If there are less than 4 rows with pieces more than 1, covering 4 rows with the most pieces will leave all remaining row each contains no more than one piece. The remaining pieces can be covered by 4 columns, contradiction. (5 marks)
Put 13 pieces in the squares as the picture below will ensure that any 4 rows and 4 columns can not cover all pieces. (5 marks) Noticing that pieces in the **upper** left

10 閱讀更多

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.

9 閱讀更多

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.

8 閱讀更多

Answer: 45 number sentences 10. An ant starts from the top left corner square of a 3 5 chessboard. It moves
from a square to the adjacent square in the same row or column. After visiting every square exactly once, it ends up at the square in the middle row **second** column from the right. How many different paths can the ant follow?

12 閱讀更多

Answer: 3600 dollars.
7. The diagram shows a path consisting of four semi-circular arcs. Each arc is of length 100 m and uses a different side of a square as its diameter. Initially, Jane is at A and Yves at B. They start walking counter-clockwise at the same time. Jane’s speed is 120 m per minute and Yves’s is 150 m per minute. Each pauses for 1 **second** whenever they are at the points A, B, C or D. How many seconds after

7 閱讀更多

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.

9 閱讀更多

【Solution】
If there is no carry when the number is added by 1, sum of digits increases by 1, contradiction to that both sum of digits are divided by 4. If there is a carry, the last digit is 9. Since the sum of all digits changes by 8 when one carry happens and by 17 when two carries happens. It follows that only one carry should happen and the original number has the sum of all digits divided by 4. In order to take the number to be maximal, the first digit should also be 9. The **second** digit can be either 2 or 6. The largest one is 969.

11 閱讀更多

8 閱讀更多

12 閱讀更多

11 閱讀更多

Thus, Max must cut a minimum of 50 × 4 + 5 × 12 = 260 cm paper strips. Hence, we select option (A).
Answer: (A) 14. The tap is leaking at the rate of one drop per **second**. The volume of each drop is 0.05 mL. At 9 pm, Wendy puts an empty measuring cup under the tap. Some time during the night, she finds the cup partially filled, as shown in the diagram. No water is lost from the cup. At what time is the water level in the measuring cup closest to the time in the following?

11 閱讀更多

9 閱讀更多

9 閱讀更多

By observation, we discover that each of the number in the first set when divided by 3 will give a remainder of 1, each number in the **second** set when divided by 3 will give a remainder of 2 and each number in the third set when divided by 3 or when each of them is divided by 3 will give a remainder of 0. Hence, according to the meaning of the problem, we must form a three-digit number where the digit in each place is selected from each set and the sum of all the digits formed when divided by 3 will give a remainder of 0, that is; the arrangement of all the three-digit numbers that formulate is always divisible by 3. Hence, we just need to find all those three-digit numbers from the above that are divisible by 2; or the digit in the units place is an even number.

顯示更多
9 閱讀更多

6 閱讀更多

Since **2017** 1008 1009 1009 1010 = + < + , suppose we take out 1009~**2017**, which is a total of 1009 numbers, we cannot ensure that there are three balls that will satisfy the given conditions.(10 points)
So suppose we take 1010 balls out. Suppose the largest number out of the 1010 balls is M , then the difference between M and the number of other balls taken out has 1009 different values, and both are less than **2017**.(5 points) Since there are only 1007 balls that have not been taken out, at least one difference M − x is the number y of the balls taken out, where x is the number of the removed ball. So x, y, x + = y M are taken out of the ball number which satisfies the condition above. (5 points)

顯示更多
8 閱讀更多

【Solution】
Consider the two numbers reported by a student as an ordered pair. The first number represents number of students in the morning session, the **second** number represents the afternoon session. Since maximum number of students in a group is 4, there is a total of 16 combinations as follows:

10 閱讀更多

9 閱讀更多

9 閱讀更多

After making your choice, fill in the appropriate letter in the space provided. Each correct answer is worth 4 marks. There is no penalty for an incorrect answer.
Questions 6 to 13 are a short answer test. Only Arabic numerals are accepted;
using other written text will not be honored or credited. Some questions have more than one answer, as such all answers are required to be written down in the space provided to obtain **full** marks. Each correct answer is worth 5 marks. There is no penalty for incorrect answers.

8 閱讀更多