Đề thi Toán quốc tế CALGARY năm 2014

One day, each person leaves his or her house, and visits each of the other five houses one after the other using paths, stopping in the last house visited. Nobody can change paths except[r]

THE CALGARY MATHEMATICAL ASSOCIATION

38th JUNIOR HIGH SCHOOL MATHEMATICS CONTEST APRIL 30, 2014

NAME: GENDER:

PLEASE PRINT (First name Last name) M F

SCHOOL: GRADE:

(9,8,7, )

• You have 90 minutes for the examination The test has two parts: PART A — short answer; and PART B — long answer The exam has pages including this one

• Each correct answer to PART A will score points You must put the answer in the space provided No part marks are given

• Each problem in PART B carries points You should show all your work Some credit for each problem is based on the clarity and completeness of your answer You should make it clear why the answer is correct PART A has a total possible score of 45 points PART B has a total possible score of 54 points

• You are permitted the use of rough paper Geome-try instruments are not necessary References includ-ing mathematical tables and formula sheets are not

permitted Simple calculators without programming or graphic capabilities are allowed Diagrams are not drawn to scale They are intended as visual hints only

• When the teacher tells you to start work you should read all the problems and select those you have the best chance to first You should answer as many problems as possible, but you may not have time to answer all the problems

MARKERS’ USE ONLY

PART A ×5 B1 B2 B3 B4 B5 B6 TOTAL (max: 99)

BE SURE TO MARK YOUR NAME AND SCHOOL AT THE TOP OF THIS PAGE

PART A: SHORT ANSWER QUESTIONS (Place answers in the boxes provided)

A1

A1 From the set{2,3,4,5,6,7,8,9,10,11,12}all prime numbers are removed

How manynumbers are remaining?

A2

A2 Alex, Betty and Chi have a total of 87 candies altogether If Chi gives candies to Betty and candies to Alex, each person then has the same number of candies How many candies did Chi start with?

A3

A3 Roll three dice so that each die shows one number from to 6, and multiply these three numbers together What is the smallest positive even number which cannot be obtained?

A4

A4 A glass in the shape of a cylinder is 10 cm high and 15 cm around, as shown The glass has a logo on it occupying 2% of the curved side of the glass What is the area (in square cm) of the logo?

A5

A5 A book has 200 pages How many times does the digit appear in the page numbers?

A6

A6 A home has some fish, some birds and some cats Altogether there are 15 heads and 14 legs If the home has more than one of each animal, how many fish are there?

A7

A7 A ceiling fan has blades 60 cm long, and rotates at a rate of revolutions per second The speed of the end of a blade can be written in the formN πcm per second, where

N is a positive integer What isN?

A8

A8 In the diagram below, similarly marked segments are equal in length Find the length of the segmentP Q

A9

PART B: LONG ANSWER QUESTIONS

B1 A truck is delivering heavy goods from cityA to cityB When travelling from Ato

B the average speed of the truck is 45 km/h On the return trip, the empty truck has an average speed of 90 km/h The total time spent travelling fromA toB and returning fromB toA is hours Find the distance in km fromA toB

B2 There are 2014 digits in a row Any two consecutive digits form a number which is divisible by 17 or 23

(a) If the last digit is 1, then what are the possibilities for the first digit?

B3 Two squares ABCD and AEFG, each with side length 25, are drawn so that the two squares only overlap at vertex A Suppose DE has length 14 What is the length of BG?

B4 We will call a positive integer a “2-timer” if its digits can be arranged to make a number of shape 2×2×2× · · · ×2 For example, 2014 is a 2-timer, because its digits can be arranged to make 1024 which is

2×2×2×2×2×2×2×2×2×2

Positive integers cannot start with a digit

(a) What is the smallest 2-timer larger than 2014? Be sure to justify your answer

(b) What is the largest 2-timer less than 2014? Be sure to justify your answer

B5 Three circles of radii 12cm, 6cm, and 6cm each touch the other two What is the radius in cm of the smaller circle which touches all three?

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