2017 Fermat Contest

The CENTRE CENTRE for EDUCA EDUCATION TION in MATHEMA MATHEMATICS TICS and COMPUTING

cemc.uwaterloo.ca

Fermat Contest (Grade 11) Wednesday, February 24, 2016 (in North America and South America)

Thursday, February 25, 2016 (outside of North America and South America)

Time:

60 minutes

©2015 University of Waterloo

Calculators Calculators are allowed, allowed, with the following restriction: restriction: you may not use a device that has internet internet access, that can communicate communicate with other devices, devices, or that contains contains previo previously usly stored stored inform informati ation. on. For exampl example, e, you you may may not use a smartp smartphon hone e or a tablet. Instructions 1. Do not open the Contest Contest booklet until until you are told to do so. 2. You may use rulers, rulers, compasses compasses and paper for rough work. 3. Be sure that you understa understand nd the coding system system for your response response form. If you are not sure, ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in circles completely. completely. 4. On your response response form, print your your school name and city/to city/town wn in the b ox in the upper right corner. Contest you are writing 5. Be certain that you code your name, age, grade, and the Contest in the response form. Only those who do so can be counted as eligible students. 6. This is a multiplemultiple-cho choice ice test. Each question question is followed followed by five possible answers answers marked A, B, C, D, and E. Only one of these is correct. correct. After After making making your choic choice, e, fill in the appropriat appropriatee circle on the response form. 7. Scoring: Scoring: Each Each correct correct answer answer is worth 5 in Part A, 6 in Part Part B, and 8 in Part C. There is no penalty for for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions. 8. Diagra Diagrams ms are are not drawn to scale. They are intended as aids only. 9. When your your supervisor supervisor tells you to begin, you you will have have sixty minutes of working time. 10.You may not write more than one of the Pascal, Cayley or Fermat Contest in any given year. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, name, grade, grade, schoo schooll and loc location, ation, and scor score e range of some some top-sc top-scorin oring g students students will be published published on our website, cemc.uwaterlo cemc.uwaterloo.c o.ca. a. In addition, the name, grade, grade, school and location, location, and score of some top-scoring students may be shared with other mathematical organizations for other reco recognition gnition opportunities.

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Scor Scorin ing: g:

Ther Theree is no penalty for for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5. 1. 2.

3.

If x = 3, y = 2x and a nd z = 3y, the value of z is (A) 8 (B) 9 (B) 9 (C) 6

(A) 6

(B) 12 (B) 12

(D) 4

(E) 10 (E) 10

The The expr expres essi sion on

(C) 8

20 + 16 20 equals 20 16

×

×

(B) 27 (B) 2766

(C) 21

6.

(D)

9 4

(E)

17 16

An oblong number is the number of dots in a rectangular grid with one more row than column. The first four oblong numbers are 2, 6, 12, and 20, and are represented below:

What is the 7th oblong number? (A) 42 (B) 49 (B) 49 (C) 56 5.

(E) 15 (E) 15

A cube cube has has 12 edge edges, s, as shown. shown. How many many edges edges does does a square-based pyramid have?

(A) 20 4.

(D) 18

(D) 64

(E) 72 (E) 72

In the the dia diagr gram am,, poin point Q is the midpoint of P R. The coordinates coordinates of R are a re (A) (2, 5)

(B) ( (B) (77, 11)

(D) (8, 10)

(E) ( (E) (99, 15)

(C) (6, 9)

Carrie sends five five text text messages messages to her her brother brother each each Saturday Saturday and and five five text messages messages



8.



If 25 (A) 4

9.

− √ n = 3, the value of n is (B) 16 (B) 16

(C) 64

(D) 484

(E) 256 (E) 256

(D) 3

(E) 9 (E) 9

If x% of 60 is 12, then 15% of x is (A)

3 4

(B)

1 3

(C) 4

10. In the the diagra diagram, m, squar squaree PQRS has side length length 2. Points Points W , X , Y , and Z are the midpoints P QRS QRS . midpoints of the sides of P What is the ratio of the area of square W X Y Z to to the area of square PQRS ? (A) 1 : 2 (D) 1 : 4

(B) 2 (B) 2 : 1

√ (E) 2 : 2

(C) 1 : 3

Part B: Each correct answer is worth 6. P QR is righ 11 11.. In the the diag diagrram am,, rightt-an angl gled ed at P and P R = 12. 12. If poin pointt S is on P Q so that SQ = 11 and SR = 13, the perimeter of QRS is



(A) 47

(B) 44 (B) 44

(D) 41

(E) 61 (E) 61

 

(C) 30

12. How many many of the positive positive divisors divisors of 128 are perfect squares squares larger larger than 1? (A) 2

(B) 5 (B) 5

(C) 1

13. The numbe numbers rs 4x, 2x 3, 4x What is the value of x?

−

(D) 3

(E) 4 (E) 4

− 3 are three consecutive terms in an arithmetic sequence.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence.) (A)

3 4

(B)

−

4 3

(C)

3 2

(D)

−

3 2

(E)

−

3 4

14. Suppose Suppose that that a and b are integers with 4 < a < b < 22. If the average (mean) of the



16. If x = 18 is one of the solutions of the equation x2 + 12x + c = 0, the other solution of this equation is (A) x = 216

(B) x =

−6

(C) x =

−30

(D) x = 30

(E) x =

−540

17 17.. A tota totall of n points are equally spaced around a circle and are labelled with the diametrically opposite if the line integers 1 to n, in order order.. Two Two poin points are calle called d diametrically segment segment joining them is a diameter of the circle. circle. If the points labelled labelled 7 and 35 are diametrically diametrically opposite, then n equals (A) 54

(B) 55 (B) 55

(C) 56

18. Suppose Suppose that that x and a nd y satisfy The value of (x + y) + ( x (A) 210

−

x y xy = 9 and = x+y x+y

− y) + xy is

−150

(B)

(D) 57

(C) 14160 (C) 14160

(E) 58 (E) 58

−60. −14310

(D)

−50

(E)

19. 19. Ther Theree are are n students students in the math club at Scoins Scoins Secondary School. School. When Mrs. Fryer Fryer tries to put the n students in groups of 4, there is one group with fewer than 4 students, but all of the other other groups groups are comple complete. te. When When she tries tries to put the n students in groups of 3, there are 3 more complete groups than there were with groups of 4, and there there is agai again n exactl exactly y one group group that is not complet complete. e. When When she tries tries to put the n students in groups of 2, there are 5 more complete groups than there were with groups groups of 3, and there there is aga again in exactly exactly one group group that is not complete complete.. The sum of 2 n is the digits of the integer equal to n (A) 11

−

(B) 12 (B) 12

(C) 20

(D) 13

20. 20. In the the diag diagra ram, m, PQRS represents represents a rectangular piece of paper paper.. The The paper paper is fold folded ed along along a line line V W so that When the folde folded d paper paper is flatten flattened ed,, ∠V W Q = 125 . When points R and Q hav have mo move ved d to point pointss R and Q , respectively, respectively, and R V crosses P W at Y . The The meas measur uree of ∠P Y V is ◦





◦

(B) 10 (B) 1000

◦

(E) 11 (E) 1155

(A) 110

(D) 105

◦

◦

◦

(C) 95



(E) 10 (E) 10



Part C: Each correct answer is worth 8. 21. 21. Box Box 1 cont contai ains ns one gold gold marble marble and one one blac black k ma marb rble le.. Bo Box x 2 cont contain ainss one one gold gold marble marble and two black black marble marbles. s. Bo Box x 3 contai contains ns one gold marble marble and three three black black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally equally likely to be chosen. A marble is randomly chosen chosen from Box 1 and placed placed in Bo Box x 2. Then Then a marble marble is randomly randomly chosen chosen from from Box 2 and placed placed in Box Bo x 3. Finally Finally,, a marble marble is random randomly ly chosen chosen from Box 3. What What is the probab probabilit ility y that the marble chosen from Box 3 is gold? (A)

11 40

(B)

3 10

(C)

13 40

(D)

7 20

3 8

(E)

22. If x and y are real real number umbers, s, the the mini minimu mum m possib possible le value alue of the the expr expres essi sion on 2 2 2 2 (x + 3) + 2(y 2) + 4(x 7) + (y + 4) is

−

(A) 172

(B) 65 (B) 65

−

(C) 136

(D) 152

23. Seven Seven coins of three three differe different nt sizes sizes are placed placed flat on a table table,, arra arrang nged ed as show shown n in the the diagr diagram am.. Eac Each coin coin,, except except the centre centre one, touch touches es three other other coins. coins. The centre centre coin touches touches all of the other other coins. If the coins labelled C 3 have a radius of 3 cm, and those labelled C 2 have radius 2 cm, then the radius of the coin labelled X is closest to (A) 0.615 cm

(B) 0 (B) 0 .620 cm

(D) 0.605 cm

(E) 0 (E) 0 .625 cm

(C) 0.610 cm

(E) 104 (E) 104

C 3 C 2 X

C 2

C 2

C 3

C 3

24. 24. For any real number number x, x denote denotess the larges largestt integ integer er less less than than or equal equal to x. For example, 4.2 = 4 and 0.9 = 0. If S is the sum of all integers k with 1 k 999 999 and for which which k is divisible by k , then S equals

 

√ 

  

(A) 999 (A) 999 50 5000 00 0000 (D) 998 (D) 998 99 9999 50 5000

≤ ≤

(B) 999 (B) 999 00 0000 00 0000 (E) 998 (E) 998 50 5000 50 5000

(C) 999 (C) 999 99 9999 00 0000

25 25.. The The set set A = 1, 2, 3, . . . , 2044, 2045 contain containss 2045 elemen elements. ts. A subset subset S of A is called triple-free if no element of S equals three times another element of S . For example, 1, 2, 4, 5, 10, 2043 is triple-free, but 1, 2, 4, 5, 10, 681, 2043 is not triple A that contain the largest number of elements contain free. The triple-free triple-free subsets of A A that contain exactly 1535 exactly 1535 elements. There are n triple-free subsets of A a b elements. elements. The integer integer n can be written in the form p q , where p and q are distinct prime numbers and a and b are positive integers. If N = p 2 + q 2 + a2 + b2 , then the last three digits of N are

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( E C n o g n t l i e s h s ) t

The CENTRE CENTRE for EDUCA EDUCATION TION in MATHEMA MATHEMATICS TICS and COMPUTING

cemc.uwaterloo.ca

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