Đề thi Toán quốc tế CALGARY năm 2014

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THE CALGARY MATHEMATICAL ASSOCIATION

38 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 9 pages including this one.

• Each correct answer to PART A will score 5 points.
You must put the answer in the space provided. No
part marks are given.

• Each problem in PART B carries 9 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 do 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.

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PART A:

SHORT ANSWER QUESTIONS

(Place answers in
the boxes provided)

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

Alex, Betty and Chi have a total of 87 candies altogether. If Chi gives 4 candies
to Betty and 3 candies to Alex, each person then has the same number of candies.
How many candies did Chi start with?

A3

Roll three dice so that each die shows one number from 1 to 6, and multiply these
three numbers together. What is the smallest positive even number which cannot
be obtained?

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

A book has 200 pages. How many times does the digit 5 appear in the page numbers?

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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

A ceiling fan has blades 60 cm long, and rotates at a rate of 2 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

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

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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 4 hours. Find the distance in km fromA toB.

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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?

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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?

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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 0.

(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.

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