Toán học và tuổi trẻ Số 221(11/1995)

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Toán học và tuổi trẻ Số 221(11/1995) trình bày về cái hay của một bài toán nhỏ; tập mở rộng một bài toán; sử dụng phương pháp vectơ để phát hiện một số đẳng thức lượng giác; dùng phương pháp tọa độ để giải các bài toán hình. Tài liệu phục vụ cho các bạn yêu thích Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 221(11/1995)

Bo GrAo DUC vA DAo rAo * nOr roAN Hoc vIET NAM tep cui RA NGAY 15 HANG tuANc E cAr rrAY ct,t m0t nAr roAN lTrro A? a A trl TAP MO RONG MOT BAI IOAI\T trl n0xo pHuoNG pHAp T0A D0 of crAr cAc nfu ToAN rixH Hyc sin.h l6p 9 torin trui-tng ChuyAn Nguydn Khuydn, huyQn Bi.nh. Luc, tinh. Nam. Ha
ToAN Hoc vA rubr rRE MATHEMATICS AND YOUTH MI"JC LI.JC Trartg o Ddnh cho cdc ban Trung hoc Co s0 For lower secondary scltool leuel. friends Trd,n Bd S, - CAi hay cira mdt bdi to6n nh6. 1 o Gitii hdi ki trwdc Tdng hi)n fip : Solutiorus of problems in preuious issue Nc;iryEN cANrn'onN Cric bii cia sd 217 4 Phd rd ng hi4n fip : o Nguydn Khac Minh - Cdc bii to6n thi NriCr t>,r'r'lt.f hoc sinh gi6i. Todn THPT todn qu6c 1995. 8 IIoAN(i ('II.IN(i o Db ra ki ndy Problents in This Issue Tu221,..., Ttol22t, LU22t, L2l22l 10 HOI DONG SIEN rAp : o Trinh Bdng Gia,ng - Tap md r6ng m6t bdi to6n 12 o dng kinh cdi cdch duy vd hoc todn Nguyen Cinh Toin, Hodng Kaleidoscope: Refornt of m,aths teaching and learning Chfing, Ng6 Dat Tr1, LO Khec DQng Ky Phong - H6a ra d ngod,i dudng Bio, Nguy6n Hry Doan, nhir cung nhu d trong 11 Nguy6n ViCt Hei, Dinh Quang o Nguydn Van L|c - Sir dung phrrong ph6p v6cto Hio, Nguy6n Xuin Huy, Phan dd phrit hi6n mOt s6 bdt d&ng thrlc lrrong giric Huy KhAi, Vn Thanh Khidt, Lo vd tam gi6c. 13 Hei Kh6i, Nguy6n Van Mau, o Kdt qud cuQc th.i gid.i tod.n ffAn THVTT 1994 14 Hodng L6 Minh, Nguy6n Kh6c o Tim hidu sdu thim todn hoc phd th6ng Minh, Trdn V6n Nhr.rng, To help young friends gain better Nguy6n Dang Phe.t, Phan urtderstandirtg in second.ary scltool math.s Nguydn Phi Phic - Diing phrrong ph6p toa dd Ttranh Quang, Ta Hdng dd gi6i cric bdi to6n hinh. 16 QuAng, Dang Hung Thing, Vu o Gitii tri todn hoc Duong Thuy, Trdn Thdnh Fun utith mathematics Trai, L6 86 Khdnh Trinh, Ngd Binh Phuong - "hd choi thay sd Bia 4 Vi6t Trung, Dang Quan Vi6n. NgO Hd.n - C6 gi6o phat ai Bia 4 Tru sd tba soan : 45B Hlrng Chu6i, Hn NOi DT:213786 Bi€n tep ud. tri su; VO Xtttt THIfy 23f Nguy6n Vi.'n Cil, TP Hd Chi Minh DT: 356111 Trinh bdy: THANH LONG
Dinh cho c6c ban Trung hgc Co s6 0 I TOAil NH6 TRAN BA Si (Thilu Y€n - Thanh H6a) Trong tAp sach Tofrn (cira I-i€n X6) c6 bni to6n sau * iEi: if,i "Trong tam gihcABC (AR > A() co gocA:a. Tr6n canh Dc thdy .ft,t u.ii EF- .48 ldy di6m I) sao cho BD=AC.l.5y di6m E li trung rJi6m AD, F trung di6m.BC. Tinh g6c BEI? ?'. Nghi6n ciru ki biri + EFI=-z:i ^Ai tohn nay ban s6 thdy c6 hai di6u diirng chrl y/ sau ; 'l'hir nhdt biri ro6n rdt phong phf vd ldi giai. Cht i Ii llAB+ 6rt:: Cich 5 Sau dAy xin rrinh b:iy ciic liti giii d6. Crich I D!|AC:b;AB:c, Goi l' li didm d6i xrlng Kt FK // ,q( * cua ,4 qua t?rm .F. Th6 thi ACb Ii'K: '22 - A('A'B lit hinh binh harnh - -: A{'/1 BA'=+ A{- : DR : BA'. KE=BE-BK: .=> A,DBA' cln. c+b c b Tu d6 y l!!S 2 - 2=1 ^chrig6^c ABA' EFIIDA 'vA Abir +KE : KF+ A,FKE cdn ::L." a + BEF: -tl - BFF = BDA': 22 2t Cdch 2 Cich 6 Goi C'lir di6m d6i xtrng K6 phirn giAc AK. cria (l qua tam E. Th6 rhi DirtBr'=u,AB:C,AC:h ACDC' hinh binh h?rnh + AC llD{*vnDB : D(' = AC KB( 'laco:;;;=- K(h - ADB(" KBC _6^:i c2rn KB+K(' c*b vtr_I), I C', : (-'l)z{ nhr.tng KBC ("DA : A((-'t) ll A() a c'tb 1^ 1^ I - B,: ; c'DAA:: iA:;.rtl ^ :, a't - A: 7a: +KB=- tl{' c *h Chir f trong AC-BC'thi E.F dudng rrung hinh ac +81; lluc'+ 6EF:6t:; Kll b +, Viv '' Alt (1) Cr:ich 3 - c b*c c-b b*c a Goi D' Orii xirng vrii 1.1 m;'rt kh;ic BE = b + )) BF:-2 qua tam F. a Til giSc D,BD'C hinh BFi, binh hAnh (2) +DB:A(-:CD' "'tY BE b*c b*c + A,,4CD'c|n. 2 KR BF X6t trtclng tij nhu cAch 1 Tir (l) vzl l2)* AB: BE+ N,BA LABK A - ta cirns "2co 6EF:+ + BLF:: II 2 Crich 4 Thil hai, b.ii todn ta vita giii cl tren la trua,ng hop d:.rc biet N6iDCgoi/trungdidmDC cta biri tnirn sau day, mA cri 16 ban hoc sinh ndo ctng dir m6t I liin girp : De rhiy EllltA(' "'I'0 gihc ABCD c6 AD : B( gqri L-, F l6n ludt l?r trung I di6nrciraDCvir AB.E{cilrAD, BC k6odai tai Kvi1. Chfing Fr minh ring 4tr: EFgai to6n ta via gi?ri tn tntdng hq,p urDB cfra biri to6n nay khi l, D, C thing hAng. Chch gi?ri crla b?ri miAC = DB=+ EI -- FI toirn ta via n€u c6 phong phti nhrr b?ri to6n suy bi6n c[ra n6 kh6ng ? Didr niry xin dinh cho c6c ban suy nghi. Do dri AE,IF cirn
Do cdc s6 hang d vd trrii d6u khOng Am n€n : fu'-5r^51 < -= hay xz { 2"n' ; . Vd.y .r € l-2 ;21. Thay cdc gi5 tri cci thd duoc vio phttong trinh da cho, ta c6 + , i 1{rOod - {rzt : i.z oo: nAnr; ttuN 747 (dpcm). Bei TZl2l7. Tiru nglti€nt ngu.y\n cio phuortg trinh : Nh{n x6t : Cic ban sau dAy cting cri ldi gini 3 (x2 +xy *y2) =r *8-1, t6t: LA Ldnr,9T, Chuy6n Phir Tho, Vinh Pht ; VO La Nant,8A, PTCS Trung TU, Ha N6i ; Ddrug Ldi giei. (dtia theo L€ Hodrug Dxong - 9T. Anh Tud,ru,9T, NK Trdn Phri, I{ii Phbng ; L€ NK Bim Son, Thanh I{oa). NhAn hai v5 cira Hbng Liruh,8T, NK Thi x5 Ninh tsinh. Cdc ban phuong trinh dd cho voi 2 r6i chuydn ve, cltroc : Nguydn Van Quang, 8D, NK Thdnh ph6. LA 6x2+6xy*9yr-2x-l6y =e Hoirng Duong,9T, NK Bim Son, Thanh Hcia ; evz + y2 + Zxy * b - 2y + I + 2(x2 + 2xy + y2) Trfut Nant Du.ng, Ngu.ydnVi€t Dung,9CT, Phan 49, *Art + 51 B6i ChAu, Ngh6 An de phdt bidu va chrlng urinh +2(.yt -7y + 4) tz - T = O o dting bdi torin tdng qudt sau : 1 1 3,- + *---+...**:;> (x+ y - t)2+ z@ + y)2+, (, -Z)'*, 17 + y2 1 12 ,ln ,z (V(z 1)- - 1). 51 Ban D6 Quynh Kh.anh,9A, Chu Van An, Ha NOi dd chfng nrinh drloc bdt ding thfc 2 tdng qu6t sau : 2
1 1 +... + t "l+ -r-+ h ,-- Phbng : Bi"ri Manh Hitng, 8H THCS Trung k. .( V(n*llA r- 1) vr'ron!, Dao pluong Boi Dito phuong Nanr. .R- L "12 \/3 vrr -*> TA THCS Bd van Ddn, Ngzzy dn sy phong gA .tr) Nc;uyEtr Chuy6n ngrt DHSPNN, .A/guydz Qudc Thang 9C, D6 Mitth Chl.u,9T DOngAnh, Hd Nfii, Mol B}ri T4l2l7 : Ch.o tanr gi6.c dbu ABC uir nt6t Ngqc Kha,8T Tr6ln Dang Ninh, Narn Dinh, rluttng tlwng d" Goi A', B', C' ld chd,n duitng Nanr Hi, Cao Qudc HiQp,9"f NK Hoing Hcia, uudng goc liin. luot lta tit A, B, C xu6ng d. Ma,i Thinh Hiap 9F, Nga HAi, Nga Son, Thanh Cltt?n.g nrin h rang gia tri crto. bidu thic Hda ; Triin Nonr Dung, Nguy4ru ViQt Ditng, Nguydn Thinh,9T Phan Bdi ChAu, Ngh6 An, A'B'2 +B'C'2 +C'A'2 khong ph,u thuQc uito ui Lgong H{Iu Klrca LuQt, d6i 3. Hi6p Phd B6c, tri cia d.uitig thang d. llanh Trnng, Nghia Hanh, L€ Quang N6.m,9'l n Ldi gini : Chuy6n Dt?c Phd, QuAng Ngfri, Trd.n Hilu r X6t hinh vE cd lrllrcn,9T Chuy6n Nguy6n Binh Khi6m, Vinh clttdng thing d khdng Long. c5t canh ctia AA-B('. V[] KIM 1.}I.IY Goi giao ctla AA' voi BCIb,E.K6BI LAA', Bii TSl217. Cho tam gid,c: ABC u6i, cd'c CJ t 44'.I(6o ddi CJ \ I dtitng l.rung tuydn AAr,BB1,CC., d,udng cit ttB' tai K. Kd AH : phdn giri,c trong cila goc ACrC cat AAruir AC L BC. Goi o ld canh 1-----'i tai P uir Q. Duitng phd.n gid.c trong cita g6c ctia tam gi6c d6u -' Il BC 1 C cdt BB I ud. BC tai M uir N. Clu|ng mirth ABC. Ta cd : LBIE LAHE. S.,y ra AII , BE ran.g ndu AP : AQ tli to, c6 BM : rBN. BI= Ldi gini. (dUa tlreo Trdn Long BC.AH - vit" C,l : CE.AH Trgng-7to6n-NK Trrong fi --AnCK = AE- Birn Sdn, Thanh n6n A,B'2 + R,C,2 + C'A,Z : BI2 + CKZ + CJ2 Hria) Tr) giA thidt Ap : A(d, ta c6 : AH2 1 _, tBE2 + BC) + CE21 rlt .tAf'}p--gan dj:r[ a AE: hay{PQ = AQE-- n6nP,:1jq'- A{Q Mb. BE2 + BCz -r CE2 = : 180('- AQP : 9r. =(BH+HE)2+BCZ+@H-Hng Hon nta Cro: Crh :2 (BHZ + HE. + BC2 n6n :2 (BHz + AE2 - AH\ + BC2 LAPC | - LCQC,, vA ta cci : :2. o:+2AEz , t/E, C,P AC, i -, (;|+oZ = 2AE2 r2t : c'-'I ( crQ cr -. I Tr] t2t va ( I ) ta cci : Tit cdc drrdng phnn giSc trong CtQ ,CtN A',y',2 +y',C',2 +C',A',2 =' t*)':+ ta lai co : AQ AC, BCT BN suy ra dpcm" QC C,C C,C NC' o Trudng hgp d c6t canh LABC x6t trrong ttr. n6n QN ll AB. Trtang tu, ta cflng cci PM ll AB Nhfn x6t C,P C,M vd suY 1) Nhi6u ban giAi bdi ndy phAi dnirg ddn '" , s c, 0,, tzl lrrong gi5c. Ban Dito Phuong Bac d6. giAi bdi ndy kh6ng cdn ding cdn thrlc vi lrrong gi:ic. K51 hgp (2) vdi (1), ta c6 2. Giei tdt bei nay cci c6c ban : C,M AC, C,B N Hdn niIa, Cr. = C,O n6n Bili Dang Quang, C2 chuy6n Tam DAo, c,N crc ctc Vlnh Phti ; Trd.n Qudc Tud.n, 10T PTNK Hai LBC.M-:- L, CCIN, suy ra Mr: Nr. {a *" -: Hrrng ; Doitn Mqnh Hd,9A THCS Ti6n Lang, gdc driy oia L,BMN tudng fng bu vdi M,N1 DQng Anh Tud.n,gCT PTTH Trdn Phri, HAi n6n bing nhau, hay LBMN cdn. Vdy, BM:BN.
Nhgn x6t : cci 103 bai gi6i, d6u giAi dring. BDT (1) trong c6ch 1 cci thd cht'rng minh b6ng Ldi giai t6t g6m c6 : Trd.n Long Trgng (7 to6n, cdch sit dung BDT Bunhiacopski NK Birn Son, Thanh Hda) ; L€ Quang Nd.nt \-@j-?-+ @ +,fy < (9T chuy6n Drlc Phd, QuSng Ngai) ; Mai Th.i {Zz+62 +t[7Taz Thiy (98 - NK Nga Son. Thanh Hda) , N guydn uANr; rrirNr; 'ulANr; Thinh (9T Phan B6i ChAu, NghO An), Nguydn B,di T7l2l7 : Cho da,y so Tu,d,ru Anlt (9 to6n, Chuy6n Phan Chu Trinh, {rn} duoc xac x; BMT Dak Lak), D6 Minh Chdu (9T, trtidng chuy6n D6ng Anh, Ha NOi). dinh bdi xt : 7, rrt -t: lgg4+rn udi ntoi DAN(; VIIIN n21. . ,Jt X\ .(,, Bai T6/217. Ching minh riing ndu. x, y, z Tint lit't't {\'l- +-x.t+... + x,,'ll lit cac sd tltuc d\i nfit klrtc nhou thi -- ) ,,...-. l, -yl l.v - "l ,ffTf Ldi giai : Tt hO thtlc xdc dinh ddy {x"} ta cci*: f1 fu, {-rTrz {T+zz l* -"1 xi tia,t (lrrj.) t :i'-,' = 1,t- .rk,r-rr: {l-+lr fl +7 rqo/Vk>le 9941 r- ,,' I -,t,- -'lu i . Ldi gini : C6ch 1 (Ctia ban Nguydn Ngoc Hd' l1A Mc Linh - Vinh Phti) Dlph Trung , 1994 ( - 1 1 , ..t(!r.{* Yk*> I - -,? t -:- ),, l,' rtt 1+r4" lt*s, l" 'tq l'- - 'L-. ; ,_) A- ^i'+l :!-+'\ -+, ^rwt, rL'.':'+ ,t 7r_ Hang (11M Mari Quyri, Ha NOi) t-*'\.+,.^Ftfrq r4 -,L Bdt d&ng thtlc tttong dttong v6i : xr .x2 Suy ra : fr, \.'t " ,ln n'u t:r\ \[6-iYlI +4 + fi/ -z)zG+E > +-+. +-:1994.(---l ,l 1 ,, i1 ii*,, > ,[@=iflt4 y\ .r, r.l l,r _l \ Jl xrt ,l /_ \t@ *tf - + @; -tf + t[1y -if +@y -::iS I > t[@ -Zf 1xy -yZf . tt) + 1ee4 ( 1 \ --xn+l/ ) TrOn mat phing toa dO D6 cric ta ldy cac Mdt kh:ic, cung til h6 thtlc xac dinh d5y {rn} didnr A(r, yz), BQ, xz) ud. C(2, xy). Ba didrn niy d6 thdy l:rr AC *-(l) o.= +o, or, : 0. MAu thuin vila nhdn 1gg4 Cach 2 (Cira ban Nguydn Tian Hba,7T Bim duoc cho thdy day tnng {rn} kh6ng bi chan Son, Coo Th.d Anh 10CT Qudc hoc Huq l/gd tr6n, vd do dd limr,, : * co. Sry ra Ditc DtLy 12CT Hai Phdng) "-* v'tt'>' 'lrtt'>' ' (.Phan Linh,9AChuy6n Ngtt DHSP He NOi) ' : o Qt' ,t\zN 'fu, t,7l' Dat x : tga, y : tg$, z : tg/ vdi a, p, y e ,,'1':r,, . \-;, r.rti dri ta thdy cdu chting minh Tt ( ra 1) vit (.2) suy *-l '-:. + - 't\ ll * r f ) ,xr ]2 r ==2 14 t\ Itsa -tgpl - teyl Its{J f limll+j+..,+l)ra rl ,,*-tJ2 J.l x,, l1/ lltlt" l,;. 4* \tt=iAz; \fTiiz4 \fr=jiz| ttr- +isT - tgyl 'rl -r' x,, 'ibat Itgc, lirn I - +- +... +- : r.r+r/\ .- 1994 ,,--\*2 x3 . \nft{; rft-Ti*i -"a'' A- lsin(a-y)[ Nh4n x6t : 1. Hdu hdt cdc ban ggi ldi giei Thdt vdy lsin(a - F)l + lsin(p - /)l > t6i Tba soan d6u giAi bni to6n theo cdch cira Idi giAi tr6n. KhOng it ban cho ldi gi6i thiSu lsin(,r -p)cos(p - y)l + lsin(f - y)cos(,r -p)i chinh xac, do cdc b+n da vQi k6t luAn > lsin(a - p)cos(p - y) + sin(p - y)cos(
,)1. \ ' _i'l riu(l'1' va 11Cl' PT'l'IINK Triin l']hri. IJrii a,,. (iltta.,\ +n,i: {T'to,,\Z 1)irirngt ; Argr.t,drr Mitt.h T'triirt t 12A, l"l'T'lJ Chrr Varr r\rr. IJa N()ii. l)ltott Lrn.h, Nguvdrz Sr e+llrf * 1 = 4[tt n:7 Illtottg, Nguvdn Vft lltng (9A, 11D PT Chuy6n = rr.,,rr o,, € R, o(, t* 0. VAy P(r:l NgrI DIISPNN Ha N6i), ?riln Tiih D{tn,g tl0T NhAn x6t r[ng da thtlc bac nhdt ndy th6a I"I'TIJ Arrrst,errdaur I{a NOi), DQttg Th.orult I'Iit m[n di6u kiOn (1'). (12C,T DHS]']I{Nl), Lttong Tu,d,n Anlt \\24 PI'Cl'l DHTI-I H:\ Noi) ; Ditth Trung I'Iang Kdt luAn : (11M PTDL lVlarie Curie l{ri NQi) ; LA Vort P(x) = o:r, n. € R. Monh. (12CT PTT]I lloing Vin Thu, Hira Binh) ; Nguyln Quong Ngttyr;rt (10A l']TTH Nhan x6t : C:ir: ban sau dny cci ldi giAi tdt : Ngtry6n Htr€, Hd T6.y) ; Nguydn. Ngoc HtttLS, Lttong Tu.rin Anh., Vft l)t?c Minh, Dinh L0 Min.h. Hidu, Plurnt. Minh Turtrr (10T, 11T. Trung Ha.ng (Hn Neri) Vu Dtrv Tudtt (Long, 12T Lanr Scrn, Thanh Hriat ; Nguydn Xud.n Son Giang, Hn B6c, Llod.ng Xtd'n Bach, Tntong (10T Phan Bdi ChAu, Nghe An) :T'ruong Vilt.lt Xudn Thutt,g, Ngttydn Xtt.d.n Son (Phan Bdi Lan, Pho,n Duy Hitng (10CT, 1lCT Dho Duy Chiu, Ngh6 An), Trd.n C6ng Cudng ("fhdi Tit, Quing Binh) ; Cao Tlf Anh., LA Minh Binh); Tr1.n Anh T'ud.n, Nguydn Van Duy (LO Tntong, Nguydn Tlti Hcii Ydn, La Anh Vu Qtry DOn. Dd Nang); Vd Qu6c Hilng, Nguydn (10CT. l1As. l1CT. 12CT Qudc hoc Hue) ; ihi ri.nt, (Le Khi5t, Quing Ngai); Phant Dinh Trinh Ai Qudc (12T Luong Vzrn Ch6nh, Tuy Trudng (Trdn Phu, Hei Phbng); La Minh Hidu IIba, Phti Y6n) ; La Qua.ng Ndnr (10CT DHTH (L4ng Son, Thanh i{cia) ; Nguydn Th.i Hd.i Ydn, Thdnh ph6 H6 Chi Minh) ; Ngttydn NhQ.t Nant Cao Tltd Anlz (QH lIuO ; Truon g Trqng Khanh ( 1 1A., Trttdng Vfing Tiu, Bd Ria - Vung Tdu) ; (1lCT Vinh Lac. Vlnh Phri) ; Nguydn Giang Vo Chi Hda (11CT PTTH Bu0n Ma ThuQt, NguyAn (Nguy6n Hu6, Hir Tdy) Daklak) ; Lang Ld.nt Huy Hoirng (12A1 PTTH NT;TIVr,N VAN MAI] tsac Li6u. Minh Hhi r Biri Tgl217 Cho tant gid.c ABC co []. Ban Luong Tudn Anh (12A DHTH Hn AB +bl : 0, NQi) da dd xudt vh giAi dirng Bii toan sau : i, tgt . ru nghianr cito, x'2 a.( tg + "()ho lmi sd thl1c o., b tltoo, Dlotl, o.b > 0. Clrc r/4.r, { .r,..,J tluoc xo.c d,irt.h boi xt : a , BC ryt , tg n td nshidnt ctio. x2 * as * b:: 0 uii, 11 r: 'Yi, CA * I l.it nshionr cio. 12 o..r 6, = $ tt t t ,, tg ,tS + h 11.-_;71.\r+l i Cluing ntinh rii.ttg tant. gto.c ABC diu nriu : N(iriyr^.N rt r.ic H,ttNlt (1 -or +bt)(1 -oz* br)(l -n., *bi) = Ildi"I81267 Tim tdt t'ti co,c cl.o. tht?c P(x) tlrn ntan d.ihtr ltiirt = 5616 - 3240f3 . Ldi giai. Theo giA thidt. ta cci : P1.r 1P(v) - P, ( ' j,' ) - Pr (lr ) rr+o,.r*b, = (*-retl (..'rsBZ), Vr,.y € R. ('') Ldi giii (Ngutdtt Tudn Dftttg, L6 Qulr f)On, DA Ning, Vft Duy Tud.n, Httong Son, Lang rr+o,r*b,: (*-tS?, (, -ry|) ;r1r Giang, Hd Biic) NhAn x6t rnng Prr) : 0 th6a rnan c1i6,u ki6n r:irzr b.ri to6n. rz +og *1,. = (x -t79n, ? -ret) , Cho x : -1, ta dtldc X6t trtldng hqp P(r) f O. tiray r = 0. y :0 : vio ('k) taduocP(0) = 0. Vdiv: 3x thi til ('r), (1 -or +6r)(1 -a2*b)(l - n-, *b-,) = t.hu duoc : P(.r) P(3t; = Pz(?^x\ - Pz(-*l :( r * rrt)' (, *,ru)' (, * rtt)' hay P(r) P(3r) + Pz(--:r) = Pz(Zr;), Vr € R (**) A B C. -) P(x) = c Yx (c * 0) kh6ng th6a rnan (**) = L, ( t +tsztczrsz ) l- ; tz) X6t tnrdng hqp dy P(x) : n > l. Goi hO s6 A ,fr B+C, cao nhdt ctta Pk) Id o.,, (n,, ?. 0) thi tt] (**), ta (Suyttttsz ts\i- 4 ) cci :
ABC l-tg B('BC 4,8 4 tS 4-tS i AT}C thay tg4 , trV , tS4 bdi. tS;; ,tSg ,t8 r; BC:13C (.n >- 2). Ban An da thu drtoc kdJ quzi sau dd-v : l-tC 4tg i+te-i+tg 4 Tam gi6c ABC l;.1d6u khi vi chi khi : ,,lBCARBC (1 -or +Dt)(1 -0t *b.)(1 -o, +b,): Tt:1g4+tS ++tgZ+tgZtg 4*tS atgZ+ CA ABC = ll+le-i-\" B.Zil t *tSZtgi= 1+r.g 4{.3) 4tSitg Khi z = 2, tn thu cltroc k6,t quA nhr.t d[ n6u tip dung BDT C6si d6i vdi ba sd. ta dttoc : trong bai lorin d trdn, N(,1 IYI':N I).\N(' l'l L\ i ABC .A B C L+tg 1tg 4 'g Z'- 3 tsZtsZrsi + Bni 'f 10/217 Tti ctian ABCD c6 d.6 rtd.i coc 3 , ,.*,A ts B t.q -iC,2 d.uitng cao lir h,(i : 1,2,3,4t. Cd.c. ntot plto.rt. +3 gidc ctta cac nhi cliin ( cfto tft d.iAn l cct.t cd.c r'o.n lt. ( i ,) d,di tuong t?ng ri ca.c d.i.inr. E,ti, =1,2, 3. 4.5,6). -AB(' Goi x, l.d. hltocin.g caclt tit E, t7Ait. lrui nit,t bin. Datr':tgTtsitg Athi 0 < x < I vi.ta lthdng chtio. 8,. Ching ntinh. : drigc : *x3 > 3x*3xl+ 884 Jl I rr. 1 x2 -4*+ 1>0
11 ,1 I . I ] r t ,.1 t,: Bdi1,71217. Hoi qud co.u,lt,hdi l,uong nt,uit E)-* (;u5. ' i \,!,7;,) -L - -l---L m ) d.toc n6i uoi nlnu bdttg nfit lit xo co cltiitt -l'iI -I:lr \ /,, )- rldi ttt nli1n I.d I uo. d,6 ctng lit k. HQ udt dtroc 7, 16 ,l 128 dQi bAn. nfit nt a.t bitlt ndnt n gang ud nltd.n. Ngttdi -2\ +) { t;a kao d,ott lit xo biing r:ach keo hai qud cdu ra xa nhou, so.tt d6 thd dbng thiti lmi qud. cd.u.. Hav 2 t,, 1)n,;' xdc dittlt td.n sd do.o ddng cila hQ. r-=l r=l vi h:ri hO tht'ic trtdng tu COng ve d6i ve ba he I{udng d6n giii. Goi d0 dan cita ld xo li t.hitc niv, ta thu dtioc Bf)T sau : r, x6t hlc t6c dung l6n rn6i quA cdu vd 6p dtrng dinh lu6t II Niuton, rtit ra dtloc phttong trinh ._ r l2E 384 ("i phdn) cta dao dQng ; til dd suy ra )... rll .*- = { -(3r i-r'i ()n,')t ()n,)' 'I tl '' u)l: u)2= l=r .r ! Ttl 12) vn (3) ta duoc bdt ding thtlc k6p ('k) Nhfn x6t. li,i giei ggn vir dr.ing C6c eur cd : r:dn cht'tng minh. Nguydn Quang Tntong 11CL, trttdng Phan Dau d6ng thrlc trong ('k) xAy ra khi vd chi B6i Chdu, Vinh (Ngho Arr), Pltont Qttang Htrng khi /2 I -lta lt.': hr -Sr = S:. = S. - S*
cAc rAr roAN THr Hgc srNH Gror roAru , THPT ToAN eucic xAM tees (BAng B) I. Dd thi : Vi v4y : c N6u k < -R2 thi qui tich cdn Ngd.y th.n nhd.t (02-3-1995) : tim ln tQp didm r6ng. Bhi 1 : Cho dtto-ng trbn tAm / brln kinh R > 0 o Ndu k = -R2 thi qui tich cdn tirn ln t{p g6m vi m6t didm A cd dinh tr6n drrdng trdn. X6t duy nhdt didm J. cric ddy cungBC cua drrdngtrbn th6l mdn di6u . Ndu k > -P2 thi : (1) ki6n : | .Ul nam trong drrdng trbn (1 , R) enz + e0 - A0 = h, v6i ft ld s6 cho trrldc. *.1 1- vdi r =;tP'.--+d Tim quV tich cric truns didm M cria BC (bi6n -)tut € drrdngtrbn (J, .R) , luan 6iitr dang qui tictr theo k vit R). (Tt dd dO thdy : & Bai 2 : GiAi phrrong trinh : ,* znt-llx+21-3 V4r-4 =0 + Ndu k >- 8R2 (+r 2|,*, ,ni qui tich cdn B}ri 3 : Diy s6 {o) , n € N, dtroc xrlc dinh tim ld tAp didm r5ng. nhrl sau i eo = 2 vd, ar, + 1: 5a, +rtTGI=66 : vdi mgi rL O, 1,2 ... +N6u 0 < ft < sn2 t"=* . ". ].R) thi qui 1,) Tim cOng thrlc cta sd hang tdng qu6t on theo z. tich cdn tim Id cung trbn fif'"riuiudng trbn 2) Chtlng minh rang : an2 2.5n vdi moi z € N. (J, r) b6 di hai didm E, F ; d day : E, F li cric Ngdy thtl hai (03-3-1995) : giao didm cta drrdng trbn (J, r) v6i drrdng trbn (1, R) cbn D ld giao didm cria drrdng trdn (J, r) BAi 4 : Cho s6 nguy6n n v6i2000 < n < 2095. n 1 vdi tia AL Dato=I ; n*l k: teeS" "e6=-* * Ndu h = 0 (
D6 thdy, day { o,,} Id dey tang th{c sU. Vi vAy, . 1995 , nt * 1. lees lit t5), (E) suy ra Vn e N ta lu6n cd f lrr*r) = (1* tgsb ),,r+t 2 . 5k - 1, trong rn6i sd d6u ccj hai chit sd I drlng canh t nhau. Khi dd : nghia lA.ta c0ng c_o bdt ding thdc cdn chrlng lnlnhvdln=k*L. 55 VAy, theo nguy6n Ii qui nap, a,, >- 2. s = lA \ uAlr : lAl - luA,l = 5/' Vn € N. i:l 5l=l Bei 4 : DAtn : 1995 *m.,v6i 5 < m. < 100. Khi do : lal -) la,l +)lAi,-n o,,l -) lAi,nAr,nA1,l l'r95 +/, i=l l
B,di T81221 : Chrlng minh bdt d&ng thrlc sau oay : az+bZ b2+c2 c2+aZ(3.- oza62q"2 \n\\ -------:-* a *b b*c c*a DE RA KI i\AY C6c ldp THCS vdi a, b, c-lA cric sd drfong. a*b*c NGUYEN LE DONG ('r'P Hd Chi Minh) B.di Tll22l : Tim s6 trr nhi6n z ldn nhdt sao Bei T9/221 : Cho O ld didm nim trong tam cho sd .1995 bing tdng cria n s6 a, , a2... , a,1 976,cABC. Crlc dudng thing OA, OB, OC cilt trong dci c6c sd ai(i : t,2,...,n) ddu ld hgp s6. BC, CA, AB ldn lrrgt tai A" B" C',.H.dy tim t6p hgp (qui tich) cac didm O sao cho : NGO HAN (He Bic) (dt LoAc)2 + (dtAoBA')2 + (dt\ocB')z B,di T2l22l : Tim nghi6m nguyOn cria phuong trinh r(1 + x ]-x'l) = 4y A + \ = (dt^oBc',)2 + (dttocA'1z 11616963|12 IRAN DUY HINH (Binh Dinh) TRINH e.i.Nc craNc (rP H6 Chi Minh) B}l;i Tgl22l : Chrlng minh bdt d&ng thrlc : BAi T10/221 : Cho 2 drrdngth6n gd., d.'ch6o .a363c3 -i_:_ nhau. l}1, N la 2 didm ldn lrrot thuQc d vd. d' \ thay ddi sao cho drrdng th&ng MN hqp vdi d b2 -bc -l c2 c2 -ca + a2 a2 -ab tbz vi d' nh[Ing gdc bing nhau. Goi K ld trung ab *bc *ca 3.--___ o *b *c di6"m cria doan MN. Tim tQp hgp didm K. trong dci a, b, c li cric sd duong. o6 rsaNu sON (u) Noi) TRAN XUAN DANG (Nam Ha) C6c dd Vat li B}ri T41221 : Dltng LABC vu6ng d A bidt canh B.ni Lllzzl : MOt v{t A BC vd do dai phdn gi6c AD. kh6i lugng m, vd vAt B kh6i vrj Hou BiNll (Ha Noi) lrrong mrn6i v6i nhau bing m6t lb xo cd kh6i luqng B.diT5l22l: Cho AABC, O ld tdm drrdng trbn kh6ng d6ng kd. Vdt A thuc ndi tidp tam gi6c. M6t drrdng thing I thay ddi hiQn nhfrng dao ddng trr do lu6n di qua O cdt tia CB, cdc c1nh AC, AJ3 ldn vdi bi6n d6 A vd tdn sd gdc luot tai cric didm M, N, P. Chrlng minh ring rr.,. Dd hQ thdng kh6ng t6ch bidu thrlc : kh6i met sin bi6n d6 dao AB AC dQng A phAi ld bao nhi6u ? BC PA.PB 'NA.NCf_ MB,MC NGUYEN VAN MINH (euing Ngai) kh6ng phg thu6c vio vi tri cta l.- Bni LZlzzl : Cho mach di6n nhrr hinh v6 : IIoANG NGOC CANH (t-ta Tinh) Bidt E1- lOV,tr= rz,Ez= 5V,Rr- 2rr. C6c d6 THCB Khi Kr ud,Krd,6ng(1)chi 3,6,4 Bai T6/221 : Cho ba s6 x, y, z nguy€n duong Khi K? ddnss)chi I th6a rn6n x4 + rt * za = 1984. f A. Chrlng minh ring sdA = 20x + 11v - 19962 kh6ng thd le tich cria 2 sd tu nhi€n li6n tidp NGUYEN orc rAN (Tp 116 chiMinh) Biri T7l22l z Cho cdc dey (a,)n e N* , (brr)n e N* th6a m6n cdc c6ng thfc saq : n(l *n\ nn (l an:l*fr +...+#*.nn\ E1, rt I ,Q,, Khi K3 d6ng@r1chi 2,5A, A,lchi 3A. B6 qua b, = ( ltlr E 1;* di6n trd cfia dAv ndi vd amoe kd. Tim"*),r(n+l) 1im bn Hiy tinh R4,rt2 vd cudng dQ qua RrhhiK, dcing. rlb eUANG vtNH (Nghc An) NGUYEN NGHlE,M TOAN (Long An) 10
For Upper Secondary Schools. T6122l The integers x, y, z satisfy the relation x4 +y4 *24 = 1.984. Prove that the number A = 20t + 11} - 1996F can not be the product of two consecutive natural numbers. For Lower Secondary Schools. T7l22l The sequences (ar)ne;v*, (br), e lv* Tll22l Find the greatest natural number re are defined by : such that 1995 is the sum of n composite n(l +n\ nn(l *nn\ numbers. a,,=l+-t+;+...+ L+;;, T2l22l Find integral roots of the equation x(l+x*x21=4y(t+\. c Tgl22l Prove the inequality 6, = ( N*. a3 63 c3 Find Jim br. "*l)n(n+t),Vn -!_ ]-_ bZ -bc +c2 c2 -ca +q.2 aZ -ab +b2 ab *bc * >- 3--------------- ca T8l22l. Prove the inequality a*b-lc - b2+c2 c2+a2 a2+b2r-]-(3 a2+b2+cz where a, b, c are positive numbers. a*b b*c c*a a*b*c' T4l22l Construct the triangle ABC, the angle where a, b, c are positive numbers. - A of which is right, knowini the lensths of EC Tgl22l Find the Iocus of points O inside a given and the angled:bisector AD. triangle ABC such that the lines OA, OE, OC T51221 Let be given a triangle ABC with cut resoectivelv BC. CA. AB at A'. B'. C' so incenter O. A movable line / nassinE throuEh that (aiea LOAC'\Z + (arca LOBA\Z * (area O cuts the semi-line CB and tLe sid6s AC, ,{B LOCB'/ - (a-rea LOBC')2 * (area LOCA')2 + respectively at M and N, P. Prove that the (arca L,OAB')/. expressron Tl0l22l Let d and d' be two given AB !- AC BC non-coplanar lines, M and N be two moving . PA. PB NA . A/C MB.MC points bn d and d' respectively so that the does not depend on the position of l. ingles between MN and d and" d' are equal. Find the locus of the midpoints of MN. H6a ra "& ngoii dutng" dNc r
Nh[.n rit. L! do khi6n : I e',Oz tdng binh phrrong tdt cl c.ic hinh chi6u cria OA,xu6ng d, TAP tIt [lIO HO]'|G 1'{OT BAI TOAN kh6ng phg thuQc vio d li : cdc do4n O.A, bing nhau vA tqo vdi d nhung gdc I{p thdnh cdp s6 TRINH BANG GIANG 2n c6ng c6ng sai (TP Hb Cni uinD -. Trong khi dd cdc canh Bdi to6n md ddu : AtAz; Ait; -..; A,it cflng PhAn 6nh mQt Trong mg phang cho duitng thd.ng d quay "hinh Anh" tudngtu cho n6n tac6 hQ qud. : tdng quanh td.m. O cia hinh uuOng ABCD. Ching blnh phuong hinh chidu crla tdt ct c6c cgnh minh tdng binh phuong cd.c hhod.ng cd.ch tit 4 cria mOt da giSc d6u xu6ng mQt dudng thing d.inh hinh uudrug ddn d chi phu thulc kich bdt kj', chi phu thuQc vio kich thtldc ctia da thudc cila hinh uuilng. gi6c d6u dd md th6i. Crnch giAi quen thuQc (xem h.1) C6c tinh chdt trrong tU nhu tlc (T) d trong Chi6u vu6ng gcic m{t ph&ng cbn nhi6u, xin n6u mQt s6 vdn d6 (A, B, C, D) xu6ng dd c6c ban tim hidu th6m : d thdnh (A', B', C', d Ia dudng thlng quay quanh tim O cta tam D'), efic tam gi6c gidc d6u ABC. d'lA mQt dttdng thing bdt k,. vudng sau dty phAi 1) Chrlng minh : tdng l{y thr}a b6c 4 cria blng nhau : hlnh chi6u tdt cL e6c canh o&a LABC xu6ng LDD',O, LOC'C, LBB'O, LOA'A d', cidrng nhrt cria khoAng crich tr) tdt cA cdc -il'fr + C'c2 + B'82 +A'A2 = ZoA2-(dpcm) dinh cria A.43 C ddn d, chi phg thuoc kich thu6c Cd,u hdi.' nhtng hinh nio cung cci tinh chdt cta LABC- dd (t4m ggi li tinh chdt (7)) gi6ng nhtt hinh 2) Cd thd thay "liiy thrfa 4' bing nhtrng loy vu6ng ? thta nio, de' khing dinh trrong tg v6n cdn dring. 1. Md nONc rRoNG uAr pnAxc 3) Thay LABC bdi da gi6c d6u Aiz. . .4 Dtt do6.n 1.Dag16cdiluArA,)...Ar,(n > 3) kdt quA cd thay ddi kh6ng ? cflng cci tinh chdt (?). z. Md RONG rRoNG xHONc GIAN Chrtng minh. Dtt dod.n 2. Dta theo srl md rQng trong m6t Chi6u vu0ng g6c Ai phing ta nh$n thdy d4c didm quy6t dinh (then xu6ng d thdnh ch6t) dd m6t hinh cd tinh chdt (?) ld c6c doqn Ai(i:1,n), dd don n6i tAm v6i dinh phAi c
StI DUNG PHIICING PHAP VECTO DE PHAT IilEN MQT SO 'IA.rjr. BAT DANG THIIC LIIqNG GLAC VE TAIII GLAC NGIIYfN VAN LQC + ---t ---t OrAr + OtB, - OrCr : 0. X6t LABC fiy !' -+ Trong bni brio niy chring ta tim hidu m6t rlng dung cta phuong phdp vectd, d o. Suy ra : 5R2 + Kh&i, Dio Ngoc Nam, LO Tdt Tdn, D6ng Quan +2t[5 Rzcos2C -2R2cos2B -?r[g R2cos2A > O Vi6n : To6n bdi duong hoc sinh hinh hoc 10. Ha N6i 1994 trang 21". hay {5(cos2A - cos2C) + "orze < f, Dibu ki€n di : Gih stt O, td tdm drrdng trdn ngo4i tidp A.ABC,,theo di6u ki6n cdn ta cci : Ddu "=" xay ra chi khi A : 15o, B= 60o, C: 1050. O rAsin2A * O rBsin2B + A rCsin2C = O (2). Ldy (2) trr) (1) rip dung dinh li him s6 sin vd 4) X6t MrB S, cd A5 454,81 Cg] 50. Theo (1) : 2OrA, +rl3OrBr +O1Cr:0. =j;0o, X6t cosin cho LABC ,RZ ':#:fr ta c6 o,-o ao LABC ---) tty-+- 9 +'nQi ti6p (O) Theo (3) : sasc (2OA +t[SOn + Oq2 > + + O * 0 n6n O.O = O do d6 O : Ort.6cld O hay 8R2 * 4r[5R2cos2C * 4RzcosLB * Rzl -th6a m6n (1) thi O ld tdm'dudng trbn ngoai + 2t[\R2cosZA > o tidp AABC. Tn he thrlc (1) d6 chring minh dtroc Suy ra : ,[5cosLA * Zcos2B + 2{5cos2C > -4 ++ dins thfc sau : bdt Ddu ":" xAy ra chi khi A = 45o, B = 600, (OAsittzAl * OBsirPB, * OCsin2Cr)2 > 0 (g) C :75o. ddu ding thtlc chi cd trong tnrdng hop 5) X6t LAtBrCtcd Ar:8, - 36o, Cr : 108o. LABC ddngdangv1i LAPPt, Srl dungh6thrlc Theo(l): _ + (1) vd bdt d&ng thrlc (3) chring ta lAp dtroc c6c (OrAr+OrBr)sin7P -OrCrsin36o = 0 b6t ding thfc sau : + + ---t -__t hay 2(orA, * orBr)cos36o - orc, = o. 1) Ndu LAPpt:6u, nqtidp trong dtrdng trdn tdm Oi thi OlAt +OtBl +OtCt = O. Khi Do cos36o : 1-{t/S + 1) n6n a ++++ dd ddi vdi L,ABC tny y ngi tidp_jrongdudng (v5 + L)(Oit+OtB) -20rCr= 0. trdn tdm O theo (3) : (OA + OB + OC)2 > O. Ttddtac6: Xdt LABC___>ttwj nQi _.!dp (O), theo (3) : t({5 + l)(oA+oB) -2oc1z >, o, do dd : 2R2$ + 2{5)(1 + cos2C) + 4R2 - cos2A * coslB * cos2C , -; Ddu ,'=,, xf,y ra - 4({5 + 112B2(cos2A * cos2B) > O khi AA-BC d6u. hay : (rl5 + l)(cos2A + cosLB) - _- D Xet LA.,B tC tc6 Ar=B r= 30o, C = L20". r - (3+r[5)cos2C
rEr ouA cuOc rHI ctAl roAN vA vAr ll rnEn rAp cHi roaru Hgc vA TU6r rnE r.rnrvr 1ee4 Vio n6m thf 30 cria minh, tap chi Tod.n hQc 3. GIAI NHI ua tudi tri dttqc d6ng dAo ban doc hrrdng fng l. Phan Hoitng VieL l2A Qudc hoc Quy Nhon, giAi bni. M6i th6ng hdng nghin bni grli d6n Binh Dinh du thi. 2.Vd Hoimg Trung,11 PTTH Chuy6n Trd Vinh Rdt nhi6u ban dE gi6i duoc gdn hdt sd bAi 3. T0 DAng Vil, 11 Chuy6n Dai hoc Tdng hop ra tr6n tap chi. M6t sd ban hgc sinh THCS da He Noi giAi dtroc cric bdi ddnh cho cdp THPT. C6c ban 4. Nguydn Vu Hung, 10C PT Chuy6n Ngo4i hoc sinh l6p 9 giti bAi vd nhi6u nhdt. Ban Pham ngir, DHSPNN Ha Noi Thi Thanh VAn (Hii Phbng) ln ngrrdi it tudi 5. Pham Tud.n Anh,9T Phan Bdi ChAu, nhiit trong sd c6c ban doat gi6i. NghQ An Cu6c thi gini bni v{t li cnng drroc nhi6u ban 6. Pham Thdi Hd,9T Pham Huy Quang, Ddng tham gia. Hung, Thai Binh 7, Phanr LA Hilng,9A Tnrng Nhi, Ha N6i Sau ddy ld danh s6ch c6c ban doat giAi. 8. Nguydn. Ba Hitng, 9A Trung Vuong, Ha NOi 9. Trd.n Nguy€n Ngec, 9K LO Loi, Hd DOng, A. MON TOAN Hd Tay tr0. Nguydn .Dic Phuong, 9H Trtrng Vuong, r. crArxuA'l sic HE N6i t. li AnhVn; 11 Chuydn to6n Qu6c hoc Hu5, 11. Nguydn Nggc Tdn,9M Mari Quyri, HA NOi Thta Thi0n - Hud 12. Trd.n Vi)t Binh, 8H Trung Vuong, Hd Nqi 2. Pha.m Dinh Truitng, 10 Chuy6n to6n Trdn 13. Trd.n Thi Ngqc Hdi, 8"1 Chuy6n LO Khi6t, Phf, Hai Phbng 3. Nguydn Thi HAi Ydn, LO Chuy6n to6n Qudc Qu6ng Ngai 14.ViAn Ngqc Quang, 8E Ba Dinh, Thanh Hda hoc Hu6-, Thrla Thi6n - Hud 15. Doitn Minh Dic, 7 Chuy6n Qulnh Phu, 4. Biti Quang Minh, gAr GiAng V6 II, He N6i Th6i Binh 5. Pham. Huy Tirng, 8A Bd Van Den, Ha NOi 6. Le Quang N6.m, 8T Cliuy6n Drtc Phd, +. crAr ea QuAng Ngai l. Nguydn Thanh HAi, i2 Chuy6n Ddo Duy 7. Nguydn La Lqc, 8A Ddm Doi, Minh HAi. Tr), Quing Binh 2. GIAI NHAT 2. Nguydru ViQt Ki€n, 11 PTTH Nga Son II, Thanh Hria l. Le Huy Kharuh, 12T Phan B6i ChAu, 3. Vu Th.i Bich Hir, 11 PTTH Chuydn Th6i NghQ An Binh 2. Nguydn Tud.n Hd.i, 11M Mari Quyri, He NOi 4.Vu Thd.ntt. Long,11CT Phd th6ngnangkhiSu 3. Phan Hoitng VieL llA Qudc hoc Quy Nhon, HAi Hung Binh Dinh 5. Nguydn Hodng Cing, 10 Chuy6n L0 Khidt, 4. Dinh Th.d.nh Trung, 10 CT Dai hoc Tdng hop Ha NQi QuAng Ngei 6. Nhil Qu!,Tho, 10T Lam Son, Thanh Hda 5. Nguydn Xud.n Thd.ng, 10 CT D6ng Hd, 7. Ph.am Minh Phuong, IOA Chuy6n Dai hoc QuAng Tri Stt ph4m Ha NOi 1 6. Nguydn Ngqc Ddng, 9 Ning khidu Thuin 8. LA Van An, 9T Phan Bdi ChAu, Ngh6 An Thinh, Hd Bic 9. Trdn Hilu Nhon, 9T Chuy6n Nguy6n Binh 7. Nguy6n Ph.i Binh, 9A Bii Van Dd.n, Hit. NQi 8. Trd.n Bd.ng,8H Trttng Vtrong. Ha NQi . Khi6m, Vinh Long 10. Trinh Kim. Chi,8 Ntrng Khidu Hn Tinh 9. Trd.n La Nam, 8T Chuy6n " L6 Khidt, ll. Cao Trd.n Kian, SH Trung Vuong, IIe NOi QuAng Ngai 1.2.Nguy1n l{bng Hit.8H TntngVrrong, Ha Noi 10. Mai Thanh Binh, TM Mari Quiri, Ha NOi 13. Nguydn Long, 8H Trung VrJong, Ha NOi ll. Phq.m Thi Thanlt. Vd.n, 6H Minh Khai, 14: D6 Hb Nga,8D D6ng Da, HA N6i HAi Phbng 15. DQng Anh Tud.n, 7T Chu Ven An, Ha Noi T4
5. GIAI KHT|YEN KHfCH 35. Triln Dtc Quyi2n,9 Trdn DingNinh, Nam Hi l, LA C6ng Son, 12 Vi6t Drlc, Ha NOi 36. Triin Thi Khu€, 98 Quang Trung Thanh H
Tim hidu sAu th6m To6n hoc phd th6ng DUNG PHUCrI\{G PHAP TQA DO Dh GIAI CAC BAI TOAN HiNH NcuvfN pur pHric (Gid.o ui*n THPT Phuong Ldm - Dbng Nai) V6i nhifu bei to6n hinh hQc c6 chrra y6u tfi "Khoing c1ch", Do M E BC+ CM cing Phudngvdi t "CUng phuong' vd d5c biCJ le y6u t6_"vu6ng goc". n6u kheo laM.oo AMsinz -cll:0 chqn hQ truc tea dO thi c6 th6i chuy€n drlQc th2rnh bii todn CB+l D -c I I ctai sd c6 nhi6u hita hgn cho khA ndng tim ra tdi giAi. D6 la =* Av1. cosa + bsina) J 6s f trtdng sit dung phudng phap tqa dQ d6 gi6i to6n. bc Trong biri vidt ndy xin trinh bdy viQc sit dgng "Phudng +AM: .:(dcpcm) ccosa + bsltu ph6p taa
:(2R\)r + (3Rr - Rx - RVJy)r * (.tRr - l{x + ltVJv)2 (1) ver (2) : + AC2 + BCl2 : 4R2 (dpcrn) TLlr -',,nr*'+ t,R2r'l + ltiRt - l2Rlx Bii 6 : (De rhi rozrn qu6c tiila'n thit23) TrCn c6c drlilng ch6o AC vi CE cira luc giSc cldu ABCDEF. - r,Rl rxl + vli+ IxRa - l2RJx : r,Rr.2Rx.{- lsR4 - l2R'x : lxRa sao.h"# : H VAv giir tri MAa + MB4 + MCa khong phu thu6c vito vi tri M. ta lziy hai di6m Mvir N =* Bni 4 : (De rhi v6 dich Anh - niurr l9R3) Bidt ring B. M, N thing hirng. Tim k. Cho AABC. Coi I la tirr) (lr(dng lri)n ngo:.ri tiip AABC. D la Lli giii : Dar 2R l:) c16 clii blrn kinh clrrirng trt)n ngoai tidp rnrng tlieur .rnh AB. E li trong"tant criianCD. Chrlng nrinh luc gihc ddu ABCDEF. Chon h€ truc nhtl hinh v6. ranglnerr n B : AC thr lE vtt,ing gtic vrti CD. Khi c16 A(0, 0), B( - R, v3R), c(0. 2v3R), E(3R. v3R). Ldi gi6i: Chon h6 trlrc nhll hinh v6 (O li trung di6m BC) M(0, m), N(x*v) viti l,5R < m < 2j3R 0). Ta cri : MN cilng phlldng v6i BM n ,,,'-r:nl : u .x \1-tn I a):0 ,I .r t' - Zrf:R,t:o I -.yr V:n | ' (nr-V:R)xt3Ry: -urR x+VJY:oR 8 trR- - V3mR " t'3nr - 2R TntR - rrV3R' \';rvl /rt. ^ l 1l v: V3nr - 2l{ o \-r. r,Rl y'Jr:rR Trnll r,y'JI{2, N , lL. D( = i-. CCJCJ VAY \ y'lrrr-lR V.lrrr-2R l - \(, -ri ^-. --l2) Zr) \2' Theo bai to6n ta c6 : + IE r DC (ttpcm) AM : ('N Ltru i' : (*) : C6ng thilc toa trong tim tam gilrc. : CIN (doAC : CE) rla) AC CE:AM Xer-n bii tAp 4 trang 33 SGK FIII10. eAM2 - CNI Bai 5 : (Dd thi v6 rlich Bungari nhrr 19lil) , t,R' - ./-l nrR , ?rilR - ('V3 R- Drlrlng phAn gi,]c trong vir ngo:ri cita gric C cira AABC €rrr-= t uL.-r* ) * i 1r"., -r* - 2v3Rl- cirl eltlrlng thing AF r'r L v:) M. C--hitng nrinh ring ndrr CL- : e 3nr' -lV.lRrrr' + l(,VjR'.nl - zl8R' : 0 CM thi AC' r B('- =-ll{- (R l;r h:in krnh rltlrtng lr,,n nlo;ri tidp AABC) e(n) - lR)(3nr' + ((, - 4r/.])Rnf ' 112 - HV3)R'rrr - 2lR l :0e11 : lP. Ldi gini : Ndu CL : CM thi ACIvIL vr-rrlng c*n (r,lo CL r AMm2R1 CM theo tinh chdt hai duitng phin gi6c crira gric) vAvk: AC :2.r3R:2.f3R :15 Chon h€ trLIC nhll hinh ve (O li tnrng rli€m cita ML) Khi dri: O(0.0). A(a.0). B(b.0). C(0. c). L(c. {)). M(-c.0) Drr f(n, : 3nrr + (6 - 4V:)Rnr2 + (12 - 8{3)R2,r', + 24R3. Theo tinh chdt riuclng phAn gi5c ta cri : Ta c6 : rBR < m2V3R+ (n+ R)2 < (2,{1 + 1)2R2 AL AC ALT Ai LB CB I.Br CBi . +-,-,.2 - 2R,r, + 12,r5;2R2 > O+ (6 - 4v3)Rm2 + 1tZ - 8v3;R2m + 24R3 > (r*o)- a-*c- > 96R3 - 48]3R3 > 0+f(m) > 0 tb - cr- h- 'f c- e ah- + ac- - a-b - c-b D6 tdt ttu-'rc, xin rniti c6c ban giii r:r6t s6 biri tohn satr e (a - b)(c- - ab) : 1; bing phr-tctng ph6p toa c16. C 1 Bdi I : Cho A ABC cAn tai A, BC : a. cfutilng cao AFI : eb: h. Goi K li hinh chidu c[ra B l6n canh AC. Tir-n di6u ki6n giita -a ,C z a. h dd K thu6c cantr AC, khi 116 tinh AK. V:lv B r)) nen li \, Bdi 2 : Goi M trong tam c[ra AABC. Chitng rninh ring 4ll AB2+ BC2+ CA2:3(Ma2+ MB2+ Mc2.1 Acr+ BC2:1al + cl; + /) + L : l-l-(1)-d I L - ) l" -\ :t't L a +++Bdi 3 : Cho AABC. Tim taD hdD cti6m M sao chcr -l AM.AB: AC AB G..ri I(x, v) h t Am rlttctng r
rAP Mo RQNG MOT BAI roAN (Tidp ttreo trang 12) Ta chi cdn chtlng minh ) AiO) ld hang sd, tt dri suy ra du do5n 2. Girii d(tp bdi Ta cci : + 1 oAt:(; -2 -r);oA.:l-; 1 2.-2) ,1 1 1. - 1,. Tro cho'i thay sd \L L + l. + I 1 l oA,: l-ZI -rI r):oAr: lZ z Z) ' Gii sir ngudi di tnldc mudn phriong trinh a*3+b"2*cx*1995:0 11 cci hai nghidm nguy6n la m va -m (rn€N+). Khi thay x-= in vd1": -m vdo phtiong trinh = A',o2: ) fi oA)t :.rr +yr * / = ) 1 ta s6 drtoc i t i:l arns+bm2+cm+1995:0 (1) NhQn rdf. Goi S, la di6n tich mat d6i di6n -am3*bm2-cm*1995:0 (2) v6i A;trong tf di6n ai I rtitrab=- 1995 A,A14At. Tr6n t Tii (1) vd (2) ta , (3) OA, chon didm M, A r,-7 ' /l ,! 111" { tri cira b d (3) vio (1) ta cci Thay gi6 sao cho OM, : 9,. am3+crn:0 Goi S; ld di6n tich hayam2+c:0(4) hinh chi6u ctia S, Hd thLtc (4) cho phep ta lim dtioc rrrol trong xu6ng mdt ph&ng a hai so a va c khi bidt rrr6t so. Tri dd suv ra cach (P) b6t ky. cci phAp thay sd ctia ngrrdi di trtrdc dArn bAo ho"so th6ng vecteur don vi lA c. \. cuoc nhu sau : Khidd: + Nguiii di trtr6c thay b theo (3), tiong dci rr s; :sr f cos(e"oM,)l - : I e. OMil Id m6t s6 nguy6n tiry chon. Sau khi nsudi di sau thav a hoac c boi rnol -:l 1 so ttv chon t6i nsudi di trtr6c thav chti con lai bang ur6t. sO sao-cho (4) dridc th6a man. Khi -)t:l s;' : t-) l fi tfu,t: tcing nay kh6ng phu dci phttong trinh se cci hai nghi6m nguy6n phdn thtr6c e (theo chtlng minh dtt dodn.2') do dcj ta biet le rn vA -rn. cci : "Tdng binh phuong di6n tich hinh chiSu 4 Thi clu. .' vdi m : 1 ta cd : mat cfra ttl di6n ddu xu6ng mat phing bdt k]', - Ngudi di trudc thay b : -1995 chi phu thu6c kich thu6c tt1 di6n dci". - Ngudi di sau thay a : rr (mQt sd tiry chon) Qua dA.y ch6c c6c ban cung d6 ddng y voi - Ddn lttot ngttdi di trtidc thay c : -