Toán học và tuổi trẻ Số 135 (1/1984)

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Toán học và tuổi trẻ Số 135 (1/1984) trình bày về thật và giả Ơclit; định lý Braove về điểm bất động; vài suy diễn từ bất đẳng thức Cosi. Đặc biệt, tài liệu đưa ra một số bài tập và đề thi Toán giúp các bạn bổ sung kiến thức về Toán học một cách tốt hơn.

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Nội dung Text: Toán học và tuổi trẻ Số 135 (1/1984)

{ i 1"iE.N Iili(it li(.rC vlF.:I l Art l:',8 ti,t\ IirtC } I !i,I,.i.:l .{ 1l >I' -- i :i1...,il,t+g;i ri::*i_ i- e:it.r RA FIAI TII A\(; Itt)T Ii J' l'$rrg 6ten ttlp . FIguyGn C&nh 'f oAn Tm sci : ? Trln HLrng Dao, Hi. N&i 'l: Phd 't'Eng ui*n !{p: Phan Dirc Chinh - : - Div ndi: 52825- t 'l N6i chu"r$n v&i c6c bqr tr6 y€u to6n r TNAT YA CIA ryCLIT /1-oArli Nri U I t:r i:,i x I t tnlfiir'theo crlLr chuy$n b$t iiirr /;' b/rr:'Totin hoc trror:g u-ng rndtalriirndt) giirii i{1r hr,rp e{c rlitm trong I t,, 'lrriri iltl .O tS't (th,ing 211S83), tdi nhd nrat irhEng vir tip lqp ,.ir, c{p sd tlnrr. s$.p thii lqri rrlnt. lri)i lroc phi) ^t ir0rrc. lilri t6i nirr tu : rtng vCvi nr6t dibrrr lli cir crl rr sri lhrtc rl uy thie nr'irc: " tai sao fai ia- o'= ui -r ct irlirrlr I.y rihiit trona- d6 sd tliiu k\ hoirnh rlQ li sfi thr'r [di Fiiaga) uri lih6ng lir al = bt - .l'.'t5i cri ban crirrg l:i tur:g t16 cria tliBm dir cho : ngrroc loi, ong i6-p dir e:ho tOi [A < dii:n r :' ( Chu Ci0n ha-v sao r'6i nr6i cirp sd thuc sip thrr tL,r (a, &) thi cti , rli,v>. Thi canh lrrytrn a ltin htrrr Lirng t'lnh g6c duv nhrit cri hoAnh 116 biug c vir tung 1:^0,, ;Iibnl. f ti, vuOng n€u;llrirr li -t- rhrr - sro tl rru.:. lli)i r16 t0i d,) bitrrg t,l. Ta cnng thidt lip tlrcre mdt sclng ,,. lihOng ilir ly le db tli-lai ban 10i nhrrn.q viin rro I nlr rr!rir r:iy oiirri tdp lrgp tiit cir erir: vong tldi hd ciirn ttriS' r&irg thii; mt"tc cirt rninlr iitr6ng phai ir\ rlirrlt hudug cri tAnr irdn nrol rlrrtitig ttrin.g cho ircAl toir; r'6 cIn cir'. ltilrttng tlirrh tlil plri) thbr:g trrrtic l) vir trip hup cric clp"sii LhLrr, sri p thri tu,: r:iiel t6i'lirc do lihOng cho itlif p tiri ili xa lrrrn ilrrtrc r'rrig rtii ril6i \'r)ng trdu ta cir riLlr st'i thrlc sip r:hirt.n5o theo htrerng Irri3r, hhic viii dibrr ilri xii-r- tliri' tq trong d6 sd clhu li hoinh d0 tim vdng r* iihi tdi thic mirc lri s:to ]ai li .sd mir 2 i'r irr',11 (s'xx lil:i tlii l,idn.drrirng thii rie I, ttrir nlr nr6i c, S, e. iliLi ddn t'F sal. lihi tiit ctl trinir aiii l
lheo & dunng hay :lm" DiErr dd gr;i clro ta f sau rliv : gr;i m&i virrrg tldrr c6 ilinh Luirrrg n6i trdrr i& ;riirt (( tii'8nr ', iruillh drl lirt: r'dng t ri)n iri {i hr;ir1h dr.} > e tia i( dibni rr vq biin kirh ' tLai s,} .'ira '.'dnd {rdn li < tuilg'' d$ > rir;r ,r rliitt:r'], iiiira rlai
.l 2 =-0 tinli theo (Z)) ; d6 ld nhfrng ( tlrrrlng tli&ng ,r bqn hly lAin .nr6t vdi c6ng trinh nghiAn eitu nho giiirr nhilng rdng tron dinh hu(rng tidp rric v
t- ho6e &.*p * t, v0 I-{"vi }::ta, o e ff, 3,.,.r {p*3i1, sd ngluy8n t6 p ta cluqc e6c sd tlu khSc nhau Ya.v'& :L 'l'r'r d6 ndu cd mdt sti uio 116 tt'oog s. F - I sr5 tg _. hihu y:1r3- .18 ab s I in:orl p,i thi rr, A C {f, f,1.., nhi6n Ji6n ii6p tdn tai iri:n chizr h6t ci:o p iiri p*21 r,'ri{+6. lidlhppvd}i rl). tarrdr vryi rnoi trong .**1 sij tr lllti€r n}.r kirdng cirn sri nno lihic chia frit clio p. .Khi d6 khdrrg ttr8 ch'la p - i s6 *6 b{t \-,i a fi }4, 1,..", p*1ll thi s,'i cri m$t r,ii chi. nrr'rl st'i o ir,lf . 1,,... p*2i vri e * {.{ db;t s I {rnodp). lrr'ulri0a dri liraiih lrai, nh6nr d3 tich c:6.c sd lr.ong Vd.v tii*r 2.:1... {p* 2i gi}nr (p *3)i;: cflp tich d6ng m6i nh
_ Xei ngrroi J ttdt h-r,. GiA si? d tn rn6t ngLr&i ntia t1'r.rng tr.on trr"rr: gilo rdi hri dfurrng tron dI. ri{u vrii .,1, ll,1 rigut'ri chrra tll,iri i,
'! Bei 41f 32, Hdg r&e dlah ad.e sd tlrpc F,4 nai .5/132. Cho aId mlt td dtong. Troag til a&c'dc. tdp *6 (t,U) tlr6s mffn dilu A'l€n lh6o md.n ctic diitt.kit'n serr dlly: I.g a -r y 2ti t' as)) l, a) cdc phtong ltinh tz * pt * 1=0' '-t.* qr '\' * ?* 0 ro m$t nghif m chung t triiy lim c{p s6 b) IDns lp l+lq I nlrir nftdf' o) Cd t l6n nhdl, l.bt gi&t. Cach t (eria lrhn Qadn 10i..T Phan b\ Co g ldn nhdl. Chu Trinh, Dir riing)' l'ir di6u ki$n a, tdrr t4i sr5 Lni gidi (rlqa theo loi gi&i c&a Hgau?n fl&ng t huc .r siLo cho e& * t'i chul'6n to6u, Phan Ciru Tlinh' Ili \ing) s-'--Pr' 'r 13+p,u-l 1o(l l)at rliiu liir;rr *t+qc12x0, ,.,:+ug){l'r? + g?:>0, rl + s2 +1' n i fr,, dr, ph,r,r,,g trinh a) 'l irn ctre clsi r: > Xet trtttlng f'.i1' '? + U3 1 7x2+(pt'q)& l3:tt (l) Klri tlo ., +aylrz+i2 *$Yo * ag r 't'z-'t (ri + co nghifm, ?:r c6 i, tam thr'rc 1:1c 2 cha y cf-l bigt sd ) 0: [= o! * 4t? * 4.t')P t1 i\=(yr*q)'--2{>0 ++lp+ql)':'1il r: {r -. 1/t;;).2 >t1/{n 1): 21i2r1' 3rlro l= 216' I-oi gidi. 'frrrrrc hdt ta chung mir:h bb dd sau: tli6 tri.nh6 nhdt ciia lp l+ lqi t*. a y'a Bb dE: ),i6u rp = 2li3{i(n * 1) vdi li v:i n la cic aat rluoc khi 2 *:f =,34=9ro * n3n" 1- k'khoirs chia her .,n: t" -1-'1) thi ,U"i:o,, ,* "O T* d6 linh duoc p,rI (nhu cAch I) S:cosq +cos2Q + "' * cosnfP = -t' 6
_-- .,n I{hu v$y'ldy 11 = :tl(n + 1}" thi theo c{eh-x[e Thr.rc v4y, rri 3 si n {r+/2).S o E 2cos Irry $in lql?} tljnh r.t;,.., ,o i, t.At, ta c6 3gffi13(/{-n + l),"', lit: *, *'*fiitn+-j), h*.t' rcri mrTi tdi(a ta c6 tr 2,rr - :j.1ti (n *' 1). I}:r ' r6.y, i"heo " (1) lihi ldy a ri=s(l(e + 1) tiii tri' {3i ta cd: I ket {',,-1"(at -- * as+ .. + oo)}n hr.s ? ! bn-rl \ .- rP " f\ -:.: .-':-..:-. tp !i sirr -L = *i,, 2 et +id! f .:. S d,,,) :n'{n * 1) {d. p.r.m} /',,, -i- t \ &tftdo ref : {,0 it trqn gi;"i tlrrng-lrni tcarr ni-r"- s.it .! :;' ro i L =14gifil:\yC*tr+elb. I cl r:os" 2 rt -i* + CGS -.Ep 1 /') I * t. (n **l **u { a! cros- J.rl + CQrrg theo trln e{J. VC c b5t dd ng thrre e&a (2i ,l I'l + ceTz *....-i do T n\ro ?, {3i II , $d6 'f i = I cos-,t"r'i I*t Ta ci, "*rn Do Slc n€n.4f dteS{ld-*S43 r'a ':hir 1' )?i*n--. I i-,.'ur2A'tr*1)* X cusA'itri)" 3{,4'M3r + t =' Pa/l{'d! ta'cd : . ,k=1 : k:=1 f*,'{if ,v + t\Ar) = {?lr } ,,')ilr, {il Liir r.ri =')ilt {n + 1J; rir r&rrc I lihdng ctriri hdi rho tn * 1i vi I { i ( n, nin theo hh iJi\ la ?'ttrrns'iry. ta ciing crl N"l/If,{ttE I + &lt "'a cftng cdr 3Ii'* n = *1 hay clrri _ri rUl/fiA! * I : p^.NAy. ta to: I pa/rfir = (24 + 6)le (r) Ii.= (n - 1)/2 v*i il1oi { E try..., n; ({)
,r' Tn ( I ) vi i2) !;r lir I r.a clur.rr. 8*C B+C B +C oo tt)S "::*** * COS E COa ' ,t -..*:.-* (bz &- ') Pu - c2) I Ji j\ E , (3) ---- (c * 2D) (o +':c) 'Dr +9 2sin I oiu-- = .o. 1 .o. i- .8- slll * S.ll] c Ttreo. b{t iling thr'rc Cosi cho 2 s6 ta cCi : 222'4 y,B - (c'F 2bl (b + :e) ( tSl.l i (b + c)2. r'a-r phhn a) drlrrc clrr'rrrg rrrirrh : 3+ (x.)s I I =.j sir'3 uir, I ':22: "n. NN > Io -{i-(l'-r') (b+c)2 : 1 -t{a--.) - . EC *9 colg *.- 11)[g]* s t I b+c a 2: Tit; Jrllrlo,tle D)rrri.rrtrfI t =GCt- GIU;\'A.v: Gqi .K li chalr dra/lrs \.u(',ng grir hg. tu I ddil true (.lr:, ta co . Tir 116 nit rr MC (213) n* 1- MC Zb BI{,CK 1"':':- = 3't rrt? ' iltr,),' I TiY , * c&a I la g). thii thi - r,(iii ,,ir.[ou dr] (.r, ()ff rtr6 * 2b). i:/S) (b - c) mcl(c G) IK - y. r.i [a cd Chrirrg rniuh trrong tu, (vtli clrri ! do L- )> . lnta .- x" = 3!1,. ncrr uri IBr * .VG + GBr) ta cri: " Yir rltri tjcir ,:iirr 1 la rirrorr! 6lip co ptrr-rong rv'f = (2/,cl (o - c) m6,t& * c) (5] tt'tl;1, \lxr la.r . trr !3). ({) r.a (F) lrr r,6 ',,2 l- 39? = 612 : 6:!.c! * 4 {}*c)rnbrnc tru hai dibrn g=0. A,M s (a*c)pu ' i !'Qli' Phiin i) c&ug duuc chiriry minh xong. tsai 9/132. Xdf edE tam thirc b4s hai f(6) * . rVfrdg rCt: erlc b4n rYgug6n Ngsc ydn -Ngay|a Khoa or2 + bl+ c ; g(r) = ctrs+Fr + T; g1(r)'- 4r2 (10CT Phan Chu Trinl Dl !iBng), Thanh * {:e + Ft. + ;r + F +'; : lr(.*)-*og(1)cs + &gr(0)r ,/d (l2C L;i ?huong Ki6t. Ili nfri)- i.-d _rrguyEn * cgtll)' , Vdn Hadg {1lCF Phan Chu Trinh, DA Nirrgic6 ldi giii tucrng ddi i6t; Chting nfinh rdug ntu cdc ph.uong trinft /(.r) = 0;9(e) =0 cd agilem, d*c)*0 trln ilaan T.l,'.r. I - t. t]. lhl phttong trintr/r(z) - 0 rffng cd nghllvl Bdi 8i132. .Trcn mSt ph&ng tpn dA t0rg) Lil gidrt (eira lrln l6p r0C!, Trudng Qudn- dibm B (--, c&o Pluan Chu Trinh, BA NEngi. i)E j, r&irg gr(r)- 'te c'i O) od dr?m'C (m". O). d ia dllnr trtt fl(,: + 1)r + p(r + l) + 1 := g(r * I) nen mQt ph&ng th6a rnd.n itiba biga tung df cria' .4 &d:rg bir lb.n tung d0 cfia tdm t dtdng ,lrdn n#i lhE r idt : 'l tiFp tom gtac AltC. l'{r) = ag(2)rt + bg( I)r + cg(o) s ?'inr qui Jlch.crttt I L.trr .4 hlln thlpn thdc mfrn = a(4q +- 2p -i- T)*2 + {z t [3 + 1')&r + cT. dieu kif,n tren, , Fib chtlng r\rirtr phuong trinlr bric hai ft(r) *C l,&i gtdt.l)at,1 I =r,''BL:::o, (;A = 1., r lr hf rr c6 nghiQnr thi ta r'[n phii rrtritng tri rilg kirh trou nQi .tiep taru giic JB{.', ta 116 tlur)-ng A "* rB(a + p .+ Tyz * .teln(4o + tp + T) > {1. (l) (o * 6 * c).r: zdt (-48{. ) .. 6di (IEll) :r(rr Theo giii thi6t ihi t! ) 4ac: pz ) a*:( {rl = ++ 6 *c = 2a e sinB -f.iinC:2siuA o 2sin {g t-rj) r:6.cphrrcrng trinh /(l) * g vi g("u) = 0 c6 n,{higm} r a o(.r) t o tr6n doan lo, 2l rren g .,_.-- l)+c R-c : 3.FC B+C g(o)gtrt * T({a + 2P + T} > o (2) 4siir 22?2 Viit lri (l) r.ltroi ,lang (* cosB*C u 2cos I +u A = (a + F +'(12(oz - Aac) * 4ac 2 .2 x [((er + F + f)t :-!t4a + rF + f)] ] o. (1'). I
i I ' Ndrr or ( o tt,r (1) ludn lu6n dung rileo'l:)), . Trncrc ti6,. to c{4.1rg rngl tir dr6n clbrr dAD.l Xe I oc j' fl. ii lri. il4 ( 1') sd dring ndu ta ehi c6 t'anh a rili dung c6-c l:inh binh hirh C8fl8, ra rlurrc ri ng . .td'#',9 v.r rl.{'I)'D. Sau dd d,rnl hinh hlnh hAntr (z+B 4it?^T(4a+:F +f))>o (B) "6S"C'C. {iuh ASf,D,A'B'C:fi' ln hinh h6p th6a mfin iElu'lii{n cria r1E bii. ({l,tre }.ran tq chung h ar' rl;uli). \ / Vi g(ri = 0 co nglrignr ','ri gir) f I tri:n r,1oqn [0, "] nAn pitrrrrttq irirrh g(.i'] = 0 ro i Ac ngirir:'m rrirn ng,:iri rlrra;1 lg,:l.7rr clria la ba trur-rng hcrp,. tlb xt't l 1) Cii.e nghi'6nr .r 1, x! efra phuorg "trinh g{r} : 0 t10u nl.ii., hr.,ir 0" filri
'n ,. * lia S- ia aiitF*n. i+l, .) r'rri gtti ur.ic 3,n-1-t ' ' rl1. i,c I 1) eh*ng minh rErig l
Tim hi6u to6n hr.re- hiQn ilqr #EHH ri BRATIE Vii affiM 86T E0NG & I{OANG EI}C TA}. --i'ROf{; ll'i.tr: , i 'l'ii ;';,' rri-r'. chir';rg t0i xirr {etr:&ng hen, rn&l hinh irot! ed rrQt dai luq'ng,( r, giui flrierr i r;i Lran ilo': bAo To[n hr:e. v]r tubi nio ilrIy haAn toiLn xi.e. dinh -*'4. ih0m nita li ndu tr'6 rn6t tlirh l.t li,li tti,r,*. rn:t n/t cd rdt ta i{l mrit dihm a bit lii- trong sd cdrc di}rn cira nhibu itng dung ttoiiE llan 1loc. 'l'rucvc htlt' 'lhring rr iEn !] rtri At ,:ho 1 ld nrdt td ,1"oog nh6 thei ta hfl1'linr tluc:ri i-rli r.r:rii .6 l.hLii rriQtn cH.n thi6t: nic di rh6ng nira ta vln lim drroe nroi s6 E ) 0 1) l{01 *.*lt.t:q lu i:iuh,{ r.rro hinh ll.' li qui lih{ bd sao sho hiQn sd hai giA tri cfra / t4i cAc Ir"iSt m,r theo ilo nr6i r,-r6t ,liEm m cira hlnh A iliBm * vi. } {G Ql llt kr: sdt vui a nrdt khoirng rlrroc ung,r'oi rndt di0nr m' -tjic l1inh cLia hinh cl,eh €, 'r-i. giit tri tur-,4t d6i ln th6a nr*n rliiiu ,{'. Eibm m'durrr. -.,.r1 1:r rirh:rla rn, ca}n fi! ililoc ti+,, t ifr -*ioi'
nhs.hon ntrl& r'.v,.. Vir nhu viv li ta nhin dtrgc di8rrr m chuybn tlong li6n ir.rt lrong hinh trdn nrnt d6.y vO htsri c6.c hinh vu6ng con ngi"t ciing {irii r-i khi'do ci d;tnr m' rirng Ci clruSbn lisn hd rtdn r1i, v* r{rdi hinh r"uOng con i*. dunc r:irua tg*.) $ng .r,#i rirSi j*,, ,, ta t"Jro cifl dni f' ciia trong trit c5 .cie, binh ruOag trucvc nd vd rn6i vi:cto dieh r:huyBn--rnrn'. 'ta th{1' . gi6ng nhu hinh vrrdng li'oug d5v dru ehfru nhfrng rliiirrr ma trulng hr:-p e&a bb r1B Vaiectrrirss: / le hnTlr li6& rq i OO /' li t n-y j- br. Dlr c5c lrinlt lrrdug rra-v ii.rc cira ilibnr m, rn'2i /' ,lau l;hGn[ 6.nt vit eri sd lhu dhu sl rtibm B cria mit'n !J. Ta sE i'trrtng nhlrng /'tr}.r, .v b6. Khi iir sg tbn tq.i ellbrn ir mi, rninh riing ',ai dibnr f, thil= 0" ThAt th€, r,i $(Q r'rng rrri rtit f x0, tri'c Iir dd rl,iri cfin mm' biug n6n tai b rlri /g hoAn toan xirc dinJr. lheo gii 0. Va vi thd lrcirg ph6p iul: xa m;i ta :6t rlilrn thi6t l; ) 0 {r,-i f' tnlr*p no'i khbng in. trirn ft). & duqi: 6nh xq lAu cliinh ni: i,ii dinh tri: i1[ drr?c Gi[ etr r&ngfr ] O. Utli riri theo I[p ludrr ] ir$n chi!rrq Illinll .tong. trcrrg nrdi hinh vridng rlt bi ,:h(ra rlitrn b sd co nhang di}ru mi t*i rlc f't, r,iy .i--brl, €ia th& 1 #" \-;1,., l:r llhiii ,'lrrrpg mirrir: {t'nn:-l s'l :'l'-' r'6c!o dibmir*ng s(i eae ctrilirr d6 IA s ri Q sao'chr: fc= ntrn' cd lhu'ng ve,,:io nfi{n trrr -j-. }ii0r: ilo t r1c 1i ttizll\ r;hilg liqn, khi.iolfh-l-l = (ti3)frlo ta 1,hrii ,'hr'rr:g ,rrinh l'.1.r;{: ririitt ;l*.i "',it tdl ci eric 1 sd
Dd lA. diEu m*u fhuAn. Vi ttr6 cte dQ diri / cria su kh6ng c6 uhftng t5 ttrring) mOt c6ch thr tien tiit ch c6c vdcto dich ehlrvbn th0ng thb I0r hon mi6n li kh0ng dtroc iim r{ch rnAng }r.ry }irn hiilg s6,t > 0 bfit itr' duoc. f ira la trorig chfing chriag dtqh I4i, lihi 116 r.6n c6 kdi l.tin r:** illnh c6 thirng v6l:try ng*.r tiry j'. {!:ith vi th$ ma ditrh if (chri d' lir urirng ctto su r,i r:& phr:p l:i€in d4rig I51 Braoe tlir 'lrt,.rt clrtiitp t,rirrh 1a plili ianr trErr r:rirt{ ntdl rrrit pi:ingr, $!nli i! Braoc SE i:ho ta mdt kdt luAn k.i ditu CQ r$t nhilu cLch rhLtng ntinh t:ho djnh l1t k;' rd linh chit'hilh.hos, n6u -i:An rlec coi hirih trou la :n,5i ming,:ao su ",a- inh -.:a ii&n tuC lA ph6p ia ni.r'" sorig ci 4d cich chung nrinh gi*i lLifu bi6n da:ig li*r; iut: i1'*ri5] cr{r) su dd {mirtg e:ao & da3, Iti rlon gii* r.it ci6 hibu nh6t. I lim hiba s&u t$i$ffi to6* hga ph& th&ng uAr suY srEH rtr sflr m&wm sffi#{ fl& sE rs vAN ]'u n sd thuc l;h0n-g 5rn C&*rg mi.nft : Theo dinh nghia v€ jps te th{-:r wu' b5i dirrg thirc sau : P. to Ck thua s6. mii ttrua sd ili I ihng /< s5 {i+as .{ .... -F aS l(tr > l/d*Ir..d; {l} h4ng rni t{p hSp ft sd hq.ng nil lir m$t tf,phrrp eon * phEn til trolg n pihBn tii' at, sp,,,,, o* Y#i Iidt dfr.ng thr'ic {!) rlircn thuQc duor ggi ia biit t4p hSp con thir i i& diing llir?c edsi clro :t si kh0rg Sm. liiy gic:'ia I ,, d,it o,,. Sei ,Dr ll'rirh t:it ei ehc ibng l* s$ hqng tir n E',, ".'., ! *'i* t+p hsp I ",, , o, sd &I...., sn. Iihi dd {a i:o : Fl = ol aE." rrn ; Fl = iol*ag)... (et*an) (ir2?83)'..-(4o*1i-a3i;,.. i ?-i=dil r6t2 ... +ork va tidi iluoe P* = I1IZ...?'x Po * *p*a2t-.., *an' \ui cac k-i l;iFtr rriy' biil a sd'i ir = e rrk. Cql Cij la rbng &- 1sn ]rirflg thu drroc fi*ng thirc ii) ciuqc vitii rlriii dang: Plin > i Pt, vi l:ihu ihi urOi 1i6n hi giita hai sri Pt r'oi Pc' tu tbng L bo di sC h4ng c1. vtli 7:1,2..., 'T* d6, nfi1' rle *.'in rlB r6ngi hon iii ngiiiAn ciru lr. Ta nifji 1i6n hQ giii'* tit* ad Pt, i = i' 2,.',. n, r'ui nha.g ! i;b girii qu1-St vrin di nir;a, irud'c ti€n ta tlB riing r,o I{i + Htz+.." + fl; = (trr * I} x ry6i riSi riua-n hS giiia "Ps vir Pl-r viri &. = 2, 3'"., r. l',ir iniii qua* hq\ r:iir iirn duoc chl ra fr dlnh lf (ni, *.., *6i*)=(/r*li ?'1. Nii*ng itreo b6t tI*ng drr6i dij ' thirc C0si : Birh ly: \r(ri n s0 iih6ng Anl sl, ozs".., an, ta L e6 bflt d&ng ihr'r* : .-, Ht* H2 + ... + Hir)- rll oi ''r"' iri I n0n r, n*i'--Fl i' nk \ k / b*| \1 ,k?,k*L o r' > .51/ F tl ,--. i ^,\ ni rrik, v+y tfrt II 1**l Y "i.'. . r.k al tl'ong ci6 Citl ra l11rp chi,p C' cira lt {C.. - .-N k- v*2{k{r. .e!(e*e)! * Pr fl tz ".'Ty1)> .t7 i=1 (*V4"fA) ,13
.-! i'[ fr& lli viSc chirng nrinh dlnh l-v. Thec, bb dE r lli L.-t vi r rii *hri ! sd thLre sd ,'ilu P,. *l lA C.^ 'ri cCr --t n ! I ",1 ,tdff^ fJ, tit* li r:ir \*-' / i*l Ii I .k k *T 'nlltlrfrc hS k Ck = (n-,rr+t) Cli*l ,r, 116 tup hgp 1 .&-1 \'n N r* *-:' I,I \t(t fr* ,1 b n llr ! H =l lL 1 I -.i t7A 4 --l fr,t! (s) l']f "'u tr'0n chi chua eJtc tirii:r --'i ri'rr P,li*r-., -i'l'l n6n la c6 trong dd k! h;Qu p ri * +1 #g,.. ff:n,Trudc khi ,i.. ,l * ,;l -f '' ' i=1 "l chirng minh ti€p dinh li-, ta churg minh bh d€ va tlreo (3) ta t'o s:tu t ..,1; n-L+l 8b db: t) fAua sti b.it ki fk_t la phen / ft-i \Ln l'xlP k cira I \ I I k':1 tr? efia,t$p hqp c6c thng If1 i.6i i :r; 1, t,".,, ]f I vi ilirrh i-,i cftng drrrrc c:hiing rninh xong. & J = 1,2''.', j{. t'ri dinh li' ia sul' ra rluo-c n;di trar hQ gi&:t c1 2) ?hua sd b$i k! cria PO*, xu{t hi$n c?acPy,li (n"-*+l) .Pt Fl-:. = 1,..., n tr'ri ni:art' Ching lian '--iin sr--r siintt vir Tn {2) su-v lzr : /i .[iln ir.ong tflp hsp c{c tbng t/'. r} tren. \.e t*ong tu trr c6 : *Thr1c vay, gqi B;i stt -1-... + er,."-,: l& thua s'J bf,t k\) cria P, ii*t , . Yc'i s,5 at,- iih6c, ta l{i-r l{ ' tt'ng: " S :r 'st.I * ut^.'l-... r +6t,K-l . = B + {rl,.Doea+ x ir6-olr,...,orU ln t{p tep 59* k phha t& trong n 'sd ,r1,..., au, n0n"tbng S ln mQt thta s6 cfr* Pt. V{-r', theo dinh nghia eiia ffi, hibn rrhi0n I ii l mdt phsn tti ciia tf,p hsp I ,j $, I = L'.., A' i j x t,2,.,.,,k vt phbu ! c{ra bli dE,duqe chung miiih. l-ai xet vui thua sd brit I B oi, *... * o,L _ , k.v' cria P,. *, li-l o thrt:r s6 B xu$t hitlii Rd rAng t t d{rng 1 lhnvi rhi xuit hign tlong c6e tbng fJ1 dugc xe1' dun$ tr'r c6e tbng-S iluoc xAc ilinh rhtr tr€u. (n*li* I lln - k.l 2)(t - k+ 3) Nhung do co a.*(kr'-1) = n***i sd al* khAc circ 8ub sd ot, r st2 t,.., JA otk*t tl'ong n sd crr crr.,,,on r16n ... gite P! r:i 8l ta cd : c6 thb ed (n-.k* 1) tbng S cd dang tren. Vi thg thrla s6 B dirqc xufrt hiQn dLrng (n*.k * i) l[rr - c\ clti, gkbk 'Pro (4) -JfJ], i= lr!,... if rA j E 1,..., &. Bh trong cac tbng LCi so s6nh Pl vtri PE4t. 'l'1r (t) thal' .t b&i dE dd duqc chirng nrinh xong, (e+t) va ehu y i l4 l
A--;- l. ---"- i:nk+l = L: .l; n In co n-*li ?i *1*2.., oo. (6) i ti ',C* p,l-k "} Dr tr$t rtirg ttric COsi ituo-ciirng dn*g hiQ* {-.: }"" h.*1 Fu. t l'+r .r qua cho rririi'u Irii toarr rtrn cci vin d'6 dLt ra niy, Ft 1A c}n tirn cirp b{i dirngthric bibu thi :ndi guan Kdo tiii q,r*" lrlni! tucing tir giira l,a Sn ta cd : :... !l hg chat r:iri lto-n bdr ding ihrr':(.risi r'.ihv vgrng c{e b{t .rllng thri'c uroi nav clro irng dring hi$u l. 1i qui hnir" malllt hun so r"r'ti b;il ding thLrc C6si' f* lt:\ {-., it P rt .r:? {k. D (s; illit dilu ret t}it ri li, ir) (6) rd r,\cq la c$ ngrtlr [i: \r, J o iiai b6t rling tt!,re : L (6i hqp ia) vrli (ii ta thu duqc ii6t qufr s*u; v t-- (rt -i-gg .J*' ... -F *r: pi!1/C't k k L- {l {r" c" in /t -{n/l''" ,ln il -.,F rk ,..,, *-n P n 1 Ti r r'alt o Ti) di$ ta Ci t1€::. : rA* P''-n > lfala+..,.4o riii nltri l'*1.2,..., _i* ii Ii ,k tin p'ilc 1l 1l t- .i, l> E, n, biBu ttii mdi quan li0 chht ch0 irnn b$.t tt&ng i! t' I tlri'r(' (:0si ci:o n sd liir6ng ln. ?rao tlbi. v*i bryrr dr.rc TU e*sT BAI TSAN THf &&E HQC P$"{RO}.IG dii to[r- tiri rio Dai lrqe & rr*c iinh & = i*8rn3)Li. Su1' ra phLrong trinh {2) e{r t plria \arn ihing 7 ndr-r l98f t-c efiu 2 ughi$in phtn },:iit. Edv gio ta chung rriirth e6 it nhd', [r!t nglriQm th6r nrfrn lX!< ] Lhi bni (ihJrlrg niiiih rdng riii nrr'i giu tr! nr phuong toirn s0 d',tue giii qtr-r'ei xcttg. 'I lrdt vt)5.'la e6: trinh: co$irE--t' n)to$3 .r = i) ludn ci; nghiQnr. X1. 13 = - 1/? (lii, Xg iA I nqhi(rrr cfra i?). E)&-1, 11 tu6t bni to6.n kba {ern gi6,n, tuy nhi6n 15r.^rzl=l'l nhiEu irr-,c siah cfia trt
r t1 i^ fo chf.r,g minh khi ry += 0 ung vdi cos* $ 0 thi 1::hrrong trinli dir chc i.irtrg lu*tt ld ngtri{*r' iy Th$t';$y,'liir phutrrrg trr'inir r.l;i tl:i.r trt r:ti tl'r'A v.i'it i lhanh:;p4ilsJ* - - ('(,1J,.. t:: tn (?('OS'l. - 1i * -'= csai;B €l ne llr-r.rszr * lll *osx =:**1ini["\o m#0 "'A cosr 1= ii 1. i]iit. .i - ,.t,:, ljri',, !ir,rri i.ii (, t ihi: t2-H"-1r,1 =-lrrrt. (5) :l"l:.{n lh$-r' i'iritg s6 ughi0rl ,:&:r phrrong trinh {j {3) phr.r ihLi0* l-ao st, giiiu diFrrr ctiu. dtrong tir,;ing I ,i i" Y *m : ii*r va phA;r tiugrng coilg U: itI" - I )/S t dunc girii trg.n bii'i hri dudrrg thing X - * I vii ,,! I it .Y m t.'lr'+rn iti', liri vd trlrr rirli pli*.rrg lil:i .Ilr i iS1' I rfi sE lninh hera iiiii tlt.r. Dudng cong 1':(2Xa'-l)r.E ert tiel'r cr'LE-11uftS lir. "E'xr (r, tiB]-rr e irn si0n tr "n2.Y. X[c dinii f 5#l; I'r, cd . C* dao hirir cip I : 1"m1i,X'--=,,. e *,t tri:r iioil! - l;r.1,l1=rizt t,/i - *ir): - t/iiz. tqi : { * iU z,o) vl rVill':, ol. -l -|{ f tcrl4i. f { -* 4} -* -. 112 { u. Nhu vi.r v6i rnoi =F 0 ttri-ducrng thing I: lldt khrio hlrn sd l' i*) i;cn trli: tr6n doan -- tfnr Iu6n tdi ria tlii I'- (NE-l)/x (i xl < 1). * [ 5f ,4, 1l;+i ndn phii trrn trii il rrir"i L I diFrn Suv ra pirttnr:g irir:lr tl'a cho ii:On eo nghiFil f rr. ruG {*$t/4,:ti't} sari clro ,/ (t'i = (i. "lirc ki ro lir iYlr$n reif : Eii-v ll rir,)L ,:5cii 3iiii kiri tll,tg d&iy ni$t lghifm ciir.r phtro*rig irirri,. 'i ii srtv r:r r'*i hgc sirL dir sir duug qua phi liidn thir,-: dd thi. i'loi giA iri cirt m thi lhrrcng trirh iu0ri r'6 nghiQnt .$61 voi l;dri tofia niy hqc sirrii tlii cln rri.lry den Qua bAi r.iei ni.v t6i rnudn trlo ill;i roi nhirn.e I i tir:h chiit c&a him sd ii6Ir tr.rc Llii bii ii,;g st hoc sinh .'liuiin bi thi niio lJ:ii ltuc ie: phai tiin l rilrrqe gili gri_v€t irh& €9p" tdi CE tim mQt,irirtlr tdi rttr cllr.r rnrit brjiLoiin, i.uch !ln'r ba.' . Lt Iii Amh lhtii ._. ...,1 Dnt I fsl = ..i.t,:.i' * *:cos?.i: (Irr&ng l).7'. T. il' An Sion :Ye1:id Binlz;" ! ffi-E-Nh{ SA#FS DU TFII G!*I YSAN E{A nSi ; },rgu3l* l*}ucl*g Cudng, E{ng Irfi Sae, &'gr98n Ytd! Quung' .\'5ugin I.fiii}u:'Brraeg, Nguy0ri ?'iPn ilir;:g, t1.i rj'i"); 'dri Qr:3j r?uoag, :VgugEn If tn,.Yum; {i2 C'f i i Iliii 'i i'lrrcltqp}. .r'gagrin -Qaang Q,io,ig IJiing. #5 r,Ji, itl Ci, DFI Su pirarrt). lYgay6n dnft .{Jinr,,r?'r'inft ?i t;pc ?-ttitu,:Y.guydn ,|'hi Ftae; Ngiiyor 'rn.i{ A[t'o,.,\guyin Ji{rn ,Stnl.-r}6 ff&.i l;rft, (t?ALe Q1j l}6rr}, 8do lllidn Crr}fig (lOt i:lru V5ii An);:ilb ?'riu P31rodg(l:trt I'gugi-n frich tridn (12I-t]. (lrrrcrng ltint;); ?hallnTrung (iiJ B -lhfng I-c;ng) : t{gri T'JranA Ph*ong tI2 P). A]gt1tr0_t Trlan,h Hd t12 q)' (L.f lhurvn.s- Ki€t): r\lgrh_?a Tia* *lqnA Crr&;1gf (t* iOanPhu,-r1g).Thinfo phd Hb Chi Minhl Piqnr TudnHug, rVgrty€n Qutng ?'iu}u, ?'G ?'r{ ,r,r&ctr:, &'g,5 Aia,l $Afigr,&IEngEn r,ia Bdo, (i{J CT'): Yd ?'ftantr Ifting, }''d Dqi. llodi.Btic, (tt Ctt; {LO II}ug ptrorg}. Tr&n ij*atro Itfirrs (n i Lrrclg \Iir. Can}; N.qd dni VI (11 46), }'r} Du.u Tdrn (11 d.?i,-(Ngui.?!n Trli); Ii,&n Anfr Sc;i i11 .\3 IguvEn'f irii ilinh)i Ngrrg3n Quaxg Thl}u ttUA t6n 'l tro luong); If,gqg6n ?'hrinh dc (11 .{s Ngu,vir,. iltitrnng I-IiEn) I t{gugfa Tirernfi Quon(12P -rVgut/a';, "girui, Yrla troc yr'te iirn) ; Xi* -(12 -qr pirt, if,ir+r) ; Ilgugr-Et la.dn i(trdi i-tt 1,r Lt Qni. DOn). Qndng.F{srn Fri }{ring: l,dtn l:urlg 6fang, Tr}n Qrrin, tr{i;agi}rr ifring San, ,Vgr:gdn P,nri Qui' Nguyla $ssc Y{in .(ii{r{r, (1{.} C't'}; yd'Xa4 *filti&:gugfn G&ng Cli, Ngug|n fdn flrng, (1i i]'t'); (Ptrau Chtr IrIuh).;"1 Ird-rt Si;rh (t? Fl,Niii I'hi1h); fr*ri tr'o* rlung t::/: r'nai PhiPn): ?'rusr,"g rni IId (10 C Ddy Xu3,,€a li. I ' (Cdn nta) In t0.$00c tai xi ngiti$p irr ?f; tidng BD HI NSi fii{i: ldoo Chl sd: 11884" Sd in 35184. Giri lLru chiFu th{ng { 19S{ j