2011COMC加拿大数学公开赛真题免费下载

历年 Canadian Open Mathematics Challenge加拿大数学公开赛

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2011 COMC真题免费下载

共计2.5小时考试时间

此套试卷由三部分题目组成

4题简答题，每题4分

4题挑战题，每题6分

4题解答题，每题10分

共计12题，满分80分

不可使用任何计算器

完整版下载链接见文末

部分真题预览：

Part A Introductory Questions:

Question A2)Carmen selects four different numbers from the set {1, 2, 3, 4, 5, 6, 7} whose sum is 11. If l is the largest of these four numbers, what is the value of l?

Question A3)The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?

Part B Challenging Questions:

Question B4)A group of n friends wrote a math contest consisting of eight short-answer problems S₁, S₂, S₃, S₄, S₅, S₆, S₇,S₈, and four full-solution problems F₁, F₂, F₃, F₄. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the i^th row and j^th column, we write down the number of people who correctly solved both problem S_i and problem F_j . If the 32 entries in the table sum to 256, what is the value of n?

Part C Long-form Proof Problems:

Question C3)Let n be a positive integer. A row of n + 1 squares is written from left to right, numbered 0, 1, 2, …, n, as shown.

Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let A(n) and B(n) respectively denote the number of seconds for Alphonse and Beryl to reach square n. For example, A(40) = 5 and B(40) = 10.

1. Determine an integer n > 200 for which B(n) < A(n).
2. Determine the largest integer n for which B(n) ≤ A(n).

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