d 4 x x 170 n 20 R 8 A 200 h S 1 y 5000 x 3240 A 243
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1 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Individual Events SI a I1 a I a 1 I3 a 4 I4 a I t 8 b 4 b 0 b 1 b 16 b 10 u 13 c c 9 c 3 c 199 c 96 v 4 d 1 d d 16 d 4 d 9 w 70 Goup Events SG s 19 G6 x 8 G7 M 100 G8 M G9 60 G10 k 1 n 8 y N 9 N 3 6 K h d 4 x 1 Sample Individual Event (1994 Final Sample Individual Event) SI.1 The sum of two numbes is 40, and thei poduct is 0. If the sum of thei ecipocals is a, find a. 4 Refeence: 1983 FG6.3, 1984 FSG.1, 1986 FSG.1 Let the two numbes be x and y. x + y = 40 and xy = x y a = = = x y xy x 170 n 0 R 8 S 1 y 000 x SI. If b cm is the total suface aea of a cube of side (a + 1) cm, find b. Simila Questions: 1984 FI3., 1984 FG9. a + 1 = 3 b = 63 = 4 SI.3 One ball is taken at andom fom a bag containing b 4 white balls and b + 46 ed balls. If 6 c is the pobability that the ball is white, find c. Thee ae b 4 = 0 white balls and b + 46 = 100 ed balls 0 P(white ball) = = c = SI.4 The length of a side of an equilateal tiangle is c cm. If its aea is d 3 cm, find d. Refeence: 1984FI4.4, 1986 FSG.3, 1987 FG6., 1988 FG9.1 d 3 = d = 1 1 c sin 60 = 3 Page 1
2 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Individual Event 1 16 I1.1 Find a if a = log. 3 4 a = log = log a = 1 I1. If = a and 3 1 = b, find b. 3 Refeence: 1990 HI1, 017 FI1.4 1 = = 3 1 = b = = = 1 1 = 0 I1.3 If one oot of the equation x 3 + cx + 10 = b is, find c. Put x = into the equation: 8 + c + 10 = 0 c = 9 I1.4 Find d if 9 d+ = ( c) + 9 d. (Refeence: 1986 FG7.4) 819 d = d 809 d = d = 81 d = Individual Event I.1 Find a in the following sequence: 1, 8, 7, 64, a, 16, 1 3, 3, 3 3, 4 3, a, 6 3, a = 3 = 1 I. In Figue 1, and = (a 9). If = b, find b. (Refeence: 010 HG3) Let = = (base s isosceles ) = b + = 30 = b = (ext. of ) b + b + 30 = b = 1 I.3 line passes though the points (1, 1) and (3, b 6). If the y-intecept of the line is c, find c. Simila question: 1986 FI1.4 b 6 = 9 c c = 3 I.4 In Figue, = c + 17, 100, 80. If EF = d, find d. (Refeence: 1989 HG8, 1990 FG6.4) Let F = x, then F = 100 x. EF ~ (equiangula) EF ~ (equiangula) E x x 100 (1), () d 80 d (1) + (): 100 d = 16 F d Page
3 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Individual Event 3 I3.1 The acute angle fomed by the hands of a clock at :1 is 1 18 a. Find a. Refeence: 1984 FG7.1, 1987 FG7.1, 1989 FI1.1, 1990 FG6.3, 007 HI1 t :00, the angle between the ams of the clock = Fom :00 to :1, the hou-hand had moved 360 = The minute hand had moved a = 90 ( ) =. a = 4 I3. If the sum of the coefficients in the expansion of (x + y) a is b, find b. Put x = 1 and y = 1, then b = (1 + 1) 4 = 16 I3.3 If f (x) = x, F(x, y) = y + x and c = F(3, f(b)), find c. Refeence: 1990 HI3, 013 FI3., 01 FI4.3 f (b) = 16 = 14 c = F(3, 14) = = 199 I3.4 x, y ae eal numbes. If x + y = c 19 and d is the maximum value of xy, find d. Refeence: 1988 FI4.3 x + y = 4 y = 4 x xy = x(4 x) = (x ) + 4 d d = 4 Page 3
4 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Individual Event 4 I4.1 If the lines x + y + 3 = 0 and 4x ay + = 0 ae pependicula to each othe, find a. Refeence: 1983 FG9.3, 1984 FSG.3, 1986 FSG., 1987 FG10., 1988 FG a a = I4. In Figue 1, is a tapezium with paallel to and = = 90. If a, 8 and b b, find b. aw a line segment E //, cutting at E. E = 90 = E (int. s, E // ) a E is a ectangle E = 8, E = a = (opp. sides, //-gam) 8 E = 8 a = 6 b = = 100 (Pythagoas theoem on E) b = 10 8 b 8 a 8 E I4.3 In Figue, = b, E 4, E 3. If the aea of E is 4 and the aea of is c, find c., E and E have the same height. 4 3 The aea of = c = 4 = 96 3 E I4.4 If 3x 3 x + dx c is divisible by x 1, find d. 3 + d 96 = 0 d = 9 Page 4
5 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Individual Event I.1 If t = 36, find t. 1 t(t + 1) = 36 t = 8 o 9 (ejected) I. If sin u = t 1 sin u = and 90 < u < 180, find u. u = 13 I.3 In Figue 1, = 30 and (u 90) cm. If the adius of the cicumcicle of is v cm, find v v (Sine fomula) sin30 v = 4 (u 90) cm I.4 In Figue, P is fomed by the 3 tangents of the cicle with cente O. If P = (v ) and O = w, find w. P = 40 OT P, OS, OR P (tangent adius) ROT = = 140 (s sum of polygon) RO = SO, TO = SO (tangent fom ext. pt.) O = 140 = 70 w = 70 P R S T O Page
6 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Sample Goup Event (1994 Sample Goup Event) SG.1 If a*b = ab + 1, and s = (*4)*, find s. Refeence: 1984 FG6.4 *4 = = 9 s = (*4)* = 9* = = 19 SG. If the n th pime numbe is s, find n. Refeence: 1989 FSG.3, 1990 FI.4, 3,, 7, 11, 13, 17, 19 n = 8 SG.3 If K = , find K in the simplest factional fom. Refeence: 1984 FG9.1, 1986 FG K = = SG.4 If is the aea of a squae inscibed in a cicle of adius 10, find. Refeence: 1984 FG10.1, 1989 FI3.3 Let the squae be E. = 0 cos 4 = 10 = 10 = 00 E 10 4 O Page 6
7 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Goup Event 6 G6.1 The aveage of p, q, is 4. The aveage of p, q,, x is. Find x. Refeence: 1986 FG6.4, 1987 FG10.1, 1988 FG9. p + q + = 1 p + q + + x = 0 x = 8 G6. wheel of a tuck tavelling at 60 km/h makes 4 evolutions pe second. If its diamete is y m, find y km/h = m/s = m/s y 0 4 = 6 3 y = G6.3 If sin( y) = x d, find d. sin 30 = 8 d = 1 d = 4 G6.4 In the figue, and N. If 3, d, N h, find h. Refeence: 199 FI.3 = (Pythagoas theoem) 3 h d = N ea of = 1 h = h = Page 7
8 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Goup Event G7.1 Let M =. Find M Simila questions: 1984 FG M = = = 100 G7. When the positive intege N is divided by 6,, 4, 3 and, the emaindes ae, 4, 3, and 1 espectively. Find the least value of N. Refeence: 1990 HI13, 013 FG4.3 N + 1 is divisible by 6,, 4, 3 and. The L..M. of 6,, 4, 3 and is 60. The least value of N is 9. G7.3 man tavels 10 km at a speed of 4 km/h and anothe 10 km at a speed of 6 km/h. If the aveage speed of the whole jouney is x km/h, find x. 0 4 x = = G7.4 If S = , find S. Refeence: 1988 FG6.4, 1990 FG10.1, 1991 FSI.1, 199 FI1.4 S = 1 + ( ) + ( ) + + ( ) = 1 Page 8
9 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Goup Event 8 Simila Questions 1988 FG7.1-, 1990 FG7.3-4 M, N ae positive integes less than 10 and 804M N 11. G8.1 Find M. 11 and 9 ae elatively pime 804M8 is divisible by M ( ) is divisible by 11 M = 11k M = G8. Find N N is divisible by N = 9t N = G8.3 convex 0-sided polygon has x diagonals. Find x. Refeence: 1984 FG10.3, 1988 FG6., 1989 FG6.1, 1991 FI.3, 001 FI4., 00 FI1.4 0 x = = 0 = 170 G8.4 If y = ab + a + b + 1 and a 99, b 49, find y. Refeence: 1986 FG9.3, 1988 FG6.3, 1990 FG9. y = (a + 1)(b + 1) = (99 + 1)(49 + 1) = Page 9
10 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Goup Event 9 G9.1 The lengths of the 3 sides of LMN ae 8, 1 and 17 espectively. If the aea of LMN is, find = 64 + = 89 = 17 LMN is a ight-angled tiangle 81 = = 60 G9. If is the length of the adius of the cicle inscibed in LMN, find. Refeence: 1989 HG9 Let O be the cente and the adius of the cicle be, which touches the tiangle at, and E. O LM, O MN, OE LN (tangent adius) OM is a ectangle (which consists of 3 ight angles) O = = O (adii) OM is a squae. M = M = (opp. sides, ectangle) L = 1, N = 8 LE = L = 1, NE = N = 8 (tangent fom ext. pt.) LE + NE = LN = 17 = 3 L 17 1 E O G9.3 If the th day of May in a yea is Fiday and the n th day of May in the same yea is Monday, whee 1 < n <, find n. Refeence: 1984 FG6.3, 1987 FG8.4, 1988 FG10. 3 d May is Fiday 17 th May is Fiday 0 th May is Monday n = 0 G9.4 If the sum of the inteio angles of an n-sided convex polygon is x, find x. x = 180(0 ) = 340 (s sum of polygon) N 8 M Page 10
11 nswes: ( HKMO Final Events) eated by: M. Fancis Hung Last updated: 4 pil 017 Goup Event 10 G10.1 The sum of 3 consecutive odd integes (the smallest being k) is 1. Find k. k + k + + k + 4 = 1 k = 1 G10. If x + 6x + k (x + a) +, whee a, ae constants, find. Refeence: 1984 FI.4, 1986 FG7.3, 1987 FSI.1, 1988 FG9.3 x + 6x + 1 (x + 3) + 6 = 6 p q p G10.3 If = and R =, find R. q s s R = s p = p q q s = 3 = 8 n n 3 9 G10.4 If = n1 7 n n 3 9 = n n n 3 3 = 3n3 3 = 3 6 = 43, find. Page 11
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