Toán học và tuổi trẻ Số 216 (6/1995)

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Toán học và tuổi trẻ Số 216 (6/1995) trình bày về đa thức và bài toán về số thực; một số dạng toán của bài toán con bướm; đề thi tuyển sinh lớp 10 ĐHTH TP Hồ Chí Minh. Bài giảng phục vụ cho các bạn yêu thích và muốn bổ sung thêm kiến thức về Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 216 (6/1995)

i ,,: -!.i f BO GrAo DUC vA DAO TAO * sQt TOAN HQC vIFr NAM r *iadt =s-,? 6ere) r995 NAM THI,32 tAp cHi na NcAv ts nANc tnAxc m D6 rn6c vfi c6c Bfil ro6n vE so rHqc A , m MQT so o4NG KHAc c0A BAI TOAN CON BUOM rl2? m D€ THI TUYCN SINH LOP IO DHTH TP Hd CTII NNINH PYTHAGORE rudr Au seu nmm wx&w uor GllI{G Beffi8B t€&ffiffi ,lnh : Trudrrg Qudc hqc Hud
ToAN HQC VA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang o Ddnh cho ctic ban Trung hoc Co sb Tdng biAn fip : For lower Secondary School Leuel Friends NCIIYE,N CANH TOAN Nguy1ru Van Vinh - Da thrlc vd cdc bii to5n Phd tdng biAn Qp : v6 sd thuc. 1 NGO DA]. TU HOANCi CHUNG o ki trudc Gidri bdi Solution of Problents in Preuious issue C6c bdi cia s6 212. 4 nOr ocir.rc erEru r4e : o Phan Nam. Hilng - MQt s6 dang kh6c cria bii toin "con bu6m" I Nguy6n CAnh Todn, Hodng o Db ra ki ndy. Chring, Ng6 Dat Tf, L6 Kh6c Problems in This Issue. Bio, Nguy6n Huy Doan. Nguy6n Crlc bii Tl1216, ... TlO1216, Lll2l6, L2l2l6 10 ViCt Hei, Dinh Quang Hriro, o Bgn c6 bidt Nguy6n XuAn Huy, Phan Huy Do you know ? Khai. Vu Thanh Khidt, LO l{ei Ki s6 Hy Lap cd 11 KhOi, Nguy6n Van Miu, Hodng o Nguy&t. Minh Hd. - Dinh Ii Ptolcmc tdng qudt t2 LO I{inh, Nguy0n Kh6c Minh, o dng kinh c,ii crich day vd hoc lodn Trdn Van Nhung. Nguy6n Dang Kaleidoscope, Refornt of Maths Teaching artd Learning. Phdt, Phan Thanh Quang, Ta H6i Am sau bdi b6o "Voi cring bing kidn" 13 Hdng QuAng, Dang Hung Th6ng, o Tidu st ctic nhd todn hoc Vrl Drrong Thlry, Trdn Thdnh B io grap hy of Mathematicians Trai, LO Brl Kh6nh Trinh, Ng6 Huu Li€n - Pythagore 14 Vi€t Trung, Dang Quan Vi6n. o Hod.ng L€ Minh - D6 thi tuydn sinh ldp 10 DHTH TP Hd Chi Minh 16 o Gidi tri todn hoc Fun with Mathematics. Vu Kim Thiy - GiAi ddp bdi Du lich xuy6n Vi6t Vu Hodng Thdi - Di6n sd vdo tam gi6c Bia 4 Tru sd tba soan : _ 45B Hlrng chu6i, He Noi DT: 213786 Bi€n td.p uo, tyi !u-; VU KIM THIIY i[r Ngv'a;nil'CJ- tF-nO Cni uintr DT: 356111 Trinh 6dy: HoANG HAI
a rh cho c6c b4n Trung hoc Co s6d Dinh DA THUC >\VA t\'CAC BAI TOAN VE SO THUC NGIIYfiN VAN VINH Thong qic ki thi hoc sinh gi6i, thi vdo c6c idp chuy6n to6n thrrdng cd cdc bii torin chrlng - (TP Hb Ch; Minh) x3 + 72a1), - 2a : O, nhrl vAy x chinh - li minh m6t s6 ndo dd ln s6 nguy6n, hodc ld s6 nghiam cria da thrlc 13 + (2a - l)x - 2a (l) ': vo ti. Bei vidt ndy xin gi6i thi6u v6i c6c ban m6t rlng dung cria da thrlc dd giAi c6c bii todn Th cri xz + 12a-11x -2a = (x-l)@2+x+2a) loai niy. x6t da thrlc bic hai x2 *x l2acd A : 1 - 8o :ft _Khia:U.1 ru c6x= Tbong chrtong trinh b6i dudng hgc sinh gi6i, '31-1 c6c ban da bi6t dinh li sau ddy vd cftc hQ qu& v6 da thrlc. I ;. I ; :, 1 - Khi o, E, Tb cd 1 - 8o dm n6n da thrlc Dinh li : Ndu phAn s6 t6i giAn " {q-tp fa "O (1) crj nghidm thuc duy nhdt r : 1. nguy6n, q ld s6 tu nhien kh6c 0) ld nghidm cira da thrlc v6i he sd nguY6n VAy v6i mqi a 2 ] t, ",; r = 1 li sd trr nhi6n ort *on-1*-1 +.., + arx * o,r(o,.,kh:ic0)(1) Bei 2. Chrlng minh ring sd thi p ld rJ6c sd c'&.a ao vd g li rl6c sd cita an. HQ qui 1. Moi nghiQm nguy6n cta da thrlc (1) d6u li trdc sd cta h6 sd tu do oo. HQ qui 2. Moi nghiQm huu ti cira da thrlc (Thi hqc sinh gi6i TP Hd Chi Minh vdng 1 v6i h6 s6 nguy6n * + an-rxn-l 4 ... * arx * ao - 1985) d6u ld sd nguy6n. Giai. Dga vio dinh li vd e1.c hQ qui ndy, cd thd X6tx3 = 6 - 1x,hay x3 * 5r - 6 = 0 rrit ra quy tic sau dAy dd chrlng minh m6t s6 Suy ra r li nghiOm cta da thrlc 13 + fu - 6 (2) o cho trudc ld mOt sd vd ti (hay ld mdt sd htu ti) Tacdx3 *- 6 : (x - l)(*2 +r + 6)Vida bx thrlc bdc hai x2 + x * 6 kh6ng cci nghiOm n6n Budc 1; LAp mdt da thrlc vdi h6 sd nguy6n cd m6t nghi6m x : a. r : 1 ld nghiOm thr:c duy nhdt cira da thrlc (2) Kdt luAn : x = l ld mOt sd htu ti. Budc 2; Chfng minh ring hoac ld da thrlc rda tim duoc kh6ng cci nghiOrn nguy6n, ho6c B}ri 3. Chrlng minh ring sd o : tt[Z +3t[4 Id trong sd tdt cA c6c nghiQm nguy6n cd thd cci ld sd vd ti. cira da thrlc, kh6ng c
Bei 4. Chrlng minh ring s6 c : rl7 + rl5 U C6ng tirng vd hai bdt d6ng thrlc, ta cci : sd v6 ti. 7
fl" *c : 1 +O (1) +lcl =1 ll"l +lbl+16-cl (2) [lo-al +lc -al:2 (3) Tt (1) vd (2), ta c6 a, b, c cing ddu ho6c bing 0, nghia ld ab > O ; bc > 0 ; ca > O (4). Tt (2) vi (3), k6t hqp vdi l* -yl < lrl + lyl, ta cd : Beri Tll2l2. Tim sd xYz bidt rang zlal +2lbl +2lcl : l" -bl + lb -cl + 3rfffi : (x +y + z)4" udi (n e N) +lc-al
c5ch lim niy ddi vd khdng phr) hop v6i y6u cdu BdiTilzlz.T?ong mat ph&ng cho hai hinh cho c6c l6P PTCS' binh hd.nhABCD uirAB'C'D'c6iinh D chung. DANG HUNG rHANG Ching minh rang hai tam gid,c AB'C uitA,BC, c6 ci"tng trong tdm. B,di T4l2l2. Hinh uu6ng ABCD c6 canh Ldi giai. Goi M vi N ldn ludt la trung didm don ui. Tr€n canh AB uit AD chqn hai didm M, cl&a cdc doan th&ng DB' vd DB, vd, G : BM fi Nsoo cho chu ui tam gidc AMN bartg 2. Chung B'N (ndu D kh6ng thu6c drrdng thing BB'). D6 minh rdng hai tia CM, CN chia duitng cheo thdy ring BM ld dudng trung tuydn chung cria BD thd.nh ba doan thd.ng md. dO ddi ba doan hai tam gtr6c DBB', A'B C' vd B'N ld trung tuye"n nd.y lQp nAn tam giac uuOng c6 diAn ich khOng chung cria hai tam giric DBB', AB'C. Vi vAy hai lon hon _.:. 1 tam gi:ic AB'C vd A'BC'cci cring trong tdm G \2 + l2)t vdi tam giac DBB'. Ldi giai : X6t ph6p quay ,Bf0' theo chi6u A' kim d6ng h6, ta c6 : B -D ; L vu6ng CBM A vuOng CDM', n6n A, D, M' thing hdng vd - BM-: DM' ; CM = CM'. Ta c6 ; NM' = ND + DM' = ND + BM : AD + AB - (AN * AM) : 2 - (Z - MN) = MN. Kdt hop vdi CM' - CM, . -\l/ = -L Y- (- a - 2M N lt Tlrrdng hgp D e (BB') thi hai drrdng thing BC BM vit B'N trirng nhau nhung ta cflng d6 ddng chrlng minh drroc ring cdc didm G, chia trong ta cci CN ld trung trUc cira MM',hay M, M'd6i doan MB vd G, chia trong doan NE' theo cirng xilng v6i nhau qua CN. Ldy didm 11 tr6n tia .1s6 I,{M sao cho NI/ =.1[A h ca{.-ddi xrlng v6i ti trung nhau, vi" do dd G, Gz: G chinh D qua CN (1) r,1n CHM = CDM' : 90o. M4t 7 = khdc, MH : MN - N}r : NM'- NB : DM' : li trong tAm cria hai tam gSac AB'C vd A,BC,. BM ndn hai tam gi6c vudng BMC, HCM bdng NhAn x6t. 1) Cci rdt nhidu ban tham gia nhau (tr.h.2, A vu6ng). Suy ra H d6i xg.ng_vdi giai bai todn tr6n nhrrng it dat didm tdi da. B--wa CIV-LU.Tr) (1I vdJ2Lta c6 : EHF = Phdn ddng c5.c ban lAp luAn thidu chat ch6 vi EHC + CHF : EBC + CDF = 45o * 45o : qu6n kh6ng x6t tnrdng hgp D thu6c drrdng 90o. VAy L EHF vu6ng tai H. Tr) (1), (2) talai thing BB'. c6 FH : FD ; EH : EB, vb. phdn ddu ciia bdi 2) MOt s6 ban st dung phrrong ph6p vecto, to6n d5 drroc chrlng minh. Dat x - HF ; y - ldi gini kha ryn bing c6ch chrtng minh rang : HE ; S li di6n tich A HEF, tacci S : Oo AA'+BB'*CC':A |*r. Mat khdc. lai co :* Z\[xy=2\[ZS vA + AA'+B'B+CC':3GG' + rtP-+T >a[xy:blzs. Sry ra trong dci G vb, G'ldn lrrot ld trong tAm cfia c6c x *y *\t?Ttr >- 2(t + {Zl{s Ma vd trrii tam giric AB'C vdA'BC', Tr) dd suy ra G' G. = bing VZ vi bang BD, n6n {s hay 3) Cac ban sau dAy cri ldi giai tdt : Mai ?hinh = #p= Hi6p, 98, Nga Hii, Nga Son, Thanh Hcia, 1 S < suy ra dpcm. Hodng Mpnh Crrdng, 91, D6ng Mi, Ddng H6i, (2 + 11z1z' QuAng Binh, Nguy6n Anh T\rdn, g! phan B6i -, NhQn x6t, Ldi giai tdt g6m cri : Mai Thanh ChAu, Ngh6 An, Dodn Minh DrJc 8D chuy6n Binh (8M, Marie Curie, Ha N6i), Hod.ng Manh Quj'nh Phu, Thai Binh, Tbdn Thi Ngoc Hai, 9T, Cuiltlg (9A, PTCS D6ng Mi - D6ng Hdi, QuAng LO Khidt, QuAng Ngdi, Ltrong Hrlu ThuAn, Z! Binh), Nguydn Anh Tl^td.n (9T, Phan B6i ChAu, Nghia Hhnh, Qu6ng Ngii, Ng6 Drlc Khrinh, NghQ An), Vu Tlung Bbn (94, Quj'nh Hoa, 10A, LO Thriy, Qu6ng Binh, Tfrrong Thudn, Quj'nh Phr1, Thrii Binh). 11A, Ca Mau, Minh Hii, Tldn Qud Ldrri, 11! p1o D_uy T,t, Qu{1sB1n!, \syvon L6 L1rc, 9A1, oANc vIEN Ddm Doi, Minh HAi, T[dn Dic Anh, 9T, Nen'g
khiSu Hn Tinh, Vri Thanh Ttr;g, 8T, Ddng 3. Do khdng hidu ring, sd nguy6n t6 16 p Hrlng, Thrii Binh, Nguy6n Hitu Cudng, 10T, trong bii ra li sd cho trd6c, n6n kh6ng it ban DHSP Ha Nqi 1, Tbdn L6 Nam, !)T. L0 Khidt, d6 cho ldi giAi sai. QuAng Ngdi, NguySn Th6 Vfr, fttj, D6ng Son, N(;trYiN rHAc rr,ttrurt Thanh Hcia, Th Thi6n Tbdn, 118. !\dn Nguy6n Hen, HAi Phdng. NGTTYEN DA},i; PI{AT B iT7l2.l2. Ching minh. h€ phuang trinh *2 = y, +y2 +y +a B,di TSl2l2 z Ching ntinh rai,4, tbn tai cac sd nguy1n duong x, !, z th.6a ntrtrt dang thtlc : !2:zl+22+z+a r* 1y)'- zP z2:x'l+x2+x+a c6 m.Ot nghiQm duy nhd.t. trong d6 p ld' m.Qt s6 nguYAn ''t ld. Ldi giei @ia Nguydn On1, |i:'ttl' - 9T Tbdn Ldi giai : (ciab4n Th.ai Minh.Hod.ng - llT Nguy6n Binh Khi6m, Vinh Long) X6t him Dang Ninh, Nam Dinh ; NguYit: Nggc Htng, D6 Quang Tlto, Tlinh H{tu Tiu:';: - 10T Lam f$) = F +P +t *a c6 f'61= $,sz +2t +1 > 0 Son. Thanh IJ:6a; Duong Vd.nY"': - 10T Phan d,o d6 f(t) ld hdm d6ng bid-n. H6 phrrong trinh B6i Chdu, NghQ An ; Tfd,n Ngr'',"2n Ngqc vi cci dang Ng6 Duc Thitnh - A,,10T CT D}j'iH Ha NQi) : = fttl [*2 t' Tb chrlng minh Bdj tod.n tdng gi'Ilf sau '. "Cho n € N* vi sd nguy6n td p th6a tririn di6u ki6n lf = fG) rul p. Ching minh rii,ng, tbn tai ' ,tc sd ngu.y€n l'2 : f(rt duong x1t x2t ...t xrt, xrt*l th,6a mt; ;, dong th,ilc : KhOng giArh tdng qu:it giA srr r l6n nhdt rtir +r! +... +rrtf = r,,+t. a) x 2 J z >- > f4) > fk) --->f(x) > *2 > y'z.N6u e > 0 thi x >- ! > z >- o Th4tviy,chotrxt: x2= -22 -x2>y2> ,2*x'Z:!2:22+f(x) - f(y): J,r(r *r! + ... +ttl = tL ,\nP rrtr'-l - -fr'z)+x:!=2. : Ndur(0+O>x>J2z---> rz0'-l)ll'-1+t111 ,.1 z.K}ri d6 y2 : f(z) < f@) : a lp - l1r,P-t + 1 : 0 (ntodp) + tirti tai k € N* +o > 0. Lai cdz2 = f(x) > f(Q1 - s + sao cho (p - l)nl- t + 1 : kp. Bdr rhd, tt (1) ta thdy, khi chgn rr,*, : nt thi ta ,.i. cci n + 1 sd =-rl"(rl"-1)20>2. y6u cdu cira d6 bdi. b) x > z 2 Y +72 > Y2 > 12 Dac bi6t, cho n = 2 vd p li s* irguy6n t6 16, Ttrong tu nhu tr6n nd.u I >- 0 hay r < 0 tt bai to:in tdng quat ta cd Bni tr,,,i.n da ra, tasu.ytax:!:2. Nh4n x6t : 1. MQt sd b4n cht: ring kdt qui Ndur > 0 > y +xz : f0) < f(0) : a cria Bai torin tdng qu6t ndu tr6n v in kh6ng thay / = f(x) > f{0) = o. Ndu r'ir, th\x > z ddi khi thay di6u ki6n "z € N*, i:, lii nguyan fi > {a -x2 > z2-x2: !2: " z'2*x = ! = z ud. n1 p" bdi di6u ki6n "2, p € N* t i (n, p) = 7" ! trdivdix>0>y. 2. Ngoni cdc ban di n6u t6n t: i r6n, c5c ban Ndu z . - {; li luan nhrr a) ta ddn ddn sau dAy cring cci ldi giai t5t : l''gd D*c Duy, mAu thudn. Phant Dinh Tluitng fil Ct Tr,"in Phii, Hei Vay hg cri nghiOm duy nhdt x. : Y : z = t,. Phdng) ; Tfd.n Tidn Dung (10T'. Amsterdam, qd., {,, Ia nghi€m duy nhit cria phuong trinh Ha Ngi) ; Nguydn Ba llitng, LA';'td.n Anh (I0 t't+t'*t*a:0. CT, DHTH Ha N6i) ; Ph.am ,\t:h Ttud.n, D6 Hbng Son, Nhu Qui Tho, Phar: Minh TLd'n, Pha.nt Manh Quang. L€ Minh jjran (9T, 11T, Nhfln x6t : C6c ban Phonr. Dinh Tl'uitng 11CT Hai Phbng, Nguydn Hodng Cdng llT Lam Son, Thanh Hcja) ; Nguy{ri H6ng Chung (10T, Phan BQi ChAu, NghQ Ant ; La Quang QuAng Ngai, Til Minh H d.i, ll D 6cLdc, N g uy 6n Thanh Titng, 111 Ddng Thrip, Pham. Qu6c Ndnu (9! Drlc Phd, QuAng Ngait : Nguydn Thi Chfnh,9T Thanh Hcia cci ldi giai tdt. Rdt nhi6u Hd.i Ydn, L€ Arulr Vfi (ll Cl Quric hoc HUO) ; ban m5c sai ldm khi cho ring "kh6ng mdt tdng Vu Dic Phi (9T, Nguy6n Du, Gb VdP, T.P' qudt cri thd gia s'(!. x >- >- z|-' HCM) vi,Nguydn La L4c (9A, D.{m Doi, Minh -y Hai). DANG IIUN(} THANG 5
Bei T8i2l2. Cho day sd a,, dttoc xd.c dinh 1*a, bdiar:1ud. I * an-, V4.y or+ t = 2 + ax- (k+ 1) [ n *;1 : dn: 7*" 2(1) -l+h- { n luaimoin> =2+ar_tk *rlL_ltl Chnng ntinh.h rang : | +2 (n -llrts4t1\ uoi nwi n 2 1. o,, - 2 * au- (k + L) : 2 + I + 2(k - 2P) - 2P+1 rA ddy tal li pha.n ngulen cio a ud { : a - [a] lit. phd,n phd.n cia a). "l :1+2(k+l -2P+1) Ldi giai. (cira tdt cA cdc ban) Th cci : = 1+2(k+ \/ 1-2llos2(k+l)l \ _7 *an_r_ 7 *an-, _l*a,,-l- Bdi todn drroc chrlng minh. l-#] hay =--r.--. [---]n, Nhfln x6t : Cric ban sau ddy cri ldi giAi tdt : Ngd Dic Thdnh, Nguydn. Bd. Hilng, Tl"d.n Nguyen Nggc (A,, DHTHHN),DOng Thanh Hd. -l*a,,-l . _l*e,,.1 (1iA CT DHSPHN1 ), L€ Dtlc (9,\ Ng6 Si Li6n, "l--;-t] =r*on-t."1- * )+" Ha NOi), LA Van Manli ( 1 1CT Hda Binh), Phqm Thy Hnng (1 1A Chuy6n Th6i Binh), Dd.a Xudn suYra' Vinh, NguyAn Thi Hd.i Ydn, Cao Thd Anh, LA ,*o AnhVu (QH HuO, TOn Thdt Th.ang (11Ar Le s rn = 2+an-t- 1 < z+ t Quy DOn, Di Nang), Nguydn L€ Luc (9A, Ddm " l-;=] Doi, Minh HAi), Nguydn Hbng Chung (lOT Yn>2. Phan BOi ChAu, Ngh6 An), L€ Qua.ng Nd.nt (9, Th chring minh bing phtiong phrlp quy nap. chuy6n Drlc Phd, QuAng Ngai), Cao Quydt Hie.p Khin:1,tacd: (9T Nang khidu, Hoing Hcia, Thanh H5a),Nhu Quj, Tho, Sao D6, Nguy1n Ngoc Hung, TYirult +2 ..In : .(n -Zlr"ul,) : t : o, n6n (1) dung Hitu I\'ung, LA Minh Hidu, Pham Minh Tifin, vdi l'. (Lam Son, Thanh H6a), NguyAn Tlong Hda (Phan Dang Luu), Pham. Dinh Ttuitng, Ng6 Gii srl (1) drlng vdin : k, t(tcld : Dic Duy (Tt'dn Phu, HAi Phbng) 0k = 1 + 2 ( k -2ltot$l) Oa, p = llogrkl NC;TIYF,N VAN uAu thi o* = 1 + 2,7,' - !P) vA / Bii Tgl2l2. Cdc didm A,,, 8,,, C,,, Dn txong p < logrk
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llA, Le Chi Nguydn, 10A, Chuy6n Phan Ngoc BdiL2t2t') Hidn, Cd Mau ; Nguydn La L+tc 9A1 THCS Voi macli dien uA Ddm Doi, Minh HAi. bAn, sd chi ct;,t uOn kd vO xru lr{(rY. ld,SOVnhung;itKrmd, B.diLU2t2 td 27V khi t ti d6ng ngat K' td.2.1\' hhi chi MQt bd.ng chuy'€n nghidng nrQt goc ct so uoi phuong ngang, dang chuydndQngu,,xudng duoi. d6ng 2 ngol K1K2, lit MQt hbn garh niim tr€n bang chuybn uit. duoc giu 18V khi tling cd. yen bd.ng mQt soi dd.y buQc c6 dinh d ddu tr€n KF{s. Bit)t kh.i d6ng KrKl{, bidn trd hinh u0. Ngudi t@ cdt d*t dny. Tinh cOng cia 3= 4,8Q uit tsubn. di€n phat cdng sudt 270W. Irlc nw sat td.c dung l.An ltbn gach cho ddn thiti 1) Tinh E. r ud. gid. tri ntoi diQn trd ngoiti didnt hdn gqch dgt duoc udn t6c u,, cila bang 2) Nhich {on chay C d Rt sang pltdi hay chuybn. Cho h4 sd rua sat giila uiAn gach uirbang sang trai d.d ,,tng sudt ntach igoiti gid.m. chuybn lit. p, hhdi luqng cira ui€n gach lit nt Ldi giai. Khi ddng K.K;K3, cr5 dbng chinh I ! qua ngu6n t"'=TE - i8 18**, 18*4* 18 18 (1) !' 'fi ,i-= o" 4s Khi chi d
rulQt Sd DA[{G rnAC CUn gil tOAt't "@0n {Bttnrn". PHAN NAM HUNG (Qud.ng Ned.i) Chic nhi6u ban tr6 y6u torin d6 drrgc lim duimg thd.ng niry cd.t hai dudng ch6o AC u?t. BD quen vdi bii torin "Con budm". TYong bai vidt keo diti tai M uit, N. nav tOi mudn dtloc trao ddi bang nhtng lAp Chtng minh IM = IN ludn hop l!, ta c6 thd di d6-n mQt k6t quA manh hon kdt quA ctra bii to6n "Con bu6m" NIM Tlrrdc h6t ta xudt ph6t til bai to6n "Con brl6m" Bhi to6n 1. (Bai to6n con budm). i Cho n&t duirrtg trin tdm O uit ddry cung I MN ; Gqi I lit trung I didnt crta ddy cung ii d6. QuaI tahi haiddy cltys !t!ac_td. !!.uo. r CD. Ndi AD uir BC hat Giii : dal,cung nity ldn luot ti Ndi OM v6i ON ; Tr] O ha OH r BD va cat MN toi P ud Q OK L AC r6t trl gi6c IHON c6 I vit H cilng Ching minh IP = nhin ON v6i mQt goc-:apnt clyrrl6n ttl giSc dci rQ. ndi ti6p viy ta c6 : IHN = ION (1) Giai. Trtt-ng trJja-gtng cd ttl giric IKOM nOi ti6p TitO tahaOK t CBvir.OL L ADN6iOP + II(M : IOM Q') vd OQ. Th,lai cd L IBD vd A,IAC cri gr5c / clqtg vn D6 thdy t{€iec OIQX-nQi tidp drrgc dudng B = A (gtic n6i ti6p cirng ch6n cung DC:) cho trdn, vi vdy IOQ : II(Q (1) ndn A .IBD L IAC md OH L BD va Trtong trl, OIBL ct1ng li trl giric n6i ti5p, vi OK L AC + H vd K lir trung didm cira cdc canh BD yiAC cho n6n L IHD - L, IKC + vdy : IOP : ILP tZ) IHN = IKM B) Nhung ta lai ccj L CIB - L AIO vd tr) + : i6il OK t CBvdOL L ADchon6nKviLldn tt (1) ; (2) vn (3) i6rt o'i, ludt la trung didm cira c6c dfur.cgLngC!=qAAD OI t MN + IM = /N (d.p.c.m) cho non L CIK ^ L AIL *AQ - ILP G) Ndu trong bii torin 2 tdi thay hai drrdng ch6: Tt (1), (2)yL(3) taco QOI = POIhaY OI BD vd AC bii hai canh ddi di6n cbn lai lh. : DC v?r AB t6i cci bni torin 3. ln phin gec QOP ma O1 L PQ cho n6n -IP = OQ (d.p.c.m) hinh - 1. Biri to6n 3. Tbi cd thd ph6t bidu dudi d4ng bai to6n sau Cho ttl giar ADCD nQi ti6p dudng tritn tdm "Cho trl g16e ADBC neri tidp dtrdng trdn tAm O ; goi I ld. giao didnt. hai cqnh d6i diQn AD ud O ; gc,i.I li giao didm hai drrdng ch6o CD vi BC h1o diri ; qua I, d4ng dtitng th&ng d uu6ng AB, MN li m6t dtrirng thirng vu6ng gdc v6i OI gdc udi OI ; Canh DC ud. AB heo ddi Lan lua tai I ; P vi Q ldn luc,t la giao didm cria hai c4nh cat d tai M ud. N. d6i di6n CB vi DD vdi MN. Chfng minh Ching minh IM = IN IP = IQ". Giii: Cric ban chri y n6i dung bii tod.n tr6n so vdi bdi todn 1 kh6ng cri gi thay ddi. NI,IM Nhrrng t6i nghi ta cd thd ddi vai trb hai drrdng ch6o thAnh hai canh ddi di6n cbn hai canh ddi di€n thdnh hai dudng chtio - thi c
Vn. >- 7 (re € N) Chfng minh ring d6y sd (.tn) khOng bi chan. TRANfr,Y*}BANG Bai T9/216 : GiA srl M li m6t didm bdt ki nim trong rn6t trl di6n gdn d6u ABCD. Ha MM,, MM-.,, MM , vd, MM o ldn ltrot vudng gcic v6i cric mat tBCDt, (CDA:, (DAB) (ABCt. vd CAC LOP THCS Goi r vD. P ldn lrrot Ia b6n kinh mat cdu n6i tidp Bli Tl/216 : Tim 6 cht s6 tAn cirng cira vd bdn kinh dudng trdn ngoai tidp m5i mdt cria -)t ^. D'' SO trl di6n ABCD. Chring minh ring : DANG HT]NC].fF,tANG ilMff+M4+Mlvrt,+Mtu4> (IId Nai) > 4f2cosA. cosB . cosC Elni TZl2lS : GiAi Phuong trinh ,r otv < {,+ (x + 2)2 + (x *3)3 + @ + 4)4 : 2 i{b oYi4Jy1,,lNri s, -r't I tot Nc;Qc ANU (Qutin'; NEAi) Bai T101216 : Gii srl P ld mOt didm tr)y y * Biri T3/216 : Chfng minh bdt d&ng thrlc nim trong L ABC vd a , p, 7 li dd l6n c5c gcic c:ia LABC.Ki hi6u Rt, Rz, Ri la khoAng c6ch 4{T--dI + 4{1 - o 4 +{1 .ul 4 3 trl didm P tdi circ dinh vd rt, tz, r.lir khoing v6i lol < 1 crich tr) P t1i c6rc canh ctta L ABC . Chr.'tng rninh NCiT}YF]N DI] (Ildi Phnng) bdt ding thrlc : Rl .sin2a + Ri. sin2P + l?]. sin27 < Bei T4l2fG : Cho hai taru glac ABC vit
PROBT.EMS IN THTS TSSUE Ben c6 bida FOR LOWER SECONDARY SCHOOLS. T1/21E., Find the last six digits of the KI SO ni Lap cd Chrlng ta d6 biSt cri m6t s6 cdc h6 ki sd drroc numbet'5". TZlzlE. Splve the equation vidt theo c6ng c6c g16 td cric ki s6 cci mit trong \x + 2)L * (r * 3)' + (r * 4]}" =2 sd dci. Quen thudc nhdt li h6 ki sd La Ma. O T31216. Prove the inequalitY ddy, xin gidi thiQu vdi c6.c ban hai hQ ki sd nua +fi-_]+a.f1-;+aft+o b'). f IIIi. 56 10 duoc vidt ld A (chri c6i ddu cria istzrc. Let (O, R) and (1, E,) be deca - mrldi). Cdc s6 100, 1000, 10 000 duoc respectively the circurncircle and the exlcircle ki hi6u la H, X, M cric chit ddu cl ,a c*c t6n goi (in-angle A) of the triangle A-BC" Prove that sd trrong rlng. Cac s6 50, 500 vd 5000 duoc cho IA.IB.IC : 4R.Rl bdi td hop c6c ki sd 5 - 10, 5 - 100 vi 5 - 1000 : F , F , F . C6c s6 cdn lai trong pham vi 10 000 drroc vidt drrdi dang cOng gi6 tri. Vi du : FOR UPPER SECONDARY SCHOOLS. HHH F f II = 357, XXHH F III : 2254, TGl?l6. Prove that for every prime p, thg number 5p - 2p can not be represented as o''', HHHFAAAAIII:393, where o and m are natural numbers, m > l. FXXXFHHAA=8720,... T't1216. Find the greatest real number k such that tong th6 ki thf III tnt6c cdng nguy6n, h6 ki Attic dd cung cdp cdch vidt s6 drroc goi ld s6 a3+b3 +c3+ kabc>*.* hG Ionian. C6c s6 tt 1 ddn 9 drroc ki hicu bdi for every a, b, c > 0 satisfying a * b * c = t. chin chit ddu cria chit cai Hi Lap : 'tgl2l6. Let be given o. sequence (r.) such a = l,F : 2,T = 3, 5 : 4,8: 5, I = 6, thatr, = 1 and (z + 1)(r--, -x-) > I *'x^,Vn :'7,r1 = 8, .9 : 9. cric sd tr) 10 d6n 90 drroc > I d € N). Prove that'the sequence (i,.,) is e ki hieu bang chin chr3 ti6p theo : not bounded. TS1216. Let M be an arbitrary point inside t=10, rc:20,,1 =30, p=40, , -50, an equifaced tetrahedron ABCD and let 6=60,o=70,lz=80,1:90. M1, M2, M3, M4be the orthogonal projections C6c sd 100 ddn 900 la chin chir cudi cirng of M respectively on the faces (BCD), (CDA), : (DAB), (ABC). Let r denote the radius of the ,P = 100, d = 2A0,f : 300, : 400, g= 5OO, inscribed sphere of ABCD and f denote the X = 6OO,V : 700, co : 800, )= 900 radius of the circumcircle of a face of ABCD. Prove that : Cdc sd hdng nghin, hirng chgc nghin drroc d rurfi + M^frr. + M4 M4 + .- ki hieu nhtl tr6n, nhrrng th6m vdo dang trrrdc ddu phdy hodc mdt sd doc. Vi du : > fcosAcosBcosC; le = 5000, lco:800 000,.., _r o)p * {, =T. Ngudi ta ding m6t vach ngang d6t tr6n c5c T101216. Let P be an arbitrary point inside ki s6 dd phan biQt vdi chtr vidt. Vi dq : the triangle ABC and a, f , T be the measures TF : 12, p€ -- 45, @tty : 883 of the angles of this triangle. Let f 1, Rz, R3 be the distances from P to the vertices and Tbong thdi dd, c5c loai ki s6 tr6n drrgc dirng t7, t2, r, be the distances from P to the sides rdt nhi6u ndi nhu Arabs, Jews vi nhi6u ddn tdc kh6c 6 virng CAn D6ng. Ching ai cri thd of ABC. Prove the relation : bi6t n
DENH rf PT0r,Enrrfi rdnrc euar NcurEN urun uA (Hcii Phitng) Dinh li Pt6lcme l:] dinh ti 'Iiong dtl : A : Arn; A hinh hoc plr:ing -Z: A,, - t ndri ti6ng rrong -1 ; Dinh ti I (Ptdl€m€) : Cho-AdBC n6i tidp drrdng trirn (0). A2na1=A,,iA,n+,:Al ,11 li nrilr didm thu6c cung itClkhrrng chiral). Khi tlri : 'Itong dinh li tren cfing nhU trong cic dinh li riip rhco. ki All.MC+CA.MB=BC.tl[A hieu (Of Q') chi g(rc tlinh htlclng grita lrai til O.t. (]y Nqdii ta cl6 tting qurit h6a ttinh li I rhdnh bdr ding thirc Dd.chitng nrinh rlinh li -1. rnrilc hdr ra phir hii'u vi chitng Ptolenle . 'Iiong bli b6o n:iy t6i xin gi6i thi€u m6r huring tiing minh bd db sau : qu:it liti;r khic cria no. Bb dE 4 (Dinh li "Con nhinr"1 : 'liong m:it plilg ,f,4, IlSv cf xem dinh li I lir rlinh li Pt6l€rn€ cho tam giric (nQi hU
Ap-tittng bd db 4 cho cla gi6c B1Bo...Bn vi hC vdctc, 6ng frfnh eal cach:eay vr5 P,d*c toen - oA2; ...; OA1; oA, d6 ddng nhfl ctrlclc hQ quA 5. Nhd he qui 5 vA phudng phdp ch(nrg minh dinh li 1, ta d6 ding chilng minh dr-tctc dinh li 3. Hbi dm sau bdi bdo Vi€c thuc hi6n chi ridt xin denh cho ban doc. Dinh li 3 cir ching lA stl t6ng Il .la "vol clrrlc niNc KIfrhr quht cia clinh li I hay kh6ng ? h6y ki6ur tra rlibu d6. (h.4) 'I'heo dinh li 3 ta cci : r _1 .. LTS : Toa soan Tap chl Tod.n hgc uit. TLdi ryl : eA,()c)f L MC '; I ,sl g.tc,oB1] + )) trd c6 nhQn duoc thu cia cdc tdc gid. Ngo I L+ I L4 . Thuc Lanh. V{r Ttud.n, Ng6 Xud.n Son trd ldi ' l,si-t - +,sl-l ; luB. oA) ..1 I ; \oc. oB)lt)I Mu I L4 L+ cac! kidn dd nAu bong biri bd.o n6i tr€n. Ddp -t (uA.OC)l +,S I .. tng quybn duoc th6ng tiru nhibu chi'€u cia - l,x L l; I-ql()8.()A)llMA ban dgc uit. thd theo yAu ciiu cila ba tac gid. I L4 .l .]) C+B A+C B+A C IB n6i bAn, dudi ddy chung t6i xin dang nguy€n -(,s z +ry 2)MC+(ry 2 +u;)ua udn bic thil d6. : f A+C 2t B+A +u-f:1 ve " f ir trr a6 d?ring suy ra : sirtC MC + si1ffi MB = sinA MA +AB.MC +CA.MB: BC .MA Nhrl vliy, rluc ti6u tran cllu cta ta di drldc thrlc hi6n. Tirv Q,fri @ cndo ode nhi€n tt6 cho lroz'ur cirinh viin rlil dang x6t. xin gi6i thi€u m6t dinh li nita. N(r li srt trI(tng trt cr-ia clinh li 3 cho trding hctp ila T!ong's6 b5,o 213 thrlng 311995 ciia tap chi giae rroi tiep \ (-1i s6 u;rnlt ch:'itt. Tb6n hoc vi tudi tr6 ban Ddo Tbudng Giang ilinh ti 6 : Tr€n mit phing rlinh hrldirg cho rl;r giac (T4,c gid bdi "Voi crlng bdng ki6n') n6u thSc m6c AoAi...A,n- , ntli tidp drldng tri)n (O) ,&1 in m6t didm thu6c cungl.,l,n -, (kh6ng chit.-r1., . ...,1:n -:). Khi d(r : vd c6ch chrlng minh tinh chdt , .l . .t ;,",^-'' oA, )) ul1toe,.. r u.t)lu,t,. + { af@)d.x: a I f@)dx (o ;* 0) (1) 1't L +> {, ryllQAro^z.oA2il ,1 trong s6ch gi6o khoa GiAi tich 12 c&a nhcim tric + t
mf cnq-rvEu oAc urA ro,iri HQC PYTHAGO&E (P5g'AGO) Pythagore (khoing 580-500 trudc CN) giric ddu, l6t kd v6i nhtng hinh vu6ng vd hinh ngrrdi Hy lap sinh d dAo Samos, m6t trungtim luc gi6c ddu, c5c hinh dy d6u cri canh bing nhau thudng mai vd van hrja bdy gid, 6 trong bidn (xem hinh). Eg6e. Tuong truydn ltic trai trd 6ng tr)ng di chu du thi6n ha dd hoc h6i cric n6n vdn minh : sang Ai CAp hoc c6c tu si 22 r.im, di tidp sang Babylone, d lai dAy 12 ndm, cd thd cdn dd sang ci An Do. Tudi ngodi 50 6ng m6i trd v6 ch6u Au dinh cu E Crotone, m6t hAi cAng vd trung tAm vin hria, thu6c dia La Ma, d tAn ctng mi6n Nam Italia. Tf,i ddy 6ng md trtldng day T?idt hoc, Thdn hoc, Dao dric hoc, ToSn hoc. T[rrdng tdn tai 30 n6m. \Ao thdi cudi, do vi nhtng biSn d6ng chinh tri, x6 h6i cria phong trio qudn chring ddi dAn chri 6 Crotone, tnrdng Linh sang Ong da ding phuong phrip hinh hoc dd c6c thi trdn Tbrente, Metapont 6 gdn dd. Khi chrlng rninh rang tdng c6c sd 16 li6n tidp thi tnldng ddi ddn Metapont thi cfrng vrla hic cao bing m6t sd chinh phuong (1 + 3 : 4, I + 3 + trdo khdi nghia nd ra 6 ddy. Tfong m6t d6m 5 : 9, ...), hiCu g&acac binh phuongcria hai sd bidn dOng tnrdng bi ddt ch6y, cu gid Pythagore nguy6n li6n tidp thi bang mOt sd 16 (22 - 12 : ngodi 80 tudi chdt trong ddm hla. C6c m6n d6 3, 32 - 22 = 5,...). 0rg cdn nghi6n crlu vd c6c cira tntdng v6 sau tAn m4n sang Hy L4p, md da di6n d6u trong kh6ng gian ba chi6u, nhrr trl c6c trtldng day chri ydu v6 sd hoc, hinh hoc, tao di6n ddu, luc di6n d6u (kh6i lnp phttong), b5t n6n m6n ph6i Pythagore. Ho dd lai cho ddi sau di6n d6u, v6 c6c ti s6, c6c cdp s5. nhrlng thuydt gleng cria Pythagore, nhu trong Cdng hi6n dec sic nhdt cta Pythagore v6 cl.c trl;6c tdc c:'ua Aristotdle, Platon, ... Cdn todn hoc ln da phrit hi6n vi trin trd cuOc khirng Pythagore hinh nhrt kh6ng dd lai tnr6c illc gi. hoAng thrl nhdt cira to6n hoc. Pythagore cci Hinh quan niQm si6u hinh v6 cdc sd. Nhtt m6t nhd Vd to6n, Pythagore nghi6n crlu 56 hoc, hoc, ?hi6n van hoc. Ong la m6t trong nhiing ti6n tri crj thdn cAm Pythagore thuydt gi6ng cho c6c m6n d6 ring cdc s6 lA nguy6n li vd c6i nhd ddu ti6n chri trudng rang Tlrii Ddt li m6t ngu6n cria moi vdt (le principe et la source de qui cdu, drlng d trung tAm vfl tru. Mdt Tidi, toutes choses). Trf c6c sd trr nhi6n I, 2, 3, ... Mat T[ang, c5c vi sao thi quay quanh T!6i Ddt, song Mat Tldi, Met T[Eng vh m6t s6 hdnh tinh di hiru tiP- m6i c6i cci str di chuydn ri6ng, khric vdi srr di 6ng ddn c5.cs6 q"', (p, q lA cric s6 trr chuydn c:iuLa cdc vi sao hdng ngdy quay quanh nhi6n) vd khing dinh ring vdi ct.c sd hitu ti ta c
ddu kh6ng dtloc srr th6a thuQn cira h6i d6ng Chti thich - Cho doan thingl-B. Didm (duy m6n, dd drla th6m nhtng di6u mdi vio bii gi6ng nhdt) D € AB ggi ln l6t c6t vdng cria AB ndu cria Pythagore vi dem trinh bdy cho ngudi .ADDB nd chia doan AB theo tisd r : -AB : khac. V6 sau Hippas ch6t trong mdt vu d6m AD . tdu bidn. C6c mdn d0 det ra cAu chuy6n Hippas bi chdt vi da tidt 16 di6u bi dn. Hon 100 nam sau nhd todn hoc Eudoxe (khoAng 408-355 tnt6c CN) giAi drtoc cu6c kh&ng hoAng thf nhdt cira todn hoc bing m6t li thuydt sd v6 ti, tudng tu nhu li thuydt hi6n dai cr.ia nhd todn hoc Drlc D6 ddng tinh dugc Richard Dedekind (1831-1916) 6 cu6i thd ki fi-t - q,6180339887... xIX. Mgt cdng hidn quan trong khdc c'(ta Ngrrdi ta cho ring cac hinh chu nhAt cri ti Pythagore vd phrrong ph6p luAn ln 6ng de d6 sd (chi6u r0ng)/(chi6u ddi) bang r ld hdi hda ra phrrong ph6p chdng minh. Tlrr6c 6ng ngrrdi nhdt. Vi udy c6.c khungcrla sd, tdm pann6, c6nh ta khOng nhAn 16 ring chrlng minh mQt vdn crta tri... thudng hay ldy theo ti s6 r. dd to6n hoc ld phAi di tr) gi6 dinh. Py'thagore ld ngudi chdu Au ddu ti6n chri trrrong rang Tlong li thuydt tdi rru h6al6t cit vdng drroc trong Hinh hoc cdn phii det ra cdc giA dinh dirng lim mdt phrrong phdp hrru hi6u dd db tim ban ddu, goi ld ti6n d6. Tt dd lap luAn chrlng didm crrc tidu cria m6t hdm f(x) dar, cdch tr6n minh drroc tidn hinh bing c6ch 6p dung cho m6t doan AB, khi didm crrc tidu n}.y khOng thd aic ti6n d6 m6t ph6p li luAn suy di6n chnt ch6 tim bing phuong ph6p tinh dao hdm. (Hdm de dl cten ket luan. don cdch (unimodal) ln hnm chi c ION = IKN (2) Ndu ta ddc bi6t h6aA : B thi khi dd cat tu;rdn Th cflng chri y A IDC ^ L IBA vd H vd K 1,4-B suy bi6n vd tidp tuydn ta cd bei to4n sau, li cric trung didm cria DC )i\AD cho n6n A Blri to6n 5. Cho dridng tr6n tAm O, d ld rye_ ^ A- IKA + IKA : IHC hay m6t drldng thing d ngodi drrdng trbn (O), Trf IKN : IHM (3) O ha OI .l. d, Qua 1 ta k6 tidp tuydn IA vit cat Tl] (1), (2) vd (3) = IOM ^ ^ = /ON mh tuydn ICD ; goiM, N ldn ludt la giao didm cira OI t MN + IM = /|/ (d.p.c.m) d v1i cdc dudng thhng AD vi. DC. Chrlng minh Luu j,; d bii todn 3 cci thd xAy ra. DC llAB IM:IN luc dd cdc ban cd thd xem hai didm M, N d xa Drrong nhi6n ndu ICD suy bidn vd ld tidp v6 tAn v6 hai phia. (vi DC ll AB ll d) nh:: vay tuydn /C thi ban cci ngay AC ll d liL mOt di6u thi / la trung didm cria MN v6n drlng. md ta da bi6t. Tr) bai to6n (1) (2) (3) Th nit ra bdi todn Dac bi6t 6 bii to6n 5 ndr ICD le dudng kinh tdng qudt : ban thu drroc bii to6n 6 : Bhi to6n 4 : Blri todn 6 : Cho drrdng trdn (O) dudng Cho dudng trdn tAm O ; d li' m6t drrdng th&ng kh6ng cilt (O) dudng kinh DC L d tai thing bdt k, (kh6ng phAi ld tidp tuydn ctra O) .I, qua I dung tidp tuydn IA t1i drrdng trdn ; goi,I ld chdn drrdng vu6ng gdc h4 tt tdm O t6i ddy cung DA cil d tai M khi dd ta c6 IA = IM. d. Qua I ta k6 hai crit tuydn IAB vit ICD tfli Bdi to6n 5 vA bdi torin 6 xin ddnh lai dd c6c dtrdng trdn. ban kidm tra. 15
oE vn DAp Aru rnl ruvEN sNH \, l6p rc cHUvEt't ToAN - TIN HOC DHTH TP HO CHi MINH 1994 (voNG r - l.tcAY so/7/xee4) Vdi li luAn tudng trr, ta thdy v6i 2000 s6 thi sau 1999 ldn xcia sO cbn lai dring m6t s6 chin. Beri 1. Sd.u d|i b6ng A, B, C, D, E uit. F tham BAi 3. Tim tdt cd. cd.c cQp sd tu nhi€n (x, y) dtt m\t gid.i uO d,ich. Dudi ddry ld, nam kh&ng sao cho y * 1 ch.ia hdt cho x ud x + 1 chia hdt dinh kltd.c nhau ub hai dQi c6 mQt trong trQn cho y. chung kdt : a) A ud. C b) V uiL E c) B ud. F d)Aud.F e)Aud.D. Ldi giai : Trl di6u ki6n cria bii to6n suy ra x*l > y vity *1> x. Kdt hop lai ta crjr - 1 Bidt rd.rug c6 bdn khang dinh dnng m.Qt n*a < y < r * 1, nghia lny chi ccj thd ld mdt trong ud.nt0t kh&ng dinh. sai hod.rt tod.n. FId.y ch.o bidt bas6r -l,x,x*1. hai dQi nd.o dttoc thi ddu trong trQn chung kdt' Ldi giei : Ndu d6i A kh.6ng lot vio chung x Ndu ! : x- 1 thiy vr)a la rr6c ciar - 1 vta li rl6c cria r * 1 n6n sG id irdc cria (r + 1) kdt thi trong ba kh&ng dinh a), d), e) phAi cci - (r - L) :2, nghia ldy : t hayl' : 2. Nduy it nhdt hai khing dinh ld dring mQt nrla vAy : 1 thi x : !* 1 : 2. Ndu ), : Zthir :./* d6i lot vdo chung kdt phni ld hai trong ba dOi 1 = 3. C6 hai cap s6 (x:2,y : 1) r,d (r - 3, C, F vdD. Suy ra hai d6i B vd.O kh6ng lot vio chung kdt, tilc la kh&ng dinh b) ld sai, cbn b6n t = 2) d6u th6a di6u ki6n bii to5.n. * Ndu ! : x thi y via ld rrdc cria r vrla ld kh6Lng dinh cdn lai dring m6t nrla. Nhring khi rJ6c cira r * 1 n6n chi cd thd le J : 1 vi khi dd d6 cA" C, D vd F d6u phAi cung lot vio chung kdt:Vdly! r : 1, Cap s6 (r : 1, J : 1) hidn nhi6n th6a di6u kiOn. VAy chi crj thd A ld m6t trong hai d6i crj mnt * Ndu ! : x + 1 thi x : ! - 1. Do tinh d6i d trAn chung kdt. Khi dd C, D vd .F' kh6ng iot vio xrlng cria hai s6 x vd y n6n ta c0ng tim 'ra cdc chung kdt. Ma B cfrng khOng thd vdo chung kdt (khi dd khdng cci khing dinh ndo ld hoin todn cap s6 tudng rlng li (.x : 7, I : 2) vd (x : 2, sai cA). Suy ra chi c
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t Gidi dfip bd,i : \.l\- DU LlCh xUvEN vlfr Nhidu ban d6 gili ldi giei ve da sd la dring. 56 d6ng c6c b4nchondi6mxudt phat la M6ng C6i, Cao Bing ho5c Ndm Cdn. Ngodi ra l) c6c d6 thi T ang Sdn, Bic Giang, Th6i Nguy6n, Viet Tri, Y6n B6i, Hii Dr.ldng, HAi Phdng, Nam Dinh... Ban Tq Qudc Hung,l6p 6, tnldng Phan Chu Trinh, Bu6n Ma Thu6t lai bit ddu ti Thanh H6a. Hai bpn Trbn Duy Hilng, Trbn Vdt Dfing,118 PTTH ff ThnY, QuAng Binh tim duQc 8 di6mxudt phSt' Cac bqn c6 liri giii gti vd s6m vir dtng I'a Nguy4n Thi Thtly Wnh 10A PTTH Da Phic. NguyEn Xudn Trudng, Tien Dudc, S6c Sdn, Pharn Hd Son, Phan Thanh Tilng lCT, L€ Thi Bich Hanh.6CT. Nghia TAn. Ha NQi, Rui Duy Hing. 9A. T6 Hi6r-r. Hii Pht)ng, Ngu.tin Li Dttng..8T NK Ha Bic. l-i Hdi Yin qT. chtri/cn Thanh Srln. Vinh Phri. NSzyin Tlti Thiitrr.98. chuy6n (lng Hda. HA Tey, Bili Thd l)ung. PI'I'H VEn Giang.L€ Dqi Ngqt€n. 11A PTTII Vin Liln. Hii Hdng. L€ Thdnh 1 C.6ng. 6T. Pharn Huy Qaang. I)6ng Hung. I 'I'h1ii Binh. Hd Thanh Tudn, Vu Trhn Caong. + ,t 7T. Trdn Ding Ninh. Nam H). Bni Thi Tu 9C Vinh Phr.hc. Vinh L6c. Thanh lfoa. Dinh I Libn 76i, Hodrtg Dirth Thi, Tq KiEu Hung gc'f NK Vinh. Ngzrydn Hau An.9A. Qr-rjrnh L.i€n. Qr,rirrh Ltttt, Ngtydn Tdi 'fhu.6,4. NK Y€n Thinh. Ngh€ An. Trhn Thanh Til. l}CT Dio Duv Tit. Quring B\rh. Nguy€n Pham Phuong Thdo.917 Nguycn Tri Phtlclng, Hu6. Phun 7 hi Thuy Hanh.l0AI fr:ln Qtr6c TtrAn. Quing. Ng5i, Phan NhQt llay 11Ar I'TTH Th6t Nr'it. Cin Thrr. Ngulin Thi M! NgPc 10A1 PT1'H LC Quy Don. Long An. Rdt hoan Dibn sd vio tam giic ngh6nh ban Npy€n Thi XuAn Nurtng l6p 7 Cho m6t hinh tam gi6c, c6 9 6 tam gi6c con VIn trlldng Chuy€n Dirc Ph6, Quing NgSi nhrr hinh v6. Ban hey clidn cdc sri tri 1 d6n 9 dA tim drl