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1 nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember Individul Grup Individul vents I Given tht + + is divisible by ( ) nd ( ), where is n integer, find the vlue f. Let f () + + f ( ) is fctr f () is fctr f () ( + )( )( ) I Given tht,, b frm n.p. nd, b, 6 frm G.P. If nd b re bth psitive numbers, find the sum f nd b. + b (); b 6 () Sub. () int (): b ( + b) b b 0 (b + 6)(b ) 0 b 6 (rejected) r b b 0 I Find the smllest rel rt f the fllwing equtin: ( )( + ) ( + )( 6) Reference: 99 FI. ( + )( 6) ( )( + ) ( ) ( ) 0 ( + ) 0 r The smllest rt. I In figure, is squre. is pint n such tht nd. Find the re f the squre. (Pythgrs therem n ) re f the squre Pge

2 nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember 0 I If L, find the vlue f. ( + ) L + ( + ) ( + )( + ) 0 r L > 0, nly I6 Given tht n is psitive integer which is less thn 000. If n is divisible by r, find the number f pssible vlues f n. (Reference: 99 FG.-, 99 FG.-, 0 FI.) Number f multiples f Number f multiples f 99 Number f multiples f 66 Number f pssible n I7 In figure, is rectngle with. is pint n such tht. M is the mid-pint f nd P, re pints n nd respectively such tht PM is stright line. If PM : M : k, find the vlue f k. rw stright line HMG // (H lies n, G lies n ) H H (Intercept therem) PHM ~ GM (equingulr) HM ~ (equingulr) P M PM : M HM : MG (rti f sides, ~ 's) H G : (HG HM) (rti f sides, ~ 's) 7. : (.) : 9 (pp. sides, rectngle) k 9 I Find the lst digit f the vlue f , 6 6,, the lst digit f 6 0 is 6, the lst digit f is. The lst digit if the number is 6 (md 0) 7 (md 0) I9 Let be the psitive rt f the equtin + + rss multiplying: ( + + ) ( )( + ) ( + + ) ( + ) ( )( + ) 0 I0 Find the sum f ll psitive fctrs f 0. +, find the vlue f. Reference 99 HI, 99 FI., 997 HI, 99 FI., 00 FG., 00 FI. 0 Psitive fctrs re in the frm b c, 0, 0 b, c,, b, c re integers. Sum f psitive fctrs ( )( + )( + ) Pge

3 nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember 0 Grup vents G If +, find the vlue f +. Reference: 9 FG G ( ) In Figure, is tringle. nd re the bisectrs f the eterir ngles nd respectively meeting nd prduced t nd. Let nd. Find the vlue f. Reference: 96 上海市初中數學競賽 (dj. s n st. line, bisectr) + ( s sum f ) + (vert. pp. s, bisectr) (bse s issceles ) ( s sum f ) G If 6 nd b 6, find the gretest vlue f b. 0 6 nd 9 b 6 6 b 7 The gretest vlue 7. G Let, b, c be integers such tht b c. If c >, find the smllest vlue f c. Reference: 999 FG. Let k, b k, c k 6 c > k > The smllest k The smllest c Pge

4 nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember 0 G In figure, the re f the prllelgrm is 0. M nd N re the mid-pints f nd respectively. N intersects M nd t pints P nd respectively. Find the re f PM. Prduce M nd t meet t R. Let. Then N N (mid-pint) ~ N (equingulr) N (rti f sides, ~ s) (N mid-pint, pp. sides f //-grm) R P M N P M t N t re f 0 60 re f 0 re f N N re f re f N 0 0..() s M is the mid-pint, M MR (S) RM M (crr. sides s) () ls P ~ NPR (equingulr) P (rti f sides, ~ s) PR NR (pp. sides f //-grm, crr. sides s)..() mbine () nd () P R ; M R MP M P MP P R 0 R R R 0..() re f M 0 0 y (): re f MP re f M 0 6 () re f N 0 0 re f PM re f N re f MP re f N (by () nd ()) R Pge

5 nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember 0 G6 In figure, find the number f pssible pths frm pint t pint fllwing the directin f rrw heds. Reference 9 FI., 000 HI, 007 HG The numbers t ech f the vertices f in the fllwing figure shw the number f pssible wys. 6 0 S the ttl number f wys Figure G7 Find the smllest rel rt f the equtin ( )( ). 0 ( + )( ) 0 r G The smllest rel rt is. In figure, fur circles with rdius tuch ech ther inside squre. Find the shded re. (rrect yur nswer t the nerest integer.) The line segments jining the fur centres frm squre f sides Shded re π G9 In figure, is squre nd pints, F, G, H re the mid-pints f sides,,, respectively, find the number f right-ngled tringles in the figure. (Reference: 99 HG) Let the shrtest side f the smllest right-ngled tringle be. Then, H,, We cunt the number f right-ngled tringles with different hyptenuses. Hyptenuse Number f tringles Ttl number f tringles 0 H F G G0 test is cmpsed f multiple-chice questins. mrks will be wrded fr ech crrect nswer nd mrk will be deducted fr ech incrrect nswer. pupil nswered ll questins nd gt 70 mrks. Hw mny questins did the pupil nswer crrectly? Reference: 99 FI. Suppse he nswer questins crrectly nd questin wrngly. ( ) Pge