Toán học và tuổi trẻ Số 208 (10/1994)

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Toán học và tuổi trẻ Số 208 (10/1994) trình bày về dùng bất đẳng thức Cosi để tìm được cực trị trong hình học; chứng minh một định lý trong bài đường tiệm cận; phân hoạch tam giác của một đa giác và một số nội dung khác. Với các bạn yêu thích Toán học thì đây là tài liệu hữu ích.

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Nội dung Text: Toán học và tuổi trẻ Số 208 (10/1994)

BQ GIAO DUC VA DAO TAO * HoT ToAN HoC VIET NAM 2 NN l0rzop ffi i# ,,:,. i iiIn;",n,.il ?illiliiiriiiiiili t rap cni na xcAy rs uANc rHAxc ln\ B$ NGRV flR SC BRCTo|lil ll0( Ull ruil lll( DRU IIfl
TOAN HoC vA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang a Brio todn hoc vd tud i tri trbn tud i 30 1 o Ddnh cho cdc bun Trung hoc co s0 For Lower Secondary Sch.ool l,euel Friends Tdng bidn tdp : Vu Huu Binh - Ding bdt ding thrlc Cdsi . NGUYE,N CANH TOAN dd tim crrc tri trong hinh hoc. 2 Phd t6rug bi4n fip : Gidi bdi ki trudc NGO DAT TTJ Solution of problems in preuious issue HOANG CHONG Cric bii cria sd 204. 5 D'Ara ki ndy Problenr in this issue HOI D6NG SIEN rAp ; Cac bdi tr) T1/208 ddn T10/208,L11208,L21208 ll Ban ui hi4't ! Nguy6n CAnh Toin, Hoing Do you knout ? Chring, NgO Dat Trl, LO Khdc Lai Iich cria cdc ddu +, -, x, -: Bio, Nguy6n H"y Doan, dung trong tod.n hoc 13 Nguy6u ViCt Hei, Dinh Quang Phan Thanh Quang - Todn hoc HAo, Nguy6nXuAn Huy, Phan vA. cdu chuy6n tinh 14 Huy Khdi, Vu Thanh Khidt, Ong kinh cdi cdch duy vr) hoc todn Le Hei Kh6i, Nguy6n Van Kaleidoscope : Refornt of Maths Teoching M5u, HoingL6 Minh, Ng:;y6n o,ndLearning Khdc Minh, Trdn Van Nhung, LA Dinh Chd.u - Chfng minh m6t dinh li Nguy6n Dang Phdt, Phan trong bdi dudng ti6m cAn 14 Thanh Quang, Ta F{dng Nguydn Vctru Mdu - Phdn hoach tam gidc QuAng, Dnng Hing Thdng, cria m6t da giric 1b Vfr Dtrong Thr+y, Trdn Thdnh Ket quA cu6c thi dec bi6t chdo ming 30 narn Trai, Ld Bri Khrlnh Trinh, Ngd bzio Todn hoc vd tudi trd 16 Vi6t Trung, Deng Quan Vi6n. o Girii trl todn hqc Fun with Matlrcm.atics Binh Phuang - Chay nhu the ndo Bia 4 Trdn Tud.n Nam. - Thay chtr bang sd Bia 4 Tru sd tba soan : 458 Hirng Chudi, He NQi DT: 213786 Bi€n tdp ud. tri sU; V0 XttVt TH{JY 231 Nguy6n Vin Cfi. TP Hd Chi Minh DT:356111 Trinh bdy; DOAN HbNC
r;fu{pg 10 narn 1994 Bri.o TOAN HOC VA ban doc t6i c1,c trttdng phd th6ng trung hoc cd I tudf tRf, aa ra ddi phuc vu ban doc ld s6. B6t ddu tt nam 1993 bao THVTT di dtroc c6c thanh thiSu ni6n y6u to5.n v6i muc dich chuydn thinh tap chi vi xudt bAn m6i th6ng "Gd.y hh1rug khi sli ndi, ltd.o hirtg hoc tod,n rrrQt kj, vdi sd ltrong phdt hdnh m6i kj' gdn trong thanh thiAu nidn, dQc biet ld hgc siruh 15.000 bAn. cac trudng phd thdng cd,p 3 hgii3, nay lit cac Qua 30 nAm hoat d6ng vi phuc vu ban doc tntitn.g phd thOng trurug hgc chuyen han)". cci thd thdy : Ngay tt khi mdi ra doi TI{VTT da duoc cAc - Nhim ddng muc dich vd ddi trrong cua ban tr6 hoan nghdnh vA y6u thich. Tr) kh6p minh THVTT da thuc su gay duoc mdt khdng ndi tr6n mi6n Bic, ban doc da grli thu vd tba khi s6i ndi, hio hrlng vi phong trio say mO hoc soan bay t6 ni6m hdn hoan sau khi doc s6 brio to6n vi giAi torin rdng rdi trong cdc hoc sinh ddu ti6n. M6t phong trio hoc to5n vi giAi toan ham hoc, ddc bi6t li trong c6c hoc sinh y6u torin da drroc ddy l€n trong c6c trrrdng hoc, xi phd th0ng trung hoc co sd vi chuydn han. nghiQp, cd quan, cdng trtfdng, n6ng tnidng vd - Ngoni vi6c dcing grip ddng kd vdo vi6c trong quAn d6i. ndng cao chdt lrrong hoc tip vd trinh d6 to6n 0 ,il ,{ BAOTOAIIIOCI,t ItIfiM[ ffiONItIil IO Nhung nam ddu ti6n, b5.o ra m6i thring mdt hoc cta cdc hoc sinh, THVTT cbn cri ddng g
Dinh cho c5c bT T*rrg hoc Co s6 T}UNG NAT DANG THUC COST DE TiM CUC TRI TRONG HiNI-I HOC eiNn 5c bdi to6n crJc tri trong hinh hoc thr.rdng canh d.y mQt hinh binh hdnh. Tint ui tri cila cci dang nhd sau : M dd hinh binh hirnh d.y c6 diAn ich. lon nhdt. Trong tdt cil cric hinh c- 4xy. BEMF, dt(BEMF) : S', dtfABC) : S (const). bdt ding thrlc ndy dttoc srl dung du6i c6c dang Cdn tim GTLN cria S'. sau : K6 AK T BC, AK c6I EM 6 H. Ta cci 1.Tim grd tri nh6 nhdt : S' = EM.HK, - Dang I : (x + y)2 > 4xy hoac x2 + y2 > Zxy. 1 S' :2 EMKH S:=BCAl 4. DAt MA : x, MC = !, EMxHKy : (dinh li Talet) - Dang 3 : Ndu tich xy ld hi.ng s6 thi tdng BC = *.+ y' AK * + y r *y nh6 nhdt khi vi chi khi r : 3l. ^ ,s' : ?.xy nen 2. Tim gi6 tri l6n nhdt : s @+rt' - Dang 4 : 4xy < (x + y\z hodc 2-xy < x2 + y2. Theo bdt ding thrlc COsi d4ng 5 thi S' _ 2-xv _. 1 dci -D?nss'(.x2- 1 _ 2. Do dd ndu S'ln hing Trong cdcbdt d&ng thtic tr6n, didu ki€n dd s S, xiy ra ddu ding thrlcld x - y vit trr) c6c dang 1 vd 4, c6c dang khSc dbi h6i r vi. y lA cdc sd s6 cdn S thay ddi thi -in$ = 2, tacci bAi duong. toan 2, Du6i dAy ld m6t sd bdi tAp tim cuc tri hinh Blri to:in 2. Cho hinh binh hirnh BEMF. hoc crj srl dung bdt ding thrlc Cdsi trong khudn DUng cdt tuydn qua M cd.t cdc canh cfia g6c B khd kidn thrlc l6p 8. tao thd.nlt. mQt tam gidc c6 diQn tich nh6 nhd.t. Blri to6n l. Cho LABC. Qua mQt didm. M Bdi torin ndy cdn cri thd di6n dat du6i dang bdt hi thu\c canh AC, lti cd.c duitng th&ng song khric : Cho gric B khdc gdc bet vi mdt didm M song udi hai canh kia, chilng tao thd.nh udi hai thu6c mi6n trong cta g
M cdt hai canh cria gdc B tao thdnh m6t tam Theo bdt ding thrlc C6si dang 5 thi gidc c6 di6n tich nh6 nhdt. :^ = xv - < ;.I Do dcj maxS' : ;l-S. Khi S' S (x*v)2 S ---" (-r +y)r 4 4 Giii : Xet;s' : *- 'zxy > 2 (st dung bdt : dd x y, tic ld hinh thang ABCD trr] thdnh hinh binh hanh. ding tht?c C6si dang 1). Blri toSn 4. Cho LABC uuing cCrn c6 canlt Cdch giCLi hhac : Tnrdc hdt dung cdt tuy6n huybn BC = a. Cd.c didm. D, E th,eo th,i tu thtt)c qva M cit cec canh cta g6c B 6 A vd C sao eac canh AB, AC. VA DH ud EK uudng g6c u(ti cho M la trung didm cira AC. X6t c6t tuydrr BC (H, K e BC). Tinh diQn tich ldn. nhd,t cio. bdt ki di qraM tao vdi c6t tuydn tnt6c hai tam hinh thang DEKH klti D ua E thav d,di t;i trt gi6c nh6, chrlng minh ring tam giSc nim trong tr€n, cdc co,nh AB uir AC. LABC cri di6n tich nh6 hon. Blri torin 3. Trong c,ic hinh thang ABCD BC ll AD) c6 dian ich S kh6ng ddi, E td. giao clidnt cac duitng cheo, d hirth thang ndo ttti LABE co di€n tich ldn nhdt ? , ,, :::, : . ,.,..............,,.,......,. . .' , ., ., ] ,,: . ,.. Giii : Goi dt{DEKItr) : S, ta cri : 25 : : {'DH + E&.HK : (BH + KC).IIK. Ta thdy tdng /BI{ + KC) + HK kh6ng ddi (bang o) n6n tich (BII + [
Ta cri 2sinacosa < sin2a + cos2a (bdt ding racci ABll cD** CD=yy MK = -1. x*h * thrlc C6si dang 4) mh sin2.r * cos2a = 1, do dd * ak ltl tnax(2sinacosa) = 1. CD:L. Nhu v6y : S nh6 nhdt
Nh{n x6t. (lii 5l bAi giii. trong s6 rl(r cri 2 bii gi;li sai. C5c tran sau dAv clt liti l4idi rdr : 'f6ng Ngi*-'l'r1 (7M Munt' - Curie, IId Nqii), Nguy6n Ngoc i)6zg (9NK'IhuAn'I'hz)nh. IIa tsAc).7'rbt Ngu,-tn Ngoc (9K. PI'1'}I l.e Lt)i. Il) 'fAv). N;rr.r'ia /-i /.rrt (li,\,- PTCS l)Anr l)r!i. Minh Ikii;. Li Qtt,tttgNitrt (xl chtrve n l)ilc Phir. Qrring Ng;ri1. T1/204 : Chfing ntinh rd,ng tbn tai cac BAri I)AN(i VI F:N sd nguy1n d.uong r,!, z, t tlzda mon dang thtlc. Bei T3/204 : Cho hinh uu1rug ABCD. Tr€n 1gx2 +Syq + lgg1xtel; - ttqq-) canh BC ud-QQ ldy hai didnl tuong ung M uit N so.o cha MAN = 45". BD cat AM ud. AN ldn Ldi gi6i : Ta sC chrlng minh rhng tdn tai luot tqi I uit K. Chung minh vd han c6c sd nguyrOn duong r, y, e, I th6a mAn dang thrlc trong dd bli. S^CIK = S 1'h4t r'6y, dat o - 2.9.1945: 35010 duoc : ^MNIK o = 1.129 (mod 1993.). Vi 1993 lh sd nguy6n t6 n6n tdn tai vO han cdc sd nguyOn duong n dd a,n * 1 chia hdt cho 1993. Ddy ring 19 +5 + 1890 = 1914. Vqy ndu ta det : rLl rr; _i:n .t = 1914r ,],: 1914" iz = l$|{t'rts. -f ,,n I I = 1914Tr thi ta th5y cr5 r'6 han c6c sd r, -),. z, / nhtt th6 th6a mdn ding thrlc dbi h6i r:ta bii to6n. BAd giai : Vi C ddi xtlng vdi A qua BD ndn Nhiin x€t. ('lc barr s:,rtr tl::rv r:r! litr giiri tdt: Brii Quung 1 \'l irtJr . t). l''l'('S (i ianc Vo : A,q, l)t'tt: I hirth . !),\. Ili: \'in I):ln. dpcm O va z > 0. Ndu Taco AMAN = r= sNiMN' r > y thi tr) 1 phuong trinh da cho, ta cd :J -xl = 2(r -Jl > 0. suy razl >.t4 hayz > -r Suy ra dpcrn. (vi r', z > 0). Chtlng minh tucrng t!r, tr} z > r ta Nh{n x6t : Ngtvin Kitu ('trng 9M. Mari Qrr1,ri. ,\gi cung cri1, > e, do ddJ > z > x >y, mau thuin. l)ut l ltinh.9,\ Ild Vrin l):rn. llii Quang Mitth TIit'S tii:ing Vo- A5zr,r'ia I')L?t l'htrrtttl4 qI I -l'rung Vttrtng. tli N0i ; Nlrnt'rr Vay r < J (1). Bang cdch giA str y > x. t,a Itoinq,1rth 9K Li: LrJi. IIi'l'A1, ; IIodng Manh Qaang. il'['l'r.1n c0ng chrlng minh duoc y > x (2). Tt (1) vd I)iing Ninlr. Naur Dinh. Narn FIii : Phutn Ti.riitt Anh.9'1. Phun (2.t, su1, l.a { = y. MQt cd"ch trtclng tu. ta ciing ]]ni ('hAu, L€ Si Dung^ 9A. 1'll('S ()uy ILlp. Nghe An;lIi cti.r' = z, do dei x = y = e. NhrJ v6y, hO trcl Si Iliiln. gNK Vinh Linh. l-t l/ro, 9NK l)irng iIi. Quing l-ri : thanh phtiong trinh.ra -2,x. +1 : O Lt Phtrit Dttnh. gi N1pv6n 'l-ri [,hri1ng. l-hit;r 'l-hi6n - [lrrd ; 2 Phant l)i?u llilng. {)i\ Qufrc Ik_r Quv Nhrln. Binh l)inh : t *r +1) 9 'l't
- Ldi giai. $heo LA Anh vu, 10CT - Qudc Ddu "=" x5y ta c+ho6c n = 2 hoac x, : x, _*fr hoc HuO : - ... - &n . o X6t a I 0, a > 2. Trong tnrdng hqP niY, S? bang phuong phap quy nAP theo z, dO dnng o V6i n = 2 hidn nhi6n c6 Qz = S, = Zn chrlng minh drrdc rn > 2-n+1 Yn e N*. Suy ra xn+ I : *n- Tn Vru €'N*, vi do d- 2 t'e ci : Y xr, xr>- 0. x., = rr,- r- 2-n+' = *n-r- 2-n+2 - 2-n+1 : . X6t n > 3.Ydi n = 3 d6 thdy cti : n-7 Qt= xfslxrxjlxrx, ( S/ q-rr *& Lt L 1x, + x, + 4)2 Sl < ...- : --; vA ddu ":" -... - IH A_ xAy ra 2, n e N) ; xt*xz*...*x,, : 1. Dqt Qn = Nhu vAy, ta cci di6u cdn chrlng minh v6i : xjx2 ... xtr_l + ,{s ... a,, * xlx2... xr*fr. n = k * 1, vi theo nguyOn li qui nap, Bii torin 1 drroc chrlng minh. I Ching minh rang : Qn: n,2. Dciu bang xd.y Tt dd, v6i Sr, = 1 ta drroc bdt ding thrlc cria bii de ra, vd ddu u-" xAy ra - 0 (n >- 2, chuydn Trd Vinh) rz € N). DAt S/, = xt *x21...*xnvd.Qn: : : xf 3...xn*xrxr... xnl...*xrxr...xn-r. Khi dri : r V6i n : 2 hidn nhi6n c6 Qz : x., * x-, - 1 SN-L _1_ Qn 4 ur- -l--. y n"' 6
. X6t n > 3. KhOng mdt tdng qudt, giA str , 1. + 1, a ;!. Thit trtrc tidp , c It2) vao phudng - xzl ...(r - rn). Vi r, + xz-* ... *-r, - 1 n6n 1 P(il : -rn -.rrr + ... rQ;r + (-1)lrl ...rn. + (-1)" trinh 16 rdng + I vd ,kh6ng bao gid ln nghi6m. Hidn nhi6n xp nr ... , xn lA n nghi6m thrtc ciia 1 P(x), vit, do d
NghiQm dci ld duy nhdt brli vi d6 thdy hnm 111* * r-) + r + 1 ln ddng bidn (bing c6ch dtng dao YqY -. him hoac chrtng minh tnlc tiSp). ,2n . ,rr, -0, sln'7 srn'7 srn'7 Nhrln x6t : Nhi6u ban gi;ii tdt bnin..)y nhtt cac banNgrl,in D{rt l'hutntg PTCS 'Irr-lng Vrt B > C. Xdy ra: (1) - \cotst , - 1) *\cotS:j - 1) * L) e + cr < 1800. H4 BH, L AAy t-6n\ +\cotSz, -r) =Z CHZ ! AA, (hinh 1). Gsi M, N, P ldn ltrot giao di6'm cria AAt, BBt, ^4n2n6w3n+2cotgjcorg * 2cotg CC, vdi BC, CA, AB, ta c6 : ,cotgn 7 BH, _so.,ur, l?n MB MB 6n + 2cotg cotg ,, = 2 , MC= -MC: -cH. = s;i = cot#or# * cots!7otfl! + -c . BA, , sin(B + a) _csin(B + a) =_._:+r b . CAt sin(C + a) 6sin(C + a) 4nT \ + cotg Tcotgi = l, dpcm. MOt c6ch tudgtrr, ta ctng NC -asin(C + a) Cd.ch 2 (Phrrong ph6p girin ti6p) g6i ' NA - csin(A * a) - ra thdy t . {+ ,T ,+lle nghi6m cra PA ______-______--_ _ -bsin(A + a) phuong trinh sinz4t = sinz3tr(1) pB = asin(B + o)'. Suv ra : (1) *'16r(1 - r) (l - 2x)2 = x(3 * 4r)2 PA MB NC _ .: . = -1. Theo dlnh li S6va, v6i r = sin2t PB MC --- NA Hay 64xj - Ll2x.2 1- 56x - 7 = 0 tacdAA, , BB1 , CC, ddngquy. cri 3 nghiQm . ,?nrx.2 stn.7 rl = sln-7 . .k . .6?t ; = ; 13 = srn47 Theo dinh li Viet thi : Iso lxrxr*xyr+x{t = 64 lz I lxfft= 6a 8
212 +.r = 1800. Ta cci AAt , BBl , CCI ch$u s' c$a v6glo kbqtg * :(Vi -!inh pOM, ddng quy t4i A. s ctOMl + + yOMT - OM : 0l6n b$t crl-mat phing nio cring li v6cto khOng : s' = +0 ). Cho O = O ' thi tii ( 1) va (2 ) ta duoc : (3) MM' = aMrM, + pM2IuI) + yM$: (a+F *Y = 1) Vi cric didm M, , M2 , M, vir M ddu nam v6 mdt phia cria z n6n cdc vecto MM' , MiIV'i d6u cung hrr6ng. vi do drj ta drtoc (4) MM' - aMtM't + PMrI['-, + yM$'t hay ln(il d = ad, + Pd, + yd, A+ trong dci ta d5 dat : MM' = d(M, n) = d., --.,, MB _MB -brir(Aj_") . MN'i = d(Mi , n) = di (l = 1, 2, 3). f, MC MCasin(B * a) t NC_osin(C+a) + a) Ndu goi 1 li tAm mdt cdu n6i ti6p trl di6n + va_PA bsin(A + a) ABCD thi, d didm niy, r = ! : z = f vd (IDA), { NA csin(A a) PB asin(B (IBC (ID B), (IDC ), ), (ICA) vit (IAB ) ld mat phAn Suy ra: PA MB :NC : giric cria cric nhi di6n ldn lrrqt cd cqnh ld DA, . -1, vi ta cci dpcm. DB, DC, BC, CA vd AB cira trl di6n. Ndu c6c PB =-MC NA mdt ph&ng phAn giric nrii tr6n theo thrl tu cAt Y€ry AA, , BBt , CC, ludn ludn ddng quy. c6c canh BC, CA, AB, DA, DB vd DC it cac Nhrln x6t : (\i .ltl irii giii tnrng sij dri chi c(r I hiri giiri vri didm P, Q, R, P', Q' vd R' (xem hinh v6), thi vi liiv : 60n. (::ic ban co l
k>-Zcotga I+ cotgp Budc 2. X6t m6t didm U bdt ki' niim trong trl di6n r,h6 DP'Q'R' d6 chrlng minh ring 6 (3) d6:x*y*z
Ctic l6p THCS Bii T1/208 : Cho cric a6 nguy6n a, b, c tlr6a san a2 = b * 1. X6t dey 6 { rn} ff - o ducr,r;j.c dinh bdi : uo = 0, un + 1 = aun + {-bZ" +-T-V Yn € N RA Bai fii108 : KI o,, Cho NAY o- li e1, ..., cric s6 duoag 16, tAt cA d6u Lhbng'cci udc,'nguy6n td Chfng arinh ring ddy un ld dey c6c s6 l6n hon 5. Chrlag minh rlng , nguyan. 11115?.. T - NGUYEN DUC TAN ola2anS -- Bii T2l208 : Cho ba sd dtto4g-r, y, zlb6a DANG HUNG THANG m6nr*y*z:xyz Ilii Tt/2OS : Chrlng minh ring ndu r vir e Chrtng rninh ring : li hai sti thuc cho tnl6c th6a m6n di6u ki6n {GTPIO+4 -l.tT+7 -{t+V + > 0 ihi phuong trinh 13 * Srx - 2s = o . "2 "3 cri nghiOm sd thuc duy nhdt ld + -{;l-\r yz a- \fQ +;TO +q -ttTlV -11717 :r ='G;.ffi l{ay 6p dgng kdt quA dd dd $hi ci.c phttong *x* 1 = o vi ryt --15x2 1 = o. '' "' -\,-1 +7 -\rTrr trinhr3 +-'fif+r|(l-+lT xy -0 TRAN VAN VL]ONG - lnAN xuAN oANr; Bei T9/208 : Cho ABC, v6 ciic phnn gi6c trongAA,, BB1, CCt. GiA sir AA, c{t B,C, tai Bei T3/208 : Chrlng minh ring, tam giSc K, BB, cbt C,A, t41 E, CC, c6L A,B,'ta'i F'. ABC d6u khi vd cfri ttrl Chllng'minh ringndu AK = BE: CFthiABC _1 111 a_ li tam gi6c d6u. sinzA sin2B sin2C ABC NGUYON KHANII NGIIYF,N 2sin , sin 7 sin , Bai T10/208 : Cho tam giSc ,43C khOng vu6ng cd dQ dAi cric canh BC : a, CA : b vi vAu Nrri oarra AB : c. Tim trong mat phing (cria AABC) m6t Bei T4l208 : Cho ndm didm phAn bi6t A, didm M sao cho cric kho6ng circh x, y v?t z tit M B, C, D,.E vi m6t drrdng th&ng d cirng nam ldn lrrot ddn cric dudng thing chrla c5c canhBC, t43n m6lmat plring.-1$si Ole did4ry sao cho CA vd AB ti 16 vdi do-dai cri--c canh drj : OA + OB + OC + OD + OE = O.GoiA1, x:y:z=a:b:c B y, C 1, DpE 1, O, Idn luot li chdn dudng vu6ng Hay chi 16 vi tri hinh hoc cria nhirng didm ha tit A, B, C, D, E, O xudng d. Chrlng gcic tim duoc vd tinh khoAng cach tr] nhttng didm minh ring : dd ddn c5c canh cta tam giric dd cho. -1+--->---.>-+ E (M, OO:.: +BB1+CC1+DD.+EE) NGUYfN DANG PHAT lE oudc HAN C6c dd Vat I BAi T5/208 : Goi R li brin kinh mat cdu ngoai tiep | ffio, frb,nLc, nt \J Ie dO dai cdc trung Bai L1l208 : MQt vdt chuydn d6ng chdm tuydn xuat phdt ldn lrrot tr] c6c dinh A, B, C, ddn ddu, x6t 3 do4n drrdng li6n tidp bing nhau D cua mOt trl di6n A-BCD. trtrdc khi dtng lai thi doan d gita nri di trong Chrlng minh bdt ding thrlc : ls. Tim thdi gian vAt di 3 doan dttdng bing nhau kd tr6n. 3 Rr lU (mo*mb*ruc+md). rlei Lztzol r"5f+Y#XTfuf [IiJu,c" ' c6 cdc ki hi6u r110V.60W.50H2. Khi ndi hai Khi nAo thi xAy ra ding thrlc. ddu ddy cria qu4t vdo 2 cuc cria chidc pin cd HOANG HOA TRAI sudt di6n d6ng 1,5von vd di6n trd trong nh6 kh6ng dring kd thi cudng d6 dbng diQn ch4y quadquatlDr0,025A. Cric l6p THCB 1. Tim tliQn.trd Ro vi hQ s6 c6ng sudt dinh mrlc cta quat diQn. - Bai T6/208 : Tim s6 c
For Upper Secondary Schools T6/208 : Find a three - digit number, divisible by 9 such that the quotient in the division of this number by g is equal to the For Lower Secondary Schools surn of squares of the three digits. .I IIAN DLIY IIINH 1'11208 : Let n, b, c be integers satisfying a2 = b * 1. Consider the sequence { rnirT= T712OB : Let a, , al t ...t on be odd positive o numbers, none of which has prime divisors dcfined by : greater than 5. Prove that : t,,, : () : u,, * 1 : ett n* {Ani+}, n : 7, 2... 11 1 15 prove that eyery utt is an integet (n --+-+...+ ar az < --:- - 1,2...) a, E NGT'YIT,N DIJC lAN DAN(; IiUNG'f}IANG TZI2O8: Let be given positive numbers r, y, z satisfyingr + ! * z = xyz. Prove that : T8/208 : Prove that if r and s are two given realnumbers,satisfyings2 + f > Othenthe * equation xlt + Srx - 2s = O has unique real yz root which is equal to \rd +eT(t+7t -ttl +-7 -\fi +7 2 -- f ___ ZX r = {s + {-/-+-F + rJs Gr --F - Apply this result to resolve the equations +\t(TTFxr +JT - \n +7 ' - \fi-Ez _____0 x:r+r * 1= 0 and 20x3 - 15x2 - 1 = 0. xy l'lLAN Xtlr\N I)AN(l ]'RAN V,\N VUONG T3/208 : Prove that the triangle AISC is T9/208 : Let AAt , BBt, CCt be the equilateral if and only if inbisectors of triangle ABC. Suppose that.A-4,, 111_f _-_ J_ _ _ meets B rC, at K , BB I meets C ,A, at E. CC 1 sin:A sin2B sin2C - ABC rneets ArB, o.t ,t, . Prove that if AK = BE = Zsin rsinTsin; CF then ABC is an equilateral triangle. NGllYfiN KIIANH NG tJYIIN DAM VAN NIII Tl0/208 z Let a, b, c be respectively the T4l2O8: Let be given five distinct points lengths of the sides BC, CA, AB of a noright A, B, C, D, E and a line d is a triangle. Find a point M in the plane of ABC plane. such that the distances r, y, z ftom which t,o D_gnote, ,by I thg pouJ "ame sugh that tlre lines BC, CA, AB are proportionnal to the ()A + OB + OC + OD + OE : O;andlet ,1 ,.- 81, C1, Dy, Ey, O be respectively the lengths o, b, c : olthogonal projections of A, B, C, D, E, O on X:Y:Z:a:b:c d, Prove that : For each such point, cletermine exactly its --?1-€-++ positive and calculate x, y, z OOt : @A, + BBt + CCt + DD, + EEt) E NCiLlYt,N DANG PIIAT Lll QtlO(l tlAN T5/208 : Let R be the radius of the circumsphere, nt o , nlh , n.1,, , ft1,7 be the lengths of the medians issued repectively from the vertices A, B, C, D of a tetrahedron ABCD. Prove the inequality : 3 *, * (mo*ntb+mr*m4) When does equality occur ? T{OANG TIOA'I'RAI L2
-. -- -1-- -, - A' r t IJI tfcll c[lA cAc D[lJ *, -, r,!, lrrrr-l = DUil0 ffiOil0 I0Ail ll0c Nam ddu *, -, X, + vd : dilng trong todn ddng thdi v6i2 ddu "#'vi "-". Trong khi dd d hoc, chdng nhtrng rdt quen thu6c v6i c5.c ban Trung Qudc nhung s6ch xudt bAn cudi ddi nhd hoc sinh phd th6ng, md ngay cA ddi vdi cdc em Thanh, trong tinh tod.n vdn dirng crich vidt chtl hoc sinh E c4c tnrdng ldp miu gi6o cung kh6ng s6 theo hing thing drlng, tr) phAi sang trdi, vi phAi xa la gi. du 3159 + 6247 = 9406 thi vidt lA : Th6 nhrrng, lai lich vd srr phdt tridn cria ncj + da phai trAi qua m6t thdi ki dai ddy gian nan khfc khujru m6i drroc nhAn loai c6ng nhAn nhrr I 63 ngdy nay" 4 27 NgUdi cd Hy Lap vi ngddi An DO dung ln 0 45 "kh6ng hgn md n6n", hq ddu vidt 2 sd gdn nhau 6 79 bidu thi ph6p c6ng, vi du 3 +] Auo" vidt le 31. Rdt 16 ring, cdch vidt ndy vr)a khd doc vrla Cho n6n tdi tAn ngiy nay, phrrong ph6p bieu khri vidt. Sau c6ch mang TAn Hgi m6i ddn ddn t.hi nhting d6y phdn sd v6n cbn in ddm ddu vdt drroc sit dung nhttng ddu "#' vi "-" nhrr hi6n ngiy xrra. nay trong cdc tdc phdm cta Trung qu6c. Vd, ndu dd bidu thi 2 s6 trr) v6i nhau, ngudi Cbn vdi 2 ddu "x'" vA "+", qu5. trinh hinh ta vi6t 2 sd cSch xa nhau m6t chrit so v6i c6ch thinh cring khdng it hon 300 n6m m6i duoc lodi ngudi srl dung rQng r6i. Theo nhu truy6n vi6t Zsd c6ng v6i nhau, vi du 6 cd nghia la | tung, thi ddu " x " bidu thi ph6p tinh nhAn dA 1 drtoc mOt ltgudi Drlc st dung trong t6c phdrn 6 -;. cria minh nhitng nam 1630 - 163i vd tt dri D Sau nAy cci ngudi drlng cht c6i la tinh P duoc srl dung cho t6i ngdy nay kh6ng thay ddi. (chu c6i ddu ti6n cta tidng Plus cri nghia ld Thbi ki trung thd ki, d6y s6 do ngtrdi An c6ng) vi M (chrr crii ddu ti6n cria tidng-Minus D6 ph6t minh ntfi d tr6n tudng ddi phrit dat, cci nghia la trir) dd bidu thi cOng va tiil. duoc sir dung phd cAp, ddu "+" tril mQt nhd . _Vi d_u 5P3 cd nghia la 5 + 3 vd 7M5 cd nghia tod.n hoc ndi tiSng th6 gi6i thdi dy cbn dung ln7-5. Thd,i ki cudi trung thd ki, thuong nghiOp theo cach bi6'u thi cta phdn sd ra, ,rfrr'r CIeAu Au tr6n di ptrEt tridn. ntOt sa- ttlo.rg "i f gia diln-g ddu "+" vidt b€n ngodi nhirng kign bidu thi 3 chia cho 4, nhin loai ddu dirng ddu hnn-g dd bidu thi trong lrrong-"ldn qu6" inOt it, . "+" dd bidu thi ddu chia theo nhi to6n hoc vi ddu "-" dugc vi6t b€n ngbni c6C kion hdng n$Ibi Anh trlng sfi dung trong cdc tdc phdm dd bidu thi trong ltrong ki$n hang ;non,; mot cria minh vao nhtng nam 1630. chrit. Cho tdi nay, tuy6t dai da sd c6c nudc tr6n Tdi thdi ki van hcia phuc hung, mQt nhi th6'gi6i d6u dirng ddu "*" bidu thi ph6p tinh nghQ si ndi tidng cria Itilia trong mOt sd tric cdng-ddu '-" bid.u thi ph6p tinh trr)1rong cric phdm cria 6ng da dung ddu "*" vI dd; ,-,. tric phdm cria minh. R16ng 2 ddu.x'bidu thi Nem 1489 c6ng nguy6n, rn6t ngudi Drlc p-h6p nhAn vi-ddu "+" bidu thi ph6p chia thi trong nhi6u tric phdm ndi tidng cria minh da chua duoc phci cAp vd thdng nLdt hban toan. chinh thrlc dr)ng ddu "*" vd u-' dd bidu thi ph6p Vi du m6t sd nrr6c ding ddu '.' thay cho ddu tinh cdng vA ph6p tinh trr). "1" O.Ngavi Dtic trong hdu hdt cac tac phdm cta minh, ho dd dung ddu " :" thay ch6 ddu _ . Sau drf, nhd cci sr1 truydn bri vA tuy6n truydn "+l' !!du ttri pn6p tinh"chia , r,,ghil la nguai hdt srlc ki6n tri cta m6t nhd to6n hoc ngudi ",; ddu chdm di ta b6 bdt lnQt gach ngang giira2 Phap mi 2 ddu'*" vd. i'-u *di drroc phd iap, xem ra thi cung hop li, do vAy ddu "+" ddn ddn ; vA m6i tdi ndm 1630 mdi drroc loii ngudi c6ng it drroc srl dgng vi ddu " :" bidu thi ph6p chia nhAn. ngiy cing drroc sit dung r6ng r6i. Tai Trung Qudc, nhi tod.n hoc ndi tidng thd Cudi cirng ln ddu "=". Ddu ti6n ld Ai Cap gi6i Ly Thi6n Lan, voi hang ding thrlCcrla vi Babilon d&ng ddu "=" dd bidu thi 2 gia tfi r_n!nh.d1 dung ddu "J" bidu thi ph6p c6ng vd bing nhau vA. m6i tdi thd ki thf 18, ddu ":" ddu "T" bidu thi ph6p trt. Song d th-di ki ney, nguoi ta thttdng dung "que tre" vd "hat d6" dd m6i duoc sit dung phd cdp, khi d6 2 gach ngang Iim cric phdp tfnh *, -, X, +, cho n6n cric ddu song song cdn k6 kha dei. Mdi sau niy mdi rut hi6u chua th6t sr-l dtloc coi trong vi truydn bri. ng6n di nhtrng vdn dAm bAo 2 gach ngangsong song vd cd chi6u ddi bing nhau. - Sau ndy, ngudi ta tta dtng ddy s6 cria ngrrdi An D0 phrit minh : l, 2, 3, 4, 5,6, LJl, 9, 0, TA QUAN(; VY lsttr uinr) 13
T HASKARA tit nhit loan hoc i'n DO tlto ki -tD ttrri 12, tac gid cudn sd.ch toan c6 ftn "Vbng nguy€t qud cila m.Qt cia m.Qt hQ tldrug thian ud.n", uidt nam 1150. 1ng cung ld. td.c gid cuott sach toan c6 han Lilauadi (Cd.i dep) ngltian ciu ub s6 ltgc. Nhibrt #-N6BaffiK cAt vA cAcH DAY HgC TOAN kidn ttfic u€ sd hoc Hindu truybn lai ddn ngiry nay dbu bat rugubn ti Lilauadi. CI_IUNG MINH UOT DINI_I Li SU tich t€n sd,ch. Lilauadi citng ntang mdu sd.c lang nran, nhx m.Qt bd.i tho tinh. Lilauadi TRONG Bfu DUONG TIFM CAN !.ir t€n ngudi con gd,i duy nhd.t cia Bl,aokara. Nitng d,ep (tdt nhiAn ! Vi co dqp nfii thanlt (GrAr ricu 12 ccGD) chuy€n), gi6i uan th.o, d.m. nltac, tlt)ng nzinh, tdi gi6i... T6nt lai, nd.ng lit kdt tirlll cita "sdc Trong phdn niy 3 bO s6ch gi6o khoa GiAi nltoc, lttlotlg trdi". tich 12 d6u trinh bAy rn6t dinh li cci n6i dung Ngity giit 16 uu quy cfia ndng dd duoc Thdn nhu dinh li sau dAy : tinh bao trudc. Ndu so,i gid gid.c d6, thi ntQt Din.h li; Goi (C) Ie dd thi cria hdm sd tai hga gh€ gdnt s€ Qp lAn dd.u ndltg. y = fk), giA sit x cri thd din t6i vO crlc. Td.t ruhian ud,o thdi didm quydt diruh do, ndng phd.i gbi can.h titng giot nuoc ti tdch nh6 Di6u ki6n cdn va du dd dr-rong thing rDt co xudng tit nrOt dbng hb nudc hinlt clt€n. phuong trinh 1 = ax. * b ti€m cAn v6i tCtli' : Mqc nudc tron.g chen nhich xttdng ddn, rdt clr.dnt. Thdi gian nhu ngilng trdi. !11tnxt-(ax*6)l:o Nltung... nhu ntQt sd phQn. dttac an. bdi, thdi didm. quan trqng nhdt trong diti da trdi qua, - Tuy rn6i s6ch gi6o khoa trinh bay ccj kh6c lic nito ntit, nitng kh6ng hay. nhau nhrJng phuong phdp chrlng rlinh nhu Nguy€n nhd.n tltQt lit. dd h.idu nltung kho crich chrlng minh trong s6ch GiAi tich 12 tru6c tudrlg tuong dttoc : Gdn gid 'G' d6, hoac ui CCGD. Vdi phttong phap d
(, ?rong hinh hoc, ngodi cdc bAi torin truy6n thdng mang tinh chdt dinh hinh vd dinh tinh sdu s6c (dga vAo c6c ddc trrrng hinh hoc co bAn duoc ph6t bidu thdnh c6c dinh li hinh hoc) cdn cd nhrrng bdi to6n kidu gh6p hinh vi chia hinh mang m6t nOi dung khric. Phuong phdp khAo sdt cdc loai. todn ndy thrrdng dtta tr6n mdt vii dac H rf tinh rdt ri6ng bi6t va rdt don giAn, cri khi chi li nhrlng suy ludn l6gic hinh thfc trrdng nhu kh6ng cri gi li6n quan t6i hinh hoc nhrt cdc nguy6n li Diricle, ti6u chudn chia h6i E' cira mOt s6 tu nhi6n cho m6t s6 nguy6n cho trudc, ... C6c bii to6n gh6p hinh va chia hinh c
mdr euA cuQcrmr DAc mrpr E Ghio mfnu 30 ntm t?p chi toin hgc ui tudi trd Trong hai sd 200 vA 201 Todn hoc vi tudi 4. GTAI BA: tr6 d5 c6ng bd dd thi cria cuoc thi dac biet. l'!ttirrll Tlrarth Tirrg.ST,THCS ChuyOn tni xa ffrai Alnn Ngay trong thring 3. L994 nhrrng bei dU thi Nguyin LA Lrt, ARt, Odm Ddi, Minh HAi ddu ti6n di v6 t6i Tba soan. Cric ban tr) c6c dia phrrong sau ddy de gui bni vd drr thi : HA ttbAnhTudn,9CT, THCS Phti Thq, Vinh Ph0. B5c, Mnh Phu, Ha Noi, Hai Htlng, HAi Phdng, Vu Quatg Ditrg. 98. THCS Chuy6n ViQt Tri, Vinh Ph[. Narn Ha, Hd Tiy, Th6i Binh, ?hanh Hcia, Nguyin Ngpc Tdn,9M, PTDL Mari Quyri, HA NQi. NghQ An, Hh Tinh, QuAng Binh, Quing Tri, Ti Trhn Tnng, 9CT, THCS Trdn Ph0, HAi Ph6ng. Thrla Thi6n - Hu6, QuAng Nam - Di Ning, Nguyin Thanh Thtty,gCf , THCS TrAn Phrl Hril Phmg. Binh Dfnh, QuAng Ng6i, Phu Y6n, D6c Lic, Luu Vdn Thinh,gCf, THCS Trdn Ph0, HAi Phdng. TP Hd Chi Minh, Bd Rla - Vrlng Tdu, Ti6n Lwtng Tudn Anh,fiCf , DHTH He NOi. Giang, Vinh Long, Ddng Th6p, Minh HAi. Cac Pham Dtnh ( hinlt.10B, DHTH He NOi. dfa phrrong cc5 d6ng ngrldi du tiri nhdt li : Hd NQi, Hai Phdng, Thanh Hcia, Ngh6 An, Vinh Phan L€,!aa, 10T, PTTH Lam Sdn, Thanh H6a. Phf, Hai Hrrng, Thdi Binh, Nam Hd... llb Ddc Phmtng,11A, DHTH, HA NOi. Nguyin Thanh tti,i, lf:Cf , DAo Duy Ttt Oudng Binfr. Nhi6u trudng cci phong trio drr thi rdt s6i ndi, Drj Id cdc kh6i PT chuy€n to6n DI{ Tdng Hop, DHSP I HA NQi vi c6c trudng Lam Sdn, Thanh Hria ; THCS Tnrng Vuong , PTDL 5. GIAI KHTIYdN KHfCH Mari Quyri, lla NOi ; Phan B6i ChAu, NghO Ngrtyitt Anh Thdi.8T Chuy6n Le Quf D6n. Bd Ria - An;LO Khidt, QuAngNgei; PTNKHAi Hrtng; V0ng TAu Trdn Phri, HAi Ph6ng; NK Hn Tinh ;f)io Duy Drutl f71111g Tnng, *T Nghia TAn, TiJ Li6m, Ha NOi. Tt, Quing Binh ; Qudc hgc Hu6... . Nguyin KiEu ('mng,9M PTDL Mari Quyri, HA N0i. hfi,iO., ban THCS de giAi dung mdt sd bni Ltodng Rhdt t nry, 91 PTTH Lam Sdn. Thanh H6a. dinh cho c6c anh chi THCB. Nguye n Quang Nghia. gA THCS Trutng Nhi, HA NOi. Dudi dAy ld danh srich crlc bqn de kung giai : (*ing, gCT THCS Trdn Ph(, HAi Phong. Lt Vdn Phan Li Mitrh.9T PTTH Lam Sdn, Thanh H6a. 1 GIAI xU,{T SAC . Dinh Thi Nhuug, 9I PTTH Lam Sdn. Thanh lJ6a. Nguy'irr Arrlt Ir,. 9H. THCS TrLhg Vudng. Ha Nai, Ngttyirt Phti Qrtirt,1,9A THCS TrUng Nhi. Ha Noi l'hant Le Itnng,9A, THCS Trunrg Nhi, HA NQi. Phan th?t Ting,9CI THCS Trdn Ph[, HAi Phdng, T"rinh lltru Trung,9T Lam Sdn, Thanh H6a. 2. GIAI NHAT Ili quang Thrt, gT Lam Son. Thanh H6a. ' Mai Thanh llinh,7M, PTDL Mari Quyri, HA NQi. Nguyitt Binh l'hmtng 9T Chuy6n LC Khidt Quang NgA. Vu Quang llda, THCS Chuy6n thi xA Th6i Blnh. Phan Neloc Lan, gCF ViQt Tri. Vinh Phu. L€ Vdn An,9T. PTTH Phan BOi Chdu, NghQ An. Dwto11 lllinh D{rc, 9A THCS Trtlng Nhi, He NQi Trhn Nguy€n Ngoc,gK La LOi, Ha D6ng, HA TAy. Vuong Vti Tndng, 9A1 THCS GiAng Vo, Ha NQi. Nguyin Thdi tla m, DHTH, Ha NOi. Tmn Dt'rc Quyin,gf Trdn Ddng Ninh. Nam Dinh, Nam Hir Ti fi6og Vt1 11CI OHTH, He Ndi. LtNgqtc: Thach,9T NK Qu!,nh Luu, NghO An. Vu Thu ttuy?n.gCT THCS Trdn Phri; HAi Phdng. 3. GiAI NHi : Nguyin t-dnh I tito, 9T PTTH Plnn B6i frar, NSf,e Rn Nguyitt Bi lling,9H. THCS TrrJng VUOng, HA NQi. Vrt llitng Minh,9I Chuy6n Nguydn Binh Khi6m, Vinh"Long Hobng LA Quang, gCF, THCS Trdn Ph( HAi Ph6ng llrhng Anh Tittg, frI PTTH Lam Sdn. Thanh HOa. Kitu Thu l/1fu, 9NK Qu!,nh Luu, NghQ An. l'ham Manh Quang,lQT PTTH Lam Sdn. Thanh H6a. Cao llbng Phurtng,1OCT DHTH. Ha NQi. Le Trorg Oiang, lOf PTTH Lam Sdn. Thanh H6a. Nguy6n Xudn Thdng,10T, PTTH D6ng HA, QuAng Tri. Nguyirr Ltit.10T. PTTH Lam Son. Thanh H6a. Nguyin Thd Trung.11 DHTH Ha NOi, Trinh Ddng Ciang.11M, PTDL Mari Quyri, HA N6i. Li lluy Khanh.11T PTTH Phan B.Oi ChAu, Ngh0 An. Ta Huy Quynh,12A, PTTH Chuy0n Th6i Blnh. Luung Vdtr Thdng.,l1T PTTH Phan BQi ChAu, Nghe An. llo?tng Thi Tuydr 12f, PTTH Larn Sdn, Thanh H6a. Nguyin ll{ru Trung,11T PTTH Phan BOi Chdu, NghO An. 16
- Gidi. Theo ding thrlc (1), thi sd c6c tam PhSn [eopch tam gi6c"." trong moi phdn hoach bang gi5.c (Tidp tlreo tra.ng 15) s:4-2+2":4g:100. Do vdy, kdt ludn cria bdi torin ld hidn nhi6n. Quan s5.t liai vi du tr6n, ta nhAn thdy r6ng ph6p phnn hoach tam gi6c cria mdt tf gi6c theo NhAn x6t reng sd 49 Id s6 nh6 nhdt dd bei m6t didm vd" cria m6t ngu giac (theo tAp r6ng) to6n 3 dring. Thdt vAy, x6t 49 didm chia m6t chia da gi6c thdnh c5.c tam gi6c kh6c nhau, dudng ch6o cria hinh vu6ng thinh 50 phdn song sd }rrong cac tam gizic trong m6i phAn bi.ng nhau (xem hinh v6), thi moi tam gi6c hoach d€u nhu nhau L: .,,,...,..;,..;.:...,.;.;:,,,;,:,',.,.,,,,.,':,,:.,,,',,.,....'...,.:.,.,,:,,.,.:,..:. iTiong vi du i ta nhAn duoC 4 tam gi6c, trong vi du 2 cci 3 tam giac). Di6u ndy, cu.ng dring cho trlldng hop tdng qurit. Bit.i too,rt I Cho n-gtac K h > 3). Khi dd s6 tan'r gi6c trong m6i phdn hoach (tarir gi6c) Nha., *Jr;;;;;;"193ng drnh nghia 1 loai cuaKddubangn-2. b6 di6u ki6n c) thi trong m6i phAn hoach cdn . DAy ld bdi to6n giin don d6i v6i chrrong li6t kO th6m c6c tam gi6c suy bidn (tam gi6c trinh phd thdng co s6. Vi tdng c6c gcic trong ccj ba didm nim tr6n m6t dudng th&ng (xem c'&.a n-giac K bang rn - 2)1 80o n6n ndu sd tam vi du 1 ) ) . Trong trudng hgp ndy, phr.rong phrip gi6c trong mot ph5.n hoach Ia ft thi tdng cdc glhi c1.c bii to6n tudng tu nhu bii toan 3 sO gcic trong cua /r tam giric ndy bang K. 180,,, Do khOng cdn don giin ngay cA khi sd didm cho vdy : K . 180/' = (tt - 2)l8U' ir.ay k = n - 2. tru6c trong tam giac khdng nhi6u. Bo,i toan 2 Cho tam gid"c K vd, nt didm phAn Bdi too,n 4. Ctro hinh vuOng ABCD ccj di6n bi6t trong K. Khi dci. trong m6i phdn hoach tich bang 1. Hey x6c dinh sd drrong o nh6 nhdt cira K ddu cci s = 2nt + I tam giac sao cho v6i moi didm M trong hinh vuong d6u tim dtroc it nhdt mOt tam gid-c cci dinh lA"M vd Thdt vAy, ndu cci s tarn gi6c thi tdng cdc 2 dinh kia ld dinh cfia hifi vu6ng ABCD vdt gcic trong cria chirng bang s . 18U,, Ydy : di6n tich < o. s. 180' : 180/' +nt .36U, hays : 1 * 2nt. Gidi Goi E ln Hodn toAn tuc,ng td theo caci giili tr6n, ta tdm cria hinh crj-ket quA. : V6i n-gac K vd, nt didm trong K vu6ng, F' li trung thi s6 tam gidc trong m6i phAn hoach ctti X didnr cira AD. Vi E theo nz didm dci bang la tAnr doi xdng -q=1r-2+2nt(l) cria hinh vuong n6n chi cdn xet M Tu1-nhi6n, 6 ddy, chfng ta kh6ng d6 cdp ddn thudc LEAD. Vi srJ-tdn t4i ciia-phAn hoach tarn gi6C cua n'-$ac thi.ng cnu:t dudng r,nal1g ouong chil:t ,t theo b6 riz didm, tlongK (Xin ban doc trr chrlng Etr' Id truc ddi xtlng minh su tdn tai cira ph6p phdn hoach). ^ n6n r I chi cAn^ ,, -i'^, xet die.m M tr6n .EF. Tr) dci, chon Ti deng thrlc (1). ndu ta goi cdc canh cria M trdn EF dd s(MBDt : s(MAD) vA" ta thu tam gid.c trong m6t phAn hoach ld "canh trong" &tac M Ii trung didm cria Ef. Khi dci idy khi canh dci khdng trirng v6i canh cua n-g1ac o, : SMAD = ll8. (cac k6t quA tr6n d6 ddng kidm K cho trr.16c, thi sd cci tra ndu ta dnt hinh vu6ng trong h6 truc toa d6 HQ quiL: 56 c6c canh trong trong m6i phAn vu6ng g6cA(0,0), B(0,1), C(1,1t vdD(1,0)) hoach cia n-gil.c K theo bO nz didm trong K R6 rdng gi5 tri o.: ll8 vdn cdn dting cho li nr6t hang sd vd bAng 2nt - 2 (kh6ng phu trr.tdng hop cci hai didm trong hinh vu6ng. thu6c vao n). Bai toon 2. Chnng minh rang vdi 49 didm triy " Tuy nhi6n, neu s6 didm ld 3, 4 hoAc 5 vd dinh cria c6c tam giSc trong phAn hoach ld cac y trong hinh ru6ng canh bing 1, trong m6i phAn didm dl cho vi. 4 didnr tny y c6 dinh trong hoach hinh r,udng theo 49 didn drj, d6u tdn tai it hinh vuong tthay cho 4 dinh hinh vu6ng) thi cdc tinh todn trd nen rat. phtlc 1ap. nhdt rnot tam gidc cci dien ticfr < fr. NCiI.JYEN VAN MATI
THONG BAO Til th6ng 9. 1994 tda so4n t4p chi To6n hgc vir Tudi. trd chuydn vd trg s& mdi. Dia ehi giri thu v6 Gidi ddp bdi tda so4n li : Tap chi ToAN HQC vA ru6l tnp Chay nhu thd nDro ? 45 B Phd Hhng Chudi - Ha NQi. Ta hinh dung sdn b