Đề thi Toán quốc tế IMSO năm 2007

注意：

允許學生個人、非營利性的圖書館或公立學校合理使用

本基金會網站所提供之各項試題及其解答。可直接下載

而不須申請。

重版、系統地複製或大量重製這些資料的任何部分，必

須獲得財團法人臺北市九章數學教育基金會的授權許

可。

申請此項授權請電郵 ccmp@seed.net.tw

Notice:

Individual students, nonprofit libraries, or schools are

permitted to make fair use of the papers and its

solutions. Republication, systematic copying, or

multiple reproduction of any part of this material is

permitted only under license from the Chiuchang

Mathematics Foundation.

Requests for such permission should be made by

e-mailing Mr. Wen-Hsien SUN ccmp@seed.net.tw

page 1 from 5

MATHEMATICS SHORT ANSWER PROBLEMS IMSO 2007

1. If 672 is divided into three parts proportional to 3, 4 and 5, what is the largest

number?

2. Anelka’s weight is a whole number. If Anelka gains 6 kilograms, his weight will be less

than 36 kilograms. If he gains 8 kilograms, his weight will be more than 36 kilograms.

What is Anelka’s weight?

3. In the figure, a circle is inscribed in the larger square. The

four vertices of the smaller square are on the circle. The area

of the larger square is 49 cm2. What is the area of the smaller

square?

4. Initially, two elevators were at the same floor. The first elevator went up 4 floors

and then went down 7 floors. The second elevator went up 11 floors and then went

down 9 floors. After the movements, how many floors apart were the two elevators?

5. Use each of the digits 1, 2, 3, 4, 5 and 6 exactly once to form two 3-digit numbers.

What is the largest possible product of the two 3-digit numbers?

6. In the expression f e d c b a + + , each letter is replaced by a different digit among 1, 2, 3,

4, 5, and 6. What is the smallest possible value of this expression?

7. Some cats and some pigeons are in the pigeon loft. The total number of their heads

is 34 and the total number of their legs is 80. How many cats are there?

page 2 from 5

8. A 10 cm x 10 cm x 10 cm cube is cut parallel to its three faces as shown in the

figure. After cutting, eight smaller solids are formed. What is the total surface

area of these smaller solids?

9. What is the largest 2-digit prime number that can be obtained by adding two prime

numbers?

10. In the figure, two equivalent circles are placed in the rectangle. The distance

between the two center points of the circles is 4 3 y. What is the value of y ?

11. Sixteen matchsticks are used to construct Figure 1, thirty-six matchsticks for

Figure 2, and so on. With this gradual increase (the middle row of Figure n contains

2n+1 squares), how many matchsticks are needed to construct the Figure 10?

Figure 1 Figure 2 Figure 3

… …

page 3 from 5

12. In the figure, three circles a, b and c touch at the point P.

The center of circle c is on circle b, and the center of circle

b is on circle a. What is the ratio of the shaded area to the

unshaded area of circle c?

13. In the following grid, how many shortest paths are

there to travel from A to B?

14. Mary has six $0.1 stamps, five $0.4 stamps, four $0.5 stamps, three $5 stamps, and

two $10 stamps. She is going to send a parcel by mail and the mailing fee required is

$38.1. At most how many stamps remain after mailing?

15. Tony is drawing some triangles and their side lengths are whole numbers. Each

triangle has a perimeter of 7 units. At most how many triangles of different sizes

can Tony draw?

16. The average of the squares of seven consecutive integers is 629. What is the

smallest integer?

17. Some people are waiting in a queue. Abel is exactly in the middle of the queue.

Sarah is number 14 and she is somewhere behind Abel. Bart is number 20. At most

how many people can be in the queue?