Toán học và tuổi trẻ Số 209 (11/1994)

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Toán học và tuổi trẻ Số 209 (11/1994) sau đây sẽ trình bày về giải phương trình nhờ hệ phương trình; định lý Trung Hoa về số dư; đi tìm một dạng định lý hàm số Sin cho tứ diện. Đặc biệt, trong mục giải trí Toán học sẽ giúp các bạn biết cách giải đáp bài toán về chia bánh.

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Nội dung Text: Toán học và tuổi trẻ Số 209 (11/1994)

,r :i ,- r' i uli =.i ;0, ' Cl |!"x"t (= " '1 BO GIAO DIJC VA DAO TAO ,r,.2 rap cni na xcAv ts uANc mrAxc 2 GIAI PHIIONG TRINH NHO HE PHUONG TRINH ,ra\n'll DAP AN DE THI QUOC GIA CHON HQC SINH GIOI TOAN LOP 9l{AM HOC 1993 . 1994 CHLA TATVII ' -i i : :::i.r.:a:j .=.::t".;";' ,.Y+i",f+,z1:i. a' ''. .5- " i '=fn
TOAN HQC VA TUbI TRE MATHE,MATICS AND YOUTH MUC LUC Trang o Ddnh cho cdc ban Trung hoc co sd Tdng biin fip : For Lower Secondary School Level Friends NGI]Yt]N CANII'I.oAN L€ Qudc Hd.n - GiAi phuong trinh nhd Phd td ng hi4n fip : hQ phuong trinh 1 NCiO DA't"lLI o Trd.n Xud,n Dd'ng - Dinh li Trung Hoa v6 s5 drr 2 FIOANG Cltt-tNc; c Gitii bdi ki trwdc Solution of Problems in. Preuious Issue nQt oOttc etEN rAP : C6c bdi cira s6 205 3 Hoitng Ngqc Cd.nh - Xung quanh mQt Nguy6n CAnh Todn. Hodng bdi to6n quen thudc. I Chfng. Ngo Dat Ti1. LO Kh6c BAo. Nguy,6n H*y Doan, o Db ra ki ndy Nguy5n Vidt Hei, Dinh Quang Problems in this issue HAo, Nguy6n XuAn Huy, Phan Crlc bdi tn T1/209 ddn T10/209,Lll2O9,Lzl2lg 10 Huy Khii, Vu Thanh Khidt, L0 Hai Kh6i, Nguy6n Van Mau, o Nguy1n Httu Thdo Hodng L6 Minh, Nguy6n Khic - D6p 6n d6 thi qu6c gia to6n l6P I 12 Minh, Trdn Van Nhung, o Ddnh cho cdc ban chudn bi thi vdo dui hoc NguySn Dnng Phdt, Phan For College and Uniuersity Entrance Thanh Quang, Ta Hdng Exam Preparers. Quing, Dang Hung Thing, Vfl Trinh Bd.ng GianC - Di tim mOt dang Duong ThUy, Trdn Thdnh Trai, Ld Ba Khanh Trinh, NgO dinh ti hdm s6 sin cho trl diQn 16 Vi6t Trung, Dang Quan Vi5n. o Gidi tri todn hoc Fun with Mathematics Nguy1n Dilc Td.n - Giei d6p bni Chia b6nh Nguy6n Ddng Quang - Ldm thd ndo ? Bia 4 Tru sd tda soan.' 45B Hlrng chudi, Ha Noi DT: 213786 BiAn fi.p ud, tri s(: vu KiM THUY 231 Nguy6n VEn CiI. T'P I{d {rhi Minh DTr 35611 t Trin'h bav: DOAN UbXC
Y tuang cria phrtong phrip "Giii hO phuong ; ,=A* f,fir =), u ), : ,)^ * )z=4a" trinh bing phrrong ph6p thd' ld srl dung c6c (vi .444N : 45o theo gia thidt) MME : I\AMN = ph6p bidn ddi trrong drrong lray ph6p biSn ddi (c.g.c) + MN:ME:u* Trong u. CMN h6 qu& dd dria ddn mdt phuong trinh chi cdn ta c6 IzIN : Ctrrt + CN +(u * u\2 = (a - u)2 + mOt dn s6. Trong bdi b6o ndy, t6i xin trao ddi *afu, * u) : c,2 - uu r,6i c6c ban con dttdng ngudc lai : ldm t6ng s6 + (o. - p)z (3) dn cua phrrong trinh. Ta dat u, * u = / thi ti (3) suy ra : Thi du I : GiAi phuong trinh : u.u = a2 - at vi tt (.2) suy ra +fi+ +{OZ_r=S at Sa,a,yry : gi6 tri l6n nhdt NhAn thdy ngay : ndu ta giAi phuong trinh , @,o dri vi6c tim niy bing phrrong ph6p th6ng thrrdng thi kh6ng vd nh6 nhdt cria S *tttN duoc chuvdn sang vi6c hi vong thenlr c6ng. Dd y ring : tdng c5c bidu tim gi6 tri l6n nhdt vd nh6 nhdt crla t). thfc dudi ddu can ld hing s6 nOn ta hiy dat T\r {..., lu*u:t a,{i=4, 11[g,]-x:ts luL - - ',^2 _ ,lt^ tu siry ra u vd u ld hai nghi6m cria phttong vd drta phuong trinh da cho v6 hO trinh bic hai lrr * u : 5 { I x"-tx+a2-at=A (4) \ut*'-'1 =97 vi dri dd (4) cri nghiOrn lA A > t) Di6u l 0 *t > 2a(.d2 - 1). vi r > 0. I [r,.u>0 Dinh cho cic a [-- Dinh cac ban Trung hoc co SO Trung hoc sd i crAr pHUdNG rRiNH NHd HC pHUdNG rRiNH ii i LE QLTOC HAN Giai hO ndy bing cSch dat u. *u : S, uu : P Khi , = 2a(,{2 - 1) thi (4) c6 nghi6m k6p vi chri y u1 + u4 = (S2 - 2Pi2 - 2Pz ta c,5 tt : t2 : a(,[2 - 1) + u : u : o.612-- 1) th6a ,ul :2 + r, = 16 man diau ki6n (1), o6il f,.,ri,,, = 2o1.t[2 - l) I i.u,=3+er:6i '+ min Sa.rrur,,, = a2(,12 - 1 ), dat dudc khi vd chi khi iz : u = at{1 - l) Ta x6t rndt thi du khci hon Taiai cci : at : az - uu < a2,ri.u, u 2 0* t 4 a. : Thi du 2 : Cho hinh vudng ABCD canh bang Khi I : o thi (4) cri hai nghi6m t1 = a, t.:0 a, vd hai didm M Alchrydn d6ng tr6n 2 canh u,:a,. u, : o BCI vd CD sao cho MAN : 45". Tim gia tri l6n * [ ' ^' u:= ' (1) nhdt vd nh6 nhdt cria di6n tich tam gyec AMN. Luz=u, 0cang th6a md,n di6u ki6n Gid.i : Ta dua viro caic dn m6i ; BM = tt, a.) CN:u(0
D i"n f Trung Hoa v€ sd da dugc phat bidu nhu sau : Ndu mr, tuD ..., mnld. n sd nguyQn d,uong nguyAn ftl cit,ng nhau tilng dOi mqt uir a1, av ..., anld. n sd nguyAn bd.t k; thi hQ phuong tinh dbng dtt x = ar(mod m|, x-= ar(r,tod _m), ..., ffi ffim x = an (mod m) cr5 nghiQm. Ngoir.i ra ndu x = c ld. mQt ngh.iQnt cia ru6 thi x = lra & EE*l ld. ntiji nghiQni'cfia hC hhi uit. chi hhi tbn tai t e Z sa.o cixt xo = c * (mrmr... nr.)i.' v6i nr6i i € {1, z, ,..,tul,d4t (Ki hi€u {nt, nr=y#. I }ftiddru, €.Fu* vd (rni, n,) = 1. t GE la USCIN mrvA ni). Vi vdy tdn t+i bi e Z sao cho b,rtr= I (mod nr,), DAtM n ) c:iula :2ofirj,Y6ij * ithia,brnri rnr.Yivdy ffi 3W j=1 M :- a,b,n,= a,.l = o, (mod m,) vdinroi i = T,n.YQyx = Ml .m6tnghi6mcuah6. GiA sr} x = c la mot nghiQm cria h6. Ndu r = ro le mQt nghiqm cria hQ thi x -ci m,Vi=T.n-. -4 " D+t *\1m1,trt2.t...,.m.,r7(nt lA.boisdchung.nh6nhdtcria ffi1,ffi,,...,mn).Vi ffi nt 1,-!2,..., nr.n nfuuydrg. r!l{ ctrhg nhau tr)ng d6i m6t nrOn tru = ntttrl2-..-fiin. vtx. -c I nt, vt: T. tt n6nr- -ei m,. Varj ton tart t' e Z:sa;chcrr. -"c - mt. Svy ra_r() = c ! ryt. Ngrrg" Iai ndu r., e z th6a tr bsE rndn r" : c I n-rt thi rd ringl : Io la mdt nghiQm cira h6. Ap dqng linh ly Tr"ung Hoa vd s6 du ta c
Bhi T1/205 : fi.nt nd.m. sd hhatnlwutrong d.ay tinh sa,u d.d1t : .(xx****xx):*t=** Bidt riing : a) Trong ba s6 hqng trong ngoQc thi c6 mQt sd hqng la BSCNN crta hai s6 hsng hia. (ac - I - f)b = a(l - P) -c (1) b) 56 chia td. sd nguyan 6 uit.lit. USCLN crta ar$ + l) - 6 = (aa - | + p)b (2) hai sd n6i tr€n. a(l +ac) -(o+c) = (c -a)P {3) Ldi gini : (tdm t6t) Ddt sd chia lir p (p lA sti nguy6n td, la(Q+ 1) - al(ac - t - P)b = . 11 < p < 100). Suy ra ba s6 hang trong ngoQc = [c(1 -p)-ct|@a-1+flb (4) sd Ld. pmn, pm, pn (v6i (m, n.) = 1) X6t hai trd8ng hgp sau : Mnt kh6c do c6c s6 phAi tim ld khric nhau i) 6 = 0. Khi dd i a, (: * 0. Met kh6c, trl (1) 6. Tt d
ai u - 1 = 0'+ u = tth6am6nu > 0. Khi Nguydn Tidn Trung, Hod.ng Manh Qwang 8T. d6 x = 0, Thay.vio ta thdy r = 0 li nghi6m Trd,n Trung Nghia,9T, Trdn Ding Ninh, Nam cria phtrong trinh dd cho. Dinh, Nam Hd. ; Duong Dinh Hit ng, 9 Nga Li6n, b){2 -uz *l = 2Q^t+ 1)=+ ,["r:AT : ht*L Nga Son, L€ Xud.n Hirng 9T, Lam S0n6n2u+l>-0)tac6 H6a; Kibu Thu Hibn 9T NK Quj'nh Ltru, NghQ 2_uz = (2u*l)Z An; La Huy, L€ MQu Tilng 9NK Ddng Hi, hay luz+4u-1=0 Quing Tri; L€ Minh. Trudng 9l Nguydn Minh I Philong, Thrla Thi6n - Hu6, Nguydn l{brug Giii phrrongtrinh niy ta dnclcu t= 'L, u2= E NhL,n,9 L6 Hdng Phong, QuAng Narn - Di f)o u 2 0 n6n ta loai nghi6m ur. Ning; LA Quang Nd.m 8T Chuy6n Drlc Phd, Ta ccir - u1,- 1 : (Lll)z - 3. = -24125. Quing Ngei ; Vo fhi Zy, PTCS Lrrong Van ThrS lai, ta thdv x = -24125li nghi6rn clia Chrinh, Phri Y6n ; Nguydn La LUc.8A1 Ddm phuong trinh dd cho. Doi, Minh HAi. VAy phrrong trinh dd cho cd 2 nghiQrn VIT KIM THUY 24 r,, : a, x, Biri T5r205. Cho duitng trbn (O ; R) ciUng du.dng trbn (O' ; R') sao cho td.ru O rtdnt tr€n Is{h&n x6t : Cd rdt nhi6u ban gr?i ldi gini. dudrry ffin(O' ; R'). DdyAB cftoduimg tri;n(O ;Rt I{du h6i c6c ban d6u giAi dring. Da sd cci cdch gidi nhrr tr6n. d.i. dQng uit tidp rilc,:6i duitng trbn (O' ; R') tqi 'rd NcuvBr.r d.idnt C. Xor d.inh ui ti cia d.d.v AB dd AC2 + BC2 dat gid. tri l6n nh.d.i. Bai T4l205 : Tom giac ABC uudng 6 A c6 .,,.,,,,.,,.i...,, AB : 6cnt, AC : 9cm. Gqi I lit giao didm cd.c l.,di giai : Goi H, K ' dudng phl.n giot, M lii trung didm cia BC. Tinh laln luot ld trung dieim ,,,, ,, sd do g6c BIM c,iiaAB vichAndtidng,':: ' , L&i g{rii mronggdc ha ,fi 0 -"6"9 ii:il,:ll.l1l: Theo dinh li O'C, ta cd 0H, AB Pitago tinh duoc "a i::,:i:::i:ii|;iiii:i. hinh chfr nnpt OACf
Beri T6/205. Cho 10 sd nguyan duong =+hdrn F(x)d6ngbidntr6nR+F(z) >f(0) (r1t ...t ar*r. Chirtg minh rd.ng tbn tai cd.c oarrfi > 0€o.> 0(Dpcm)E' sd cre {-1, 0, I fi = 7, ..., 70), khOng d.bng thdi Cdch2(ciaPhanHod,ngVi€t,Qudc hoc Quy 10 Nhon, Binh Dinh) :V6i m6i k = L,2, ..., n det : bang 0 son cho s6 \ c,a, chia hdt ctto 1023. 11 qL* s tl . LIAJL i:1 A.: K - ). cos--- - . n*I Ldi giii : (theo Nguydn Thanh ThiQn, 1=0 10CT, L6 HdngPhong, TP Hd Chi Minh). Cci : 10 , h,n : 2sinhn_ * X6ttatcacac.64 : 2 b,o, trongdcib, e {0, 1}, ( 2sin ,rJ 1 ) .Ao l: I i = 1,...,10. Cd tdtc.h.ld2lo - 1024c1.cs6A,, \kn kn *sin \kn *sin --"n*l sin ---n+I j : | ...,lo24khac nhau nhr.t thd. Khi chiaA, - n*L- - -- \kn (2, + t)kn (i : 1 , . ,, lO24) cho 1023 cci thd drroc c6c s6 drr . Ia 0, 1, ..., lO22(1023 s6 du). Vay theo nguy6n li -sin n+I- +...+sin n-rl- - Dirichl6 5t pnai cri hai s5 A* * Ah khi chia cho . (2o -l)kn : Sin kn - * - StIr 10 n*l n*I ----:- lO23 cd cung s6 dtr. GiA srl dri liA* : D b*fli (2n + ltfu kn i_l *sin:=sin n*l n*l-* 10 - vdA,, = 2 bnioi. Ta ccj i ,kr \- -j"n*l \t = +sin ---- 12kn 0. i:1 l0 l0 kn A* - A,, = bnpi: R6 ring, sin, * OVk e {L,2,..., n} n6n }btiri-Z -.,. 1 i:-t i:1 Ak : 0, Vk :T, n. Do dci : 10 ,tn _+ $^^^^k.2tr_ : EiL ?= )'t p /-+\ ., @ti - bo)o;: 1023 :) ) o,.os'"-"', - i:1 l=tt \n +l/-.L.L',k"',"n+1 l.0k-0 Det c, : bri - br,i, € {0,1} n6nsuyrac, € {-1,0, 1} Vi6ki,bhi =i,o,i k:o ++ 0"o" tLT1= t Vd viA* * A,, ndn c6c c, khOng d6ng thdi bing TI khdng. Dpcm. * 2 oo Au: Qt * 1)o,,. = (n * 1)o,, NhAn x6t. Cdc ban sau ddy crlng cci ldi giAi k- | t&. Le Minh Hidu, Hoitn.g Sao D6, 10T, Lam YiP(x) > 0 Vr C Rn6n ? > 0, hay Son. Thanh II5a,L€ Huy Khanh,12T, Phan Bdi (n * l)ar.> 0 = r.,, ) 0 (Dpcm) [I. Chiu, Vinh, Ngh0 An ; L€ Trudng Giang,lzCT, Nhfln x6t : 1. Cung giAi theo c6ch 1 vi cho DHSP I; Hit Klfinh Liruh 9A, Chu Van An, He ldi giai dring cdn cd c6c ban :Vu Thitnh Lotug, Vu N0i Iluy Ph.uong (11CT PTI'{K l{ii l{ung) ; Dd Le T6 NCiL}Yil]N Tdn (lZK PTTH Thang Long, Hd NQi), td nii fZl:tlg : Clto P(il : a,, * arcosx * Trudng Giarug (12 CT DHSPi Ifn IVQi), Nguy4n * +... + ancosnx nhdrt gid tri duong arcos2x Minh Tri (11CT DHTH Ha N6i), Dinh Trung Vr € R. Chtir'g nrin.h : o,, ) 0. ilazg (11M Marie Curie, He NOi), Nguydn Van Ilidu (12 A DHSP Vinh), I-a Huy Khanh (12T, Ldi giii : Cr{ch 1 (cria Bii Quang Minlt, PTTH Phan B6i Chdu, NghQ Ln) ; Nguydn Van PTCS GiAng V6, He NOi) : X6t nguy6n him Hodrug (114 QudchocQuiNhon, BinhDinh) vdVo dl o'2 an Hoitng Tiung (1 1A PTTH, chuydn Tte Vinh). F(xt : o.1 + --sinr* ^sir8r+...+-sinnx ,'lzn 2. Rdt nhi6u ban d5 giei bdi to6n theo phttong tr6n R. Cd : phdp cira c6ch 2. MOt sd ban cho ldi giii sai do mic F'(x)=P(x)>0Vr€R phii c6c sai ldm cd ban trongbi6n ddi hlong girlc.
3. Bing phrrong phrip cria C:ich 2, ban (ft-bdn kinh dttdng trbn (O i OA) n6n Il cung Nguydn Vu Hung (10C PT Chuy6n ngoai ngu th6a mdn phuong trinh d6 cho vd ta c6 H e d. DHSPNN Ha N6i) da giei dring Bii torin kh6i Ndu MBC kh6ngd6u thi H + O vi tAp hop didm qu6t sau : "Cho P(x) = ao* arcosx *b rsinx * ... * P li drrdng thing d. Do P nim tr6n drrdng trbn ancostlx, * brrsinnx nhQn gid. tri. duong Vr e R. Ole n6n ld giao cta drrdng trbn dd v6i drrdng Ching minh : oo )0".Tuy nhi6n, Bii to:in vr)a thing d vd c6dfng hai vi tri ctia P. Ndu LABC n6u cbn cri thd gitri m6t crich ngin gon theo d6u thi moi moi didm tr6n drtdng trdn Ole ddu phrrong ph6p ctra crich 1. th6a mdn di6u ki6n di cho. 4. Ngodi c6c ban dd n6u t6n d tr6n, c6c ban Nhan x6t. Cri 20 ban giai bei ndy vi d6u giai sau dAy cflng cd ldi gi6i t6t : Vuong Vu Thdng df ng. Ldi gi6i tdt gdm cri eic ban : Trd.n Tdt Khiant (9A1 PTCS GiAng Vo II Ha N6i), Pham Huy (10T Lam Son, Thanh Hda), Nguydn Thi Qu\,nh Tirng (SAPTCS Bdven Ddn Hd NQi) ; Trd.n Van Hoa (1lT PTTH Nang khiSu He Ttnh), Phant Ii Binh (LlTK4,Bdc Thrii) ; Nguydn Trerug Nghia Hilng (10 CT DHTH He Ndi), Ifb Si Thd.i rcT, (11A1, PTTH Vi6t Tri, Vinh Phf) ; Ngl Dtlc PTTH, DOngH4 QuAngT4). Duy, Vil Hoa Mai (10CT, Trdn Phu, HAi oANc vreN Phbng) ; Pham Manh Cuitng (1t1, Chu Van An, He NOi), Nguydn Quang Nghia (10 CT Bei T9/205 : Cho didm M bdt ki trong tant DHTH Ha NOi), ViL Ch; Cxdng, Trinh Thd ABC. N6i AM, BM, CM cdt cac canh ddi gid,c di€n tuong thtg tai Ar, 81, Cr. Chtmg minh mng : Huyruh (10A, L0 HdngPhong, Nam }Jii ;Thanh Huong (11T, LrrongVen Tuy, Ninh Binh) ;Nhz T_ fcM Qui Tho, L€ Minh Hidu (l}T,Lam Sdn, Thanh \l uc, '4 H 3r'[Z . d, b, c. Ching minh rang tuAru dudng tritn O-le RAt hoan ngh6nh c5c ban sau ddy di d6 xudt cia tam gidc c6 tbn tai dilng hai didm P d,d : vi6c khai thric bdi torin T3/199 cung nhrr bdi P42(bz- c2) + PB2(cz - a2) +pC@z - bz) : 0 Tgl205 d tr6n theo hrr6ng khrii qu6t hda (thay Ldi giai. K6 h6 truc toa d6 vu6nggdc Oxy vdi can bAc hai bing cdn bdc n) hay dat bdi torin A(xr, !1), B(xr, !2), C(xr, y) vd. P(x, y). Tt gia tudng trl trong kh6ng gian : thay tam $5,cABC thidt ta cd : bdi tf di6n ABCD thi bdt d&ng thfc cdn chrlng [(x - x)2 + 0 - y)21 (b2 - c\ + minh s6 thay ddi ra sao : LA Quang Nd.m, 8T chuy6n, Drlc Phd, QuAng Ngdi, Ngzydn Vu + l(x - x)2 + (t - y)2)) (cz - ar) + Hung 10C PT Chuy6n ngrl DHSPNN He Noi, l(x - xr)2 + (y - yz)2 pz - uz1= 0 (t). PhqmThy Hilng llAPTTH Chuy6n Th6iBinh, Sau khi nit gon ta c6 cdc h6 s6 cta x2, y2 ddu LA Xudn Hing, 9T Lam Son, ?hanh Hria ; bing 0 n6n (1) li phuong trinh bQc nhdt vi li Nguydn Thi Hdi Ydn I0Cl, Qu6c hoc Hu6, L€ phuongtrinh drrdngthing goi ldd. Gsi O .FItheo Quang Minh, To5n 11K5, PTTH Ndng khidu, thf tu ld tAm drrdng tron ngoai tidp vi truc tAm tam Bic Thdi, $6cABC.DoOA=OB = OC =Rn6nOth6amdn NGUYEN OANC PTTAT phrrong tdnh da cho vi O e d ; hon nira Bei T10/205. MQt cd.u nQi tidp fi diQn HAz + a2 : IiBz *b2 = HC2 + c2 - 4R2 A\A3A4 tidp xrtc udi mQt ddi di€n d.inh A, t
tai Br. Gqi dr lit dudng thang qua trong tdm Son, Thanh }{6a; Phan Hoitng Vigf, Qu6c hoc m.qt d.6i d.i€n dinh B cia ti di|n BrBprBouit. Quy'Nhon, Binh Dinh ;VO Hod.ng Trung,llA, uuOn g g6c udi mst d.di di4n dinh Arcia tt diQn PTTH Chuy6n TriVinh, L€ QuangMinh,llKt A.A/3A4 G = TJ). Ching minh rang cd.c PTTH Nang khidu B5c Thrii ; Pham Huy Tirng, duong thd.ng d dbng quY. 8A BdVan Ddn, D6ng Da, Hd N6i. Ldi giai. (ctranhi6u ban). Goi1, G vdB'r(i = Ldi giai tr6n hoin toin d6p rlng drroc bAi to6n 1,2,3,4) ldn lugt la tdm mit cdu (/, r) nQi tidp sau ddy, mOt dang tdng qu6t h6a cria bii to6n trl dionAr,4"44o, trong tAm trl di6n BPP3B4 TlOl2OS d tr6n ; "Goi B, ld hinh chi6u (r'u6ng va trong tAm met ddi diQn vdi dinh Br cria trl di6n gric) cria mOt didm M cho trrJ6c trong kh6ng gian B.B2B3B4. Thd thi 1 cflng li tdm mat cdu (9) tr6n m4t phing chrla m4t ey+#t, d6i di6n v6i ngoai tidp tf di6n B,B zBzB+ vd, theo tinh chdt dinh A, (l = l, 2, 3, 4) c:&a mdt trl diOn AiA$+ Chrlng minh ring c6c dudng thing d. trong tAm ctra trl di6n, ta c6, CE' i: -* *1, di qua trgng tdmB'rciametB{LBt, d6i di6n vdi Do drj B'ile. Anh cira R, trong ph6p vi trr dinh B cira trl diQn B ,B p rB o vit vudng g6c v6i V- 1tAmG, tis6h= -: mat phBLngA/ y',, ddngquY 6 m6t didm". G-l NGUYEN OANC PUAT Lai vi d, ll IB, (v\ cirng vuOng gdc v6i BiiLi LllzDl. Tran hinh ua du6i, xy lit truc mat A/ir, d6i di6n chinh cia m1t thd.u kinh ; S ld. mQt didm sd.ng v6i dinhA, cria trl di6n thQt, nd.m ffAn xy ; S' la d.nh thq.t cita S qua A44y4i vid, di qua thd.u kinh ; F lit. ti€u d'idm uQt cia thda kinh. 82 B', n6n, tt dci suy ra d, Ia Anh cria drrdng thing A, : 118) trong Bang ph.€p uE hinh hqc, hay tim ui tri quang ph6p vi ttt V 1 ndi tr6n. C6c drrdng thing A, t1.m O cfla thd.u hlnh. G-- -) Hu6ng d6n gi6i. Dua vio c6ng thrlc thdu ddng quy tai tAm l mdt cduS@rAprBo) ; suy kinh cri thd chfng minh cOng thrlc (hdu h6t cdc ra cdc drrdng th6ng d ddng quy tAr m6t didm 1'. bei g'Ji ddn d6u lim drioc) ,362 = sr. ss. vai Didm d6ng quy f ' niy ld Anh cira 1 trong ph6p vi ttr 3 didm S, fl S' cri vi tri de cho, dr)ngph6p ve hinh ncii trOn, vd 1' cring chinh ld tAm mdt cdu hoc dd dgng didm O. C6 nhi6u crich vE khric 1 e'(B'tB'28'3F'o\ cd brin kinh ,' : gr. nhau, nhrlng uich v6 cria em Nguy6n Trung Thdnh li don gi6n hon c6. Dtrng drrdng thing Nhfln x6t. D6ng tidc cci mOt sd ban vi khOng x'sy' vu6ng g6cvdixy t?i S, rdi ldy 2 didm M vi biSt sit dung v6cto (hodc d6 dii dai s6) n6n ldi N d hai b6n didm S tr6n dudngthing ndy sao cho gini thiSu chinh x6c khi kdt luAn 1'ld m6t didm SM = SF vA SN = SS'. Sau dci v6 dridng tron c
Biti L21205. Cho m.ach diQn nhu hinh ua DINH r,i rnUNC HOA . . . Rr: 2R, - 6R ; Rr: 2Ro = 8R. Hai dOh Dr (Tidp theo trang 2) ud D,gidng nhau, c6 hiQu di1n thd dinh m,*c Giai : V6i m6i s6 trr nhi6n m, det Fn = 22^ + I ld 24V. 86 qua gid. tri di|n trd ctra ddy n6i, thi m6i nt : 0, 1,2,3, 4 ta co {,, ld sd nguy€n t6 kh6a K ud. cd.c ampe kd. Khi K d,6ng G) chi conF. =22t + I :232+1=64lpv6ipldsd 1,5A; (i) chi o. U/ nguyilir t6, p > 216 + I : Fa. Do @, 232- 1) = 1 \') n6n theo dinh ly Trung Hoa v6 sd du cd v6 sd s6 tu nhi6n k thIarrrdn k = 1 (mod (232 - t)64D va tu - - 1 (modp). Ta s6 chrlng minh vdi k > p thilfr.2n + I (n = 1,2,...) d6u ln hop s5. Thnt vdy sd n vi6t drroc du6i dang n : 2111(2t * I) v6int., t > 0. Ndu nz : 0, 1,2,3,4 thi vi k.zn + I = 22^(2r+1) + 1 (mod (232 - D), 232 - | i Fn vd zz*(v+ t)+ 1i Frrrrlnk.2,\* ., ".n + 1 : :tr'or. Vi k.2tt + | > p > f'o (do h > p) it ndr, k.2'1+ 1 ld hop s6. Ydi m. = 5 thi k.zn + I = 2n + 1 (mod 641). Matkh6c2n+l i tr'r+k.2n+li 641,md 1. cA@ua@ehi Kmd. Hay mr dinh sd chi h.zn+ l rFs >641+k.Zn +1lehgps6. YOim > 6 thiz = 26.h vdi hldmot sd tu nhi6n. 2. Khi K nt6, hai dZn D, ua D, sd.ng binh thudtlg. Tt day ta c6 k.2\ + 1 : 22uh + 1 (modp) hayh.Zn+l: pi226h-Li 22o-li Fri p. Hay tinh : a) HiQu d.iQn thd ngubn U ; YQy k.2D+ t ldhop sd (k.Zn+ I > k > p). b) Cang sudt dien fiau thu di,nh milc crta T
0 SCN l6p 9 cri bdi totin : Cho LABC dbu XT]T\{G nii tidp trong dudng trdn td.m O. M lit didm SUAI{H bdt ki trOn cung nh6 BC, ta c6 : MA:MB+MC (1) Moi ngUdi giAi bni toSn niy m6t crich d6 ding. Song sau khi giAi bdi to6n niy cdc ban da cd suy nghi dd ph6t trien bai to6n. Sau diy QUEI{ TH{-lCIC t6i nOu ra mdt s6 suy nghi dd khai thac bii HOANG NGOC CAXTT to6n. Goi a canh cia L,ABC, D ld giao c&aAM vd BC (Ha) MAz + MB2 + MC2 : 2az (5) Lai binh phrrong 2 vd o a (*), mt bdc b6n hai vdcria (1), 6p dUngnhu cach tr6n ta lai cd hO thfc MA4 + MB4 +MC4 : 2al (6) (C6c h6 thric (5), (6) rip dqng cho didm Mbdt ki trOn (O),hai thrlc niy cring cci trong mOt s6 hQ s6ch chon loc, song crich chilng trinh cdn ddi). Quay lar bai to6n A (H.b) ta cci clrc h6 thrlc sau : 1 IAz +IBz +ICz +rgl]+rc1+rc| = a2 +bz +c2 (7) IA4 +IB4 +rca 1 5tte! +rcf{ct) : o.4 + b4 +c4 (8) Apdungh6thrlc ( 1) vichfngminh AMBD ^ LMAC (Trong d5 a, b, c canh ctla tatn giec ABC). 111 Ta c
BAi T9/209 : Dudng fidn (I,r) n6i tidp tam gidc ABC tiep xuc vdi c1.c canh BC, CA, AB ldn lttot o cric didm M, N, P. Dung c6c drrdng trbn nbi tidp cac tam gi6c congtdniANe, AfM vd, CMN, va gqi c6c tidp didm tr6n cAc canh AP, BM vi CN ldn lrrot ld H, I, K. Ddt AH =r, BI : .y, CK : z. Chrlng minhx * y * z > 2r. DE RA KI I{AY Khi ndo thi xiy ra dang thfc^? Ltr OUOC HAN cAc r6p rHCS -Biri T10/209 : Ki hi6u goi BC, CA, AB, Bai T1/209. Chrtng minh ring, phrtong DA, DB vd DC ldn lrrot ld sd do cdc gcic nhi trinh x3 + y3 + z3 - Sxyz : l9t c6 nghiOm di6n canh BC, CA, AB, DA, DB, DC cira m6t nguyOn x, !, z e Z cho moi s6 trJ nhiOn.n. ttl di6n ABCD. ld.dy xdc dinh hinh tinh cia tr1 DAM VAN NT1I di6n ndy ndu biSt : B,iri T2l2O9. Giii phuong trinh BC : DA CA DB --_: = ya +4y2x -l ly2 +4x y -By*^.*, -n^* * 5.2 : 0 sin BC sin DA ---:-_ sin CA sin DB AB DC Bei rg/2oe , ri,,,s'i}t),il1).1?i1,, sin AB phuong trinh : "*" sin DC (r2 +y) (r + y:) : G -y)3 'i RlNr I B.4.NG Cr ANCi DT\NG IIT-ING'IIIAN(; . cAc oii vAr li Bei T4l209 : Cho tam gl5,c ABC vu6ng d A. Biri Lli209 : Mdt hat kh6i_lrtong nt : 10 3 kg, Dridng ttdn (I) n6i tidp tam gi6c tiep xfic v6i tich di6n duong a : 3. 10 5C, duoc treo bang AB vit BC 6 P vA Q. Dudng thing di qua trung dAy rninh I : 10cnt. tao thanh con I5c. Con l6c didm I, cu,a AC vd tAm I cat canh AB 6 E. ndy duoc treo trong m6t tu di6n, khi tu di6n Dudng thing di qua P vd Q cit dudng cao AH chrra tich di6n thi ddy treo con l6c vu6ng gcic vdi tai M. Chfng minh : AE : AM c5c bAn tu Khi di6n tnrdng trong ltrng tu di6n v DAO TRIJdN(} (iIANCJ. dat ctrdng d6 E : ,O' Bei T5l209 : Cho dudng trdn (O; R) drrdng ; thi stic cang cua ddy kinftB. Goi m6t trong hai cung nta dridng trbn treo (hic con lac dfng y6n) vir chu ki dao ddng ld AmB. Dung drrdng tron (1, r) tiSp xric vdi cung cua con l5c la bao nhi6u ? R Giai bai to6n cho 2 tnrdng hop : AnB vd tiep x[c vai AB sao cho ,' = - Bin drrong tu di6n d tr6n I - Bin duong tu di6n d dud.i oANc vti,N Pi{AM llUN(i QL}YIr'f Biri LZ|ZO9: Cho mach di6n nhrt hinh v6 CAC,LOP THCB fJ = 30V, r": 2Q, Et = 6Q Bei T6/209. Tim tdt cA, cac sO tu nhi6n /z R: = 2t) M / lA nrdt thanh di6n trd dong sao cho tinh tiet dien d6u co chi6u dai 20cnt vA tcing tu,+l + @ * l)r chia hdt cho 5 di6n trd la lpQ. Ddn D : 15V - 37,5W 86 qua di6n tr6 cria ddy ndi vi ampe. he. TRAN DL]Y IIINII ,2Og Bin T7 : X6t dAy sA {r, J ,>t xtu dinh b6i x' : 113' (2n * 1)xr. = 2t) + 2n x,r_, (n > 2) Chrlnr€i t6 rang : *,, :2 c|/ek + 1) vdi moi n >- I k-" NL;trYr N Ir' I)t NLi Bdi T8/209 : O m6i m6t 6 vuOng crla mdt bAng k6 6 vu6ng Con chay C cci thd chay ttr N din M. . hinh chrr nhAt kich thu6c 1964 x 1994 drroc 1. C cha.y til N den M, xac dinh so chi cua ghi m6t sd c
that the surn of the numbers written in the rectangle is not greater than 3334. t-F] THON(i NI IA'I' Tll2O9. Prove that for every natural Tgl209. The inscribed circle (l r) of a number z, the equation triangle ABC touches the sides BC, CA, AB 13 +y-l + z3 - Sxyz : l8t1 respectively at M, N, P. Construct the has integral solution (x, y, z € Z) inscribed circles of the curved, concave DAM VAN N}II triangles ANP, BMP and CMN and let H, J, K be respectively the tangent points on the sides TZl2O9. Resolve the equation AP, BM, CN. PutAl/ : x, BI : y, CK: z. Prove ya + 4y2x - l1y2 + 4xy - 8y + +8x2-40x-t52:0 that r * y * z > 2r. When does equality occur ? 'IItAN XI]AN DAN(i TJ|2O9. Find integral solutions (r, y) of the ^ Tl0/109. Let BC, --1"19{'-jII CA, AB, DA, DB ---- equation and DC be respectively the measures of (r2 +1,)(r +y2) = (r - y)3 dihedral angles with sides BC, CA, AB, DA, DB, DC of a tetrahedron ABCD. Suppose that I)..\N(i I ItJN(i'fl{r\N(i BC DA CA DB T4l2Og Let be given a right triangle ABC, ----_l right at A. The inscribed circle (1) of ABC sin BC ^: DA sin CA sin DB sin ^, ^= touches AB and BC respectively at P and Q. _AB DC The line passing through the midpoint F of AC ^- sin AB sin ^DC and through the center of (1) cuts AB at E. Determine the form of tetrahedron. The line passing through P and Q cuts the ]'ItINI I BAN(i (;I,,\N(i altitr,rde AH at M. Prove that AE : AM I)AO TRT]ON(J (iIANG T5l2O9. Let be given a circle (O ; .B) with a diameter,4-B and denote by AntB a semi-circle deflrned by AB. Construct the circle (1, r) which is sua r,4r tangent to the arc -4rzB and to AB so tttat r : Rl3. Trong TC s6 208 (th6ng f0-1994) DANG VIEN o Ddng 8 cdt 1 trang 7 dE in Q,, For Upper Secondary Schools ,i *i...x],_t: ; T61209. Find all natural numbers z so that nay srla lai ld : ,n+l -',r (iz * 1)n is divisible by 5. *i .fltAN , *i., ,,. , xi,,t DLJY IIINH T7 l2OS. Consider the sequen". {*r) nr t ( rr_t < ,i_r < rr, (1). Theo 4 defined b.y x,.J= - dinh Ii Viet ta cri (2n + l)xr, : 2n + 2nxrr_, (n > 2). rT*xl+... '-r ''t I xi,-t : ;n-1 Prove that for every n > l, *rr r:) L tct vn ri . x) ,,, xi,_t : Q, i A:(i-"2h +lt o Trang 11: Ho6n vi tdn 2 tidu muc N(iTJYL'N LE DUNG cld.c l6p THCS vd. cic ldp THCB cho T8/209. In each 1 x 1 - square of arectangle nhau. of size 1964 x 1994 is written a number the Thnnh thit xin l6i ban doc ! absolute value of which is not greater than 1. THVTT. Suppose that the sum of the numbers written in every 3 x 3 - square is equal to 0. Prove 11
pAr Ax oE rur eucic GrA cHeN Hec srNH Gror ToAN LoP q xANI Hoc 1993 - 1994 Ngiy thi 3-3- Lgg4 (thdi gian 180 phrit, kh6ng kdthoi gian giao db). r - oii rnr sdNc a a) Chrlng minh ring dddng trung truc cria ML di qua trung didm D cta c4nh AB, Bdi 1 ; a) Tim c5c nghiOm nguyOn cria phtrong trinh b) H6i v6i di6u ki6n nio ctia tam g;d,c ABC thi trung tnlc cria ML cing li trung tnlc cria : 7x2+73y2=1820. c4nh AB ? Chrlng minh di6u dd. b) Tim t* ce c5c s6 nguyOn td p sao cho 4 : Cho tam giac ABC vu6ng d C vd cri tdng cta tdt cir cdc rt6c s6 trr nhi6n cria sdpa ^ {5. b) Ttm ti sd giita cac cqnh cria tam g;dcABC. 2) Tinh gi5 tri ctra tdng (a * c)2 + (b + d)2, khi cho bi6t S : {5 DAP AN b) Giai h6 phrrong trinh v6i dn dn sd x, y, z BANG A sau ddy : Biti 1 : xy yz zx x2 +y2 +22 d\h 1820 i 13 vit 13y2 i 13 nlnTxz i 13. ay*bx bz * cy cx * az a2 q62 4 12' Vi (7, 13) = 1, n1n x2 i 13, tt dri ta c
Theo kdt quA d tr6n (2n)2 = 4n2 = COngttng vd bi6n ddi dr.toc : i :4p4 +4p3 +4p2 +4p +4 (2) 1a2 + bz) 1c2 + dL2) :I Tt(1) vd(2) suyra ip2-2p -3=0. Trl ('r) : a2 +62: "2 a42 n6n suy ra Til dAy tinh drroc : Pt: -l,Pz= 3..Y6i P Ii s6 ng,:y6n td n6n chi l5y gi6 tri p = 3. Ro @z +b212 = (cz *rt\ =*3 rdngp = 3 th6a mdn di6u kiQn cria bii torin ?t dd tinh dtroc : I + S +82 +83 *84 : IZI = Ltz. Biti 2 : a) 1) Vi (acl. - bc)2 * (ac * bd)? : a2+b2:c2+a, - lE:L = Vs-..I5 4 -az d2 + b2c2 - zabcd. + + a2c2 + b2dz * Zabed. Tta:-rb? +c2*cF=T5"a _ azdz+ b2c2+ azc2+ b2cl2 : @z+ b4{cz+ d\, V6i gin thidt oc - bc : J. thay vdo bidu thfc 2 -;vo 2or +Zbd = - . Co.ng tr) vd, iridn ddi duoc : tr6n ta cci : 1 * (ac + bilz = (a2 + 621k2 + d\ (L) (a+c)- + (b+d)- = 4222i8 V6i S : a2 + b2 + c2 + dz + or: *bd, rip dtrng lE"*r- l/3-) = T-t = -3 bdt d&ug thrlc Cosi ta cci : 1a2 + b2) + 1c2 + d3) > VAy S:lr5 thi 1a+cyr + \b+(t)2: (dpcm) > zrt@'+6\17 +d\, do dci : T ' s > ac 4bd +2 {d@T+ bT;r-+75. @ Tn b) Trttdc hdt ta chtlng minh m6t dn sd tidt (1) vd (2) suy ra : ki khong thd nh+n gid. tri O. S >- (ac + bd,) + Z ,[T + lac-+ bdf ThAt v8y, ndu s : O thi t'i phr-tong^trinh dd cho ta cri ngay yz = a vd ri * yt + z: : A, . D6 thdy S > O, vi riag tr) dci suy ra x = 0, y = O, z -- 0 di6u niy llm 2\tT + @clbAY >lac +bdl cho c6c bidu thrlc cira phrrong trinh v6 nghia. V6i Dd cho don giAn bidu thiic, ta dat r: or *bd. ! = 0 ho6c z = 0 cfing cirdng minh trtong tu tr6a ta drJoc : Tha;z vi-o nhu tr€n, Vayr * 0, y * 0, z * 0. S > x + z {TTii. !'i vd phAi dirong n6n Theo chfng minh tr6n ta c6 xyz ;e 0. Nghich binh phuon g 2 vd dri.oc dAo eic phAn'"hric cfia phrrong tnnh de cho dtloc : s2>(x+2,[T+if= abbcca o2 462 q"2 :x2+4(l+x2)+4xll+7 _J_-_I_-_;_- xyyzzx --iy" x" *z- (1r : (1 + xz) +4x \|TTF + 4u,3 + 3 = Ta cung chrlng minh dttoc c6c s6 a, b, c ddu vi ndu mdt sd bang 0, gtrh s& a = A = $lT +7 + Z:qz +3 ph6.i khric 0, va)L,s: > (dt {;r +zr)2 +3 > s. Tfi dd ching han thi ti{ (1) ba cei : L =! *9 = 9, t,l S > {3 (dpcrn), yyzz' dAysuyrab=0,c:A 2) Theo chilng minh tr6n, La bi6t ring Didu niy cring idrn cho cdc bidu thrlc d trlng + t|: (*) vd c:&,a phrrong trinh v0 nghia s = dE xdy ra chi khi , )1:l!'-:,c: +r- *2r = 0(**) L!1 Ydy abc phAi khd.c 0 (sbc * 0). fi: fi +lz+ 2x = 0, suy ra ngay Tt (i) ta co : x < 0. Chuydn vdlJT +F = -2.x. a b c 3(o:+62+.1) Binh phuong dudc : 2t:+**:)= 'X ,,;;;j,hay .V Z XLl-yL*ZL I *x2 : 4x2 > &x2 = L > -t2 :i *: - a_+_+_=__,_^___;_ b c B 1o2q67q"2) (2) = I * L Y z Z xL+y;+z!_ Vir < 0 n6n chi lay gi6 tri, : - 16g. Tt (1) viL (.2) bidn ddi drioc : Kdt hop v6i giA thi6't bei to6n suy ra o b c I a2+b2+c2 /e\ .1 a . \u,/ iaz +az = s2 62 q (*) 2', Jt'+y'+zL S - {3 chi khi : lr lr= ac *bd : - xvz : D4t:aoc ="7 ^= i, tacd : x = oi, J : bt, *1(*n) I t nl lad-bc:L (*x'") - Tt (**) vd (***) cri : ?hay cdc girl tri niy vio d6ng thrlc : L b l. *21624r2 a.2c2 +b2dz +1abcd =-.) Y 2 )ct+yt+?t' b : I oz462qr2 1 I *Zabcd -. h^v 2t! : i' ? = 2rz o,2d2 +b2c2 1 6aY bt 2lo4 b2 + r: i3
Suy ra '. LIAO = Ttdaytinh drroc t: 0, t= l.Z U, t*0 non LIBA, !*l o, hay o -f c -a c D6 thdy : APOX ^ LPAI] vn AOXY -" ;lAlJC b - o > 0. do dci --=;- '22 > ; cho n6n N phAi XY OX nen:Bc = AB: Ap OP nam giila O vd B. Do d lrn6n -: -: ^ B\Y cri : YC > YL = OQ (3) Tt (1), (2) ve (3) suy ra : ,2a2 +2b2 +c2 -2ac -2bc. a : BX* XY + tsC > Op + Oe + OR (dpcm) r-l \4t T BeIi 4. a) GSi dQ.dei orta cAc canh AB : c, AC=b,BC:a,.YiA
Ta cci h6 phrrong trinh ddi vbi a, b, c I : =Ap a,+ Nt:3c (**) ME 2 = DF (.1) .) , o.2 + D'=c' LF: =BP 1 : DE (2) Z a r Dat x: vA.y: -- L , dtlclc h6 phrrong M6t kh6c to ja: c trinh r,6i c5:c dn r., y * A c ^ _lrio : _z EAi{ = z reL -\\ epL : "" DEP : EF!& gcicylQlhinh 4qb ha$fvi vav jr+3i':3 (l) ta c6 a : DEP + PEM : DFP + Pp,L G) l.r2 +yz: 1 e) i tt uy giri tri cta x tinh theo y fr phrrong tr] dri cti DIUI = DL trinh (i) vao pt (2) vn rirt, gon drroc pt bic hai LDLIL c.dn 6 D non trung trtic cria M-L phai cldi vdi y : 5;v2 - 9y + 4 : O. qua D (dpcm). : : 4 (*) Ghi chi : Co thd chrlng rninh ria EP Tinh dr.ioc ngay.yr 1,.y. _ nim giua 2 tia ED vit EM nhd sau : r) Vi P trong AABC n6n AP nAm giita 2 tir+ Thai' vAo tinh -r r,=0.xt:J tL5 : -4B vi AC do dci : AB, PC nim tr6n 2 nita mp 4 o cd bd chung AP. LiJ dd D vd M cung.nim ir6n y : i th6a m6n 1) Chi cd nghi6m r = =. t;) ' 2 nfia mp r:ci bb chungAP, ttl cki ED, EM ndnt tr6n 2 nrla rnp cd bd chung AP. do dri EP phAi di6u ki6n bii todn. li tia nam ${ta 2 tia ED vir EM. 3 b 4 * 'lrtong ttl Cil,l cho l.'P nam giua 2 tia ItD Vayx: c = 7, o -v- -= c =5 vi FL. a D b) DII li trung tryp_ctia M!_gtrt g li, trung Tri dri : tinh drroc trtic cira AB ktii ADM : BDL {.v\ d5 cci b 4 ab L), = Dr). \{ttdn vAY cdn c6 b-4DM = LBI)L, T
T.orrg thi todn tuydn sinh vio dai hqc Y Drioc c:ic tinh phia Nam dd ZI (l$. i nanr 1993 c
ddi mdt canh ndo dd ; linh c6m todn hoc s6 AB AC mdch bAo rang dci chinh Ii canh BC, VAy ta sin(AB) . sin(CD) sin(AQ . sin(BD) drr do6n : BC sinrBCt sin(BCy . sin(AD) Vasco - k. Sanc. Soa,,.-- (sd Ldy g,6i han khi D ti6n ra vO cuc voi chrl y : k tim sau) Chring minh du do6n : (lim sin(AB) : Iim sin(BC) : lim sin(CA) = 1 D+q I)+q f)+a lim sin(DA) : sinA D-a lim sin(DB) : sinB D+q t: lim sin(DC) : sinc - --- ltA', D-a - 0 AB BC CA Ta cci : sinC- sinA sinB drj chinh la c6ng thrlc sin cho tam giSc md chting ta d6u de bidt. Gid ddy ta cci thd khing 1 v^sc D_ BC dtDHA'= dinh ring : c6ng thrlc sin cho tam g;ac ABC chi rf ld tntdng hop ri6ng cta cdng thr.lc sin trong trl 1 n DH . HA' . sinlBC.; : 1 BC' di6r,DABC, trong dci D chay tr6n truc cta dudng ,f J trbn ngoai tidp L,ABC vd tiSn ra xa vO tAn. 1 sint BCt (.BC DI{\ . (KA. BC1 = Liti binh; Mac dn ldi giAi ban ddu cho bdi 6 -BC sin(BC) torln khdng d6 cira ta d6 Id m6t ldi giAi dep, '\^ 2 q 3 't)H( '"11t(-', BC -mac du y dinh ban ddu : dinh lr:ong ding thfc 2 sin(BCt (1) (tim ldi giAi dai sd) gap phni khd khan l6n, vAeco:3szurc srr, * (3). vi trong kh6ng gian it cci nhrlng ding thrlc bidu Tr) (3) ket hop vdi.(l) kh6ng nhung c6 (2) di6n m6i li6n hO gitta cdc canh vA cdc gcic vdi mi lai cdn cci dal.dang thrlc "rnanh hon" (2) : nhau, song v6i m6t vii suy IuAn cci Ii, kh6ng AB CD AC nhrrng chung ta chi dat duoc muc dich da d6 sinlABl sinr('Dr - sinlACl - ra : "Tim th6m trdi giAi nrla cho mdt bdi to6n" BD AD BC : = si(Ar) : SrrlaD;sirr(BC) mi cdn gFft hei drroc nh[tng kdt quA dep ddn bdt ngd. Dti cung Id m6t cdch hoc md tdi cho Kh6ng ngrlng lai d drj, md chri y rang tt (3) ring c6c ban tr6 cdn ltru y. Tr6n dAv chi Id cdn cci : mOt dang dinh li hdm s6 sin d6i vdi tf di6n, _4 : v:tu, t, s,rR, So,'t, s, tr,. Sl,ot, 9 tudng tu dinh li hdm s6 sin trong tam gi6c cd thd cbn m6t vii dang khdc nrfa cira dinh li hdm ., sinlBC) sin1AD.l s6 sin d6i v6i tr1 di6n. Tdc 916 bdi ndy xin .BC AD nhrrdng lai dd c6c ban tim hidu th6m. Chric Tt drj ta duoc c6c hO thrlc sau dAy, ggi ld dinh cac ban thdnh c6ng. li harn s6 sin cho trl di6n bdt ki (dang thrl nhdt) AB CD AC BD Nhq.n xet th€nt (ctra tita soan) : TrOn ddy sqAB) sln\CD): tinleCl' sitL(PD) = cung chi ld m6t phrrong phSp chr.lng minh dinh : BC AD I hnm s6 sin trong trl di6n (sii dung thd tich). ttn,rpq "ln(AD) Ngudi ta cring cd thd suy ra dinh li ndy bang Ndu chon D nim tr6n truc cria vdng trdn cdch srl dung cdc dinh Ii hdm sd sin d6i v6i ngoai tidp LABC + DA = DB : DC + tam gi6c vn ddi v6i gdc tam di6n.
Gid.i dd,p bd.i CIIIABLNII chiSc b6nh th6a m6n y6u c6u nhu sau N(}T]YEN I)LTC'fAN Lim thd ndo ? IB4n cli hidl cror rsmu cAc xt THr TOAN c0 Qu0c rE TrOn tubng c6 LE FIAI CFLAL- m6t cria hinh Ky rivo dic- toa^ q:oc:e ,llO -:e"a:io'a, Mat'e-a:rca Oy-c,ao a kv:' c.gc -r' -g-,ri "d oieirdiin nnie- ^nir. kv trr ?a rJoc:a a'e,.- ie-:-a- o--vac:'a'c l'a-'91t ,a'a'c'L- ti' r,rr6ng, cao 1m. teg'* .l- Oq it i tr'nuy Nhurg ngoai k! tr i loan qucic t€; IMO riy cbr co ';r-g ki ii :oa' co quoc te sau oay Nay cAn thu hep : 1. Thi todn ducic td vino Bantic. Kv tri "av ro chLrc lan da- !e- ra- '990 o l ca--a:via, -oi cfra sd lai sao cho dgi dut'igo-5iocsirhvbp-l"ai giai Z0riai :oi^'hong5gioo".r-go'o'g:-'e^goie: t! di6n tich cbn m6t 2. Thi todn qudc td (AMU) Ky thi td chuc cho cdc nurrc ChAu Phi An Cau lien nam 1 987 o Rabat-Maroc nria, cao vAn 1m, g. Thi todn qudc ld Oilrityila " Nam Duvng To ch"rc lan dau tien tai Canberra, ngang vAn 1m vh ^ OstrAy ia ndm 1976. Thdti"Binh v6n hinh m6ng ! a..Thi toan qucic te ving Bancing @MQ Io c'ic vAo ri'g 5 ra^g 'er a'da. der .a- '985 oAter-HiLap va -A' phier rai cAc'-roc vLng Banc;i'rg. NCUvin DNG qUANG 8. Thi toan qudc td ving chdu A " kei Binh Dvong. Hier -ay co'2 ^,:oc liar 9r; rc^o cot ra nuoc OstrAy ia roc si^n o lLra t-or'5 - 18 ri 9. Thi todn qudc t€ gitra cfle thitnh ph6.Ia Lh,rc tu'nam 1980 va deln ndm i9E8 Viin har lam khoa:ocrugaquy6tdirnao'gngkiit'ic'ocacr.roct'ert'dgioi Hocsi"o-i:'o jP. l-oi '3't6.vri so 1-ong 5000. ma nbng cor la nuoc Nga. lQ.-Thito6nqudci{vnngBdcAi(i/#C/ NbngcdtlaThLryDien,vois6|-rsngthisinhkhoAngTc. ll.Thitoilnqu6ctdvungNamMi.HqcsinhthamgiadLr6i l6tucii Nongcdt ldnuocAchentina. 12. Thi todn audc td trinh do tiiu hoc MOPil Kv rri ddc cierav to chuc ci-o hoc sirr tie.r r,o.o-r,troig- iim1 .o.i.ofaoi:'Jvir'vittti--'zirri.sril.rongnoc'sinndr1 thi rardbag 13500. n'rungd,rrctoc"icll-anhS[i, -5; thAng'mgrkyttrt"ang5ddnthingg.m6i k]/grim5oai roar' Nhu vriy chL,ng t6 toan hoc dr-rgc cac nr-r6c ch0 trqng, thanh ni6n hgc sinh dugc dong vi0n, khuydn rSSN : 0866 - 8035 Clai sd: 12884 Gi6: 1500d Ma sd : 88T111\T4 S6p chu va ln tai Trlng tanr vi tinli MQt nghin va Xucrng Chii bdn in Nhir xuAt b&n Giric duc r-rim trdrn rl6ng. Ir, xonq i; g""i luu chi6u t'hhng 11"1994