2012COMC加拿大數學公開賽真題答案免費下載

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

真題與答案下載

翰林國際教育全網首發

力爭超快速發布最全資料

助你在升學路上一帆風順

為你

千千萬萬遍

2012 COMC真題答案免費下載

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' Solutions:

A3)The answer is r = 3.
Divide the hexagon into six regions as shown, with the centre point denoted by P.

This is possible since the hexagon is regular. By symmetry, note that PA = PB = PC = PD = PE = PF and the six interior angles about P are equal. Then since the sum of the six interior angles about P sum to 360°, ∠APB =∠BPC = ∠CPD = ∠DPE = ∠EPF = ∠FPA = 60°.
Therefore, the six triangles ΔPAB,ΔPBC,ΔPCD,ΔPDE,ΔPEF,ΔPFA are all equilateral and have the same area. Let K be the area of any one of these triangles. Therefore, the hexagon has area 6K.
Note that the area of ΔACD is equal to the area of ΔPCD plus the area of ΔPAC. Since ΔPAB,ΔPBC are both equilateral, PA = AB and PC = CB. Therefore, triangles ΔBAC and ΔPAC are congruent and hence have the same area. Note that the area of ΔPAC plus the area of ΔBAC is the sum of the areas of the equilateral triangles ΔPAB and ΔPBC, which is 2K. Therefore, ΔPAC has area K. We already noted that the area of ΔACD is equal to the area of ΔPCD plus that of ΔPAC. This quantity is equal to K + K = 2K. Hence, the area of ABCDEF is 6K/2K = 3 times the area of ΔACD. The answer is 3.

Part B Challenging Questions' Solutions:

B1) The answer is t = 59 or t = 61.
Since Alice ran exactly 30 laps, Bob meets Alice at where Alice started. Since Bob started diametrically across from Alice, Bob ran n + 1/2 laps for some positive integer n. Since Alice and Bob meet only the first time they meet, the number of laps that Alice ran and the number of laps Bob ran cannot differ by more than 1. Therefore, Bob ran either 29.5 laps or 30.5 laps.
Note that Alice and Bob ran for the same amount of time and the number of seconds each person ran is the number of laps he/she ran times the number of seconds it takes he/she to
complete a lap.
If Bob ran 29.5 laps, then 30t = 29.5 × 60. Hence, t = 29.5 × 2 = 59.
If Bob ran 30.5 laps, then similarly, 30t = 30.5 × 60. Hence, t = 30.5 × 2 = 61.
Therefore, t = 59 or t = 61.

Part C Long-form Proof Problems' Solutions:

C1)

1. (a) The answers are f(2) = 4 and g(f(2)) = 4.
   Substituting x = 2 into f(x) yields f(2) = 2² = 4.
   Substituting x = 2 into g(f(x)) and noting that f(2) = 4 yields g(f(2)) = g(4) =
   3 · 4 ? 8 = 4. 
2. The answers are x = 2 and x = 6.
   Note that f(g(x)) = f(3x ? 8) = (3x ? 8)² = 9x² ? 48x + 64 and g(f(x)) = g(x²) = 3x² ? 8. Therefore, we are solving 9x² ? 48x + 64 = 3x² ? 8.
   Rearranging this into a quadratic equation yields 6x² ? 48x + 72 = 0 ? 6(x² ? 8x + 12) = 0.
   This factors into 6(x ? 6)(x ? 2) = 0. Hence, x = 2 or x = 6. We now verify these are indeed solutions.
   If x = 2, then f(g(2)) = f(3(2) ? 8) = f(?2) = (?2)² = 4 and g(f(2)) = 4 by part(a). Hence, f(g(2)) = g(f(2)). Therefore, x = 2 is a solution.
   If x = 6, then f(g(6)) = f(3 · 6?8) = f(10) = 10² = 100 and g(f(6)) = g(6²) = g(36) = 3·36?8 = 108?8 = 100. Hence, f(g(6)) = g(f(6)). Therefore, x = 6 is also a solution.
3. The answers are r = 3 and r = 8.
   We first calculate f(h(2)) and h(f(2)) in terms of r. f(h(2)) = f(3 · 2 ? r) = f(6 ? r) = (6 ? r)^2?and h(f(2)) = h(2²) = h(4) = 3 · 4 ? r = 12 ? r.
   Therefore, (6 ? r)² = 12 ? r? r² ? 12r + 36 = 12 ? r. Re-arranging this yields r² ? 11r + 24 = 0, which factors as (r ? 8)(r ? 3) = 0.
   Hence, r = 3 or r = 8. We will now verify that both of these are indeed solutions.
   If r = 3, then h(x) = 3x ? 3. Then f(h(2)) = f(3 · 2 ? 3) = f(3) = 9 and h(f(2)) =h(2²) = h(4) = 3 · 4 ? 3 = 9. Therefore, f(h(2)) = h(f(2)). Consequently, r = 3 is a solution. From the result of part (b), we also verified that r = 8 is a solution.

 

完整版真題下載鏈接請注冊或登錄后查看

文件為PDF格式

推薦使用電腦下載

 

2012 COMC加拿大數學奧賽完整版真題免費下載

請點擊下方鏈接訪問

2012 COMC真題題目

 

如需打包下載，請聯系小助手了解更多

翰林學員全站資料免費打包下載，專享高速下載通道。

[vc_btn title="查看更多COMC加拿大數學奧賽相關詳情" color="primary" align="center" i_icon_fontawesome="fa fa-globe" css_animation="zoomIn" button_block="true" add_icon="true" link="url:http%3A%2F%2Fwww.linstitute.net%2Farchives%2F99767||target:%20_blank|"]