英语数学题及参考答案

2009　English Day of Beijing Mathematics Club

PartⅠ: Questions 1 to 10, 10 marks each

1. At the right is shown a 4 × 4 grid. We wish to fill in the grid such that each row, each column, and each 2 × 2 square outlined by the thick lines contains the digits 1 through 4. Some grids have already been filled in. Find the number of ways we can complete the rest of the grid.

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2. The areas of the faces of a cuboid are 84 cm2, 70 cm2 and 30 cm2. Find the volume of the cuboid in cm3.

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3. The fraction can be wrritten in the form where the greatest common divisor of m and n is 1, Find m+n.

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4. Find the sum of all the integers N > 1 with the properties that the each prime factor of N is either 2, 3, 5 or 7, and N is not divisible by any perfect cube greater than 1.

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5. A large fresh water reservoir has two types of drainage system, small pipes and large pipes. 6 large pipes, on their own, can drain the reservoir in 12 hours. 3 large pipes and 9 small pipes, at the same time, can drain the reservoir in 8 hours. How long will 5 small pipes, on their own, take to drain the reservoir?

Answer： minutes

6. At a local village gala, the entire population turned up, 500 people. The event raised £3,000. Tickets were priced as follows: £7.48 per man, £7.12 per woman and £0.45 per child. How many children were there?

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7. Each of the distinct letters in the following addition problem represents a different digit. If A=4, find the number represented by the word “MEET”.

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8. Let two 8×12 rectangles share a common corner and overlap. The distance from the bottom right corner of one rectangle to the intersection point along the right edge of that rectangle is 7. What is the area of the shaded region?

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9. A spy had to send the 4-digit code to headquarters. For security reasons, he sent instead the 9 separate 4-digit codes shown. In each of the 9 codes, at least one of the digits a, b, c, and d occurs in its correct position. What is the value of ?

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10. In how many ways can one arrange the numbers 21, 31, 41, 51, 61, 71 and 81 such that the sum of every four consecutive numbers is divisible by 3?

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PartⅡ: Questions 11 to 14, 20 marks each

11. Town A and town B are connected by a highway, with a service station at the midpoint. Mike and Sam start from A to B at the same time. When Mike reaches the service station, Sam is 16 km behind. Mike reduces speed by 25% after he passes through the service station. When Sam reaches the service station, Mike is 15 km ahead of Sam. What’s the distance between A and B?

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12. Given: ABCD is a trapezoid, AD∥BC, AD:BC=1:2, , , Find the area of ．

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13. In how many different ways can the seven empty circles in the diagram on the right be filled in with the numbers 2 through 8 such that each number is used once, and each number is either greater than both its neighbors, or less than both its neighbors.

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14. How many rectangles are there in the diagram on the right such that the sum of the numbers within the rectangle is a multiple of 4?

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