Toán học và tuổi trẻ Số 151 (5/1986)

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Toán học và tuổi trẻ Số 151 (5/1986) bao gồm những nội dung về kỳ thi vô địch Toán quốc tế lần thứ 27 tại Vacxava, Ba Lan năm 1986; đề Toán thi tuyển sinh đại học - cao đẳng tháng 5 năm 1986; sử dụng giá trị lớn nhất và giá trị bé nhất trong chứng minh;... Mời các bạn tham khảo.

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Nội dung Text: Toán học và tuổi trẻ Số 151 (5/1986)

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8ii Tllst : Cho il BCD IA iri gi4e lhi i 6 c6 c dinlr hh6n j . nf;m Ir0n rrr0t tludng trdn. Goi r4'B'C'D' i& t; giA+ e6 r&e dinh ,{'. g:, C', .&, l& t&n.r rfia ctc,, drr&ng trOn ugctri t!6p c6c t*m'gl*e tj(:I), ACD. :t€p vt 3B['. I'q vi0r T( qBCDt: A' R'(: D, Lri t{y .dl'8"D"'C" =T{A'B'C' fi't. . t\ Bai UIir . a) Hey eh6.ng minb AB(:D vA a"B"rj'"I).' dAog ria ng Cho dIr sfi n1.. oZ,. ., on.. e6 r-Ar r lnh ehHi b) H+ sd {l}ng dang phr.r thrrQc v}o d$ I&n do--- dsn. n)' 1: oir;+1 * I ; fi.,r+Sr s 0 voi nroi " e*e gde eiia r* gi*c JB{;'D. XAe diuh hf s6 r:Ay. rtlfr. I frb dr.t tagin tht tedn qa6c t( IIn llrri ?? Chr'rng minh dfi3.s6 kirq:ng r* eh,u ki. Tq ydn fry B&i 8;l5l : f,ei 2 r5I Cho hai rlilrn C v| lr:4n n&a elrrong lrdn .D l flr.)mrr-8, r..* r:? vi n Jir rd nguy&n dn&nslinh'AE, A li m6r dibm di Oqni rrtn - Chc ed dinlr. *oi *rlng fr iy,t .At doan {tO rqi S. l{B .at aos,, == af1{1.} da{,r\,.. d1,at {r} l* }ilu_ digq 1i,gp /{.r,} 3."1*3 plAn cria f(*) Ta rtr*atr t*p tl.D t+i r PIny dr.rng vi tr! cfrn itibrn.$f sso cho Fh*u sd th?p p1"{n tloan S/' c6 ilQ d*i ian nh6t. { : 0.1 dl (?}rlr (?i... d;'2, {Zi jr{at... Bri?I"in f,franh ..,d,.1s {3,d1i l),r2 (i ).,. Bir grlsr: , Ph*n s6 th* p phsn a r.6 lA hrjr: f i. d6i r-6i GSi /(n) ls s6 nh6 nh{t ege di}m kh*e nheu gi& tri n nlo dl6 irav kliCaq ? Irln nr4t pbfing sao eho ildi v6i m$l t:7,2..n fif ,{tr tuuT:: tht loan lufrr Ie fin t,nrr 2T
' i ,. tr3t ; ffC = {.P': FA - l:. !.i5r rgr dinii k r!} drfa tlch A i/ffI, ir&ag (r z$-rt;go-tirh A ,r,8t;. i' ':1 ,{hs tam grilc ,,udng & ,4 t:ti r{ ft, Al) "1BS Id rlirong cno r'* ph&o 'giit. Uur r, 11 r2 li ll{n [,8p emsi cdp $kb r,h6xg c€]' *cr kluh eria drrdrng tri'ri ntli tidp rAe {nm-gig.e lunng il'ng ,1 B(l: A tl B: AHC- Chr'rng rrrinl, r&ng : I. rlch ra thua sd: PlrAn '*',, I ,-leu = tlAft.+ r,f .4(. ,l . 4'.ir.2 * 6{c * n)i 161,e + b\2:4abe a(b , b] 1/.1 ilz *.11 ABz r llAi-? , :t a(b * c)'hz - ct) -1, i,(c * a?isz - a2) :" + e{a * a)taz -t2}. e) r' : r'i * r.t ?' Che ?r,S * g: * la7 :: (t. Clir'rng nrinh r$ng. i'E Fr{n ?'tr |szus{* 4 i;) + psrz{z- y} * t,zT*il;+ 3i = +: 3 Cho ailbt=aafbg,,x. .qall6r: jr ih sd n'guy4n Lep 11 drron g lrro- i,r,,. 'ii'' , .1"'50 ^ .-" (' : t. Chung minir rilf e1 {:, .' ek 11 {i' er I*, la r&e e&e s$ trir, k1' sS b5{r kir4ug rrrr: rs,]+- + re?*. 4.tgz-I*2-= t c 1! ltt-L:- i : fi" 1/ ril t/ y --',i' t'.'i l/-t--:t u,/-t-r =, r F cho r*1ion AacS',' .^*n ,,',,: ctt: h, c:t:e n. L,' ' I/ t. t" i eon lgi bi.ng c ;. *,, ., ^".*;:;;,:,i,,*^^.. c?nh _ ''___ -" .__***-* rri'firrr A, h(} Itliir,'giire ! tr?1 lre ,r, ,I C hlr,. grur il, c.i'F rlilrr |6 rit dl{rn .'!p tir .6 agc rAr .r'-i* ,2 11 .,2 i. .l-;a ,,3 ,l; eip cunh ,!6i rriong gdc vtir n.hr,ri. u; b)'l'irn hS l!;r'rc giia o. h c rlt :f dr{n ,.6 e{c. _ ,r /,.., _ nL.* . -j..:F :,,n, teol,n|,i,'a;il, bfrng,,r,ru. lttr i t;J:. 1/ .i3 -;,., .t- 4 =r it e| Tie.r h+ thf; giUa o, i,. r dE tr.r d,(n cr! ri,.**j; i. '3'rSo l.qr tr'$. ^SC vri f-'"{ e,rja i\,461{ -e*c t&*'r ,:.*u r:6i ngcai ti{1.r trirlg rileru. r- l{y .:irc rlrini Jl .\ r'} -{, :,: u tro .4,ii I .tlf," :* l"r'y,6il;r y rt n .i,l t.),t t0
t l i sE r{}^&H TmE TU?EN sIIuffi , EAI HOC vA CAs il.IHG: Yr$Atrfr a - --___- -- &-gs&s (180 pltfit,- giun dpc ur) chip db t.ftii khlnq: kb,tktii -/ ' NGuYfiN rinn rai $ t. neu a* Ctlt';:. - Thi sinh khdi Ii vA D khing plui, , G.Hip.h&.n ldm: I) CAu.Iylvi cf,u lf.. Cdtt I: 1)-Glli phucrg triuh ; Thi cinh' kh6i cea d8r:g" KgO v* lrop'lf,c { sin.r{cinr -i- i xtr drr thi cdr, khdi A, E, tt iiang pluii f,tnr,'pfrJ. "osc} - =Q cosSr * sin.r, -l- 1 3) cAu IL phEn 3i elu Iy. cAu V. t
Sing hi#n ihi6a "1) D& thi cfia hrm s6 (i) ai qua iltlm p(a,6) I f khi vl chi khi: r?-m lrJ' -{x ? = ,3 Th{m* i ur + (q' - 4m+l}a-l(a2 + t) i ie1 (a*2)nt2+1o a-: m+;s .:nilo_i_u=n vi i fri. ,l -r *L I ,.56 it_b rh-i erla b_9 hAnr s6 {l) iri q,/a di}m r .d*-l bEng sd nghiQm e sia phuong trinh (r)" {xern I phuong. trtnh aldi v6ll m). li I i * iol*r +"- vi I Gil *fr n :2. Khi d-v hni s6 liang diu rria n6 I /-\ ,7 lrii c*a (t) b&ng 0, ,lo rt4 pt,rulg-;-f;nh (3) ji _OS rluge th6a nrf;n vdi rngi m n(u -2 lt; ro'* o -- 2 - brair e 0, nO vO r:ghi6m [6u 1;f, b+0. ,)* Z. Xtri ,4-v (:) l{ pi,,r,,i,s rrigh b{c t ,h-ri,Giin6"rri.r-* rd 3 nghigm,,i nghifm f,u-, O nghl;h"ij; 11 Ut : theo bigt thrbe. A',1* rluong, bigg a'la-v ern. Te o6 4':u'(o * e)z-{a*.21(a3 * 2a2 * a 2 bi (e-2) (-a*:+b,. - =- _Ndu }o ttri^&'}> O tfr; ?{ r: ( 6*2. & r1.=o *5#itt n{u a:[r+2. ,S'( 0 ncu o < 2 ioa. 'r]f +:. f"*,Gr * 4, y''= 6 khj I = 2lS ve d6i d{u khi rr 6i qua gi{ rrf Ul. {lb thi c6 ililrn u6a t+i J| = 2is. {}a thi rl?* hlrir s6 c5.t trul tung tgi ry=-2, .rHt tr{c hcAnh tai c * 3. ti: Jr'2 * {rrn l)r. + .,r,, - im * t, B6 tt.t)T91-trm lh{re b{* -hai. Do {16 {t&Lr . ],idu D ( 0 rhi a' > o khi b +2 (2, *,1 n6u a : b -- 2.4'
f;&ng ri h*.m eS lelr,u( r-*4) dsu di4ii siim, ne* - N{u EF i Sa. rhi ri F ii itiBrn gi&* e&* S.B, btl .toAu -qui r,& t,inr ni dl eho phuong-triah (B) tern gl*e SEIJ *tn +,!ff * ffA *f hai .lnm gi*c .16 2 nghi*m ph&e hi*[ l6.n h$n* 1. -]Iudrr v{y vu$u.g .9;tl', .8CE b&eg nhau * BC : SA x a. \ta ph&! cti : llio lai *{u l:.i;r rli}r:r C tr$n drr*ng trSB sao cho A ='-rr.+- t)? - J(rr * :}Xm + l) = 8r}r +}i> { (nr 0 IJ(: = S.l - a thi h;,i .tx nr giai: r.uOrig SAt,:, tjcE .} *;,)l(- r): "11p {r:r,.8) } u _. b*.ng nhau -) S'il = 4rl'* d,-rring trrag f,ur*n .&i-F. j S,'2 . (,l = (:ar r l)i2(m - li) + t = 18a |anr gi$* e&* SfB cirng 1* du&ng"*to.* EI-' -.L .\8, t = irin _i,)i2(m."S)>tu \'+y *!il* < r: { o boee nr } .3. ,r J\ il+fptf##*% %Hff+t'"fi# 05 @,fu -F- 7 C.?& l) f i'8C1.{, , bLl s.r + BC l_(Srf,,} l.l,'. ,+ #.C u+i hiric Sit .ix *,rr ,[- (iect. sB i_ -li ', str i -|(rrr) +o-Bli{ + -L,r,fi. {} *r. l/A' ln gde t*r bii liai m{t ph$ng Sg.r, SBC. 'l'6rn lsi, dih.,. kiln ctia {&u' bii {trge th6a 'f'a c6 sin ,ilF: AK!At Troog renr gifie rusng nr6o, udu l6-v drlrn (," rrOa elu&ng rrdn *ao cho d.tC, ta r* -qfi = 45. dCiSC. rrong lam Ei&c BC c n vi $.{ a ( 2l? n&n *J'z ,i-iJ *,i" e = vu6og dSB. AI = .!S. JSlSt. Nhtrng d-B = :n, uhr v$-v rrGn durlng 116n. xc*2nr:o*e, s's ={lW 5c*{}fr}{[r[ lrdo' :) Oit,a if vua rhuQe- m{t phiug r:rie ttrr0ng vra tlru$e m*{ phSng di que f ru$rg g6e . rljlr ' sirl :t: - 1l/l;1' -.{ft . x cu&fl v6i S8, nta id nlm tr6n giao tu,r6n ,(g) *6u i:ai v ---=;: o'-,i.i.fi" cos'.I mgt phiing dy. R6 reug (ri) di qua ,{. }Is* n*a (d) vu6ng gde r-tii Si? n0r n6 cOng r.u6ng g4c v6i hirh chi4u cfin SB lCn. rnlt ph{,ng cfi* dur}ng trt)n, rue t* {d} _L AB a (,1) chioh Ia ti€p:uy6ii v*i dtrr*ug trda du&ng ki.nh jB t6i .a. I)&o lgi, t$y ragt diEm # bdr k$. tr+* (d). Ndi ,tJ8. il6 gEp tlrtdrng irin ir L', n6i JC, n6 g{p ttusng rtr*n} iru & /{. yi BC ;L,S,{. aC JL dC * "BC (Sfc) *+RCl.{f. T* lai er} Sfii(arlr} *+ Ss '[- .ar( + je l- {ssc) -L "* dtr -L.rs. Vgy qu-? tich ph[i ri ru li duung thlr,g (d). , Cdr f . ${r gl.r.l = *{rt11lx * rn, *(.} du{}e. riu €inh.uor rnqii r 9'{"2} =, *&r -{- 2. g'(r}:g khi r: - il4. AB tinr gi& rri tO& lir6r r$.-nh6 ahEr c&a g(li.tren deriirr [ * i, :], te p]rii r!al: i1( *1], gt1ll), g(:)..'t's r:{r g( -l) = *o*ri. _q{r?+} *- 1!4 t m. S(I) s *l: + m. }du la thav aot rlr thi db ttri *ria harn g(;) e€ tlnh tidn theu n:6t veets :) li ti trrirg iurr,6n crla. te.m ,gi&e Strg vu*ng Bgf {- rong r*i trle r}y, l}o d6 rnu6n cho giA gdc & : + dlr = il? " SB, e{ing r{y rrgng,rapr !r'i-l6n rrhAt rria harn g(c) tr€n dogrr [*f , l]-tf nb6 .nhflt, etri viQc chen rn eEo *ho irri glA lrl ginc SCB vuOng g$c & a: is *d CF = 1i2. Sf e 11* * m vd *l?-irn l& dr.5i nhau, tirc lA {{- 1 df,.lA?rong }'E clug tanr gitc cAn d*dng e ar) F? I'f ["{FC, rrung iny(n ,ti;. "ot&+ld4 + ?rn €e 0 * rn e {?18, 1i4
'( f.rea &bi $ra.ihqn dp* su sut{G GIA 'Tsl tffN 'fiHdT vA Gr}{ TRI sf NHf,T \ \ TNOilG cHurc IbTIT{H \/ >. l0 rinm rHrnE- ?Grdi, T,a c6 g'- rz * rn. DL &ONG cae hy thi rvdo Bq.i hc.re throng gEp lArn s6 ilbng' ,. *&c trfii ioiri rn& n6u driqg 5i& tr| ,!&n nh{t bi6n khi r ) 5. e5.n va. .hl lA g')O'Vt' ! f. v[ gi& tri bf ri:fit eira h]rn e6 g {gqi t6L ]ludn vA.v U n,rn ) 0 khi r ) 5 la dt'r. \'i g' Ia il. ilnax iA r7,r,,o)' thi e{ch gliii bti to-*.n sd lrir para.p0u,qusv irE l6'm vE phia ir0n, 16 dinh tqi " ',leJ ,1ira siii, li{rn uhiBu; ", I diln r = 0, n*n y' dbng bidn khi r ) 5 vn , , ''" Thi arl 1" iOe ihi D4i h?911 - B,'1l11985t. l'.irn nt ,lb , hu irirn sd 'Th{ dB 5. C.iAi prruong rr;nh -: rnsins;cosi f 1/{ x cus?ztr' ' ' ll = lnle *'3cor2.t' ,:\ . (2 + s,nrri(tt - sioc) = 16 .r1l:*l lu0n lu0n tlb.g brEn. ' Giriir Sq:i ld' tr&.i l&, g i'5 Cl{t-I : ainr, #lJr. Liprn sd xAt rlioh r'6i rr:gi .r' - 1
V; Iiait. Ii r:i'i,: i:l*lii,;';,r;, ;:hr;rrS rlrll'. irlli i:!-.i ti n Jr lt B + :f ;: '::?-!ti:t l- ,Vr;; i- iltl ,, lt *hn x:ie {Tinh iri*t} i;irdp cd t&ng eas i:4n}r {ft ii,ir4 *-i,o[r,;: cAtir tu l::,1t,ji&rn {rung ?Am sn + sl, -i- sll b6 nhsi.. l& tlri, Gr2i t:/ li r:h$u gi6e d0u ,1tia.c$.c rr"ruir kh0i,g tlhi, b&ng chibu ,-'ar,, ruu rom giSr. ,6n Ll',ting c*.i:h ir! mdl diElr riuung cair'SI1, khi ctf ia chi er$n tini rrirrh ehdp 16 t6irg c'''c hinh chi6u H,r. t HB*lILl bf nh{t lgoai lrirn gii,:. ciiu ildu eili: c;rnh lr!n hon e:hihr: la tt ii. i:ao tsrr giic.rlin. Ban il,;c tLi [ru:,g rt,inh b*lg ' f i ciir gt1t .r:ur. ,\ g) piirrr;rg ph;lp riign ti, h), {iucii *ring. rn&i cfre b4nrtu gle: m'il sS b$i ianr 4;16.+,1Ef d&r:r (Ap sau dir' : bC irtrn.'120'. n€n t.6n iBi rl;:y lririi t. Chuug nrlnn lf ng -hirrr sd mOt rlihiu O lroirg ll = 7 t't' l/?. sii,r8.u t "lsin.lc f sinr tarn gidi:. nhin Iu0n luSn cldng bi6rr, c6e canh rlbu duui 3. t{Ay tirr: rdt ci cAc giA tr! e&t *t tIE mOt g
y0 cut'{c ?HrroNG Tr[c GrAo s{"r T& QUANC BUU =\ ' Gi6c su Ta Qr:asg li&n, sinli ngsy 33 rhirg ? nfirn lglo, ds rrt rr;n hbl I t gi3, gt) phtr o6a1 3! :hinq I 16rrr 1986. lni H& x-Oi. Sinh lhdi, giJro ru T4 g6,u quarr liiru ldii xny dqrng vi phAt triBa 'ng*t1k rehr ]ge Iri4t lr{aflr" eunl9 "ai* \ "i+c ' v(ii t&ri$ u* il d** tltit ddi voi nh*ag nrng khidu toh:r hua ir6, Gieo sr d* co .rX6rg (6P qusn lr{is* trsag iiqc sir-}g l4p ua xr:, d$ns.beo ?o;1a hgo v* T.u}i:16. Th*y eegt'F.toi ilbng bi6n tgp, Toa uogn, cgng ri,-c trrrr rii Baa dgc e1a bi* ?o3n hqc I a I u]i trd, rhung rOi rin'g&i tOl thEB ,1ayir* girt su itl! ciris b116n sdu siic rrlr ii. i E,ir) 'r*jts Hgc liA r uht lRi:: t ''t'h+i r..4y. irdu trai rAu dirp el;,, h*,i c*u b6i 1 v*. 2 dbu ill, gQl iroftr, d0u lA /,jr_.. l'fim rri6t n&ty ln-ng cgrcri tri l&i I Srn ir r:g*iri.cria I*p 9C tihi d4 rr0u ciir d6p cho c*r h6i4 lt Id; *glria l* c&l-r t36 n*i itdi, pu].r:i c&u ii{p cho e*rr liirr 2 lA a*i thiil V.{v n$u e&u ril;r c*ri t &i ! le 9{l tlii I'drrr LhSng cAq tri:.i *iu ir:r,- irr ir*r. I€tr ll Jr?e ti:i Tfrr:r pli&i i:or ciu i vlr bi#t chie q BA vl TH"{N :'*ag *Au rlapl*. u6j tir*t. f iu eAu dep eho c{u hoi 3 lS ge?l :igtria ia ciu dr: nSi rh*.t, sr-1 re e&u irir ba vi thlo rgdi ai.{;; th+ *iu h6i 2lA n6i ci6i. )i{u rtu ilAp eho SE:i v IIgt it]f,n l)& {rri ru6r dsr o6l rai rdu l,.r6i 2 ia Ide tiri TAm hhi)rg ein h6i r:Au * A,ltcn ,\gatrr lhtu t6't, t!* hai rifr.r. Ndi; la gQt thl 'I*rn pHii Li;i e rlu 4 r.i trl6t S6i u6.rig ktri dr3*g, h,hi sri kh6 iu*ngl ring riu '$llr cho c{r.i hfii ni,r' 15 n6i d$i. M,)t *gircli r*g d6o ra ptruong ii*u hai *A* d$p rh* ?