Đề thi Toán quốc tế IMSO năm 2014

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__________________________________________________________________ Mathematics Short Answer Problems

__________________________________________________________________

Country: Name: No.: 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.

Page 1

International Mathematics and Science Olympiad 2014

SHORT ANSWER PROBLEMS

(1) Calculate 9+99+999+9999+99999+999999+9999999+99999999+999999999.

(2) The income of a taxi driver is the sum of the regular salary and some tips. The

tips are 5

4 of his salary. What is the fraction of his income which comes

from his tips?

(3) The base of a large triangle is two times the altitude of a small triangle, and the altitude of the large triangle is three times the base of the small triangle. What is the ratio of the area of the large triangle to the area of the small triangle?

(4) In an election, only 80% of people planned to vote. However only 85% of those who planned to vote actually vote. What is the percentage of the people who actually vote?

(5) INTERNUTS company offers internet service with an initial payment of 300000 rupiahs and a monthly fee of 72000 rupiahs. Another company, VIDIOTS, offers internet service with no initial payment but a monthly fee of 90000 rupiahs. Johnny prefers INTERNUTS company. What is the minimum number of months he should subscribe in order to pay less than the subscription with VIDIOTS company?

(6) Twelve identical squares are put together in a 6 × 2 configuration to form a rectangle. If the perimeter of each square is 6 cm, what is the perimeter of the rectangle?

(7) The sum of two 2-digit numbers is also a 2-digit number. What is the maximum value of the product of those three 2-digit numbers?

(8) How many positive integers less than 2014 such that the sum of the digits of each is a multiple of 5?

(9) The diagram below shows two 3 × 4 pieces of paper, part of which is shaded. A student copies both figures on the same 3 × 4 piece of paper. What is the fraction of this piece of paper which is shaded?

(10) In total: Alice and Brian have 377 cards, Brian and Colin have 685 cards, Colin and Alice have 546 cards. How many cards does Brian have?

(11) Write down six positive integers whose sum is 100 such that each integer contains at least one digit 6.