Microsoft Word - HKMO1986final.doc

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1 Sample Event (Individual) (i) In the given figure, the sum of the four marked angles is a. Find a. 附圖所示四角之和為 a, 求 a a = (ii) The sum of the interior angles of a regular b-sided polygon is a. Find b. 一正 b 邊形之內角和為 a, 求 b b = (iii) If b 5 = 3 c, find c. 若 b 5 = 3 c, 求 c c = (iv) If c = log 4 d, find d. 若 c = log 4 d, 求 d d = Bonus

2 Final Event 1 (Individual) (i) The given figure shows a circle of radius 18 cm, centre O. If AOB = and the 3 a = length of arc APB is a cm, find a. 附圖所示的圓之半徑為 18 cm, 圓心為 O 若 AOB =, 且弧 APB 之長為 3 a cm, 求 a B P O A (ii) If the solution of the inequality x ax + 4 < 0 is 1 < x < b, find b. 若不等式 x ax + 4 < 0 之解為 1 < x < b, 求 b b = (iii) If b(x 5) + x + 3 5x c, find c. 若 b(x 5) + x + 3 5x c, 求 c c = (iv) The line through (, 6) and (5, c) cuts the x-axis at (d, 0). Find d. 過 (, 6) 及 (5, c) 之直綫與 x- 軸相交於 (d, 0) 求 d d = Bonus

3 Final Event (Individual) (i) If the equation 3x 4x + 3 h = 0 has equal roots, find h. 若方程 3x 4x + 3 h = 0 有等根, 求 h h = (ii) If the height of a cylinder is doubled and the new radius is h times the original, then the new volume is k times the original. Find k. k = 若一圓柱體之高增加一倍, 且新半徑為原來之 h 倍, 則新體積為原來之 k 倍, 求 k (iii) If log log 10 k log log log log 10 5 = p, find p. 若 log log 10 k log log log log 10 5 = p, 求 p p = p cos A q (iv) If sin A = and, find q. 5 tan A 15 p cos A q 若 sin A = 且, 求 q 5 tan A 15 q = Bonus

4 Final Event 3 (Individual) (i) The monthly salaries of 100 employees in a company are as shown: Sa1aries ($) No. of employees If the mean salary is $m, find m. 某公司的一百個員工之月薪如附表所示若平均月薪為 $m, 求 m 月薪 ($) 員工人數 (ii) If 8 sin (m + 10) + 1 cos (m + 5) = x, find x. 若 8 sin (m + 10) + 1 cos (m + 5) = x, 求 x x = m = (iii) In the figure, AP // CR // BQ, AC x, CB 1, AP 10, BQ 15 and CR y. Find y. y = 如圖所示,AP // CR // BQ,AC x,cb 1,AP 10,BQ 15 及 CR y 求 y B 1 C x A 15 y 10 P R Q (iv) Define (a, b, c)(p, q, r) ap + bq + cr, where a, b, c, p, q, r are real numbers. If (3, 4, 5)(y,, 1) n, find n. 定義 (a, b, c)(p, q, r) ap + bq + cr, 其中 a b c p q r 為實數若 (3, 4, 5)(y,, 1) n, 求 n n = Bonus

5 Final Event 4 (Individual) (i) It is known that 已知 If n = 0, find n. 若 n = 0, 求 n n = (ii) If the lines x + y = 3 and nx + my = 4 are parallel, find m. 若直綫 x + y = 3 及 nx + my = 4 平行, 求 m m = (iii) If a number is selected from the whole numbers 1 to m, and if each number has an equal chance of being selected, the probability that the number is a factor of m is 39 p, find p. p = (iv) 若由整數 1 至 m 抽出一個數字, 而每一數字被抽出之機會均等, 被抽出數 p 字為 m 之因數的或然率為, 求 p 39 A boy walks from home to school at a speed of p km/h and returns home along the same route at a speed of 3 km/h. If the average speed for the double journey 4 is km/h, find q. q 某小童以速率 p km/h 由家步行上學, 並依照原來路線以速率 3 km/h 步行回家若來回兩程之平均速率為 4 km/h, 求 q q q = Bonus

6 Final Event 5 (Individual) (i) a A die is rolled. If the probability of getting a prime number is, find a. 7 投擲一骰子, 若擲出質數之或然率為 a, 求 a 7 a = (ii) In the figure, x = a, y = 44, z = 5 and w = b. Find b. 如圖所示,x = a,y = 44,z = 5 及 w = b 求 b b = w x z y (iii) A, B are two towns b km apart. Peter cycles at a speed of 7 km/h from A to B and at the same time John cycles from B to A at a speed of 5 km/h. If they meet after p p = hours, find p. A,B 兩城相距 b km 彼得從 A 城以速率 7 km/h 踏單車往 B 城, 與此同時, 約翰從 B 城以速率 5 km/h 踏單車往 A 城若兩人於 p 小時後相遇, 求 p (iv) The base of a pyramid is a triangle with sides 3 cm, p cm and 5 cm. If the height and volume of the pyramid are q cm and 1 cm 3 respectively, find q. q = 一角錐體之底為三角形, 其邊長分別為 3 cm,p cm 及 5 cm 若該角錐體之高及體積依次為 q cm 及 1 cm 3, 求 q Bonus

7 Sample Event (Group) (i) The sum of two numbers is 50, and their product is 5. If the sum of their reciprocals is a, find a. a = 某兩數之和為 50, 其積為 5 若該兩數倒數之和為 a, 求 a (ii) If the lines ax + y + 1 = 0 and 3x + by + 5 = 0 are perpendicular, find b. 若直綫 ax + y + 1 = 0 及 3x + by + 5 = 0 互相垂直, 求 b b = (iii) The area of an equilateral triangle is cm. If its perimeter is p cm, find p. 一正三角形之面積為 cm 若其周界為 p cm, 求 p p = (iv) If x 3 x + px + q is divisible by x +, find q. 若 x 3 x + px + q 可被 x + 整除, 求 q q = Bonus

8 Final Event 6 (Group) (i) If = a10 p where p is a positive integer and 1 a < 10, find p. 若 = a10 p, 其中 p 為正整數, 且 1 a < 10, 求 p p = (ii) If (p, q), (5, 3) and (1, 1) are collinear, find q. 若 (p, q) (5, 3) 及 (1, 1) 共綫, 求 q q = 7 (iii) If tan =, 90 < < 180 and 100 cos = r, find r. 4 7 若 tan =,90 < < 180 及 100 cos = r, 求 r 4 r = (iv) The average of x, y, z is 10. The average of x, y, z, t is 1. Find t. x y z 之平均數為 10 x y z t 之平均數為 1 求 t t = Bonus

9 Final Event 7 (Group) (i) In the figure, QR, RP, PQ are 3 arcs, centres at A, B, C respectively, touching one another at R, P, Q. If AR r, RB 6, QC 4, A 90, find r. r = 如圖所示, 依次以 A B C 為圓心之弧 QR RP PQ 相切於 R P Q 若 AR r,rb 6,QC 4, A 90, 求 r C 4 P Q A r R 6 B (ii) M, N are the points (3, ) and (9, 5) respectively. If P(s, t) is a point on MN such that MP : PN 4 : r, find s. s = M N 依次為 (3, ) 及 (9, 5) 若 P(s, t) 為 MN 上一點使 MP : PN 4 : r, 求 s (iii) x + 10x + t (x + a) + k, where t, a, k are constants. Find a. x + 10x + t (x + a) + k, 其中 t a k 為常數, 求 a a = (iv) If 9 p+ = p, find p. 若 9 p+ = p, 求 p p = Bonus

10 Final Event 8 (Group) In the given multiplication, different letters represent different integers whose possible values are, 4, 5, 6, 7, 8, 9. 在所示乘法中, 不同字母代表可能為之不同整數 1 A B C D E 3 A B C D E 1 (i) Find A. 求 A A = (ii) Find B. 求 B B = (iii) Find C. 求 C C = (iv) Find D. 求 D D = Bonus

11 Final Event 9 (Group) (i) 7 oranges and 5 apples cost $13. 3 oranges and 4 apples cost $8. 37 oranges and 45 apples cost $C. Find C. C = 7 個橙和 5 個蘋果值 $13 3 個橙和 4 個蘋果值 $8 37 個橙和 45 個蘋果值 $C 求 C (ii) There are exactly n values of satisfying the equation (sin 1)( sin 1)= 0, where Find n. n = 方程 (sin 1)( sin 1) = 0, 其中 0 360, 共有 n 個根求 n (iii) If S = ab + a b 1 and a 101, b 49, find S. 若 S = ab + a b 1 及 a 101,b 49, 求 S S = (iv) If d is the distance between the points (13, 5) and (5, 10), find d. 若 (13, 5) 與 (5, 10) 兩點之距離為 d, 求 d d = Bonus

12 Final Event 10 (Group) (i) If b + c = 3, c + a = 6, a + b = 7 and P abc, find P. 若 b + c = 3,c + a = 6,a + b = 7, 且 P abc, 求 P P = (ii) The medians AL, BM, CN of ABC meet at G. If the area of ABC is 54 cm and the area of ANG is x cm. Find x. x = ABC 之中綫 AL BM CN 相交於 G 若 ABC 之面積為 54 cm,ang 之面積為 x cm, 求 x A N M G B L C 3sin 5cos (iii) If k = and tan = 3, find k. sin cos 3sin 5cos 若 k = 及 tan = 3, 求 k sin cos k = (iv) 1 If S = , find S 若 S = , 求 S S = Bonus

Microsoft Word - 0.5bh.doc

Microsoft Word - 0.5bh.doc 198 FG7. 199 HG8 E 圖中,DE 為一正方形, = 及為一邊長 1 cm 的等邊三角形, 而為此 = 90 若 DE 的面積為 10 cm, 三角形內的任意一點 ( 如圖所示 ) 若至三邊求的面積及的垂直距離的總和為 x cm, 求 x 的值 In the figure shown, DE is a square and is an equilateral triangle