Đề thi Toán quốc tế IMSO năm 2018

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MATHEMATICS SHORT ANSWER PROBLEMS

Name : Index Number :

Country :

15th International Mathematics and Science Olympiad

Zhejiang Province, China

29 September 2018

Instructions:

1. Write your name, country and index number on both the Question Booklet and Answer Sheet.

2. Write your Arabic Numerical answers only in the Answer Sheet.

3. There are 25 questions in this paper.

4. For problems involving more than one answer, marks are only awarded when ALL answers are correct.

5. Each question is worth 1 mark. There is no penalty for a wrong answer.

6. You have 60 minutes to complete this paper.

7. Use black pen or blue pen or pencil to write your answer.

Page 1 of 5

International Mathematics and Science Olympiad 2018

SHORT ANSWER PROBLEMS

1. A job at Hai Liang Education Park can be done by Alex alone in 6 hours and by Bob alone in 10 hours. Alex works on the job for one hour alone, then Bob continues to work on the job for one hour alone. If they repeat the pattern, in how many hours can the job be done? Express your answer as a common fraction.

2. In the figure below, E is a point on side AD of rectangle ABCD. Points F, G, H and I are midpoints of CE, BF, CG and BH respectively. If the area of triangle BCI is 1 cm2, find the area of rectangle ABCD, in cm2.

3. In a sequence, the first two terms are 64 and 36. Each subsequent term is the average of the preceding terms. Find the sum of the first 2018 terms.

4. In the figure shown, the distance between adjacent dots in each row and each column is 1 cm. What is the area of the shaded region, in cm2? 5. For any positive integer n, we define the function ( ) f n to be the sum of the digits of n and the number of digits of n. For example, (218) 2 1 8 3 14 f = + + + = . (Note: The first digit of n, reading from left to right, cannot be 0). What is the sum of maximum and minimum values of n such that ( ) 6 f n = ?

C D

B A

H F

G E I

Page 2 of 5

6. A rectangle is divided into 9 smaller portions as shown in the figure below. The perimeter, in cm, of the 5 known portions are also given. Find the perimeter, in cm, of the original rectangle.

7. A cylindrical water tank, with diameter 2.8 m and height 4.2 m, is filled in by a pipe of diameter 7 cm, through which water flows at the rate of 4 m/sec. How many minutes will it take for the pipe to completely fill the tank?

(Take 22

7 π = )

8. We want to divide a square into obtuse triangles such that every two triangles meet at a common vertex or at a common edge or are disjoint. At least how many triangles can we have?

9. What is the last digit of

2018 2018 2018 2018 2018 2018 2018 2018 12 14 16 18 20 ... 2014 2016 2018 + + + + + + + + ?

10. How many 3-digit positive integers have the property that the product of all of its digits is equal to 18?

11. Two overlapped equilateral triangles are shown in the figure below. The sides of each triangle are parallel to the sides of the other. The perimeter of the two triangles are 744 cm and 930 cm, respectively. What is the perimeter, in cm, of shaded hexagon?

11 20

12

11

8