KVS Junior Mathem Mathematics atics Olympiad previous papers 5th KVS Junior Mathematics Olympiad (JMO) – 2001 M.M. 100 Note : (i)
Time : 3 hours Please check that there are two printed pages and ten question in
all. (ii)
Attempt all questions. All questions carry equal marks.
1.
Fill in the blanks : (a)
If x + y = 1, x 3 + y3 = 4, then x2 + y2 = ……..
(b)
After 15 litres of petrol was added to the fuel tank of a car, the tank was 75% full. If the capacity of the tank is 28 litres, then the number of litres in the tank before adding the petrol was ……
(c)
The perimeter of a rectangle is 56 metres. The ratio of its length to width is 4:3. The length of the diagonal in metres is ……..
(d)
If April 23 falls on Tuesday, then March 23 of the same year was a ……..
2.
(e)
The sum of the digits of the number 2 200052004 is ….
(a)
Arrange the following in ascending order : 25555, 33333, 62222
(b)
Two rectangles, each measuring 3 cm x 7 cm, are placed as in the adjoining figure : Find the area of the overlapping portion (shaded) in cm2.
3.
(a)
Solve :
log 10 (35 x 3 ) 3 log 10 (5 x )
4.
(b)
a b b c c a (a b)( b c)(c a ) Simplify : a b b c c a (a b)( b c)(c a )
(a)
Factorize : (x-y)3+(y-z)3+(z-x)3
(b) 5.
If x2-x-1=0, then find the value of x 3-2x +1
ABCD is a square. A line through B intersects CD produced at E, the side AD at F and the diagonal AC at G. B
A G
F
C
D
E
If BG = 3, and GF=1, then find the length of FE,
6.
7.
(a)
Find all integers n such that (n 2-n-1)n+2 = 1
(b)
4ab x 2a x 2 b If x = , find the value of a b x 2a x 2 b
(a)
Find all the positive perfect cubes that divide 9 9.
(b)
Find the integer closest to 100 (12- 143 )
8.
In a triangle ABC, BCA=90o. Points E and F lie on the hypotenuse AB such that AE=AC and BF = BC. Find ECF. A F E
y
C 9.
x z
B
An ant crawls 1 centimetre north, 2 centimetres west, 3 centimetres south, 4 centimetres east, 5 centimetres north and so on, at 1 centimetre per second. Each segment is 1 centimetre longer than the preceding one, and at the end of a segment, the ant makes a left turn. In which direction is the ant moving 1 minute after the start ?
10.
Find the lengths of the sides of a triangle with 20, 28 and 35 as the lengths of its altitudes.
6th KVS Junior Mathematics Olympiad (JMO) – 2002 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks. 1.
Fill in the blanks (a) Yash is carrying 100 hundred – rupee notes, 50 fifty –rupee notes, 20 twenty – rupee notes, 10 ten –rupee notes and 5 five-rupee notes. The total amount of money he is carrying in Rupee, is …………. (b) In a school, the ratio of boys to girls is 4:3 and the ratio of girls to teachers is 8 :1. The ratio of students to teachers is ……… (c) The value of
0.5 1 0 . 5
2
is ………….
(d) (123456)2 + 123456 + 123457 is the square of ……… (e) The area of square is 25 square centimeters. In perimeter, in centimeters, is ……………. 2. (a) How many four digit numbers can be formed using the digits 1,2 only so that each of these digits is used at least once ? (b)
Find the greatest number of four digits which when increased by 1 is exactly divisible by 2, 3, 4, 5, 6 and 7.
3. (a) If f(x) = ax7 + bx5 + cx3 – 6, and f(-9) = 3, find f(9). (b) Find the value of
(2002)3 (1002)3 (1000)3 3 x (1002) x (1000)
4 x 4.(a) If x > 0 and
(b)
1 1 3 47 x , find the value of x4 x3
If 82x = 161-2x, find the value of 3 7x.
5. A train, after traveling 70 km from a station A towards a station B, develops
3 a fault in the engine at C, and covers the remaining journey to B at of its 4 earlier speed and arrives at B 1 hour and 20 minutes late. If the fault had developed 35 km further on at D, it would have arrived 20 minutes sooner. Find the speed of the train and the distance from A to B. A 6.
C
D
B
The adjoining diagram shows a square PQRS with each side of length 10 cm. Triangle PQT is equilateral. Find the area of the triangle UQR. P
Q
T S
U R
A square of side – length 64 cm is given. A second square is obtained by connecting the mid points of the sides of the first square (as shown in the diagram). If the process of forming smaller inner squares by connecting the mid points of the sides of the previous squares is continued, what will be the side-length of the eleventh square, counting the original square as the first square ?
7.
Seven cubes of he same size are glued together face to face as shown in the adjoining diagram. What is the surface area, in square centimeters, of the solid if its volume is 448 cubic centimeters ?
8.
10.
Anil, Bhavna, Chintoo, Dolly and Eashwar play a game in which each is either a FOX or a RABBIT. FOXES’ statements are always false and RABBITS’ statements are always true. Anil says that Bhavna is a RABBIT. Chintoo says that Dolly is a FOX. Eashwar says that Anil is not a FOX. Bhavna says that Chintoo is not a RABBIT. Dolly says that Eashwar and Anil are different kinds of animals. How many FOXES are there ? (Justify your answer). The accompanying diagram is a road-plan of a city. All the roads go east-
west or north-south, with the exception of one shown. Due to repairs one road is impassable at the point X, of all the possible routes from P to Q, there are several shortest routes. How many such shortest routes are there ?
Q
x P
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7 KVS Junior Mathematics Olympiad (JMO) – 2003 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)
Attempt all questions. All questions carry equal marks.
1.
Fill in the blanks (a)
The digits of the number 2978 are arranged first in descending order and then in ascending order. The difference between the resulting two numbers is ……………
(b)
Yash is riding his bicycle at a constant speed of 12 kilometers per hour. The number of metres he travels each minute is ………..
(c)
The square root of 35 x 65 x 91 is ………..
(d)
The number 81 is 15% of …………..
(e)
A train leaves New Delhi at 9.45 am and reaches Agra at 12.58 pm. The time taken in the journey, in minutes, is …………..
2.
3.
(a)
Find the largest prime factor of 203203.
(b)
Find the last two (tens’ and units’) digits of (2003) 2003.
(a)
Find the number of perfect cubes between 1 and 1000009 which are exactly divisible by 9.
(b)
If x = 5 + 2 6 , find the value of
(i) 4.
(a)
x
1 x
(ii)
x3
1 x3
Solve :
x2 1 x2 5 x2 2 x2 6 2 2 2 2 x 4 x 8 x 5 x 9 (b)
81 49 25 9 Find the remainder when x x x x x is divided by
x 3 x. 5.
(a)
OPQ is a quadrant of a circle and semicircles are drawn on OP and OQ. Areas a and b are shaded. Find a/b.
(b)
Assuming all vertical lines are parallel, all angles are right angles and all the horizontal lines are equally spaced, what fraction of figure is shaded ?
6.
Alternate vertices of a regular hexagon are joined as shown. What fraction of the total area of a hexagon is shaded ? (Justify your answer)
7.
In a competition consisting of 30 problems Neeta was given 12 points for each correct solution, and 7 points were subtracted from her score for each incorrect solution problems not attempted contributed 0 points to the score find the number of problems attempted correctly by Neeta.
8.
A cube with each edge of lengths 4 units is painted green on all the faces. The cube is then cut into 64 unit cubes. How many of these small cubes have (i) 3 faces painted (ii) 2 faces painted (iii) one face painted (iv) no face painted.
9.
Let PQR be an equilateral triangle with each side of length 3 units. Let U, V, W, X Y and Z divide the sides into unit lengths. Find the ratio of the area U, W, X, Y and Z divide the sides into unit lengths. Find
the ratio of the area U W X Y (shaded) to the area of the whole triangle PQR.
P U
Z
V.
Q
10.
Y
W
X
R
Five houses P, Q, R, S and T are situated on the opposite side of a
street from five other houses U, V, W, X and Y as shown in the diagram : P.
Q.
R .
V.
.T
X.
Y.
20 m
20 m
U.
.S
.W
Houses on the same side of the street are 20 metres apart A postman is trying to decide whether to deliver the letters using route PQRSTYXWVU or route
PUQVRWSXTY, and finds that the total distance is the same in
each case. Find the total distance in metres.
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8 KVS Junior Mathematics Olympiad (JMO) – 2005 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1.
Fill in the blanks: (a)
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is ………….. 4 4 2 2 0 then equals……… x x x
(b)
If 1
(c)
If a=1000, b=100, c=10, and d=1, then (a+b+c-d) + (a+b-c+d) + (a-b+c+d)+(-a+b+c+d) is equal to …….
(d)
When the base of a triangle is increased by 10% and the altitude to the base is decreased by 10%, the change in area is ……….
(e)
If the sum of two numbers is 1, and their product is 1, then the sum of their cubes is ……………
2.
3.
8 2 log 2 8
(a)
If x = log
(b)
4x 9 x y If x y 8 and 5 y 243 find the value of x-y. 2 3
(a)
Find the number of digits in the number 22005 x 52000 when
find the value of log3x.
written in full.Find the remainder when 2 2005 is divided by 13.
4.
(a)
A polynomial p (x) leaves a remainder three when divided by x –
1 and a remainder five when divided by x-3. Find the remainder when p(x) is divided by (x-1) (x-3). (b)
Find two numbers, both lying between 60 and 70, each of which is
exactly divides 2 43-1. 5.
In triangle ABC the medians AM and CN to the sides BC and AB, respectively intersect in the point O.P is the mid-point of side AC, and MP intersects CN in Q. If the area of triangle OMQ is 24 cm 2, find the area of triangle ABC.
6.
The base of a pyramid is an equilateral triangle of side length 6 cm. The other edges of the pyramid are each of length
15 cm. Find the volume of
the pyramid. 7.
Chords AB and CD of a circle (see figure) intersect at E and are perpendicular to each other segments AE. EB and ED are of lengths 2cm, 6cm and 3cm respectively. Find the length of the diameter of the circle.
C
A 2
E6 3 D
B
8.
Three men A, B and C working together, do a job in 6 hours less time than A alone, in 1 hour less time than B alone, and in one half the time needed by C when working alone. How many hours will be needed by A and B working together, to do the job ?
9.
Pegs are put on a board 1 unit apart both horizontally and vertically. A rubber band is stretched over 4 pegs as shown in the figure forming a quadrilateral. Find the area of the quadrilateral in square units.
10.
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The odd positive integers 1, 3, 5, 7 ….. are arranged in five columns continuing with the pattern shown on the right. Counting from the left, in which column (I, II, III, IV or V) does the number 2005 appear ? (Justify your answer) I II III IV V 1 3 5 7 15 13 11 9 17 19 21 23 31 29 27 25 33 35 37 39 47 45 43 41 49 51 53 55 . . . . . . . . . . . .
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9 KVS Junior Mathematics Olympiad (JMO) – 2006 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1) a,b,c are three distinct real numbers and there are real numbers x,y such that a3+ax+y = 0, b 3+bx=y = 0, c3+cx+y = o, so that a+b+c = 0. 2) The triangles A,B,C has CA = CB. P is a point on the circum circle between A and B.(and on the opposite side of the line AB to C). D is the foot of the perpendicular from C to PB. So that PA + PB = 2 PD. 3) Given reals x,y with (x2+ y2) /(x2 – Y2) + (x2-y2)/(x2+y2) = K. find the value of ( x8+ y8) /(x8 – Y8) + (x8-y8)/(x8+y8) in terms of K. 4) In a traingle ABC right angled at B, a point P is taken on the side AB such that AP = h and BP = b and AC = y such that h + y = b +d. Prove that h = bd/(2b+d). 5) P is a point in the traingle ABC. Lines are drawn through P parallel to the Sides of the traingle. The areas of the three resulting traingles with a vertex at P have areas 4, 9 and 49. What is the area of the triangle ABC?
6) A lotus plant in a pool of water is ½. Cubit above the water level. When propelled by air, the lotus sinks in the pool 2 cubits away from its position.
Find the depth of the water in the pool? 7) Let C1 be any point on the side AB of triangle ABC. Join C1C. The lines through A and B paralle to CC 1 meet BC and AC produced at A1 and B1 respectively. Prove that 1/AA 1 + 1/BB1 = 1/CC1. 8) The triangle ABC has angle B = 900. When it is rotated about AB it gives a cone of volume 800 ∏ Cubic Units When it is rotated about BC it gives a cone of volume 1920∏ Cubic Units . Finmd the length of AC. 9) A number when divided by 7, 11 and 13 (the prime factors of 1001) Successively leave the remainders 6,10 and 12 respectively. Find the Remainder if the number is divided by 1001. 10) Two candles of the same height are lighted together. First one gets burned up completely in three hours while the second in 4 hours. At what point of time the length of the second candle will be doubled the length of the first candle ?
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10 KVS Junior Mathematics Olympiad (JMO) – 2007 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1.
Solve | x-1 | + | x | + | x + 1 | = x + 2
2.
Find the greatest number of four digits which when divided by 3, 5, 7, 9 leaves remainders 1, 3, 5, 7 respectively.
3.
A printer numbers the pages of a book starting with 1. He uses 3189 digits in all. How many pages does the book have ?
4.
ABCD is a parallelogram. P, Q, R and S are points on sides AB, BC, CD and DA respectively such that AP=DR. If the area of the parallelogram is 16 cm2, find the area of the quadrilateral PQRS.
5.
ABC is a right angle triangle with B = 90 o. M is the mid point of AC and BM = 117 cm. Sum of the lengths of sides AB and BC is 30 cm. Find the area of the triangle ABC. (a x ) (a x )
a x
6.
Solve :
7.
Without actually calculating, find which is greater :
(a x ) (a x )
31 11 or 17 14
8.
Show that there do not exist any distinct natural numbers a, b, c, d such that a3 + b3 = c3 + d 3 and a + b = c + d
9.
Find the largest prime factor of : 312 + 212 – 2.66
10.
If only downward motion along lines is allowed, what is the total number of paths from point P to point Q in the figure below ? P
Q
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11 KVS Maths Olympiad Contest – 2008 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1)
Find the value of S = 1 2 – 22 + 32 – 42 + ……………….-982 + 992
2)
Find the smallest multiple of ‘15’ such that each digit of the multiple is either ‘0’ or ‘8’.
3)
At the end of year 2002. Ram was half as old as his grandfather. The sum of years in which they were born is 3854. What is the age of Ram at the end of year 2003?
4)
Find the area of the largest square, which can be inscribed in a right angle triangle with legs ‘4’ and ‘8’ units.
5)
In a Triangle the length of an altitude is 4 units and this altitude divides the opposite side in two parts in the ratio 1:8. Find the length of a segment parallel to altitude which bisects the area of the given triangle.
6)
A number ‘X’ leaves the same remainder while dividing 5814, 5430, 5958. What is the largest possible value of ‘X’?
7)
A sports meet was organized for four days. On each day, half of existing total medals and one more medal was awarded. Find the number of medals awarded on each day.
8)
Let ΔABC be isosceles with
ABC = ACB = 780. Let D and E be the points
on sides AB and AC respectively such that BCD = 240 and CBE = 510. Find the angle BED and justify your result.
9)
If , and are the roots of the equation. (x - a) (x - b) (x - c) + 1 = 0. Then show that a, b and c are the roots of the equation ( - x) ( - x) ( - x) + 1 = 0.
10)
A 4 x 4 x 4 wooden cube is painted so that one pair of opposite faces is blue, one pair green and one pair red. The cube is now sliced into 64 cubes of side 1 unit each. (i)
How many of the smaller cubes have no painted face?
(ii)
How many of the smaller cubes have exactly one painted face?
(iii)
How many of the smaller cubes have exactly two painted faces?
(iv)
How many of the smaller cubes have exactly three painted faces?
(v)
How many of the smaller cubes have exactly one face painted blue and one face painted green ?
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13 KVS Maths Olympiad Contest – 2010 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1) Let a,b,c are real numbers a not equal to zero such that a and 4a +3b+2c have the same sign. Show that the equation ax 2 + bx +c = 0 can not have both roots in the interval (1,2) . 2) a) find all the intgers which are equal to 11 times the sum of their digits. b) Prove that 3 (2 5) 3 (2 5) is a rational number. 3) A circle centered at A with radius 1 unit and another circle centered at B with radius 4 units touch each other externally. A third circle is drawn to touch the first two circles and one of their common external tangents as shown in the figure. What is the radius of the third circle?
4) Given three non co-linear points A,H and G. Construct a triangle with A as vertex, H as orthocenter and G as centroid. 5) A triangle ABC, angle A is twice the angle B. Prove that a 2 = b(b+c) where
a, b and c are the sides opposite to angles A,B and C respectively. 6) The equation x2+px+q = 0, where p and q are integers, has rational roots. Prove that the roots are integers. 7) Triangle ABC is right triangle with angle C is 900. Measure of ABC = 600,and AB = 10 units. Let P be a point chosen randomly inside triangle ABC.Extend BP to meet Ac at D. What is the probability that BD >5 2 . 8) Prove that the sum of hypotenuse and the altitude of a right angled triangle dropped on the hypotenuse exceed the half perimeter of the triangle. 9) How many times is digit zero is written when llisting all numbers from 1 to 3333? 10) Find out the remainder when x+x9+x25+x49+x81 is divided by x3-x?
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14 KVS Maths Olympiad Contest – 2011 M.M. 100
Time : 3 hours
Note : (i)Please check that there are two printed pages and ten question in all. (ii)Attempt all questions. All questions carry equal marks 1) Show that for any natural number ‘n’ the fraction
21n 4 is in its lowest 14n 3
term. 2) a) Factorize:
x6+5x3+8
b) Prove that 3a4-4a3 b+b4 0, for all real numbers a and b.
3) M is any point on the minor arc BC of a circum circle of an equilateral triangle ABC. Prove that AM = BM+CM. 4) Solve the inequality,
x
1 x 1 4
5) a) Find the square root of
3 ( x 1) 2 x 2 7 x 4 ,x>4 2
b) Given real natural numbers x,y and z are such that x+ y + z = 3, x2+y2+z2 = 5, x3+y3+z3 = 7. Find the value of x4+y4+z4 ? 6) In the triangle given each side is of length 4 units. If the length PQ is 1 unit and TQ is perpendicular to PR, find the ratio of areas of triangle PQT and the quadrilateral QRST.
7) Prove that for any natural number ‘n’ , the expression n
n
n
n
A = 2903 – 803 – 464 + 261 is divisible by 1897. 8) Find the number of odd integers between 30,000 and 80,000 in which no digit is repeated? 9) Find all the integers which are equal to 11 times the sum of their digits? 0
10) Prove that in any triangle ABC is one angle is 120 , the triangle formed by the feet of angle bisectors is a right angled.
