Tạp chí Toán học và Tuổi trẻ: Số 228 (Tháng 6/1996)

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Nhằm giúp các bạn chuyên ngành Toán học có thêm tài liệu phục vụ nhu cầu học tập và nghiên cứu, mời các bạn cùng tham khảo nội dung tạp chí "Toán học và Tuổi trẻ - Số 228" ra tháng 6/1996 dưới đây. Nội dung tạp chí bao gồm những bài viết chuyên ngành như: Một chút sáng tạo từ một bài toán, họ đường cong tiếp xúc với một đường cố định, viết số trong bàn cờ,...

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 228 (Tháng 6/1996)

- I rr-:l L .,.) r BO GrAO DUC vA DAO TAO * HOr TOAN HOC VrF.T NAM -oA Nryfrd.3*, { =O€. rap cHi RA NGAY 1b rIANG uraNc I MgT Ctrqr sfil{G 160 ro mQr afir TofiN @, rdr ouA KI THI HOC SINH 1 GIOI TFfiPT 1995 tee6 @ He DUONG CONG TIEP XUC VOI t{ MQT DUoNG co DINH@ . vE vOr PHUONG PHAP CHUNG o MINH DANG THUC VEC ro 6-i A. ^ \--l o€ THt TUY€N SINH VRO C6C LoP CHUV€N Dnr HQC TONG ^? HQP slEnrr@ 1X cun uOr Ncuor @ 7/qt7 so 7Ro"rE ei"teo @. Th$t vd tro truong THPT Qu6:c hqc Hud.
TOAN HQC VA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang s Ddnh ctu cdc bqn Trung h7c Ca st For Lower Secondary School Level Friends' Nguydn Drtc Td'tt Tdng bidn tdP : - MOt chrit sSng tao tr] mOt bdi to6n' 1 NCUYEN CANH TOAN Phd tdng bidn tQP : t Gidi bdi ki tudc. NGO DAT TIJ Solution of Problems in Previous Issue HOANG CI]ONC c6c bdi cias6 224 2 c Db ru ki nd1 Problems for this issue HOt DONG elEN rAP : "t11228, ..., "ltol228,Lll228,L2l228 8 - 1996 10 Nguy6n CAnh ToDLn, Hoing o Kdt qud. hi thi hqc sinh gi6i THPT 1995 Chring, Ngd Dat Trl, L6 Kh6c o Ddnh chn cdc ban chudn bi thi vdo Dqi lqc' BAo, Nguy6n HuY Doan, For College and University Entrrance Exam Preparers' Nguy6n ViOt Hei, Dinh Quang Trd.n Phuortg - Hg drrdng cong tiSp xtic HAo, Nguy6n Xuin HuY, Phan v6i mQt drrdng c6 dinh' 12 Huy KhAi, Vtr Thanh Khi6t, Lo Hei KhOi, NguY6n Ven Mau, o Tim hidu sAu thim todn hoc phd th6ng' HodngLO Minh, NguY6n Kh6c Helping Young Friends Gain Better Understanding Minh, Trdn Van Nhung, in Secondary School Maths Nguy6n DEng Phdt, Phan Nguydn Viet HAi - V6 m6t phrrong phSp Thanh Quang, Ta H6ng chring minh ding thfc v6cto. 15 Qu6ng, D4ng Htrng Thing, Vo Drrong Thuy, Trdn Thnnh o Db thi fiq&t sinh uin cdt l6p cfuryen DSt hP tdng h'Sp Trai, LO 86 Kh6nh Trinh, NgO o Gidi tri todn hqc Vi6t Trung, Dang Quan Vi6n' Fun with Mathematics. Binh phuong - Didm cria mdi ngUdi Bia 4 Ngd.n IIA - Vidt sd trong bin cd' Tru sd tita soqn : 45B Hirng Chudi, Ha NQi DT: 8213?86 BiAn @P uit. tr[ s4; VU KII TEII 231 Nguy6n Vin CiI, TP Hd Chi Minh DT: 835611 | Trinh bd'Y; QUOC HONG
Chring ta b6t ddu tt bdi to6n rdt quen thu6c sau ddy- : ^ Blri to6n. Chrtng minh rd.ng udi rtgi sd nguy€n n thi nt *n * 7 khdng .-:,r-M -...qS"hE chia hdt cho I Chdc rang nhi6u ban da bi6t cric ldi girli sau : /::M '-*W Cach 1 : c X6tn = 3h (k e Z) thi n2+n+l = 3k@+ 1) + 1/ 3 non n2+n+L / 9. z^-c' x6t n = 3h + I (k e z) th\nz +n + | : th(h + L) + 3 / 9. @ i< .,8 .l\- Ho X6t n : 3h +2 (k e Z) thinz +n * ! = 3(3kz + 5k + 2) + 1 / 3 nen n2 + n + 1 / 9 Yqy n2 * n -l I / 9 v6i moi n e Z Cdch 2 : frs >{ C6 n2+n* 1 = (n +2)(n- 1) +3 J!{ in F. (n + 2) - (n - 1) = 3 n6n z * 2 ; n - 1 ddng thdi hodc kh6ngddngthdi chia hdt cho 3 Z n*2i 3;n-li Ssuyra (n+2)(n-l)i gsuyra (n+2)(n-l)+32 I n+22 3;n- 1/ Ssuy ra(n-*2)(n-7\ / 3 suy ra (n+2)(n - l)+3/ 3 + (n't 2\(n - 1) + 3 / 9Ydy nz * n * I / I v6i moi n € Z Cach 3: (PhAn chrlng) Gi6 str r*'+ n + | i g. bat n2 + n + | = 9m (m e Z) n2 + n + I - 9m : 0 (*) L, = 36m - 3 = 3(72nt - l) i 3 nhrrng 3(l2m - l) Z I A ld sd khdng chinh phuong n6n (*) khdng c
a) Ndu bd, = 0 ch&ng h4n b 0' Tt (2) ndu d-c d * 0-,[n :; € Q miu thudn v6i (1). VaY b=d,=0---a=c. b) Ndu bd * O. Til (3) 1'l*" € Q' Tt (2) + o{n +b{tl1n : dn * c'ln - + O -c\r[n = d.n -b{nxn € Q. Ndu a * c+ Bidi TLl224. Tim cd'c chil sd hhd'c hhOng a' irt e a tr6i gia thi6t, Y4v a = c + b, c, thda md'n a66Z : dE x ac x 7 ' br{m : d,[n. Ldi giai zCd.ch 1:Tac|ffifr =d6 ' a'c '7 (l) Di6u kiQn dtr ld hidn nhi6n' (1) 0 thi vd tr6i cria (*) lu6n kh6ng n
Miruh (9A Luong Vnn Ch6nh - Phf YOn), Ddo ThuQn, 8T Chuy6n Nghia Hdnh, Vd Trd.n Th.dVu (BAr PTCS.Gi6ng V6 II - Ha NOi), NguyAn LQc,9T Chuy6n Le Khi6t, La Hoitrug Nguy1ru Dic'Xud.n Binh (9T PTNK Hd Tinh), Dic Khd,nh, 9T Chuy6n Nguy6n Nghi6m, Drlc Mai Hdn Giang (8T Chuy6n Le Khidt - QuAng Phd, Qu6ng Ngdi, TrdnVidt NhQt,8/1 Chuy6n Ng6i), Nguydn Thd.i Son (9T NK Nga Son - Nguy6n Khuy6n, Dd Ning QuAng Nam - Di Thanh H6a), Trdn Van Nghia (9A Luong Van Ning, Le Ch; Thd.nh,9l Nguy6n Tri Phrrong, Ch6nh - Phf YOn), Nguydn Tud.n Anh (9 Todn Hud, Thrla Thi6n - Hud, Nguydn Thi Sen,9A Phan Chu Trinh, Bu6n MO ThuQt - Dek Lak), Trrrdng Thi, Trd.n Khoa Van,8A NK Quynh Nguydn Thd Vinh (9II THCS Tnrng Vrrong - Lrtu, Nguy4ru Tidn Trung, 9T, NK Vinh, NghQ Ddng Nai), Trd.n Khoa Vaz (8A NK Quj'nh Lrru , An, D6 Thitnh Trung 8T Nguy6n Hi6n, Nam - Nghe Ant, Nguydn Ch; Thdnh (8T, Nguy6n Ninh, Vi7 Thi Thiry Duorug, 8T NKY Y6n, Dodz Binh Khi6m - Vinh Long), Vo Ch; Thd,nh (9 Nant Thd.i, 7T, N guydn Trgng Kian, Phqm Thu Toan Chuy6n L6 Khidt - QuAng Ng6i), Hoitng Giang, Nguydn Tu(in Anh, 8T Trdn Dang Phuong Dong (9A PTCS Cdc L6u - Lio Cai), Ninh, Nam D!nh, Nam }Jd, Mai Xudn Truarug, Hod.ng Qudc Hoa (9A Chuy6n Lrtong. Vdn 9T Chuy6n D6ng Anh, Nguydn Ti^tn g, Vu Manh Ch6nh - Pht1 YenS, Truong Quang Trung (84 Tud.n 8CT Ti Li6m,.Pfram QuangVinh 9A,Bd Trung Hoc Chuy6n Kontum), Phant Thu Ven Den, He NOi, Bili Hdi Nant,9A Le H6ng Huorug (9Ar THCS Hdng BAng - HAi Phdng), Phong, Ngd Quy6n, HAi Phdng, Biti Thu Nga, L6 Tdm (9 To6n THCS Chuy6n Phri Thq, Vinh 9A Chuy6n Phong Ch6u, Vinh Phri,Nguydn Thi Phu), Hoirng Trung Tuydn (11A PTTH Ha Hdi Ydn,9 Neng khidu Bdc Giang, Hd Bic, Trung - Thanh H
QD+DI B,di'171224 z Tim tdt cd cd.c da thrtc P@) uoi QI JA+AM +N B +B E +tr C +C P +Q D +D I 4(o) = o, P(x3 i l1 =rd1x; +l vr G R. JM+NE+FP+QI Ldi giei kira nhibu ban) : Tl} cdc gia thi6t :' D -m (l), Chrlng minh trrong tu, ta ctng cd cfra bii to6n suy ra : P(1) = P(d + 1) = n : F(q * 1 : 1. Tr) day, bing phrrong phap u:+(2r. D -n Tt (1) vd (z),ta"ory =+, quy nap theo z, d6 dnng chrlng minh drtoc : P(xr,) = xnVn € N (1) 6 ddy xn ld cric sd hang p-m-n p-n-nL cria dey xn ' dtgc xl.c dinh b6i : suyraL = * ,hdnnta,hidn _ nhi6n p >m*n n6n p -rl-n + O, \r|y : xr=O,xnt.: 4r*t Vz € N. Vi rr,*, >r, m =rL,dWT\. Vz c N (dO thdy) n6n tt (1) suy ra da thrlc Nhfn x6t. Cdc ban sau diy crl ldi giAi t6t : Q@) = P(x) - x cci v6 s6 nghidm thrrc. Do vAy Nguydn Thi Thanh (9TL tx Thdi Binh), TrZiz phAi cci Ngqc Anh (9 torin Trdn Dang Ninh, Nam Hd), Q(x) = 0 Vr € R, hay P(x) =r Vr C R. Nguydn Mqnh Hd (9K PTCS LO Loi tx Hi D6ng Ng,rq" lai, d6 thdy da thne P(x): r th6a - Hn TAy), Nguydn Tudn Anh (9 Torin Phan m6n tdt ch cdc di6u ki6n cria bii ra. Chu Trinh - Buon Ma Thu6t - Dak lek), Viy cri duy nhdt da thrlc cdn tim, d0 (1) n+- 6 Ldi giai. (da s6 cric ban). S{r clung Dinh li n4Nc Hur.rc.THANG him sd sin trong tam gi6c, ta cci : 4
(1) 0 Quing Binh : Nguydn Trung Kian, Nguydn Nggc Tfud.n, Dinh Trung Hoit'ng, Nguydn Trung Ki€n, Nguydn Httu Cliu, Trd.n ,$ eiae * sin,B,+ sin^c) Hrtu Lutc +sirfA +sin2B +sinzC > Hlr Bic z Nguydn. Ngge Son, Pham. Vi|t > 2rlSsinAsinBsinC (2) Ngec, DQng Hod.ng Viet HiL 3F. Phri Y6n : Phqm Nggc Td.n, Trb.n Dinh Sit dqngbdt d6ng thtlc : sinAsinBsinC * 8', Ld.m, Trd.n Minh DOng ta c6 : sin2A + sin2B + silPC > Ddng Nai : Pham Hodi Lang, l€ Khic Huinh > B3r[@inAsinEin$ = Th6i Binh :?hirng Thanh Tilng 3(sinAsinBsinA SsinAsinBsinC 3{siffiinasi* - r'tJg{E DfucLEc zvo Ch; Hba 8 NGUYE,N VAN MAU : 2r{ 3 sinAsinBsin C, dpcm. BdiTgl2z4. Gsi a, b, c ld. dO ddi cd.c cqnh Ddu d&ngthrlcxAyr:a khi vdctri t
a*b-c Ldi giai. @ia La Van Cuitng, 11T, PTTH co' : u,, Thd thi : Lam Son, Thanh Hda vi Nguydn Quang O'1, < OI, rdi stt dung cdng thrlc Ole : Nguyan,llCT Nguy6n Hu6, Hn Tay) DUng hinh l4p phrrong ABTCDTCTDATB oP = R2 -zlrta drroc < R2 ry -2Rr ngo4i tidp trl di6n dduABCD, hinh lAp phuong Kh6ng mdt tdng qu:it, giA srl o < 6 ( c, ndy c(c-af+10-a12 B(0,\[2,0), c(0,0, {z), D(,[2,r8,'{v) vd Tt dd ta drtoc (*) cdn tim. ^,r[2 rtz D(; ,t, - e, ,, ) \ 2, 2) MOt s6 @n srr dung dinh li hAm sin vn h6 thrlc Sry ra : R ,.A.B.Csint *: : 4sinfin = cosA * cosB * cosC - 1 ["7 i +, -+, -Y), e ?+,*, -Yt r6i chuydn vd trrii cta (x) v6 dang lrrqng girie ; bi6n ddi vd sir dgng cdc h6 thrlc vi bdt d&ng thrlc t - +, -Y,Yl, n (Y,Y,*l lrrong giric trong tam gi5c, cring di ddn kdt quA t"= cdn tim. Goi e(x, y, z)ldvecto don vi cta drrdng thing Cdc ban sau dAy cd ldi giAi trit. a, thd thi : QuAng Nam - Dn Ning : Ld Tri|u Phong sA' = lffil , sB' = lBEl , gg' : IZBBI vh QuAng Binh : Trd.n Hftu L4c, SD': lffil tt dd ta dugc (v6i chri y reng Ha NQi : La Tu.d.n Anh, Ngttydn Vu Hung, Vu Dtc Nghia, Phan Linh, Triiru Phuong, x2 +y2 +22 = l); Nguydn Anh Ti, s = SA'4 + SB'4 + SC'4 * SD'4 = Vinh Phri : Trd,n Manh. Tilng, Dd,o Manh I Thang = Ut{-* -ty * z)a + (x - y + z1a + Thanh Hcia : Trinh Hilu Trung, NguYdn Anh Duong, + (x +y -44 + @ +y +z)4) NghC An : Phqm. Tud.n Anh, Nguydn Thinh, = I + 4(x2y2 a, tzrz + z2x2) < Truong Xudn Thung, Nguydn Xud.n Son, L"-"7 Duong van Y€n' >1+;(rr+y2+r2)z=g NrcuveN oANc puAr 7"a)1 Bni T10/224.7i di1n dbu ABCD c6 td.nt lit. Viy : "**:B + x,':Yt=r'- g o S ud. dQ ddi cd.c canh bitng 2. M|t dudng thd.ng lrl = lyl :lzl:llg 1 h, quay quanh S. Ggi A', B', C' ud, D' ld.n luot lit. S*yrac64 dtricriaA hinh chidu cila cd.c dinh A, B, C u?t D ffan L.. th6a m6n di6u ki6n tdng s d+t giri t4 l6n nhdt, Tim tdt cd. cd.c ui tri crta L sao cho tdng tudng fng c
2) Ngodi phuong phrip toa d6 dd n6u d tr6n, Bidi LZl224. MQt khung dd.y siau dd.n hinh c6c ban cci thd giei bai todn tr6n bing phuong uu1ng cqnh a, khdi luqng m, dQt nilm ngang ph:ip vecto. trong tit truirng khdng dbu theo quy luQt t B* = NGUYEN OANC PUAT -ax; B, = 0 ; br: Bo*qz.Wtung c6 dQ tr"t cd.m Bdi Lu224. L, kh1ng bidn dqng. Ban dd.u tdm cila khung Thanh AB truot trirng udi g6c tga dQ, cd.c cgnh cila Ehung song lan hai cqnh cila song udi cd.c tnq.c Ox, Oy. Khung duoc thd. tt! do g6c uu1rug (hinh uit. trong hhung khOng c6 dbng diQn. X€t chuydn ) Thanh c6 chiAu dQng cia khung sau hhi thd.. ddi l. Dd.u B tl Hudng d6n giai. Vi day si6u ddn R : 0 n6n chuydru dQng u6i uln tdc khOng 1 ;e 0nhrrng E = RI : 0-lEl : # = 0-tD C tr* ddi Hay xd.c ur.,. = const. Khung dAy roi tinh tidn n6n O" : Q vd dinh phxong, ?t u' O = Oz *@r: a213, * ar) -r L,.Ban ddu t : 0, chibu ua d0 lon crta uecto uQn tdc z=0,i =Q+O: uir uecto gia t6c ,t/ a2Bo --- a2Bo = a213o *az)*Li-i=- a?az L cia trung didnt Luc di6n tt tric dung l6n cdc canh ctra dAy : lgc C tai thiti didnt thanh lirnt udi canh thang ding do B, gdv ra thi triot ti€u ; lUc do B* Bay ra d hai ntOt g6c a. canh hai b6n truc z cung phrlong, cung hudng Hudng d6n gi6i theo O, vd c
k=0 m(m-l)...(rn-s+1) d64n= 1 .2 .3,... s Tim lim n+*o 'itq rncs CAc r,6p H6 QUANGVINH BdiTlizbS: Tim m., n € N dd B}i T9/228: (Neh€ An). A:3h +0r -ol + 4ld s6 nguy€n t6. Chrtng fi rd.ng udi n udc to don ui cila -mQt NcuvEN ooc rAn (TP Hb Cht Minh). ph&ng ia tM tho.n ra duqc cd.c chi sd i 1, i2 ..., iO sao cho 'Bdi T21228 : Tim nghiQm duong crta ha phuong trinh: ,+ + 4, nk x, *'xr: x\996 lo,r*oir+...+oirl>4V DAMVANNHi x, ].xr- x1996 (Thdi Binh). Bhi T10/228: xtsrss * xtsgo: x1996 . Hinh ch6p SABC c6 tdng cd.c mQt (g6c d dinh) cfia tim diln d.inh S bd.ng lSOo uir. cd.c x,se *xt- xlee cqnhbAn SA = SB = SC = l. PHAM NGQC BOI Chtng minh rd.ng di|n tich todn phd.n chip ndy hhOng ldn honr{3 iai tslzzs : rim cd.c "tol{,*",#;)tan k, u) NGUYENLEDONG eia phuong trinh. (TP IIb Cht Minh). x* cAc nii vAr r,i B,di Lll228 z Thudc 96 AB, dd.u A duoc BUI QUANG TRUONG xuyAn qua mQt tryc quay, truc ndy d phia bAn khi thudc mQt nudc. Thd thudc xudng nudc,'trong ff r+tzze z chn tam gi* r#\Xh < AC)uit. d.ing yAn th) phd.n thudc"ngA.p nudc cdc tam gid.c cdn BAD, CAE (BA = BD, CA = CE) q d. sao cho D nd.m khd.c phia udi C djLqdi,4B-E CB = AB . Tlrn I4c ddy Acsinwt tac d+ng uiro tlurft . nam hhur phiau1i B ddi udi AC uir. ABD = ACE. ; Tlm. lqc dp cila thudc leru truc quqt (trqng luong Gqi M ld. trung didm ctra BC. Hdry so sd,nh MD thudc lir P. Masd.t 6 trq.c quay duqc b6 qua). udi ME PHAM ntlNC OUv6r. NGUYEN KHANH NGUYE,N @a Nai) (Hdi Phdng). tsdiT5l228 z Troryu.&phang cho tam giac ABCcdAB : 3,AC = 4, BAC : 7500, ffAn dudng R2= R1,R, = 3R, ' U kh1ng ddi' 86 qua gid' trurlg tr4c crta doq.n BC, ta ld.y didm D 6 ci^rng tri d.i\n trd cia dd.y ndi, hh6a K ud Antpe kd. phia udi A so u6i BC so.o cho AD = 5. Hd.y tinh cd.c g-6c cila tam gia.c DBC. oiHl'rH,o#fr*" cAc r,6p rHCB Bei T6/228 z Cho p lit. m|t sd nguyan td. Chtng minh rd.ng udi sd m nguyAn kh6ng d.m. bdt ki, tbn tai m1t da thrtc Q c6 hQ s6 nguyan sao cho pm lit. udc sd chung ldn nhd,t cia tdt cd cd.c sd ar: (p + 1)n + Q@) (n = 1,2,3 ...) Khi K m.d dbn D dqt cOng sud.t fiau thu c4c TRAN DUY HINH. d,q.i, (A) cni lA. (Blilh Diilh) 1. - Xd.c d.inh s6 ch; 6) khi K dong BdiT71228 z Gidi phuong trinh : 2. - Vdi U = 750V, hay xd.c dinh Pa, Uo hic 729x4 + tll1T.: u_u K m6, K d6ng. TRAN VAN HANH. LAI TH6 HI6N (Qudng Ngdi). (Thdi Btnh)
ilFd{} Eit,F:lhI S I,'0 R 3'fi i S I S S ti 8i. : 5 and D and A are the point D such that AD For Lower Secondary Schools. on the same half-plane bounded by BC. Calculate the angles of triangle DBC. Tll228. Find nz, z € N such that A=3!nz+ott-61 +4 For Upper Secondary Schools. is a prime number. '121228. Find positive solutions of the T61228, Let p be a prime number. Prove system ofequations : that for every positive integer nz, there exists a polynomial Q with integer coeflicients such xr*xr-x1996 that pm is the greatest eommon divisor of all xr]-xa- x1996 numbers ar: (p + l)n + Q@) (n: 1,2,3, ...). ! T71228. Solve the equation , * xtgge= x1996 xtggs 729x4+8f[:lz:86 .t xtsgo*rr- x)ese T}lzz8.Let S, :i where TBl228. Find integer solutions (x, u) of the k=0 "*,, equation ru(m-l)...(*-s+1) u^=T ^s xt :U. Find lim ryS, 71+ *o T41228.I-etbe given atriangleABC,AB < AC, Tgl228. Prove that for n given unit vectors and the isoscles triangles BAD, CAE (BA = BD, ---t CA : CE) so that D and C are on distinct larl , ti : 1,2, ..., r1,), in the plane, one can half-planes bounded by AB,,E and B are*on choose indexes i1, i2, ...,1^ such that distinct half-planes bounded by AC and ABD : ,+ + +. nk ^ Let M be the midpoint of BC. Compare : ACE. 1o,.,* o,r+ ... + aiRl >- 4E MD withME. TlOl228. Consider a pyramid SA-BC, the T5l228.In the plane let be given a triangle sum of the plane angles at vertex S of which is ABC with AB : 3, AC : 4, BAC : 1500. On 1800andSA: SB: SC: 1. the perpendicular bisector of segment BC, take Prove that the total area of the pyramid is not greater than {5. HO DTIONC (](}NG "", (tidp theo trang 11) NCUOI GLII BAI CdT CAN CHU Y - Ldi giAi ciia m6t bIi todn viSt ri6ng tr6n (3nt *l)x-m2+m mQt tb gid5,. N6lI bei dai nhi6u trangthi dinh 7. tD667Il Choy - x*m (m. * O). chring vio nhau" CMR: D6 thi hdm sd lu6n ti6p xric vdi hai - Tr6n m6i bni giA.i ddu ghi hq t6n, l6p, dudng th&ng cd dinh tnJdng, huydn, tinh (thAnh ph6) vi ghi sti 8. [D6 141Va] : CMR Ho drrdng thing cira dd ra (khdng cdn chdp lai d6), C(a):(x -l)cosa +g/ - l)sina -4:0 lu6n - Ngoni phong bi cfln d6 rd 6&l gidi cfia tidp xric v6i m6t drrdng cong cd dinh sd bdo rudo (kh6ng gui bei cira nhi6u sd b6er vdo ci-ing m6t phong bi). 9. CMR: Hq dudng thhng D(m): y : mx+ 9 ... Chi gti bni giai vd dia chi nL : (m * 0) luOn tidp xfc v6i mQt parabol c6 dinh. 458, Ilang Chudi, HA NOi. roan rrqc vA TUd;t TI{.t
t I rfr euA Ki rHr Hec sINH GIoI I THPT ru0x roAx xAM Hoc lges - t99G I NGUYEN VI€T HAI Cu6c thi chgn hoc sinh gi6i Torin THPT n6m Nguy6n TrongHAu (He Tey), D{ngThanh Hd, I nay d6 duoc tiSn hrinh trong hai ngiy 15 vi 16 Deng Vi6t Dung (DHSP - DHQG He Ndi), thdng 3. Thi sinh phAi ldm 3 bni to6n trong 180 Nguy6n Tri Phuong (Hd Tinh), Nguy6n Xudn I phrit m6i ngdy. 397 thi sinh cl&La 55 d6i de dq Th6ng (QuAnS T4). thi, trong dci cti 5I d6i cria c5c tinh, thdirh phd Gid.i hhuydn khich (tt 20 ddn22 didm) NgO vd 4 dOi c{n cdc l6p chuyOn ToSn thu6c cdc Dfc Thinh (DH KHTN - DHQG Ha NOi) - tnrdng D?i hoc. Cdc.d6i drtoc chia thinh hai Trlnh Quang Khf,i, Nguy6n Trung Ki6n (Hd Tdy) bAng, mrlc d6 dd thi cria bAng A khd hon bAng - Hodng LO Quang, Nguy6n Thi Thanh B. BAngAgdm 31 d6i vdi253 thi sinh, bAngB Thty (nrr), Ph4m Dinh Trrrdng, D4ng Qudc g6m24 d6i v6i 144 thi sinh. M6i dOi cir tdi da 8 Drlng, Ph4m Drtc Tung (HAi Phdng) thi sinh, trt mOt s6 d6i manh duoc cir nhi6u thi - Nguy6n Minh Hi6n, Ph4m Thny Li6n (nu) sinh hon. Didm tdi da cria m6i ngdy thi la 20, (Nam Hd) cua cA hai ngAy thi la 40. Kdt quA thi nhu sau : O Uang A c6 2 gi6i nhdt, 11 giei nhi, 21 gini ba - Bni Thi Ngoc Tudn (nt), Pham Thdi He, vd 33 gi6i khuydn khich. Tdng sd giAi cria bAng Nguy6n Quang Trung (Thrii Binh) - Trdn TiSn A la 67, chiSm ti1626,5q,, tdngsd thi sinh, trong Dring, Vu Tdt Thing, Trdn Minh Anh (HaNQi) dd sd giai ba trd l6n ld 34, chidm tila 5},7,lo tdng - Ng6 Thi Hai (nir), Bii ?hi ?hu (nr), Pham sd gini. O Uarg B kh6ng c
NGAYTHI 15- 3- 1996 (180 phrit, khdng kd th6i gian giao db) md. mdi chinh hqp (a1, aL, .,. ak) th6a md.n it BANG A nhd.t mQt trong hai dibu kiQn sau : Bei l. Gid,i hQ phuong trinh : (1) Ton tqi s, t e { 1, 2, ..., k) sao cho s < t (1 lrh.(1* . \:2 \ x+Yt uit, a" > ar. J' t.-. (2) Ton tai s e t 7, 2, ..., kj sao cho (a, - s) " \11--*) lrlzy x+Y/ : 4!2.1 hh1ng chia hdt cho 2. I Bei 2. Cho g6c tam diQn Sxyz. MOt mQt BANG B phd.ng (P) khdng di qua S ud' cd.t cd.c tia Sr, Sy, Bei nghi|m thrlc cila 1. Hdry biQn luQn sd hQ Sz ldn luot tai A, B, C. Trong mqi phA.ng (P) dung ba tam gid.c DAB, EBC, FCAsao cho m.6i phuong trinh udi d.n x, y : fy -Ya : oa tam gid.c d6 hhing c6 didm trong chung udi tam x2y t2xy2+f=b2 gidc ABC, uit. LDAB - AS/8, LEBC - ASBC, theo cd,c tham, sd thgc a, b. LFCA = ASCA. Xdt mQt cd.u (T) th6a nfi.n dbng Bei 2. Cho hinhhldi€n.ABCDnQi tidp trong thiti hai dibu ki€ru sau : mQt cd.u td,m O uit. AB : AC = AD. Gqi G lil (1) (T) tidp xic udi cdc mQt ph&ng (SAB), (SBC), (SCA), (ABC). trgng td.m tam gid.c ACD, gqi E ld. trung didm (2) (T) niim tron g g6c tant dian Sxyz uir. nd.nt cila BG ud, F lit. trung didm c&a AE. ngodi hinh ti didn SABC. Chrtng minh rd.ng OF uuOng goc udi BG khi Chirug minh rang tdnt duitng trdn ngoai ud.chi khi OD uuOng g6c udi AC. ti6p tant gid.c DEF ld. didm. tidp xic cia mQt cd.u BAi 3. Cho n sd (n > 4) ddi mQt hhd.c nhau (T) udi ntat phang (P). e, a2, ... an. H6i c6 tdt cAbao nhi4u hod,n ui cia Bai 3. Cho cdc sd n guyan duong k uit. n udi n-s6 d6, md. trong mdi hod.n ul khdng c6 ba sd k < n. H6i c6 td.t cd bao nhi€u chinh hqp chd.p nd.o trong bdn sd a1, e2, a3t a4 nam d ba ui tri k (a, a", ... a*) cita n sd nguy€n duortg ddu fian, li€n tidp ? NGAYTI-il16- 3- 1996 (180 phft, khdng kd thoi gian giao db) BANG A . BdNG B Bai 4. Hay xdc dinh td,t cd. cor hd.m sd f .'N* *N* thm, man dang thtlc : dinh td.t cd. cd.c hd.m sd BDri 4 : Hd.y xd.c f(n) + f(n + 1) : f(n + 2).ftu + 3) - 1996 udi f : Z -Z thda man dbng thdi hat dibu h,iQn sau : fitot n e. N' . (1) f(1995) : 1996 (N. td tQp hqp cdc sd nguyQn d.uorug.). Bni 5. Xdt cd.c tam gid.c ABC c6 dQ diri canh (2) udi mqi n €. Z, ndu f(n) = m thi fht) : n BC bang l uit sd do goc BAC bd.ng a cho trudc ,df(m*3)=n-3. ? ,3) ruai tum gid.c ndo c6 khodng cdctz tir (Z lir. tQp hop cac sd nguyAfi. td.nt duang trbn nQi ti6p ddn trgng td.m.bd nhd.t ? Bni 5. Xdt cd.c tam gid.c ABC c5 dQ diri canlt Hay tinh hhocing cdch bd nhdt d6 theo a. BC bang 1 ud. sd do g6c BAC bdng a cho trudc xi hieu f@) ld khod.ng cd.ch b4 nhd.t n5i ffan. H6i khi a thay d.di trong khod.ng (1, , tni ? r3) uoi tam gid.cnirc c6 khod.ns cd.chtit ") tdnx duitng trdn. nQi tidp ddn trgng td.m bd n hdi ? hdnt sd f(a) dq.t gid. tri l6n ruhdt tqi gid. tri niro Hd.y tinh hhod.ng cd.ch bd nhtit d0 theo a. ctta a ? BAi 6. Cho b6n sd thqc khOng d,m a, b, c, d B}ri 6. Cho ba s6 thqc khOng d.nl x, y, x thoa thoa mdn dibu hien : md.n dibu ki1n: 2(ab + ac * ad * bc * bd + cd) * abc * abd + xy*yz*xz*ry2=4 *acd*bcd = 16. Chrtng minh rd.rug : 2 x*y*z>xy+yfxz. a *b * c * d, * ac * ad 4 bc +bd @b + cd). i H6i dd.u bang xd.y ra hhi ndo ? H6i dd.u bang xay ra khi n?to ? 11
Danh cho cocbqnrhudnbi+hivao he" $l fuQ mry0md rsmo ffip ffin uffi m0u Mre t6 pilmm rnAN PHUoNG Tlongc.ickithituydnsinhvdoDaihgcvitrong v6.i 9a9 bdi torin chua cho bidt dang cria drrdng BQ d6 Toian ta ttrriang gap dang toSn sau ddy : c6 dinh (D) ',Chringmintrhgalang coig(C^)c
4.2 Ndu -B(mt phq thugc vin m tht (Cm) x^: -l:::J Ph.uongphd.p 2: Didmcddinh Mo(xo,!,,) e " d(ml b3+G-rn)x,,*L*m ludn tidp xuc vdi @) {i cac Aidm di d6ng tr6n (D) dd thi *!o = xo-ffi *(xn -!n - l)m + x()r v() - ?-x.2 () x.- -1- 0 Ym -,xo -1 eJ lro-yn-1:o €P V= l*oyo-bl,-*" -1=0 I -@o,+1))2P -0 P,=-t €2 l^,_o 2 Ito- 2x2 o - 4mr() +L*2- -2m-_1 \ LtE/ M4t khricy'(ro)= (*o - *)2 (m + 1)2 _-r -(1 - P1z %v dd ftri nam sd lu6n tido xric vdi duons rh&ns"c6 dinh le tido tuvdn vdi a6 thi tai ti56 didm" (-1, -D ^cci " phrrong tiinh 6 y = y'(x){x -x) t!,,ey - x - I 4.J Phttong ph6p 4 thudng dtroc str dung khi Phuorug phd.p 3 : (C,,,) le ho dudng thing cbn (D) li m6t drrdng @2 + ?s + l) + x2 - (m + l)x + rn cong cci "tinh chdt cria hinh ldi' VD : (D) la v- x -m drrdng trbn, Elip, Parabol, ... Ndi chinh xac hon k*l\2 dridng cong (D) chia mat phing toa dO thnnh 2 = i_x-m ---i + (.r - 1) mi6n trdi ddu theo phuong trinh ctra (D) (x i l\2 Vi phrrong trinfr f_;- +r - [ : a - |e (x * lt2 x. -m lu6n ti6p xfc vdi dudng thihg ! = x - I t?i didm c6 dinh (-1, -2) Bdi 2 IDA 921.3] : CMR : Tienl cAru xien cia hdm. (m + 1)x2 -2nm-l*r -m) -2) sd. y=-- itioil x-nL tidp xilc u6i 1 parabol cd dinh . Sau dAy In mOt s6 vi du minh hoa cho c5.c : Bi6n ddi y=(m+l)x+m(m-L) + -2 Gid.i binh luin nAy ,, Bdi I : CL{R db thi c6.n xi6n y = (fir * 1)r * ,rfnr'-Tl - Ti6mphtip 2i+(1 -m\xl-1*m Phuoig I : GiA itr tidm can'xicn tiion tiali J- xtic vdi parabol y = axz + bx * c kht dd phrrong x-m tr\nh ax2*bx*c:\trL +l)x+ru(m- 1) ha), ludn fiAp xuc udi 1 duitng tharug cd dinh tai 1 dlem co dtnn- *1!\b,-,1.- nlx *c -m(m - 1):0 cd nghi6rir k6pVm' Phuorug phdp 1; Ho d6 thi hdm s6 d6 cho #a + 0 vd A = (7+4a)m2+2(l-b -2a)nt+ luOn ti6p iri-c v6i drrdng thhng'y : o.x * b 2t2+tl-n)x]-l+:nt +(l-2b+b2-4oc\:0Ym '11+4o:0 ephtlong trinh : ax * b c.6 [o:-ll4 --lll1 .-]t-b-za=o *)b:Bl2 nghiOm kep Vm *(a - 2)x2 + l(l - a\m + + 16-1)lr - (b+l)ni - 1 = 0 cci nghiom k6pYm It -ZO *b2 -4ac :O lc : -ll4 €a * 2 vd A = 1t-a)2m2 + tz(l-a)(b-l) + *Farabol phAi tim te y=(I\41x2+1llZ1x - U+ Phuong phd,p 3 : Buoc 7 ; Dtr do6n Parabol + 4tu-2)(b+t11m + (b -\2 + 4@-2) = O Vm (LAm nhrip) , ,*9 :2m + (x - 1) : 0 l1t -o1z : 6 ' dnx -]z1r -q@ - 1) + 4(a -,i}(b + 1):0 **= 1 z-r Ito-\2+4@-2)=o la=1 | -x VOy dd thi hdm s6 lu6n tidp xric v6i Ttrd m : lao phttong trinh tiC.m cAh xi6n - lb = _t z^ drr&ng th&ng ! = x - 1 tai didm crj hodnh d0 +y: (.l--t l-r .l-x 'r\ - +1'lt r+ 2 \/ Z - *o : _B : \ L 2A - -t -!o - xo I -2 ---Tqa d0 -*2+Gx-l ti6p didm ld (-1, -2) 4 13
Bu6c 2; chtlng minh di6u du do6n xcow *ysina *2coYx * 1 :0 haY [: trinh +Z)clw *ysina : -1 vd nghiOm a ; *6t Phuong @ I -tz + 6x | I (m+l)x-tm(m-1): 4 - (x +2)z +yz < (-1)2 = 1 -DU do6n Id drrdng trbn (r +212 +f :1 I -- mz +(x -t)m *+: Budc 2; Chfng minh gidng brt6c2 d tr6n Bdi 4 : CMR t Hg drrdng cong -- .*)z : o c6 nghiem k6P' vaY Q -r+ -^,2 I l* xi6n ludn tidp xric parabol !:x5 -4x4 +3x,3 -5x2 *mx lu6n I tiorr, ti6p xric vdi 1 dudng cong cd dinh "a" +(312)x :71t41x2 L 3 : Budc 1 (DU dod'n) : Phuong phd.p I Phuong phaP 4 : Buoc I ; (DU dodn) : Didm dv l,t (x, Y) # Ti6m c6n xi6n nio € a-:x-*-O*nt-2x' nL I o*rnu I y-{nl+1)r+m(m-l) nr'= 2* vio bidu thrlc hdm sd * I nav nJ +@ -|\m + (x -Yl: ov6 nghi6m nz I o>a=(r-112-41r-y; ! : x5 - 4x4 + 3x3 - 4x2 * f ,u dudng cong c6 I -----rl+6r-l -:f > y + Du do6n drrdng c6 dinh dinh drr do6n I Bu6c 2 r Chrlng minh di6u dtt dorln' Illa -*2+6x-l X6t phrrong trinh | 'o! = ---4 -i Auoc 2 : Chtlng minh di6u dU dorin x5 - 4x4+ &t3 - bx2 + rux *' I I X6t phrrong trinh (nz+1)x + m(m-l) : ,. tr-l)Z =.r5 - 4x4 + 3x3 - +r' *X I :--12+6r-1 --n-*nL2+(I-l)nz+--it:O I - r-'l-. : o cc; nghiQrn k6P v+Y ey2-mx+ft: m2 nL') I - +:U:]" (r-;)-= 0 cci nghi6nt I - lnt k6p. VAy ho dudng cong d6 cho lu6n tidp xric v6i I parabol efin tim liy : (-ll4)xz + $121x - tl4 3 &' * Z adi i tDb 19val. cMR : Hs du?tng thang dudng congy = -r5 - 4r! +'3x3 - I I O(a): xcosa *ysina *2cosa * 7 =0luOn tidp M6t sd bd,i tQP dP dttng I ric udi 1 duitn.g cor"g cd dinh 1. tD6 41l.2l CMR Vm. * 0 d6 thi hdm s6 I (D{ dodn) : Giii ph.uong phup 3 : Bu6c 7 : (m*I\x*nt L lftlrt'"1=' [r"o",r?sina * 2cosa+] :: g "y : * x +nL luOn tidp xric vdi 1 dttdng he iI 1! o " -1*=in,, +ycosd -2sina o th&ng cd dinh ld"+ =2;coscr l *Ysina : -1 2. tD6 t02l.2l CMR Dd thi hdm s6 I tlx *]y"o* -(x *2;sina:0 ! : x'2 + (2nr + 1)r + m2 - | luon tiSp xric v6i 1 dudng th&ng cd dinh Ilcos41 : x +2 -=--=- 3. tD6 4l[Val CMR: Hq dudng thing €
NGUYEN VIET HAI (Hdi Phdns) Dd chring minh m6t ding thrlc v6cto ta thudng durig cdc bidn ddi v6ctd vi str dung c5c ding thric vectd cd bin. O ddy dtta ra 1 phuong ph6p chfng minh ding thric vecto khri thuAn loi trong nhi6u trudng hop ; phrrong ph:ip sii dung hinh chiEiu vecto tr6n I truc. 1. Hinh chidu cia mOt udcto tr€n mOt truc : TrOn mqt phing cho r-= AB vdmQt truc x. GoiA', B'ldn luot ld hinh chi6u vuOnggdc cria A vd B tr6n r (Hi4 1)Ja gSi d0 dai dai s6 A'B' In hinh chiSu cria a : AB. tr6n trucr : ki hi6u : AB' : "hr(6: "nr1fr1 Hinh 2 fi Cric bdi to6n sau dny ddu cd thd giAi drigc kh6ng cdn dirng chi6u vecto. Ban doc cci thd so sanh -ach giAi d day vi c6c c5ch gi6i d5 bidt, dac bi6t 2 vi dU 3 vd 4 ndu kh6ng dirng chidu vectd, c6ch giAi s6 khri khan. Vi du l. Cho If vi O Hinh I Iir tnrc tAm vi V6i dinh nghia tr6n ta dtla ra 4 tinh chdt sau : tam dudrg trbn l.l cb@): I ol . cosp t trong dri 9ld g6c ngoai ti6'p LABC. Chrlng tao bdi o vd chi6u driong gtia trUc_{. murh -Iang i o 1.2 ch, (a + b) -- ch,(.a) + chr(b) OH OA! ') 1.3 vft € R, : n"4(6 . _ +oB= +oc (1) "hlq 1.4 a:b ech*(a)=ch,(lb) uir chr(a):chrlb) + Gidi. e Chondttdng thingBC ld tryc, ntrt, Hitth 3 trong dri r vd y ld Ztruc khOng song song. Chrlng (hinh 3) nrinh c6c tinh chdt 1.1,1.2,1.3 khOng cd gi khd chi6u cria truc + c:&a BC '+ BC ld chi6u khan. Ta chi n6u ph6p chrlng minh 1.4 o ch^^ (OA + OB + OC\ = Dl\++'++ Didu ki6n cdn : hidn nhg" qg rip dung 1.1 = u r(o Allch BC(o B +o C) = ch n c(oA) + c h Dj6u kieg dt : D{!o : AB : b = CD, giA srl : +c h : r(2O lvt) c h e c(O A) +0 c h n c(O r : A chrAB : A4r;ch*C! = CtDt i o Tt/onE trr vdi AB ld truc thi +o:++'+ "!i!: IEyWD sP, theo giA thi6't choo(OA + OB + OC) = chou(OH) AtBt : C,D1;AFz: = CPz. Theo 1.4 ta cci dine thfc (1). Vi du Z (xem h'ii tdn 3". 4 6n t6p cu6i narn HH10). Goi E ld giao cria AArvit BB, F ld giao cria Cdn'vd dri ddJiiinJ le tAlrr drtgng frbn nQi CC ), DD | (Hinh 2), bang c6ch x6t 2 tarr, glirc ti6p AABC ld aIA + bIB + cIC = 0 (2) vu6-ng bahg nhau ta suy ra EA ll !C, EA-: 79, 'Gidi: vQy EACF ld hinh binh hdnh + AC = EF . Di6u kiAn cdn : tV EBDF ln hinh binh hinh "+ GiA str 1ln tdm dudng trdn nQi ti6p L'ABC -Tuong BD:EF (Hinh 4). o Ldv truc r li dudne thins vudng g6c v1i DELng thrlcAB = Ai cinnng,tg- IA tai A, cri'chidu nhu Einh v6-, H4 BBt, CCr AB:CD,haya=b. vu6ng g6cvdix. 15
+E +r7;) chur(d\=+ = "hnc(* rti o Cfing ttldng trt nhrr vdy khi chon AC li --'v;vd ta suv ra (3). 3 canh MBC, M vd o lit truc ai- a ,h. b, Id " :l"o"sgbJ1:*8lfl :#:Hff u:'i8"ffi"1 vi ldn luot vuOng ggp v6i B-q CA, A'8,- cirng ;irt&;"6ii; " iiJto1l , IE , I:F trong vi du 3 Itlnh I Chrltgminhging:- - --?^ +++- cn.lelA+ btB+ clc)=0 * ch*lblB* clcl : d7, + Ot4 + c3ru'., = 12S.MO (4) : bMt + cACr99rA, ld giao c:&aAI vb'BC,ta crj nhdn x6t : AB, ud. AC, nguqc htrdng vi AB, : AC, = AtB : ArC = c: b (theo tinh chdt cria dr;}ng phAn gi6cAA,), b6i v4Y : bABt+cAC,= q suy * ra ch*1alA + bIB + cIC) : o : chx(o) -4+t+ . yu:f u" u t"i ; tr'ilr,,? E iii'i'iri{+ fil#,?;*-it#".'-:Hi ,V1 = ci-( .,8 " Theo 1.4 ta c
DE THI TUY N SINH VAO CAC LCTP CHUYBX 2 DAIHQC TONG HeP (DAr Hec eudc cIA HA xOO ltAtvt Hec 1995 - 1996 vONc r (Thoi gian lim bii : 180 phft) Cdu I : CiAi hQ phrrdng trinh el lx=o llJ lr: -3 )^ -Y2:1 1^,+l:2 C ihzt Ddt a : ,{l=x,, I l^r = {+-+x ta thu duoc hc Cdu II : Giii phttong'trinh (u+v=3 ,lT- +.tT+x:3. l-'r , , . suy ra v :3- lU-+V':5 u.uz + (3- r)2 = 5 Cfui III : GiA s-t a b lA cac sti nguyen drJdng sao cho l' lu:r a*l *, b+1 ldmQts6nguy6nGoidldu6cs6cfra avdb.Chitng e 2a'-6u+4=0olr:z = r f,:b U mintr ring d CAU M: < ,la TT . Cho hai hinh chtl nhAt c6 cirng diQn tich. Hinh * fvr- lvr-- =2o lx: -z chrl nhAt thir nhdt c6 c5c kich thttcic a va b 1a > b). Hinh chrl L L oz+b2+a+b nhat tht hai c6 c6c kich thr:6c c v a d 1c > d). Chrlng mi nh ring Cdu III (1, 5 didm) tU -- ob li m6t s6 nguy6n n6u a > c thi chu vi cira hinh chrj nhAt thu nhdt ld \ acl Cdu I (2 didm) : Tih€ +t'=2 s-rv rar Ta Co a- c > 0 (theo gia thi6t). Vay .s : lt1 )q + ? : G - zy' ** - xy - zyz = o " lr) (a - c) (t -;) , 0 (dpcm) l, lr:- 1 V (3 didm) : 1. (l di6m) AE : AF, ta ' - N&ry : 0 thu drJoc j 2 He vo nghiCm Cdu c6 AEz : AB .AC sty ra ti: I z AE:,fAB .-Ae khong d6i+4 F rhu6cvdng trdn raml -Ndrv-0tacn: b nkinh{AE:Ae 2. (1 di6m) Nemdi6m4 41, O, F nim tl9[-{rJdng,!qn dgg!& ,,,*, - (;) -2: 0e kkhAo (AEo: 4Q: A!A_ [r,)' = 90o). S:uy ra AoF : AIF (cing chin clng A\ (1) g6c 1^ k:v AOF =: EOF : EEI (2) z , 1ri -n': I [J,: liJl), llBC. (2) suy ^ triF-: lz x : ---tt * llv:t : -, : EE'I+E'E 3. (1 didm) Gqi K li giao di6m BC v6i EF. Ta c6 rfi )7 I (hQ vonghiQm) II' gi6c ONKI li t0 gi6c nQi ti6p. Suy ra vdng trdn ngoai ti6p tam crha LP: -, lr'-r'= I giSc ONl li vdng trdn ngoai ti6p t0 gi6c ONKI.^Ia c6 LANK- A,AIO s:y raAK AI AN. AO : C5chl:Difukie.n-4
Giii ddp bii DIE,M CUA MOI NGI.JOI s6 di6mvd titng ngtldi trong c6c cAu trA ldi vAo bAng "urlir*I, XuAn Ha 'l'hu D6ng vlllT sCi rnorc BA\ cO XuAn tri lr,i 8 9 7 Bdn cd vua cd 64 6, trongdri s6 c5c 0 tr6ng lla rre ldi 9 8 10 bing s6 cdc 0 den vd xen ke nhau. I'hu rri Idi 7 '7 7 C6c ban hay vidt vdo m6i 6 cira bdn cd mOt Da)ng tra ldi 8 8 I s6 tt 1 ddn 16 sao cho th6a m6n c6c tinh chdt : Can c;r vao didu ki6n :. Kh6ng c6 ban nAo dtlQc 2 ban n6i 1) 56 cdc 6 drroc vidt tt 1 deh 1 6 Ia bang nhau. dtn Klv ra : rltng vdi s6 di6m cua mrnh ta K-ry cunr:i: dune 'ltak 2) Tdng cacs6 viSt trong cdc 6 tr6ng cria m6i 1 Ia khdns dudc diem 8 Ttiu kh66e dir.1c di6m 8 hdng ngang hoic m6i hdng doc d6u bing nhau l)ring khd-ns Donu du0c diem khd-ng duoc 7.- di6m 7. . : -