 3 .

If AB = m and AD = n then the value of m 2 − n 2 is

A

m n 2 B

A) 4 12.

C) 16

D

D) 25

C

E) 36

If 1 × 2 × 3 × ... ×199 × 200 is calculated, then the number of zeros at the end of the product is A) 42

13.

B) 9

3

B) 43

C) 46

D) 49

E) 52

A tank that is in the form form of an an inverted cone contains contains a liquid. liquid. The height h, in metres, metres, of the space above the liquid is given by the formula 7 radius of the the liquid liquid surface, in metres. metres. h = 21 − r where r is the radius 2

}

h

r

The circumference of the top of the tank, in metres is A) 9π

B) 12π

C) 15π

D) 18π

E)


21π

14.

In this diagram, there is a total of 14 squares of all sizes.

What is the total number of squares of all sizes on the board below?

15.

A) 49

B) 63

C ) 77

D) 91

The fraction

1 53 can be expressed as 3 + . 1 17 x +

E) 105

y

If x and y are integers the value of x + y is A) 8

PART C: 16.

C) 10

D) 11

E) 12


Two numbers numbers are in the ratio 2 : 3. When When 4 is added to each number the ratio ratio changes changes to 5 : 7. The sum of the two original numbers is A) 20

17.

B) 9

B) 25

C) 30

D) 35

E)

40 40

A local council council consists consists of 4 female members and 3 male members. members. The number of different 3-member committees consisting of 2 female members and 1 male member which can be formed by the council is A) 18

B) 15

C) 12

D) 9

E)


6

18.

Lee gave Petrus a 10 metre metre lead in in a 100 100 metre race and Lee was beaten by four metres. What lead should Lee give Petrus in order that both finish the race together, if their respective speeds stayed the same in both races? A) 5, 7755 m B) 5, 9 m C) 6,1 m

19.

In the addition problem problem TSR TSR + PSR + RSP, Themba Themba substitutes the four letters with the four digits 2, 7, 5, and 3, in any order. Different letters stand for different digits. The largest value of the sum TSR + PSR + RSP is A) 15 1579

20.

D) 6, 2255 m E) 6, 5 m

B) 1499

C) 1571

D) 1701

E) 1537

Colleen, Jakes, Hendrik, Vishnu and Tandeka play a game of cops and robbers. The robbers’ statements are always false while the cops’ statements are always true. a) Colleen says that Jakes is a cop. b) Hendrik says that Vishnu is a robber c) Tandeka says that Colleen is not a robber d) Jakes says that Hendrik is not a cop. e) Vishnu says that Tandeka and Colleen play on different sides. How many robbers are there? A) 1

B) 2

C) 3

D) 4

E) 5

THE END


     

THE HARMONY GOLD SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SOUTH AFRICAN ACADEMY OF SCIENCE AND ARTS in collaboration with HARMONY GOLD MINING, AMESA and SAMS

FIRST ROUND 2002 JUNIOR SECTION: GRADES 8 AND 9 19 MARCH 2002 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions Instructions : 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules : Each correct answer is worth 5 marks. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, a ruler and a rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams are not necessarily drawn to scale. 6. Indicate your answer on the sheet provided. 7. Start when the invigilator tells you to. You have 60 minutes to complete the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR DIE AFRIKAANSE VRAESTEL PRIVATE BAG BAG X11, ARCADIA, 0007 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail:[email protected]


1.

Find the missing number if 182 × ∆ = 2002 . A) 8

2.

3.

B)

C)

10

D) 11

E)

12

17

D) 19

E)

20

E)

50 000

The answer to 5 − 2 + 4 × 3 is

A) 12

B) 15

If

the value of (7n − 5)( n − 5)( 3n + 5) is

n = 5 then

A) 0

4.

9

B) 50

C)

C)

500

D) 5 000

An ant covers a distance of 90 metres in 3 hours. The average speed of the ant in centimetres per minute is

A) 30

5.

Given that then

0,9 0,25

A) 0,036

B)

9 25

=

40

C)

50

D) 60

E)

70

C)

3,6

D) 36

E)

360

0,36

is equal to

B)

0,36

2


6.

A square is divided into 4 identical rectangles as shown in the diagram. The perimeter of each of the four rectangles is 30 units. What is the perimeter of the square?

A) 36

7.

B)

40

C)

44

D) 48

E)

52

Allison, Nomsa and Jan shared a sum of money in the ratio of 4 : 3 : 1 respectively. Allison received R70 more than Nomsa. The total amount that was shared initially was

A) R640

8.

B)

R560

C)

R 480

D) R 400

E)

R 320

NOT form a closed triangular prism? Which of the following nets will NOT form A)

D)

B)

C)

E)

3


9.

The product of two numbers is 504 and each of the numbers is divisible by 6. Neither of the two numbers is is 6. What is the larger of the two numbers? A) 48

10.

B) 84

C)

72

D) 60

E)

42

tank. One windmill windmill pours Two windmills pour water into a 1 000  tank. water into the tank at a rate of 20  per minute. The other other windmill windmill pours water into the tank at 20  in 3 minutes. How many minutes will it take to fill the tank?

A) 37 11.

1

B)

2

75

C)

112

1 2

D) 150

E)

175

The Ancient Romans used the following different numerals in their number system: I = 1 C = 100 V

=

5

D

=

500

X

=

10

M

=

1 000, etc.

L = 50 They used these numerals to make up numbers as follows: I = 1 VI = 6 II

=

2

VII

=

7

III

=

3

VIII

=

8

IV

=

4

IX

=

9

=

10, etc.

V = 5 X So, for example, XCIX is 99. What is the value of XLVI? A) 42

B)

44

C)

46

D) 64

E)

4


66

12.

Did you know? •

An equilateral triangle is a regular polygon with 3 equal sides, o

and each interior angle is 60 . •

A square is a regular polygon with 4 equal sides, and each o

interior angle is 90 .

If a regular polygon has n sides, then the formula to find the size of each interior angle is

(n − 2) × 180 n

. If each interior interior angle angle of a

regular polygon measures 150°, then the number of sides ( n) is

A) 6

13.

B)

9

C)

10

D) 11

E)

12

E)

44

A solid right prism has a square base. The height is twice the length of the side of the base. The 2

surface area of this prism is 160 cm .

If 1 cm³ of the prism has a mass of 250 grams, grams, then the mass of the prism in kilograms is

A) 28

B)

32

C)

36

D) 40

5


14.

The diagrams show three scales. On each scale there are different objects on each side which balance each other, as shown.

Diagram 1 How many

A) 3

15.

1 3

;

a;

Diagram 2 -shaped objects will balance a

B)

b;

Diagram 3

4

C)

5

-object?

D) 6

E)

7

1 2

These numbers are arranged from smallest to largest. The difference between any two adjacent (next to each other) numbers is the same. The value of b is A)

16.

5 12

B)

7 18

C)

4 9

D)

5 6

E)

1 4

Given the set of six numbers below: 11; 31; 19; 3; 10; 6 Three numbers are selected from the above set and added together. The remaining three numbers are also added together. These possible two sums are then multiplied to get a product. The maximum product is

A) 1 200

B) 1 400

C)

1 500

D) 1 600

E) 1 800

6


17.

The digit 3 is written at the the right of a certain 2-digit number forming a 3-digit number. The new number is 372 more than the original 2-digit number. The sum of the digits of the original 2-digit number is A) 4

18.

B)

5

C)

6

D) 7

E)

8

Matchsticks of equal length are used to make the following figures: i)

ii)

iii)

3 matchsticks are used for figure i). 9 matchsticks are used for figure ii). 18 matchsticks are used for figure iii).

How many matchsticks are needed for a similar figure which has 10 matchsticks along each side?

A) 84

B) 108

C)

135

D) 165

E)

7


180

19.

In a certain town some people were affected by a ’flu’ flu’ epidemic. In the first month 20% of the population contracted the flu whilst 80% were healthy. In the following month 20% of the sick people recovered and 20% of the healthy people contracted the disease.

What fraction of the population is healthy at the end of the second month?

A) 0,68

20.

B)

0,60

C)

0,52

D) 0,44

E)

0,36

Mpho, Barry, Sipho, Erica and Fatima are sitting on a park bench. Mpho is not sitting on the far right. Barry is not sitting on the far left. Sipho is not sitting at either end. Erica is sitting to the right of Barry, but not necessarily next to him. Fatima is not sitting next to Sipho. Sipho is not sitting next to Barry.

Who is sitting at the far right?

A) Mpho

B) Barry

C) Sipho

D) Fatima

E) Erica

THE END

8


     

THE HARMONY GOLD SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SOUTH AFRICAN ACADEMY OF SCIENCE AND ARTS in collaboration with HARMONY GOLD MINING, AMESA and SAMS

SECOND ROUND 2002 JUNIOR SECTION: GRADES 8 AND 9 21 MAY 2002 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. 2. 3.

4.

5. 6. 7.

Do not open this this booklet until told to do so by the invigilator. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. Scoring rules: For each correct answer in Part A: 4 marks in Part B: 5 marks marks in Part C: 6 marks marks For each wrong answer: –1 mark For no answer: 0 marks You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. Diagrams are not necessarily drawn to scale. Indicate your answers on the sheet provided. Start when the invigilator tells you to. You will have 120 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. DRAAI DIE BOEKIE OM VIR V IR DIE AFRIKAANSE VRAESTEL PRIVATE BAG X11, ARCADIA, 0007 TEL: (012) 328-5082 FAX (012) (012) 328-5091 E-mail:[email protected]


PART A: 1.

The value of 1 0 00 01 × 99 − 99 is

A) 99

2.

The value of

A)

3.


1 2

B) 990

C) 9 900

D) 99 000

E) 990 000

3 1 1 − × is 4 2 2

B)

1 4

C)

0

D)

3 4

E)

1 8

A truckload of books contains x cartons. Each carton contains y boxes and each box contains z books. The number of books in the truckload is

A)

B)

x + y + z

x( y + z)

C)

xy + xz + yz

D)

xy z

4.

The smallest integer x, for which

A)

−29

B)

−14

C)

15 x − 1

−4

is an integer, is

D) 0

E) 2

2


E)

xyz

5.

14 $ S U r e p d n a R

13 12 11 10

X

9 15/10/01

Oct ‘01

15/11/01

Nov ‘01

15/12/01

Dec ‘01

15/01/02

Jan ‘02

Feb ‘02

Date

The above graph represents the value of one United States Dollar (US $) in South African Rand. At point X, on 15 October 2001 one US $ cost approximately R9,31. Use the graph to determine the approximate Rand value of one US $ on 15 January 2002.

A) less than R10

B) between R10 and R11

C)

between R1 R11 and R1 R 12

D) between R1 R12 an and R1 R 13

E)

more than R13

PART B: 6.


Of the 28 T–shirts in a drawer, six are red, five are blue, and the rest are white. If Bob selects T –shirts at random whilst packing for a holiday, what is the least number he must remove from the drawer to be sure that he has three T–shirts of the same colour? A) 4

B) 13

C)

9

D) 19

E)

7

3


7.

Virginia is making 5-digit arrangements from the digits 2 and 4. An example of such an arrangements is 22442. She was told that the t he first digit cannot be a 4. How many such arrangements are there? A)

8.

16

B)

24

C)

20

D)

8

E)

32

Two friends Petros and Sammy have pocket money in the ratio 3 : 5. Each one spends R30. The ratio changes to 1 : 2. The total amount the two friends started off with is A) R210

9.

B)

R240

C)

R270

D) R300

E)

R330

There are ten learners in an environmental club. They have decided to go on shell collection trips. The vehicle with which they undertake the trips, can only take eight learners at a time. Each learner goes at least once. What is the minimum number of trips the vehicle must make so that each learner goes on the same number of trips? A) 2

B) 3

C)

10.

4

D) 5

A

E)

6

Q y 2 x y

P C

B In the above diagram,

∆ ABC is

equilateral.

∆ APQ is

∧

isosceles with

∧

AQ = AP and AQ || BC. If P A C = 2 x , then the size of Q in terms of x is A) 60 D) 2 x

+ x

B) 180 E) 60

+ 3x

C)

120

−x

−x

4


11.

The units digit of 2 2000 A) 0

12.

+2

B) 2

2001

C)

+2

2002

is

4

D) 6

E) 8

A

The total number of different triangles in the diagram, including

∆ ABC ,

is

C

B A) 21

13.

B) 42

C) 63

D) 84

E) 105

A 3-digit number has x as as its units digit; ( x − 1) as the the tens tens digi digitt and

1 x as its 2

hundreds digit. The number in terms of x is

14.

A) 5 x − 1

B)

7x − 1

D) 61 x − 10

E)

5 x −1 2

C)

111x − 1

A company makes solid blocks with square bases. The volume of this block is 640 cm3. Its height is 10 cm. The cost of painting all the faces of this

10 cm

2

block at 15c per cm is

A) R64,00 B) R67,20 C) R70,40 D) R73,60

E) R60,20

5


15.

skills”. Nurf klar In an alien language, jalez borg farn means “good maths skills” borg means “maths in harmony” harmony” and darko klar farn means “good in gold” gold”. gold” in this language? What is “harmony gold”

A)

klar darko

B)

borg nurf

C)

jalez klar

D)

darko nurf

E)

farn borg

PART C: 16.


As a result of poor attendance at soccer matches it was was decided to decrease the ticket price by 20%. At the next match the number of tickets sold increased by 20%. Compared to the previous match, the income from the sale of tickets

17.

A)

increased by 20%

B)

decreased by 20%

C)

increased by 4%

D)

decreased by 4%

E)

remained the same

The mean (average) of

n

numbers is

p .

When the number

q

is removed

from the list of numbers averaged, then the mean (average) increases by 2. The value of A)

p −

B)

p − n +

q

is

2n 2

C) 2 p − n D) 2 p − n + 2 E)

p − 2n + 2

6


18.

1

R

14

G

2

B

15

C

3

X

16

T

4

5

S

17

O

N

18

J

6

P

19

F

7

8

E

20

D

21

U

H

9

M

22

V

10

Z

23

W

11

L

24

12

13

A

25

26

I

K

Q

Y

In the above table each letter of the alphabet is given a value. The algebraic expression 4 x − 3 is used as a key to convert the letters P S R X B O E into the word H A R M O N Y. Which one of the following

keys is used to convert S R X B into G O L D ? A) 19.

x + 2

B) 2 x − 1

C) 3x + 2 D) 5x + 1

E) 2 x + 1

C

ABC is an equilateral triangle with sides of 2 units. Using A, B and C as centres of circles, arcs BC, AC and AB are drawn.

The shaded area is

A) 2π

20.

−3

3

A

B)

π

−

3

π

C)

2

D) 2 3 − π

B E)

π

Five children, Amelia, Bongani, Charles, Devine and Edwina, were in the classroom when one of them broke a window. The teacher asked each of them to make a statement about the event, knowing that three of them always lie and two always tell the truth. Their statements were as follows: Amelia: Bongani: Charles: Devine: Edwina:

“Charles did not break it, nor did Devine. ” “I didn’t break it, nor did Devine. ” “I didn’t break it, but Edwina did. ” “Amelia or Edwina broke it. ” “Charles broke it. ”

Who broke the window? A) Amelia

B)

Bongani

D) Devine

E)

Edwina

C) Charles

7


THE SOUTH AFRICAN MATHEMATICS OLYMPIAD organised by the SOUTH AFRICAN ACADEMY OF SCIENCE AND ARTS in collaboration with OLD MUTUAL, AMESA and SAMS

SPONSORED BY OLD MUTUAL

FIRST ROUND 2001 JUNIOR SECTION: GRADES 8 AND 9 28 MARCH 2001 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth 5 marks. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 60 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. KEER DIE BOEKIE OM VIR AFRIKAANS PRIVATE BAG X11, ARCADIA, ARCADIA, 0007 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail: [email protected]


1.

When 2001 is divided by 200 the remainder is (A) 0

2.

( B) 1

( D) 10

( E) 99

( C) 9,399

( D) 93,99

( E) 939,9

If 241 × 39 = 9 399 then 2,41 × 3,9 is equal to (A) 0,09399

3.

( C) 9

( B) 0,9399

A solid triangular pyramid has six edges such as AB. Each corner is cut off. A (see new figure)

How many edges will the new figure have? B

(A) 24 4.

If

( B) 9 2

a ∆ b= a −b

(A) 2 5.

2

( C) 12

( D) 15

( E) 18

( D) 16

( E) 9

then 5 ∆ 3 is equal to

( B) 15

( C) 4

E

ABCD is a square and EAB and CFB are equilateral e quilateral triangles. A

B F

∧

The size of BEF is D

(A) 7,5 6.

( B) 10

( C) 11,25

( D) 12,5

A sewing machine stitches 0,6 kilometres of cloth in one hour. The rate of stitching of the machine in metres per minute is (A) 0,01

( B) 0,1

( D) 10

( E) 100

( C) 1


C

( E)15

7.

The value of the fraction (A)

8.

If

1

1

( B)

30 +

7 3x

x

(A) 2 9.

=

5 6

1 2

10 + 20 + 30 + 40 +... + 400 30 + 60 + 90 + 120 + ... + 1 200 ( C)

1 6

( D)

is

2 3

( E)

1 3

then the value of x is ( B) 5

( C) 6

( D) 4

( E) 3

Did you know? A palindrome palindrome is a number number which reads the same forwards as backwards e.g. 35453. Next year 2002 is an example of a palindromic number. What is the difference between 2002 and the number of the previous palindromic palindromic year? (A) 10

( B) 11

( C) 101

( D) 121

( E) 1001

10. 2001 people people stand in a queue at a voting station. There are at least 3 women between any two men. The largest possible number of men in the queue is

(A) 500

( B) 501

( C) 502

( D) 667

11. When a die is rolled the chance of obtaining obtaining a 5 is

( E) 668

1

6 When two dice are rolled the chance of obtaining a sum less than 5 is

(A)

1 6

( B)

1 3

( C)

5 6

( D)

2 9

( E)

5 36

12. The volume volume of a rectangular rectangular prism with length 3 a cm, width b cm and height c cm is 240 cm . a + b + c = 19 . Each side is 3 cm or more in length. a, b and c are whole numbers. The largest possible area of a face in cm 2 is

c

b a

(A) 15

( B) 30

( C) 40

( D) 48


( E) 60

13. The value of 499 − 497 + 495 − 493 + ... + 3 −1 is

(A) 2

( B) 250

( C) 496

( D) 498

( E) 500

14. Did you know? The sum of the lengths of two sides of a triangle is always greater than the third side.

4

Twenty matchsticks of equal length are placed to form a triangle, as shown. The total number of different triangles that can be made with a perimeter of 20 matchsticks is (A) 9

( B) 8

7

9

( C) 6

( D) 7

( E) 10

15. The sum of two consecutive consecutive numbers is S. The The square of of the larger larger number number minus the square of the smaller number is

(A) S

2

( B) 2S

( C) S

( D) S + 1

( E) S − 1

16. The The sum sum of the the digi digits ts of the the prod produc uctt 999 999 999 999 × 777 777 777 777 is

(A) 54

( B) 63

( C) 52

∧

( D) 48

∧

17. In the adjac djacen entt figu igure C + D = 150 ∧

∧

∧

( E) 50

B

∧

The value of A + B + E + F is

C A

(A) 210

( B) 300

( D) 390

( E) 570

( C) 360

F D E


18. A solution containing water and a liquid concentrate has 60% concentrate. By adding 20 litres of water to the solution the concentrate is reduced to 40% of the solution.

Concentrate

water

How many litres of the original solution is concentrate? (A) 36

( B) 30

( D) 16

( E) 12

Solution

( C) 24

arranged in a triangular triangular pattern pattern as shown. 19. The odd integers are arranged 1 3 7

.

13

.

5 9

15

.

11 17

.

19

.

If this pattern continues the first number in the row which has a sum of 1 000 000 is (A) 99

( B) 990 001

( C) 991

( D) 9 901 2

2

2

2

( E) 99 001 2

20. The number of terms of the sequence 4 ; 5 ; 6 ; ... 39 ; 40 that have an even digit in the tens place is

(A) 29

( B) 28

( C) 27

( D) 26

THE END


( E) 25



SECOND ROUND 2001 JUNIOR SECTION: GRADES 8 AND 9 29 MAY 2001 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: For each correct answer in Part A: 4 marks in Part B: 5 marks in Part C: 6 marks For each wrong answer: –1 mark For no answer: 0 marks 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 120 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. KEER DIE BOEKIE OM VIR AFRIKAANS PRIVATE BAG X11, ARCADIA, ARCADIA, 0007 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail: [email protected]


PART A: 1.

The number 36 is 12% of (A) 250

2.

n

3

+

n

3

+

n +1

(A) 3

3.

(C) 350

( D) 400

( E) 450

n 3 equals

( B) 3

3n

(C) 3

n+3

( D) 3

3n + 3

( E) 3

3n + 1

( B) 98

(C) 42

( D) 204

( E) 168

The greatest number of Fridays that can occur in a 75 day period is (A) 15

5.

( B) 300

Ansu works in a bookstore bookstore and has to count count a number number of identical books. She arranges them in a stack 4 wide, 6 deep in 7 layers. The number of books in the stack is (A) 17

4.


( B) 13

(C) 12

( D) 11

In the diagram, ABCDE is a regular pentagon. DEF is an equilateral triangle. The size of angle AEF is

( E) 9

A E

(A) 168

( B) 150

( D) 132

( E) 170

(C) 120 F

B

D C


PART B: 6.

In the weekly Lotto six different numbers are drawn randomly from the numbers 1, 2, 3, 4, … 48, 49. Mpho’ Mpho ’s parents bought a ticket with the numbers 2; 17; 26; 29; 30; 43 on it. The first five numbers drawn were 17; 26; 30; 2 and 43. What is the chance that the next number drawn will be 29? (A)

7.


1 2

(C)

6

1

( D)

30

1 44

( E)

1 49

( B) 15

(C) 20

( D) 24

( E) 30

( D) 100,5

( E) 300 ,5

Observe: 1,5 × 1,5 = 2 ,25 2,5 × 2,5 = 6,25 3,5 × 3,5 = 12 ,25 × = 9900,25 is The value of if (A) 33,5

9.

1

The extra extra time, time, in minutes, that it would would take to cover cover a distance distance of 120 km travelling at an average speed of 60 km/h instead of 72 km/h would be (A) 12

8.

( B)

( B) 66,5

(C) 99 ,5

The entrance entrance fee fee at a concert concert was was R5 per child and R16 R16 per adult. A total total of of R789 was raised. The maximum number of people who could have attended the concert was (A) 37

10. Given that ( 21)

( B) 38 4

=

(C) 138

( D) 149

( E) 157

4

194481 then (0,21) equals

(A) 0,000194481

( B) 0,00194481

( D) 19,4481

( E) 1944 ,81

11. ABCD is a rectangle such that AD AD = 2AB. If AC = 5d , then the perimeter of ABCD is

(C) 0 ,194481

A

D

5d

(A) 4 5 d

(B) 6 5 d

(D) 6d

(E) 8d

(C) 10d B


C

12. How many non-isosceles triangles of perimeter perimeter 23 units units can be be formed with sides of whole number units?

(A) 6

( B) 8

(C) 13

•

( D) 23

( E) 66

•

4587 means 3,458745874587 ... 13. Note: The recurring decimal 3,4587 1 1 1 1 If recurring decimals = 0,1428 142857 57 and = 0, 3, what is + as a 7 3 7 3 recurring decimal? •

•

•

•

(A) 0, 4 42854 •

•

•

•

( B) 0,14286 0

•

•

( D) 0, 4 7619

•

•

( C) 0 ,4 7619 0

•

( E) 0, 4 76 2

14. The chicken on Thabo’ Thabo’s farm can gain weight at the rate of 20% per week. Thabo wants them to double their weight before he sells them. The minimum number number of weeks he needs to keep them is

(A) 3

( B) 4

(C) 5

( D) 6

( E) 7

15. Two athletes, Johnny and Sarah, are running running in opposite directions on a track after they have started at the same point. Every time they meet one another, Sarah gives Johnny R1. Sarah runs three times as fast as Johnny. The number of laps Sarah has to complete to make sure that Johnny collects R120 is

(A) 30

PART C:

( B) 40

( C) 50

( D) 60

( E) 90


16. By placing a 2 at both both ends of a number, the number number ’s value is increased by 2317. The sum of the digits of the original number is

(A) 9

( B) 8

(C) 7

( D) 6


( E) 5

17. Six numbers are represented by a; b; c; d; e and f . The average of a; b; c; and d is 10. The average of b; c; d; e and f is 14. If f is is twice the value of a then the average of a and e is

(A) 10

( B) 11

(C) 12

18. Water flows through a network network of pipes in the direction shown in the diagram from A to B. Five taps are on the network as shown. Each tap can be opened or closed to let water through or to stop A 5 the flow of the water. There are 2 = 32 different ways of setting the taps. How many of these 32 ways will allow water to flow from A to B?

(A) 17

( B) 16

(C) 15

( D) 13

( E) 15

Tap 2 Tap 1 Tap 3

B Tap 5

Tap 4

( D) 14

( E) 13

19. ABC is a wooden wooden equilateral equilateral triangular triangular block block with P as its centre. The block is rolled clockwise on a flat surface such that one side touches the surface each time it is rolled. A

B

P B

P C

A

If PC = 2 units, what is the length of the path of object P in the above diagram? (A) 12 3

( B) 2π

(C) 8π

( D) 6π

( E) 14 3

20. From the numbers 1; 2; 3; 4; … 500 a sequence is formed by deleting numbers so that no two remaining numbers have a sum which is a multiple of 7. The maximum number of numbers in this sequence is

(A) 216

( B) 217

(C) 213

( D) 287


( E) 284



FIRST ROUND 2000 JUNIOR SECTION: GRADES 8 AND 9 12 APRIL 2000 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth 5 marks. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 60 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. KEER DIE BOEKIE OM VIR AFRIKAANS P.O. BOX 538, PRETORIA, PRETORIA, 0001 TEL: (012) 328-5082 FAX (012) 328-5091 E-mail: [email protected]


1.

1 2

−

(A)

2.

1

×

2 1

1 2

equals

( B)

2

1

( C) 0

4

( D)

1

( E) −

8

How many boxes identical to Box B will fill Box A exactly?

1 2

6 cm Box A

(A) 24

( B) 10

(C) 12

( D) 18 10 cm

2 cm

What number should be in the square in the following pattern of numbers?

(A) 5

1 2

1

( B) 20

1

(C) 21

1 1

( D) 10

1

( E) 25

4.

6

3

1 3

1

4 6 4 1 5 10 10 10 5 1 15 15 6

In the figure, CBA and CDE are straight lines. 0 0 If ABD = 110 and EDB = 130 , then C equals equals 

1

A



B



(A) 30 0 ( D) 60

0

( B) 40 0 ( E) 70

o

110

(C) 50 0

0

o

130 C

5.

5 cm

Box B

4 cm

( E) 5

3.

1 cm

D

The value of 0,3 × 0 ,4 + 0,3 × 0,9 is (A) 3,9

( B) 0,189

(C) 1,3

( D) 0,78


( E) 0,39

E

6.

If a

b = (2 × a ) + (3 × b) then the value of 2

(A) 30

7.

10

( B) 19

(C) 41

( D) 21

( E) 67

( B) 2

( C) 1

( D) 625

( E) 4

equals

(A) 16

8.

5) is

4

4

5

(3

The number of times the hour hand and the minute hand of a clock form a right angle with each other between 06:00 and 12:00 on the same day is

11

12

1

10

2

9

3 4

8

9.

(A) 12

( B) 11

( D) 6

( E) 5

7

( C) 10

5

6

The average (arithmetic mean) of three numbers is 18. If one of the numbers is replaced by the number 38, then the average of the three numbers is 23. 23. The original number number that was replaced replaced is (A) 38

( B) 23

( C) 15

( D) 18

( E) 33

10. If the the following following pattern continues and the th numbers in the 100 row are added, the answer will be

1 5

3 7

(A) 1 000 000

( B) 1 600

(C) 1 000

( D) 10 000

13

11

9 15

17

19

( E) 100 100 000 000 11. It takes 8 hours to fill

4

of a container with water. The time, in hours, that that 5 it takes to fill the remainder of the container, at the same rate, is (A) 1

( B) 4

( C) 2

( D) 1,6


( E) 2,5

12. Two ants start at point A and walk at the same pace. One ant walks around a 3 cm by 3 cm square whilst the other walks around a 6 cm by 3 cm rectangle. What is the minimum distance, in centimetres, any one must cover before they meet again?

(A) 18

( B) 72

(C) 216

3 cm 3 c m

A

( B) 32

( D) 26

( E) 28

m c 6

( D) 36

13. The letters A to I represent represent numbers. The numbers are added a dded vertically and horizontally to give the numbers in the last row ( 17 ; P; Q) and the last column ( 20; 14; 12 ). The value of P + Q is

(A) 29

(C) 31

( E) 108

A

B

C

20

D

E

F

14

G

H

I

12

17

P

Q

14. Four cubes of equal equal size size are given. One is coloured green, one red, one blue and one yellow. The number of different ways they can be stacked one upon the other is

(A) 4

( B) 9

( D) 24

( E) 64

3 c m

Yellow Blue

(C) 16

Red Green

15. Mary was given given the task of removing removing all multiples multiples of 2 and 3 from the set of numbers from 1 to 100. The number of numbers that remained was

(A) 17

( B) 33

(C) 18

( D) 34


( E) 26

16. The size of angle x, in degrees, in the regular octagon is

(A) 90

( B) 67,5

( D) 60

( E) 75

(C) 45 x

D

17. ABCD is a rectangle. E and and F are midpoints of AB and CB respectively. respectively. If the area of the shaded ∆AEF is 7 cm2, then the area of rectangle ABCD, in square centimetres, is

(A) 28

( B) 49

( D) 35

( E) 56

C

F

(C) 42 A

B

E

18. Three rulers and one pencil cost the same same as two erasers. One ruler, two pencils and three erasers cost R25. If the price of each ea ch item is a whole number of Rands, then the price of an eraser is

(A) R 2

( B) R 3

( C) R 4

( D) R5

( E) impo imposs ssib ible le to find find 19. Increasing 800 – 10 x by 10% gives 600 – 6 x. The value of x equals

(A) 45

( B) 40

(C) 56

( D) 80

( E) 50

20. The sum of the first n odd natural numbers is 2304. 1 + 3 + 5 + 7 + 9 + …

= 2304

n numbers

The value of n is (A) 123

( B) 50

( C) 46

( D) 48


( E) 69



SECOND ROUND 2000 JUNIOR SECTION: GRADES 8 AND 9 6 JUNE 2000 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: For each correct answer in Part A: 4 marks in Part B: 5 marks in Part C: 6 marks For each wrong answer: –1 mark For no answer: 0 marks 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 120 minutes working time for the question paper.



PART A: 1.


The units digit of the product 11 × 13 × 15 is (A) 1

2.

( E) 9

( B) 2

3

×

5

3

2,001 ÷ 2 ,00 0 00 1999 , ( B)

1 4

( C) 3

2

×

5

4

( D) 2

4

×

5

3

( E) 2

3

×

( B) 2

5

4

is closest to

( C) 2

( D)

1 2

( E) 1 8

11 The same number is subtracted from the denominator of the fraction. 2 If the resulting fraction is equivalent to , the number subtracted is 3 ( C) 3

( D)4

.

( E) 5

If 100 is divided by the positive integer x , the remainder is 2. If 198 is divided by x , the remainder is (A) 1

PART B: 6.

( D) 7

A natural number is subtracted from the numerator of the fraction

(A) 1 5.

3

The number

(A) 0

4.

( C) 5

The number 2000 can be written in exponent form as (A) (2 × 3)

3.

( B) 3

( B) 2

( C) 3

( D) 4

( E) 5


A cube cube--shaped aped water ater tank ank has dim dimens ension ions 1 m × 1 m ×1 m . Water is flowing into the tank at a constant rate of 2  /minute. The The rate at which the the water level is rising is [Note: 1  =1 000 cm3] (A) 0 0,,1 cm / min

( B) 0, 0,2 cm cm / mi m in

( D) 1,2 cm / min

( E) 2 cm / min

(C) 1 cm / min

2


7.

8.

You wish to travel from A to B along the lines as shown in the sketch. You may only move downwards. The number of different paths from A to B is (A) 9

( B) 10

( D) 12

( E) 13

( C) 11 B

How many many numbers from 11 to 99 have the the sum of their digits digits a square square number? (A) 14

9.

A

( B) 15

( C) 16

( D) 17

( E) 18

The product of all the natural numbers from 1 to n can be written as n!. For example 5! = 1 × 2 × 3 × 4 × 5 = 120 . Find the smallest number n such that n! is divisible by 990. (A) 9

( B) 10

( C) 11

( D) 12

( E) 13

10. It requires 12 12 litres of paint to paint paint the 6 sides of a cube cube with dimensions dimensions 2m × 2m × 2m . The number of litres of paint required to paint the outside of a recta ectan ngular lar box with ith dimen imenssion ions 4m × 1m × 8m is

(A) 40

( B) 44

( C) 45

( D) 47

( E) 49

11. Just before her last mathematics test this year Thandi’ Thandi ’s average mathematics mark was 89%. She scored 97% in the last test and her average is now 90%. The number of mathematics tests she wrote this year is

(A) 5

( B) 9

( C) 7

( D) 13

12. In the the diagram, diagram, ABCD is a square square with diagonal AC. BCE is an equilateral triangle.

( E) 8

A

D

B

C

∧

The size of angle ACE is (A) 105

( B) 60

( D) 135

( E) 120

(C) 90

E 3


13. Each side of a regular hexagon is extended extended by a length equal to its own length. The end points of the new segments are joined to form a new and larger regular hexagon. What fraction of the area of the bigger hexagon does the smaller hexagon occupy?

(A) ( D)

1

( B)

3 3

( E)

8

1

( C)

2 1

1 4

6

14. A class survey survey found found that 25 learners watched TV on Monday, Monday, 20 on Tuesday and 16 on Wednesday. Of those who watched TV on only one of the days, 11 chose Monday, 7 chose Tuesday and 6 chose Wednesday. If each learner watched on at least one of the days and 7 watched on all 3 days, how many learners are there in the class?

(A) 61

( B) 45

( C) 41

( D) 39

( E) 38

15. The length length of the diagonal diagonal of a square is d. d. The area of the square is

(A)

1 2

d

2

(D) 2d

PART C:

(B) 2

d

2

2

(E) 2d

(C) d

2

d

2


16. Suppose two natural numbers are represented represented by n 2 and n2 – 1. Which of the following must necessarily be true?

(A) Both are odd numbers. (B) Both are even numbers. (C) Both numbers are multiples multiples of 4. (D) Exactly one of the numbers is a multiple of 4. (E) The sum of the numbers is even.

4


17. A box contained 31 sweets. sweets. On Monday Sam ate

on Monday. On Tuesday Sam ate

3 4

of the number Pam ate

2

of the number Pam ate on Tuesday. 3 Then all the sweets had been eaten. How many sweets did Sam eat? (A) 6

( B) 7

( C) 10

( D) 11

( E) 13

18. A slow train travelling travelling from Springs to to Soweto arrives 9 minutes late when travelling at 36 km/h. If it travels at 27 km/h it arrives 39 minutes late. What is the distance between Springs and Soweto?

(A) 54

( B) 90

( C) 48

( D) 36

( E) 64

19.

A coin with a mark on it as shown above is rolled without sliding along a straight line. The path the mark follows during one revolution is: (A) (B)

(C)

(D)

(E)

20. A sealed envelope contains a card with a single digit on it. Three of the following statements are true and the other is false. I The digit is 1 II The digit is 2 III The digit is not 3 IV The digit is not 4

Which one of the following must be correct? (A) I is false.

(B) II is true.

(D) III is false.

(E) IV is true.

(C) II is false.

THE END 5




FIRST ROUND 1999 JUNIOR SECTION: GRADES 8 AND 9 29 APRIL 1999 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer answer is worth: 3 marks in Part A, 5 marks in Part B and 7 marks in Part C. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 60 minutes working time for the question paper.



PART A:

1.


1+ 2 + 3 + 4 1× 2 × 5

is equal to

(A) 0

2.

( D)

1 2 1 6

( E) 4

( B) ( E)

1 8 1

( C)

1 3

4

( B) 6

( C) 5

( D) 8

( E) 9

For long distance telephone calls, Telkom charges 30,9 cents per metering unit of 13,6 seconds. The cost of a 3 minute long distance call to the nearest rand is (A) R 3,00

5.

( D) 3

If two dice are rolled, which total is most likely? (A) 7

4.

( C) 2

Noni knits a blanket. She uses 2 colours and follows the pattern as shown. The fraction of the blanket knitted in the darker shade is (A)

3.

( B) 1

( B) R 7,00

( C) R5,00

( D) R6,00

( E) R4,00

2

The leng length thss A rectangular carpet has a perimeter of 16 m and area 15 m . The of the sides of the carpet in metres are (A) 10 and 6

( B) 15 and 1

( C) 8 and 2

( D) 7,5 and 2

2


( E) 5 and 3

PART B: 6.

A five sided regular polygon polygon is drawn with dotted diagonal lines as shown from one vertex to the other vertices. If a regular polygon has 50 sides and diagonals are drawn from one vertex to the other vertices, how many diagonals are there? (A) 46

7.

8.

( B) 47

( C) 48

( D) 49

( E) 50

Suppose there are 20 million learners attending schools in South Africa of which ten percent are in Grades 8 and 9. Each learner in Grade 8 and 9 must receive a Mathematics text book costing R30, a set of language books costing R70, a science text book costing R40 and a history book at R60. What is the total cost for the Department of Education if they have to purchase all these text books? (A) R40 million

( B) R4 million

( D) R4 R400 mi million

( E) R4 R4 000 mi million

(C) R2 million

Approximately Approximatel y how far away did the lightning strike if I heard the sound of the thunder 12 seconds after I saw the lightning? (Let the speed of sound be ±332 m/sec.) (A) 1 km

9.


( B) 12 km

( C) 6 km

( D) 24 km

( E) 4 km

Dr Maharaj Maharaj travels travels 45 000 km in his his car. During During that time he changes changes the tyres, including the spare, regularly so that at the end of the 45 000 km journey each tyre has the same amount of wear. How many many kilometres kilometres of wear does each tyre have? (A) 45 000

( B) 9 000

( C) 36 000

( D)18 000

( E) 27 000

10. Anusha has has 3 questions questions correct correct in section A of this paper and 7 correct in section B. She hopes to score 60% to advance to the next round. The least number of questions she must get right in section C is

(A) 1

( B) 2

( C) 3

( D)4

3


( E) 5

11. Circle, centre O, has triangles AOC and COB COB drawn inside it. AOB is a straight line. ∧

B o

58

∧

If OBC = 58 then the size of OCA is

O y

(A) 58

( B) 60

( D) 32

( E) 30

C

A

( C) 64

If a pen costs costs 20 cents more than a 12. The cost of 2 pens and 3 rulers is R1,90. If ruler, then the cost of 2 rulers and 3 pens in rand is (A) 2,10

( B) 1,90

( C) 2,50

( D) 1,50

( E) 2,00

th

13. In the 20 century (1901 – 2000), how many years are divisible by 6?

(A) 15

( B) 16

( C) 17

( D) 24

( E) 96

14. A birthday present is tied with ribbon as shown. The bow, knots and ends used 47 cm of ribbon. The total length of the ribbon used, in metres, is

(A) 1,57

( B) 1,67

( C) 1,79

( D) 1,37

m c 0 1

c m 1 5

20 cm

( E) 0,92

15. A target consists consists of rings of width width 1 cm. cm. The radius radius of the inner black circle is 1 cm. Determine how many times the area of the outer black ring is bigger than the area of the inner black circle.

(A) 9

( B) 3

( C) 7

( D) 4

4


( E) 5

PART C:


16. In this addition, addition, different different letters stand for for different different digits, digits, but each letter represents the same digit each time it appears. If the letter O stands for 7, what digit must U represent?

+ =

TWO TWO FOUR

(A) 5

( B) 6

( C) 4

17. The areas of the faces of a rectangular 2 2 2 box are 84 cm , 70 cm and 30 cm . 3 The volume of the box in cm is

(A) 96

( B) 420

( C) 5 880

( D) 3

( E) 2

70 cm2 2

84 cm2

( D) 88 200

18. The diameter diameter of the the big big circle circle is divided into 5 parts. These five parts are the diameters of new circles as shown. If the circumference of the big circle is 30 cm, then the sum of the circumferences of the 5 smaller circles is

(A) smaller smaller than 30 cm ( B) 30 cm ( C) 45 cm ( D) 60 cm ( E) 3 30 0π cm

5


c m 3 0

( E)176 400

19. You have 3 weights: weights: 1 kg, 3 kg and 9 kg as well as an equal arm balance, as shown. How many different weight objects can you weigh with these three? [Remember the weights may be placed on either side]

(A) 13

( B) 15

( C) 9

1 k g g 3 k g g 9 k g g

( D) 11

( E) 7

20. In the star, ∧

∧

∧

∧

∧

∧

B

∧

A + B + C+ D + E + F + G =

A C

(A) 720

( B) 540

( D) 300

( E) 900

( C) 360

G D F E

THE END

A man is like a fraction fraction whose numerator is what he is and whose denominator denominator is what he thinks of himself. himself.

The larger the denominator the smaller the fraction. In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1989.

6




SECOND ROUND 1999 JUNIOR SECTION: GRADES 8 AND 9 22 JUNE 1999 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer answer is worth: 3 marks in Part A, 5 marks in Part B and 7 marks in Part C. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not n ot necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 120 minutes working time for the question paper.



PART A: 1.


If (1 053 × 15) − 450 = ❤ × 15, 15, then then (A) 603

2.

3.

4.

(A) Mo Monday

(B) Tu Tuesday

(D) Th Thursday

(E) Fr F riday

( D)1 023

( E) 60 ,3

(C) We Wednesday

The figures ABCD and BGFE are overlapping rectangles rectangles as shown. The area 2 of rectangle ABCD is 80 cm and the area 2 of rectangle BGFE is 60 cm . 2 The difference in area, in cm , between the black areas ABED and BCEFG is (A) 40

( B) 24

( D) 12

( E) im impossible to determine.

A

B 80 cm2

C D

6 0 c m 2

G

E F

( C) 20

To pin up a rectangular picture Sam needs 4 pins, one at each ea ch corner. For two pictures he needs only 6 pins since he can overlap pictures. What is the smallest number of pins he needs to pin up 10 pictures? ( B) 22

( C) 20

( D) 18

In isosceles ∆ABC , AB = 2BC. If the perimeter of the length of AC in millimetres is (A) 120

6.

( C) 1 050

Freedom Day, 27 April 1999, was celebrated on a Tuesday. Tuesday. On what day of the week will we celebrate Freedom Day in the year 2010?

(A) 24 5.

( B) 105

❤ =

( B) 40

( C) 50

∆ABC is

( D) 10

( E) 16 200 mm, then

( E) 80

1 000 dots are evenly evenly spaced on the circumference of a circle. They are numbered from 1 to 1 000 with dot 1 000 opposite dot 500. Which dot is opposite dot 657? (A) 156

( B) 157

( C) 158

( D) 159

2


( E) 160

PART B: 7.

Three parties parties contested an election with 100 000 voters who all cast valid votes. One of the parties won the election, obtaining more votes than either of the other two parties. What is the smallest number of votes this party could have obtained? (A) 33 333

8.

( B) 33 334

( C) 50 000

( D) 50 001

Three figures consist of alternate black and white square tiles as shown. If this pattern continues, what percentage of the number of th tiles in the 50 figure will be black? (A) 49

9.


( B) 50

( C) 51

1st

( D) 66

( E) 66 667

2nd

2

3rd

( E) 75

3

If the the length length to breadth ratio is the same whether a book book is opened or closed, find the ratio of length to breadth. (A) 2 :1

( B) 2 : 2

( C) 2 : 1

( D) 4 : 1

( E) 8 : 1

10. The LCM of 42 and n is 462. Then n cannot be

(A) 33

( B) 66

( C) 88

( D) 231

11. In rectangle rectangle ABCD, AD = 12 cm, cm, AB = 7 cm and 2 EF = 5 cm. The black area, in cm , is

(A) 42

( B) 35

( C) 49

( E) 462 B

E

F

C

( D) 56

( E) Impossible Impossible to find with the given given information. information. A

G

12. In the following calculation, calculati on, a and b represent missing digits. If 79 287 ÷ a21 = 2 b7 , then a + b =

(A) 3

( B) 4

( C) 7

( D) 8

( E) 13

13. Each male honey-bee has a single female female parent whilst each female honeyth bee has both a male and a female parent. In the 10 generation back, only, how many ancestors does a male honey-bee have?

(A) 89

( B) 144

( C) 10

( D) 512

3


( E) 233

D

A 14. In

∆ABC AB AB =

AC aan nd BD = BC.

∧

∧

If ABD = 36 , the si s ize of ABC in degrees is (A) 45 ( D) 54

( B) 72

D

( C) 75

o

3 6

( E) 60 B

PART C:

C


15. Which of of the following statements is TRUE with regard regard to a circle?

(A) The area is sometimes numerically larger than the circumference. (B) The area is always numerically larger than the circumference. circumference. (C) The area and circumference are never numerically the same. (D) When the circumference of a circle doubles then the area also doubles. (E) The circumference is always numerically larger than the area.

16. A solid solid cube cube with 3 metre metre side lengths has square square holes cut through it from the middle of each face to the middle of the opposite face, as shown. The three holes intersect in the middle of the cube. The square windows so produced have side lengths of 1 metre each. What is the total surface area, in square metres, of the new solid?

(A) 72

( B) 76

( C) 78

m 3

( D) 80

1m m 3

3m

( E) 84

17. A woman arrives arrives at the the station at the same time every day. Her husband drives from home to the station to pick her up. They always arrive back at their house at the same time every e very day, driving at the same speed. One day she arrives at the station one hour early and starts to walk home. Her husband meets her along the road and they drive home together. They arrived home 10 minutes earlier than usual. Assuming all speeds of walking and driving are constant and all transfers are immediate, i mmediate, how long had the woman been walking when her husband met her? (answer in minutes)

(A) 10

( B) 50

( C) 55

( D) 70

4


( E) 90

1999

18. What is the unit digit of 2

(A) 1

( B) 3

2000

+3

?

( C) 5

( D)7

( E) 9

19. How many numbers are there in the list 1, 2, 3, 4, 5, …, 10 000 which contain exactly two consecutive 9’s such as 993, 1992 and 9929 but NOT 9295 or 1999?

(A) 280

( B) 271

( C) 270

( D) 261

( E) 123

20. An ant walks walks around around a triangle triangle with sides 5 cm, 6 cm, cm, and 7 cm so that it always stays 1 cm from the outside of the triangle. When it returns to its starting point for the first time, in cm, it has walked

(A) 19

( B) 21

( C) 24

( D) 18 + π

THE END

5


( E) 18 + 2π



FIRST ROUND 1998 JUNIOR SECTION: GRADES 8 AND 9 10 MARCH 1998 TIME: 60 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth: 3 marks in Part A, 5 marks in Part B and 7 marks in Part C. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not necessarily drawn to scale. 6. Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. prob lems. You will have 60 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. KEER DIE BOEKIE OM VIR AFRIKAANS P.O. BOX 538, PRETORIA, 0001 TEL: (012) (012) 328-5082 328-5082 FAX (012) 328-5091


PART A:

1.


7 is equal to 5 (A) 1, 2

2.

(B) 1,5

12 12,34 34

(B) 2,5

−

5.

(C) 25

(D) 50

( E) 2 500

(B) 11,114

(C) 11,006

(D) 11,116

(E) 11,104

(D) 27

(E) 24

What mass, in kilograms, is indicated on the scale? (A) 42,0

(B) 40,2

(D) 40,4

(E) 45,0

The value of 1 × 9 × 9 1+ 9+ 9 (A) 1

PART B: 6.

( E) 1,4

1, 23 234 equals

(A) 11,106

4.

( D) 1,6

50 50% of 50 equals (A) 1

3.

(C) 1,7

(B) 4

(C) 44,0

× +

8 is 8 (C) 0


The desks desks in a classroom classroom are arranged in straight rows. John is in the third third row from the front and the fourth row from the back. He is also third from the left end of a row and fifth from the right. How many desks are in the classroom? (A) 24

(B) 30

(C) 35

(D) 42

2


( E) 56

7.

In a contest to guess the number number of balloons in a bunch, bunch, Sarah guessed guessed 25, Betty guessed 31, Nomsa guessed 29, Jakkie guessed 23 and Edith guessed 27. Two guesses were wrong by 2, and two guesses were wrong by 4. The other guess was correct. The number of balloons in the bunch was (A) 29

8.

(C) 25

If 3 ! means 1 × 2 × 3 , thus 3 ! 4 ! = 24 , then 6 ! is equal to (A) 120

9.

(B) 27

(B) 384

=

( D) 31

6 an and if 4 ! means 1 × 2

(C) 720

(D) 1 008

(E) 23

3

×

×

4 , thus

(E) 5 040

How many different rectangles, with natural numbers as side lengths, can be constructed so that the perimeter of each rectangle is 16? (A) 7

(B) 5

(C) 4

(D) 2

( E) 1

(D) 30

(E) 29

10. Sipho has made a cylindrical clay pot. His sister, Thandi, wants to decorate it with a triangular pattern as shown in the sketch. If the triangles are equilateral, of side 5 cm and the circumference of the pot is 1 metre then the total number of triangles in the pattern is

(A) 20

(B) 40

(C) 39

11. Which one of the following fractions lies between 1 and 2 ? 2 3

(A) 17 24

(B) 1 3

(C) 1 4

(D) 7 12

3


( E) 3 4

12. In the diagram the lengths of some of the sides are indicated using letters. The length of p in terms the other letters is

(A)

w − x − y

(B)

w −

3x

−

2y

(C)

w −

3x

+

2y

(D)

w + x + y

(E)

w +

3x

+

2y

13. The greatest number of Mondays which can occur in 45 consecutive days is

(A) 5

(B) 6

∧

(C) 7

( D) 8

∧

14. If AOB = 60 and COD = 30 , the fraction of the area of the circle which is

(A) 1 3 ( D) 1 5

(B) 1 9 ( E) 1 4

(A) 26

(B) 32

( D) 38

(E) 45

shaded is

(C) 1 6

15. A beam of light shines from a point at point P, and reaches a point T so that Then x is

PART C:

( E) 9

S, reflects off a reflector (mirror) PT is perpendicular to RS.

(C) 37


4


16. A march goes through the streets from the School (S) to the Community Community Centre (CC) If the the march march can only travel East or South, South, number of different possible routes is

(A) 6

17. ( 23

=

(B) 10

2

×

2

×

(C) 4

5

2 an and 2

=

2

×

2

×

2

( D) 8

×

2

×

then the

( E) 9

2)

If 2 x + 3y = 41 , where x and y are natural numbers, then the value of x + y is (A) 9

(B) 8

(C) 7

(D) 6

( E) 5

( D) 7

( E) 9

100 18. The last digit of the number number 3 is

(A) 0

(B) 1

(C) 3

19. Saul plays a video game in which he scores 4 for a hit and

6 for a miss. miss. After 20 rounds his score is 30. The number of times he has missed is (A) 5

(B) 10

(C) 6

( D) 15

−

( E) 4

20. Solomon went to the shopping centre ce ntre to buy supplies for his mathematics project. He spent half of what he had plus R2 in the first shop, half of what he then had left plus R1 in the second shop, half of what he then had left plus R1 in the third shop and, in the fourth shop half of all he had left. Three rand was left over. How much money did he start with?

(A) R32

(B) R48

(C) R56

( D) R64

5


( E) R72



SECOND ROUND 1998 JUNIOR SECTION: GRADES 8 AND 9 26 MAY 1998 TIME: 120 MINUTES NUMBER OF QUESTIONS: 20 Instructions: 1. Do not open this booklet until told to do so by the invigilator. 2. This is a multiple choice question paper. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 3. Scoring rules: Each correct answer is worth: 3 marks in Part A, 5 marks in Part B and 7 marks in Part C. There is no penalty for an incorrect answer or an unanswered question. 4. You must use an HB pencil. Rough paper, ruler and rubber are permitted. Calculators and geometry instruments are not permitted. 5. Diagrams Diagrams are not necessarily drawn to scale. 6. Indicate Indicate your answers on the sheet provided. 7. When When the invigilator gives the signal, start the problems. You will have 120 minutes working time for the question paper.

DO NOT TURN THE PAGE UNTIL YOU ARE TOLD TO DO SO. KEER DIE BOEKIE OM VIR AFRIKAANS P.O. BOX 538, PRETORIA, 0001 TEL: (012) 328-5082 FAX (012) 328-5091


PART A: 1.


The value of 5 × 4 × 3 × 2 − 5 × 4 × 3 is (A) 1

2.

3.

( B) 2

( C) 20

( D) 60

The polygon polygon has sides which meet at right angles. Side lengths are as shown. The perimeter of the polygon is (A) 44

( B) 34

( D) 22

( E) impossible to to find.

If a  b (A) 39

=

ab +

a b

5.

2 8

( C) 32 12

, then 6  2 is equal to

( B) 15

( C) 12

1 3

∧

4.

( E) 120

In the diagram P = 90º and PQ = 8 cm. 2 If the area of ∆ MPQ is 24 cm , the then n th the perimeter of the triangle in cm is (A) 24

( B) 40

( D) 54

( E) 48

( C) 18

( D) 12

( E) 11

M

P

Q

In an orchard with m orange trees, every tree produced p oranges. Some 3 oranges were lost due to a hailstorm and mp oranges remained. 4 The percentage of oranges lost was (A) 75

( B) 40

( C) 30

( D) 25

2


( E) 10

6.

One day I noticed noticed that my newspaper had 24 pages and that page 6 and and page 20 were on the same double sheet. Which other pages were also on this sheet? (A) 7 & 19

PART B: 7.

( B) 5 & 21

( E) 8 & 22

Two gears, one with 15 teeth and the other one with 20 teeth, fit together as shown in the figure. Each has a marked tooth as indicated. After how many rotations of the gear with 15 teeth will the marked tooth be together again for the first time? ( B) 2

( C) 3

( D) 4

( E) 5

Which one of the following is not a prime number? (A) 2

2

− 2 +1

( C) 2

4

− 2 + 2 − 2 +1

6

−2 +2 −2 +

( E) 2

9.

( D) 7 & 21


(A) 1

8.

( C) 5 & 19

( B) 2

3

2

5

4

3

3

− 2 + 2 −1

5

− 2 + 2 − 2 + 2 −1

( D) 2

2

4

3

2

22 − 2 +1

The length and width of a rectangle are both doubled. When the new rectangle is compared to the original rectangle (A) the area and the the perimeter perimeter are unchanged. unchanged. (B) the area and the perimeter perimeter are both doubled. doubled. (C) the area is 4 times the original area and the perimeter perimeter is doubled. (D) the area is doubled doubled and perimeter is 4 times as large as the original perimeter. (E) the area and the perimeter perimeter are both 4 times times as large as the original area and perimeter. 3


10. A fruit fruit vendor vendor makes a display display with her peaches in a pyramid structure. structure. The pyramid has a square base. She manages to make a display six layers high. The number of peaches she needs for this display is

(A) 21

( B) 84

11. In the diagram the angle m is

( C) 91

2 3

( D) 72

( E) 36

the size of n. m

The value of m is (A) 75

( B) 60

( D) 96

( E) 84

30

( C) 54

n

o

12. My teacher had 3 big boxes of sweets. One box contained red sweets, one green and one yellow. Each learner was given 5 sweets of two different colours. Every learner in the class received a different combination combination of colours. What was the maximum number of learners that could have been in the class?

(A) 10

( B) 12

( C) 15

( D) 18

( E) 60

13. In the diagram, the sloping line divides divides the area are a of the rectangle in the ratio 1 : 4. What is the ratio a : b?

(A) 1 : 1

( B) 1 : 2

( D) 1 : 4

( E) 2 : 3

( C) 1 : 3

4


1 4

a b

1998

14. The remainder when 3

(A) 0

PART C:

is divided by 5 is

( B) 1

( C) 2

( D) 3

( E) 4


th

15. How many black tiles will be required to build the 15 figure in the given pattern?

... st.

nd

1

(A) 403

rd

2

( B) 365

th

3

( C) 481

15

( D) 421

16. Two lines AB and CD intersect at 90º 90º in O and start rotating around O in different directions as shown. AB rotates at one revolution every two minutes and CD at three revolutions a minute. After how many seconds will the lines be concurrent (i.e. on top of each other) for the first time?

( E) 225

A

C

O

D

B

(A) 13

( B) 4

1 11

( C) 4

2 7

( D) 5

2

17. ABCD and XYCZ are are squares. squares. The area inside inside square ABCD, but outside square XYCZ is 30 square units. If DY = 10, then the length of CD is

(A) 35

( B) 65

( D) 10

( E) 8

( E) 6

3

D

Z

130

( C)

X

2 A 5


C

Y B

18. Danie has three three types of marbles: marbles: small, medium and large. He He finds that 18 small marbles and 6 medium marbles weigh as much as 5 large marbles. He also finds that 2 medium marbles and 1 large marble weigh as much as 10 small marbles. The number of small marbles that weigh as much as 1 large marble is

(A) 7

( B) 6

( C) 5

( D) 4

( E) 3

19. The houses and trees in the diagram are all in a straight line. In each of the the six houses lives a child. At which tree should the children meet so that the sum of the distances they walk to that tree is a minimum?

A

(A) A

B

( B) B

( C) C

C

( D) D

D

( E) Impossible to determine.

20. Three girls A, B and C run in a 100 m race. When A finishes, B is 10 m behind A and when B finishes C is 20 m behind B. How far in metres was C from A when A finished? (Let’ (Let’s assume all the athletes run at a constant speed)

(A) 30

( B) 29,5

( C) 29

( D) 28,5

THE END

6


( E) 28

SAMO Junior 2015 to 1998 Questions

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