If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a

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1 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Events I P I P I P I P S S S S Group Events G G G G c 8 c c c d d 6 d 5 d 85 Individul Event I., nd c re the lengths of the opposite sides A, nd C of the AC respectivel. If C = 60 nd = P, find the vlue of P. c c c = + cos 60 = + + = c + P = = c c c c c c c c c c c P = = = c c c c c c I. Given tht f() = + + is the common fctor of nd If f(p) =, find the vlue of. Let g() = ; h() = g( ) = = 0, ( + ) is fctor of g(); division, g() = ( + )( + + ) h(.5)= = 0, (+5) is fctor of h(); division, h() = (+5)( ++) f() = common fctor = ( + + ) = f(p) = f() = + + = I. Given tht nd =, find the vlue of. ( + ) = + + = + = 0 ( ) = 0 = = = I. Given tht nd + = S, find the vlue of S. () () () (): = 8 S = + = ( + )( + ) = ( + ) = Pge

2 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Event I. Suppose P is n integer nd 5 < P < 0. If the roots of the eqution (P ) + P P + 8 = 0 re integers, find the vlue of P. eference: 000 FI5., 00 FI., 0 FI., 0 HG = (P ) (P P + 8) = m m = P P + 9 P + P 8 = P + 5 < P < 0 < P + < The onl odd squre ling in this intervl is 5 P + = 5 = 5 P = I. AC is rectngle. A = P +, A = P + 6. AE nd CF re perpendiculrs to the digonl. If EF =, find the vlue of. A = 0, A = 0, = 50, let A =, cos = 5 E = A cos = 0 = 8 = F 5 EF = = A E F C I. There re less thn students in clss. In mthemtics test, of the students got grde A, 7 of the students got grde, hlf of the students got grde C, nd the rest filed. Given tht students filed in the mthemtics test, find the vlue of. = 56, let the numer of students e, then is divisile, nd 7. i.e. is divisile, s < 56, so = = numer of students filed in mthemtics = = ; = 7 I. [] represents the lrgest integer not greter thn. For emple, =. Given tht the sum of the roots of the eqution is S, find the vlue of S. eference: 99 HG9 [ + ] = + + = + +, where 0 < = 0 <.5 + <.5 + is n integer + = or = 0.75 or.5 S = = Pge

3 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Event I. AC is trpezium such tht AC = C = 60 nd A = C = A = C. If this trpezium is divided into P equl portions (P > ) nd ech portion is similr to trpezium AC itself, find the minimum vlue of P. From the grph, P = A C I. The sum of tens nd units digits of (P + ) 00 is. Find the vlue of = , where is positive integer. = + 5 = 7. I. If sin 0 + sin sin 0 = cos 5, find the vlue of. 7 7 = I. Let nd e the roots of the eqution 8 + ( + ) = 0. If of the eqution 5 S + = 0, find the vlue of S. eference: 996 FI = 0, =, = 5 + = S = = S = nd re the roots Pge

4 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Pge Individul Event I. Let 7, = nd =. If P =, find the vlue of P. P = = P = = = = 7.5 = 5 I. If regulr -sided polgon hs P digonls, find the vlue of. eference: 98 FG0., 985 FG8., 988 FG6., 989 FG6., 99 FI., 005 FI. The numer of digonls = C = = 0 = 0 I. Let = nd =. If = 0 6 6, find the vlue of. = 0 = 0 = 0 = = 0 I. [] represents the lrgest integer not greter thn. For emple, [.5] =. If S = , find the vlue of S. S = = =

5 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Group Event G. Given tht ( + + c) = ( + + c ) nd + + c =, find the vlue of. Su. () into (), = ( + + c ) + + c = 8 () ( + + c) = + + c + ( + c + c) = 8 + ( + c + c) + c +c =8 [ + + c ( + c + c)] = ( ) + ( c) + (c ) [8 8] = 0 = ( ) + ( c) + (c ) = = c + + c = = = 000 G. Given tht, find the vlue of r r r r r r r 8 r r = = 000 = Note tht nd 000 = = = = = 50 = = (000 terms) 000 The given eqution ecomes: = = G. A si-digit numer is divisile oth 8 nd 9. Given tht + = c, find the vlue of c. eference: 00 FI., 07 HI The numer formed lst digits must e divisile 8 nd the sum of digits must e divisile 9. i.e is divisile 8 nd = 9m 0 + = 8n (); + = 9m 0 () () (): 9 = 8n 9m n = or 0 When n =, () hs no solution; when n = 0, = 8, = 0; c = + = 8 G. Suppose log t = 6, log t = 0 nd log z t = 5. If log z t = d, find the vlue of d. eference: 999 FG., 05 HI7 log t log t log t = 6, = 0, = 5 log log log z log log log z,, log log log z log log log z log z d = = log z Pge 5

6 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Group Event G. Given tht = 7 5 nd, find the vlue of. eference: 99 FI., 999 HG, 0 HI7, 05 FI., 05 FG. eference: 99 HI9, 000HG, 007 HG, 009HG = 7 = = = = = = ( ) + = 0 5 = = = G. E is n interior point of the rectngle AC. Given tht the lengths of EA, E, EC nd E re,, nd respectivel, find the vlue of. eference: 99 FG0.-, 00 FI., 08 HI7 Let P, nd S e the foot of perpendiculrs drwn from E onto A, C, C nd A respectivel. PE = p, E = q, E = r, SE = s. Using Pthgors Theorem, it cn e proved tht p + s =...() p + q =...() q + r = 6...() r + s =...() () + () () (): 0 = + 6 = G. Given tht = c(c + ), find the vlue of c. eference 996 FG7. = ( + ) = = = = = c = G. Given tht cos 6 = sin + sin d nd 0 < d < 90, find the vlue of d. sin d = cos 6 sin sin d = sin 7 sin 7 7 sin d = cos sin sin d = cos = sin 6 d = 6 A S A E r s q p P E C C Pge 6

7 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Group Event G. Given tht the solution of the eqution 6 is, find the vlue of = + + = 8 G. Suppose the eqution = 0 hs onl one positive integrl solution ( 0, 0 ). If =, find the vlue of. ( ) = = = = = + = 8, = = 0 G. AC is cclic qudrilterl. AC nd intersect t G. Suppose AC =6 cm, C = C = 8 cm, G = cm nd G = cm. If nd re integers nd + = c, find the vlue of c. As shown in the figure, let CG = t, AG = 6 t. Let CG =, AC =. 8 Then CA = (eq. chords eq. s) A G Then CG ~ AC (equingulr) 6-t t : 8 = 8 : 6 (rtio of sides, ~s) t C t = 8 It is es to see tht AG ~ CG (equingulr) (6 t) : = : t (rtio of sides, ~s) (6 ) = = 8 Assume tht nd re integers, then possile pirs of (, ) re (,8), (, ),..., (6, 8),..., (8, ). Using tringle inequlit + t > 8 nd 8 + t > in CG, the onl possile comintions re: (, ) = (6, 8) or (8, 6) c = + = log0.5 log0 G. Given tht 5 d, find the vlue of d. log 0 log 5 + log 0.5 log = log d 0 log (0) log +( log )( log ) = log d (log + )( log ) + log log = log d log + log log log + log log = log d 0 log d = log + log = log d = 5 Pge 7

8 Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Group Event G. = 00. When n >, n n. Given tht 0 =, find the vlue of. n = = = = 6 6 = 5 6 = = 7 8 = = 9 0 = 0 9 Multipl these equtions gives = 0 = 680 = 0 = 80 G. Given tht the units digit of is, find the vlue of. Arrnge the numers in groups of 0 in scending order, the units digit of sum ech group is the sme (ecept the lst numer, 00 ) (mod 0) 5 (mod 0) (5) + (mod 0) So = G. A nd rn round circulr pth with constnt speeds. The strted from the sme plce nd t the sme time in opposite directions. After their first meeting, took minute to go ck to the strting plce. If A nd need 6 minutes nd c minutes respectivel to complete one round of the pth, find the vlue of c. c 6 In one minute, A nd rn of the totl distnce. The will meet t the first time fter 6 c 6c 6c c 6 minutes. After more minute, (i.e. totl time elpsed = 6c c 6 c point. So 6c + c + 6 = c + 6c c c 6 = 0 (c )(c + ) = 0 c = 6c c 6 + minutes), retuned to the strting G. The roots of the eqution 5 + m = 0 re prime numers. Given tht the sum of the squres of the roots is d, find the vlue of d. eference: 996 HG8, 996FG7., 005 FG., 0 HI6 Let the roots e,. + = 5, = m The sum of two prime numers + = 5 =, = ( is the onl even prime numer) d = + = + = 85 Pge 8

1 (=0.5) I3 a 7 I4 a 15 I5 a (=0.5) c 4 N 10 1 (=0.5) N 6 A 52 S 2

1 (=0.5) I3 a 7 I4 a 15 I5 a (=0.5) c 4 N 10 1 (=0.5) N 6 A 52 S 2 Answers: (98-84 HKMO Finl Events) Creted by Mr. Frncis Hung Lst updted: December 05 Individul Events SI 900 I 0 I (=0.5) I 7 I4 5 I5 80 b 7 b b 5 b 6 b 8 b 4 c c 4 c 0 x (=0.5) c 4 N 0 d 9 d 5 d 5 y d