2016COMC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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2016 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

完整版下載鏈接見文末

部分真題預覽：

Part A Introductory Questions:

Question A1)Pat has ten tests to write in the school year. He can obtain a maximum score of 100 on each test. The average score of Pat’s first eight tests is 80 and the average score of all of Pat’s tests is N. What is the maximum possible value of N?

Question A4)Three males and two females write their names on sheets of paper, and randomly arrange them in order, from left to right. What is the probability that all of the female names appear to the right of all the male names?

Part B Challenging Questions:

Question B2) The squares of a 6 × 6 square grid are each labelled with a point value. As shown in the diagram below, the point value of the square in row i and column j is i × j.

A path in the grid is a sequence of squares, such that consecutive squares share an edge and no square occurs twice in the sequence. The score of a path is the sum of the point values of all squares in the path.

Determine the highest possible score of a path that begins with the bottom left corner of the grid and ends with the top right corner.

Part C Long-form Proof Problems:

Question C1)A sequence of three numbers a, b, c form an arithmetic sequence if the difference between successive terms in the sequence is the same. That is, when b ? a = c ? b.

1. The sequence 2, b, 8 forms an arithmetic sequence. Determine b.
2. Given a sequence a, b, c, let d₁ be the non-negative number to increase or decrease b by so that the result is an arithmetic sequence and let d₂ be the positive number to increase or
   decrease c by so that the result is an arithmetic sequence.
   For example, if the three-term sequence is 3, 10, 13, then we need to decrease 10 to 8 to make the arithmetic sequence 3, 8, 13. We decreased b by 2, so d₁ = 2. If we change the
   third term, we need to increase 13 to 17 to make the arithmetic sequence 3, 10, 17. We increased 13 by 4, so d₂ = 4.
   Suppose the original three term sequence is 1, 13, 17. Determine d₁ and d₂.
3. Define d₁, d₂ as in part (b). For all three-term sequences, prove that 2d₁ = d₂.

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