Individual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of.

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William Lewis
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1 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July Individul Grou Individul Events I If = 5 = 0, find the vlue of. Referene: 003 FG., 00 FG.3, 005 HI9, 006 FG.3 log = log 5 = log 0 log = log 5 = = log + log 5 = log (5) = log 00 = Method = 0, 5 = 0 5 = = 00 = 0 = I In figure, ABC is stright line, AB = AD, BDC = 3, CD is tngent to the irle ABD. Let BCD =, find the vlue of. BAD = BDC = 3 ( in lt. segment) ADB = ABD (se s isoseles ) = (0 3) (s sum of ) = 7 = ABD BDC = 7 3 = 33 (et. of ) I3 If = 0 y 5 y, where nd y re rel numers, find the lrgest vlue of. Referene: 999 HG7, 0 HG5, 0 HI = + 0 ( + y + y ) = 7 ( 5) ( + y) 7 = the lrgest vlue I If the following three stright lines interset t one oint, find the vlue of. L : 6 + 6y 9 = 0 () L : + y + = 0 () L 3 : + 3y = 0 (3) () (3): 3 = 0 =.5 () 5 Su. =.5 into (3): 3 + 3y = 0 y = 3 Su. =.5, y = 3 5 into (): = 0 = htt:// Pge

2 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 I5 It is known tht 6 os = 7 sin os, find the lrgest vlue of tn. (sin + os ) 6 os = 7 sin os sin 7 sin os os = 0 Divide the eqution y os : tn 7 tn = 0 ( tn + )(tn ) = 0 tn = or The lrgest vlue of tn = I6 The totl ost for tikets ws $ 93. Beuse the rinting mhine ws not funtioning well, the first nd the lst digits of the 5-digit numer were missing. If the ost for eh tiket is $P, where P is n integer, find the vlue of P. Let the totl ost of tikets e , where, re integers etween 0 nd 9. P = = (3 + 56) + (33 + 6) + = (3 + 33) = multile of One ossile guess is =, = 6 P = = 7 I7 If is the ositive rel root of = 0, find the vlue of. Let f() = f( ) = = 0 ( + ) is ftor f( 5) = = 0 ( + 5) is ftor By omring oeffiients, f() = ( + )( + 5)( 7) = 0 The ositive root is 3.5, = 3.5 I Two ersons A, B n omlete tsk in 30 dys when they work together. If they work together for 6 dys nd then A quits, B needs 0 dys more in order to omlete the tsk. If the roortion of the tsk A n finish eh dy is q, find the vlue of q. Suose B n finish the tsk lone in dys. Then + =...() q () q From (): =...(3) 30 q 6 Su. (3) into (): 6 q 30 q q 0 5 q q = 75 htt:// Pge

3 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 htt:// Pge 3 I9 Let,, e three distint onstnts. It is given tht r q where, q r re onstnts, nd s = 7 + q + 9r, find the vlue of s. Theorem If,, re three distint roots of d + e + f = 0, then d = e = f = 0. Proof: The qudrti eqution hs t most two distint roots. If,, re three distint roots, then it is identilly equl to 0. i.e. d = e = f = 0. Now multily the given eqution y ( + )( + )( + ): r q Put =,, resetively. r q r q r q q r q r q r, nd re three distint roots of (r ) q = 0 By the ove theorem, = 0, q = 0, r =. s = 7 + q + 9r = 9 I0 In figure, ABEF, BCDE re two squres, BE = 6 m, nd AE nd CE re the rs drwn with entres F nd D resetively. If the totl re of the shded rts is S m, find the vlue of S. (Assume = 3.) Rdii of the two qudrnts = 6 The two unshded tringles re identil, se = height = 6 sin 5 = 3 Shded re = re of qudrnts re of unshded s = 3 6 = 36

4 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 Grou Events G The time on the lok fe is now one o lok. After minutes, the minute hnd overls with the hour hnd, find the minimum vlue of. In one hour, the minute hnd rotted 360, the hour hnd rotted 30. So the minute hnd is 330 fster thn the hour hnd for every hour (= every 60 minutes). At one o'lok, the minute hnd is 30 ehind of the hour hnd. After minutes, the minute hnd will th u the hour hnd. 60 By rtio, = G In how mny wys n 0 identil lls e distriuted into 3 different oes suh tht no o is to e emty? Referene: 006 HI6, 00 HI, 0 HI Align the 0 lls in row. There re 9 gs etween the 0 lls. Put stiks into two of these gs, so s to divide the lls into 3 grous. The following digrms show one ossile division. The three oes ontin lls, 7 lls nd ll. The numer of wys is equivlent to the numer of hoosing gs s stiks from 9 gs. 9 = 36 The numer of wys is 9 C = G3 Let = nd y =, find the vlue of y. y = = = 6 + = 0 G 6 If 6, find the vlue of G = In figure, ADE is right irulr one. Suose the one is divided into two rts y ut running rllel to the se nd mde of the wy u, the rtio of the slnt surfe of the smll one ABC to tht of the trunted se BCDE is : k, find the vlue of k. (Referene: 000 FG5.3) urved surfe re of smll one : tht of lrge one = 3 : = 9 : 6 urved surfe re of smll one : tht of the frustum = 9 : (6 9) = 9 : 7 = : 9 7 k = 9 7 htt:// Pge

5 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 G6 If ten-digit numer 6m6m is divisile y 3, find the mimum vlue of m m m = 3k 0 + m = 3k m =, or m =,, or 7 Mimum = 7 G7 Find the re enlosed y the -is nd the stright lines 3y = 0, + y = 0. From the figure, the verties of the enlosed re re + y - = 0 (0,0), (, 0), (3, ) A(3, ) The re = = - 3y = 0 G In figure, PQR is tringle, S is the mid-oint of PQ, RQ = PS = SQ, nd RQS = RPS. Let PSR =, find the vlue of. Referene: 99 FG3. Let RPS = y, RQS = y (given) QRS = QSR = 90 y (s sum of isos. ) PRS = QSR SPR = 90 y y = 90 y PRQ = PRS +QRS = 90 y+90 y = 0 3y Aly sine formul on PQR PQ RQ sin PRQ sin QPR sin0 3y sin y sin y = 3 sin y sin 3 y sin y = sin y = 0.5, y = 30 P y M S B(, 0) N R y y = 0 (90 y) = 0 (dj. on st. line) Method Let RPS = y, RQS = y (given) Let M nd N e the feet of erendiulrs drwn from S on PR nd Q from RS resetively. QSN QRN (R.H.S.) RQN = y = SQN (orr. s 's) PSM QSN QRN (A.A.S.) MS = NS = NR (orr. sides 's) sin MRS = MS RS ; MRS = 30, MSR = 60 (s sum of MRS) 90 y y = 0 (dj. on st. line PSQ) y = 30 = PSR = 90 y + 60 = 0 Q htt:// Pge 5

6 Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 G9 If stisfies the eqution =, find the minimum vlue of. Referene: 99 HG, 00 FG., 00 HI, 00 FI.3, 00 HG6, 0 FGS., 0 FG.3 Method Cse : 3, =, = 3 Cse : 3 < 5, =, lwys true, 3 < 5 Cse 3: 5 <, =, = 5, no solution Comined solution: 3 5 The minimum vlue of = 3 Method Using the tringle inequlity: + + = = Equlity holds when 3 nd 5 the minimum vlue of = 3 G0 3 shoes re hosen rndomly from 6 irs of shoes with different models, find the roility tht etly two out of the three shoes re of the sme model. In order tht etly two out of the three shoes re of the sme model, either Cse the first two hosen shoes re of the sme model. (Proility = ) or 0 Cse the lst two hosen shoes re of the sme model. (Proility = ) or 0 Cse 3 the first nd the 3 rd 0 hosen shoes re of the sme model. (Proility = ) Required roility = 3 = 0 htt:// Pge 6

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

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 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.,