Toán học và tuổi trẻ Số 145 (5/1985)

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Dưới đây là tài liệu Toán học và tuổi trẻ Số 145 (5/1985). Mời các bạn tham khảo tài liệu để hiểu rõ hơn những nội dung về kỳ thi Toán quốc tế năm 1985; áp dụng phương pháp lượng giác để chứng minh các đẳng thức đại số; khi nào nên sử dụng phương pháp đồ thị.

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Nội dung Text: Toán học và tuổi trẻ Số 145 (5/1985)

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-==- f"r q"" &"8 YFHE YffiAffi ffiL$## Yffi ffiAffi ffi-&Y EGAFi eUtNE" 1=friC ebc brn thee r10i iri.o *tr! v*' tl*i dS bi6i ihi dilh cAc bni d6 ra nrQt tiong bdn ti6ng \r ki) thi tcfrn qutlc td (l-\IO) rrtrm na-} (ihn chinh. thue dtng trong cudc thi. lt Anli, Dri,c, thu 8{ii e!& iluge tr6 chi're tai Ph*n Lan Nga, Ph6p. m6i bAi tht co di&m t6i da Id 7. trong nhfrng ngd,v dh.u ihing 7 vri:i qull. Ooiu X#t gufr etra rn6i bdi dtsu cirrnc cOng bS ng*1'.-sau hgc sinh pr::}e t* gbnr 6 l;gn Ngay#n ?-CJn Oi?ng khi ch{nr v{ iiuoc trinh lug.r sOi nbi ndu k#t (chu;'*a tsAn flai trgc 'l8ng hop lli Ngi), flrrynh quA tdt. .lfintr Yff {Truirng CLu \'frn i\n i'li Ndi), i-rirn S{] rlibri cdc bdi crla doin ta ln: bai 1: 43 Tung Girsng ("1'ru,lng Phan Chu Trinh, flu&ng di?:rn tteii r:ta), uiri 2: 35 diarn, bii 3: 2 ci.Fm, i\jan: * D[ Ndng), fr6 Dug Khdnh i,n Hril)nft t'riii bii I : 25 clibrn, trii 5: 28 riibnr, bAi S: 11 r{iBm, Ikinh (tr:u*ng Ngu,v6n vnn T"r6i. Fhri I{h*lrh}. 1'bng *ong dibrn dcdn ta iluqe lr14 ilibrn (dibm Chd.Quaaq Qrr.gEn (tnrong Long 'I'h:1nh, Odng trli ,:h l:\ 252j. Il6e .dll cu6r: thi c6 ttnh chflt cS. Nai). 5 han (trir ban QuyBn) l,oc i,',.p I1. ntn tilLr nlrdr:, lhung ngu$i ta ctrrg so s{rnh tl.h0ng chinh khi lr&. thr?r::):ti6t qu& e .&a e ric cloin so s,d,i 31 d.odn cci dri 6 Trua rtg{v i lh*ng i rloln ta di *{n s:,1rr ga hn* siuh dr thi thi iloari ta ilrruo x€p thu 5 flhilu tdi tAu l'i6a l{elsinsi {rli tr'r .\Irr:liva) r'a dr.rrr l-rsrr i f7 klit bi6t C.roc ttiBu dtr, riiriBu driir* di rlirn chirc rb chuc drra xe hoi .,'t rrrSt khu nir:I rr61hi l:Cn rilt'rn6 do6.r.r t:i vE the.nh tich Lii tlai rllro.....NhuDg ta hb lr*ng rnSt khu rr'trg tlrrr,a gltr: t_hi tr{n .-Ior:isri. I,i-il phtn !ir.1, tliini kii r;u& riia rioirn ta. cAe h llelsiul.ri khc&ng 3Cii xm r +i phiu bEc. PirBr: I'nri?,: h€i, v& rbffi sd tlitsm, k6t quS doAn ta Lan lA rii0t rud{: m6i 6' E[e Au, r6rrg 338 C00 kh6ng t6t. rrhu narir rrurlc (14{i252 so v6i lS?/ kms (vb i;e,., ti*i,, Sfrns i.liir 5 i, *heri A;. ;t l5!). .\lrur,g cDng nlri[u rtrcin lihAr: nhu vdy ; d* rQng hon niroc'ta rnst it), e5% d[t dai l] r.r'rng, e16 nhilri ngir&i cbo r&ng clb rrr\nr nay -kh6 hon t0% ln hb, 8%' lS df,t caah trlc m{ cirl c6 4,9 nlinr ti'iir-le *ilr:r ttji d* cfra rni:i bii thi cho trif r-r dAn (mflt dd; 15,8 ngurii trfla mtt hm2). toAn thb hcc sinlr d1.r ihi li 7 X 209 := 1403 t.hi 'frong nl'ii?:rg ng*\"v rtlu th6ng 7 vl"ra. rbi fr FL[n tbng sd di€:n riia c6c bii dii ctr{m theo thr tg Lan, nhigt dQ khoirrg i50 *25c; urii clung lroi tti irdi l,ddn bni 6 tn 86t, 7$7, ltil,504,381,,12J, trotrg s*Lug, rn6.t rrrB li thi6u l-r(rng trii-1i:)n di$l ain c6 tl:3 rii: r:hc bii ii? tid ddn khfl (dua l.rt)n (m6i ng*r. -.- dem rhi :nd t6; Lhoing tr'r irh 4, n olr,. iidt qllii ci:a roAn cuQc thi) ld bAi 1,3,4,5, ti,3: Nhn fll e i'ra cAc' dolr, h
a n c{u trrirc d*i s6r,. (criu tnic nbdm) cria v{n Chrirrg ta hoar nghe nh ii:rinli ti,:h cira c{t: ban dA" Bei 4 lrr"y ciodn 1a ilrrnr. 25,diFrn thr\j rrhrrrrg rlrr ihi vS ehr'rr: c{ic }trn cd g{ng hg.:1$p gi6i :;ori, chi ihua 4 dodrr. Ccrr; br'ri 3 thi c6 ti{t ca 14 lr:ru ddi riao rlric" elb sriu n.i.,,'pirge vqr tri:. t *oAt il:it t'ii' l ilii.inr r16 lin (tro;rg do ? doirrr cd qu6,'{u. tii?:nr triru ilririg irilli 1A ,\ii 3+ iii'6m, i-i&* Xd 21 Snu rll3 Ii 6 ,ii lirri i'r"ru 1..; li'. l+:n qr,o' ri r{ifi11, r 'i r)rrg c,',rrg r i lr;r 'li}i 3, 3. 1 drrin i:t rlrrr.,'r' rr.{rrr rr&"}; r:**: !:nn h5i' cd g.{n1:11i&i crie }r i ti2 ,lrE,r,, I hul 'j tf r itrr Iti:Jt, (,lt'ii: ,'o niirr,rr iilri, ;:ri:', Ctrf:ng t&i s& gioi ti:ipl i:'r: glii! ,:rin hii r:;,i, J.-: I!i/ r:t. .ljE;rrl. i';rJ. ri, tl,b ry,ri I r.loirr rlr.r,,ol r ilSnr !lri s:rlr : '!a chirir iihji Li Iri :"8 c1?.i st}, sfi lrac. 'tlhri r&rrg .t j3;ii l. (rrrrrli: Ar:ir, ri€ , r ,1,i , : (l i',-r lirirt i!ui'u..j ci r 55i l*]' Li;iu (16 d,'(.:rp rIl:n li'uun{ fi; ri:.i trr'rn tSm f; tl'i'r* a;iui.r ',{"li *::;: rri0i irl' gier: tdi co hai pliErr t& 0.tr 1l rn$i irhLi nidrrr lien r;uan tri,:l{l/f ve tif p lri3'ln lui ba t'a.iri) ,r(i:i tti' ehrng rlttr niriEn ilrnl' tir;rr; trr.rrl hot: (rn,iy trnh, titi nrinll r*.*g" nFrl tn' 5i&c ,'1BC{t i',3i !i.'i;':. tt! {r{i : h t{;,.. .r .4!) .+ BC ='. -1 I-i. B:'r iii Lr ;;:,,1 irrri r','r 1j,'tt rnu ildn { giii iich D 1. I
i.r .t -: --; ; ' .l 'i' .'i-"iii ffifurs'# ,tf -t' rltr E .:i +;,* i] {J # I 6i*i :rkdr I erAI TIr[Io.Nfl Lr;p 1O - P,h*:rg it6 - Hdr,.ith6i chuy0n ro&n 04.r rbnc sd *)rfiftr flACI l. Ogi hgc TAng hop lle NOi. Ldp 1I * )'gf) lloitig ilrrr,y, l(htri r:iiu..,6n rohn, Iqhdt: Kli6i chu-r-0n to{*, $*i it,,:ri:,TSng hop Dlii hgc Srr lrhaiir l{i l,ldi i. l{t N6i. Ifhi : Khdi c.ilrr.v€n to&.rr, l):ri liri.. Su plram HA Noi 1. I.r;.p tr{h -* Pl;qrri rJr:dr Ou,r*g, Hhdi chuy*n to6n, Ba: 'l'rrr
I"dnr {*ibw s&c* t$6w *at*n fl*pr pl:i& a&,4reg &w.mrywffi ffiffiffi'qrffiffi pffi&ffi tffi'&'ffie ffifr&tr w& egg#ffi6 ffififfiffi t& eA# ffi.eNG Tpq t6e It Y' I tr*Af ' 5# i{$uYfii+d vL*- xi.t: TROir; ch:r.;:,;r llirir {o&* ph& lirSng trr.rng li:ii re,6* ?: ili:ri;r6: uir:h rl:gg rliiu i.!il t'&n ' hQ. r'hir,:r ia lhtri,ng g,r1, nhfr';rg bli loin i i. il ii ;1' ' r"E e hrl'ng niilitr *lac d&ng thrc ed rli6u liif n. rgJ igl? ii tgil ig(. -l- lg{. igA -- 1 it'l Xudi prhfit,tu rniit loai r&ng' Lu$c btn r:lhir t;r- {.i L ii -r t: = r:,2 1 1,.: efln phii kiBrn tra tixh rtirng diu cfia ahiing d$.ng 1,, i thrire ktr&e. Cdi: 1:!rtrorig 1--hdp lo b&n db'gi&i {.,1, B, t; =F rl2 * nr r:hirng thutirig dua vAc c6e hlng tling thire 6fing ciai (2i(9 tg8 itge -F tgL',i =' 1 - ig"'t i€a (2') nhi,r, eflc dlBr:g nhfit thri'e, c6c phdp biCn d&i r) "'{,Yu tg,4 tg(l: I thi iuir irgA tgC; = fr' dai s5 lrrong duong (nhu pltfln tieh ra tlirlt s6" ii lii 'ilo dlt ttrta sd cirung, mir h64...)' Trong nhili:.g tsc = 1 ndr:r eudl b{e phb thOng lrung'hp*, iiatt khi de it tga f :gc -- tt , khor:s xa-;- ra thhltr th4o cic r:6rrg thire l:ridr: dbi irrqng gi{*, \:j-, i'))€i ilLll^{ thi ehring ta l+i drsc lrar:g,bi thdm nrQi phrror;g phnp higu Ir;c irir.rrt, i16 IA phuong plt*p lugng i1 ,r*1gc:1-,|.uril*l;)=o ** [i tc.Ii :[l lrir i; =iL gi{e. Firucrng plr{p nAy r5t tla r}1ing, trry trhiSn. viQe sri drlng lhirng kh0ug 1:iriri lt d3. Crii iiltc t.r -- i:: i 'i -: tii: & dllv h viQe eirgl;;hrrorq 6n x& l-v. '['r-or,g rrri,: ls : ni 'hAi lodr: cr.r lhE' tiry iirei: trlng darr thi ?i6rrg' :';l':tnfi e&n 1;h&i biit. tlrol ni0i krar i:i€iu diii tir!i'!r lroi, + I 't-C ,(|ihfrn11 h&i ij{:) tr} hai loai bi6n ribi - trir iert. ia+ a * c : rl2 r ln-r,li,t:. ttrur'rlg dro. c':crrt i*. lrrtsi i{r{i$ khd)' 'l'rt'n t;A' Lrj Nt-Iu tg.i irc f I thi trliEg bAo *.'I'i;*r ltttr: v.il 'iubi tr;: r, eiie bq.ri dI i2) f) rgll (rg.-l + i3r-')i(1 - iga tgc) - 1 'lJri"t sitc v*i nlrldrr' cd niiiSu dip iirn .que':l la e tgBtg{A * l,'i : 1'+ i'os(a I B -r C) = o b&i tods r;1 r:h*tg Eli vidt vE tic {rng rlting < \'& Ou (1rl 2 'k - r{ -1 -i-tg 6''t tq (; - tg,{ IgB tgc {i) l* .{ .+ B t C: il; : ,{,.9.C. }-r!:. . tc )< !'0 /)+C -.1 ! ,*--*-** {u S r {3) Giii. Trr r'6 2 {lr *+tg,t + iirl, * tgc tl - Ig4. tga) = {) u tat ' fr :: {3i i,'ost A'r tt) * cos{lll.os{.i*'8}+r'r-rsCl=6
. -{+B+C -*,.rJS .{+It-(.' Y --.-^-* .'t,: sz *-+3 ,{+c-a * {os B+c-.4 *--.*. - *.--- I '* 3.c? 2_ = .i dpcnr. L3*.ug phrr,ing ptih r; t,rrrng trr. ta dE d*rrg ,:iiLrlrg = 1l--:-'- 1, i#:--{u , ;- 3.vz I * saz -' ilzt ^i 1. i8) minlr r,Ac lr+ tlruc crr b.ln setr dA) l tt6i roan 4: Chr'rng r,:rinh r&nq SiFu kign'' eSn Giii: dirt c : tgi, y = tgB. : =, l{l{' thi 17) fi tg:r + rgB + igf"r 'ttreo lrri r i tl(].tlE rg-{ { tgl} * ig,C -* tg.{ tg8 rgc = = tg.{ tglt rgc j tri{n I ihi d +,'J + C = /jzr var' 3.1 -{s,BI + ;ie =: l:l/rn i: = I -. tq.t ryrS * i;.8 ti.C tgC'tgii Ctng tl.*c; l:di toirn i lhi €; li:1 * q+C =?i1,4+1rr; J. B,.C+riiZ+ftn. ti1ii,{ * tg3B .t tg3C.= tjq.: r rg.:tr rg,la' Bii rorin S: Chr?ng nrirrh rEng drBrr .ki$n c*u S&'dung L:6i;g tLrlic khLrr tlilrr g6,.nh*n bs !a tir dir ,ii, dugc {8r I+is.1 I*rc8 -r t+ruc , Yfdu 4: Cho I - tgd 1 *.rgB ,1 "- tgC = ' s+ g * t* rfz = i' * sll * g, - .:ri; s,g3.f-g (9) --+- = 11tg-i-".t *leB I*tgc_ Chr?n4 ;ni,,lr rfing 1"-ted 1-tt8 _1-tgC l*r9 l-lJZ ln d+ B+C:rl4*rtri; rt,B,C+nl2+nz". r-.--.---r **lE * rm. l l-a2 t-yz U l.*rZ Bdi toin 6: Chrlus rnirrh rlns diEu ki6n cEn = . ----i- r l0) vir rlii rlB sin2,4 f-si,rza + ,,i2{a + B) = 2 ld 4 ll 3-. cos:l eosB cos {;l * B) - €. GiSi : tiJ I r' .= tr . ,. = tgB, : = tg{,1 thi ,1 3. Ap dqruel (g;.+C+B+C=nz'.i * /sr (liai io:1n d). Vi d.n 7z eho ob=t - r, ,bc.$ -1 ; rs * -I" Dn v*v: Cllt?16 m!nh r.dn;: 3,{ + ?A * 2C =. n.'2 * ?hr u-b , lt*r , i' o 'l'heo bii todn I thi l*ah 1*&r: L* ca ettg2L * cotg?11 + eotg21-. =.co!g2L cotg?E i:o?g?C c"-b 6 *c c*0 t1) Iihai tlibn c0ng tl*uc 96c nhAa d0i ta 11uoc.(101. lfab 1*&c 1*ca vt dit 5: Cho * y, +:2 * jru: = l: .r;? Glei: DH.t 0 : tgd; b = tg g; e * tg{. v:\ qt dt;ng clo,rr-'-{},-}0 (l-il edngthnc (tgr - tlt]')l(t +rgA tgs] ra 16 tb0 vijr ({) du0i dqrag : = tg(A- b) Cirirng minh r.ing.: tg{d * e) + rS({ -C) + rg(C - s).= t*ogz =,r'i/ (t * gzl(r;e*)* =iJ .l tB{B*C) rgt6 A) * uV FFNTIS+ -fn:7.,fi (1 2) 'Oing rirur. -B)n:1y crung vi (d* tl + {B - C) + ciai: 'l$ ('lt) *,ry.'r.i tl(,r:{ I ; ri
A+B+C ;-;--- YS-v ccrs "' "' "- =xl2 . B{y gi& mdi e6r ban h$y v0n dr.rng phrrong :* - 0 ha.v A+B+C ?2 ph6p dd trinh bh-y elE tqr giii mdtrs6 biri io6n onu dij: =,)cos(a+a*c)= - 1 . +-+ *. co5.4 eos8 eos/_ -- cos.,l sinB sin(,' l) Cho.I'1, trg, .rS ti c6c nghiim cria phrr,,rrg --cosBsiui.isinf,'-- iosCsirr.{ sinB .' -. I llinh 'I'ri dtl , t* nlrin rlur.r{: { 121. . o3 + a.,'2 * c -l b:0. C),,rng l;'i'" i" no Vt d4 6: Cho rg * g: * --r: I {f:) * i/rt' - tlr:z'; i {rz * I/re) (rs - tlq}+ {cr i.t'z Cht'rng minti ring + {,'s l/*s) (rr * I/r1'} : ,{ r'+yt:-3r'9:= s) ciro : ,\u2 + zz) + g{rz *,r*t} + z{*2 + gz) {r+) -l-.ri*a', 1*r 9 ---:---- + -:* 1 ._.;+ ry= Giii: fllt r = tgl. y = tgB. z-lgC ta duoc r5 L*; (t3t** A + B -,,-,: : ;,t2 4 l;;
_-:---r/ ___Et!l I 11 DiEu nAy_ kirSng thB ed cliq*, vi X960! + I Zi Bdi 411,e.2. Giii uri biQn tu|n hg lrong khi l-rv, S. NI:Lr ili6, n6u :t) c6 nghi{m nguVtn thi ta pir&i c6 s ) 0, g>0, ,:16. fif,i -/.--._.l_- do t* (I) ta duqe :- : .+ Vor*VW -d ,rX . -v (3) h I rr {3) su-! ra 5Y > t.. kf o tiicr U }.,. Ltit giitt {,,fr'a tec ban fritn vdn.!}ilng;.Bdi _,Y6i S),2 ve y e ii&n, lric g = ?k, k> I r,a c6 M_anh .Tudn, €gyUA4 Nug -'t'ar}n, Nguy€n _Iloang 2r : 5r * 1 = lbk**1 : 4{25k-l+25x*a+...*ti. .I#rda (Qu.y Nhr:n),.tC- r,dn Dqc.*phon gju6c{hani, Suy. ra 2" : B *- ,ltot rliErr ktrong- tlf rly ra Le Qucttg g!1,,.t \-il,arrli ll6:il, ) gugin Hdt l,inh, t1o Ngug\n Quatts 'iltii. ti gutj\n ffu ffidu (lle Nqii 11,;tr: - 1- vii nhrFir barr lii;i;r]. V6i g-)2 vA i: 16, tir.,: li= Zk + i, A.) I ra t* .t')':5Y -lo>l2E*+r)>l i4\ ' 11 uat r":rre,' ; 1. = 5'u, i -';'= ; zaf,'* t: :b(2gk t)+4 -- s.l/ J- { (rf ta mOt srl ngu-vd n d uur,g). Stty 1s 2*;8,* .r ( 3 n,Ju rir,-:e,, v6i (4). g:1. fih, d,,. trr (:lj ta Nhrr v&.r, chi co rhb drrue r :2. \'4J r:2. U : l, i: 0 ii rrgh:gr.:r (t) tluv nhiit c&a pl:rrcrng trinh dii cho. V,r+ VF-=-, !{hQn s:€t: Cic ban Tr&n An yiet (12 CT phrn i \6u b ( o hoflc a ({r thi h6 dd cho r.0 IJ6i Ch6u -" sshe tinh) va r'.an guir, trt c-r' prlnn nghiQm. Chiu.'I'r:inir Qu&ng Narn Di \Eng) ctng c6 2i x6t b> o, a)7 0. Kiri -d6 t,, i giii r6l N.X,&t 1r) +=+ I ( ld--a tsi i 371 ( V7-+ Y7 = b 11. cr,gr,.s tt,ii:1 r itti:t :,ii pharoag d,-ir;it 'a) tibi n ,F ^t-:T- , V b" lrq dtr r ho r.6 n3bigm. .,.4+.,.'+&s,3*cr* l _U c'i ryhi\m ihi az * (a - z)? ) f . i b) Ntri ,, ='/ l3 tr,i {r} * VP + V.r- =e (2) Su;r ra {,bi qi{:i (crie. e6c :,irii : r} , \,d.t: L}iit,g, i.u 3r.- f*rr 'I*ri::ie Tr:ifu1 B;n7 ...,ghTc, *)t1 t;g Drry f',;r.*1jr, Y;r*-(0,.V ((lrl-r{VaB I '31- Dlrong Mintr Phtrang, Huni S,tr,, p;tti'i, euiinit"t:,tr, Cam ,Duy - .ifg&i,r, Bii Bric j kt..,.) I Vs*.3 (trice {s}}.1 ++ 8rro. orol) + 3lr" { r. Iorod.,l } = U n5u ding tlr re khr)ng ru.r rr,. o(r * 2a cosc) {+2+;?arcosAJco2.u*ZA - 3r,2): g NhQru.uet: Ndu {'rn rlr.rr.:g bat tling. tlrric (J)
-,E7l_F---. : -'\'JrAn rdf : h{rii s5 b+:r sr! dr.rng cdng thri'e Khi {ir: : * Cae dtiu,5 dt sg;li riirl pir&i hii:p'.ibi rucm r.\. SMnFN : Sssr' * sgelt *:-',qrsx:.LsluEx -- : l'B t j' D.B i(. = ""'*-* .it,r;8 $uRr ,f li2"1,lB'F (:) . siru ,'[*r,' -' 11,.i 611{3" .'t'im !dt cri cdi' bi, rl,; lhri i: { tita ) rri_v ra i.:l:i ;,.,.i rp ".'J't. Iirc dd ddtr ( * .mdl lifn sS ilrigr:) f , g, h i/rtio rltdn ,r?!i=3:rl2-il {- ,,r ,'i I gf r i i-, /i{.r i -' r'2 . t) clfu (; )) xiV rli kbi ,-,: :, --- t''i liic C5 \, !t llrIr- *r' :q\ \lilN =. (i -1- r:l:. \ f ., ';"i el i ;i:ri ," ,;{.t) * lii:rz} = .7 ? (:,, t t>, = s--I-ll* \yzcz * BDT|'a S -l- r ar sr-) 25 >,!'+ 112 at:.il!";ina = ii{ b) Sl + t$ ^ft1'ii-r,tl uli lrii ctl.c d*trrtg chda tita ift 1c ld g;',c gi&n 2 tl'rrins liico Af r a iil) ci;; gide ban dhu 8&rrg nhctu ud utt$ng gdc rrrli Ji&au. trl gi6c \ru.-Ol. d6u bfrng r'i,t' iu khi vA chi kl'i l-ot qtar : . ACL BD r) s:n cl: 1. haI 4t: = IID v0 l --:r ? , K1i iri$u itri; N, P. Q. iirn irrr'i ld iurrr eria c{9 Ndi c6rh ktrfte , Sr: 25 klii vl'r:hi khi cic drroil{ Iiinh vu0ng. ti\rn$l vE plrla ngoAi lil gi{c .4.BCl) ch6o cfia tu gitq rtJi{lli i;d.eg ni'au td vu0n4 g0e tr'6s sdc; citntr AfJr BC, l:{),. EX' K}* hi$u truug r,6i nhau (i!prr1 -: ditm efr:r c&d cgnhi AB, ljil, CB' $A, l$n itrqt li Nftdn e'di : C[e bg.u ?rBlr ,{-ru yif i '18 CI' Firan BOi Cb&u. Ngh$ 'Iinb) '.'Pa' i:lgu11Tit Qnaag 7'lttii I Si * Suspo * SmrcFI * Svssp *,lNrep * SrcHQ 'i' tllCT, DIISP HI Niri li +6 li;i 6i{ti tuong e16i t6i' i b'pHn,u il) in sE ti;r'r SntnrN. l*ii q :':: 116L" l'I.li. ill . 10
Bei 8,142 Triti nifit (!$ nt S,hdng dudttg thfrw rong sotr$ z:r.i nittiijng lfrrirag crit tugt:t d0i m6,t - d"8i *.dr.fgrs,ai+3$c*o'-+**o'd.+oop :,u'2 Nhrrnq ctit ;tltott, 8i-ff1 trtt,q .,t;d.rlq cd 3 drrdng thdrtq naa dbng quLj. lti;t ilzt1t p,'rdng itvqt cltta th,,inft n,dq ,;,1',;i t)'.1?+ssz =.: ){ir,z i:2112 ph,i tt'.' . - i * fi'rr i.2lii =,{& {ri * n') + ?fi,2 {}i 1,&i .qirii. CS d;nh m {u&tg thflng sor:g eong, Ntld .,{rrl + ,ir i ss" irlr.rnr: .iR iJ{ * /t') trr.li! hiQu *6 l*, sd pirirri bi cliir r.a trl nriii (hi nX ,iri i,l;6rrg ;;irr.r t i,': 'c J:) r l) phi:rg tng r,6i n *5t luy6in. Ta rd uo ==-m * I !) 'i'rrx*ng iroll ili?::r iir)1; rirc ?" tr.*ng r.6i nrS,t vlr rft = 2m r |. Gi& sfr ta tjfl c6 n e6.t tuydn iiirrh.ll uur, .?t-r,,ria ,tr,r gii,.. Khi d6 dOi vCri cilc i;i.i n.,l-rl pirr'rlg ri.i riugc ehia ihirnh ar1 ptrr\n. ,\6t rtinlrAs, k * i ta tl:i ,,r'r ( i l. RiFnp ,tai v rii ilin6 .i; c{t trr_v;-1n thri'n * I. Drtlrng thing uAv c6 l'6i ln c6- m * a,drrolg thring dn rho .rn * n giao dibm, cAc giar-r i'tiErn ttr i:iria drrolg ttiing rriy ihinh -*rBf + ,{i +sc* B\+,t:t =,\t.;r.,,;21:"!+9e2: m * * * I ph[n iir.cn;1 116 Lu'j rri'ra clrrring th&ng =o'A?+ssz*-H'3=(rrt *n';2 fr'r==aE{/?-ft'} v* rn * n - 1 tl.oan th.{ngi. Vi nri-ri ttoaa hav i{ghia ln tldi i..ryi 31 ra c0n6 co {l). ridc tliing nira d.ron;1 tlring lrri clria diing I trong s6 al lhr'rr. droc ehtrr g rninir itoin tor)n. pirEn n.r{.l piriqrg iim d0i *}n v,-ri cs.t tuy€in tin: ' n * I s6 ph0n mlr ph$ng rlroe tfing inern te m*n*1. Do d6 &.*1-t=os*rn*n F1. (1) Vi6t lai {l} iln lrrryi v6i n -.0, l, 2,.,. n-l - rBi cOng l*i, db f r&ng {}o * }ii * 1, ttr ll:u tltro-c 'Cong thur' -*--l--*- 00' 1 A,=T L- 6{5 s lrri * m * 1 * r:{n*1)1.2. D.B.Ii 'Aii 9/t4g..Cho ftrii udrrrl frdo(O,R) ud J'n'J tlip:tuc iraag udi. nhau. Girt.&i ft > R'.'l* c4c dinh ctta da gitic dbu AtA2... .dt9s4 nili tt6,p tro*rt uiittq lt'in 0 tc k6 cdc ti?p tLtgtnA;B; tl ( i ( lfftdn cil ; I{hitl-r haa kh0ng d} $ truorg h
) \ Bb bli tlr-rng irOl ii*o sd 14? cliua "tliy d3' Ida ABy: OBlsin (a.t-i-a.e*as),'tio {se * as} sor;r:r t}:rinh ihit xin l6i han dec) Tu
& ,f .i:,.* \ ,l' +" *-f +, b * e) +_v, E*i 6ita5. Tr$n rr:{t phln$ hor ritsrn (i '.' ,t' = lAotl, *o1J' '= lBoH, i : {;r: - 2)z "1. {y *- .,2. N1go:iri:' m. cJ nam so- .r-/1:r, 17 ,j , oC' : 2e6I-l, DoD' :2DoII. v#i mqi clirlm {:r. gi ciirr G dEu thi irllrE rl Iie bEit t)[ ITtit l1::; i'l .l (!iinF tl] ti,' 'q *th tt: tr ig 'l ,, -r- i ,. ),; 'i ') #
'l . {: 'rr,,13 I L{p b&ng bitln t}:ien = i-3(l - sin22rll= l* lri,r?!, : 4l-Il j { nen ta vi6ir 'phucrng trisrh dut?! d4ng I - 3/,1 . sin22r. -+ sin2r -: nrrg2r *'nt . cos?:t- , ' ,._ .cos2.r. 'aqt \'*-y cln di$u kiea eos?r $ 0 v A, vl - dosZl."- * sint-lr = 1, n6n suv rr siu2.re * * t. Vdi diEu kign fiy ta c6 ' I f- -. 9 osin?l.u = nr sinlr ' Y+,v: d,b ir +1) (: *3i {*2 + *r + 6}> rnv { 4 stt j sinlr * l- t ta cd i r'& chir .cin phii cd m( - 2. sirt2J, = I -" t2+mt*i =o ?fui drr 3.jDO ihi klr6i A n:irn 1(t76). I/'' lri rtrio r:ito a thi Jrdat sd Phuong tlinh lib$ng e6 tghi{nr 1 * 0 V m Chia J r.i' i:ho I vi chu.v&n vd la rl*qe : lt: i;F,t .',"q;: ;sinrcosr' .1r ^._ 1li tz ... t.rr q l :* ..- I= y I *" d*1tt .tdt dlmtrf o. i ,all = Girii: DB y ric dinl, V r ttri sir,or + eo36:l'-*' tr: --*:,:-*-{cl -*. t DB ph,rong tri:rh r:6 ughi$m' thi rltr&ng th&'ng * csirrr eosr)> 0 V r, 1'Leo ihi d9 i, ta c6 niirr ng'tng g = Il p}iri e 5t db th! : a- ' sindr *.on6.r=1'1 ' .ri:,?2.r" ,J = -:-- * .-:* r ir&n ilo4n *1 oVt #r 4. 7 'l .: a,ia Zmar == Ztl) = * tlz, vh\i ,).' * .: .Q2 1f2 Vl I ;4 -i- I. aeu dE phuong trinh c6 nghr&nr I(hi l{r
r' : TEr co i2 :, tg?.- r,- ,ri:rtgg;r-;2,n8n phirrrng tr-inh l) r*et rs ;= ?','lihi:rtr$-O { r d {#aiva;r; = 2I'** +3r+ll8:v.,' f6.i. rn, 1- 1 = 0 Bt phrrrng lrirrh it,lY c,5 'ri.: :|i+:n irl'&i c6 !) \'6i rn=5/.iur r:6 t2" -l . t -F 1:sU.+ {imin(m(r;*n*. lla !6 y'=s'{f * 3.= 0 F+ f : -"3/4.'[,4p * 5lz 4- 312 .*i d1 = - I/: (loBi/, n&ng Dien I hlen =tsf: tZ = - 2, do vAy tgr * *.1., x - .* ril4 + nx, n aguv6n. Sgllr0rn niy thleli hop, l 2) Yi l=0, ta c6 1 = 0 v0 l-i:. v{.y t:0 kh0ng phii iir rrghi0m cria phrrr:lg trinh'fm, {lhia? ; vi1 chotf0tadugc il?= - {l?+ l\.'i : *I . l/t =1y ,, V{-r. dt pilut}lg ,.i616' c6 ngh{bt;l'o , ':i'n pi'&i ' , I-ib phirong trir:ir cd ngniQur tiri iirrong ttr&ng 'nfi.m I/8
l- --.T-.-_-.-. 'fheo hr,ic ! ta sry rl sd dd nghi plrli chftrr 2. .....r.i..-r :i .,.r,. .,.-,.... :;:i:i".:a.., {:{'ifi ptr}"i "r*c Liinil l'! tri.t-'ho c6u ch'i sd tlri. .'ir'jr.,.r'r:.-- : .' ii,',:.._:r , .- ,.:.::. sd lA s6 .iil$r: dting (nh$.n kdt qu6 1*) tt Afr sri-!"r';i rE^ng 6 li,h{ing ti'rB rlrinq 6' hirig cl;ui' citr;lj nlru harr..J cj,,n , , 'i'1,l-i, hrr rlr 4 si [i r ilni: k!r''rr,* , - iltB$ efip ?fi$ fi$gst s{lAri $& iji: I:;e,.:l;,1",i J;:;ri ,iu'l',,1i;,"01;,:;'r;r lhE tlung i! irirnc lgiri'r' 'i'iier; I J'";ti': 2 ' A.bu6r ij \t'!', ' \tig; i . tt\v/ lt i' i;eoS tiri 2lihiir:g !ri.3du;rg'r ir:;,-.1, ir'[n ,,i d*rro',, iriit',r,a ir.i,tr l:rro;rr irlng chtr,.. Y4.r, 2 plrii ,lrrng ri hrirr*,1 ."i1 1,i. llu r'16, ,i buu. I 'l''rrr,ic tr6t de i ri"ng sd c5.rr iirn ph6i cirua 0 cLft s6 2 lth0ng ptili la chdc ding, il-lEu ni,v 't'hi.t viv, ndu rid l{hsng chri'a 6 thl iheo broc 2 k6o theorG dirng & hing chgc sE ll clric dtng' - Sd dl nghi c6 d4ng. ??62. Theo budc 4 ihi I ' iiet qrii 3...) ta .uy r;o- 16 phli chita 3,2 ..'i 5, Giti"r nil' r'nAu tt,u5n voi llrrry* 3 {sd tlrem do kh3ng thQ dfrng & hArg rghil.','$.y. chi con e:,} ktri cilng chuu 3,'j vir 5 nhtrrrg kii ri.,i chl c6 l- i. n[ng 5I6'J' DAy lA sfj cltr ti:n' {r i.i x it. Budc tldin do q1!a g,,Lg "'g'*-.{i{ K? E S I 2466 :^ 'f-.t d:1v i'l,r:;,': t:1 r \hir: ir1 ,ci.r;; v6i l-ran li 11 2 J*Ct) v c.rr:g i. r p ;r".i l* l1;rrng. iirrrl'ng cti ch! g6j 3 I arJir, t r) i t,:r: Ii, i:,i.,, ',;tli '. :ii :i,r it i)dn3 vA chi 4 1653 di: t0n 1a ):i:lir. liirin rnuen l;rt:t iribi r:*a tung .,ngrrr;r ndn iri.ri iiudng. .\'htilt: llu(r;rJ ui. ir: ii .,':u irir' rtrlr.. r'lri irong gia Einh bqn niiu iil.r nlr.".' Ndu s$ {5 nghi e ir*a 3 thi theo birire ? ehtr iludng': Tubr ciin ciri ThiL lirg tb;ig c*c s6 3 irhaog thO dung & hdng hgbin, vi kdt qu6 cirl} s6 n[rir nirrh cfra .sinh niinh. con iuiii eiia a'nh kh6n"g r:d; clSu * r:io. 'Iirong t*, theei brrilc 3 DAng b&dg fbng 'ngi3 r i thi'ng ririh r:ria vii tiuric 4 ia th[;,' 3 r:h0itr,l iirB rlirng ti hrlug n:iuh, hsn n*a, l{.p pirur-r;rg hif Lr hai s6 d6 ingirv tilm {lo{e hirrg rlon lI' iLi i'*n rrrit' liti&. tl[ng ,sinh vil th*ng si*h) b*irg tiibi r:&a chi diu ;ninh. 1A 3 rlirrg dr h;ing chqc '*A sd r11 ugl;i r i, darig ?.'l 3:i. !'i1qri-r pirin.irEn, si'i d* rrghi pl*i irlri';i $. Hhur:g 6 ,!hdn: l{hrrrg ininh.riirri ;lr,,rpg ciiinh ^;ic ng*y' iirdrng-r:i.ni sidh e::i* ]:ar. iilr0ng tirb rlrrag i; i;lll5'i'lr.'ri n;! {iLt* b*,rr:. 2) .^-/,rci,rrg; 'f i:i: ihi ltan thii ilArg lrich rliil'l l'r'r Ir'rn$ '.LAtrl 1i!i,r'r, L,r'':': 4). ili.i ulti . cir r l;8. , 6 Nf;.r,iqinir cfia rri*ir l:i ruitt con rd l]: thu c6 brjn tt4ng 6? i,'/, '{'heo },,uiic i lhi 6 Lh6ng phii la s6 ch&,: tlrilg do de1 *iri cl)n c6 th0 r:*i 4 i:r stl chir si): H;ni: phr,rotre eil:i thng hai ch* s,i dSu' t-,lng thlg cir irinh pi:tro i .1 cui ltrii cirir sd cudi. r:hi.c dring, r,iv -qd dn ngiri, cd d:rt:rr ii ,i ll'? Sr: sr':uh r,cri btrivc 1 t:I tilril s,i rll n:qhi i;lrr:ng uiirilr 2' Ciro hnn bidi lh&m mi:t dii;rr iii anh .Dtng vi ,i. *.,-, Cili \itilll r -. ltiOl iiiirlg l: Khi d6 the,.r,brruc 2 sur r i sii rI:i nglri 1ri,ii chr'ru 5 C+l,c ban c6 ihb r:;i r:iro i,ihdii bi6r aghy tiiirng, r.* e6 rl:ing 0, 4 il a, []ii:u q]"v rnsu 4hLri.n I'ivi trrrire 3 (s,5 ttrXrn tkr tung .hii* 4,..i.5 nhirrig nir:$ siuh uii* llrrrrng "r'i tubi ril,c :,ah ehi trr.ing * gia dinn iitiring iuoc khdng? \ '- k6t qufr chi b6o 2*). Vg-v sd r1.t nghi ktrdng chria 3. Pit r,.oiic f ilas* $:iu./rr&y, d* e$e l-,+.ii l*;ra b6t ki mgt i'6ng tic :r*o, n6u o&e b?a r:tda dEt $hr}ilg thArrh rith tqoug cu{rng vi oOng tfe eaia minh thi & il6o cae b4n efing ehn phfli c6 rth.&rg ki6n rh&e' ia6a &9o ' ' . : KAl,lliIN i* ll,*frfl 669s" T.]i ei ngiri$1; in ?S il*rp nh -ll& \+i. "+g* in i?$l&&. 14 qt*x6 r-& Esl, iiru qir;Eu ihsilg !:l 1$sF tihli ee' t&e{t.s #i& i So,? *$ag ,- a. :; ..l',.1 t'