Đề thi Toán quốc tế IMSO năm 2015

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SHORT ANSWER PROBLEMS

Country: Name:

ID: Score:

Instructions:

 Write down your name and country on the answer sheet.

 Write your answer on the answer sheet.

 For problems involving more than one answer, points are given only when ALL answers are correct.

 Each question is worth 1 point. There is no penalty for a wrong answer.

 You have 60 minutes to work on this test.

 Use black or blue colour pen or pencil to write your answer.

“Smart, Skilled, and Creative In a Joyful Competition for Excellence”

Page 1

International Mathematics and Science Olympiad 2015

SHORT ANSWER PROBLEMS

(1) Anne asks her teacher his age. He replied ‘My age now is a square number but after my birthday it will be a prime number.’ Assuming his age is below 65 and above 20, how old is he now?

(2) Given that 240 84 234 56 90256 a b   . What is the value of a b  ?

(3) If I place all three operational symbols +, –,  in all possible ways into the blanks of the expressions 5____4____6____3, one symbol per one blank, each resulting expression will have a value. What is the largest of these values?

(4) In the diagram below, the regular octagon ABCDEFGH and the regular hexagon IJKLMN are centered around the same point O such that AB // IJ. If the measure of 56 CBJ   , find the measure of BJK  , in degrees.

(5) A boy has a cup of tea and a girl has an empty glass having the same volume

as the cup. In the first step, the boy pours 1

2 of the tea from the cup into the

glass. In the second step, the girl pours 1

3 of the tea from the glass into the

cup. In the third step, the boy pours 1

4 of the tea from the cup into the glass.

In the fourth step, the girl pours 1

5 of the tea from the glass into the cup.

This alternate pouring continues such that in each step, the denominator increases by 1. What fraction of the tea is in the cup after the thirteenth step?

A

N

M

H

L

K

J

I

G

F E

D

C

B

O

Page 2

(6) There is a committee of 5 members. The chairman will be seated in a permanent chair at the round table. In how many ways can the other 4 members be seated at the same table if there are exactly 8 chairs?

(7) In the arrangement below, each number is the non-negative difference of the two numbers above it. What is the average of the eight possible values of z?

(8) How many positive integers from 1 up to 2015 are not divisible by any of the following numbers: 2, 20, 201 and 2015?

(9) In a survey of 100 students, 84 said they disliked playing Tennis, 74 said they disliked skiing, 62 students said they disliked both playing tennis and skiing. How many students liked both playing tennis and skiing?

(10) We know that 0, 2, 4, 6 and 8 are even digits. How many even digits are used from 1 to 100?

(11) Two numbers are called mirror numbers if one is obtained from the other by reversing the order of digits. For example, 123 and 321. If the product of a pair of mirror numbers is 146047, then what is the sum of this pair of mirror numbers?

(12) In the hexagon at the right, 1 is placed in the top triangle. In how many different ways can we place 2, 3, 4, 5 and 6 in the remaining empty triangles, such that the sum of the numbers in opposite triangles is 5, 7 or 9?

◎ ◎ ◎ ◎

◎

◎ ◎ ◎ ◎

◎ ◎

◎ ◎

◎ ◎ 0

w 6

x 28

36 y

◎ 76 z

1