Toán học và tuổi trẻ Số 121 (5/1981)

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Mời các bạn tham khảo tài liệu Toán học và tuổi trẻ Số 121 (5/1981) sau đây để nắm bắt những nội dung về hình lồi và định lý Helly; phương pháp quỹ đạo; bài toán vận trù. Với các bạn yêu thích môn Toán và muốn bổ sung thêm kiến thức về lĩnh vực này thì đây là tài liệu hữu ích.

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Nội dung Text: Toán học và tuổi trẻ Số 121 (5/1981)

1fil$ ,J V ilutS dr t; iv HOt roAu FIuc YIh,T:YAM ffi ffiiffitu tn1i',,,i.[ffil,.S(. ,r:,lri, T I t I aAc RA HAI rHANG uOr Kt cnfi nhi€m: :lcUYAx C,lFlH f'oAn Ihl' kg rd* rpan , HOANC CH$I{G -fru s&: 7il Trin * t{a *N6i H'ung l)ao $Ay ur.ri I SZBZ1 N6i ehrrygn vir,i e6c b4n tre 16u toAn HiHH l&x YA Bgffiffi r,'f ffiffi&ty rroANG ouc raN C6c ban ilr$n rn€n lr -* tr, i-.". * i"r .-' i t*- i1''") I *. C6 l0 trong toan hgc li- tlirr-r,€t (alirc bidt lA vA' lr,U: y''1r'r-11 - ... i ./.r.t.r.tt ) hinh hoc l6 htTp) r'A tr.ong toAn irrrg ciung (tldc biit lli to[n kinh tg) fhi hirh liti li nrQt trong tlorrg t1o {, li rn0t dii\m bSt k;i' tr6t m[i phing (Jntr tronri hlrorig uia rr). r:ac rtt)i iusng crl tiirri quan trong (lic bi0t, d*ur: rrhiBu ngudi quari tam ddrr, tlo no e0 nhiiru ririg (;h*ng lifnh: T'a ki irj0u G lrl uriiln Ifii cfia rl urg dirc si.r:. 'Irr:ng bii vi6t nay tdi rnudir girii thit)rr viri lirirr doc rndI sri iinh cirst l"i th( ci]a cac rlii-rrn ,11,,... -4r v* kri iii&u i l* t,np tf,t c& hirh l6i. crir r'1 iErn [ht?;il niltr circ quan hd (r,). Hirrh ,,ll (lrdn mit pirirng ha I' trong khillg t{i8n nhi0n ta c6 ,{1,."., ,f l S fi Chir;g hgr, tu i giilrr) fl1y111 ggi l{ trinh tri)i, n6u v6i n-rr.ri crip ' --| --i, *' l$ di6in ,1, E € rU thi clr-rau ihiilg CB nirrr horirr toirn 1'i.0--l- .". *. {i= 1 i{)Ar= I . oal +- t}.0.42+ ... + trong lI (tl viit l:i l,tE I q,tf .l. 'fa qui ir6r gt_,i {). OAu [a sn-1' ra .{1 g 6;. I nri6;i li)i ,'irt r:*r rli6nr C1, ,{2,..,,1, la hirr!r i6i cri didn ticlr uhil rhrit chua chc diiir.n J1,,.", An, T'a chr'rtrg nrinh G iiji. T.hfil viiy, gi& sr| 3f, lV 'la. co el! r:h l_.r.. s.ru tlf,y: Ii ehc diiinr brli k--r' cira f Xfl, itr'r t6r: tai e{c Binh iti.'1'ri.ri rntr phing (tray tlolg trtrO,,g s6 ttrui khS:rg :irn ).1,.., I* r.A l"ll ..., !lr. thija gi;rrr) t:iro li cliiirn At,.,,, -4i" (k l{ srj tu, n}ri0r.r miirr r.ac rluarr hd: n&u iiri). Khi d6 tii6rn .lI Ia thiigc ruidn ldj sia cAr: iiiEm,.l1,..",.4rr khi vA chi khi tijrr tai eic ir-i- .." ,i- l,* *-- t, m: ir da].+ ... + i.1ifr ; sd thuc hhdlg riiu l,1, L2,..., Ar sao chcr r:o ! l-lr + ... * Fr - 1, O.l :tlr Zr-ir + ... +- pl5b-nt.
= Ndu T::'ute. "*t,0" doan MN tlri ta sE c6: ia gil thi6l rring t16i vdi k di6m ,4t... A Op:OM * MIt:OM + \J,l/fl (renr lrinh l), ri thi G : G (th6a rnen (x)).134r giu ta ch{nrg minh d&y 1: li cti thuc th6a nrftn : 0K v -"-( l. nri aO : rlng di6u d6 cirrlg ilring cho (ft + l) cl iltm dt, ,{z'..., Al, Ar+t. Gi& s,i ,t e G tric la t6r tai tr1;..,, l'p11 ) o siio cho 1.1 +1 .. + Ir. * Ir+t:1 v& i.a c6 V ftlt : tq O,ql + ... + ir f;-fr * Ar,f1 rlgu+r. * Trong- circ s6 tliuc ir, l,z, ..., It, l'1a1 c6 it nhrit mdt sd kh6c 0. Gii sr! so do la 1.1 1c O. tini ito ),r *...* li.}0. t'r, hv hi6u l.: trr +... + IrlO, Sau d6 ta ky hi6u f: ti diFnr saci cho: kk ON-X ("Lil?\) {),1i. Vi I ti,ii'):1. n€. theo l1 giri thi6t qrri nap ta c6 N phrii thu00, l-- Vzz>0 vi Itroi i.1, F;)n y n€n p1 ) 0. Nsr)ii ra v6i moi thudc i ttri Mtingphrii rhuOc 5m kk k G. lri'c Ii G C {.;. Ep,='JXfr, + (1-V) X I: *1: . I -+' (l-\r). i - i, 1l I EiEu d6 ch*ng tii ring tlibrn P lhiia rrrl r) qlril ti he(x), ttc In P€ 6. Ndi rr}$t c{ch kbAr-: la trip (, chira toAn bg doat tiring MN. tlo df,7le ti:,i' Vi G i6i dr,.',. ,4rC G. NOn G chira rniBtt vA l6i cria cfre di&m ,{t,..., Al. flay lA ta d[ chrlng ? minh cl uoc G G: G. DA chung niinh clirrh li ( tuc Au T, $ lA chting rninh G: G ), thi tr.r chi cdn phii t,hri'ng Il inh 2 jI nrinh GG G tA .rong. Sttu ddy ta,se chr'rng miah t)inh I'i.ilggc chrrug rninh -rong hodn toAn. iliBu d6 birg gr,i nap theo n' (s6 diSnr .41,..., .'ls). Dinh lj tr0n dAy cho ta nhi€u kdt qu& dgp Khi ,{ = 2 tht ddi v,ii crp dibnr ,11,. Ai ciia flE trong te trn hoo so ciip. Cliing han Ilan cloc hiry dua vio dinh li tr6n db tinh khoitng cach G tr?c ld lAp t6t c&, c{c di6nr i14 mn If i- 1,2 = I gita tArn cAc rn{t ciu n6i, ugoai tidp cria mgt ve7fr = Ir d7i + ha o^lz (vdi h1, i,2 ) 0) chinh l"u diQn cho trufc (theo c{c clnh cfra t(r dign). li do4n tlrfing ztt*{z v* r'0 rAng lA no trrirrtg vdi YAo ndm l$2l Helly di ap dqng rlinh lf trdn miBn l6i S sfia 2 diBm .41 YA .{!, dB ehring minb m$t k6t qufr nbi ti$ng vd c6 rdt q &,
nhi6u *ng dung vE sau nAy irong to{n trgc. BAy giir ta chuyBn tqong h€ thrirc (2) tdt cri D6 ld dinh i_v mang t€n Ong. cic phhn t& nri cr.r hg s6 Am sang v6 plrii. Gil sii ta xf c dinh duqrc d"t, ctz, a3> 0 cdn ctc, cx5 < 0, Einh lf Helly. N6u trong kh6ng gian Oclit llii la sF nhAn duoc: .Rn cho trudc mdt sd hii'u han cac vtt thE ldi mi trong il6 m6l bQ n * I r,it thF trong c{c ct1 11 + ct rZ * d3 13 : 1-a4\ 14 + (-es) rr) (g) vnt th6 di cho c6 chung nhau i di6m, thi khi rJ dri-v tdt cfr, cic hQ s6 a1 , dz, d3, (-44), (-45) dti s€ t6n tai mdt ili6m chung cho l6t ci circ r'&t tlOukhting Am, vi ct"t*dz * a3 : -- cta * a5.B&i f rhO lbi dl cho. (6' oav khi n:1 thi 111 lA dudng thfrng; khi vi ta c6 s6 T = at+eZ +.t3 drrong n€n cri 2 v6 cira h€ thric vecto (3) ta co thb chit cho {. Lric r= 2 thi .R? le nriit phirg; .r,n iihi n - 3 thi rlo ta nhAn duo. c: fi$ le khd,,g gian (3 chitu). Ftrt + Fzrz + Fs"e: pa1a * p5rs . Oh*ng ntinh: Sau ddy tdi sd churrg nrinh dirh 11i- Heily trorrg truong hop n=3 (khdng gian 3 tla,v BOAr + PfiA, + PgO-A3 = Fuo.ar+ Fsctrs chi6u thdng thrrdng mi chLing ta itang hoc trong to ttdy [3t : ar/T. Fz : o,1lT,Pa = cts/T, $4 : -aalT, trudng phb thdng). f rrrong hqp n :2 m[t phElg) Fr : -rs,i T). Ilri ring id. tfit ci cAc hd s6 chi cich chirng minh cfrug hoan toau tuuug tn Fr;Pz, i3a,9*Fs ?0va Fr + Fz +Fs + Fr * Ps = 1 kh0ng kh6c gi lim so r-di lrudng hr:p n :3, (iiA su ^lJ la 11i0rn. sao cho: Tru6c h6t ta hel' giA thidt rn nq ta co 5 vAt thil 16i Ft, Fz. Fs. I'r, FE. 1'a cl:on lriy 1 rlidm chung cho tit ci clc hinh dang rrit. lrt vnt thE ,nt : gt.,s* PzO-ar + proAs - p*Trt + Pro.rs. F1 vA ta ki- hi6u di6m chuna dc lA lli (6 dAy I(hi 116 do rlinh lf ta d[ chinrg nrinh i'y ir'€n, i cd thl bing 1, 2, :j,4, 5). I(lri clri ta rihiin cluoc thi ta cri tli6m If phii nim trong nri6l l6i cria 5 di6m At, Az. As A4, A5. X6t cic v6eto. c{c di6m .4t, dl, ,-li vd ngo[i ra nri phii rr.'rm +4 trong lri6n ldi 'cria c5.c di6m Aa -ta A5 nfr.a. et = .45J1, e2 = A1AZ, eS = 45A3, e4 = AiAq, Nhnng nr6i m6t trong cic vat th8 l6i F4, Fs B6i vi 4 vecto il6 nim trong khdng gian Oclit lai chria t{t ci cic i1i6m,41, Az,As vA t{t nhi6n 3 chrigu n6n chfng p]iq thugc tuy6n linh, tuc li c6 di6m M nta. Vi mdi nr6t trong c6c vAt lA. t6n tai 4 sd thuc ct1, ct|, Cxi, aa khdng ddng th6 l6i F'1, F'2. lra ch*a t{t ei cie tli6m A4, As thoi bi"ng 0 eao cho : vI t{t nhi6n cii dii)m ,}{ niia. Yi vAy tdt ci cAc +E vAt thb F1, Fz, Fe, F+, Fs c6 chua mOt di6m cl et + d2 e1 * a3 e3 * dA e4:0 .(l) churrg lil M. (Di",* ld 1 trong cic clinh lli co birr cria hinh , Ti6p theo ta sE kitrn tra birng qui n:rp theo giai tich rnd cic ban cri th6 lu ki6m tra Iiy drro. c). sd vAt th6 ilang x6t. Gi& sii rlluh 11' Fleil-y BOy giir ta chon trr.rrg hhdng gral nrdt dilrn r) dirng cho m vit tha. c) dAr m ) S. 'l.a clir?ng niro do va k"f ir:, minh 1,, cirng tlung cho m * 1 vAt tlr6.1;'t, F'2,.,,; I'mll. K,r' hi6u @,=F:AFm*l (i-1,2,,.,,nr). 11 :Otl1, rz = OAz, r: = (1.{:, 14 = ()A4, rs =. OAS" - 'l'a nhf,n duryc m v0t thi! Idi O1,..., O,1 mi. trong Khi do rir rilg 1A: d6 rnSi m6t bd 4 vet thii Ii c6 I di6m chung" s4 Theo giri 1hr6t qui n;rp thi m vAt tir6 Or,..., (D- el : .4s,{1 = OAt -- O.lE: rt -- rE. c6 mQt ili6m chui:g. Nhrrng cliiiu c16 co nghia lA t6n tai I di3m chung cria c6c viit th6 i-t,..., es : AsA4: (]r14 * ()Ai = 14 - 15 fr6, Fmgl. Dinh li- ill tluqc chung l'rinli.xong. Cho n6n quan h6 phu thu6c (1) ta vi6t tr€n c6 CAc ban thAn m6n I Ctich cirung mirrh tr6n 1 thb vi6t iui du6i dang sau: tliy r$t ha1 vd dgp. I'I6t di0u mai ban dr.rc chn a1 (rt - r'5) + 32 (rz - rs) * rs (rs * 15) + lrru -t ld clinh Iii I{etlr c6 rdt nhi€u irng dUng la li; thir. cri th6 trong mOt vAi s6 hAo nira ldi sd 7a4(rt*rs)=0 gi6i thigu rndt s6 ung dung ha-r cira tlinh l{- IIelly. flav .I11't e2r2 * crsrg + ct4r4 + Nbfrn dn1' chi xin n€urnit vi du IA t* dlnh I1t * (at + ch +. s3 * flr) rr: (l tr€n ilAy Young cltr rrit ra nr0t irdt qu& dle li6i crleh khirc ta ludn cli qua;r trt: sdc snu : cx,1rl * axr2 l ssrs * e4r4*q.5r6:i) (2) Einh lf Young" i\6u X lir mOr vflr rhb lbi & dAv as - -* (at * az+ o3 * ct4), rigoili ra trong trong lihfllg giau 3 chiBu c
Bii 2lll1. Ir€n gid11 k6 d ou6ng nqttbi id ui|t tdt cii cac td tq nhi€n ttt nhd ddn l6n, mdi sd trong ngt 6 theo lii?u xofrn 6c nhtr hinh 1. H d.g tdc dinh cdc sd 1930 pd 1931 ttrlm ir r&m mmm tlr)ng, cgt thri ntiiu r(u coi sd 1 nrim & tLong 51, Sry dr*r#e tit 51 (ddntr sd /ring rlrin lr| frrii sonq phiri, t* dtroi ldn trin). Liri gitri. K-i' hiSir ir u ,lr,rro llrti .r, col tirr'r y BAi 1/117. Chirng minh r&nq lA d (r; y). a) f t 6 ngrbi tiru g ta cd thr' chptt ra ho(t: ddi mdt quen biEl altau, hodc 3 ngtoi dAi mgt khAng -*t- ?- I | quen lrtlt nhau. I ta cd tlft clwn ra 4. ngtrbi b) ?'rI ?8 ngtr'di ttig iy I hodc ddi rnQt tlucrt hiit nhou, hoL
\- , Di6u ki€n (1) tuong dunng vdi n I981...1981 3*::Y I tgro" IvIe S(A) - E a, -- XI r-r l9ri4...19ti.i \,6 phii ding thric trdn l0n hqn 1, suv ra :118i...1981 1964...1904-1 n .a= 0. Ta ciing c6 g* 0 vi irdu U = 0 l.hi v6 'Xi plrii lrring 19302 + llll- 2 (mod3), trong hhi ili; Ii vL{ [r'ar chi:r hdt cho 3. I1 Yi r.6 trfi cira (t) (ktri r, !]=- 0) le mot sd 16 s(d) : 0i2) [1e81Jeil (1s81-J18, + 1) - ndn su1, rri i = O. Yil' phucrrtg tr"inh rIf, r:lro tr& thd nh - 1064...1961 (1964...1s61 - 1)l .3x*1--i. 2Y (2) I1181...1981 : 1tt8i ( r + rOa + 7$4'2 1...rr 16*'*9a; a) Ndu r 16 thi $x ='' 3 (nrod1)' Do do .2v -:2 19ri1...1964: Ig6,l {t 1 + fi4.2 +...+ 104.4e4) to4 . (,.,.,o4+1. Srty ra 17= l vi r:6u g) l' thi 2) i+' voi Ye5' s(A) i (t + toa + 104.? +... + t04'4e4). g: 1, tt (2) sur* ra r: I' vi I01e80 (/i: - = (104 - 1)(I + t04+.r. + 104.4e4) 1 b) l.i6u r:2k I' 2, 3,..') thi hhi 116 (2) rr6n d ; (i + to4 +,...+ fi4'4e4) tun:ug rlu,rrig r,6i 3Ek- 1=2v +4 (3k - t)(st+r)=zv' \ri rndt s6 chi ci.r du1' nhflt cich ph5n tich ra Nhu v6v cac s6 nhirn ilucc. theo c6ch vi6t noi tlrua sd rr{rrr'€'tt l.o rr0n rti; r-d trii Irirrg r'6 phli trong tl:iu bli c6, rnQt trdc chung li ii,i:ln - t t:*. 3k 1l:2'. \'i 3k - I va 3k + I I + 1oa + roa'z l-...+ Io4.4e4 lr) lrai sd chir, litrn tidp ndn chi cti th6 nt:1, ,r: !, trie [d k: ]. Iihi d6 I : 3 r'i tri (Z) ctro chi chria toirr sd () r'i I trong cAch vi6t phAn. thdp .r' = 2. 'I6rn Iti J.rltrrortg trinh dld cbo c(r hai bf rrglliinr: P.H. (r: l. q:!, ::0) vir (x:2, y:3. ::0). Circr bart ?'ri,ri Iurin .Iiinh (q'tlurr litru 7" Itiri Eei 5/f 17. Cho 1la+1lb+t c:e (1) Hung) r.d Ffrnrn Xudn Hit (6A {lhu lirn i\n. I:lA (tla * llb)fila i t'ld (tlb + l/c) = I o) NQi) c6 ldi giiri tu0ng tu ldi giai tren- tr{ot s6 harr tim ra nghiQr cira bni toin song khi bi€n abt=T (3) luii cdn thi6u ch{t cli€' Tlnha+b{c the.o d, F,T (a*Ot. - N,K.lt{ Lbi gidt. 'l'ri (I) rd (2) sulr ra @ - 1lc) @ * tlb)(o( *t/o) = p. 'BAi 4i117. \!l*!ti!_ce cdc.sd.q*t?ql jxl'16 Sau khi khai tri6n vd rdt gon ta c6 biit dil.u r* sd .196,1196.1...1964 d.dn sd 19E11981...19E1 e3 - at (tlrr + 1lb * 1ld + d.(tlab * tlbc.4llea) lidn Lidp theo hdng ngang 'uo'i ffiri' fqr tntl t' _ ljabc: g Chi:ngt minh riLng tdt cA cdc sti lhn ifu{c theo
7\ Ndu a :0 thi f (a)_* 0 ndn tu (1) su5, ra o=0. 7, A khdng ndm trEn dnbng lhcing @t Oz. DO tbd-r, ndu o'* O tni .r + 0. (1) ttrcrng rlrrtrng Phd.n tich: GiA sr? tIf; duirg iltlcrc dtuong. tro n (u) cfrt rlrr&ng trdn (ar) tai hai tIr6ur xuy€n [.dm voi f{x\la + {(a)lo:0 (2) 'Ie l4i r16i 81. Cr vi cXt (ur) tai hei cliirn xuy6n tAm Ct.r ddi Az, Cv. AO1 c[t dudng trdri dli dung tai ,lf r. r *,f(c)= .r7 -r.,.8-p21..5 - 3ir,4+ljr3-21,.2 * i r- | AOz ctrl dirrlug trdn df, tl.ung {ai M3. .6 +21.,.4+girl +7 Ta c,l : (e - l)? (3) A01.O1.111= ()tBr.OlCt =r? '401)l 1= ,\l AOt (1) ,---+zi-r+3b7+z TrrrvrS iu : ,7 +7 r,6 +zr*5+3brn+ 35,. o - 21 12 +r r + O*iIt * y?!,+Oz r * f(c)= 1 (Zt ro + 21..4 +B5rP -t7 Nlru r ir .Ut vi Mr ld trai tlibrn x6c dinh dugc. (#*1)7 (a) ,n6+ 21.14+ 3512 + 7 (n) Ndu o:1 thi /(a) */(t) : 1. Do d6 (2) €/(rr) *.u = (). Yi vAy 1* (4) ta suv ra :r: * 1. 'Iucrng tt ndrr a= - l thi rr::1. D6 rhdy a++ 1 rhi r+:L1. Do d6 z -i- /(:"-) f 0. Vi vey tri (3) vn (4) suy ra lr - ftx)1lir, +/(r)] :(,t' -'D7l(r * l)7 (5) D[t (,r - 1)/(r f 1)= y ta (a - t)ltu a 1) = b. Ttr (5) su\ ra {-""--:y)---/ r-l(r):U?.r* g71b) +)r(1 -g7)=y(illi *97) I{inh 1 # f.:.)lr = (l - s')l{t + s7) ("i .r f o va Cd.ch dryng: Tr€n AOl ldy Mt vd trdn doz l{y 1+y7+o). llt th6a man (1) va (2). Ducrng trdn (u) dung Trong d&ng thuc tr€l thay r b&i a thi dugc qrra -{, )It, M2 lA. rtudng trdn ciin dung. { k)la = (1 * O7)i(t + t7l Cht?ng min.h: Ta chung minh (rr) crit (r't), (uz) Lric ndy (2) trtong tlucrng viri i tai c6c tli6m xuydn tim d6i. Thqc vflv, -giir sfr' (r,) c6 I giao tliilrn vdi (ur) 0-u7)10 +e7) + 0-b7)11l {-}?)=0. ld (s1), Btol ctaL (r,) tai rri6rn cl. Th6 thi ++(r - 17') {t +b7) + (t - b?) (1 * g7) =u
/.\\ KV hiQu ba kich thrtdc hinh h6p 1A r, u,z Viy: fi-MtMtl?sir OtAaz (hinh ?) ta c6 circ kdt qu&: -r- R: V !t1+Avi-ZlMt A Hz r:osOrl"t0yl2sinttlAo2 -\ tlrrng il6 ,ll,It. ,\]lz dlrroc tiuh baus ('t), (5). ?' A ndm- trin d$dnq thrtng qua O1O2. Ndu (u) ln mQt drdug tron b.irl ki n:\o cl6 cffL (ut), (ur) tai e irc di6m xuy€n t.im d6i vn (u) ? cit rlud.ng OtOz ttli dt vir Az (hiuh 1) thi Ot,lt. ()lAz: ''i L t)2At, rt2Az = ,22\'*' Nhn vri.t' ,11 vA .t.J Ii cri alinh viri ntoi (u)' (D6 x5e dinh ,1t, -12 ia tlqrrg cAc irAn hinh OtDt. ii-- l'E O2Dz vuOng goc v(ri OtOz' Dudng trdn cti fi-nr *dn atOz, riung qua Dt, Dz se ciit OlOz t:ii ,tlt vd lz'). \IAy rrdu rI nirnr trdn Ot(tz tlri chi cir ^l trriirg' vdri ;1t ho[r:.4. trtingydi Az thi moi dnrrg dugc (u) th6a rnitn di6u kiQn bii todn. Hinh 2 . Cdch dgng: trirc ililh dt, '12' Dung tlr,rting trdn tt1 r' qirr '11, '12. 1) tioi R l:\ birn kinh rn5.t citr ngo4i ti6p tu (ihtrng rtinft: Clrrine nrinh ftrcnrg trr trtrri'ttg rlii'n vA Irirrlr hdp lhl lrrlp i. Itz=(.r'2 rttz+:zlll 3, KF-t ludn : Ndu A rr[rfi ngoiri cltrtrng t]iirng O102 thi bni toirrr co m6t nghiAm hlnh. Nhung *2 + az + z2 = (rz + bz * .2)/2 n6n - N6u A trung vdi .11 bo5.c J2 xirc rlinh b6i (*) thi bdi toan c6 v6 s6 nghiQur hinh. riz = (o2 + t2+ ct)la - N€u ;l nim trdn OtOz n.rA lihdc ll, dl ttrir 2) Gqi r Id bin kinh rrirrt chu nOi ti6p tri bii toSn kh6ng c6 nghiQm. di6n, 7 li.th6 tich vn S ll dign lich to&n phin trir di6n, tIi ta c6. -Vrlf,n rtil : Rit ii ban tlram gia giii bAi ni1'. Tron{ stl cac bii girii cira cirr: i;arr giri al6rr tr)a 1.: (i/S) Sr soan khdre cti hdi nio i:irn ilrinq phii.n biQn -3 r = 3l'/'S. hrAn ; t{t cir il6u li6t luAri trong tlucing 'hSp, tl, Ot, O2 ttrirng hdng thi bei toir.r r'0 nghiQm T'a co: (trong khi & vd i2 hi6n uhidn idrr tai (l)). ona,oo ? 'fl BAI 8/113. Cho ttr di(n girn rtdu ABL'D ci li,,lu-t :',, * I;; ::';',; *62 BC : DA = a, CA = I)I): b, AB: DC = c. Cltittg '' (=- *.,=h2 lr':o'+nZ -c a minh ring , a ,., , , l, : *grl\: (az + bz + cz)z (2a! b' a':bz iY -lzc2 a2 -,,4 - t,r -ca)- >r) ? + zbaca -f zc4o4 - 2o4ba ,rt -- D* - --, =/@'r+t.r*rrlts d Theo c0ng thrtc H6-r0ng : i9dng thr'rc t.ilg ra kii pri chi klti AIICI) ld {t1 cliin dbu. s = 4 t/ (a * b+ d(a * b * c)(lt* c - a)lc+ a- b) 12 Liri gidi. Quir c6,' canh tldi iliQn r:ria trl. didn \',j)'r=itlr'S:1/4X dung circ miit phang song song vtii nirarr, ciic n'ri t phfrng do cfit rrhau tireo mQt hinh hgp c]rii nhflt * uL)(cz + o2 - b2) oz + bz - cz) nl.rAn chc canh tri di0n li rlucrng ch6o ciriL 16c zfirz + "2 pirt, h*b * d h+b - r)(a * c * il(c* o -D)
3) Nhu viy ta c6 th6 vi6t : Ldt gldt, Goi Or, Oz, Os,li tflm ba tludng tron b{u kinh a n6i trong c15 u bAi thi d6 chirne Rl r = rninh duoc O,OtOZ Or I[ gric Iam di6n cti 3 nrirt _ f a? + at + ( nt + c)( qt * gl( rij c * g rz_j bri rrg tiOo. c? )1 o a a) t ) + cz * az + b\ * (Chirug han .mudn tinh goc t)tQot Ia khio sit ( ar + be + czl( bz )( c2 + o, - bz) { o2 t:r ) tr€n mirt phirng @OO). ]Ioi duong tron tAm Or i (l) vA Oz c6 rn6t tlu&ng liinh nirl trtn rrr{t phirng BAy gid ta chitng minh cI i; hai rluong kinh clrl liE nharr li Irip r-oi nhau' (a* b -- c)(a + ir - c)tb * c-rr)(a + b - : g6c 120o +6rdoz:600). ft c) =2rlb2 *Zbzc, *Zca? - a4-ba - c4 (!) f ThAt vAy: pfu,, b. c) : 2ozbz * 2b2czf lc?rr? - n4 - ba - c4 : - tr4 1 2(b2 + c2)ct2 - \hz - /r2 VAy P (a,b,c) li. tam ilrue bf,c hai d6i tcri aI c6 nqhicm = bl t "2 ::*2bt. = (b= c)2 * o2 ++ rr=b.1- c.b-c,-b:.-c,- h*c. Do rlo rd th0 plrAn Iiclr d rrr.rc P(c,b,c) = (a * D -1- c) (a * b * c)tt* c -- a) X 3 X (c * a - bl, tirc la (2) clLrnc chirng minh. .Hinh 1'u (1) r'd (2) tu c6 C6 hrii ttudng tron ii6p f,ric v0i ci ba tluong Rzlrz = irtrn b6n kish o; gi.{ s& duong tron nh6 co {dnr ft, b6n kinh rt vd dunng lron lon c6 tAnt Iz' (az + bz + i)z (zaz bz + zbl e, + zcz a2 - oa - ba - ca) bdn l
I t) AB ad + oli I ( Bc.D,D + oczI rca.c B) - l o A2 I 2) (HB.[rc] I GB.ac) -r (uc.t{il I lBC.n}.) + + (ru .,y.u s) I (;tt.c B) : t ndu A, Ij, C
Tim hiDu ceu. th6m to*n phb :Ulrg ,ha_e PFIUONG PHAP QUY EAO P}IAN DI-'C 1'IIA\*IT ISAI ne1'gio-i thi6u rn0t trorro lhii'ng phrror:g Sii to6n 9. Cirrirrg minh rins lg piuip c(r liiou qu6. d'6 gi&i'cAc bdi to6n tb hop, tlo ld. phaonq ph(ip qug'rtqo. NOi dung s. rm -m-1 T. L- Ln ' Ln-1- /-m-lr rl-, 1 .. :* \-m- f-m-!I cfra phuo*ng phAp qu] dqo trong c6c bdi lo{n tb hop IA chi ra cich giiii thich hinh hoc cho Gidi: 1'r€n rn:r rs lrriri. k6 6 rrrong ta h[y xet phep dua bii toAn 'r,E vi0c tinh s6 tluong rli lflt ci circ drrtrng gf,p kht're ngin nn{t noi diim (hay s6 qu! dao) c6 indt tinh chdl" xAc ilinh uio iIo. O = (o, o) r.rii diilm M =. (rn, rr-ni). Tbng s6 c{c Ilu di6m cfia phuong phAp qrrri ilao lir tinh cluong gtip klruc d:' Ia CX'. tlor Hs la l6p circ [ruc quan trong c6ch ch*ng rnirrh. dudng grip khirc cirt cludng tlr&ng .r = 112\qi ril6rn < B{i toAn con 6ch ,) (tuc bdi toan sd 6 trong (ll'1,*) (It = 0. 1..., n - 1). R6 rdug ,8g c6 Co-1 1 kli thi 1'o6n Qudc t6 lhn Lhu '21) c6 thb gijri ducrng gdp Jlhr'rc. Ti' 116 su). ra bing phuong phap quj- clao (xenr Toin hoc vd [uAi rre sd 2/1980). Bai niv gi6i ttri0u ruirt sd bAi to6n diEn lrinh cr kxo :'-i cl;t-, c6 th6 giii beng ph ucrng phir p qu1=' d4o. Rii toSn 3. Chri;rg minh rirrg Bdri to6n 1. Ch{rng rninh rang ' Clll Cfl -1- CT:l Cl+r * ... Ci,F (C9t2 + (cl)2 + ... + (Cl)z Girii : Ta hdy x6t mgt m4ng tu6i kF 0 vudng cd n X n. 56 duong di ng6n nhdt (theo nrang * Cl-,, CF+* = Cf+rar Iudi) tu tti6m O = (o, o) d6n dibnr M = (n, n) birng Girii : Ta hdy x4t tdt cir cac dudng rli ngin nh6t Ci". Chri f ring m5i dudng tli {y di qua mdt tri di6nr O = (o. o) ddn di6ni -lI = (n-m * k*1. m). I'd chi mdt di6m lu : (k, n-.lr) nim trdn iluoog ch6o .BC'. Sd du'ong di tt O ddn dx bHng Sd cAc cluong tlo bnng Co lr+l C[r''t'-1t : Cfi' Goi Bi (i = 0, 1,..., ni) la lOp nhii.ng drdng g{p Sd rluqrrg rli tt ,{s d€ln M bing Ch-t+k = Cf. khuc cirt dudng ihing :t = A'* 1/Z -tai di6rn Do d6 sd cluo-rrg di tu O tl6n M di qua .4p bing (k + 112, i). 'la nlrAn thdy rang rn6i dudng gdp Ch. C* = (C*l?. Tu d6 d6 cldng suj ra hQ tht; khdc thudc l6'p E1 gbrn 3 phAn : phiri chtng minh. Ph[n I : l)rcrng gdp khric n6i (o. o) voi (fr. i), 'on uira ta c6 hQ th[rc tblg- quAt hon: Phbn 2: Errnng ninr rrgang ndi'(/r, ,) vdi (k + 1, i) cf; Ch + Cl" CFt +.,. + c[ C,% = Chnn Phfln 3: Dudng gdp lihirc n6i (k + l, i) vdi (n-nr*k*1,m) Tu cI6 ta thdy ring 't6ng s6 c{c drrcrng gdp khuc thuQc lCrp Ri IA { j i m---i (,6*t. ( o_i t Tr! d6 su-\r ra hd !.huc phrii chring minh. Biii to6n 4, Cb rn * n ngucli sip hdDg mua r'6 trong il6 cci n nsrroi mang tl6n ioai 5 hlo, :iA m ngirni nrang ti€n loai I il6ng. h{6i ve gi6 5 hAo. Trrr6c hic liAn, ngrrcii bdn ve khdng c6 ti0n. H6i c6 bao nhi6u cdch sHp r6p m * n ngudi mua v6 dA khdng c6 ngudi ndo phii ch& tra tien thrla ? 10
E Gidi: Gih s& c6c ngudi mua r.e ile tluoc sXp Bii to6n 5 (nai hdng theo rrrdt cich roAn v6 bir phi6u). ! nio d6. Ta rl[t Trong m0t lAn bdu cri., ung cir vi6n zl drto,c o ., : I I nrru ngrriri mua r,6 thi,r i c6 ' I - I ncu ngu,o-i mrru r-e th* i lri 5I hio ddrrg phierr lrlrr, il.rrg crl vi6rr B tlucyc b phidu bhu (a) b). Ct} tri bo phi6u li0n ti6p. Fi6i c6 bao Khidy 56=e1 *...*cr nhi6u c6ch b6 phidu 116 r'tng crl vi6n ,4 lu6n lui.rn lA hi6u s6 gi&a s6 luong lgrrcri eo ti6n 5 h{o din dhu vE sd phidu bhu cho lrinh ? vi s6 lrlnng ngtrdi cir tiBn I tt6ng khi c6 k rguoi s*p hArrg. Girii: 'fa dat Tr6n ma.ng luni kE d vtrOng ta r,,:) c6c di6iu ..- [+ t rrdu plridu ihu { bhrr cho,{ -{r: (k. Sk) (k: 1,2,..., nl -f ;r) "r.a xet iirrcrng g{p { - I ndu phi6u thu i bhu cho B khric n6i O = (o, o) r,6'i rliFnr _lm*n = (m*n, n_m) Ort .fk : dl + ... * ,,k. di qua cic cliEm ;1r,..., ,{.+r_r. .l.a goi drrdng ;x6t gip khric nhu vd1- 1A rnijt Qttij dao. ii.o,,c ,irp .(1c,I'a ... tju_!- dao r,6i cric cIi6rl O; (1, Si); ... vdi cAch s6p hnlg cira ngrriri mua vr,r. To,,.g *E Sr) ; ; (a * D, S"15). c{c gu} rlao bring C.** t-t'OarSa+b=o-D. I10 rAng rn6i c{ch b6 phi6u trrong ring vryi rndt a) Cllri f rirng cic qur- tiao, trrtrng ri.ng.v6i qu! dlau xac. rJinh. 116i qu j clao qdrn o-o A Ao,,n cdc^h siip hinpJ cira nguiii mtra vi t16" lih0ng ai thing trong ilo eo a (toarr huong fen ,re,, .ia,l, phii chr} tri ti6n thuzr sE khring c,itdr.u,,e th?ing u_ _ t- i? cec qui rlao bang ci+n. Ung ct? vien .1 lrroi dan clhu rrdrr' qrr! clao trrong ung rli gua cli6m Thuc viiv ndu tloi r-ui A. nAu dJv .S;_1 6, (1. 1t r,-d khdng cirt truo hodnh. Sd qu! {!ao nhrr St - - i thi cii6u ilo c6 nghia ia trolg * _ 1 rgri.yi = r..1y bing sip hing rliu tien srj irrong ngrrO.i cii ri0n b hrio vi s6 iugng ngrrrri cr_r tidn lding Id trhu nhau, n* 1- tn ^m ngudi thr'r /i cri 1 t1i)nq ne:r phiii cho trri tidn thi.ra. nf1 m*n b) Ta st: chung niinit r[ug sr5 qui- ilao cilt trongdo n:a-l duo'ng th&ng u: -1 binq CHt". Voi m6i quj- m=b. dt9 A oHt clu
trong do a) dirng Bzo._z gu! dao nSm tr:6n truc hodnh vd hhdn'g c6 ditirrr chung r6i trLrc hriirnh (tlu cAc t*' _t-ei-i_r. diFnr O vi 7'r. 'i-Si-':"'- b) tlring 1l2n qu! clao khi,ng t'ri itirth naiu ilrtdi Sx : I/. truc hoAnlr. Gqi N (c, ry) la sd tiit cir cAc qui dao n6i di6m (0,0r viri rIi'6rr (r, U). Ta cO ChLrnq minh: a) chf j' ring tdt ei circ qui dlro r-.6i L) voi 'l' nirn.r tr'0n lrue ho*nh vi lih6ng IUDNn r)[i I ] 'c6 cAc.eli6m chung khhc vrii Iruc hodnh nhdt thidt phii tli r1ua. tlibm /'r : l2n - 1, I )' Theo m9n1r A (.r. y) = I C, 2 6 -r+r-- niu r va y currg rlE 3 s6 qur- dar.r nrii 0 v6i Tt vr'r khfrrrfi c."rt truc chin hoirc cung 16 hodnli liing $ " 0 tr,,ng lruirrrl hop cdrr lai 1 i\'(2rt - 1, 1)=--{ lr1 Thuc vAy gih sr? qui rl:ro gdrn p doan hrrtiog zt1- I * ln 1 2n-1 lin tr0n I'a q dorrr lirrOng ru6ng tlr
Vdy ta di cL"tt ilucrc 2* * 7g :998 (lonn 0,7rn vi 12.r*5y:1992 doan 0,5rn n0n chi chri cit thcm TAAN TTOC 2 tloan 0,7rn vir 8 cloqn g,5rn. 1'a ciri cin e5t th6m 1 ihanh 7,4rn theo kibu thri, nhAt nta. V{y de ding tfit cir ii:121 + 108 + 1:230 Ztrff- tltarrlr 7,Inr. Ta cdn phii chung t6 t'eLt'h ciLt trOn la ti6t kierrr nhAt. That v,ly, ta thiy tbng sd dQ dni eria 1000 ilo?ir 0,7m vb 2000 CIoan 0,5rn le 0,7.1000+0,5.2000= :l?00m: vffy phiri dilng it nh:it iii 1700 7,4*1 th:inh VAI BAI TOAN VAN TRU 7:[,rn ha1' 230 tbanh 7,4m (dpcrn). 1'6m lai ta chi ciix irit 122 tiranh 7,,im theo vdi bii toan lii6u thu nhdt vd 108 thanh 7,4m thco ki6u thtl IEA'-i:'.f,i:J iiil ;;i ,';';u:*' hai ld rLlng. i \loi clc b;.rn gii.i lrii trr:'rtt srru Eii toail l. \'oi i1 th{rc tidt kiem ild.t li6u, : a'nh hag chi hAq tinh ren cdn ehtinq bao nhi6.u Bdl to6n 2. Chn e &t it nhdt bLLo nhiau thcrnh thanh s&t mdt thanh ddi ,:'.4n r12 c&t thdnh 10C0 sdt drii 3,5m fhdnh '100t) &tqn mii docn ddi A,4m doan, rttdi doan rtrii 0,iar. uri 2000 do7n tnei d'oen ud 330A dsan ntdi daan ddi 0,,$n., ci thri'ng 16 tlroc rdng"citch diti 0,5m. .lrth htig 0,( a *('tO 71-7a l+2u '+1*2a15. vi nguy6n li6u Sd lLrr;ng tion Loai vi b= = j-o- _ . nguytin D r.r chi phi cho mQt don vi i) D trir shn phhrn iQu mi I { t * 2a ;21 vA (1f2n) i6 n0n suy la: I i * 2,r :5 ltt,;ic I * l,i : 15. Drr drl hoirc a = 2-+b = il hoiri: a - 7+D - 5. I 8 2 \riiv tl t:riJrai circh cit m0t thanli 7,,[ur Ir:i rrhiit: II 14 l) Ciit lirinh 2 doan 0,7m r-i 12 dr.ian 0,5nr. III 2i 4 3) Ciit tlrIlh ] doan 0,7ar v{ 5 doarr 0.5rn. Goi'r thrrnh duoe c:il theo 1
NgoAi ra, cdn c6 dibu ki€n trr nhi6n nta lA ilihu vay v€in dE r14t ra li tim e.tc fct| sao cho .'t:, ll d6u nguydn khdng im. Vty chn tim n' Mttx(Z:5r t 7y) vdi di6rr kr6n: : bi,'i (I) rl:tl ciir trr nlrr', nhit. f:.r+ri(s (r) i:I ,,\ (r) !r, ( + zg \< r+ (2) Nhirr x6t ring, n6u i ),i yp ci) ci thi (vi n'+ oy ( 25 (s) 6i),.zbi) (ci* r:1) (bi I cib1 * cjh. - b1) ) A hrL1, clDi * ci hi ) (.t.y)0 n(ttr'0n Tu (1) va g2o+ r(a, 'Iri'(3) r,'n r)0 Yi ya]'lrng v6'i hoan r-i cL,e2,..., cncb ci)cj + y--(4. v6'i i ),i ta cd hoirn vi ci, ,':i.."., ci trong d6 (l) ta e6 : 6c * 3y ( 11 + 6:r' + 4A : 6"r * 39 Tr'r c; = cj, c}: ci ri : .k vdi A': r. 6 43 t)ri: + v ( 24+U (fs +3:r -F 29 -\( l,t. Tuc ta ttr nn (l) => (2). : bici7- I bi r'i. ,-=1 i:l Nhdrr thdy phtty(392{15*1{=29 irdi sang ph&i, thi tuong ring lo3.i hiug & cirt r = 2 *) y n{ 2-+,2 ( l0 * 14 = 24 chira bi sd ttugc ch& d6n dia dibm thir i. r=1* gy(a.+z(Zs. ol * 5, az = 20, as : 7. 04 = XB vS. cu6'c phl vrrn Y&y ta suy ra Max (z - 5r * 79) - 29 klri chuyBn m6i clon vj hrrng t16n ilia di6rn thu i ln r=3vir A=2, bi, D1 = 25, b2 = :10, b3 = 5, b4:9. Do d6 phiii stin -tuf,t 3 sirn phhn'r loai.A vi 2 Tr Iip bing : srin phhm loal B, khi {y ti0n ldi sE nhi0u nhdt 1A 29 dbng. 4 Bii toan 4.Tqi mit kho cd chfta n laqi hd.ng cd trong bqng miti loqi ld at, a2,..., an. Cri?n b; I 2i't :i0 chuydn ch(t cuc lo(ti hd-ng ndg rltn n diadi?m,bilt. rdng m6i dia diDn chi nhdn nAt bai hdng ud cadc phi chugtn ch6' mlt don ui hdng d.dn dia aj 2l 18 dibm thtr i tri bi. ttdg tint cach chullOn ch& nht thd ndo db tbng cac caoc phi lt) nh6 nhd.t, Ldi gidt. Kh0ng mdt tinh tbng qutit la c6 thb gii thi6t vfly phil chuy6n ioai hing thr'r nhiit ddn dia t,r ( bz q,... .-( Dn r at> oz)>...) o, di&m thu 2, lo ri hlng thir 2 d6n dia di?tm 3, Ta th{y loai hAng thu i cl,roc ch6 ddn dia lo?i bang tlru 3 d6n dir di'Om 1 va loai hang tli6m thu 7 sB c6 cu6'c phi li. aib;. Do d6 tbng thu { ddu dia di6nr thu l. s6 cu6c phi ctruyOn ch& n loai hang itdn n ilia - B+n hay giii bii tohn sau: I di6m lA X oibl trong c16 t6ng X aiDi g6m n s6 hang va c6c chi sd cri:i ai tl6u khdc nhau. chc Bdi to6n 5. Co n ki(n hattg cing Loai cti khdi chl s6 c&a bi dEu khic nlrau. luqng 01,o2....,ar, cb"n brin cho n dSa di?n, bftt Gqi lcil i = lr....r ld mQt ho{n vi cfia day [ail rdng 6 mdi dia di?m cht mua 1 kiQn hdng ud, r1 i= 1,2,,..a theo mft tht! tp, nlo i16. itia di\m th* i ngudi la chlu mua u6'i gid bi dbng t4
mii ilon ui hr)ng; Ddg phAi hrin nh* the ndo dl Ta chung niinh nghi4rrl ri cira bili toan sdu ldi nhibu nhdl;. (cu6c phi chu-u6n chrl nriri r,lun cttrg nA-v thoa rnln : vi hdng {l6n c{c dia di6m nhu nhau). rrli\ E .ui--lrin{lr. I *,} nroi j = 1.2....,n. Bii toSn 6. Mit mdg bt.tg co tfri trong M.'Cd n i:l \ i=t I 'Oi Ioqi hdng cl\ r1p LAn may bag d6. T'rong ltopg 1'a ehung minh diEu ki€n bing qui nap theo loai,hdng i ki a; ud gid Bt dbng, ciiu x?p rnSi loqi s6 hn s6 n. hd.ng bao nhiiu don ui d'A trpng luong tbng cQng k!6ng utrot qud M ud cd tbng giri tri t6n nhdt. Iiir rdng khi r:1 bdi toAn clring. Ta giir thi6t # ktlt IuAn ctia bAi toin tr6n cirng dlirng r"di (n- 1) Ldi gidt. Gc.'i llfi li trong ,luong loui hing thrt iin s6. i Lluoc r6p 16n lrhy l:a1,. 'Iheo dd bAita co: Xdt'bni to6n tr6n hhi c6 n ftn sd. 0 ( ri s{ ar, rt -r:rz -i- ... + r, (,11. 'frL c6: Gi6 m5i tlou vi hirng loai i li pi/ai ddng. n6n n /*. i,r,",: r, (.fi ,+; t ,,-,-*" tbng gi6 tri hnng chf,t IOn nriir- bay sd lri : .--, sl .r-.., - ) n X. rpi di) .ri -. I': ; r;r- : njt/ii \ '; i "--. 1 :T, II :-r .-,\ (+-r)rr ln / J I YAr vAn ili o-it r;, li: ' t-l l-t. n n-1 :T"M * E (Ti-T,,)ri . (: = 1=:_ l inr -\lrr .I r. tpi ai).r,1) srr o cho i:l 0(,u ( ai i= 1,...,n Ndn bAi toarr.dsx 1'.r . n n-1 E_ ri' -(.1/ Tirn Max(z-IrM= X 1Ti-To)g11 ,:I i=1 voi tliEu kiQn : D4t 11 = $ilai thi fiua r,o | 0 T;>..'>T|;20 / o< r'i Di1' chirrh ln bii loin (n- t) fin 1" 1 r V{y theo giA thi6t.qui nqp thi: vo.i s6 =^t ] r;t ii ! *i:Min(Ii', ! or) uu, rnQi 7= t" 2,..., n-1. Irr]fzV...]7,]o _ i=f i=l 15
n *'1 .ly'cdn t8ng s6 cAc trong lucrag hing & nCt J Nhung x ailX a1 Vj = t,..., n-l 1, 2,...,i kh6ng b6 hon M. : i=1 i= 1 Ktri {y tilt ci c6c iu.qrig heng o cAc c6t nCn 1,2,..., i-l rlugc clr{t hdt l6n m6v bay, vA chl n*1 i .i chrit thtm loai hang 6. cQt i mQt lrrong bEng Min (Min(M, X ai), X ai): Min (,I1, X qJ lri6u s6 gifra M vA t6ng cic ^ lircrng- bAng 6 cAc i=i i=l I i=l eQt 1,2,..., i-1. I Y7- 1,..., n-t 'Con ndu khdng c6 cQt i nrto th6a min tiuh Tri'c 1A , ' thi ta chdt trit cri cac lo*i. hine *;r', r.i. Jen . i -t J Thi dn: cho c[l =.5. Ct2 =7, d.l - 2, cta- 10 tr{in (iU'.-X o,).=X,Iin (111, X .qi) VJ= 1,..,n-1. Pt-7,F2=4, Ps:8, Ft:2r: vit. M=22. .* i=l i=l : , ':l Dc d6 ta co: Lnp bing: , t jj . I *i:ltin (ff, t ei) V.t:1,..., n-1 i:l i=l Kdt irup r t-r i nn X ci=.tf =NIin(Ll ,Z a1 i =i i:1 ta ilur2c : j .t 'X ri:I{in (.&{, I o:) Yl:1,...,n (dpcm) l-l i:1 Ta c6: ct3 (!{, o(s+a4 ( M, e31an*ar ( M_ Tri dfiy ta suy ra thuAt toin .dB eh6t hAng nhu va ds'.{- aa + clr + cr2l> M. sau : Lfp bing gbnr 2 hlng vi n cQi. t Viy ch{t tfit ei cdc loai hnng thu l, 3, 4 c6 tlrroc vi thefi 22 - 77 = 5 trong luong hilg lo4i Hing dhu ghi gi6 ti€n cfia m6i atan vi cria thri 2 l€n m6y bay. tung Ioai hdng tuc Ia 1i so Pi/ai *uo ch,r no giim tri trii sar-rg phii. Khi do tbng sd gi6 tr.t sF lon lhLit lA : I'iing thu hai' ghi trong lurrng cria loai h:ing (8/2).2 + (20i r0).10 * (7i5\.5 + (117).2 lol tunrrg u'ng vni gifi liEn t16. = abng. t;9i i I* cot nh6 nhr'it cii tinh chit: 1'bng s6 cAc tlr-rn8 lrron.e hriuj ii cc,t l. 2. .. i - I be hon N6O DUY NINH ', In tqi Xi nghidp in 75 - 77 ildng Bb He NQi Chi s6 : 12884 Gt6: O,2OII -,: 'Sdin l24l8l In xong vd giri lrru chi6u thAng 11-1981 1 ,,i ' .. ...4 ,il i otii