Toán học và tuổi trẻ Số 142 (2/1985)

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Toán học và tuổi trẻ Số 142 (2/1985) bao gồm những nội dung về tích ngoài của hai véctơ trong mặt phẳng; dùng trật tự để giải các bài toán có các yếu tố bình đẳng; Toán học và Cơ học đối với ngành Hàng không và du hành vũ trụ. Mời các bạn tham khảo.

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Nội dung Text: Toán học và tuổi trẻ Số 142 (2/1985)

r- ;:i-:Er t. I I I I I [=". I t!-, & [ '' ffiE ilr I fl $Ae tEA..HAI ftralru_ M$T x,* Tbng biitt tdp : I{gui-Sn Cfri:h To*u I0n9 ,:r ru so : 70 -i'r'drr ;l,rrrg Dao * Le NOi Ptrd bitr,t lrjp: l!'96 Set Tir Eiu ntli t 8281f, t -,/ *#'iffire .i,nt ellr pFroNG It} ngAnf vIEN KHOA HOC *,I*T t "l 5i *"rss5 *;; 43 1S75
kt I i t r ff6i, chuydn tJri }6n er$ vS* {,*stktt {*e sg{Tff Tfi*tre p4&s sps#6 I rlryq p{6#&s $f;&t trGrYf,n* T'flil(i IIi(} 1. 'Irong nhilrr s6 cia br{o Tr:{5 iruc vA 'i rrbi 'Nirtr vi.r'iA nltt6n lii'lti:iir iir'ii vo hlr6ng' tr6, c{c tran.
t." I la co: rd (g), ir{i), {11} ta suy ra: ;,, l; ,;i FnF ?-B 4 lr.. Ul - l:ria 1 .rzu, ri lE + glol lr.ql= (.r'taz - r,ryr) [u. r] € g '= ig leosa ' u f- lr7 isina ' u Ilo do +:* 4 _-- l;:;l = ittyz - rssl) 5 il2 ) . (r" g),= [r I. ly l*oae "l'l:ti ta dtii yriyi ticlr ngoAi e tia hai vect,: trong $frn6 th{re {r0n clro ta .{' nghin Linh hoc cira miit phing, chi eln cho giA tri Df 0 bfia n,lr tinh v0 huilng Vi ctng chinh tl&ng ttr*c fl1", t'oi hai verlr: chon tri1, y, kh6c khOng v& l ti vs rnang ri:l u*. n6rr (r'j-u .r:.rln -l- rzu " thi iihi dr) [* qi li didri ii,:h c,i liasr,] cria liini.i =-({-l:} y a17p * 11zu bintr hirnh.
l. f6nr l*i. trong'irinh lror: prhing, v#i hai Fiai tich v6 tru6ng.vi tieh rgorli i[Eu-co'ung dr,rng rQug rei trong hinh hoc vccto ptriing' N6i telto tirv ]' r. I/, ta l'cAc !nh tioh r0 hrrdng r'd t1 ctrung, lilri ngrroi ta chi sei 116n,,nhitng-tinh chrit tich r;4oni lA: *h,r,iibrn thirlg lrnlg, direing thing"dDng ilui' G}= l.lJA "r , r J il{) rlu*ng thing siing so;rg' doarr ilring rhia tleo lj- 1... ,;1.- lr I ir;t sin s l " i0. r-i's"{ di{rt tir:lr" tr'tc lir nhiiug tinh ch{t gr.ri Ii Itl stir t:r :. ilnh chit a fin. tiri ngudi ta cl:i chn dling ddn vS lr: hsi biiltr llrt'tc 1r&ri, *t tit'h ngoAi rua tlr0i' t.eir l. lri rrgtirii lu xCt ci ddrr 7'*^',., -*t ,1. l,l)" -tl lr" g l' =. u' . !t' ( l5) nhirngli,ntr cL{t r:tg li€n clriarr den dd eldl vir sd'' l)irrX tl'rri. ni-i'r:hr: ih.{r'qLl a$ .lr* giire 1i, h thi ngudi ta r:itt dring ddn tich vd hrrdngr mi f nguiri "ri Iirlr Id lrut';rig. dei khi cfr n'g ndn tlrirrg.,:i tich lgorii nita' ilii Hs{# ':["R&?' Yff ffifr \-rJ1\-' BAI 'i*A}I #iA *nr"r ffi& qAc wffiru wffi srroffi ffiemG vi qiras{; S*il .", [1 lli L,urn iL)li rr. ' i:ri. cli,' iran g[;; iil;3rru !l triritrtg Zht. 12b -i- 6r f 2c -) l/c{ iib > t/2Eb S.)'1, : -: l'"*ry '' iU * .r - I f a-v u2,..., #o ti,r* gis lirr.-r nrc! lr*i lr-rirn iriuir r.lfing 116i uhnu, thi l;at: gic' ta'eirng cri tlrh gi6ri bAi ii=(.r'- t)(!, -. 11. rri r - lb y- l. l6n e' - 1- to6ri vdi gi& tili{r i,; t( a: { I < an }io[r: a1>' = 11, g - I *- I vir-:r; :"::12. g = 2, .'= l. a?pt..Pan 1a>y),)>j. 'Trtr6r: i'rdt ts'gili i,li ira; lrlfln I'ttn niu ri tlan' Eet u =a,-'j, u = ll ^3. a. . : * 2 + a> u) ,u ) 0 "r'e ta c6: Bii toirl 1. f inr irli , ri ctlc sd rrgogdn {r} a. li'i rr f o * rr' = (rr + 2)(u +2)(u; * !1 = i;. c ,lgo tho abr, {. ib.;- l,t + r t. J oru, * 2(,ap ,i- u,y .t- g,u) * 4(rr a u *_ru) f I Giti: Yi eAc sG a, b, 'e binh ili'rng r'6i nhau Suy ra : 7:u.utli-2(uo*'ulu*uru)+ 3(u-1-u*rrr) (*) nen khOug rn{t tinh t"bng'qu5t ta Si{ tbi6t - N€u ta >0 thi v6 phli cria (*) ldn hsn hoic e(e(c. b&ag 16, n&n ding th&c {') khdng xA1' ra. !r{6u Suy ra cb r' iri' * c,r { 3&c. rr,=0tr'7-2uu*3(a+p) r,i ia ls.i xtl 2 tludng hqp: NEu ap$*$br(ale '|cb i lle -ica(oeb, 0 kh0ng th} xi.v mAu tbu$.n v6i lidi ra. V*y a = ? ( ri e ngulBn = 0 =* '7 ='$u ra, t'1). Do dd : u } 0 + u-? *bI + ?r{ol-l(u*p) }E : vd lri
T6rn lai nghi0nl ciia pirrrorrg irinh la: Giii: lt thAl rfing { ed rltli' ni,il lihrrt ;rlrau 'J', -" tz, u -- z, : - I \ lt eac fto;r l) \ i. at, oz, dg, q4 thanr gi.a vao bii loArr i'inh iliing Bii toAn 3. Iim t2 s$ ngtulin Liw,tr;,1 . ,i,j,) v6i nhau, nGn':khCng niSl tinl: tbng rpril, tri gili il,i6! cho tbng crie e/rrin.g &dng llrlr cria clrin17. Gi&i: Gt:i cAc s6 phiri tim li xl. ir,z,... ,?ri 2' at)ar)rg>"ue Rd ring :rt, a'?....,.tta trinh d&ng r.6i rrh.au n0n li h i rlo li6 ( li trrcrr-rg i!rong vl!i : la gi6 thi{t c-t} rz...2 t,p} t (r) i (ol *d2).r2 .{n1 -os).rr3-t (cr1 -44}.rr= } (u) Ta gii i phuong llinh J lrr- o2t.tt+ {oz'-os)r3f (u2 - o4).r'+ = I (b) 1r1 .'1 .r1J2... rl2:.1'l *::z -'f .. +.1112 (?) ) (at ,r;l),r'l-r(/rZ -ol).r'z'rt,tc-u4) x4: I (,') '-' il 'I'rI (ll w" *1r2... ;12 ( 12r1 rl iot- ua).ri'i(n2-o-l).l.e*(ar -era),rs* I (tl) hay t2... r:tz --( tZ \\ ri (3) Tu hii ('l). tl,Lre hii'i, 0ic"r,lr(,p tinh (ai -- (b) I Vi 2a-= 16 > 12 nen lu i:)) ra rhdy trolrg il (l;) -'(,'): (c) - 1,ir ',, illyua i:0 irr,rrrg rittr"nq: s6 ca, .ls!...,.r1a iitrOng thA cei lron 3 s6 lorr hcln 1. YSv trong t2 st1..r'1,.r'2..... .1rt* khOng thb f (rt '- a2] { .r'l t- r'2 - .1 3 !' ,r'tr) = it \ cd hon 4 sd l6n hrrl i. Do d ,12 ): a3 ) aa thi nghi{lr tira h6 id : 'ltrel bAi loAn 2. la c(r .t 1 = 12, ra = 2. .r'r : r : .r? =.Tlr'= ii" r'I =.r4 = |l{at *,i*') kh6ng th6a mln vi rs ) L Cliir r' riing n$r. (ti. ltz. r]S, tl.1 c{r rtr,it lltri ltr 5. t,tb.L-7>.rB2. u2 rs:...-= r'r* * l. 'I'r'r khAt' 1hi nglrignr ring cd rntit tht'r iu trrrrpg 1i11g, (2) su.v ra rlrzns:rae 8 * -rt * 12 f rS * rl (4) Ching han ,*,, a2 ) a4 ) nt )ns D{t tlt: .r 1 * }, !12: *z - 2, r7g = rB -. ?. thi: .rl = 1:4 ,::0 r,A rA - rS = llteZ - uS.) !4as4* 2, su1' .r^ yrb 0 Vi = I. X.3. '[ r:,r tar 'f r'€n day, la ile rlt nrdt sd LrAi to.{.r vE giii e15 {1) ttrong ri rir7ng 1'{.-i ohLrong lrini;. Ii*v gio ta xet m0t sd .bii loi n l6*yr*ilz rus+ lli: {!.lt.r2) lrJz*2i (Us- 2} t g++2t iogi khite. =, tri * g(gr * yz +"43 * ya)*... Bei to$rn 5. {ldc sd dtong a, b. c /hdo mdn h"r 0=711il I llz *,t3 * u+) * ... di'6u kiQtr gi, d? odi mqi sd ttt rthii:t n. cde sd Suv ra oE, &4, c: ld dd dr)i ed,c cqnh cfia m$t la'n gltic? !j1= !]2 = y3 : y4 = 0 -> :rl = t2 = t3 =;rtE2 6lii:' Khlrng indt tl*h thfrl lhng quA:. ta gi& T6m iqi cAe sd etrn t im lA llri6t rAng olc)> fil = I3. 12:2, ,. t) Fi6r"r t)>b)2 c -)> 0 llii I -)$ia )tla Yr c|t .rB =.r4 =.tS =... = X"1g I s0 hta. citr d0u nhir- hsu I nln l6rr lai sd trt ho{o nhiirn n rli lOn sart eiro- (bio)o +'(c,'n)o'{ t khi' ,1'I:.1 2=.l'B :- f* = 2: 15 E .r'6 : ... =..r,ll - I at6 ,n + co {.i no r*n \'ui n i!qi, rru. Iro, .ro khf;ng 8Ai toan 4. C}o a1, a!, a$, d4 ld, 4 sc) t/rgc phLi i;i 3 raoh ciia ir;di i*;rr gil,.c. rldi nt.ot khac nhau. Gttii t* phunng trinh tctr: 2i 5sri ,.r : b>r-')r0 thi oh = b*>',")-t1 I \ lrr - aalrg*-lor - oelrs +lor - arl re = I ltii nroi n st1 {r nhiEn. vir r5 rilng rhrirg ia tl6 llrr. a1l 11 -5 !o, - nrlrs .i- loa - d.lr{ = I ,,, di i 3 canh cria rn$t t$rn giic. ) Iaa - a1l.r1 * log - ael rz + iaa * otlcl = l "' \.i vg1' di&u ui4n chn tirn c&a a, i, c l{ hai i lal - oil.rr * lnt azl..z + la+* asl13 = ! e6 l,in nhst phii b&ng nlrarr.
[' Bdl to6n 6" qhu.*t rninh ldrr.g lranrr i dotln \ral 5012 d 5. *st,., ra :r2:{ li 10 (dpcni) th&.ng,m6i dsan edtlp dai t iiai, unr I ( I :t.l-tz'za i{ I -i: Irt} t2+ t3} l II=3, ha.r' n(r,37, --, t)-i- {x2+yz +;2 * t? ) -2(t':1yt,\:k" itlt ts + tA>:i + 3 = i'; ia-P ia 'i I5 .i> 3 * r. =' :i, ga e,2 * !!z + :2 + f2 = uz + bz * cz :'rJZ a6n i?',> it 4- k,2 5 * I =- :il. gia tri cria n(;', lJ,:, l) plru thuoc tin gii tri iJliiu rrd-r, rnlu thuin ltii gii thidt 17
b) i rrrr r'il c sfi ttgu_r i:ri ilrrtrtg .f , i/, ;' I lxr) j;i f iru tlorig iuiitr s6, dirr titn r&i .:i t,irt bE cho .ra il 1 .:' i / ..r'I:i, L;a i;ij suo cho thng cAc irinh plirrrltg i'ris "lrirng ehia hdt eho lictr cfia ct:Lirg. 10. St sd Jlrcrng.r':,.rJ,.T5 tho* rtrl.ri..{ l..3.irB :} :i va rl * eg't- .r:g "i i/rr-i" l'':r'3 "i Lr.r6 12. ChIrng nrinh r*ng, tl'r 25 sr) dLrarlg iihic nlilLr cri th& r:hrin le hai sd, sa+ ctro c*e sd cdn Chlrng ;r:inir rir n{: l*i ir iiring thh bing thng ho{e hif u t:ria irsi -sd a) ri*i voi nroii:1,t,3 r tl'r, r'iroti. l-l)'l'rong 3 so rt,.rz, .r:.T to tI trng i r0 i.;c iior: I :3. r.lrrn'g r:rrrrh lrrr', ri:ing lltut: sarr: ll.;r) \'oi g;a iii ugrt-r'0ti nau tiia .< li: i iili: l,;i -. uil{ui * egl ."' (at *rr;}*(oz- qiiidt * d3).." lai e{e s,i ngtt;.6ti ri,irrrt3 r.,; -' ll.rir :ni n .ro-: . {,r;'-ir5,}'{- .,. *iug*- *1)(*t,*d:) ... ius-u+)} 0 -r yl + --3 = iasrr^-. 'i'rong rto ol,.nz, .."" o5, li c*c sii lhui:.b5t ki' L,,. 4.' 1! 1(rtt+^''}.niil-.t)+r) r * il;;, r r:ri E -- s11k iA sd ngu".,,** io,' [& H ffi&ffi x.ri.\,.' ta #y {ysmi.* Sei 2 l3'il. Iiu"r o;ir tri idn fli#i pd nhi nh?i! ,'6e Etiu t.liri"s r '+- i; "r t. btdt rfi.na .r'. S, : tJrda rnd.n .phtro n5q lrtnit. 1,'' - Y: "' '2 ,r, 1 -' 3,Z 12 ? Bil l/l:|9. I'inr sS tr njli,;r: n nho.'rhfil *u+ cho tltEt p/:rl* cd eilil d{t!t ld l*i *rrf * : L#i siiti : il(n198*+ 19li{ n-i3); riilic*+ +' ii} 1 r: * i ir... '" i 198{r' {nlptl+ + l9'\ I n '$ l$8ti)' ti{&i Yinh, l&4, Pi"l'H ',l'*y sr-rr, ryi !'i+i) tsl i' tlz i-:? o lci- lt't'21? * 212 -r ?'yz + :zl + g.na Ltri giii : Ci,c pirtn si di 11..,, ci rirr r;g : --t ti (n138a + 198.1a * I' : i.] :,t i- - i;;f'r. ++(:i, -r g + z)z +{o'*:)z -t (.r * giZ =: Cie ph:ln *6 dr1 ii ltli g:1n i.hr r * r:l:l klri : tt6i vli-r' (r -i- g -l :is {l vi dtai gi8 iri (1i, ntgsl - t9E{r } & . l) a ; .;r' 1 *1ffi (r i-g+:J2m2 khi **: = 0 rA .r'-r; =0. iuong dironu vdi Yr.ry * , S + ., i:6 gia tli l0n nbdt l;, /: tt,i (A. ,r19st * ltrilln 1-ll) e r, t;5 I,lg8{, r,,'-f i .r .. U - .- - :: vi c$ gia tri nhr., nhii ta 'Ia rlli*n ilii.r ng:r-r' firtg' '-'li; fi* { iiii ni9SS -- -- -i' I ;ii; - _l -t 198 ln * j= tSE:, la s$ rrgir-r'r'i, ltl rdrr I i . ,:t.t \. i V I = 1,19li4. \-'i1rr viiv gi{ tri ft... I lim e ho cri,: Ftk8n xet; l{t ci cac lqi giir gL?i sE dtsu phAr sd tl[ elrt l& l#i gi&Lm vir hcrrl eirs do I lrt dung. rd ti: nhi*n mlrii nhfit tloug t{p sd; irr nli!*ri n&n li.Q.1' n-1 ehinh li gi& tIi c** rirn" Fthtrn xot I i,ii. t,';ir i,.r,, V i i:ia, fi6.ng Vil ir"c It6i $,'139; {l}rri"ng rtti,rr r,,illj-I*no IiBC (1i Cl', CIfi'i'H tii !tl.,i): ir{, l)rrv XtrAntr (11 f,'i'" giuo 4il*'; rli* t/rrring pid* 6idc A&lI ptti t{rtsng lrgul,dn 1'*n 't'r11i, \'h*'l :'*rrg-i, L.{rir l'ung 6i:ira" tru*c iitiir .!firl$ sont tr*; r{lr. r,d ede d.i?n litp ,\gu.v6n l{b .{.rih irigu.vlr:. 'i.r& e t tu&n, hglu.t iri "r'r-ie e*i; ,irirrxg trrirr rqii fiip &..tSr. utzi i* olt 'i'r"iril;, ['lrlng Snr: (1: C'f! Fhan Ch*u Ea lirirgr; -'1 {. ri,im frdr: rr.:r rtrdng !hino HBi Virih {lilA,'l'li-y Son* Iid Xa;i}; NSuvin i.i;.r:rr \.ii,'i r[* {)ut' tiini.l {ili;. i2{i 'l lrrng Vu,,r:8. Ut:r Ltii gini; i:ri:r !&rr 't'ring tiiang, t1(.;'tr, trhan Nircrx) t* Pirtr;g ,\ul,n l,*i: (ff i,1', l,aor.!*n. 4'1,1u '! rinh. fla NEng. 'i a phAa th&nh : truong Thanh il6a) cil 'l{ri gili t(,t. B4n Ngrir,{n Yarr Ir r.I trl : Quaug {1lc'l', l."ir rr; Scu. 'l'h*nh H*a.! rr'un e trirng rninh du{rc rtriEu lihirng dinh n-rgah hon : ( n = I Phirn gi&rr r:ria 6 r:*t d*6ng irur$ bilh li Ilr s{l tr.r nlii*n ahfr nhst i*rri €.h$ *r phin td rru trd5tei i) ngt:fti dcqn M:\. S+i f lA lAln dlu'trn64 li tdi u r&;. : tr*u n$i tiip, il r'* B lar e*,-' rfilm ti6p rrit crla
?' / duong tr6n n$i ti€p roi iC yi Dlng th-ai do d" I BC, ar,lCt vi dc_Ld6 "1g. Kh, d6 I'a ANtiiia. €Ailldr r.a Op//d.. -* n6n I JL tc, /x -L a3. ,r'a c6 : ,il]? = &+ Iff) ifr = .ffi $11-y 'G'ci = R,CIZ. Vrir rlrp,rlc yuA*tg_ lai f -"i-fi,f if lA. gia.e d.ilnr .:fia t{e r.l dr. Iet hei gi*c IttDC noi ridp. iI'* r.IS nfiC = JfC tain gi6e "trlf viL aitrlr ta ta{y. = ={fi.-Fr')ilft) DE''MN EK!lltO tdo EK nixn tr0n dr) KDil*ll (da tiD ndm tr6n d*) Suy ra$xff .A* - AA,lfX + !)H;MN = IiBl0N, fiBE *E OMN (2) KDI6M t KilF = oMP * hxnr*t:.o.liP (3) Ta c&ng i.:i A$ = 1s *tjffi = [i +"'6 i , ,r, (1) vi {t) ctro r;r thdr oui:triN. viv D. r{, suy la F{llptl. l{4. de 1{ fuang thi-ng di qr,ra Il th9ng hir:g. F ri song song v6i OF n0n suv r.r .Fl( nim 'l) IlrBm cit D ninr ti.ons,loan iI&; ir€n dc. BiEu niy chr'rng t6 d r, dy, dc rtbng '\n.- qui.{dpern) Tactng *6 t*rrr gile EBC vutlng gdc t4 i lL 4I H r,){. HT It I ri gil:c rrrl; ii6ir, 4' Tu 116 EHI] E! -\Ci, 1. \ til (a 4 c; i2=a.gK ltay D, l't , K, thdng hdrg. N&6rr x4t. Ban T,r.*r.r An- Vj*t, ttcl.. phan Brli.ChSu, Ngh6 ?inh cri loi gi&i t6t. NhiEu irAi xdt thi{lu tr'*}ng hop. DIc bi{i hai lran phan Cdch 2: {erla lVguyEn ltuuIodr, Jy'guyEn Holt -Phuong"Eat, 8I{, PTCS'I'rung yunng, ijs N6i \.a Vfi, l0A ud 11 B Trung }'-aong Qay Nhon, Ldm D5 Thenh Ssn, 8A, PTCS Bd Van Dan, I-ie N6i 'ting Glang. 1|CT Ph,ta L.hu Trinh Di .\&ng). cd gi,i tr.ri giSi; t Gqi O, G, Q lin lu9-t la tlm riuring tldn n6i N.Q.T. ti6p A3-BC,-troos tAm A*llp vA diBm gi&ri eira ,\'P. Da thd-r Q la auh crja'if qua phep 1,! tu tAm BiLi 4ltS9. Cho A rSC. (,pi lt..V, F tri cdc t!U! qilrn e*.a drdng tritt nt)t ti|p u6t cric cqnh G, ti s6 (- :). oorg thoi do Q nlm rron da BC, CA,- dB. Gi{i sri: riu /d cl*o'ng lhrigg qwa rtttm \ 2i 11itra e&a PN ud uudng g|c i,oi BC, ai li ,fuar,g t,b, dri iOM (r,i etng vudng gdc vdi 5fl) nln thdng q*a dibm gittt cria !,]I ua uu6ng goe uo.i suv ra .ds chinh te lnh cria dudng thing OM C4, d, la dtong th&;zg qua tlt?nt.gitra eri,i-"y-t ud qua pliep ii (tr vr}a ntu. HoAn toi.n tuong tg ta uu6ng gdc uoi A8. Chung miuh .riing d.6, d6, rlc cirng cd dl vA dc lhn luert lt inh cia c6c Sbng qut. dudng th&ng ON vi OP qua phep vi tg t{m Lrri gini: Cricft 1 (c*u Ngugi:n lib Auh.lgugcn, G ti s.i l-qu.v tlzl. &{t e{c rluong thEng OrV, ON OF dbng le.i O n6n suy re cdc ilud,ng thEng 1-l Phan {hduV.'rixk flrt xdng). Ggi /), E.,--F-lEn ds, db, ds ciing dbng qui t4i O'*inh cfra O qua lugt li fiiEm gi&a cria ni,€, pil vA" ilfrv D6 jtng, ph6p vi tu tAm ti ehting rninh duoc * I,fffp * Anpp. G sd (-1l2) (dpem)U ?u dCI bnlet:: - ttf)lpr, i'i} =;tr; (r) Nhgn x6t: Cde ba.,n Nguv6n Birh f'n, Trln (ICCT, 11CT Fhan ChAu Trinh, Di Niue), Gei O Ii ti:n dutrrl trirn n{i tidp AdgC. Quan Khi d6: Ph4m Xu*n t,ec (tt8to. Ngul'6n Ydn 'fr$i, liha t').v LBC. oNIa,4 vr op.l,_AR Trang), D6 ,vgoc Minh Ch&rr (10A, PTTH SeikuI, Oia Lai * iion?um) e.tng cd l2i gi&i t6t": NhiEur 8- li i ';
bgn kh{e c6 lr'ri giili dirng nhur:g ruom ri, phuc Bei 6l I'3$. ltgt tri' gide gh1nh co eac 11(te d6 tap. Ban El6 Ilu-r. Khhnh (1lCT, i\gui6n Yln 'ir6i, rli$rr &dn.g nfura.(ltfing minh rang rdc t'qnh dd likr Trnug) q6 .\i irav nhuog db vQi vl nOrr J$i cita t* lSiac.bd*g nhau. gi&i lhi6u chinlr rac. N. K.tr. Otii: Cdci i'(cfra Dang Va \on. liCi, BHl'Il lln r\Oi, Triin l,)u.r t-iinh. 12G. Tlung Vrrong, Quy Bii 5r'139. c) Iinr phu'qng trlntr b{c J cei Nhon vd rnbt s6 ban htrAc) nghiQm ld cos (r/7), eos (3rs/7) r.i cos t5rl7). Xdt tu gi{c ghiinh ltsL'/] c0 TI} : ;A;ut r''_-\ j-- b)Tir di hda tirth I6ng I,zt:os irlT)" +: li cos (3m/z)1 ADC = AIlr.. -i- t.'co1 (3;!i). Lai giii : Gi& sir ,{A { l.l-}' Vi 2 drrring tron ngaai ti6p. a) edch l (crja b+u La6 DuSz liS6nnr i,l fT, Ngu.r,Fn \tin.'116i, Nha 'l'rang). Ta th{y r/?' cic tam gi6c ,l.RIl vu B'*D l:ir'g nhau (do BD chung vi ^ 1: 3n 7, 5Tl7 la r:!cr nghi{rn r:ia phutnrrt trinh BAD::: BCll) ntrr tir' ,1 1J (CD ta su1' rt ra Cl) --\ ,r;(,'. I xi -tit ! ilrr.trng ti.i)n ngoai ti6p 3r * 4,.r: .= (24'* 1)r: (I: ngrrr'€n) n€n'cos 3r: * 1- cos 4r = ll. Mtr ta cri tarn giac ACI) ,t W.l,jX-S ,:ting hing nhau eos 3r: J46536 - 3cosr (do lC'churrg r.i lBC'= -'lDr.), tu r{D} f '; ta. su-\! ra CD { e,g, l'6 lj. Ttrirng ttr ndrL nB >C, cos 4r : I (? c(.)s.r * 1)2 = 8cos4c * Scc,sBr *' 1 ia cfing su-\' r'e rrrlu thr.riu. \a_r lO = L'D. tuong Suy ra 8 e oe4.r'* 4 cos3.r, -'8 coszr * 3 eosr* tu ta cirng cb AD = BC. *!=0 hay (cos r * lXB co33r * 4 cosz*-4cosr*1):=0 Nhung eos;: * I f 0 ncn cos (tl?), cos (inl7)_, cos(6rl;)t$ 3 nghigm cfia phrtong trinh tlrs * 4yt *4u* I =(t. {deh 2 icrl:r nhibu ban). ila c6: s6s (c/7) * cos (3riz) * coo {5rr7} s - lsin (.]ri z;*.in tl +sin (4n/7)*sin(2?r.'7]* + sin (69,7i '-- siu (lrl;)lr:sin (rl;) :: siD {ori7)l2sin (r'7) = 1i2 66s (rl7) . cos (3r:l;) { cos (3r:i 7). coc (5i/z) + 2: (ciia Pharn Hai i1l2 Phan CtrAu'l'rrnh, La,:h +,'os (5ri7)cos (::,ii = Dn Na"gl-- = i'(cos (nlZ) + cos(g::/z) -F eos (sa17;z Durrg C81 : CDr- AD. I(hi d6 ErCllr --11i. ^ *. *.,*z1n17) * ,,ou2 (en77; *" 1srr771172 = : 6CIl * Rr.; D rLt D0Dt =(.1)A-: ORA = BCIIr = 1/6 * (cos (:m/z) + I {- eos"ou! (6n/i) +1 + . -[ rEn caL: rlgrrng theng t]D I i (-'lJt i1{t cac rToan + 00s.(1{}n,,i7} + 1)/.1 = - 46, f;i - i-8, l'a suy .r:r (1.1, = 118 * (:l : cos ("|'ij * eos (aniZ) A {-']fI = 11 t;BrB Ic. g. c) "-? lllN = llljr - cos(snl:i) lt * * tlz -\ I-'BJI : A CNBI (c, g. c)-*) BM =.lJrN r:os {r/7)costBrrT)cos(58i7) = cos (n/Z)[costsniz) + . * cosi?a tl)]ll: Ilu {g}'81 la hinh binh h:lnh *=i I(lF= Sfr :AC=D[]t +.U=D.,V:El]1 ulABeCD i = lcos +rii\ ^ cos.(ZriZ) + cos (B?E/?)li4 = AD =CD1: ii8 itlpetn) * = (112 rli+" = =.1/6 (n/Z), cos(3c 7), eosgnlT) la ba nghiQm .Vf,r' cos PI r tla plruong trinh. *.rz,'f .- : .: * l,:g = rj 13 \t b) l'a co theo clirrh li V iet : \1 \A j' t/costelT) * l/cos(3r;1'Z) * rr'cos(5nlt) * r' ,q *4 cos (r/i) cos(srlz) * eos(3riz)cos($ntz) * * aos(5r/Ztcos(rlZ) c"oa (xl7) eoc(S,!ii) e og(trn/Z) -*(1i:)'(*r,8) = 4 , D.B.I{ D.B,X
8ii 7/L 39. t:ho tam th*c blic tai Ndu .t'o la nghipnt cira (l) tiino{Y, Gii sii l(rr) : bl * c. ()hung minh riing rttu m, rt, k ,= *^,o tA ng-liQln c'irt (l) (rn n€N, (rn,n)=1) "?.+ &a rd ngug|n dt6i rrrrt ,khric nfiau J&i llri ta co fd kr , -' .k-.. i - -nt, n]' * uii * rn tt)* *' . * o1l . r .k , ., I t u'[ao( -- n 'n,r* l11(rn)|,lt'to)|, lf,(A)ll V-;' (1) h n-v .q. [*i ,rot-m)k f ,ii("'nt)k-i.* * . I aa.nk^: u gi$i. Cricfr , teiia Darg Yfr Sol, Cao Yi' Ba, O6 Dq,r. Khith, lrlguriir: ll-,ii Vfr, f-f,nr Turig T u' div srtr) i'a 06 : tt ; it. I nt, do v{1' Giang). l-(irr(9r t(n{9. .Gi&.s& m{n(trtri k*m'bz. {2) li-(' hiQu v6 trAi ,'tia I l) ix l'{r') thi pr-= P i16; N6u l/(mll( tt2: lf rn)l( 1/2ilf(r)l i ilz ri: rhi l/ t/i) - / (nri g lf (*tl -r lfrrll < r ' rrk.P(.r,): ,,ki.Fi.") * P(-atrn)l = (rr.r' * m)g(r) hr:'l{l'- n) (.k + n + b)l( t trong dd g(, ). la da tlrrt to-i hi s6 ngu.r'en, Suy ra,ltc + n +bl t/2. cii ,.i' n,a".lll'(m)i, ll (n)1, ii(r)li < rlr. Bei 9;139. Goi llt ld bon l;irth dvang trbn ngoqt tidp tant gldtc cri..i anh lA 3'du*ilg trilng l Xdt him bflc nh6t h (x) = r (t) - g (x) thi e li*n erta,L:lBC ; R td bdtr kinh dlrdng trdn ngaqi , /r (m)',= I {ni .* g (irr} -{< [ ( nt) * 1/r < 0 tdp btgc. Ciiing rninh rdng: , rr (n) ,= f (n) * g.{n) = 1(n) + ;/2>a firlfi } ({/J)eor{li z} cas (ttt2} cos(Ctil. A (/
\ (tri a, b, e l.i 3 canh lutrng ung dinh -l-.r .4, fl. L, Sei 10, 139: Clo hrim s6 l: @,*)-+ (9,.x.;. ,- ira A,{ #r. . e,, : t'.iubng arlrrh rting iat khd.ng dinh silu tu$nq 1,v =d16zt J *i2,t*i ie+ 7-+:a. .*l r '!u.rng t tdlnh iy harn t.nsin) ) V':fr. tr I ."r.if= u) /' (i(/(#)) : l'(rt v y r,u }, o l,)l (r.v) :7'1r).7'(y) : * cos (3,r2) /e. ;' (;t"r!) = r' f ,r"g ) 6 I'uong trr ta r:iing c,i, BNi cos tnll) .l.ii, I.o'i giA,; CF}r'o. tr:tzl .tll| " Iri ir) *- {rr: 'Ihry .r g i::}ng I ttri /'(l"r l)i= r,A Val t:{t /'(r) (lr lit - , 'l'ira-i' r1 = i tlri /(.r/.1)) = /'i ,r) el r .ii.\ . {.1, 4 .\ n^ 8,1f.ry - .1.11 'I'ha.v.y=7'it) rhi I'i,rl'(,r{t)t,.. (3), 7,1t) . l1r) t" * ^l 11 1-,'-\ . {- /, t rA,, u,.. Ti)' i11.,(?). {lt} ra ,.ti i'( r) -= i tr}. [r)) ma l,:)'>0n;rr l{Ii - t. }i,1y1 f1 rhi y(/( r)) = > {1,':l}n. ilrf) . i b, ,rcs {l,ll) " }l;; ;t, 1'{r\ ,- r.lit;-.. 'I'u drt {g,1rs I'lr) f(y) - . r,rrs {i .:' . ,ila) i ot, = tt.v.fi,r{r})) : / (.r, yr "> , | :ri, ,.,,s(,{,,1} , ,. i ' V.., ulu '. i; . _. t:os(I,.r,?). irlpr:m) . ftt b +*): (lr l'(.,-.1(tr)li =,"i.r)/'(/(u))=S/(c) '\ir{n xet: Cic }ran N-q,,., i;, ,t1., .Arr!) \'cu.}.€rr ll:tC'f, I'hrrn Ch*u 'l'rinh, ita ltHng) Xgrr.O,, - .,u "\ 0, ttn*i 'l'rrrtr iai : a) i-a L) li tuo rr,, il 11ry11g. c(r Iti (.iiE. '{lrunS Yuong. !}rrv Nlron) i:o" ir;i g.itn giting i: tr€n. Igoi.i r.a cAr: bqn LSrn.firnil 6iii I'idn f;d, : Cic ba n Tlin erri ii, .\grr.r in I[i, tjiang,'l'r.{n QuAu, Ngu_v6n }lt\ng Srrn, (rft.f, Anlr Ngu.t'6n, l,im l trrrg (iilrrr6, .\grrri:ir Ilrrrrg _ Phan CIr;1q l'r'irLlr, Di Nf rrg); Ld II0ng IIenh 5sn. \gur.6n Binh lrr (plrar, r,lr,1 ir l r inh, I)n 'l'rhn i)u-r Iiinir ii? G. l rulg TrL,rn$I, t P t"I'l{" Ngu"1'6n I'rri i, IIa Na:i) cit:rg ,:ti i,i1 I:rgl: eu.r. I rl (] nE d6i *'u, lrilirrn), I-)ii rrg Vij Sori, Cao Yi B; (l,ip it A., l0 t. Llll I'H tt$ Noi) t,(r liri giii r6t. I .\"7 Il-r' Bel it.'t4$. fiiii r a hii:n lLran h9 I .l i . l'"1rl: - ,, - i'#,, ;- t ll ,r o t1 il' I /'-;- 3/-L- \ !te- , 1/y' :fi lu gugir, I'd ri .ryd u i'teill,, l4O-.-ChrLngominh ring ndu n ii s6 tg nhi6n 16 thi ns * I Lh"6ng thE ld s; chinh phuong. Bii 5i t{0. Giii Jrtr utrng trinlr Sd lif c 7'lld, .3 * .3o'r, * !,r9r,oc 3r = o Eit 2lt40. Tinr i{l ri c6c sri ngur,6n E, !,!, z vrii a lir iharr; sfi ducng..1 Itr tjoc cho trrrhc. th6a mEn phunng trinh : , ?t * iy + 1960, =._ 0 Pd Bd Khangl Ph*ctng Thoo Bei 6,, 1,4O. l'inr tit cA cue bi cla thrrr cia. 8ii 3/1{O. Chung rlirrh ring n€u phuong mdt lri6n sd 1.huc f , g, h, th6e min h{ trinh ' l',.r2) 'r !t{r\'i }r.r') : .,2 .r'4 .i- e.r5 + brT * ar -l- i= n I'te) * g(.r'2) + frr.r') = -! cd nghiQm lhi f(x\ * s(J) * rr(r2)= -. 'r2+1t,-2)s>3. v6.i rnoi .r. Nguudrr Vtin. Mciu ,\'guuin Ntnh Dte 1t
l,-' t./ B&i ?i tr 4.O. 'i"r6n c{c canli c,la tu gi{c l6i eir Ber l0iI4S. flir,, .[ rluirrr.4 rlrar)g rlr az, d.t. . diQ,ntir:h S, vB. phia ugoil i, dr,rng c{c'hinh vu6ng. ria-dbng qu,rr tii {,r. l'u urdt.clibm 81 tr0n r11{ TSm e6c hinh vrrdnS il(r tno thlnh rn6t tu gi{c th6c rn$n OB 1* l, ta k6 during thSng song sorlg r(rr c6 di0n ticir Sr. Hfrr' clrring nrinb ring da cfit dz [r ,B2, ttr 82 liir ttrucrng thing song song e) sr.>2"$. vdi d1 c{t d3 6 83. tri f}3 k6 du
[.'.6 ron_ rin] hrr.og c6 rh] de *,"y aai. ci. cho ta gili phip lA dung bOt kirn Ioai pha d6ng I :,, ldn, t$ru vb tru crrn i,.6 vAn t6c l6n hoq g{p 6p- thinh vdi phun, Khi e h*v, cAc hgt r'fDng o6ng I oFn ,:Ac phi L'Orrg \.fi rru kh0ng iti khi chl.y trudrc, hav hsi vA lAm nguQi khung-ngoai rr:ing dicu l,,hrln cr,n tlu rrong m0t s6 tiuh hurlng cfra viri phun, gidng nhu mb,hOi gi{t clJco tnB I I l,irrtc ia1r. t-)E liir&c phuc l.h6 khlu dti, irgur)! ta c6 rrhi0t dS i) dugc llD I Erl yA troRg quA. trinh d6 chunp rninh arta BC ud lD D. dusc EC J- m{t phfrug (dnI) vi d@ do Bb LIt { le r*,
l'' B&-y gi+ xirt iiit) csu h6i 1. Cic hqi gi&i thli.orig Tai sao lai phii lOi thdi lhu I'h"v ? Pl,:1 ii ti-ir I ldm iheu mfit trrtrlg hs i c!1r,ir sa ri : lfeil fiu e lriruii to il .lu J,r. 1tr-r' rrftiin, t,',i rirrri, n Cde,,r thri lii?lr: Yi dri cti fJ _i_ 8r-l n€n nrri,ln ili rrrlhi cit ilan sui' nghi therrr !'6 hai thi dri L/ Li t1 vge irrr 1iC v:i ib iiri r.hi ,,An /.1 r.: s:i ll : nt"r*. l.rii: rltl trLm uiric ilI) cri I./ vr'rn lii 'L.{/r trrrrig 'i-hi t!rt 1. 1iii,r. linr d!iiiu kril ilii lrri tarrr giac tii-r itt r.ria li dtrring- (.it f) nin 1S tiinr qieic t in vi\ i:rl diBn tirh *iing nhsu. \t:rr tr',i ltli r ( Ilai !em r:l1i ii,i Il = LD. Srrr 116 h:ri iarii giiri: .4_IB vA fi.r: [fi,,* n]rau thi co diiln lirir i,;1ng nhau,,, iiiii lrrQng 6 l, iei rti 18 chrrnq- r.ri 1.1 = Ifi rri rt\ng ld, rr)ri rth s(rl 1{1 llljirir truirng hop ii:ri ne:r r,irfrng lrin0 nhllu. Oc riir .111 = 1] jJ hal tanr gilc ir',,1;,'ir ii,'ir ir,it.,- nl':rri. &.c, T'ht dtt ?. 'l'inr rliiu kiFn ,13 :rr, l ritlong thirrg '['rii tli-r i'r'r l;1 I r.lirlg l:ri, i,ii li:ii vi6t tlrdrr; : vnBrg *tllu r.rii rri6t m;t I)lr:tnr.i llibn nhitln Ii Vii,r /./ lr) rir-qc t.ri:,i lJr'. v:tr .{1) lilii [:'r. rruiirt rltrr)'ng ttl'ing 1) r'rrirng grir: r'i'ri mit phi.ng {lach !hr'r hti:.'lrtrr'rc lrii lhtin r(:t: Ii.lii i=i (l',i tlr.i /) pirii rru0nS g/rrt vi;i it irhiit milt ilrrdtg, tlri "I.l 1r\ tl i g1' 1.i1i1 iJi.' r d .,t/) ri)i t:lrr'rng nrinh. liririg eira rnirt phling (fl. Ne,, trr d,i ta k€t lu4rr ,.iiir sr', i) c ltri.1Ji:: /)li, srrr r',r lr:ri trrrrr;i,:,: :ing tiiEtr ki0n.,it rrrdt. r!rr,'rn;f tl"nr ',rrr)ng qrir vrr6ng ,iI8 r.it 1,ti 1i b.iirg nir:rn ivir:r) r.nnh huy€rr vrii trtit' rnIt phirrg 1& nl' r'rii,iig f,i.. ',-iii !t nhdt L:ilti1 irlrlru vi 1Ji , lliitri), rio rlr'; .i.{ .: ^il}. sil}' trrrli rlrrrilrg ttrEng erl:: nrirt phins iir i rii rirrrg Lu l:rri 11iirr: .{i/) i.liri. \:rl irltDg trrlrrr L/ r"rinq I; liiri.:q ilting. iiii rlrr'iiiril i'rro, r:..1iiiil ii 1/ i" ,{/). ir,:l lrrp. r'i;i hii ( .lirn diPir };iin r,i i r i c dt I,i .[ r; rir ili r.ri rr Liiiu ilr-i cltr.rirg ;r:i1i; ir it'tl,r, l,/ l_ St; la su.\, t.d. li dlgc crirL Ii: ','i dI] a ciis ir:rri pi,ii i:i3u n6 rri U lA iivgc. .U(- \'i .1 D. nilhia iA : i'ir.,-r rliEtr lii0n vb lr I'i. c e iu vi dri dP Viv lilrr lhr:r.i i rii'h ;ii)o l.- ,ir'l irg '., /./ li di g.r .'iia Bf-' i,i ,{Jl. i(hi lunr niii .rrr: ban rhi ilir. gi* r1.1, xong k-i ti;i. Ilirrl: linir nr0t oiirit tirir r:L rrliil iiibl [i$n dri thi eing chrrt r11ng mii r:i r:, Iig rr :i t lr]r' r'i ng : phii l:1nr thd ni\o tim dluoc hit t.'1.,: tliDLr kiQn rli. :t) I ir,t Ilreir t.il,. li llrrt rrlr.', I trri' i.r ,.irr'irrg rninlr Y:it- trr rflt re nhirn x$t slru : Carr h,ii r Tinr di?:; rltsc rrirrg nilu 1./ la rlvgc crla #/-.' vi ,.1,I.} tIi ti6n uP... de... )) tlrrlc rhit li r:r,",t c..iu hi,i yE ir=c. lrjc li rnlri r.hirng nrirrh tlrro,: i'lii:u hiSn rii tllEu l;it n fit cit ud rlit. cri dB /./ l:\ dvgc cira fiC vA ,{1) li } - c rni. Cirng e6 nhin xd.t irr,-rng tu ildi v,;i (4Lt ltdi i clrrru hi r.l;i r10ng ridn nrgnh rlB ( lihj h = e thi I./ tide-_
f -;a-*" l: -q l* I I I I cnn sd ud ba sd eri ? chr?' sd soo chci iich efia khi drlng brlt in* c6 l.hB. n6i ring: < Rir i toin I h,ri $ ndu brtn6r llrir cfio hal ad kio. ahi cl ng&n :{y nghiQrn D' yd.them m$t ai&nr vui: ( tlong tl6 e6 rn$t nghigm crls minh ). I ngnl 6y, ttri .,uri r;'l shu rac ban. \'A lrri roin I ,ii h{u tlir, tOi r,ir rirOt sr}c manh rl:,1 ,rtvur. l)irng trn&e bii toin c6 diQn rflt r$ng,'lai c6 | :(u!.t trign tu lirr nirrrng rhr* duoc giii qu3.&i tht *l.r nhi&u rghiQrn. r'igc tim duoc mQt hay l_ tld)n vetr, ud thu,.stlJI trO thilrrli rri,.,i tlrfluh hai rrghiQrn lA clang t1 iri : irhrng d{t ilugc co s& I thric I ft.ong cudc ch1.t, dua tirrr nghiQrr: cho iieh tj' ir.:&n tl't tri d6 qu6t sr,:h rrroi ng6c ngActi, trflt tun hon li.hi cli,-ra i:o r-:;,rt:h gi ,18 i.irn iir.rr.y,:: turin b& drr*i: toi.n !0 e5c nghif rrr .'drr qul hul. B&i vi I u6 drii hiii phi.i e6 rnr't l;.i:r linh vurrg vAng, I rie nghigm. ng-uii t* dS phit higri drrr;,.: t:lrin nghiFin sarr dir': I ilr$i cieh rrhirr vu& buo qilrii, r.'rjra siu sdc. 06y ll chinh li rtiFu toi ruu6n n(ri voi c{c ban. s 71t .!.1 :it) >l :l;i r:: l',;lil{} L.t'tir H6m rral'. tiii lai gqp nrQt bAi to6.n: < i. .. , - 3S8,,, ff .f{ry rnr.rrii nzfrl rriii rri tro*g !.uEng bting nhau. Hdg f E4 )< 2jl :: tlli X 58 (:: .tr0[:) ding *dn hai diu. kh9ng cd qud cdn ud chi biiag t4ti >< !!i :::: )' l.J i t-- be--I&n ciin; lim te uQ, khda itd ud. -eho bidt nu f 'll 1';^l +2iJ,tr) n$ng futt1 ftirr ilrrn ), XtC0 X ::: i{':- 9i (* 502ti Ndu bau d$ dqc b{o fo{r hgc tubi trr! ahiEu, [:an rif thdy bii io{.n niy c6 hg h0.ng v&i c&c y t74 x.3: . Sii . sti { = 5o6si lr*i (i ll:irrq b* ifln i'ilr r.
lr rj t'I 0 i tri,)t s6 ban doat'gii-i ri c&c thAy .1uh !i'6tt Atih tr$n: CAc han dcet gi6.i i:hi grar., i.;.i,,1i i.irur.ru Ir..rrr lrr,ong Dai hcc lglrg Lop 16 : 1. \grivdn (Jriirrg {-r.r1.:is Iir.) Ll tl:] ,\ O t. ?. Fnng \ih Si:n -lnli dulri: irrir bErr doat giii r11 r6c rlri-v 3. \guyirl \3,r.'r'ir |rbu: cO giAil trrolg Pfub ttr6ng trueg ligc Tlung {.9p :l1 {. Iltri,nh il{ri: lh6ng !'rrrrng, Qu.t \hon,:\gl,la B,inh. .. 5. fi6 'lrong Yinh l.0p 1l : 0. lt{guy3n Hoanh Currnq ri** e{"rc Tr*Afvr GtA eusc TH} GtAt T{,}eN -.a €ffi&ffi fr*rFffie *&{ ffiffi&y r* rffr* Hffi#ffi& ffi&$fl rqs$ In- 15.000 sd t+i xi rigbiep in 75 HAng Bil * I'l* Nqi. Sd ia ?4/&t, Iu xo6g vn grti ruu ehilu th6ug $-xBB$ &hi adr tI&S4 6i6, * dtsng