2012COMC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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2012 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 A3)Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose angles are equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD. Determine the value of r.

Question A4)Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactly two other lines. Furthermore, no three lines intersect at a point. Determine the total number of intersection points among the twelve lines.

Part B Challenging Questions:

Question B1)Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactly two other lines. Furthermore, no three lines intersect at a point. Determine the total number of intersection points among the twelve lines.

When they meet for the first time, Alice has completed exactly 30 laps. Determine all possible values of t.

Part C Long-form Proof Problems:

Question C2)We fill a 3 × 3 grid with 0s and 1s. We score one point for each row, column, and diagonal whose sum is odd.

For example, the grid on the left has 0 points and the grid on the right has 3 points.

1. Fill in the following grid so that the grid has exactly 1 point. No additional work is required. Many answers are possible. You only need to provide one.
2. Determine all grids with exactly 8 points.
3. Let E be the number of grids with an even number of points, and O be the number of grids with an odd number of points. Prove that E = O.

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