Toán học và tuổi trẻ Số 214 (4/1995)

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Toán học và tuổi trẻ Số 214 (4/1995) trình bày về hàng điểm điều hòa, chùm điểm điều hòa; ma phương trên máy tính điện tử bỏ túi; một tính chất đẹp của đa giác đều. Đặc biệt, tạp chí còn đưa ra một số đề thi Toán học trong các kỳ thi Olimpic Toán quốc tế; đề thi tuyển sinh 1994 tại ĐH Xây dựng.

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Nội dung Text: Toán học và tuổi trẻ Số 214 (4/1995)

fu, q4 so crAo DUC vA DAo rAo * Her roAN Hec vrEr NAM 4 pr+1 rep crii nn xcAy rs uANc r:uANc tr rrANG ordnr orfiu nor, cnilnr orfiu rr0,r tr MA pHUOtvG rnEX ptAy rixru ptgtv rt sd rut tr THI IITIMPIG IIIIN KHU UUC CHAU A.Hft BiIIH DUONG rG.tjffit!1q:i@q*i*l; Hgc sinh gi6i tocin trtrdng Qu6:c hqc Binh Dinh
roAN Hec vA rubt rRE MATHEMATICS AND YOI.JTH MUC LUC Trang c Tim hidu sdu th|m todn hoc phd thdng Helping youllg Friends Gain' Better (Jnd,erstanding in Secondary School Maths Tdng biOn tdP : L€ Qu6c Hd.n - Hdng didm di6u hda, chim NGI.JYEN CANIITOAN 1 di6u hba Ph6 tdng.bi4n tdP : o Gitii bdi ki tutic NcO oAr rrl Solution of Problems in Preuious Issue HOANG C}IUNG C6c bdi cta s6 210 2 o Ban c6 bidt ? Do you know ? nOt oOttc atEN tAP : Nguydn Van Vinh - Ma phtiong tr6n m6y tinh di6n tt b6 trii. 9 Nguy6n Cinh Todn, Hodng Chung, Ng6 Dat Trl, LO Kh6c o Db ra ki ndy Problems in. this issue Biro, Nguy6n HuY Doan, Cric bii fiTll2l4 ddn T10/214,L11214,L21214 10 Nguy6n Viot Hai, Dinh Quang HAo, Nguy6n XuAn HuY, Phan o 6ng kinh cdi cdch daY vd hoc lodn Huy Khii, Vn Thanh Khidt, L0 Kaleid.oscope, Refornt of Maths Teaching and He.i KhOi, NguYOn V5"n Mau, Learning 11 Hodng LO Minh, NguY6n Khac Nguydru Dtc Tdn - Vai y nh6 trong m6t bdi hoc Minh, Trdn Van Nhung, d s6ch dai 7 11 Nguy6n Dang Phdt, Phan o DQng Hitng Thd.ng - Thi Olimpic Torin Thanh Quang, Ta Hdng khu vrJc chau A-Thai Binh Drrong 12 QuAng, Dang Hung Thing, Vfl Drrong Thuy, Trdn Thdnh o Biti Quarug Trudrug- Dd thi tuydn sinh nam 1994 Trai, L6 86 KhSnh Trinh, NgO trrldng Dai hoc xAY dung. 14 Vi6t Trung, Darg Quan Vi6n. o Vu Qudc Luong - MOt tinh chdt deP cria da giric d6u. 16 o Giii tri todn hoc Fun with Mathematics Binh Phuong - Gi6iidap bdi Thay cht bing s6 Bia 4 Vu Kim Th&y - Du llch xuy6n Vi€t' Bia 4 Tru sd tba soan : 4bB Hlrng Chudi, Har NOi DT:213786 Bian ft,uit' tri sr/:'trU XlVt THIIY 2Bt Nguy6n Van Cil. TP Hd Chi Minh DT: 35611 L Trinh bitv: TRQNG THIEP
*I5:itEii+$Bel qrich girio khoa hinh hoc l6p 10 hi6n lrru hdnh kh6ng trinh bAy RSs[fi.il* \\r:iJiilsiiliJP "cdc vdn d6 tr6n, cbn sdch bni tdp hinh hoc ldp 10 thi gi6i thi6u z.,Gf+rf$rx @6i;ffirm phdn "hdng didm di6u hda" m6t cdch so sAi vd b6 qua phdn "chirm WiI5il;*r di6u hba". Di6u d
.47 Hir, Nam Thanh, HAi Hung. D6 Diau NgQc, Nguydrt Hbng Dung, 8T, Trdn Ding Ninh, Nam Dinh, Nam Hh. Trd.n Qui Ban,6T, Nang Khidu, Th6i Thuy, Thrii Binh, L€MinhThitnh, La Dinh Duy,9T, Lam Son, Thanh H6a. Pham Xud.n Thnnfu 8D, Nhng Khidu ThAnh ph6, Truong BAi T1/210 : Tirn x, Y € Z sao cho Ngqc Tuyan, 9T, Nang Khidu Nga Son ; La 8r3 = +997 3iy Xtd.n Trung 8T, Nang Khidu, Tri6u Son, Thanh Minh, IJ6a, Trd.n Nam Dtrng, gCT, Phan BOi Chdu, Ldi giei Cd.ch 1 (ciaDQng Th! I{6ng Vinh, Nghe An, Triruh Kim. Chi, 9T, Ndng 9T, Nang Khidu, Ha Tinh') Khi6u, Hi Tinh, Truong Vinh Ld.n, 9CT, Xudn 8x3=A+997 (1) Ninh, Nguydn Nggc Linh,9r, DOnB Mi, D6ng Tt y e, z + 3iv+997>0. Tr] (1) + Hdi, Qu6ng Binh, Li Quang Ndm,9T, HuYnh 8r3>0+r€N* Minh Soru,7T, Chuy6n Drlc Phd, QuAng Ng6i. Tt dd suy raY € N. T6 NGUYEN - Ndu y > 2 + 3ir:0 (mod 9) ;997 :- 7 Biiti'lzlzl} z Ba sd thttc x, y, z d.6i mQt (mod 9) khd.c nhau thda m.d'n dibu ki1n (1)+8r3:(Lt)3=7(mod9) ltT7 Ttff 1T-7- : r,y ) + @ -*) + 1* 1) o Di6u niy mAu thudn vi vdi ru e N thi z3 = Ching minh rd.ng 0, 1, 8 (mod 9) -x3X1 -y3X1 -23) = (l -xyz)3 (1 -Nduy- 1+8r3 = 1.000+x:5(th6am6n) Ldi giii : (cria s5c ban Dd.o Trgng Thd 9T - NduJ : 0 + = 998 + x. #N (loai) 8x3 PTNK Hi Tinh, Nguydrt Ngqc Linh PTCS VAy cri duy nhdt m6t bQ x, y th6a m5n (1) li D6ng Mi Ddng H6i, L€ Huy Binh 9T Hd Tinh (5, 1) (x, Y) : Philng Thanh Til.ng 9CT ThSi Binh, D87Lg Ngqc Minh l6p 9 To6n HAi Hung) Cdrh 2 (cia Mai Tharth Binh,8M, Mari Ta d6 ding chrlrrg minh dugc : Ndu Quyri, Hd NOi) Ta cd 8r3 = 3[ + 997 :0 (mod 8) a*b lc = 0 thi a3 +b3 *c3 =3abc + 3/ = 3 (mod 8) =Y 16. Ddt a = U-|)TTry, b = (z -41[l=f D4ty =2h +t + 8r3 = 3*+ 1+997 vdc= (x-y)1[t=7bc6 0 :3(xt)0 -z)(z -x) @ (1) =+ (2r)3 - ld = 0 (mod 3) Bidn ddi vd trrii ta drrgc v6 tr6i bing : + ?-x = 10 (mod 3) 3{t -xyz)(x -y)(y -z)(z -x). + 4x2 - + 100 = (k, - lqz = 4Ox 0 (mod 3) Y\ x, y, z ld ba s6 khric nhau n6n tt dd suy ra = 4x2 * Zox * 1oo : o (mod 3) * (Lq3 - 103 = 0 (mod 9) l-xyz=@ Nhrrng (# - 1) / 3. Mdu thu6n hay (1 -xy43 = (1 -r3)(1 -yr(L -?.3) x X6t h = 0 trf (l) + ?^x = 10 + x = 5 Nhin x6t. Tdt cd edc ban tham gla gl.ei bdi nny ddu lim d0ng vi theo c6ch nhrr tr6n Viy ta cri cflp nghiQm (r, Y) = (5, 1) DANG HUNG THiNG NhOn x6t. Rdt nhi6uban gif,i drlngbni ndy' Sau diy I} c6clr4n cd ldi giAi t'6t Phgm Hrtu BAi T3/210. Luu, 9T, Nang Khidu Gia Ltrong ; NguYdn a, b, c ld. ba s6 til,Y ! thuQc doan [0, 1]. Nggc Dong, 9, Neng Khidu, Thudn Thdnh, Hi m.inh rd.ng Bic. Le Tidn Dqt, 9, ChuY6n Van-To5n, "Chnng Thrrdng Tin, Hi Tdy. Nguydn Anh Dfin'g,9H., l) a2 +bz +cz < 1+ a2b +b2c *c2a Trrrng Vrrong ; NguYdn Thu Hd'n9,9A1, D Zla3 +b3 +c3) - (azb +b2c *c2 o) < 3 GiAng Vo ; NguYdn Si Phong, 9A" ChuY6n abc^ Ngt, He NQi, Phil.ng Drtc Tud.n,9NK, Thanh 3) *1
Ldi giii: 1) Ta c az 1t -01 ruongry;fn*h=#; b(t-c)>b21t-c1 Mitkhric ' o.. , - 0 ta .6 (t - a)(1 - c) > O ---+ TuongtVBD=p-b. Ta cri : CA. CB : 2DA. DB 1*acZa*c bbb(--(- *ab-2(p-a)(p-b) Khiddca+l - a*c -b*c *2ab = (2p -2a)(2p -2b)
+Zab:(b+c-a)(a+c-b) . duitng trdn ngoai tidp d bdt crl th.iti didm. nd.o, tant gid.c ABC ludn ludn di qua mOt didm cd +Zab : c2 - (o -b)2 dinh O niro d6, khd.c C. *c2: a2 +b2 aL,ABC vu6ng tpi C. Nhfln x6t : Giai tdt bdi ndy gdm cci cdc ban : Nguydn L€ Lttc 9Al Thi trdn Ddm Doi, Minh HAi ; Nguydn Phi Khdtlh trttdng LO Dai Drrdn-g, TAn An ; Vu.Dtlc Ph.i,9T Nguy6n Du, Trg, Long*fq Gb'Vdp, H"ynn Duy Nguy€n 92 tr''tdng Colette, Q3, TP H6 Chi Minh ; Tr?in Phi Hilng, 8T, tnrdng B6i dudng Girio duc Bi6n Hba, D6ng (b) u Goi A va B ld hai vi tri x6c dinh ndo dd cta Ayunpa, Gia Lai ; NguYdn Thi Nhu QuYnh, hai dQng tit ; ch&ng han, dci li hai vi tri xudt gT, Le Phu Thd.nh, 8T ChuY6n L6 Khi6t, gA THCS Nguy6n phrit, ring v6i thdi didm f., : 0 ; Ar vd B, Ii QuAng Ngei ; Le Tu NhiAn, Lai vi tri ttlong rlng cria A vd B d thdi didm VEn Tr5i, DiQn Bdn, QuAng Nam - Dn N6ng ; Ngo Tidn Dilng,9B Trdn Cao VAn, Quy Nhon, t = tt,Thd thi ta c6 :'*:#3:'; Binh Dinh ; DiruhTrung Hoir.ng 9APTTH Phu Bdi, Thrla Thi6n - Hud ; Nguydn Ngqc Linh (khong ddi) 9, D6ng Mi, Ddng H6i, QuAng Binh ; Duong Gqi O lDr giao didm thrl hai cta hai drldng Thu Phuong 9TNK Hn Tinh ; Trd.n Nom Dfi.ng t$a u(ABC) vd ur(A.,BrC), (x. Hinh v6 b6n) ; 9CT Phan B6i Chiu, NghQ An ; L€ Minh d6 ddng chrlng minh ring : Thd.nh,Vfi Thi Lan Huong, Hoitng Vd.n Hit'ng, LOAA, ^ LOBB.. Tn dd suy ra : Le Dinh Duy 9T Lam Sdn, Mai Thinh Hia.p, .^' ^: 98 THCS Nga HAi, Nga Son, Nguydn Khuydn. lApBt: AoB- ACB:e Ld.n 8D tnrdng N6ng khi6u Thanh H6a, Trd'n OBt OB BBr uz ; trlc li : AnhTud.n 98 Xuin Ngqc, XuAn Thriy,Nguydn OA, OA AA1 u1 Ngqc Hung 8T NK Y Ycrr, NguYdn. Anh Hoa 9T, Nguydn Hbng Dung, Hodng Manh Quang, 1da, o'a1 : (CA, = rp (khongddi) (1) Trd.n Tudn Cudng,8T, Trdn DSng Ninh, Nam OB ' u2 "h) Dinh, Nam H.d Dod.n Minh Dtlc 8D ChuYOn *: OA ul lkh6neddi) (2) Quj'nh Phg, Thdi Binh ; Nguydn Ngsc D)ng, Nguydn Dang Thd.ng 9NK ThuAn Thdnh, HA Suy ra O cd dinh vi, theo (1) vd (2) O le mOt Bic; Phirng Dic Tud.n,gCT NK Thanh Hi, trong hai giao didm O vd O' ctra hai dridpg trbn Nam Thanh, Hai Hrrng ; Pham Thu Huong xric dinh : drrdng tfin u(ABC) vi drrdng trdn 8A1, THCS H6ng Bdng, Hii Phbng ; NguYdn ApOldnift (a), qu} tich nhiing didm M sao cho Minh phuong 9A PTTH Vi6t Tri, Vinh Phrl ; *l:". tchf v rang ndu 1dA,o-81 = Nguy\n Quarug Thang, gC Hei Bdi, D6ngAnh, ttt,q,- u.' Lvrru r Phan Linh 9A PTCNN, NgO Vart Sdng 9A Chu +-!9,+++ Van An, Pham HuY Titng 9A Bd Van Den, - (CA, CB) = gtltt (O'A,O'B):-(OA,OB) : -P vd do dr5, giao didm thri hai O' cira hai drrdng Nguydn Tidn Dung 9Ar, Giing V6, Biti Viet trdn tr6n khOng th6a man (1)l Hd 9C Gia LAm A, Triin Thu Hibn, NguY67 Ta drroc d.p.c.m. Tudrug Minh, Nguyin Anh Hoit'n,, Nguydn Nh?n x6t : 1) Sd ban tham gia giAi bai Hbng Hd 9lfl^, NguYdn Quang HuY 8H PTCS torin niy kh6ng nhi6u. Di6u dd c6 lE phAn anh Trung Vudng, Mai Thanh. Binh, Trd,n Trung su lfng tung cria nhi6u ban vd phuong phrip Tidn 8M Mari Quyri, He NOi. gi&i bdi torin kidu ndy : ching minh s4 tbn tai VO KIM TTI{JY ctra mQt hinh (trong bii tozin niry, hinh 6 ddy li mdt d,idm) th6a min m6t sd tinh chdt dac B}d T5/210. TrAn hai dudng th&'ng a ud' b trdng ndo dci. Dd chfng minh drrdng trdn cdt nhau d didnt C c6 hai d1ng tt2 (ch&ng han', (ABC) lu6n luOn di qua mOt didm cci dinh, hai bO hd.nh) chuydn dQng thdng dbu nhung trong dd A vd B ln hai phdn ttt chuydn d6ng, udi uQn tdc khd.c nh.au : A c6 uQn' tdc u, trAn ta phAi c6 dinh h
thrl hai O ctrahai dudngtrbn (ABC) va (AlB1C) Nhgn x6t. Cric ban grli ldi giei ddn d6u cd ld didm cd dinh bing crich chi ra cac tinh- chdt chung m6t c6ch giAi dUa tr6n vi6c phdn tich dec tnlng cira nci mA ta c
+ dey {u) ld day tang. Hdn nfa, theo giA d vd tr6i cria (3) lu6n duong hay (3) khOng xdy ra. Vfly (1) kh6ng xdy ta vd ta cci dpcrn' thi6t, {u) ladeybf ch4n' Suyra, t6n t4)limun' n+& NhOn x6t. Cri it ban gui bei giAi trong dri 9 thidt bei giei dring, phdn ldn kh6ng dirng phAn chrlng Dit lim un= u'Tt gia n+@ n6n ldi gili dni dbng. B4n doc cci thd vAn dung ldi giei tr6n dd md r6ng bii todn ctro (2n+l) un+t(l -un) > 1nVn e N* ta cci didm chia drrdng ttdn, (Zk*I) didm cirng mdu 1 ld c6c dinh cria lz tarr, gi6c cdr, vdi 1 dinh chung Iimur*, 11 -limur) > n,hal * o- aK n+@ n+6 duy nhdt sao cho u 1 1. ^ 1 [-;) u(t -u), 4o(, -;\' * 0eu = r' oANc vteN Bed T9/210. Cho hai dudng trdn (O ; R) ud' t::""= 2 (O' ; R') udi R > R' cat nhau tai hai didnt A 1 vcv ua B. Hai tidp tuydn chung ngod'i MM'uit' NN' NhOn x6t : Cci rdt nhi6u ban, tt kh6p mgi (M, N e. (o) ; M" N', e (o')). DAy MN cat OO' mi6n cira ddt nrl6c, giri ldi gi6i cria bdi to6n' ri C ; d.d.y M'N' cat OO' d D..Ching minh tt2 Trong drj chi cri : 3 ban (hai ban 6 trrldng gid.c ACBD ld. hinh thoi. chuven Phan B6i chAu, Ngh€ An vd 1 ban d TH chuyOn Nguy6n Binh Khi6m, Vinh Long) Ldi sini. ru.a ffi :# = #,tt c6c cho ldi giai sai do chtra hidu dring v6 gidi h4n ti6p tluydr, MM', cira day s6 vi do ng6 nh{n trong cdc ddnh 916; NN'ta lai cd mOt ban (6 trtrdng Dio Duy Tr), Quing Binh) oM ll o'M' ; cho ldi giii kh6ng ddy dt, do qu6n kh6ng chrlng minh srr tdn tai cialirnun. Tdt ch c5c ban cbn Suy ra l, Iai d6u c
Bei Tf0/21O. Cho hinh lang tru ti gid.c NhQn x6t : C
vd gsn Nguydn Trsng Nghia, PTTH Vi6t Tri, IrANc DI6M DIiiU HOA... Vinh Phri ; Nguydn Quang Tudrtg 10L, Phan ('lidp theo trang 1) BQi ChAu, Vinh (Nghq An) ; Vu Th,i Lan Huong, 9T, PTTH Lam Son, Thanh Hria, Thi du 2 : Cho chirm di6u hoa S(o, b, c, d.) Nguydn Hilu Hd.i, 12A Honi Nhon II, Binh c6 c t d. Khi d6 c, d ld phdn gyitc c(n c1.c g6c Dinh ; Nguydn Thi Hoiti Thu, 8CL, PTCS tao b6i a vb.b (kd cA g
€q-"o e A€* ! ffiffi. WffiWffi Wffi MW WffiM ffiMffiffi W ffi& WM a, Ncuytsw vAN viNn Tp Hb Ch; Minh E EE tSon n^rquan srt bin 1) Tdt ce cic s6 t4o boi ba chr: sd rl tren cirng m6r ddng, phim ctia m6t m6y tinh bo ttii. mdt c6t, m6t dUdng chdo dEu chia hdr cho 3. C6 thd hinh dung, cric chfr : 3.41 E E[il sd crla bin phim duclc sip xdp 123 456 : 3.152 258 :3.86 159 :3.53 trong m6t hinh vu6ng gbm ba 32t : 3.107 654 = 3.218 852 :3.284 951 : 3.3t7 dong. ha cQt. har drtong cheo. : l4'/ = 3.49 789 = 3.263 369 3.123 357 : 3.119 : EE NCu tlfii voi c:lc ma phuong cda ngrlrii.'I-rung }ioa vA ngucti 741 3.247 987 = 3.329 963: 3.321 753 :3.251 T t An D6. t6ng c6c sd tren cirng m6t dong. tong ciic s6 tr€n cing mOt c6t, tOng ddc s6 tri.n trl Ndu ban sip xdp c6c thr.tOng s6 nhan dudc trong 16 ph6p chia tr6n day (s6 chia Ii 3) rhco rhi rrr ti sd nho ddn sd l
B,iti Tgl214: Trong m6t phing, tr6n hai dridng trbn d6 cho ur (Or , Er) vit'u2(O2, R2) cd hai dOnE trl chuv6n dOns ddu theo cirns m6t chi6u'(c[ing h4fi,'ngubc"chidu kim ddn{h6) : MyfiAn (v1) va M2ti1n (ur) . Chfng xu6lt ph6t ldn lrrot tt hai didm ,41 vd A2 cho trudc tr6n (u1) vd (ur) sao cho O, Aj{O2A2vit sau mdt Cac tdp THcs vdng, lai trd v6 A, vd A, cirng m6t hic. B,iliTll2l4: Vdi m6i sd tu nhi6n z l6n hon Chfng minh ring trong m4t phing, ncii chung 1 sao cho 2n - 2 chia hdt cho n, hdy tim UCLN kh6ns tdn tai mOt di6'm P cd dinh IuOn lu6n c6ch l22"-t-2.2n-l\ -l d6u h-ai dOng tir'Mr vd. M2 6 riroi thdi didm; trr) \- -,- VUKTMTHUY hai t^rdng hop dac otrt Bdi 121214: Cho rlB ldnghiQm IId N1i crlaphriongtrinh Sfr?fs i? *.'tI^, Hd N1i. x3 + ax2 +bx * c : 0 (a, b, c e Q) BAi T10/214 : Cho trl diQn ABCD v6i dO Tim cric nghiOm cdn lai. dai cac drrdng cao ho , h6 , i, , h4 , bdn kini NGUYE,N DIJC TAN hinh cdu ndi ti6p r. Goi do dai doan ndi dinh TP Hb Chi Minh r.l1 di6n v6i trbns t'am' mat d6i di6n la Biai TBl2l4: TrOn mnt phing cho 5 didr4 A, io , m6 , ffic , ffid. doi trong tain, tam hinh cdu B, C. D,.E sao cho kh6ng cd ba diSm lio thnng nOi ti6p tri di6n lb, G, I. Chrlng minh rang hine. Nzudi ta n6i tdt cA cdc doan thSng cci hai ddu'1a h-"ai trone cdc didm dci rdi t6 cdc doan , m, fiLb ffi, *d, GI max ifta"n niv hoac"xanh. hoac vdng, hoic dd, m6t cdch'hu hoa. Chrlne minh rane lu6n lu6n thu {E' l%' 4' ,rof' g, drroc it nhdt mQt trl Sdc (ldi, l6mtroac kh6ng-don) DAM VAN NTIi v6f sd mdu ctia cdc-canh bgt g vudt qua 2. Thtii Binh ANc rY PHoNG Ha N1i c6c db vat li B}lj T4l2l4: Cho tam gi6c ABC n6i ti6P Ll l2l4 : He vdt duoc bd tri nhu hinh v6 : B,iti /O). trons do B, C cd dinh. trr la tia phAn gi:ic rn, duoc tieo bang dAy mhnh kh6ng ddn, ddy troirE cril s6c A. M : N theo thrl ttt Ii hinh duoc vit qua rbng roc c6 dinh gin tranm.r, ddtt chidri cria E ; C l6n'Ax. Tim qu! tich trung didm 1 MN' kia cria ddy g6n vdi m3. Budn$ tay kh6i m, tt.\ 'ua sb euaNc; vrNu hO vAt chuvdn An NghQ d6ns hm cho ?Av - Bei T51214: Cho AABC co c6c gcjc d6u nh6 tieo"mrbi l6ch 30b hon 1200. Hdy dung didm M trong tam gi6c so vdi pht ons th6a min : t[a"n atrnl. cr,o' ' MA.BC : MB.AC = MC.AB m, :"Or2 kE ; NGUYEN KHANH NGUYEN n|: O,4hg. 86 -r- - "-- " Hdi Phdng or'rl srit. Tinh j tcrdi-ulrrongm, ? " C6c l6p THCB - Gia t6c cAcvdt? TBlzl(: Chrlng minh rdng B,di : . ltu. (xt)Q - @+i) < 2tn(1+a) vdi x > Y > o PHAM HUNG OUY6T NGUYEN PHU LOC Hd N1i Cbn Tho BdiLzlzlf: Didm sSns S d tr6n truc chinh B,iriT71214: T6n tai hay kh6ng tdn tai hdm cria flrons cdu I6m G. ben kifih n : 6cm. cbch dinh s6 f(x) x6c dinh vd li6n tgc-trong (-co ; *oo) vA guofis 3Zcm. Dat thdu t
I.SS.IIE T71214. Does there exist a function f(x), defined and continuous on (-oo, *co), satisfying For Lower Secondary Schools the conditions : Tll2l4. For every natural number n > L a) fl1995) < b) f(f(x)) = I995-x for all r € R ? such that 2n -2 is divisible by z, find the ^1996),. TBl2l4. The sequence {r,r} is defined by : greatest common divisor of 122"-t'-2,2"-l). : : 17 - \r-l-xtr l,xn+t 1r--_ 1I T2l2l4. Suppose that {5 is a root of the x2 . equation a) Prove that there exists lirr, x, and find x3 + ax2 *bx * c : 0 (a, b, c € Q) ; this limit. n+6 find the other roots. b) Prove that there exists a unique number T3l2l4. Let be given five points in a plane, no three of them are collinear. Colour blue, A such' that L = lim a ,, u finite number yellow or red at random all segments with distinct from 0. EJb;":# in terms of A. these points as extremities. Prove that we Tgl2l4. In plane, on two given circles obtain at least a quadrilateral (convex, ur(O.,, Rr), ur(Ot, Rr) move uniformly two concave or no-simple) such that four sides of niobiles iL ttr"e (for example, which are coloured by at most two colours. counterclockwise) "i*"'di"""tion : Mron (ur) and M, on lu r). '1412L4. A triangle ABC with fixed B, C is They start from two given priint A, on'(u , ) aid inscribed in a given circle (O). Let Ax be the A, on (ur) so that O,A, is not parallel to ' O y4, and after a round, they return to A, and Ai it inbisector of angle A and M and N be the same time. Prove that, in general,'there foes respectively the projections of B and C on Ax. not exist a fixed point P equidistant from M, Find the locus of the midpoint I of MN. and Mrat any moment, except two cases whicli T51214. Let ABC be a triangle, every angle must be determined. of which is less than 120o. Construct a point T1O1214. Let ABCD be a tetrahedron with M in the interior of the triangle satisfying : altitudes h^, h,n, h., h,l ; the radius of the MA.BC:MB.AC:MC.AB. inscribed sphere is r ; the lengths of the segments joining each vertex with the center of gravity of the opposite face are mu, trl6, nLr, For Upper Secondary Schools. mo. Let G and I be respectively the cen[er o]f that T61214. Prove : gravity and the center of the inscribed sphere of the tetrahedron. Prove that : (r - y) (2 - lx + y)) < ,,"1ffi) fro nlb tu, md, GI forr >y > 0. max { 1'tu , h,, _thrJ' 3r. Q,"g ki,"h ciii ciich dAy ua hgc tofn *, i ' i',1? jii.i "f,F?{'}i'rl' tl 'ir $ i {q *. : {$ Se#hq ffi,&! "i '1 @nrorr* I muc 96 dai s6 7 phdn 1 li ludt a t (-a+ b) : [a + (-a)J * b : 0 * b = b. giAn udc v6i n6i dung nhrr sau. Cci ding thrlc R6 rdng 6 dAy ta m6i thir lai hay chrlng t6 a * c : b t c, tacd thd xda c dhai vddd dtioc -a -t b ld m6t nghi6m ctra phtrong trinh mi a = b. Tric gii chrlng minh tinh chdt ndy chtra llm 16 "cd m6t" vi vdy ta n6n trinh bdy bang crich c6ng -c vdo hai vd cria ding thric 16 hon lA c6ng -o vdo hai v6 cria d&ng thrlc a * c : b *c nhung suy cho cirng cOng vdo, a*x:btac6 xcia di li m6t. Nhrr vdy d ddy c
sffisr,mryE$B$AN _: E{IltI lIlIC I Cllfill il.TIlf,l Blllt! Dll0lls (f,Ptll0] DANG HUNG TH.6,NG thi Olimpic To5n cria phong trdo Grti he't qud. ud' thu hang cia 10 thi sinh {$rr^nay hgc sinh trung hqc d5 phat tridn manh tr6n dat d.i6m caa nh(it uit' kbm theo biti thi cia thd gi6i. Ngodi cu6c thi Olimpic To6n Qudc td thi sinh dung thi 7, thi 3 uit. thu 7 vd :nttdc truydn th6ng hdng n5rn mi chrlng ta d5 quen dang cai. bidt vn tham gia trf ndm 1974, nOt s6 hhu vrJc tr6n th6 gi6i de td chrlc c6c cu6c thi Olimpic 4) Nrr6c dang cai s6 di6u ph6i cric kdt quA Torin khu vttc.'Trong bdi nly chring tOi mu6n ti cac nu6c vd xdc dinh giAi theo c6c nguy6n gidi thi6u vdi c6c ban cuOc thi Oiimpic ToSn tic sau ddy : khu Wc chnu A - Th6i Binh Drrong goi t5t ld APMO (Asian Facific Mathematical Olympiad) i) Tdng s6 huy chrtong khdng vudt qua Cu6c thi APMO duoc td chrlc hang nam. f g=t (6 di n ld tdng sci thi sinh) bit ddu tt n6m 1989 ddn nay l}, ldn thrl 6. Ldn l2t ddu cri 4 ntr6c tham dU (OxtrAylia, Canada, ii) Didm dat huy chuong vdng 2 m * 6. Hdng KOng, Singapo) nay de c m - fi. Trong dc; Malaisia, Dii Loan, Th6i Lan, Hin Qudc, Singapo). Vi6t Nam ctng dtroc mdi tham dU m : didm trungbinh vb.d : dO lOch ti6u chudn. nhrrng chrra nh4n ldi. CU thd goi r, ld didm cria thi sinh, thf i thi Ddy ld cuQc thi khu vrJc l6n nhdt tr6n thd gi6i vi cd nhi6u didm ddng chri y trong cdch b., i:1, rn- vi o.: 1,?,t'' -ntth td chrlc vd xdp giAi. Sau ddy ld m6t vii didm chinh trong di6u 16 cria cu6c thi APMO. nim vAo ng?ry l4ll nrl6c dang cai 1) Hnng iii) D6i v6i m6i mQt ntr6c s6 huy chtrong gtti cho cdc ntt6c tham gia d6 thi, ldi giAi vd ving ( 1, s6 huy chttong vdng * huy chrrong d6p 6n. D6 thi gdm 5 bii to6n, m6i bni drroc 7 bac ( 3 vi tdng s6 huy chuong (vhng, b4c, didm vi thdi gian ldm bAi la 4 gid' C6c bii ddng) < 7. to6n niy drloc chon trong sd cric bdi torin md Quy dinh niy nhim tao ra m6t stl c6n bdng c5c nrldc tham gia drroc y6u cdu gtti ddn. tudng ddi giita c5.c ntl6c. 2) Vno tudu thrl hai cria th6ng 3 m6i nti6c tham gia td chrlc ki thi APMO ngay t4i nrr6c Ngoni ra nudc d6ng cai cd quy6n t{ ddt ra minh. Sd luo.ng thf. sinh tham gia ld, do mdi mgi s6 ti6u chudn dd trao giAi khuy6n khich nu1c quy dirth (khdng cri mOt han ch6 nio) (Ti6u chudn ndy c
Kdt quA cu6c thi duoc th6ng brio d6h c6c tiSp bdt ki la duong vi tdng cria 11 sd li6n ti6p - nrl6c vdo cudi thring 4. bdt ki ld dm. Dd minh hoa, dr.tdi diy li d6 thi vd kdt quA Kdt quA cu6c thi drroc cho d bAng sau. cria cu6c thi APMO ldn thri 4 {1992). 't iing 1.' Cho tam gi{.c v6i ba canh a, b, c. Cd.u N u(,c I I(] IIC IIC KhuyOn s() (l tr'm vi ng bac d6ng khich Goi s lA nrla chu vi. Dung tam gi:ic m6i vdi canh ld s - a, s - b, s - c. Quri trinh niy Inp O.xtrivli, 2| I 4 3 Iai cho ddn khi ndo kh6ng dung dtroc tam giric Canada 2L8 I ) 4 mdi nrfa. Vdi tam gi6c ban ddu nhu thd ndo (lolOmbia I t).1 I 2 2 2 thi qu6 trinh ndy cci thd k6o ddi vO han ? I lirng KOng 262 2 .l -t Cdu 2; Cho vdng trdn C tdm O b6n kinh I ndOnCxir 62 3 r. Goi Cv Cz ld hai vbng trdn tdm 01, Orb6n kinh 11, 12 tudng rlng sao cho m6i vbng trdn Nlch ico 62 3 C, ti6p xtic trong vdi C tai Ai Q : l, 2) vd C Niu ZilAn 90 r, Crtidp xric ngoii vdi nhau tai A. l'hilippin l-56 2 5 Chf ng minh rang ba dudng thing OA, O tAz [)e'ri Loan 288 2 4 3 vd O.rA,, ddng quy. I I:jn ()u6c 2t5 z 4 3 Cdu 3.' Cho n ld sd nguyOn duong n > B. Singapo Chon ra 3 s6 tt tap hSp {2,9,...,n }. Tt ba sd 2t). 2 4 , ni.y ta srl dung d61u nhAn, ddu c6ng vd ddu 'l'hiii i.an 133 1 5 ngoflc dC' ldp ra cdc bidu thrlc trong dci m6i s6 chi cci m5t mOt ldn trong bi6"u thrfc. NhAn x6t : Hai dac didm ndi bdt cria cu6c thi APMO ld : a) Chrlng minh rang ndu ta chon 3 sd l6n n - Cu6c thi duoc thidt kd dd vrla mang tinh 7 thi giri tri cta hon cric bidu thrlc ldp ra d6u chdt thi qudc gia (thtrc chdt ld hoc sinh ttng kh6c nhau. nudc thi ddu vdi nhau) vrla d6m bAo mOt ti6u chudn qu6c t6'dd so srinh. b) Gie st p ld sd nguy6n td < 1/t. Chi phi cho cu6c thi d mrtc rdt khi6m tdn. Chrlng minh rang sd cdch chon 3 sd sao cho s6 nh6 nhdt ld p vit gid tri cria cric bidu thrlc Theo y kiSn cria ngudi vidt bdi ndy, Vi6t kh6ng phAi tdt cA kh6c nhau chinh bang sd Nam nu6c ddu ti6n trong khu vtic chdu A - cdc rtdc dtrong cria p - 1. Thai Binh Duong tham gia IMO - rdt.n6n tham gia vd.o cuOc thi APMO. Ddy ld m6t cd Cd.u 4. X6c dinh tdt cA c6c cap (lt, s) c6"c s6 h6i t6t d6'n6n gido duc Todn hoc phd th6ng nguydn duong ccj tinh chdt sau : Ndu ve h cria ta hda nhdp v6i n6n gi6o duc cta cdc nrl6c dudng thing nim ngang vd s dtrdng thing trong vung, bd tro thOm cho chfng ta trong khric th6a md.n di6u ki6n : vi6c thanh lAp dOi tuydn thi To6n qudc td vdo i) Chring kh6ng nim ngang th6ng 7. Th6m vAo dci cudc thi lai kh6ng ddi ii) KhOng ccj hai dtidng niro sorig song h6i nltdc tham gia phAi chi phi ldn v6 mat tdi chinh. Ndu Vu Giao duc phd th6ng dd bAn quri iii) Kh6ng trong sd /z * ccj ba drrdng ndo s nhi6u vi6c, thi n6n chang H6i Torin hoc Vi€t dudng d6ng quy thi sd cdc mi6n tao bdi /z * s Nam s6 drlng ra td chrJc dd c6c ban hoc sinh drrdng ndy ld 1992. gi6i To:in cta cA nudc cci m6t dip thrl sfc minh Cdu 5. Tim m6t day dei nhdt g6m c6c sd trong m6t cu6c thi qudc td cci chdt luong cao nguy6n kh6c kh6ng sao cho tdn g oia 7 sd li6n td chrlc quy cri nhrr APMO. 13
l i. ,IT I I I ffi ffir Mffi sffiffi ffitm u $$4 ffiiffieffiffi Wffi ffiWffi rtq}Il[$ffiG MOn thi : To5rn (thoi gian lim bdi : 180 phft) I Cdu I 1) KhAo s6t stl bidn thi6n vd v6 d6 2) Cho AC = b, AB = AD = x (x thaY ddi). x2-x*l a) Tim di6n tich cria trl gSac AB'C'D ' theo thi cria hdm sd Y : a, b, x. x2-l b) Tinr x dd ti gi6c AB'C'D ' Id hinh vuOng. x2 2) Chrlng minh : cosr > I - 7, voi nQi I oAr Ax x*0 Ciu I (3 didm) 3) Tim giri tri nh6 nhdt cria hdm s6 : 1) (1,5 didm) a) (0,5) Hdm x6c dinh vdi moi y:lx -11 +l*-21 +lx-al, vdi a la x kh6c + 1. tham sd. x2 -4x *l COu II 1) Giai bdt Phrrong trinh : y':$:0khix=2+t[5. lx" - L)' ltx b) (0,5) I tC 4 +2,x+3 \[4=ir -x -----A 2) Giai vd bi6n luAn h6 Phuong trinh : It lqt t,rv -- I v6ialA t,hanrsd. t )o lx*2v:a ' CAu III 1) Cho phuong trinh sinor*cos(t: a.sin2a. a) Tim nghiOm khi o: 1; !(:D : YQ -'{5) : b) Tim a dd phuong trinh cci nghi6m. J(:7':Y(2 +{5;: 2) Chrlng minh rang m6t tam gi6c ld ddu n6u ba c4nh o, b, c vd. brin kinh drrdng trdn n6i tidp r th6a man a2 + b2 + c2 : 3612. CAu IV Thi sinh chi lam m6t trong hai phdn A, B sau ddy : Phdn A 1) Tinh thd tich cira hinh xuy6n tao n6n do quay hinh trdn x2,+ (y -2)z < 1 quanh truc Ox. 2) Vi6t phuong trinh chinh t6c cria drrdng thing qua didm ntl(1 ; 5 ; 0) vn c6t cd hai dudng thing : c) (Ti6m cAn vd dd thi 0,5). Ti6m cQn dring (dr) j,lu-r-l:o v: -|- t" *v-4:o 1 ; Ti6rn cAn ngangy : 1. .. . [9, *v -2: o (oz) 1v - z -2: o t: 2) (0,75 didm) x * O, cosn > , -t * Phdn B Cho hinh chdP SABCD, B vir D lu6n nhin AC drt6i mQt gtic vuOng ; SA : o vi f(x)=cosx- 1 +,x2 > o. SA vu6ng g O. 1) Chrlng minh rang trl giric AB'C'D' nQi tidp dtroc trong mQt dtrdng trbn' Y6,y f '(x) > 0 v6i moi r > 0. Do f'(x) 16 n1n l4
f'(x) < 0, Vr < 0, suy ra v6i noi r * O c6 CAU III (2 didm) - f(x)>f(0):O. 1) (1 didm) 3) (0,75 didm) X6t hdm a) (0,25 didm) sin6r * cos6r : * 3sin2r x1 y : lx - m.l +1, * nl + l, -pl, (vdi nL, n, p .3 li ba sd thuc mi nL < n < p). Ta ccj x cosrx - 1 - sin2 2r, phuong trinh dua v6 7 3t2 + 4at - 4 : O v6i t = sin2r (1). lnt+n+p-3rne'ux -2, thi h6 v6 nghi6m. dang A(2x - z - l) + B(x + y - 4) : 0. (Xem tidp tran? bia 3) ,U
nnor r[run cnAr DEP c0n DA cnc pEu wr QU6c LUoNG I thd rip dung cho da giSc d6u e
Do dri : (**) duoc tdng qurit thinh : VAy ta ccj kdt que )t Dinh l;f : Cho da gi6c d6u ArA, . . A,, n6i 2UO? : n . 2(r2 +/?2) : 2n1r2 + R11 tiep dttdng trdn (O ; R) M li mOt didrn tuy y Tt kdt quA tr6n, ta cci thd chuydn sang thu6c dtidng trdn tAm O brin kinh r. ta luOn chrlng minh duoc cho da giric d6u cci sd canh tny y nhd mOt ph6p drrng hinh kh6o 16o sau : cA | rull ld m6r hang s6 kh6ng phg thu6c vi x6t da giac d6u n canh B rB z . . . -B,, ndi tidp (O ; R'). Dung da gi6c d6u 2n canh AtAz. " . A2r, nhdn tri crfa M vd hdng sd d
*l'? .J I \ -/Hc Lonq- I i Phong oinh Bdi to6n tuong HOI duong v6i Tn d6 ta x TI *** I th6Y ngay l H:1,8>I>2, *** I O x T I:2hodc ?\ (l)=O=0 + Khil:2 thiu:2 : t (loai) i\ * Khi 1= 8 thi 108 x 48 c6 5 chri t \ so (loai) ,1 o -N6u I: 6 => I =4vi (1) = O=0 \ Khi dd U : 6: I(loai) \ _ I= 9.>1: 3 hoac I : I vi Ndu I (1) = O:0 I;ll +Khi1=3thi U= 5vb'A:7 1 + Khi 1: 7 thi 107 x 97 cd 5 chri so (loai) r''r Vay ta thay nhu sau : 11 : 1. O = 0. t. I = 3. T = 9. U : -5 vd A : 7. Ta dusc iJ con tinh durrg : t25 Ti'? H0 9579 : 103 :93. T}iN}I PI{TIONG ) Cdn'Th d ( #ra #e& Mi$Ar* qW Nhnn d1p 20 nim giii Ph6ng mi6n :'/{iii Nam vi 50 nim ngdY thdnh l{P nudc. / : {to.%ring 2, *rrr!llan'.t : JTtrii' ljTuyht, I : "ll'ie'Ltuug"; lt' : 'ldrbto, b : J-uu+tt Zrrrn4, 7 : 7;i/ 'l-r;, A. : ;'.;il 'q t /ii'(lrin' moi c6c b4n hiy tham gia chuydn du Q'i6r,o,'Bi6n,Pii, tt,96,9o,,/2:.//ta ilJiJt, "B'ii, t."J ; J(i ''l'i'n, ///: ''li'r'l llch xuy6n ViQt. Ban hiY tim mqt ,B)>,/,, t;, {/ot .A Jko, 6: Qing,!t2, t7: lluri,tgt J-u, n: |ft'Qi 'riln, t'e : duong