Toán học và tuổi trẻ Số 114 (3/1980)

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Toán học và tuổi trẻ Số 114 (3/1980) trình bày về chuyện lạ mà quen; bài toán trên các mạng lưới; tiêu chuẩn chia hết cho 7; toán học và đời sống; làm bất đẳng thức Cô-si chặt thêm. Tài liệu phục vụ cho các bạn yêu thích lĩnh vực Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 114 (3/1980)

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l\- ta. cirl ]' r.iir{ ah ha -: u yi tr.bns t.i{p rlir ta tlttre hiir: idnh i,.,, ) ddi r+i i .i,ai1i 01,,. phip cbi6rr vu0ng gde =rrrdng hrr 6r:.lrrI + \r: d t veclrr Il vi d*c;c ri-ee tg I-' c6 rga d th&ng vu$ng g$e v&i ulrau. tl lh.rrr: hi*n p&*p ghil tatl 9,r;# ehj$u-nio trri.rc oArS dirqt, kt,t quii,liri Arrrrc 1 1"cct\, u .- )ihrrng khoi,y phii hao gi# cirng r:rv. e * ths ra], re r:^i, : ri nlrung tro -i::i. i:i:*r,! trurr u$r.r ir lir 1*xr!"r :.e}i{q r.g6*9lO* 1"*o,n,} ,oO,,g 'l'itay;r-: ri rr'iriug ri,. pir rust dur,rng tlirrg D l*rc r,;i truc ho*nh gire c tt,i d f5). tr,lr:ge ,r, In }*ah ; . ?o.ru,trri* r:r&t ehi€u rong song I h.,cr 1ih u.rrig r1:r -- ji = ,.i.i.as - + l"tr.i. 1- da) bllJ ,r l} drrrlng thii;g D'ien, v,ii'trr," ho*nir e '= {r!0 + iri, }..r, r \ha' "l rlb,:l:g- A"ic ii (!. 1) thi r"6 ring it ul:::A ri fi:ld rn6r !' = t (nt. I ht1, - r/'{r,-r a ;/.rr,) yeels lim tr*n dtur:rng th.,iug D n6r.r ,ztf-fr U x -, cd'):,: +. i6r' .i- ,lrl.' rLi. vail,r ri \irrrng aei r:hung bu,.l I, vI aia nCi . {1c' t ' t y.]i.p (h. chrtn€. klrdr;g vs$;1# gac 1r, tr.ji trBOr-,S t^ \ut hgp F -' a. = izk -r I t?tl:n"' (,i ,,:, ) (::) .Diy.giri iill r!S rhly 'rl'rig, ,viii r{c ldnb tn,}i iri_r csl tbF riv r.r r.r/, _ r, tSi r., ,; r!. l, *- 0. yi.Cu .: ,*lI 1',: I ri , . t,-'I I:' *'l:'l\ I, r :r t,.l l'l, \ - r ': l':t .l Crt vi0r tlem rrtr*a'h*i lduir a*y thea qri ielc {ii} thi thly ilCr, : ,th =1. \rr kirrr ntr- tr. 1a chlr f rir,g xg..1.r, ,:h.i r*, qi trr .= bi' uhie& Ir Linh ciei{u vtrgng guc eri* l: l*rr pii:rn giic thr'r Lri, cclu T' - ,,"o ch:rrh i; ,hi.+tr chitu vudng gtit.: ct'i* F f*o phin 4i,u rhi! Ei*h 1 nh{ t lit) r,i ng !ir nFu te rhi{rr r ur)Hfl yFr.:td r[i]rig nr{t ilu&ng t hiirg 61lo *6t nio O,:,. I i+p *t, lai utri,:u.vu6Btr grle.hiriir chiiu niy ,ud,,g'n,qi t.hiu &.ho tinh toin cu thF, b1u rr thdy rang rlrdrn3 tbiog rhir hri vur.i4rg g6e v ji A",rug '6 &iy: -9. t-t,r, th{t thi liF!, qui cu{,i cring il tl ,.nr,, .y&i -l eb{rr r*t D*L tq cri th} llnrlnhi}a vi rt.u Ln{I, rioziesp -rr,r?i.'os$ \ ,/ --.- chi vite yhi;;r tSy liri. 6rr$xg ti*i,rg -,,uing I,o. J siu'.a -- r ' .. r ini r. P) I t&t h.i {i', thrc higrr }i*a .riip lirri -phei'..rrl€,, .r:- I Ii '. i I rur',ng gSc theo th ri tu ru8nf dlo. :rilr".rinF I -"i"i-pi - cr,s':tsini I \6u b*i drrrlrrg eltl Iiru vi.?i trur,heink \ r*(r:F, I goi' l. ri 'xi': -t * ,n:,OU ,trn5 tiirtr r".;;;;,;' *or u* |7 - coi'a- lioaeoss \ { I 1 .or3* r irAlc€ie \ . \ air?eot:r' rin*'j{ / Il=l I ... \ siraetse rialt l' vi ir*r rd thF tbir trr:e ti*p liSng c*eh ap dgng ' qui i*c (Ot tit tbliy all = 0 cd1r- 0a -: U. . 1.=-i trrl ' *r -. _ i in?r.rl;a, 1 \ -- !tinfir.,]d" ,,orl* t r tlrl ring ir cley, ehi rr:gt v*a it* ilhtr vf;r de khi nlo rhi crl I L?i ,:hu.r,irr tu rJ) :ang.{t) ciin* ulu a *:1 , j. .tr d,ro+ ni ri {u & rr"cr: \pu d:-t khi pl:r'rr.trp. t hic cic b1n ,tiuu kh,',nH rrgli .l Cho 1:: rl [h Ju',r. : tap Sri lEri trdri tsi lU fh$r dEu rung sir'rn{ ih$y h* nri' rn n)i)t chtn trrii n-r#i. alr:i v.$i ri :5 0: "- (i ;) u: r:' - i 1' - - rl rrhurr g brr .: $ eEDI l4 dtr nhrtng ci-rrrr1 quett dii_v. C'ieh ehtn vi t&m obin quan ti.r.rn.E \u11 .t , e:')
4 / YA$ s*r r,#&s gn$m cAC H*lqG ttl$l PHIN g#c rgrl..,l'rr lEDat ney gioi tlii{u mqt Ya.i bhi tolrr tr&n eie rao chu kiroing c{e} tri n*t ilo ffdu tia ,Y sE b€ W mang lu,ti ph&eg v{ rn$t sS v'f,s d} e6 li*n hon inOl sJ e ), u cho tranc. quar fl{n bAl toar-i 66" \ii c.'rc tlnli'ch{t trerl ltsn giin lrin Lltn doc . e,', thb tr.r eiruirg mirrlr l:i-v. - 1 $.i. *Ac II4.N{I r-u$t 'rnhr n+t rflaxc llryn n&e c6 tbB chring rnin!: durve r&n51 lr'ong erie khoin6 c{e}r cia ti}ng cirp c
.! Thi dU S. Sga ginr dEu khnns n4i tr6p dqqre Binlr aghi* '9. ra ndi ring mgl d* gitu hha vio nr$t n:r r{ lu&i rtio. nhr r$i liii1r r/rrgc.vl'o Il$t m5ng luOi ndn n6 c - Thrrc vrr l gia sti od rr$t rnqrg ]rr&i al{. n;.ii giic ngi ti6p dirtre. rlcr rn4ng lro'i dld vtvi rnrli e )>0" r18u n$i tiip. Ta Ddi t(t ca cri c litrirng clrdt,. 'l'he t H6 riug ii n:ru rii(rf tIa gif,c hrii ti6p duqc (m(,t hldcredinhthrl 4 afra hinli l,lnh irat:ii. c;ic giaa cir:h ehlnh -5{q} ihl tr,iA h}n'nhu n0i tiCp duor. .dilrn cfir ngil giae ri&u ef,ng li riru rrtii rnir nt;rng. Tu,r' :rhi6u m0uh dS ngrr$c l1i hldng {tuog. t-itc giao di6m nir. crirr-lg I&p tlriirrh mit ug* gii;- 66u. 'f i6p tr.re qud triuh ni1- ta. ttru durrc cie ngii gi{e tlFu h6 rliin, cr} c{nk b6 hun kho&ng eleh tSi EINI{ Lf , Tam giuc dEr Jriiri n}ru nr4i'ri6p drrEc tbilu gifra ei* nitt cia mqng. Do it diEu kh')ng lili L;do mong ltroi ki c) uudng, duqc. r-hring irrinL. 1'a hd, dnt tsm gid,c dEu 16 nanh $l+l €6ch ti'rng qr-rAt cd th& ehring n:inh drt.rc bring 2 trdn lr;6i ki 6 vudng l*hrL hiuh 'r'8. r&r:g: T'hi d rl 6. ',\l9l da giie. diFu ,,, )sa.nte ln)>il kli*ag n$i ti6p iiuoe yi.$ nrqt rri4*g hy4i rae. Fhuong phip cliung mlnh le: gi&/thi6t ring tbn ';4i fia gi{c dEu n cgnh e+ dinh tai ,:*e diBm nut c[e mang va xe]" CguS da gi;ie r canh be hurr.co tllatr t*i eje uut. 'l'i€p tuc qrr; ilinh ifo. cu6i cung'ta di ddu rn$t da gife rl&u l ca-nh c6 cAnh bd hoa kho{ng c*.eh t6i thiSu gifta.cac srit r.:iin :rqilg. D6 ]t diEu khdng thE.tluqc. \hu vAy bii toan v! c6.c da giae tiFun$i ti€;: j' v*o c{c rngng'lu$i dE r{ur.rr Egiiiriu-v{t. Cr.r th.6 la : / -Ygodi tan qidt;, h.inh uttAa! uti [rtt': girit-.thi kltoug cd d* gitic d$u ndo n$t tiip drrr.tt' tatt mgt uwtt-g fuoL nda dd. lA, fi Hoa n*a mrit 6a tiic cho trutic nu dang luy t etins lih6ng tb} rdi ii€p ,jaqe z?rc radt trang ludi nao. Nh'd phep vi tV tdru i? r'A h$ sd I+ ttt nhi0n Nhrr v$y trong phirr: isti r{c tru'dng hr.rp, hhi ta:rsgid'r- lrigli thililh t*m giac ;ltEl{-'1, ;11,rg1Cl tolin rrdi ti{p m$t da 6t*e urgng lu&i d6. dr?a trong dS As rii .By ll eAr: niit e&* nlgng. son Ch "eo tt$ bnt tsan kh6ug d$n kdt rtruri phri rlinir. Yt v$y c6 trlx iiQ tO, rt':.) rti .lvva Xt li'c*c nrit i:ria t{i yaa ng.ti eirt, 16 tq uhiin trr dlua i,,r6t y&u chtr nr*ng rtri. mgi. A' a€n hii torin c'iua vE' ri$r-' tim 1'6rr hon, v&i .y6u eiiu n&y rT+j hiii eie diuh,crla .L' sao r:irc khoirng ctieh t* t.'1 ddn nrit.g.il rrhpl tla giar: fr-.rj 1:.1s nrit m$t e*.eh giin tlrinr. ?:'r dd ta {ltra vi$ : ,eue mang 116 tur-ri s.tut l* sao trh.e f iJil"i U"- : mrr! sd ngu-r:6n rriro do rnqrl oo be tr*n.g. 'Ia 1i5 Dl nh nghia;1. Da. siir drr^c goi ii r * nf s# tu' irbi*ir ;\l oro cho N
., ' 855 gic ta eht't4g'r1rrnh r;i.ng ros l?r 1l ta ii r',ii 111,.,1 ..d tu uhiln q > l. Lii"n qua4 idl trii toan'aQi ti$p rt* grac dilu trfn. m+.EC lLr*i, lronl'mur; niy rrhaqg ta # chrrng 'l hr,renly, gi* sr? col{tI ',1} :: tt1!.nt h&,_r ti. t'rlin,h r.$69 : ,, ' 'l'a t rla gi[r: rJ]u i:u i'li, rliuir .4o. .{t,,.., ,{rtl__l ,rf .nLm t.rdn drrrrug tr'6n rl,rn Ti ri co t,irr'g,-i,. ltan3 \:r';i uliung ,t -L'i.fr "), {SAl -,* Ir rt ts thl eLi${l ILng ]i t, ,t,rt|,..., {2q -- 1J jl:q. (if'oag do ,r palq) lx qS t,1 ti. = I-r.rs fi5 , irr dilnr .1k lr ' Vrii n!ru ng r/ .-- LJf l?.*jk -1. l)rt,/4 lht slna fit p:t;r7) lr sfi vfr ti. lt,r\ .-: X.:lir ..- n t nhii*g {r + ,/."?f"' + 7ti4 -l- &?r thl 1ga {rr 4r= {1'r'i\ I _\:rii \.ttll TrIf 'q) f ;1 T6 r e, t i. f i qoa (r q) irti.,r rf nlu hoinh do cua jr BB r:brirrg t'nioh m6nh dtii irtn {a e}n i1iu r:ir r'frrrg hiru ti Tui ruoi s4 tr.r rrltiln n, h*rr cosNr li 1'. {:r'..r ili qt*,nt (.,,, a l ds ;j I',1k,/.lll. th*r tr{i: n;ctir (osr vi'i iri sd nguy64 \ q/ ) (',lg/lJ = /.g (r,og#l lg bang tir'tr ii'r,sci sin f Ltiiq dp r,ta (7f rqt 1'_\j ;d $ ,r tr\ trong ,io P1 {rir li ,la lh*c blc roj hurr :i {./r o',-- rr hE s4 nguran. ' {, I =-,-o uu. \ tl I l, V$i sd tt.t ahiGn ir 1 \ii,r. ,l siJl ll.t. : sin.r.!r*.1 (ratr) ' { 't!L f,\, troilI do Q",-t 69) li 11u tlrti,. hl. .t ,.t 1,.,s iA ,{L={**r*rs!}1 "k ngulen. sr{ \ nl .{i. ";) Cl bai tinlr chel do rl.rr.,e lhuni mtnh t,arrE Ggi .S ii ur&u,sdi chung *fi,e cre pbln s5 myi n*., quy n+p thes rt. Tu e6e tiub clr{t 1 ra ? ta sur ,ris1-, khi do ra hai h{ qui.tlon giln: ' ro'- 3. Nriu oo$r le rr) h&u ti thi caenr cflng'htuu '{1 = {\ o 't}u ,iu a ) ti, cdu cdr: e6 Ei&s y* rinnr tt6ng uric {vol D ql rinne l'). (-Uk. \L - ngr:vln, A:1.?..... 2.i-1), - {. .\drt (p,
-- j-- s : Khl rlo d F56, b - ii via=67, b=54. I Cae sr5 chlrrh ph.tong e$n tim,Ii lli6 r.e 2011i). r-i &r mmx (:i . ("1481t 1156)" IlJ ,r.t(, &:$ trxrdf* Sei SiLl l: Gta.sri ,'t I), Bti, {F Ie,crir pirdn gid{, tLan{ cfru ltLttt gieie J.S{-' Uri ^1{, lr, P tftonq i?ttg lri- tac. iti"i.at gitra cia cltittttl., Cl.r'r'ttlg minh' riing tti{n lieh lam.grdc ;Vr\I bttilg 1 4 1i€n tieh FA.i 11L Ll . llirt tit :r:ery lrjcng cidg sd iaol ur.r lrrut qiit' DDI'.. ,' " lu harr). Goi l', ft'. C' lln Lo'i gitti ttua Irul'. lriu-t ii trung c1i6n: cac c+-nh BC' {::1, dB vI O ex l:r-tiil cluon.q trorr rroi ti6p larn giac,d.RC' Dq 4n=l' "1-.r l.liil: #, ,T.'.P'tr.inr {rr,.,,,g rlng igtrn B'' ',$,{.''4'. (n l' ls' c:an sd ?) cd buo nhiirt sr) /ri sd ngu,"icn td, L&i qirii., f 'a t:o trl =: l- 5 --' 7 li. sti irgrrtgn / ta,. \'oi ri .i'j tlri - {{r =-. 13ii 4- r + lk;i. li = (.j -l- l}k -l, 5 .- ;r .. i:: -iinr *,0 = tl {mod ,ii}. ,{ j tr ,,i ri' ,r1 '' I lt .,' r:gu. itr li dur \'fr'\ dal irn{ I nrai, I I' Bni PrI 1.'1 . 'T;nt hai rhith l:htr;n-g kitir se, I nh$u alttztrSu& ad hlbzbeha tiittL mdrt (lieu kren' . al * l]:= e?,.. &2 :dB- 6; =u4* b4. Lo.i ,rir?i (r:i:, .\'iltllrr l lti tlttt/Dtr lluung * 'I'r'u'r\rg cif I Xrilin Sonq", Nghi .\Lral, Nglrg 'l'inh r-i iirqit. sri f4.ir llrr.c)' . : Irhi,rri r,tal lllrlr I;,i'l,l1ril. .rir .ir :'b1b?,1.3b4. 'rn I)iitg .l di k;r hi€u diBn tich tam giic ghi 6 Di1 rfi{r:ilJ(/,{ :: {i h1lr2S3[1 : 7" chi rr,. lr lo rrl.:&1:,,-;g'- ir2:a3--fi3':a.1'*!.9=1. IiIi tlr; ta cir lt B' p: -UB'.PB'.-cin&'12. {t{r-,c --b'...i.) lir .uB' = il{,; 2, 1t6'= r1pi2..6''-B, nen 1a 11 / ,\ rrF'p = ilrr ..lI: sirr816 ulfirdzu4 l,lb.:blba',!' -.tr' =' 1111.r "- }j D} (,,{ B *AF Jsi n.8'l8 .-* {8{_1 fr.al . L,t ^lrll n I /, 1 - 111,'{' Il.Irrl.1'. ;- S{....18-i-fi t}.Et; -.- Ri:.BF* Bn. d 8.sirBl8 Vi t1 r'n lltl ln h:ri sii rtgi11'61n Io r-a vi ' ..i( ( r., 1Br .l-( ' '6.i1l_.--\' , B( l:-., {314,,* r/, c -1- &
. , :r,ii:-.*.,i :--_ -l---_- 3VFB (2) = 3tg6,o"l2 ( ftjr * rgE n r-rc).'2, Nh$n 11) i.oi (2) vd voi ve! 'ta aluoc tric la . - ./' fi * a t il{ab* a -i- A b *l"u.;V]%r=sor., 2(sin,l * sirr8 * sinC) -{ tg.t * tgB * igC. Dirrg lhuc xit1,' rn khi r-a chi khi ianr giac dBC Bei 5/lLl. ftnn gii tri citct,ltibu."thrc l'r l;ttrt gi,it, rTitrr. * sin{a;i5-(sin;: *siu2r * sin4"r.)B tta { lr hac ta lo "t =(sinr*sin3r * (sin2r*sinlr* sin,r'!5* (si:r{o*.si,n.l *sin?r)5 --\+ -8r.1rer,- ^lf-="t Ar0Br loBrCr : t,a'i t' = ?0o , = do drr Liri gi&i (cria '!{gugdn }igqc Khanh - 9C Doan : orrtsf;}; I!6't, H, NQi). .,{orl =r tg.d1 \r ,r. = Ll0o rren toi r 1a bau kinh dunrrg tron ndi tidp.tam gilrc Io si nc f si nl;l; = lsi n(Jri2) cos(r/2) = eos(r.l2) - Si r+r. \riy .4 tf6 thauh _. .i Ao .4 =2"{ {sinz* sinl.ir}t* sirrslc-'sin5rl 3].r.) = !5.5[si n.r'si r:'-1r(si r r',r, +.ri n .i- + 2'i rr2r, .i r.,2:r (si ri.r *,inl.r") I E =2D.5si rrr :i :,1-u' ( si nr -{ si nlr)X {si rr?r * si nr si r,?;r.Xsin22*) =25.5 sinr $in2r cos{r/:)1c,.,s2(.:ri2) -;ilr si nkl" I btn#"sinfo, : (cosr - 112)/2" Ftn A\, 6f t " : cosc cos(:ri2): [eos(*/?)+sl,2iiz. { . Ta cd p, - ArBl * Eoer + tsdt l- ,/ .- r(lnJr * tgBr * rger), A - l-.5(cosr- 1/2)eos(acl3) . = Btct r- {.'1.{1 i-.1181 X [cosz(.r'12]' cosrr/?) 1 l'{i 2p1 = 2r(si rr;!1 * sil,Bl * sinCt). I - J+.."r(g65.x - I iZ)r,os(.2/l). (:l/t) q- = 22.3.5[ct,s( rlill) * tllit - c.,sgzr:)rz] ,Nhung theo tr{t tl8ag thric (1) ta c,: t- q = it.5. Y3 = 15V3. ?(sinA: f siu8t * siuCr) { tg.{r + tgBl * tg0r. o 6 -l
.-...@ I Tu alo suy ra 2pt --( fo. Tucrng tE ia co Ttgo. tlinh Iy Pt6ldmd : A.C.8D = ae * bd" 2pr+t ( pi (i lla sd nguydrr khdng am). Tu dd . : .}} tf'e .': i ' ' {81fts J- esfr:},. (firFa -t- fizftr}4sitrie/ft? y1 * p, +.... * po * ...1'.-( = (fi1fts +rqgftl){sin"a. d poi2 -, potlr + ... + px/Zt -f- ,.. p-12 ca = - 1:'-: r l'): =Po^ iuc ln .I p,.{ po. .f = V{fttEt*rtsltr) (firft* * F.Z.kg)l(f,rft3*Eefta) - t- I Ding thuc xi1' ,r'a khi va chi khi tiit cri Sae Bni S/1I1. Cha hai n;tm ra:fri vd',q(*).idc tarrr giAu Ii dpu l;r tam gilc deu: diiiu na1- r[_r dinh u&i ntqt giti tri eita r, th6la nzan Aftu ktln ra khi va chi klri tam giac Io Id tani. giac d6u. lon lai i'd,ihryr' a suu cho . :\r' {(r -i. a) -f(+) a- gtt) ,_ . ' Bii 7.,lll, Cdc dc6ng ch6,o ctga ttr giac ngi tiip , * AECI) ciit nhatt & y. Cna bitt bdn kinh t:ac dttbngl Dti i lrdn tryoqi tidpc&a cat tam.gitic lt.\8, -llRt;"|{CI) r gl'Ll *a' 't chdn ud NDA, hfrg t{nh bdn kinh etra dudng trau ngaai ti6 p'Iri' giite -lBCl). o{.r = I( * dr) + na) nta a li f{$=1:n6u 0{*(n, ,;l-q; si&i}[.'le ='a) BC =-b, CB = c,'DA = d, : AllB = {t..{B}f :. t6,. Ggi Rt, rtl" Rg, fta, .ft tlreo thri tr,r li b{n kinh cric duong tron ngoa.i ti6p. eirc tam gi{c.}fJ8. MBC,.IICD..UD:t li iu gi{c 0 { i(r) -x . .--ttt\-/,1 (1) LBCD. Ta co: Lbi gitit (,:.&a Iv'grugln Thi tlugi:t Hwong * Sg-hi '-Xu:iu. Nghq -l-Ii:h). Ta eo: 1-(c 'i- ita) = 1[tr -i- a] * s1 , :; it r i- a] * gb * e] lli fb * a): f(il * gtu), gb * a) = * gfu) ndn . fb a2o1= lb). Vay f(r) l['ham sci tuin hoiln p6 ehu kj. 2o, Te chi cln chrlug rrrilh. bdt dilig i.h*c (l) irong hhoing0(r{:o, ' Trong litLoing 0 { , -( a., theo gia thrct f(fi = 1 nen (t) dtrcrc th6a rnln. Vg;i a{* (:e, d"3,t y =tr.*al ta e6: ': fG) ; f(s * ,1 = f$) * s(y) ': t -F gk). - olBt = blR2 * clRS =.dlR+ = Zsincr. llat khec, vi lg(y)l ( 1 nen troug khoi.ng niy I trf:l = 2&isirrp - 2Rtdiz,E = rld/tr. 'Nhungvi d !ff4sina ndrj o{lrrr{:, = ta c6: t iJ) dufc ch*ng minh. B-f,t tld.ug thLrc .* UA = 2Br&*sinrlft. . eic b4l: Ngugfa Vdn .\"icrc i1lCT Brii Tbi Tuong tu; Xu*ir, T.P; Hb Chi , Miuh), i/gugdn llaai Nam (10A Huin Kiem, Hi NQi) fforing Minh Quiic (lzC \ trlC = 2ftekesinalE. Bni Thi Xuin, T.P.'Hb Chi lliuh) vb, Bijng Tdt Tr d6: .: thlttlt (phuong lli, quan 3, T P. flii Chtsi\'iinh) ': ca toi gi&i tu;rrg tLr.' dC = 3.fl * MC = (ftrE* * ftsRs)3sin*l-8. Tuong tu: Eii 9ilt'1. €ho hinh ua6ng co cqnh b&ng I ilaga chta tnadr t+'A uudng ilon u!. Cd m6t ioat BD = (ErFr * niRr)esina/n. e&e udt cdt &ira tho khdng ea';n$t 6 uttdng ndo
'a' rgugin .rren nhtttl$ .ed it nh?itlm(tt' dinh c&:a hiah Eii l$iltt" biit: cha t,aitr'lt l:Iwrt:l ci t,di cilt niw Ci rjua. chia h.inh tu6ng thriith nqit sti'
I €Ae o* roAru efia eu6c THi elAl roAH reas ' ! :t; ,! .,: , , '..': : Bii l,lI-14. Lho dir r,-, iLr nhien {ui. 1r...., r-ttc), r,ui ,r "d or 'd ,.. ..( ol r-i ii.rg ,.ae !r:i {l = l,!,.... Ai triqs 1r(1. r^ar. ,ri(i:j l.1i.,.., 1i.t. tlii sir li .,;, I, chirnq llillh ierii ti_,r_rg 1r, lir,ir L:"rL: ti; ti-'r lutlt ludn b. h+ r- huic L,al,g :^. .l.rrr.r ,::r-t 11{i - l,:,..., k) tr,,.-,r::. Lr-ru ni i11,,1i 1irr,; r i lriir car' ui barrl 3,t,. t\ 1 tit Btl 1r; i\ if r luarr ir,,,! c Bil*ZillI,tt. Chr t,ir: s,,\ uAuvrn ({i, l. i,., . :r). H*1.: iiru rur)t hoi,r ,i r,iia ear: sci .ngu,:*f ra,,.- hlrorrg t'biu, , tp 5i, 1.rin-. .rrrrr .t ,t; 1,,,,,,,. Nqetr$ri { ,:t1t L1irt7 \i;ci flqt I'it Bei 3rj114. Gii thri 1at. r:t,.,,,crlre {&1, gs,..,,i.,. Brr ', \14 r,tr,'G. t1,i, li hri !roi*'ri eia ,JXr {1. l.:i.;,...,!'n,}. ihia nialr fi.r,.l : it.l.t =.,i 1.,,r? ,. c,:r + d. diltr k i.n iltd* i::It dilu k{*r; il{.,-1r --_ i hh; ,.,.( t. nf i- ) *l -j. bz ..r. ..,) o,,, br, 51 {ei ,l,rrug r,:r.1 rl;rg L.ii !.,i !,.1r r4 l'.hLlr,-, nliirh ci4g r{.i 11,,.; i.- 1.2.....1, t 1lk * ,-.. -,tli'. 'l"rrrr 1!ae i'E'C' g,ri li lrr:ri 1rid,, l,lrr"ra ,:Lla ,lllilr Lr Ira.E lril] Iilr ,{Fl,. Flttttt Ft:rre Ch-ittlr r) f;htlng rni*h rlng ri*i i n,i r*6i b6 bdr di*ri ! {DH?l{ IJa N,.,1r ;:,hT-ag t-{, }i:, {'. I}}, b6r rrni g!:-e ri,iiu cir* b$n t$i&ilr d..r tr.*n **c raal 4rrc'riJ U;*1., hi_}i.h* Bal 4ill{; {the t li mgi *d tr, nhi€r ls* ',.u lai ir lr.:rr tu;n gi*c it,ug utar,E. U,On, ho* 1. (iif r& co t{t ei t p&in *fi rdl jjii.; ,}6u,! . elb y*i I ( a { rr {- l. Chirag *i*rh ierr{ ,rriu - h) Ecr dltsrrr phixg t. Lt, {), O l,lril li:,:a wIn Cr&r bign gi tlB hfia {rm gi*e ,}e liag nhau r, 1 rtar;1 b1
.: Bii,gill4, Cho ni{r clu't&rn O h*n kinb ft tb8 tieh ltiu nhit va nh6 nh{t tl6 the-"o R vi. rn6t AiSm P ed tlinh nirm tro*g ra{.t. clu vid. -. ' l {O;p=d
b) 1$8492 + * lu = lgtu59 ry 193i + 45 = 108.19 edrr ndu n = 10a * 6 -< 4$, iue ld ri { 4 r'* + 0 = 203 -"|20 * 15 = 35. Ynr. 2030 --t. 2ffi ;' c d r,{\l g, rhi: , 19849: ehin hel r:ht, 7. 0,{n =a*rl}d++4i=4{,. Yirr dung Iitrrr li6p ti6u chrriu ll'i.rr rl*1' lre,) Ti€u chu&n cliia hdt cho 7 vri'ri tiri iluEr, thpc: gid ta cirrrg ftulr tlttr,rc. ri6c rdt tinh chir hdt ch,, ra kh(rng ph:ii ln du-1. nhflt, Ch$rrg han chring 7,efia rngL a6 trr nhi€n n |rf,t ki. r,E vife rel tiritr ' ,'hir lrtlt cho 7 crilr sr, lrr rrhirrr ..r'gr',m rihilLr ta c6 ti$ir r:Irnhu sm ,. , l$m li.2 chii sfi.7'I'hAt vil: n.-. l{lit*-lr:?#}n --il -.iItgr \drr n = li)il -l b ) ftt, luc j:r n -. Yi dq: i1;iy iai vi ilr,1 Lr) dir n$u r'i tril'r,): rr --"' 1 , ', Tt < Z,( 9. thi : $c -> +l! 1B+1,]>r:*ob. lils+{-i2 *-+ l-i}.e{l} -- { : }L}Btra}-, ltlg{ - i0 = ttl;4 -"r nghia Ii t* itu:* r.i6c tFt rl6i t:o tu nhi0n & 1{}; * E: Isil *'|. l.v - j;i =. rr- '- -r n = lttc. :i' t, ) lg V".iv 19$iii! r:hi:i hdt clro 7.. - r.E xit sd tu nhi8n {.lirririg 10i rin irh{ra,irr t,ho har ilc,,e t1t, r:hiirig . n = rl + .rf, r_\ I.t, mi..,h l,i.r' tipiL r'hulrr r lrr rrdrr. Li tlitii gi..*.i-i eiill thr.gt dA gia e6|rg r:hi tjdt : tr6r: r#/gtrt'J#t' m.{.r' ;r, Hd-r' x{c alinh trllir trr gia c0ng .rac r.hi ti61 {tr€n mdi m{y gia roirg chi ti€t liir tru&r:. r:hi ti$( nlio sari) il6 toiirr bd thui giau san xufrt -dr*- r*.e chi tiflt l{ i:}rfi,nh-6t. r'r}i circ tliEu i;ign r:t rr ,l$J : - UOi vrii rrroi t:hi ti€t thi ti{c gia odng nd tr*rr nr6.;' 1r chi c{ thb tlu'tyc lr.{t {Idu sau khi chi .TST ti6t 116 d:i iluut: gi* c0r:g song tr6n m{y7- I; TRINH Tu H&CI t& H$gT --' - 1'r€n mr-ii' nriy cring inot lric chi r,6 th0 gix cirng nhi€u nhfrt lir mQt rhi tir{I, khi rn{v tlir qii( r:Ong chi tift nao thi phrii lam rrrng chi tigt,di) fnOXfl eur7r. s0rrg hing ngur chfug ta tlrrr.r'rnr rrir'ii 6J11o,c cliirvin sang gia c'tirra chi tiit hhiri.. I phii tinh tqin. sfrp dft c6ng vi$c sno eho hgp 1-v, vifc nio n0rr' l:im tru'6'c. vi0c niir; Bai tiin niy thr,ri'rrg g,ap tr'(,ng r i*c Llp kE nerilim sau..'Irong thq:c ti6n sln su{t cirng vri1.. hi-rar:h sin xtifrt & r:irc nhi ina_v*, Iflp lic.h rhay tAu, trong m6t sn&ng m{y ha3, trong rn6t rl6i srin lip thoi gian hlFu lioc tflp il c{c trut}ng hoc v:i xuSl. thrrrinu cr) rrhiBu cOrrg I i€c'phii Iim. moi trong nhiiiu vlin'r-IE khfc. D6 L\ ioai h.ii tr')rirr ripc ciin triii que mdt s{i lhao tic hrr-t mf,y moc liur trinh ttr"t6i uu, tirtcr-c nghibn ciru trung mOt rlhrit rtifih. thtli giari liim mo(i vi€c tr:0n u:6i mftl- m6n to{n hoc 'trng dung rflt rnfii m6 r:6 t8u-gr;i iai rflt'khic,nhau, r'S.y n0u s*p -x6p: tr:inh tr.r lsrn l$ r
s tt.r iii* l:rillg ("itc r'.;* tr&n criu tit;i t khitir chi ti$t -il. \.' s il] gi*,.'O*g khlrir rrl.iatr' Lr6ri i-l; Ii nir)[ hiirig sii rr*ii:r i lr jiri ILrir .'lru. uhit,i, ,:ri thii r:ii tr;.r,n "'. j: r,iri r nr{rr"cii i{t r:[ im!]8i"i':rl"l sip rip tri:rh tr.r' Fd Ii ;liit ,riin sii htitr h'.rrr. l-i thd {16 gi;ii l}:li rE Loiut n*.r i:ri ttlf t i rtt:h q rlrrrr giln >r r:lrdt l;'i {ring tiFu Im (,s) qiii r i hiri toirn tiirl'c-r-rq ti3,.: X .r-' tl Irhrr)tig phiiii {( dtrvii.'l.r;;irt hil v" tr'ri: li ki€m il a li l,ilr tlriri ,;i-ir, ,iclreI B eria rrril f't. iii:r iri,;l tLirrg i:it,'h 1 sitp r[p ri.)i r:lron ra c:lrc'h s:lp rdr, t.i5t rlrut (ivlg i-ir'.i tfrtii [lrrr'hoiiri thirl]r' 1ri.t,iit.(rhi iii:l ph:ii --ia..L,rrq Iin luot trCrr t,rriir hi, caiirg ', i6e il'r sri'tti rihiiii. \hrl:g tidr .tr'::ir- I rili nrjir'B radn thdiCirrr ni:iy E chir gia lhr.r'. viiri ,r, ','r, , hoi lrin rltirt i'irrit thi'(rn i)4 ci)rs {ihi tidt t rting{bing- thiri gien ei.a r0nq chi .li it ,.,rf,f t:tltr's{i I'lhiirig 1il. i:}rir dir i'i1 t'ling trrrir lit t I fl rr' rnav -4. lri,. lir I = {rl. i ,.r tinli iliEtr t ii
:ft*r i Ii. 'hai 1rilt, it,. .f,; kfurt nharr i/ ,.uc vi rt.i h*i t:hi *qia ;i. \5rl .,; t1 r r.lr,.,,r l:i ei;u 15i h{t d:rrrg 1;6ng ti6i. / vii /-f t. l;i rh; f,nl,o: .t., f,,iir,, tlirir (* t th[rr'..i-;ii.;11ith l.hr)i h*ir ]-ii;drr l]r;ir:h
,\ - :.,Ey-r+Y -rE- , :... - I \ ai-\' -Fsfl t'ix rrlt :- ta c6,: E, I n I r stfT.r3r-...tT*o rtm rf,r sflN6 TH8€ c6-sr 1+tlz+t!2+...+tl? . , clt&T THlffi ,_ ? ------- n-2 1+ 1/2+. .+l/l . lili ri.u tr:inh'b{1' l'tti cie hgn 're$1 UAt Ahrrg a-l -eF-b 1rt th{rc: rCho rrl, t"Ig,",. dn IJt cae sfi a'rrong" Hey ehurrg minh lhuel: '.li of +or r-..*ar,,}1'tl.1d2...d71 X, :l:T n + l f$-r...t n,' I E ++ 2 T "n NNDE E n!z x.{ilffi+1'*[*.. +d,r.-r,o"+i/ur'nrl (1] n,i 't- I ilZ al? , ra lil:i (11 : {?& =..... = {lrr D&ng thr'tc x&1' v L13 ... :1]n = f rJ rS "." it's Bdt d&ng thu'c rri-v chdt ho-n bf,t ding ihric Inr .i 1 s- n ] 13,.,r!.. A nt[ l'r-' -x tg3 f ... -f -1-r, ? T.,r1 COsi vi ia lu6n *ir v0i &1. fig:..., a*'duong ihi ')-.8- g, 2 - n . n ..:tt q T'E(erflg tE', ta c(l: ,, dil,,rEf * r/errlar+,.. + -r,'rr"-*r" + i';ilqr. l zl.-r I n 1 a- '. C{eh erhilrg rninh b$t d&ng. th*'c {1) nhu e: l .t., 2 rt ...*-Tro * Tr, a SSLT : i ltn t t2 14 ... :cetl D,.r1 u1 = ,-riq (l = l, 2.,... r)..xiE U (l) +. 2