Toán học và tuổi trẻ Số 109 (4/1979)

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Toán học và tuổi trẻ Số 109 (4/1979) giới thiệu tới các bạn về bất đẳng thức Éc-Đô-Sơ; những bài tập về mặt cầu; công thức Sin; vấn đề về so sánh các số; một bất đẳng thức đã được giải quyết. Đặc biệt, những bài tập được đưa ra trong tài liệu sẽ giúp cho các bạn nâng cao kiến thức về Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 109 (4/1979)

t: Hor rOAN lISe YI$T NAil i € 5t r Y -t . BAO RA i-,rAr THANG MoT KY chfi nhi|m: NGUYf,N cANu T0AN 't'ha. k$ lda soqn r HOAXC CEtiNG *** " DAY n6i:, {2825 lr Trr.r s&: ?0 Trin Flung Eao - Hir N$i # .t., ,ty.. Ii$I 'cac ,1, { r. wfu cauv rY BA^N 7'N YEU TAAN i^ j" PIIAT TRIhN THfiM gfiT PAgC EC-B*-S# $,.' i{ TE{TTC , ,i^r, a NcirvfiN cONc QUt .: tf}AT d&ug ttrue'lic-d6-ss dn duqc gi0i thrQu c{c ttsng th&c & (1), (2), (3) vn (4) x&v ra khi lp lrong !r5o 'lo6"n hr.rc v& tu&i tr6 s6 I * 10 vi chl khi tam gidc ABC li tam giAc illu vA il/ n{m 1Q74.Cd ihF torn t6.t'nQi dunu bdi bio tem cha tam gi6c i16. !6. d0.uhu sau,- NLrn tgfs nfre tohn iqc Hiig-ga-r'i p0 th{y rn6i qu*+r tr$ gi&a c6c bdt d&ng thtc i)- 0c-,10-scr, ilor:g khr nghi0n irlu c6c itnh chf,t .tr0n, ta cla bi€t th6 nio lri $i6 trf trung binh cria'c6.c hinh phfrrg r1e kE xu{t m$t bfrt dfthg cria c6c s6 duong. Igudi ta ggi gi{ tri trung - ' thr?c (sau niry Inaug te rr 0$g) nhu sau: blnh bqc kr(k lil mQt sd thgc) cia eac s6 duong !- ,1, *f?l ! Rc> 2 ({3 + 6t6 I dc) (tl f,I,:1ra,..., t'o II su trguil d; ni, Et, /ic l1i khoang ci,tb t* mEi alEm M x' (rr, xz,'.., ,,)-?GT;E;J;TI6'- t M n'im lrong rnXt ph&ng cda m$t lam gi6c ;{8r' \rdi"dinh nghia d6 ta'th{y ngay bf,t d&ng thric '?6n cAe rlinh cira tarrL gi6e: du, rlb, r1,' l[ khoing ' eich t& dibrn M ddn crrc canh cirn trm gi6c dd" (l) c6 tlrB viCt dtr0i dqng il ,rAm sau ugu{ri ta d* chftng miuh dtrqc }rit' ,lrr (1i., lll, Ro) > 2 Mt k)r, db, dc)' (1) dfi,rg lhitc nr\y. (Trong b[i llAo c6 cho m$t chttng . m!nh,.srv c{p cria b[t diing tit.uc). rt6 sau \r'gu]i Cdu c6c uf;tvEd[tS tbtrc.(D,{3), (+) cnng dE ddng ta uott chitiiE mlnh duqc nbtng'b{'t dfirrg th&r: itrra ilufc daug i .. ' co litn cluar €dr b{t dlne thuc (1) ; fi6 la .Mt(Ra, Rr, ic) V z Mx(a., du, dd. (rr tro-ng iI6 "k lBn luqt nlrQLn.cfe gid tri C,I12 r'l l Nhu v[Y, mQt gi& tliiEt rluqc d{t ra ' b?lt d&ng ii*" til ,i afins ubi mqi sd fft+' ''6t:
t. Tr&i qua nhitng sg tlur iOi nghi&n crlu, nhd }l{t kh6c ta.ed MA.MA' :1 truy ftqR'2: i, to$n hqc Ao Florlan nam 1958 di chung rninh tir d6 co .Ri -'llR, hay : r&ng b{t tl&ng thuc (5) d6ng vA chl r}rtng veli /li = IirX,/It*/?ri?c \ nhftng gid tri c&n /c € l. * t, t.l, tir dbng thoi e &ng 'iro'ng Pul/?"Brll" ,t, chung minh duEc ring vr'ri nhfi'ng s6 tirqrc t u fiL = E, [ /c khdng.thu$" [ * 1, 1] ttli tn lu6n lu0n e6 bdi /ii = ,liufiu lla,Rnllc t Vt .4{p {d", dr, d6). (6) b{t d&ng t}rL'rc giita .Ra, Rl, fi., cls, d5, dg' gi& sri lar a ) $ (9). Yi ,{BC ld m$t tafir gi6c b.4.t k] ;} Ti6p s*tl ddr bf,t rt*,ng thttc (t) d& tlugc m& n€n b{t d&ng thirc aIE iir;r ra ei.ng dlirng cho \ r$rrg vir* cta gi{c lili vir trcng khOng gian, niir tam gi{c A'B'C'. tsXrg cAch thay cac giA tri c&a u ilo ta SA tilrs tlirqc nhirng k6t qu& ttrrJr vl' rri, rli, rig vn I'li, iU, n; & (i) vii (8) r'?re l;{t d&ng thire (9) ta s0 thu dnc.rc mOi b{t,3&ag lhuc Ntli tiip uh&ng ;d1 ng,hi !r0n. sart dtAy.si n0rr moi. I;ib don gi&n c6c ptrep tinir ta nliin :et l0n m&L ph*otg hu
\- Ddi soi rrh&ng s6 ihge k)1 hole Jc< *1, vi thrpe hidn nguydn t6c dl n6u tri:n dii ph6t thrrc hi€n ph6p bi6n iiisn d8i / rsri hiit d&r-rg hiQn th€rn nhftng quan irs hoi . Lh&.ng'hgn ed thri,: r6) r(t l'luriau ta cu th6 n€u [Cr b{t ding thric sali riirl': ,r,ih(Hr,P". "&,,tig, ,Qaftl) ) do.ltu -f drftir i- d*I?. F 2(ddc * d.d, * d"d6)' Itl (14) Cic b4n hfly chirng minh tr6t ti&ng thr?c tr6n 7{i.rwv(r{alia, drnl, dc[i"), sau. d6 6p drlng nguy6n t&c d[ n€u 116 .tirn ra Ch&ng h6n vfi lt':2 vir k: -Z ta lhn l*qt cir rnst tr6t d&ug tht'c m4'i. 1{6t c&u h&i duqe ilSt ra: N6u ta thr.ic hi$n nin? +niai+r3ni> phep bi$n dbi / tren rndt bdt dfi.ug thr'rc n&o dir (15) li0n ti€p h-ai ho[e nhiEu lin ihi ti6i1 ehirlg ta ri\ ), tiri?;nf, + at ni + {i3fi*2). r:6 thu atg-c ltdt qu& gi mni nira kh0"ng ? Criu tr& ldi iI: lihorrJ. I'hEe hi€n phdp bien tli5i / ti0n ii{p hai lS.rr thi ta sB tr6' vE b{t d$,ng { /,ilrr-i -i'r r,ri;;?, + (r li,Jlry?). ( } ti) thuc han tI.\u" iirlLr]'llu irh{n cira di6ir ir'r iiuqr: B6t rlirg thirc (i-i) cung- t,Ltr *ae tr'-trong hr.r1i tim 1lr.{1: & tinh ririt ddi trgp c{ra 1,hep nghich dac bi*i e&a nd la cdre bat d8.ag ihfte (i5), (ltjJ rlao; n6i rii horr 1*: u;iu la thgc'hi.an iitu ti6p khdng th6 ,< tang cuorg > itoti duqrc nir* ; nr,ri hri iln mQt plrep nghii:li d&s ni* dil reis rnSl c*,ch lihir, d5i voi bIt d&ng ihitc (14], ti.'sd hrnh thi ta iq.i tr& vC hi;rh d6. C.tdi cirng xin n6rt, mot gt.r'i i khAc. Ctrrg ihr.rc ,'l{r (liufi". 6"Jf", Iir.fia) I ,t/r tr{o}iu, r/ufil' dsl?e) lu'l-' hierL pbdn nghich d&o pqc ilf , phuo'ng tich 1'
\ .- Chring minh ring ti' s6 -tt/r bd nhdt (:3) SAi !.. Ihlng rn{l chu tli qua 'J di8nr A' }f'C- khie)(7') lil"d6u, trgng iio fi i f{n,luqt ti bin tU .io ""fl- ,+i *e t rri{l.eH*-{{-i) cho rufrc; hinh m{t. c},u n$i ti6p vi rtgo4i ti6p. tsAi 2' M$t rnlt b[u hin kitir r ti6p'xfic voi Hly thu tiep bnl-toAu tr0n vE m{t ph&ng -a" ;a\lrra" *au t-r"r, ch6p lam gi{c tei tty: ta}] ;;;; *ut ev. rbng 3 g*c 6"dinh cill :ti"T4t BAi .8. G*c tam di$nr dinh D trong ttt diQn' ;;;;r-; " riel' tinir cantr u6n crla hinh'eh6p' ,,IBCD lil vn'Ong. Gqi O IA thm m[t chu ngoqi , tsa! 3. Di6n tieh xuug quanh ":: ,*?-t lll^1 ti6p tu &i+, ay vd l1 li b6n kinh eria u6' chOp tam $i{e hfrng S, ehu vt 66y }ing 3c' }t$r' 'ir) Chrlrg minh-riug tbn lrri m6t tam giie mi l tqi ,i+t'.i"-it[p xric v'*t- 3 t"4nlr c!a.tt"ar ts"i troog rung ai6,r, *6i cauh vil cit efc cgnh bdn iie* *At cinh. ch&n$ minh. r5ng hinh cbdp ld b) Gqi S lA. diQn tleh cria tam $i6c gy'D,1^ n: i- ae" ve ttml'b6n Uinh.m[t s[u' : nn-d DC= c. Chitng miuh ring aDc = 4liS' Bsi 4. Cho hinh btrop tu giaee'encn vil hinh (dB BAi 9, L6y hai diph d6i nhau trong 'htnh lap *a" "Oi-tre;. il, ABdD ta uinn I th*ng can uhuons. rbi, n6i e{c trung di6m cfia c{c canh ii ,,*4, nti6) co canh h€n bllg vir g6c nhoa inono?i huu hai ttirrtr iiy, tqo thlnh mSt hinh ;;;i,.'ru.J *+, ii€n IPD vi rlpc l! d6{ .!at tam tge gia. d€u : thi6t di$n nAy chia hinh i{p phr:ung ;i# ";t.l ct.( Atr'"Ilnh = PD; BP = PCr*tao .'.6'" :l-'i ra ilm hai phhn. Tim l:6n hinh m5t chu nQi ii6p I i6t gO. b6n kinh r tta hinh eiiu r81 trong m6i pira.. tlr"o cqnh tt c&a hlnh lSp phuong' ti6 p. Bei f,O. ba mdt ehx c6 chuug 6ibm '{' Bi6t XUi 5. Mqt htnh ehu (S) tern O ti6p -xrlc vdi rilng kh6ng ed dudng thEng nio qua { la^i ti€p srlc M tren mat pU&nc (Pl tai f' Tu rnOt iiiEnl. bf,t.kj. vdi-cA I m4t cau 6y-, hEy chung mintr rtng 3 miLt t,iP eit (P) t+i A vi B' ;;, ;ffhei tuYdn clu di oho'cda c6 rn$t dibm chung thu hai nira' 'a) So e0nh hei g6e Ail[B vd AIB- t.Uac;. b) csi I' le diEm &di'xriug ciiriMr qua n[m d8' Bii,l-1. 'tt m6i 6inh c&a-mQt'tir diQn d\rng i.-:hunp mich r&ng #;;ii"; A,'{' 1" "'i'*tg m{t ph&ng chua t6m tlucrng tldl ngory ti6p eia ' ,t"ou'*0, met ptr&ng vil IIJ' luOn lu6n Ci qua *at ifOl d-i€n vd vuQug gdc vdi m4r 6y' Chtlug *oi-u,E*'"0 glrit, ttt''i rr rli chuvBn tren (S)' giao nhau tqi. 'minn riog 4 m[t ph8-og thu trdn SSi 5. Cho i{u&ng trdn (C) iA'm O'.d.u6ng kinb t$m rn{t eh.u ngoli ti6P tfr rli$u: (cl tlm'dudlg BAi L2. Trong tanr'giot vu6ng, ta c(s r:p-rt'- 'an::zn. eqi (S) la'Ltlt chu nhanp:i"c-(P).tai a; i.-ur, ls;t-("sl ti6p xrlc v&i m4! v6i r ta bin.- kinb tlqong tr0n nQi ti6p' o la ! ;";;-tt,hrg (d,'nnrn trong (P)' c{eh I rr0t tloan bhngl*o':I'L tuong tg trcng:khbng gian. (d}' riru "a) Gei L{ lt mOldiqm $1"}-1-'ET tran . BAi 13. u) Troug ,tam giilc '{-BC' chrtag minb' qot ni;n giao di6ni M,' crld *U.t trdn hQi ti6f tlnng v6i tAm ring ndu tflrn &udng I.,r) Quf tirb c'l a il1' li' nrQt '*.": :u"._, dudng tit)l' Hiy auing trdn ngoai tiAp.thl AB'- RC = C:l va trcng tAm cIa tatr giic ciirig trring voi hai tim noi tr0n' tiJ; #; kinh i cfra no rir tirn tfirn I cila dudng trr.ln ii )'. h) }[iv l0ag eho lrud::g hq'p tu cliQn: r"6ri tim' cdu I c) Tinr'qrii tieir tira t ktri (cl) tli'ctrirlFn sao ii4t i'Lu tOi-tiep lrirng--v0i li*."1'rr{t + . t'i$'p x*c't1*i-rlrtcrng frdl tam fJ ]:6n ngogi"iiJp thi c6c [n6t cria tu iliin blng nhau va "lr;'";'"A; kinh c. tl!ng ta'n cia tr! tli6n trirng v6i.hai't&rn n6i tr'0u' 'i,i u bal z."cpi.o ie iam rlrat eiru n{oai ti6p tu e) ,Dune tit tli"l* ntr4n nr{t clu (si vdi bti* rlign {?'},: '' d'ecn: G, "Gr, ti5, G"' Ga l}rr luq! li !irirh r lirn mEt chtr uQi ti6p vA nh{'rr. m{t +hu (r) cira d';i iS) vsr birn liilh It I&m rnlLt chu ngoqi ti6p' trang ' ;;;s';d,r', c,ra v'ir "'-*ttYlat ;irfn.-riintr ,f,'g,i,13, ro }ir t*tn qr{t c*3 di qua G;GuGrGa;' tlo l? ) :ir. ;;;il; chi6u vuQ-ns P,oc cria o't:'' I? l+n m{t Ent 34.-r:nq'tu diEn A1:4,z.43Aa Chtng minh pl;-rr! ,tse I&n luqi ia ua, Gl' r':s' Ila; giiro 6!3trr' rd *g e6c cgnli' f.i,44, As 'te; vI ti trerng thlng ndi I *,1*-iil*,-.&i ni6t e$u ii -q; -+ i-r,i..,iig;lflrn G voi i6r* rl$t ehu nfi tiip e ring soug n). Chrlng rnierh ring u)o'd : t lt!llt)" .onj'rui' rnQt rn{t Pd&rig' ' ' b) Ctrung minh rhng ndu {r''r) iidp. xqc v&i mQt 3. Ilh&ng ,{ea gi'&i ragt'sd &'*i tr:{q ban *4,'"I; ji: ,rrr ar,'t iil* t" c6ch dlu b tlou "'ia irlY t1s vE hinh)' :r tir.-:o xric. l,l6u i*)ti6p xirc l'Si:2 mXt (o) cria (?) hfng nhau;-ndu ti6p Bai 1, ]-am .t nfr.m tren lrl.r7 (d) ctra.tam giric 'e'"rg tT) c6 5 cqnh (1r) ii tu diqn XAC,'trcp D:ii1m trern O1' U4t.ph&ng (O,tl) i ui+t thi dEu' 1]ibm
ii tritt pli&ag tlsi .";iing. r:&i nr5t cdu tlf; r:1:r.i theo s* /c. \"fl5r q:r! ii,llr e*a l ld ilurlng tr.&ir inh,:r,a ' dir
-=- I { I { J< .{CB f4 /\/\- = ADB = 3. Ari] -'A,CD:=l', BCD*BADE?'"' ^ /--2\6 4\ m{t phi.ng,tidp xu* v*i (s) C&tlt rh,rng: 'Dqtirg rai 0r n$t t3, m[t niy eit t,S) thr:o (Ct). TrBn -\ B.{r:i'. Trong c{c'ta"nt gi[c;ABC, ABli CBil= ..-..^ (Ci), ldy di$m o l41*!:i rbi lEy G1 trong ({i1} \ BCD, ACq ta co lhn l.t9j' !l Z + t: *3;' .Q); --2iJl)'-1'+2 sao cho ArGt:'tl A2 - gr2l3. Snu d6. ldy -EI sao it+?+i::g'(2); 1* 2'+ -S=zo. 3' + cho cll: (3.':l) r:ril { t3i2t ((:it * ofit} - (3,'2)>< (t) (?) 1{i. Tn ua ta IrGl: !j 2= 1'+ llrr + (r/g)-/7'i -- r'+.;z]{ 21it. Tri II rlun[ AB ]- ? tsl: tri (3) vl (4) l'r cd: i '.--:J-:3' 9' OiIJ. lisi G1 vcri Ol cilt (S) tai l)' t&i to$n c6 Cuf:i--.*ung 111 ii') va 6 srr:' ra : 1:1" 2:2" -i:?, vd s6 nghiQm hiob- '- tir* cic rr;iit b&ng -nhau; khi do lrong Bei'14. E{t 51 :s, 5z -- s'i-d. [)'ung d6d Ia iim G trring ro'i ilm m{t c}u n$i ti€p vir rdm ,r,5r ngi-,r; tiSp. e) Fftdn ficir; '\'f 4t phBng e0rg sa! cr3a crip sd eOcg. G1:6 lir tr-gng tArn "4,, {lEC) clt .tiai "}o r:goqi ti6p therr- -dlrldng lrdn cria tu diQn, ta c6l. 4{){} -O-1r i-,"'+ t.}'{4'Ta (Cr) c6 txm or. b6n'k{nh nr =/n2 --r2' Qua D da bi6t rHng vtli .1 diBm 3i meqg c6c..--.> "h0 s6 cri f, i* A,lr**it ptriins s(lng soilg vdi nr[t ph&ng \ABO, cit rnAt chu ngoai ti6p theo ilu&ng trdn (Ce), tht t6n tai di8m / du1," .nir$t sao eho c.t|At * e6 rAm Oz, bAn kinh fir. Cbt f r&ng At Oz'-Ot . -?, '-> 1 thfrng hAng. Trung tu]0n^ DO cl't m{r -ph&ng a usIA2 * ct3 1A3 + cr4 /-'t4 = 0\ trong dd khr (,{BC) tai t.rong lenl G1 eira tam gi6e ABC n4n di6m I trtrng v0i t6.m'O efia :r:flt ehu n$i ti€p hai tam giAe OOIGI vd OOzD dbng dqng theo thi a:/sr ,- .... : d4ls4. 'f* dd sB c6: t!' sd 3. Do dq, "l -O9Z=3r,nr:y'nZ-taJz o1G1 -= Rzl3 = \t Rz - gr2lg ( Rr. tri'c Gt nIm i:+.!4 +(.s* a (ilOG + i d lr..ta+ :" d Az-1s = 0. trong du&ng tron (Ct). ; 222a a Thgi: hi€n phdp ttnh lr6n, thu ti€n lim ra ..* =.{, ttray nn & sd iri oh&n birg 4 ta tim ra rln dn-1 = 8,... l&s mmt Cr't tiOp t4c iam phep tiltr cho ddrr khi con frerbc sd & s6 'tr! nhAn lir sd'l, vii sd & kOt quh le &dg ' sd B. Khr d6 ta iluqc .k6t qu6. cria phdp ttnh - nhf,rr lA ,4 81012€5822?81: -\ V&i at :9 'o ta thqc: hign ph6p tinh tsei I/f 06. Tirn cac sb iqt nhtln to tinh rhdt 1rr3,..on9 kfut chugbn cir& sd dhuliin cfia n6 uiro ui fri sau X o cing thi ta.dtgc nt6t sd ktm S lhn's'd ban cldu. g 1tt3,;,an {,bi gii.i (ciia Fiam udti Hiin - A^8 Dqi hQc sc t cl ugc lic i ({ua .g TBng hqrp llI Ndi). Gqi s6 phi.i tim la R : 9t 1392{050032. il, N = aiatas, {ln L' &6 ri.ng li cdrc phdp tlnh tr€ii ndiu ilugc tidp The" dfru bAi ta c,i lur thuc hi€n thi'& tdr qu,i 'sd ta iai thu du.oc *dc con's6 l{p I4icic,cou di thu duEc. V{y l. \ lV.: 8 uxca... oor.r1 circ s6 N th6a rn5n dhtr bi j la cdc sd co cdc d4ng: Ta cc nirln xet,1 l& at e hi cd th&.li*ng 8 ho6c :.{.".4 blng I cun c2 = 1. Nt v* Nz:= 8....B Ydi ut .. ti ta ,u phdp ilr.h n"'11 - '!ag...orrg Ttli n =r1. 2, :1,,.. -. \V g SAi 2/LCI6. Cio !,o = n4 + 4a, HAg tim tdi c& ill cac 11ta tri cito n db Pn ld sd ngullen t6. a3' "'r. *.
Lbi giiri (ct:a Trtottg Dinlt ?riln -- 9E c{p 3 --lt,I \ \ t.t 1 - llring Vdrrng. \irrh I'}liri). Tu nh{n th6.v rEng ndu n tA sd chEn ttrt Pn: n4 + 4n la s6 chB.n l6n '*(t*rT*") \A*F'" j. hnn2. Yg-r. ta chi xdt c6c n t6 i ry:2k -f- l, v0i 1lz =..7-u2' _*"tE__ z 1 d=- 1 k ngul'dn r'il khdng Am. q a T-- 1.-112. 3 3 3 l'a c6 i Pn = r,4 -r- 4n - (nt +- 2o\2 - 2n+lnB M{t khdc tsr cd: r : h2 + 2")z - 22{u+11. n2 I , Lffi--*X=- = (n2 + rn + 2k*1n) (nz -l zn-zk*1 .il. liul (-1;n*t-rl ---:- = 0, ' n-+Oo ^n-I Z , 1) Trrd'ng liyp k) 0. i. ' Vgy lim ua=lir-n (aDc)"r .,lim ei{*I) nr.1 ' a f Ta di darrgJ irhfln ttiay Jn ) Br. ' 'II4t khac E =n2+2n:2ktr.n:rr2* 2ft oo n-> c€ n -+ oo k :nz +zzu-2.2k.n+22k =V}it I :(n-2k)r+z2E>1. M{t cdch hoAn todn tuong tu ta s€ tim duoe f'Ighia ln khi /r > 0, Pil khOng th& li s6 Iim un = Iim arn =Vobc oguy6n td. n---+ Oo n ---> Oo .il Tradng hqp k-0, a L. H' I(hi iI6 n = 1 vA Pt = 5 la s6 nguy€n t6, \'?y. Po li s6 nguy6n td khi vI ehl khi n = 1 Bili 4/106icidi phwong trinh: z7'dru* - 6.6.r'o*u* :612 -rI*, g'oe6* Bei 3/1O6. u, b, c. Ngurlr Ctto ba.s6. ctuottg . Ldi g{di" (cria nhi&u bpn). ta !qip cdc diiu sd lu,f , lu,l, lru,J rthlr- stu: a1 :atDl =b, llrl-e:v6in>2thl Ri6u kien d6 loga c6 nghia li r]>0. uo c--6jos6= n6n pbuong trinb d[ cho c6 th6 vi6t drr0i dang ,,, - 1',,,,]iil.r, :.r'I"1il], -r:l/il;A; o ^_1og^x 27 logox log^r log^x log^x 9 =6.36 -t' -11.0 -6.8 -tj Tim rlio'i han.ctta cdc ddg sd d6. B'idt rdng -6.6.1 logox - log^x log^x I o88*:0. hay 27 6.8 ; o + 11.4g o -6.01 a+0 I.oi gitii (c rla N g ug 6. n T hantr N/rn l0 p 1 0c,trudn g r Chiq hai v6 cfra phuong trlnh cho O,*'ouu* - - ta eluqc ptruong trinh tuong dqopg c$p 3 L6 HbngPhong,Tbinhph$ Hb Chi Minh). , Sheo -ithu bhl ta c6 : *u"+ /-.---- ,--"r+ (+),,"*"_r(+),, r, (3)'"'.* _u.o llnunrln- vun-1 un-.l .11 Urr-1 tur-1.vr'r-1 uo-1 , = lln-llrn-lltr.ro_l J*I UoUirUro:Ila-11;a--!&.ra-!=...:ItIDltrJ 1= abc (1) D{t y: (+)'o*u* ,, .n Hf, t khac ontln :/rr-1r"-./r"-lrrJ ,g3-- 692 * 11g - 6,:0. -r/- . Iln-lUn*1rua-1.V ua-_1 - Giai phuoag trinh nd1. ta', ituqc cic nghiene ?:! , abclf u"_t e) iJt = t, gz = 2, 93 : 3. (Chrt 3t ld t&ng c{c h6 TiI (1:) r'i {2} ta duse: s6 efia phuong trtnh bEng 0 nEn n6 c6 nghiQm bEng 1). * ud:(abc\r.r2. u:'': ,- (i) -6 : t ta nm'duqc rql i,,- (B) 7, 1 Iog^x n._i1 Theo (3) la tlur duoc: 1111 1 , ,2 22 23 ' -;*... 24 "' | (-1)" \ ".2n_1 t- (+)'"t6x :2 ta tim dusc r-6t'e3i42, AI*\ADC) X (-1)n+l o 3 ta tir4huEc , * log^x 1 "- (;J ' , on-1 = Xa f*s'o%r*'. r11 VAy eAc nghifm crla phuong trlnh ban dtu. Ii D{t pr, - - * + ...+ (* 1)"*+ 2 2" ?n-r . Khi n-> oo ' rl=1,0?:6 .ilog^t-2' Ji4,ca=6 -J"-13' log^ ' tItl pn-+p
w1@v .i : I t a,r. ts" +rst( E r/1t < troo (d.p.c,m). . i=\ aess( x r/t/i(asoo i=1 R i 6/f06 . Chang minh r&ng uoi mpi s6'tr"r nhifn n ta c6 nhieu bsn). rrufc hEt ta chrlng ,,r"[, #i.,;fia \ l--:-- i i- 11rn+ t,.2r + i/r: Vdi m,oi sd tg uhitn n tlii vv . (cria .I.iti giiii r 'Irdn llug Li6n - Ao8 Eqi hq" -'' a ;[u-tt*-vFJ. ffi .+[vns vG=] Tbng hqp Hn NOi). Bdt d&.ng ,thrlc chn chrt-ng mith tuong iluong voi,. r a.' . Chtrng ,minh , 1 l---.-:* l /----- t !// vrn+1)1' Vtn-+-tl: & *[v*-vF]. ur (tt ' V6i n :' 1 ta c6 tting ttr*c 1 - 1. trr&i n ) 1, e (n f 1)3/4 - nala'( 31Anr]a 6p dung b{t dfr"ng thr}c C0si cho tr sd duong +l(r+t/n)s/a{t+elan kh0ng bing nhau ta c6.:
__E!iF ff,- ,!*=yl+ yI , Cirrril, ha d6ng thuc trdn vt\ r,oi r,6. la rlirsc : ,67"o*t2a +b61"or128 + : (ei*s2) (yi-l +si-1) * atw kti-' + yi-zt "8 I "nrlz'y == a Ap dung bdt ding thirc C6si ,l'6i 3 s6lcltrcrng, llir Trto,ng tu : Sx-'t : 9Sx -z - iSy-3 Itrql la cd : Suy ra , o n - :l -a-q-a_ o'.-+ ['* ) tJ\ u"b'c", S' =70S,-z -*55x -q:4i1.9 Ss-z-1lSx-s)- 5, -il. , .,6 "" -, 6r ,,t 2,2 2 . .ttta6 -.r,oi{-tlc6}'.)l azbzcz , Nhu vi1' 51 vi S;1-3 ho5c cring ehia'hdt cho .l 'i ir d.i.1- slr y r.a ho{p cring kh0ng chia h6i cho 4. . & Ndu,S*chia irdt cho 4 lhi S:-s,5=-6, 5x-g.,, tlEu (a2 r bz*.:i6 (1,,,16 - 1i1,6 r i I c6t ) chia h6t cho ,tr. do cI6 nrd{ 'irong cAc sd 51, Sz, 5s ^7AO 2'2cz zrs; i''. F ph&i chia irdl. cho 4. Nhung ,1 - Sr:Ur*Ue-9. lidt hop virl i{frng tltric (*) ta suv rn cliiiu c,in olrttnq mi rrir' ,\rr = yi+vi : (yr* uilZ - 2gtgz T T. -g?-2.5=71, Ss - 9-sz -5St:9.71 - 5.9 = 594 aIEu ln cic s6 khOng ehia h€t cho 4. VAv Sxkhdirg Bni 9/106. Chti.nl1 rninh r&"n91 chia hdt clro 4 .vui moi s6 trr nhidn c. (Chri y: vi St, St la c{c s6 ngg1,'€n n6n tu c0ng thirc Si = k . k , ,t /----;i- ,i-il 9Sx-t - 5Sx-2 suJ ra S* Ii.sd nguy€n r'6i moi sd tlr rhi6n e). \ti th6 phuong tlinh dd cho kh6ng t1t nb V , +fi;-V ,,^li. c6 rrghieur nguy€n duong. Ltoi p- vit k !1, sd lr nlt'iiu. . I..T. ' Ldi qii.i (eia ban N.qrrgin t\igat:. Khrirth* 8" Or,fin : Bni 8/1O6. Chring minh r&"ng ndu'hlnh hQpchit \6t Ill Trung, Ila N6i r3 Ngd llulS Ninft;* !2.. r:iip lir Qtrang err-r; Nlrcin). nhQt cdba elnh ld a,b,c, thi tieh V, ba g6c tuc lflurrc h6t ta chirng mi:rir hdt rl*ng thri.c b&i du'dno chto ud ba canh a, b, c trcong ang td .a,F,t thi ta e6 bdt d&nq th:rc .rlt+6k i a i-b \k (*) o6.'corl2cr +- &6/.o112F + 16/*or12T > 21s7v2,. .2\2! ld'i k lri sd la rthiitt ud ,r, b ) U. ->l:--"-i. . Voi /,':1 bit d.ing ltrric (*r tr.,; thAnh m?t d&ng thitc dirng. Gi'i *,i' lat d&ng thyc (*) clirng vr5'i A-n, Itirr la la c(r, .& a"_ -_ 1*Lt 6 J,,. 2\2/ tr \hdu hai vd cfre hlt ri6ng t.h{rc nAy v
j: SB ctrirng n:inh b$t d&ng thue trAn te chl chn Ldi gidi. Ta chung minh ij6 dE: chrin:1 trjrrli: .Ndu da thftc fb):snrrn*a1cl*1*... jr:o (uo4=0t (a -- Ir) (o" - I;" I '7 *. c6 n$t nghiQm td ei thi nit it€ tiruc nhy lrr6n lu6n. dring (do a, Din n.i rcrl=i1+max la;ia.l (i 1,2,...,;r). nin u,:.lr tlrl att.:1,". co11 n;,1in(a tni a'(D'). rtEtt - Clzi,.t{r .minh: Do rt Ia nghiQm eria ph uoug . :.*.hu r.&1'. bit d*.ng thue {*) ciing dtng v$i trinh ; \-J):0 nen /i n*1. 'Ihe 1,2 (q= l,?,...,I$28) , Yfi Quang $ri.E {fr.n.S.P. \rin}.r) i,'.I]. L ffiis'l
.i l./ Bdi 3i i O9. Cho r.,dn ,l t\r'nlridn q.b,c,d RiF.t.' I Bai S/109. Trtrrs mat' pheng hiv tlm mor r ldng tbng cta bOn sd ch-ia hdt cho eir b6n die'm (r, ur *'1 116 sd vi. qnQt scj bAt ki trolrg b6n sd nh6 ho'n tbng .a. roi itd r:u.vi y'cia ch{rrrg.th6a aldn phuong trinh c'irh brsci con lai. Chung minh rfrng cd it uhdt hai s6 bing lhau, .r,,., At'!l--*'-!l''-o I I rhn Xudn Arinq (D.It.T.tt) Ilu6) sruo cho biBu thrtc"/c + rz d?r gia tri Idn rihst Bet.4/109. f,rin diry"lCrn fff vr\ gi6 tri nh6 nhdt. hinh A.RCD cho mQt di6m llf."Tim tr&n ",i*ila_v'Cllrhrng m6t Ngu116n Z'hdi I/oe (S.H.S.p. Vinh) trilm .l/ san cho. di$n tich cria iu giic lhAn dugc clo cdc duong tiiflng ilN, fiA', CM. Dtr{ cht nha.u - Bai 91o9.^'fron3 khfrng gi:rn cho m{f md,t tao thinh li lon nh{t. eB.u.J li cac di6m ed dinh rlfr. M2, ..,H, (n ll rn6t .1 it .+ l'gu.ydl Ydrt Ldc (D.HS.p. Vinh) s6-tt.r uhf 6n cho trurrc ldn ho1 l) Ung vui m6i - BAi 51109. l'im nghi6m nguy0n cluo,ug cria h{ di6m,l/ trong khtlng gian ia dung vecto, lq. bdL l,huong trinlr sai *+n '[2. .: , rt{K: 'i*1 X 'MMr . -i!]}-ul + t> o f z* + ly-r; < s. Tim qui tich rli$m K klii diBrn ,rtf chav tren ,Eio .euang .BiBn ([iai Hrtng) m[t eEu S. :" e - BAi 6lL09,.Giii phu.ong uinh ,, Ngug|n C6ng etiil sin6i - Bsin8* - llwa-o ..,-i (D.H.S.p.I'.p. rI6 Chi Minh) vdi o th6a mfln dibu kidu lsinrl < VS-IS. ' fi do euang DiEn t. Bni 7/f 09. 'Gi,ii v* biin lrrin h9 phuoug rrlnh Frqc srNH rim rol sau theo thdng *c nr 2r?-i.osnL.rr*l ,/Aai lO t.09. Cho mQr ramgiic diiu lIlC n6i *t - 2cospr. ltz tidp trong rnot'ilulng trdn (c). .Tn'mSt dibmM trit''1. ^ trdn dudng tron (c) kA IIff, MI, MK vudng g6c 2ri-4cornr.rz*1 vdi .{8. RC, CA (ff, I. ( theo thu tp.n&m tr€n AB, _s3 12 - 2cosm BC.CA). Tlm -vi'?tri. &ibm M tr0n d-u&ng trgrr sao cho Ir, d= MA +IIi+irB+ tril+.11C+MR an-l:.2cosrn ' eo gi{ tli nhd nirtt va l6ri n}rdt. + 2,rfl +.4"ou*.rn * 1 ,Q = fl1. - NguyTn Qudc Qudn .rcn - 2cosm (12(ir hrgrr.v6n Diuh Chi$u, Duong Qu6e ViQt (10*3'Tey B&c), " .B€n Tre) V Lf y!.t1 -:------ urfru. s;iu ia\u rrsAi , :1' in6'fudt.c uqc -- i' , !'-i,., r;'" t- v& HoAuc rxir fe ae bi6r troug hlnh hoc ph&ng, ri$u 6ho rngr Ndu La I'f lrigu C6o goc A, B,CiHnlugt ld (&, c) , r tam gi6c ,{BC, goi c6c cqnh d6i didn v6i (a, c), {u. b) thi ta co thE vi6t c:Orrg ihrlc (1) nhu p{c dintr A, B, a' tat tu$tlin_o,. &, c thi ia S..Ui , cd cdng thric sarr i - \_ "b/sirr(a,c) oi sin(D,c) : - cr'sin(o,b) (I,) d/sinA=Diein8=c*in,, (rl E6 lI cdng th*c sin eho tum giuc.
t , ,' ,i Cdng thuc il6 eho ta thfiy ntdi lienalt€ gis.cr citc ChiS c& hai vd eho sin2 (at,oil sinP (at,as)X cardt uir cdc gbc tronq m|t tam gidcT X sin2 (at.uil ta c6: f.ihu .v{y, m6t vf;n d0 duac dlt ra lA: -trong ,, r r sin"a1/sinltsz,os) - Lsin'(at,a:,) * sin'(at,os) * hinh hgc khdng giarr h[y. tim mcii li€n h$ giira * sin2 (a2,og)l/sinz (at,az\ sinz(ot,a3) siu? {az,as) *' e6c m{t vi e6c g6c nh['di6n pria m$t g6c tan, , + [2cos( ar,az) cos(a1,a3) cos(ae, as) - 2j/si n2 (a r,az)x di{n, cflng ntrs'm6i li6n hQ gifra c{e.mf,t va e6.c ,: g6c crla mdt tft di0ri; >< sin"(atica) sin(,.rz,ort) (4) Trong khdng gian, clio m$t gdc tam di0n Ta thdy d&ng thuc (1)- ddi xrltig aOi voi cric Oayt2us (hiah 1). Ggi c{c m$t ddi di{n vdi cic rn{i (oi,az), (a1,a3\ vil (a2,a3) va vi vai trd binh e+nh al, az, os lin iuot la'(az, os), (ql og), (or, cz) il*ng gida cic g6c *ti dz,
'f C/Ong thuc tld ciro ta th{y rndi li6n hC gifra cac s2 cr!.nh ud eae goc. cfia m$t tam gide chu. ., 81 420 411 slnct2g ab atl sinc[lq Bi.-v gio ta tim,mOt c6ng thuc cho thdy mdi li6n hQ gifta c{c m*t vA c6c g6e cfia mgt hintr (1 1) i rr di0n. a91 ao2 sin(["3 Gii sfr tlong khdng gian, chb binh tri diQn Tar (g') ve (10)*ra'c6 AoAtAzAt Ggi c6c eqnh ddi dien ;6i c*c e{p So 2,, dinh rli, ,41 Ii o;i (holc o3i) vil cdrc goc ph?ng aol aaz nol sinagl sina62 eiia cac 96r rghi diin cd cf,c canh ,,ti Ai li o;; I ,}, (ho[c a1i). ' Sg Ap dBng c6ng l"hirc {5) cho g6c tam di8n dlnh 430 {r5l o}g sine31 sina32 At la c6: Chie c& hai y€ cho sin eroa ta sin dlq sin arz "1' sJ sin (cosags) sin (ao2, 432) --= aol aOZ do3 sinaol sinag2 si noigg siD c[o? (9) s3 sin (ao2, ao3 ) (12) 430 o31 432 siirct30 sinfl31 sinctg2 Tuong tr1, tu (9') vA (11)'ta,c6: a _.' so a0t.a02 o0g siuq 0t sincxo2 sinlo3 ! $2. t (ts) 030 a2! q!3 sinc!20 9ina21 sina23 Yi cEng nhu v{y, tt (10) vi (11) ta c6: .2 -Jo o01 a0? 403 sina0tsinq02 siDa03 'b' : aLO alz al3 sinc[l0 sinotlz sinctl3 TrI eAc tlEng thuc (12), (13) v& A2' so 2 Nhfln ihu lugt ci t& s6 vi rn6u sg cira phAn all a\z oo3 sinc[01 sin02 sincxoS s6 tiru nh{t vdi ao3 vi o2s, phAn sd thr hai. i. s1 v6i onz'vi.o23, phAu sd tht ba vdi aoz vi aos vir goi s5, sl, st, .ss lln luEt li c6.c diQn tieh efia an qrl al3 siriTlg sina12 siucxl3 cec tam giirc it6i r"li6n v6i c{c tlinh ,{o lr, lA,.{: S2 c&a hinh tu tii6n thi tn (9) ra c6 : aLo azt fl23 sindzo sind,ai sirrd23 ,Ca sl .il SS diqa3t siBc[3r a2ra23 si[i[21 (15) 430 o31 432 aifl0l36 sincrt3l sincr32 s2 (lo240.3 sitrctol (9) ti cd tlt6 vi'6t (15) nhu ,1. sau : "' : So 'rl Tuong tqi, {p dgng cdng ttiire (5) eho g6i t"m 3[4o; sipCtoi == ?fflti sirrdij di$u dinh 3z ic cd ; i:il.Z,3 .i-0,2,3 ,s3 s1 r) 2 . .S2 alo est sin ct3z dlOA1g sinctll 3ta2t sinazr 3[ agm sinct3rn J6 (10) /r=0, 1,3 tn = 4,7r2 at.laogrin@6 2 Ed th cdng l&*c sin'-e/ro hinh lir d,i$n; Cfing {p dgng c'6ng thue (5) cho gde ra m diQn thuc tl6 'cho te thly mdi lidn hQ gifra ct ilin?r -4s ta c6: ade cqnh ud cd,c goc, efra rndf iinh ttl d.i|r
,l .! a -a {^Fi^t^ VAI VAF{ NE VE SO $ANII CAC 56 .I,IGUYE\' \:Ifi X. THANTI ,q (FO s6nh cAc s6, nhfi.t lI cdc sci l6n lA mQt viQc D r'{t hir6 khlrr vi ph*c iqp- Eac biQt chc s6 Gfi)ro + ioir)'o 2 Cic bsn h{y tlm I6y tru}ng.h.op n16. Theo b{t ding thuc Beenuli, .ta dlnqc (Trfr I&i: 16 + 26r* ...fus ( ro6t" '10 10 /'r \rAJ ta .Ii d6u'06p sti lir ' ,, ' Glil = (1 "r 116) ) r +rO;rio>2, _ - ( dd lt di6u phii chung rrrinh - v6i 1-( n 5 thi 1n + 2n *.;;+9o ) ltlo - vdi n ) o ttri - 1' +.2'*..,.* l]n < 10; " (;deh i: Ta c6 53 + ti3 = 341 ( 343:73 tra3r ' l*tt)B -r (o/z)3 { I B*i to4n 4. Sd .'uiro I6n hcrn 'trong hai sci '2 'rit khirc ,2 {b/?)10 < (br7;3 vrr (6/7)10 l rcltf r+22 + 33 + ..+lotlolooo ur f
/ P i t. I ! LOt gtat. 1.'a cd B-e.'.gi& ban hdy chon t{y rnQt dfry s6 i$ , n , n-hidn li€n ti6p /., A.+ I,.,., k -f ' {r ot . ., "' lap cdc s6 tbeo quy IuAt sau: n vOi ilZ va t* q.*^a ot oru \ . YI ' :lo '= 102.t ) tos 'k +'Ir k +') k 'f vir 26: 6.r o6n 216 > o4ooo \ On , k(k+l) vd (&+lxk+!)' 'r3 --- Bai to'68 6. Hai s6 vira $:nghla la- a -)l ^g4ooo '. ndo'l0n l&p'dugc rhl sd -* -:r hon. :qYv I 'M+t nnac ; - Lbi giiri. Ta s& t + zz + :r3 dgrg hEir df,ng rhuc : a...+ 10001000 < 1000.10001000 = 16691001 --: (2 0)1001 = 210010 rkn Do 264000:_. 210010 v6'i k{n thi 1/; > /; mi dd. cri lhn chung rnintrrir0n b6Lo To6u hgc vfr 22 . ru6i rr.a. 2 N€n 1 2'+ 3, r-...+ I0l(,looo ( 22 F Bsi to6n 5, s6 nao lon hon lrong hLri s6 z1 : t - (Jc*I)(k+?) a ," I n 2' zz...z. thi bei todn (lue v6 so sarrh hai sd Aa va ak ! lrA 22 ... 2--Y- vdi..k,a}I. i-- n Rd ring /.'" ) ak (vi nd tu.r.ng dirong vrli { Ldi giiti. Ta hn-v dflt kn; .P I ttT > iit. i ')n gid ctng v&l dEy s6 cI6, song ta lgp c6c ?; an:22 ) s6-BAy th_eo rnQt quy luA.t phrtc tap hcri, t,' vi ' 8Ai tohr_ bn= 22 ... w z?U ) V . 56 nAo I6n hou trong hai sd '- J? .k+n t+o Khi d6 ,i I .. ur,- ,, uZ:1, = 10, A4 = 6EiB6; bt:=4. Cg {sr1-1,p(t+i+tj (r+r)(k+i+ri' . Nhu v&y aO thfly ring (t+r)' ar{Dr, az{bz, rg(fs, a4 jr)24 = 2lo .'t ndn 2l22yzz"' .) > - Z6F5c6 ,rtojzzzzz - ,zlzzzo 6 = P(k*11 r'[ 6=(i.+ i1 t){k+,i*a; tuc lii thi rd ring a)2 vn D)?, vn bii to6il (tua r€ f. . fa hdy-chrlng minh r&ng v6i n)6 thi qnlbr. bs ). os. so s{nh hai s6 { 'fa cd 22 ... 2 < 10tr ' . Jl+r+r)r (s1i)b ! \fi€f n (t+i)", * vi (tr+i+r)" Tr116! ]r6t Ia hIy ch*ng minh rang vdi t.)>i, n€n a,' < fto")l,o'= lgnlo! ,rarttt24\n- a)2,6)2ttrt ' .-b b * ,*,r.2!!- - -\z nzs" , o1L*i*t; ) Zo(k+ii . 5n Thf t vfly, do &'*i+ l)b ) G+i;b11 o6o tirc ti L un\ a4 I .. .h --- ,,.,b .b b M{t,khde ta lui)n cr.; un*p} 5n v6i n }, ti a(k*'+') ) a1t,+', tr = a,a(kr,) > ,r1outr" 5n Tr& Iai hAi totn ra ihi5,: do dd en: 2'"dn.-z.> Z? > b,,. Vi,y ta co k6t qufl Jx+'+r).o -.(r+,)b (k+i)a "- * (k2-p2ftirr-1zyq (o+i)o .' -v6iI("(5rhra,(Da ,, ),tl1i)!il *VOi6(nthiau)ba. (t+i)o >(k+i+1)" . Bdi roin duq.c gili qu16t :
-T- :-:a.--' [-;::f.--'>o---, .1 r ..- toin 9. so n&o ldn.'nhit BAt trong nhftng 'I'rong cl6y q6 tr'r uhi8n {6' b+n hiy c6 g&ng vil gi&i-quydt'no' , s6 sgu rl*lli""tui6t iap c6c biri toirulirm'v0.ich',Sau 'eu'x.rri"g:&i ie mdt vi&c iI6' 10 mltt bnn h1v chou d:'y sd ot' "z' "' an thiia 5' ll ... 1 .tn10 -5 o. :l