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2015加拿大国际高中数学竞赛试题I


The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca

Canadian Intermediate Mathematics Contest
Wednesday, November 25, 2015
(in North America and South

America)

Thursday, November 26, 2015
(outside of North America and South America)

c 2015 University of Waterloo Time: 2 hours Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. Do not open this booklet until instructed to do so. There are two parts to this paper. The questions in each part are arranged roughly in order of increasing di?culty. The early problems in Part B are likely easier than the later problems in Part A. PART A 1. This part consists of six questions, each worth 5 marks. 2. Enter the answer in the appropriate box in the answer booklet. For these questions, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet. PART B 1. This part consists of three questions, each worth 10 marks. 2. Finished solutions must be written in the appropriate location in the answer booklet. Rough work should be done separately. If you require extra pages for your ?nished solutions, they will be supplied by your supervising teacher. Insert these pages into your answer booklet. Be sure to write your name, school name and question number on any inserted pages. 3. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks. At the completion of the contest, insert your student information form inside your answer booklet. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location, and score range of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some students may be shared with other mathematical organizations for other recognition opportunities.

Canadian Intermediate Mathematics Contest
NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write solutions in the answer booklet provided. √ 3. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as 4.14 . . . or 1.41 . . ., except where otherwise indicated. 4. While calculators may be used for numerical calculations, other mathematical steps must be shown and justi?ed in your written solutions and speci?c marks may be allocated for these steps. For example, while your calculator might be able to ?nd the x-intercepts of the graph of an equation like y = x3 ? x, you should show the algebraic steps that you used to ?nd these numbers, rather than simply writing these numbers down. 5. Diagrams are not drawn to scale. They are intended as aids only. 6. No student may write both the Canadian Senior Mathematics Contest and the Canadian Intermediate Mathematics Contest in the same year.

PART A
For each question in Part A, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet. 1. Stephanie has 1000 eggs to pack into cartons of 12 eggs. While packing the eggs, she breaks n eggs. The unbroken eggs completely ?ll a collection of cartons with no eggs left over. If n < 12, what is the value of n? 2. In the diagram, ABCD is a square with side length 4. Points P , Q, R, and S are the midpoints of the sides of the square, as shown. What is the area of the shaded region?

A S D

P

B Q

R
100?

C

3. In the diagram, line segments AB , CD and EF are parallel, and points A and E lie on CG and CH , respectively. If ∠GAB = 100? , ∠CEF = 30? , and ∠ACE = x? , what is the value of x?

G A B D F

x? 30?

C

E H

4. If 12x = 4y + 2, determine the value of the expression 6y ? 18x + 7. 5. Determine the largest positive integer n with n < 500 for which 6048(28n ) is a perfect cube (that is, it is equal to m3 for some positive integer m).

6. A total of 2015 tickets, numbered 1, 2, 3, 4, . . . , 2014, 2015, are placed in an empty bag. Al?e removes ticket a from the bag. Bernice then removes ticket b from the bag. Finally, Charlie removes ticket c from the bag. They notice that a < b < c and a + b + c = 2018. In how many ways could this happen?

PART B
For each question in Part B, your solution must be well organized and contain words of explanation or justi?cation. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks. 1. At Cornthwaite H.S., many students enroll in an after-school arts program. The program o?ers a drama class and a music class. Each student enrolled in the program is in one class or both classes. (a) This year, 41 students are in the drama class and 28 students are in the music class. If 15 students are in both classes, how many students are enrolled in the program? (b) In 2014, a total of 80 students enrolled in the program. If 3x ? 5 students were in the drama class, 6x + 13 students were in the music class, and x students were in both classes, determine the value of x. (c) In 2013, half of the students in the drama class were in both classes and onequarter of the students in the music class were in both classes. A total of N students enrolled in the program in 2013. If N is between 91 and 99, inclusive, determine the value of N . 2. Alistair, Conrad, Emma, and Salma compete in a three-sport race. They each swim 2 km, then bike 40 km, and ?nally run 10 km. Also, they each switch instantly from swimming to biking and from biking to running. (a) Emma has completed has she travelled?
1 13

of the total distance of the race. How many kilometers

(b) Conrad began the race at 8:00 a.m. and completed the swimming portion in 30 minutes. Conrad biked 12 times as fast as he swam, and ran 3 times as fast as he swam. At what time did he ?nish the race? (c) Alistair and Salma also began the race at 8:00 a.m. Alistair ?nished the swimming portion in 36 minutes, and then biked at 28 km/h. Salma ?nished the swimming portion in 30 minutes, and then biked at 24 km/h. Alistair passed Salma during the bike portion. At what time did Alistair pass Salma?

2015 Canadian Intermediate Mathematics Contest (English)

3. Heron’s Formula says that if a triangle has side lengths a, b and c, then its area equals s(s ? a)(s ? b)(s ? c), 1 where s = 2 (a + b + c) is called the semi-perimeter of the triangle.

A c B a b C

(a) In the diagram, ABC has side lengths AB = 20, BC = 99, and AC = 101. If h is the perpendicular distance from A to BC , determine the value of h.

A
20 h 101

B
(b) In the diagram, trapezoid P QRS has P S parallel to QR. Also, P Q = 7, QR = 40, RS = 15, and P S = 20. If x is the distance between parallel sides P S and QR, determine the value of x.

99

C
20

P
7

S x
15

Q

40

R

(c) The triangle with side lengths 3, 4 and 5 has the following ?ve properties: ? ? ? ? ? its side lengths are integers, the lengths of its two shortest sides di?er by one, the length of its longest side and the semi-perimeter di?er by one, its area is an integer, and its perimeter is less than 200.

Determine all triangles that have these ?ve properties.


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