Đề thi Toán quốc tế IMSO năm 2005

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International Mathematics and Science Olympiad 2005

SHORT ANSWER PROBLEMS

1. Complete this magic triangle so that the numbers along each side give

the same sum. Use each of the numbers 5, 6, 7, 8, 9 and 10 only once.

(You are required to give only one solution.)

2. The height of the ground floor of a building is 4 m. The height of each

of the other floors is 3 m. The total height of this building is 61 m.

Inclusive of the ground floor, how many floors does the building have?

3. The composition of Scotty Cake for 6 servings is shown below.

How much rice flour is needed to make Scotty cake for 10 servings?

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4. If a positive whole number B is divided by 2, 3, 4, 6 or 9, the remainder

is 1. Find the smallest possible value of B.

5. During the first five months of 2004, a company suffered a loss, then

gained profit in the remaining seven months. The biggest loss, occurred

in March, was 10 million rupiahs. The lowest profit was 9 million rupi-

ahs in June and the highest profit was 15 million rupiahs in October.

At least how much was the company’s profit during the whole year of

2004?

6. The average score of a mathematics test in a class of 48 students was

80. Changes were made to the scores of two students. One score was

changed from 86 to 93. The other score was changed from 85 to 84.

What is the new average score of the test?

7. Ms. Olmer pays the employees of her company every first Wednesday

of the month. She goes to the bank to get the money for the salaries

every first Tuesday of the month. One Wednesday morning, Ms. Olmer

realized that she had to pay the employees, but she had not yet gone to

the bank to get the money. What day is the fifth day of that month?

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8. The figure below shows four equal circles. Each circle touches two

adjacent circles. If the radius of each circle is 10 cm, find the area of

the shaded region.

9. Mr. White multiplies the first one hundred prime numbers. How many

consecutive zero digits can be found at the end of the resulting number?

10. A, B and C are nonnegative whole numbers less than 10 and satisfying

the following multiplication: Find one set of values for A, B and C.

11. Andy multiplies the first fifty whole numbers: 1 × 2 × 3 × 4 × · · · × 50.

Counting from the right, what is the position of the first non-zero digit?

For example, in 205000, the position of the first non-zero digit from the

right is 4.

12. A circular bicycle path is 1 km long. Dodi rode a bicycle for two rounds

at the speed of 30 kph. If he wants to average 40 kph, what should be

his speed for the next four rounds?

13. Each letter represents a non-zero whole number less than 10. Different

letters represent different numbers. Find the four-digit number STNA.

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