2010COMC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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2010 COMC真題免費下載

共計2.5小時考試時間

此套試卷由兩部分題目組成

Part A共8題，每題5分

Part B共4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 :

3)In the diagram, each of the three circles has centre O. Diameter CD of the largest circle passes through points B, A and O. The lengths of the radii of the circles are OA = 2, OB = 4, and OC = 6. What is the area of the shaded region?

6)On a calculus exam, the average of those who studied was 90% and the average of those who did not study was 40%. If the average of the entire class was 85%, what percentage of the class did not study?

7)In the diagram, rectangle ABCD has AB = 20 and BC = 10. Points W and K are outside of the rectangle with WA = KC = 12 and WB = KD = 16. Determine the length of WK and express your answer in the form WK = m√n, where m and n are positive integers with m > 1.

Part B :

1)In each part of this problem, each of the variables in the grid is to be replaced with an integer. The sum of the integers in a row is given to the right of the row. The sum of the integers in a column is given at the bottom of the column. For example, from the grid to the right we can conclude that X+13 = 30, Y +11 = 23, X+Y = 29, and 13 + 11 = 24.

1. Determine the value of C.
2. Determine the value of n, the sum of the integers in the second column.
3. Determine the value of P + Q.

2)The parabola with equation y = x² - 4x + 12 intersects the line with equation y = -2x + 20 at points A and B.

1. Determine the coordinates of the points A and B.
2. Determine the coordinates of the midpoint, M, of the segment AB.
3. A line parallel to the line with equation y = -2x+20 intersects the parabola at distinct points P(p; p² -4p+12) and Q(q; q² -4q +12). Prove that p+q = 2.
4. Point N is the midpoint of PQ. Explain why line segment MN is vertical.

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