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 5 7 0 0 S S S S Group Events G G G G 80 00 0 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 + + 5 + 6 nd + 7 + 9 + 0 If f(p) =, find the vlue of. Let g() = + + 5 + 6; h() = + 7 + 9 + 0 g( ) = 7 + 6 5 + 6 = 0, ( + ) is fctor of g(); division, g() = ( + )( + + ) h(.5)=.5+.75.5+0 = 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 = + = ( + )( + ) = ( + ) = http://www.hkedcit.net/ihouse/fh7878/ Pge
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 = 50 8 8 = 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 = 0.75 +.5 = http://www.hkedcit.net/ihouse/fh7878/ Pge
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. 5 00 = 00 + 5, where is positive integer. = + 5 = 7. I. If sin 0 + sin 0 + + 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. 8 + 5 = 0, =, = 5 + = S = = 9 5 5 5 S = nd re the roots http://www.hkedcit.net/ihouse/fh7878/ Pge
Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 http://www.hkedcit.net/ihouse/fh7878/ 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 = 5 5 70 = 0 = 0 I. Let = nd =. If = 0 6 6, find the vlue of. = 0 = 0 = 0 = 0 5 5 0 = 0 I. [] represents the lrgest integer not greter thn. For emple, [.5] =. If S = 00 00 00, find the vlue of S. S = 000 00 00 00 0 00 = 00 + 0 + + 0 + =
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 5 999 00 5 999 00 r r r r r r r 8 r r = 5 999 00 = 000 = 5 999 00 00 00 Note tht nd 000 = 5 999 00 8 8 5 8 999 00 000 000 = = 8 5 999 00 8 00 000 000 50 = = 50 = 50 00 50 = 8 00 00 00 00 50 00 (000 terms) 000 The given eqution ecomes: = = 00 00 00 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. 00 + 0 + is divisile 8 nd + + + + + = 9m 0 + = 8n (); + = 9m 0 () () (): 9 = 8n 9m + 9 + 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,, 6 0 5 log log log z 0 6 0 5 0 log log log z log z d = = log z http://www.hkedcit.net/ihouse/fh7878/ Pge 5
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. = ( + ) = 000000 + = 000000 + = 00000 = = 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 http://www.hkedcit.net/ihouse/fh7878/ Pge 6
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. 6 6 6 8 + 8 = + + = 8 G. Suppose the eqution = 0 hs onl one positive integrl solution ( 0, 0 ). If 0 + 0 =, find the vlue of. ( ) = + 7 7 = = = = + = 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 http://www.hkedcit.net/ihouse/fh7878/ Pge 7
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 = 6 5 8 8 = 7 8 = 8 7 0 0 = 9 0 = 0 9 Multipl these equtions gives = 0 = 680 = 0 = 80 G. Given tht the units digit of + + + + 00 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 ). + + + 0 + 8 + 7 + + 5 + 6 + + + 9 + 0 (mod 0) 5 (mod 0) + + + 000 + 00 00(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 http://www.hkedcit.net/ihouse/fh7878/ Pge 8