2013COMC加拿大數學公開賽真題免費下載

歷年 Canadian Open Mathematics Challenge加拿大數學公開賽

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2013 COMC真題免費下載

共計2.5小時考試時間

此套試卷由三部分題目組成

4題簡答題，每題4分

4題挑戰題，每題6分

4題解答題，每題10分

共計12題，滿分80分

不可使用任何計算器

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部分真題預覽：

Part A Introductory Questions:

Question A3)In the figure below, the circles have radii 1, 2, 3, 4, and 5. The total area that is contained inside an odd number of these circles is m for a positive number m. What is the value of m?

Question A4)A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer b, the number of bi-digital positive integers.

Part B Challenging Questions:

Question B3)Teams A and B are playing soccer until someone scores 29 goals. Throughout the game the score is shown on a board displaying two numbers – the number of goals scored by A and the number of goals scored by B. A mathematical soccer fan noticed that several times throughout the game, the sum of all the digits displayed on the board was 10. (For example, a score of 12 : 7 is one such possible occasion). What is the maximum number of times throughout the game that this could happen?

Part C Long-form Proof Problems:

Question C3)Alphonse and Beryl play the following game. Two positive integers m and n are written on the board. On each turn, a player selects one of the numbers on the board, erases it, and writes in its place any positive divisor of this number as long as it is different from any of the numbers previously written on the board. For example, if 10 and 17 are written on the board, a player can erase 10 and write 2 in its place. The player who cannot make a move loses. Alphonse goes first.

1. Suppose m = 2⁴⁰ and n = 3⁵¹. Determine which player is always able to win the game and explain the winning strategy.
2. Suppose m = 2⁴⁰ and n = 2⁵¹. Determine which player is always able to win the game and explain the winning strategy.

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