Tạp chí Toán học và Tuổi trẻ: Số 226 (Tháng 4/1996)

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Suy nghỉ về hai hằng đẳng thức, đề thi chọn học sinh giỏi Thừa Thiên Huế, suy nghỉ về mở rộng một bài toán, cắt hình chữ nhật,... là những bài viết trong "Tạp chí Toán học và Tuổi trẻ: Số 226" ra tháng 4/1996. Mời các bạn cùng tham khảo.

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 226 (Tháng 4/1996)

\6 \o[)tt_ eo crAo DUc vA DAo rAo * ner roAN Hec vlFr NAM rap cni RA NGAY 1b IIANG TTTANG * soy l{Gui vE ner niua oina rndc * I r^\?r^2 DE THI CHON HOC SINH GIOI THUA THIEN - HUE * suv NGHI vi vr6 nOrue MOT sAt roAr.l * cXr nixn cnCr xnAr * odtr cte 116r xcuor i,l t I t t Thfu;, trd Chuy€n toan DHSP Hd N)i I trudc mia thi 1996
ToAN Hoc vA rueit rRE MATHE,MATICS AND YOI.JTH MUC LUC Ddnh cho cdc ban Trung hoc co s& For lower secondary school leuel friends Hod.ng Nggc Cdnh - S.ty nghi vd hai hang ding thrlc 1 Tdng bi€n fip : . NGTJYE,N CANI I TOAN o Dd thi chon hoc sinh gi6i tinh Thrla Thi6n - Hud 2 Ph6 td ng biAn fip : o Gidi bdi ki twdc NcO oar rll Solutions of problents in preuious issue HOANG (]IIUNC C6c bdi cias6 222. 3 Db ru ki niy Problents in this issue nOr odttc etEN rAp : TU226 ... Tt01226, Ltl226, L21226... 10 6ng kinh crii ctich day vd hoc todn Nguy6n CAnh Todn, Hodng Kaleidoscope : reform of Maths teaching Chung, NgO Dat Tr1, L6 Kh6c and learning 86o, Nguy6n Huy Doan, Phan Thanh Quang - Nguydn Dlc Tdn - Nguy5n Vi6t Hai, Dinh Quang 1 HAo, Nguy6n Xudn Huy, Phan !r-3=Ol' 11 Huy KhAi, Vrl Thanh Khidt, Le Dd thi tuydn sinh vdo dai hoc nam 1995 - 1996 Hai Kh6i, Nguy6n Ven Mau, tnldng Dai hoc xAy dung Hoing LO Minh, Nguy6n Kh6c 12 Minh, Trdn Ven Nhung, Ddnh cho cdc ban chudn bi thi vdo Dai hoc Nguy6n Dang Phdt, Phan For college and uniuersity entraruce q,o.nt preparers. Thanh Quang, Ta H6ng Doitn Quang Minh - V6 m6t bai toan nh6. 14 QuAng, DAng Hung Thing, Vrr Tim hidu sdu th€m torin hoc phd thdng Duong Thuy, Trdn Thdnh To help young friends gain better understandirug in Trai, L6 85 Khdnh Trinh, NgO school Maths. Vi6t Trung. Dang Quan Vi6n. Le Qu6c Hd.n- Suynghi vd mdr^6ngmOtbdi torin 16 o Gidi tri todn hoc. Fun with Mathentatics Binh Phuong - Gidi ddp biri CAt hinh chir nhit Bia 4 Ngdn llb - Didm cua m5i ngudi Tru sd tba soan : 458 Hhng Chudi, Ha NQi D?: 8213?86 Bi€n tQp uit tri su: VU Xllt THIfY 231 Nguy6n VEn Cir, TP tld Chi Minh DT: 8356111 Trinh b&y : TRQNG THISP
c, L C,; n"i ningding thrlc rdt-quen thudc v6i oic ban hoc sinh gi6i to6n, chring dugc drra vdo trong chrrong trinh phd thOng nhrr ld mdt bai tap, d
t, Dii THI cHoN Hoc sINH GIoI TINH rntre rHmN - sud xAM HQC 19es - tss6 wION THI : toAN rz v0Nc r L\-r 111 I : (180 phrlt, kh6ng kd thdi gian giao db) "' 1t=ir3A.sinB 1+sin3B.sinc' 1+sin3c.siM- B}ril:(4didm) 111 L- f- = I +sinaA' 1 +sin4B 1 *sinac GiAi phuong trinh : logrrrr(tgr) = cos2r. Bbi2:(4didm) BAi6:(6didm) Tim tdt ca gi6 t4 nz dd h6 bdt phuong trinh : Cho trl diQn ABCD c
*s2s2162 _e _6oz16z -e - - 13162 - o) + bc(bz - a) = O e(bz - a)l(azc2 -baz) - k3 - bc)l = 0 *{bz -4@2 -b)(oz -c):0 Tt'dd suy ra mQt trong 3 s6 o, b, c phAi ld binh phrrong crla mQt sd htu ti.' Bdi'tU222. Nhan x6t. C
Nhin x6t. C6c ban d6u giAi drlng. Cric b4n sau Ttnh ; Nguydn Huy Cuimg CT DHSP Vinh, Lc day ui ldi giAi tdt : Vfi ?ry (9 Nang khidu, Phir Ti6n Thi Td.m 9A PTTH Hung DOng, Phan Tharuh HAi Hung) ; Tftn Thi Bich Lien (8T Chuy6n L6 Trung, ST Qurnn Hdnh, Nghi L6c, NghQ An,.L€ Khidt, QuAng Ngai) ; Nguydn Nhu Chudn (8 Nang Hodng Duong, Nguydn Th! Nhung, gT NK Bim khidu, Thuan Thanh, He B6c) ; Phan D*c Phnng Son, Dd lhi An,9A NK D6ng Son, Thanh Hria, (11& PTTH Cam 16, QuAng Tn) ; Phqm Hoimg Phgm Dinh Qudc Hung, Trd.n Drtc Thinh,7T, Anh (9 Chuy6n V - T - TT - Ha Tay). Trd.n Minh Tod.n, N guydn Tro. ng Ki€n, Vit. Trdn DANGVI6N Cuong, PhqmThu Giang, TrdnDinhHitng 8T, BniT4l222 : Cho tam gid.b ABC udi didm M Trdn Ngoc Anh, Mai Nggc Kha, Vit Thily Nhu, 6 bdn ffong. Ggi I, J, K theo thit tU ld giao didm 9T Trdn Ding Ninh, Nam Dinh, Phqm Anh crta cd.c tia AM, BM, CM udi cdc canh AB, BC, Tud.n 98, Truong COng Luong, 9A Thanh Lrru, CA. Du?rng thd.ng qua M ud. song song udi BC Thanh Li6m, Bili fhiVd.n Anh, Phqm Thi Mai cdt IK, IJ tqi cdc didm. tuong ilng E, F. Chrtng Thdo, Biri Thi Khd.nh ThuQn, g NK'i YOrr, minh rdng ME = MF. Nam Hi, Trd.n Vd.n Hd, 7T NK Kidn Xrrong Ldi giai : Theo dinh li Ta l6t ta c6 : Trh.n Cuitng 8T Ph4m Huy Quang Cao Thi Ly MP IB ME CI MQ BC T9 NK Vu Thu, Thrii Binh, Ds.tlg Trd.n Dung 8AI Nguy6n Trdi, Nguydn Thu Hit.,9K Le Lgi, MQ -: IC' MP CB' MF BI HiD6ng, Pham.HoirngAnh, 9 Chuy6n Thtrdng Tin, Nguydn Hd. Duy,9T Chuy6n Phri Xuy6n, Hi T6y, VtL Thi ThuQn 9T PTNK Th! xe I16i Dtrong, HAi Hrrng Dod.n Huong Giang, 9A THCS Nfi Ddi, Kidn Thuy, Nguydn Minh KhOi, Trd.n Huy Hoitng, ST Chu Ven An, Nguydn Thanh Vdn, 9Al, Trlln Quang 8A,, Hdng Bdng, Tq Thitnh Dinh, gT Trdn Phf; Hei Phdng, Nguydn Minh Hidu, T8 NK B6e Ninh, Phing Drlc Dilng, 9T NK Bdc Giang, DQng Dinh Hanh, gCT NK ThuQn Thdn h, Hd BdE, Nguydn Hoitng Lam, 8AL Chri Van An, Phgm Quang Suy ra : Vinh,9A, Triin Thanh Vinh 8A Bd Ven Dan, MP ME MQ IB CI BC Pham Tudn Anh, 8A Lriong Thd Vinh, Khudt MA' MP' MF: IC CB BI Minh Duc,7H Trung Vuong, Nguydn llbng ME Quang, TD Nguy6n Du II, Nguydru ViQt Hirng, lhay ME = MF (dpcm). 9CT Chuy6n Ti Li6m, Nguydn Tud.n Aruh,8C MF = Do d
!1.2:3r t r{5. 0 Trung Hd.ng, D6 Thanh Hdi, Nguydn ViQt DoO 0. Hud : L€ Anh Vu, Ddo Xudn Vinh, Nguydn Thi Phuong Chi. -Ndub < orhi f(;) : -l +6 < -1 Hir. Ti.y : Luu Tidn Dung, Nguydn Vinh Quang, Nguydn Xudn Vi|t. -Ndub>0thi f(-;) =-, -b 2 ;; . 1 . Khi dd tdn tai ro Dit. Nd.ng : LA T?iA.u Phong, Le Anh Khni, Nguydn Duy HiQu, If fU Nhian, Dinh Bda Khoa. dd cosr,, : -9NL re Thanh H6a : Nguydn Thd.nh Trung, Nguydn Hfiu Hd.u, NguydnKhd.nhTilng, Doitn f(x)= cosZxr,+ acoff,o= -l*Zcos2x,,* acosr, Thanh Hdi, Nguydn Huu Chil, Nguydn Van BiQn, Tq Hilu Hung, Nguydn Van Thdng, LA :-1 +2.a2"--:-1t-.(2-ml
. TiI dd, d6 thdy, ndu (r, y, z) ln nghi6m ctia h6 ,a*d. , L -bc)l * )=(1-"d)(6*c;: (1 da cho thi phAi c6 x,y,z **+ . Bdi thd: = (1 - bc)(b +c) (iabcd = l). 3z -23 (1) oNdubc=l*ad=1(dpcm). J_- | -3zt a*d I I oi.[dubc*l+-=b*c:- - ad acd abd 3x -x3 (4: "v- (2) (rD ' 1 1 : 1,1 -; ( | - bd) A \;-r ) : Albd -t) | -Axz 1 -y3 3y (bd-lr __ - t-w2 (3) *_;aa- _ 1 t). a@d Dat r: tga, vdi q. € ( -;,;) (4) vd sao Ndu6d = t-ac: 1(dpcm). N6u bd * 1 -ab = I -cd. = 1 (dpcm). cho tga, tgtu.,tg9a* * (5). Khi dd, tt (2), Cd.ch 2 (ciaban Ddng Thanh Hd 12A DHSP, + (3), (1) sd ctiy : tg3a, z = tg9a vdr = tg27.Ttt Ha NQi vd Vu Dilc Soru 12T Luong Van Tgy, day de dring suy ra k,,y,z) ln nghiQm cria (II) khi Ninh Binh) vd chi khi r = tga, y : tg3a, z = tg9a, vdi a D4t Sl=o+b*c*d drroc x6c dinh bdi (4), (5) vi tga = tg27a (6). Sr= ob*ac*ad*bc*bd*cd hn- St= abc*bcd*cda*dab 96,1($)+2(b = ktv,h ez+cc : eZ. I 26] ,h So= aSssL - | 'fi thda th6a d6ng thdi (4) ve (6) khi ve chi khi Theo dinh li Vi6t o., b, c, d ld. 4 nghiQm cta phuong trinh : kn ft nguy6n th6a : -12 < A < 12. D6 +s*'-srr* o " ilvdi x4 -srx3 1= (1) ding kidm tra drtgc ring tdt ch cdc 915. tti a \ttabcd=1vAo+6*c*d: : - +;1 +:1,+; n6n s6 thdy Sr : 1 drroc x6c dinh nhrr vila n6u ddu th6a (5). 1 VAy, tcim lai, hQ phtrong trinh da cho cri tdt Sl. aocd cA 25 nghi€m, d
chuydn L6 H6ng Fhong, thdnh ph6 Hd Chi HQ thfc (6) chtlng td IJ cQng tuydn vdi IK ; Minh vi mOt sd bah khric). nrii khric di ba didm I, J, K thing hing, ctng Dat I BC = a., CA : b, AB = c thdthi : ttlc lir (JK) I. = BD = p - b, trong d6 2p -- o, * b * c. Nhfn x6t. 10) Rdt d6ng c6c ban tham gia GoiE li didm d6i xrlngctaD qua trungdidm giei bai to6n tr6n, trong dd cd nhi6u ban l6p 8 J cltaBC ; CE = p - b (ctng vdy,BE =p - c) thd vi ldp I PTCS (vi chi sir dungdinh li Tal6t, tam thi -O lA ti64 didm tr6n canh BC c:iaa drrdng tron gi6c ddng dang vA dinh li v6 drrdng trung binh). (I a , r a) bnng tiSp g6c A c&a tam $6c ABC . 2''r; Sd ban sir dqng phuong ph6p v6cto kh6ng nhi6u, thrrdng chua gon, cbn dii dbng. 3()) Tdt cA ddu giAi dring, trit hai b4n, tuy nhi6n c6c b4n cdn tim cdch trinh bay ldi giei thAt rnnh mach, khric chi6t, chat chd viL ngin ggn hon nta' N.TYEN oANc pHAr B,di T101222. O ld n'tQt didm bdt ki 6 ben trong cia mit tt diQn ABCD. Goi Ar,B |, C, uii Dr ld. hinh chidu cia O ldn luqt tr€ncd.c mQt phd.ng (BCD), (CDA), (DAB) uit. (ABC) sao cko : OAr+OB1+OCr+ODr=g' (*) K6 ti6p tuy6'n B'C' ll BC cria drrdng trbn (1), Ching minh rd.ng : ti6p xuc v6i (1) 6 E'le, didm xuy6n tAm - d6i ctia D tr6n (/). Thd thi drrdng trdn (.I) nQi ti6p tam OA, + OBr + OCr + ODr < 4r, (**) gi*llBc trd thinh dttdng trdn bAng tidp grfc trong d6 r ld.bd.n kinh mq.t cd.u nQi tidp tt diQn. B'AC' cria tam gSi.c AB'C'. Ph6p vi t{ tim A, Ldi giAi 1. (ctra Nguydn S, Phong, ti: sd k : *, AB AC BC, (: o", = B,.) biSn drrdng 10,\ chuy6n todn, DHSP, HnNoi). trbn (/) thinh dridng trbn (/r) bdng ti6p gric GiA sii mqt cdu BAC c;&.a tam giric ABC, do d 2aIK *2(p -a)U =aIA +bIB *cIC;(4) Mat khric, vi .I li t6m dudng tron n6i tidp tam > (O\+OB.+OCr+OD)z; Q) gSdcABC n6n ta,cci hQ_fhrlc ($ay clrlngminh !) Tt (1) vd (2) ta drroc BDT (**) cdn tim. aIA+b IB +c IC = A; (b) Ddu ding thrlc xAy ra khi va chi khi Tt (4) vi : + __, (5)rsuy ra %t =!A, 4Cr = OD.*titc ld O = I o aIK*(p-a\IJ=O (6) IAo+IBo*ICo*IDo=O (3)
Nhrrng, ndu ggi Sl , 52 , S, vd So ln di6n tich bing crich sit dung ph6p chi6u vu6ng gc5c : cric mat cria trl di6n, ldn lrrot d6i diQn vdi cric Chidu cdLc v6ctd th6a m6n (*) l6n mQt trong c6c dinhA, B, C vd.D, theo dinh li "con nhim", ta cd c4nh cria trl diQn ABCD, ch&ng han, chidu l6n (vi4 = IBo -- ICo = ID4 - r) -_- --_, dudng thhngCD. 30) Ban Nguydn Khanh Quynh, ldp 11Ao, Sr IAo + S2.IBo+ 53. ICo+ S .ID,: 0 ; (4) trrrdng PTTH Phan Ddng Luu, Y6n Thinh, Suy ra : (3)
Do1, - Iz-Zr_= BiiL2t222 : zr-ar: ar. MQt con ldr don duoc k€o ra tdi ui tri A mit Theo giAn d6 v6cto d.A.y hqp uoi ph*ong thd;ng ding mQt 56" g vd theo d6 bai thi d0 rbi bu1ng ra (u,.: 0). Hdry xtu dinh cdr ui tri ";, l6ch pha girra i, vd i, li crta con lac md tai do gia tdc cia uQt nQng c6 %tn? Ll gid. tri nh6 nhdt, ldn nhd.t. Lg: o o, = -o, :8. Hu6ng d6n gini. Goi a li gric gita dAy vi T (2) + Z,-: Rtga, = 60 {5 Q. (1) + phuong thing drlng. Chon hQ truc toa d6 cci gdc tai vAt nQng,Oy hu6ngdoc theo diy treo vi Or .UMN: tr,lfl +ZI = t2OV hrr6ng theo tidp tuydn tai O v6 phia phrrong DoUr* s6m pha hon U, g6c ar: nlT ndr- th&ng drlng qua didm treo. Ap drlng phtrongtrinh UMN: l2O{Z sin(lO0rl +nl4) (Ir. Khi r li dinh luAt Niuton, nit ra a* = gsina or:l I lf ur* chi6u dii dAy treo). Ap d\mg dinh luit bAo todn , rrrU 6 (3t : iZr: Zc-1'3----oAR cd neng, suy ra au : 2g(cosa - cosrp) vi l;., vai urrcingpha o=g (hiCn tudng cOng Arrdngt. V+y Dd x6t crlc tri cria a ta x6t iz: z{2 sin ( roo"r +i) fe>. y = flcwx) 3cos2a - 8coq2cosa vi tim = drto.c : o NhQn x6t. Cric em cd ldi giei tdt : Phitng Duy - Ndu con l5c dugc tha tt vi tri 9 > orro, f, Hyle _B-!18 CL, DHTH ; Nguydn Duy Kiat, (= 0,723 nad) thi gia t6c nh6 nhdt nhtr gric l6ch 12Cr, PTTH Li TU Trpng, NhLTrang,-Kh6nli cria dAy ld c, = arcos(4l\coqp) ; cdn ndu con l5c F_ag_, Nguydn Vu Hung \2D chuy6n NN, DHSPNN, DHQGHN ; Le rvi,12B, PfTH 1 M6 Dfc, Qudng Ngai ; Thdi Thanh TLdn l2T, drroc th6 tt vi tri I < arcosf ,ni *t" tdc nh6 chuydn Li Khi6t, QuAng Ngai ; Can Ngoc Tudn, nhdt tai vi tri cAn bing. 12T, PTTH Ddo Duy Tt, QuAng Binhl Lo Dinh B_d.o Khoa,11Ar, PTTH chuy6n Le Quy DOn, Di - Ndu con l5c drro.c thA tt vi tri g > arcos(0,6) thi gia t6c l6n nhdt khi con l5c di qua vi tri cAn Nn"g; Phan Ai,h Chuy€n,12A PTT'H phd Ycn, bing ; ndu con l5c drroc tha tt g < arcos(0,6) B_tcTlarr.VA Cdng Phuong, L2"12, TH chuy6n thi gia t6c l6n nhdt khi cosa : cosrg titc ld khi N_gy6" Binh Khi6m, Vinli'Long,'Vu Phuong con lic t6i 6e vi tri bd ; v6i cosp : 0,6 thi d d \inh,_l2A,, Vung Tiu, Bi Ria - Vrlng Tiu, tri bd vi vi tri cAn bing cd cirng girt tri bing Ngyya" Vai Thudn, 1 18, PTTH Neng khi6u Ngo Qr_!ig", Hn Bic ; Pha,m Minh H6dng t28,, nhau vi l6n nhdt. PTTH Kim Li0n, He NOi ; Truong Phti Thid, Nh$n x6t. Em Pham Minh Hod.ng L2A,, 12C3, PTTH Nguy6n r"r,"l,XI:l;, PTTH Kim Li6n, Ha Noi cd ldi giAi khd tdt. MAI ANH SIIY NGIIi vii u,tr rrANc oiNc THIiC ftidp theo trang 1) Bni 5. Trgc cdn thrlc d mdu s6 cria bidu thrlc : A : ---=--a-:- 414 +212 -t6 Liti Sjdi;^Ap drlng (1) coi miu sd cta A cd dang a *b * c.Khi dd nhdn cA trl s6 vd m6u s6 cria Avdi(a'+b: -ab -ac -bc)tac6: 2.- 1- ^ 16tro * +W +286+16-64W -szW ^= 272-60v4 15va-68 to -3056 764 Sau dAy ld m6t sd bii tAp d6 ngh! : Bhi 6 : GiAi c.ic phrrong trinh sau : a) Gr3 * Br - b = 0 b)o3+bx*c=0(o*0) Blri 7 : Truc cdn thrlc d m6u s6 cta cdc bidu thrlc : " - V;JE+ D_ i[ Bei 8 : GiAi cric phrrong trinh lugng gir{c : a) (sinx * sin2x * sinSx)3 = sin3x + sin32x + sin33x. b) sinSx = 6sinx - 1.
BAd T8/226 : Cho o, b, c, d, e, f ld sdu s6 thtlc th6a m6n di6u kiQn : ab * bc * cd * de * ef = | Chfng minh ring: a2+b2+c2+d2+e2+fr, L , cos 7 r-E vAN QUANG CAC IOP TIICS (Thia T'hi€tr - Hud ) BdiTll226. Tim sd dffiltronghQ ddm thfp Bdi Tgl22S: Ggi G li trong tAm m4t ABC t,f dien ABCD vd M ld niot aiAm bdt ki - th6a m6n : phAn "ri" thu6c thuQc mi6nn tam g16c ABU. tant giric ctiLcABC. Dudng tnang ABC. uudng Dudn quaM thing ela Y1 1) 56 6c x 4 tQn cirng bdngcd. chtc5cm4t vAiDG caf, phingDBC, DCA cacIIIaf, PfiaIrBjJDv, songsongvdll,Li - D a6in - 6c x 4 ld s6 chinh Phuong. A' B'vdc'. R'wh A', B' vefrABd A', v?r a' C'. a,hfins {-ht1ng Chtlng mrnh rins:: ring: minh rang o6 rHnuH unN DA', + p3' +DC' > SGM (Milth IIdi) DAM VAN NHI BAi T21226: Giai phrrong hinh nghiQm nguyOn : (Thdi Bttth) 4y2 = z+{Tss --F*u BAi T10/226 : Goi h^, h6, h. vi lu, .16, l, BUI QI.IANG TRT,ONG trtong rlng li d0 dai c6c dtrdng cao vd c5cdrldng Ha Nai) oiran siac dtroc k6 tdi citc c4r,}' a,6, c ctra tam BidiT}lzz6 : Tim tdt ch a e N dd Phrrong g;tac. r", R li b6n kinh v6ng trbn n6i,tidp va ngo4i trinh12 - a2x + a * | = 0 cci nghiQm nguy6n. tidp tam gi6c dd. Chdng minh ring : DANG HUNG THiNG L"LhL*i uIa Nai) Bidi T41228: Trong c6c hinh thang cAn cd chu vi 2p, gdckd d6y l6n banga (a < 90o), dqng hinh thang ccf di6n tich ldn nhdt' ;.i " pHAN,I Hl6,N e.i.Nc tBdc Thdi) v0 HOU siNH (Hd N1i) BidiT51226: Cho hinh vuOngABCD. M, N cAc uii vAr li Ialrai didm tdn lrrqt nim tr6nBC vdCD sao cho BdiLlt228: fnly'Jl : 45o. Hdy tim vi tri etia M, N dd diQn MQt quA cdu kh6i Warrgm dang chuydl dQry tich tam g15.c CMN dat gi6 tri l6n nhdt' tr6n i ariang thing nim ngang vdi van t6-c u thi NCUyE,N'xLlAN uuNc va cham vao cdu kh6i quA cau mQt qua vno mgf, r{'nur ruvu6 \ vatt6 lugng rul- dang (Thanh. Htia) chuydn dQng cirng chi6u tr6n dudng thing dd ;i ""a; tac ilg. Siu ra ch4m qua cdu m chuvdn CAC I6P THCB dQng v6i vfn t6c u 12. Coi va ch4m li xuy6n tAm' Bb q"ua ma s6t gitra hai quA cdu v6i drldng nim BAi T6/226 : Chrlng minh rdng vdi moi s6 t-6 ring Chring^ t6 nsans. Chrlns ch4m hai rnng sau va cham qu6 cdu hal qua cau ttr nhi6n n > 2 ta cd : "S"irg tiBp {t" chuyEn dong theo hudng cri. Tim di6u n-l xay ra. }m xAv ki6n dd va cham 20 c'*1-[> . C],-ochia hdt cho 4n- 1 NcuvEN DLIY TRUY k= (Thdi Binh) Hb ortRNc vtNll Bicil&lzz& z S, M, N, P la cac dinh cira mQt Nsh€ An) trl di6n d6u cti tAm hinh cdu ngo4i tidp [i O. Ddng BidiT7l22S: Cho Phrrong trinh i , diqn trrong dQ / chay theo dudngMSPM (hinh 1) xr3-x6+gx4-3r2+1=0 a) Chrlng minh ring phuong trinh dtj cti J\ dring mQt nghiGm sd thrrc. b) DAt xr= | vArn*, : (x-113 + l)-3113 ,o Vdi moi s6 nguy6n dtrong n. Chrlng minM ring day sd {x,.,} c
PROBLEMS IN THIS ISSUE a) Prcve that this eqrration has a trnique real rcot. For Lower Secondary Schools. b) Put xr =I and ro t: (*r1/3 + 1)-3n3 for Tll228. Find the number oEcd (written in every positive integer rL. Prove that the decinal system) satisfying the conditions : sequence {rn} has a limit and ro = 1) the number 6c x 4 ends up by the 2-digit l,Y,t" number cil,2) a6cd -6c x 4 is a perfect square. the above mentioned real root. 121226. Find integral solution of the equation : c) Use calculator to find approximate value 4yz = 2 +t[Tds=7=g. for this root to two decimal places. T31228. Find all o € N such tha-t rhe equation T81226. Lest o, b, c, d., e, f be six real numbers satisfying ab + bc * cd * de * ef : l. x2-a2x*o*1=0 has an integral root. Prove that a2 + b2 + c2 + d2 + e2 + f -- + roY T41226. Construct the ismceles trapezoid with given perimeter 2p, the angle adjacent to the great base is a (a < 90o), which has greatest area. Tgl?26.Let G be the center of gravity of the T51226.I€t face ABC of a tetrahedron ABCD and M be an be given a square ABCD; MandN eplwo points respectively on BC and CD so that arbitrary point inside the triangle A-BC. The MAN :45o. Determine the position of M and N line passing through M, parallel to DG, cuts the so that the areaof triangle CMNhas greatest value. planes DBC, DCA and. DAB respectively at A', B', C' .Prove that DA' + DB' + DC' > \GM For Upper Secondary Schools. . TlOl22S. I-et ha, hb, h, and lu, ln, /, be the T81228. Prove that for every integer n D 2, lengths of the altitudes and the angled-bisectors the number rt-l correspondingto thesidesa, b, c of atriangle, and let r, R are respectively the radius ofits incircle > cP"Ji'.-!) ' cj-o is divisible bv 4- k:o 1. and circumcircle. Prove that : ho hb h, T* u* r," T7.226. Consider the equation xt3_x6+3r4_Br2+l:0. 6) 0 ry 6i,rl c;i dal dqr oa bo ?u.o 1 5 Thily : Em l6n bAnggiAi bii torin sd 11 trang 7)u : Thtra th!y, cdc udc sd cria4ld +1, +2, +4 36 Dai sci 9. cho A : laj-l Thdy : V4y t/:q - 3 cd thd bang bao nhi6u ? ?ro (vidt) : Vr -3 = *1, !2, +4. {7-s Tim moi gre tri nguyOn cria r dd A nhAn gid lri nguyOn. Thrra thdy ndu rE - s = | tni #=: # ?rd; Thrra thdy, bni ndy em thdy ki cuc... = 20 cring ld sd nguydn a. Thily : Em vidt V, + t : {7 -3 +4 Thdy (lnng tring) : Kh6ng thd duoc... Nhung trb khac : Ifdu Vr - 3 bing ?ro(vi6t),A=f+=##* 112 ... thi- 4 cflng ld s6 nguy6n a 4 i,1 , E rE-s .f_ _c A= 1-G= Thay 1dm bdm) : Ndu r li sd nguy€n thi : 1 ln sd nguy6n, vdy mudn choA ld s6 Thay rE - s cd thd tang ... unu.,* ? c6c em v6 4 ], | nguy6n thi phAi ln sd nguy6n. nghiOn crlu bii nAy, h6m sau thdy siAi ti6p... fi:: A Ldi blrn cria Mao T6n Quarig-: Thilry'lilng Khi ndo thi td s6 nguyen aifiSJA chi phd,i, ui chua c6 d,inh li: Ndui e Za ? -1--B t/r thi lx chi c6 thd ld sd nguyOn hoac lit sd uO ti, ?rd: Thrra thdy, khi 4 chia cho {7 - B ld sd hhdng thd lit. mQt phan sd thgc su dttoc". nguy6n a ( !) Chttng minh dinh li ndy citng dd, nhurtg ndu hhnng chilng minh (hny it nfuit lit. nau len), mit. ctl Thdy : : " s6 nguy€n khi r/t - B ld tr6c xentruhu c6 sd.n m|t cdch hi.dn nhi€n dd.dp d4tng, tlr -3^la thi c6 duoc kh6ng ?'xin cdc ruhd. siflm,"#!ly, cl&a 4. 4 cd nhttng tldc sd nio ? NGUyEN oucrAN l1
\A !r -.-- Dii THI TUYEN- SINH DAI HQC NAM HQC A 19es - tss6TRU0NG DAI HQC XAY DUNG HA NoI MOn thi : TOAN (Thdi gian lim bii: 180 phft) m'x2 +.x + m, 1) Tirn quj tich cria didm K khi I chay tr6n c6u I : cho harn sdv' : r. +nL ntld doan BC. tham sd. 2) Tinh dO dai O/ theo a vb' x. 1)Khim=l: 3) Tim r dd dd dai O/ l6n nhdt, b6 nhdt. a) KhAo s6t stl bidn thi6n vd v6 d6 thi cria him scl y. DAP AN b) Vidt phtiong trinh cac elrrdng thing di qua Cnu I (2,0\ didm (-1 ;0) .,a tidp xric v6i d6 thi ctra hdm s6y. 2) Tim giri tri c'0'a m dd him s6 dd cho kh6ng ' Khi az : I 1). (1.5d). x:*x*l 1 dci ctrc tri. Y: x+1 =x*x+1' a) (1,0 d) CAuIIi r +J) (0,25 d) Mi6n xric dinh r * -L ; ti6m cfn l2srruz(.r =1 1) Giai hO phuong trinh I - . Z(xt + : 1. drlngr - -1 ; ti6m cAn xi6nY - r I Yz1 2) Cho bdt phrrong trinh': (0.5 d) *, * r*.: o khir : o, x: _2. atl**
tn : Othiy' : 0 vdi Vr * O + y = hingsd t1).#(x) = + him kh6ng dat cuc tri. (2a.2 - 1\r2 - 2an + tu2 > O (2) (0,25d) m * 0, y' = 0 cci 2 nghi6m phAn bi6t (2) drlng v6i moi r < -o khi : x= 0, x : vi ddi ddu n6n hAm cri cgc td. -Znt noac [zor-tro Kdt luan : Hdrn kh6ngcci crrc tri dri }rhi m : 0. ---lL=a2(7-12a210 'lZGziyz)=1 " ' 2h2-1 lry:--: 4, It>o vdikeZ.' [ I - \a2 + 2d + Ga2 > o (0,25 d) H6cci nshi6m khi lff-"1 = 12a2 ,2kz-1." lslz=al2a2-1>-a L,: k/-4 (.;\ = I - k2 > o sin2x +vdphAi>2. rdj_.- lzxz+4x-6>o , t*> _112 a lVcosx > cos'x (0,5 d) Giai ra 1i , , ;;" *. _r:, , 1 cdnil+sirr cw : 7+{12)sin2tr < BP VQy vO nghi6:n b)(1d) t 2) (1 d) (0,75 d) Yi sinB > 0 , sinC > 0 n6n : Cd.ch 1: (0,25 d) dua (l) re #- > o. ,1 \2x'17 - 1 , (#* 1 ri*-q = cotsB*cotgC+ \0,2s d) y' = $ff ({z?i - nz^= o khir = + t[21 ; 2 (sinB *sinC -,[ 3 sinBsinQ < < sinCcosBlcosCsinB* y'>0khi-{21
Ddnh clto cdc-bqn rhrdn bithivaoDqihec I ffiffiffin MNNffi DOAN QUANG MANH (Hdi Phdng) Trong bO 'D6 thi tuydn sinh vdo c6c tnrdng B}ri to6n 2. Dai hgc, Cao d&ng vd Trung hoc chuy6n Chfng minh ring tam gi6c ABC ddu khi vd nghiQp", c
+ t*q .EB .tgC + tg2A .tezB . tgZC < O MQt cr4ch trJ nhi6n ta di ddn bii todn sau : + tg"{ + teB + tg? + tgzA + tg2B + tg2c < O Bhi torfln 4 Mdu thudn v6i (3) v4y tnrdnghop niy kh0ng xAy ra. Chrlng minh ring tam gi6c ABC d6:u khi vi chi khi : 2)c
I fa Ufinir ilfu tuit firyki'tliry o A I ' L- -_- L- -\---4-- swffiGrilucrcmorcmffimmffi[ oo r-E oudc rraN (NghQ An) C6ch ddy ba mrroi nim, b6o "To6n hgc vd Cho O ld. tro. ng tfr.m cfia ha didm A, Az, , ., A^ . Tudi tr6" cri dang d6 toSn sau. ud. M lir. mit didtn tiry ! tron g hhnng gian. Khi d6 Bii to6n I z Cho tam gidc dbu ABC canh bd.ng a ud. mQt didm M bdt ki ban duimg trdru Mo2 : (U@) M4 - 1um21) ey'l ngoai fiAp tum gid.c d6.^Ddt^MA^= x, MB : !, i=t i,.j:.| MC : z. Chilng minh xz *yl I zz = 2az. t
suy ra di6u phAi chfng minh (chri y : dridng trcn Theo bdi to6n 5'. ta cci : nay cd thdthu v6 m6t didmhay ld"dudngtrbnAo"). yMBz + zMCz = rj + zlMAa aa{52 + zA,C 0) Tr6n co sd bdi tod.n 5', toi da d6 xudt. Ta lai c6 OH : OA'. cos HOA' : Blri to6n 7 : Cho tant gidc ABC uit nfit didm = OA' . cos(B - C) M chuydn dOng ffAn duitng trbn ngoai tidp tam +OA'=OHlcos(B-C)- gioc do. Xdc dinh x, y, z dd gid tri bidu thic rfuIA2 = R . cosA I cos (B - C) :OA.cosA I cos(B - C). Mat khric : +yMBt * zMCt kh6ng phu thu6c uin ui tri M. sin2A l Gin2B * sinZC) : Sau ddy ld m6t trong c6c ldi giai cria bdi torin. = 2sinAcosA lZsin(B * C)cos(B - C) - "faldy x : sin24 ! = sirflB vD. z : sin2c. Khi dti : cosA I cos(B - C) ,\t_t_yun2 + zMC2 43 tz.sc. : TiI dd : OA' I OA = cosA I cos(B - C) - +sin2Q| = sinA I rsinzB c)(y+z)+0 + z)OA * x.OA:0 + + Dd bai brio kh6i phrlc tap vi quri Lai rlp dung kit qud \hi giAi bni todn 5' dai, t6i xin n6u ta cci (y * zlMAt * xMAt = c5ch giAi khi AABC = (x*y iz)Moz+(y+z)oA'2 +x.oA2 (2) COng ttng vd (1r vd, (2t, ta cd : cd ba gdc ddu nhon. Kh6ng mdt tdng x.MAt*v.MBz*z.MC2 = : (x * y + z) iWOz + y . A'82 + z . A'C2 + qu6t, gii sir M nim *(v+ztOA'z+x.OA2. tr6n cung nh6 BC. Do do x . MA2 +v . UA + z . MCznhrin nrot GoiA'ldgiao ciaAO grri tri kh6ng ddi, kli6ng phu thu6c veo vi triM. vd,BC.IGOH tBC. Cho M : A, ch6ng han, ta duoc giri tri dci lA Vi OB : @C:_Q@in kitb,fulngtrbn ngoai tidp 8R tsinAsinBsinC : 45 n.qnt.. AABC) 'rasin OAts: sin OA'C n6n bdn kinh c6c Dd k6t thric, tOi xin d6 nghi c5c ban girip t6i ,ba vi6c drrdng trdn ngoai ti6p ci.c tam gi6c BOA, vd. : CoA'bengnhau,suyra, .'{=, *'sin A'C - 1. Tim c6ch gi6i dep hon cho ldi giAi bdi to6n BOA - sin COA' - 6 d tr6n cria tOi. 2. Tim cl,c 916,tri x, y, z cta bdi to6n 6, tdng M sin2C z qurit cdng t6t. A'C _ sin CO{* S1g2B y 3. Md rdngbiri to5n 6 cho hinh hoc kh6ng gian. + .y . A'B + z.A'C : O. Chric cric ban thdnh cOng DE TI{I TUYEN SINH . . . Phdn B (Tidp theo trang 13) 1) (l d) (O,L5d)Tamgr.EBtlder nen trung luyenEF L BC ; rm ggc nhi 2) \2 d) di6n g[la ($ ve ([AQ lA wfrng nen E.F -r 1P). a.) (l d) (0.2-5 d) lci( Ie hinh chi€i-r rar6ng goc cua EK tr6n (p). 'Iheo (0,5 d) * (d,) qua Mr(0;-t ; l), c6vecrcJ chi phucrng{ dinh ty.ldUongw6nggoc thi t:K I At : {l; l;2} (0,25d) Goc AKF lu6n w6ng.,4io c6 dinh n6n qui tich ( (d,) quaM. 1- 1 ; 0; 0). c6 vecld chi phuongi : {t;2: t} khi l dich chuydn lir phdn cung trdn dudng kinhAF. YA, gging song song voi (d1) r,a (d.) phii co vecrc, ph6p (0,25 d) Do / chay trCn BC vA g6c BAC : z/4 n6n clui rich -rt: S, A.tr : t-3 : 1 ; l) va mit phing c6 dang : lA 1/4 dtlong trdn di rrr B qua 1t d6n giao di6m voi AC. phdn -3x+y+z+D:0.(P) clAo hi6n nhi€n. (0.5 d) * Dd mar phing 1p) ti6p xric v6i met c6u 2) (1 d) (* - 1)2 + (y + 1)2 + / = ltthikhoingcachtirtamrlt ; - I ; 0) (0,25 d) 1'rung tuydn O12 = ul+ ze i2* ar2 trdn (P) phdi blng ,{Tf -#2: fi1 oD : 15 hoic (0.25 q AFz= 5a2 ; EF: 3a2 D : -'7. (0,25 q AE2 =lf + tf = 8a2.rti : taz + x2 l)udc 2 m5r phing ti6p xric vtii mdt cdu vir song song v6i (0,2s d) (dL), (dr) : --l,tly +z + 15 = 0va -3x +y + 7 / : Q. q --1[1- 1jP - b) (r d) "r 3) (1 d) d) Mir phing 1Q) chtta {d,rva / tn mir phing qua (0"25 or:{Q;;DP;7F, {i,-t.$)va\^r6nggocvtiivocrcr4: q M:t : {- t;3;-1}+ vdi 0
Gidi ilip bd.i cAr nirun cH0 NHAr Ta th6y di6n tich ctra hinh cht nh6t Ie I x t6 = 144' Ti dd thdy c4nh cta hinh vuong m6i tao thinh Ld12 (12 x lZ :144). Vi vay ia dua ra cdch cit nhrr hinh v0 ,f . it (Theo Pham Thnnh Vinh, 5T, NKY6n Kh6nh' Ninh Binh)' Nh$n x6t : MOt s6 b4n glii giei d6p ddn d6u dting' BiNH PHLIONG DrdM c0a u6r Ncr.rdr NGUdI GT,I BAI CulT CAN CHU f B6n ban XuAn, H4, Thu, DOng nhQn duoc didm cta tai kidm tra to6n cu6i hoc ki' Ban Lan - Ldi giAi cOa m$t bei .to6n vidt cirng I6p mu6n biSt didm ctra m6i ngrtdi ' Khi ri6nq tr6i mQt to gidY. Ndu bdri dAi h6i ;hi drioc cric ban dci tri ldi rip md nhtl sau : nhidu trang ttrl dinn chring vdo nhau' Xud.n n6i; Ban D6ng drrgc ?, ban Hq drloc 8, ban Thu drroc 9. - Tr6n m6i baigiai d&r ghihqt6n,l6P, Ha n6i: B4n Thu dudc 8, ban Xudn dttoc 9, truong, huyQn, tinh (thdnh ph6) vA ghi ban DOng dtroc 10. sd ora dd ra (kh6ng cdn ch6P lEide)' Thu ru6i: CA ba ban XuAn, H4, D6ng ddu drioc 7. - Ngodi Phong ni cdn dd ro bdl D6ng noi: CA ba ban XuAn, Ha, Thu ddu oiai cia s6 b6o ndo (kh6ng gtli bei drroc 8. iua nrridu sd b6o vdo cing mqt Bidt rdng kh6ng cd ban ndo drtoc hai b4n ndi phong b)). cirng dring didm cira minh vd m6i cAu trA "ai "a tai d- trOn ctri n