Toán học và tuổi trẻ Số 120 (4/1981)

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Toán học và tuổi trẻ Số 120 (4/1981) giới thiệu tới các bạn những bài về trở về quá khứ có gì lợi? Đường đối cực của một điểm đối với hai đường thẳng đồng qui đối với đường tròn. Ngoài ra, tài liệu còn nêu lên đề thi học sinh giỏi Toán toàn quốc năm 1981.

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Nội dung Text: Toán học và tuổi trẻ Số 120 (4/1981)

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I Ldn rdn 3. So sAnh nr (aa) vd m (o4). Ndu rn (sa) = m (ual thi .r = il+; rrdtt nr /,43) * rn (or) Chd 1i ld p ngu,vOn td cung nhau. vrii .4.,.8 Theo ilinh l.,i nhd Phec-ma ta cd thi hdn niio uir ng lrrrn cirinh lA rc. aP-I-Iip, Iin-l tip b) \0u nt (ot, Dl' rr1) )> rf (nr. lt2. Ds) * . ,' jp*l -F Rp-,l ,2 :' p. thi e' € [ ar, lz. l,s i. Lhn tdn i. So sinh rr (62) 'i,a nr (bs) ndu blrg Vi alfl c6 ,,1I,'-1 * ltp-l' le bQi ciia p. suv ra 0- nhntr thi x = ol, ndu lihriri rrtratr thi hdn ta-o 2 i p. t'tl lv. \rlir. chi cri thE k t:.h:iri, r,a rn6nh H I nlrc hrrrr chi nh lir r. il6ritttg vrii n-N+1. fr c) Ndu nr (n1, fil" ct){rrr (ue. 52. be) t hi, r ef ar, arf . ,\lirin rril: Circ bitn ^lfryul/in 'l lt,tnh. NftA (12C7 L.)ln can 3 sp sanli moI tr',,ne |:ri.h6lt dir voi o Lrl Qrri D0ri, ihinh phd frd Chi \{inh). A'rrttgdn si tittt t',: ,r'. Dug 7'ttai S'an (12C: Phi C:it, l,ighia Rinlr),Ngrrgalr li. m'(-{) { nr (B). Loi giii tur;ng tr tlrrurrg I/tin lry {,'/rdrr (40 liguytln Du, l.)ui NIion) co loi hc,p t. girii tot' Ir.H.. Bdr ?/ lL6. t,hring llrr('\,, tt..tiltl ,,nr1 ' A'+;.lndosL.t) , tLt Sri,longUqtn wr ntinh rdng sa 100 .01 c/ri cri t Bdi 3ll16, Tim t(rt rd ho tiing cac sd ngtrud.n {d r sao tt; ruutl p11111 t','ri s,) 1.2.3,..'-ntr. lt = l! il * [t/z] + ... | 1/.r:1i Liti qiAi. l'lrin 2 eirrr lrdi trran lA lri6n nhi6n efing li s,\ 11r111r1itt td. t i rnr,l lror' sir rigu\-an 1r-i cira sij ta sit dFrr eri Lo'i'qiiii (cria ?'riin Bat Dug -- 1iC7. Bni Thi ,1,,,,,I 2'il l; .' t ! 2n ' 1. .\ulrr, lh:irrlr Plr6 lJii Ciri NIirrh). '''l'rr rrhAn xrlt thfly rlng Yi 1t10.,"01 = i02', 1nin brii toirrl ,,iio ,u,U \!,\rV 2t'+l [rtn, 1 - t/F+]l : ... =: t/i;l lF:Jt:,, trrroug'hop li6ng cua mgnh r1i;: + t/^'t + [1"r's-+ Ii + ...,. ( Voi rt, B ln c.{c s6 tn nhien vA (l,B) : I ' lt'tlllf - Il = (2n + i)r. thi s5 -.1t -l- l-?2nchi 2t*1 k + L. cri uoc si; ld dq.rrg Do d6 , = tr1,/ri + l{i1+... + 1y'"--11 = Mfnh d6 hil\n nhi6n dung n6u ta chung nrinh r-1 x-l - r--l cho c{e tr/rc sd ngur'6n t6 16. ,'Ia dung gui nap. I It2n+l)n = I n=1 I nz +'I n n: u:l - \'rii n = i,, Iri0tr nlrii'r: ;rrqri wrc ngur.t\rr tri :: (,r l),r.(o * l)iii +.r ( r * 1)1i! : ,- I ii d * ll chi cii r-lang 2A + l\ = 2"+1 ft -r,- 1. I I "fta : r.(.-l,r'2 * ll.r * I )16, Cii.ri mf:rh ,1,',lttttq ri..\. riuhirr la rrri Hrl . I I (A, B) e 1 thi tr6c ngur-rn lo lrl .riu .{t* + B'^ .. 4,.2 -- il.r. -. I - tili,'r (1) I I +bl/: I chi c6 r.lang 2N*l /'' r- I r. (21 l)o.r, { lir hni sd irguv€n td ndn ta cri (2 khi va Yrri 1.1. B)=; tlr i tA2,BZl=j ,,.{"-*'r B"t' chi iihi 5oic 6 I .r,holr: .r. = !1.T.ir 6 : .r. *).u = 2 \N : (ll)?' + {lJt)2 rritrr utic s6 nguvtn tii ld' r,ira hoic :r: = 3, lihi tlri tr (l) =n y = 3 hr1i0 g: 1"J, ni; lhi er1 ddnrr 2N*l 1r + l. CIri coir ph6lrlrurrg Vr'ri ,r::gtLi (t) trcbu: -1vig-7ril, lai ii rni rrlr k chi I:. gia tt'i kiri'ng th-'r'r nritn r:eu cIu ciia tlAu nir,r, bii. Vl1- nghidm cira biri ioan IA r'-- 2 ve .r= 3. s C iA sri .c(r rriot.. rr/'lc ' ugul'Srr td ld p .cira .\:.hf,n .zril; \,k)t. s5 h;rn sau lihi r.li dirr ra ?;r.. [-t ,N+l p : 'J]- Fl J' | !1" Ia k'+'1 rrl\'/, lii. 1rd tlrr drrtrc (1) cto khirrrg loi rlurrg r'lidu ki0n r" vri u 1a'lrai sii ngui'0n t6 nen bi sa r,io lrip lurin rli i rl, ing. .1P-l -F llp-t - .rt*o1. + fi,n*'o Bai 4/116 elrri .S ki ,sd nhrl nhd.t trong c(Ie , =(r,"u,)**(ur-*,)- sd r, !11.x, tl t,tlq, 1ir. t,!ronq rli r. t1.:.t lti ddc I sit lhqc dtonq" lliiy tlnr mi'dn qid tri r.lia ftIti .S la hoi cri:r .4'n-"t * B'no' ,uc tA boi cuD p. ,'ir: sd.rr, r/, :, t thaq d'bi. 0
nrirl l&, 1, 1, llk" lllr. 1c I = ft. Y{y khi .r" g, ,, t thay d6i trdn lAp eac sd thuc duorrg t^ stl ulirin nrli giil l.ri tlotrg nrla kholr:g t$. 11. l{/r(in e:ril: Nhitiu b:rl v0i vr\lg k€:t lrrin rni6n A IJ B giir tri ciru .S lA t0. t I serr lilri m6i clii chirng nriuI S (; ((]. I i rn* t:]rulr rrhriug rrrinh S nhin Ta co: ! r'l'),.1i gia tri It'r)rig Ii. lrolrrg tlc. tr,-7 r- 2 .\ ( p - d) tg (ll2) BAi 5/f 16. Chftng mtrtlt b(tl dang thttc. sint (.1 2'i P s ins ("li 2) .,rur -*-lI*- 1 -- 3I'* 1 ... -r. ' 25 - f:-- ,ttla 'Jl---i rn*1 "o.2 2n+1 =-;;il:i,,f;,,,(;CI -. ;;;-= - '+ .or* ?'lfi- n -:zR ha tp*illit -+ CtAzitt, =. 2,8 (p *a)/p < 2n l- 1 2 iuons tr i ";,';,;:,':oi",ii,ru,Dtuu J.::r,!u:!,'!{',o':''^1,'r='2titp- illp+ rhdiron-rdp 12c2 '*.:', II o-r' :\ tl O l2 O uz)iiz ti' not'" Drit st= X cosz --"::- " Eris {+ l" ll lrc ;> 27 rt,* e$2 r)cz. k=1 2n | | D:rng thuc. (1Lic tlik^cira dr'iu -r&y la khi biri),:hi Cht j'ring 2cosza= trl-cos2a, ta co O.12 hr:()82 ht):()C2.'lt"*(p-rt),'p=lp:t) p= ' t) :(p-c)l pe) 6=fi=t"dj^r' = B': C:6()o' 4rk 2st:=n *,1^2, trong d6 52 = [ .or _ ] \'+:' tam gihc dBC th6a tnin ditru kiQu bii ' li=l 2tI * 1 r'a c6 t:ire g6c bing nhau .r'i birg liflo. Ta c6: Nhdn ril: FIhu hit c{e b:rrt giri brii dlrr c6 ltri Zr n I t:.:t + l)llf uiai ,iring. L"r'ri giai tlrn n{iin grr,t ltcrtt cii . si, u "z ;; = ,,1, L '"' - ,. \.\r. (2A. - l) 2; .1 " (lri -: l) 2lx 1lai T 1116.Trttrrg lttur giac At)(.. tn ,!vtt(l ,ltrdng & * *in -2fi . - -.1,, J-. baig nhar,. Grd sit rrl ,v ri chcin d*&ng cac gZc * 2n*1 22*1 oudng ooc lttt tttotrgy t?ng tit '1 ddn C.u. B ddn {.g'. $ ,51 : * 1i2. Do c16 2St=n-*-Sz-2*l/2+St: Cht?tt11 tninh r&ng 4 dt?m rV' Ar' $' I' nim trln ni2-1l41nlZ. / m1t dtrdng trin. = N/rcin rct: I{hu hdt c6c ban grri ldi giii bei -.- t-oi qiiii. 'l'a chi ciirt chtng rttilh ba tlu}ng niy dBu cti lrii gifii giitn t,r;ng"'irrnh,r't.en. trurtg trrrc-cr"rf 1li5 c13+3 If,U. :lN, HL glp nhau t+i rnQt tli6ml Gifr sir Cr vA Cy ldp tuong rtng BAi 6/116. l'inh eac 1tt)e bfia tam giac ABC, v6'i C,4, CB clc g6c bi"ng- llrau vA birng a. (1x bidt rt\ng , bb sung vtro ClAtr b&.i eire g6c n{y ph&i r}gugc "8ltt LolroL. *27 oA2. unz.' ocz httcrng nhau vtr khac glz: neu & .l"l c(/f ri i 10
!L \, ).' chio ehia tn gidc ra thdnh hai tarn giAc vdi e{c ,{s riffu tlirng thiie s&.y 1;1 €;6:lr=c=tl, tJ--:D:'i1 f2' t rl.f := .Y = c, hdt luAn li hitn n$i6n)' {i6c lut' lu giie AB{. l) llt hirrir vtloni{. dilnr C, ,11, ,1, H nrir:r trin rn$t ilrrirng lrirrt tirn li i1i6rn ciita 1t-irr C-1 : circ tlibrn (1,:\'. /l' 11 Iliii '.tri6n rluoc clt{tng lrirrh nilrm trtn Intt duong tron tLlm ,.I lir ilibnr gifta lla rr l,d .V,,,uqrin 1i rii (12C2'l' ll a nh \l j''I'd.v' th d rrh cria CIJ. Tr'r tl6 sul t'a tt'ttrrg truc cria 1l ll di phd Flii Chi llirit) tdl cAclr giiii lrr,..rlg lu lrdn {{ua 1. trurtg truc crlrr llN ili qua '/ vit cac trun{ luy trirlr irrrl dLltri di,rnlj hcri lihar:. I tluc niy lAp vrr^i -lB cr)ng rn{i g6e lstt:*r' N,,II. {nguoc }:rr6ng nhau). Nhu vl1,s n6u gqi O la giao cirr hai trung lt.tr,-' Bal tlri 0-lB vA dt' d6 OtJ lil tanr giAc ctin. I)i: thiv tlrrtrg trLtc e fra I11- Bii.9i116. Cha hai dtrbnq tron dlrng nhau ddng thcri ctirrg li .trung truc crja /,/. r-41' tlung (O, ll) uri (I, r). Ci*nq tninh rdng neu cd m1t truc cfia Il1, phli tli qua O (d.p.cm). tum gitic .ABC uri'a n$i tldp rlu'dttq lrdn ndg uia Cic ban D6 Quang flqi (7K \gu56n Tr'[i' II] ngogi ti|p dritng trdn kia iii cring co od s6 tant .Dong, Dtlng Trwbng scrn (8c.'I r)HTH' rln Noi)' giac nlut u|g, Bdt cti tti?m nrio cric rttitrtg lrdn Dd Hrru Tidrr (9A Hoa Iiu, PlAycu)' NguyTn E&Lr i&n cfin{t lii m$l rlinh ud. bdt ctl dLlg cu.ng ndo cfir'r (itif Phan Chiu 'frinh D& N[ng), Ngugbn'l'hotth rltrbng trdn l*n titp rfic orli tlrt'd'ttg trdn nhl cfing r\'I&, (12 C7 L0 Hbng Photlg, thdnh ph6 I{b CIri td mt)t cqnh efia mdt tant gidr. nhrt thd. . llinh), Ngug0n lllin Xgi Cftriu (Quy Nhon) ct5 ld'i 'l'r
1'I '-.f NEu ret tam gi6t: -{C./ thay cho tam giAe ABJ' ta sd fluoc cac h0 thric iirrrng trr (8) va (9) trong d6 ctri vi6c iha.v l8 l-,iii A(;. ljdi va-'; 'ld dei cf,a car.r canh -'{.8.;i 11. Ii hai nghi'drn cira plr Lrong trinlr biit: hai srru rla,-r' : r,' -z-----'z , ,.2 * . -2 *.r'- '' ' " l-..i44 - +,r2 + .1fir1=0 (1ll) .,! \',r do il i,: )P,. '\C=: tz + lfir. (t tl Di{n lir:h 5 rria A l1l(,' dtrot tinh theo cdng llrrtc : (12) = ( 1/2) ,lli. ,l(,' si'r ,l 5 (7) rA (11) T jrav giA t li ria sin,l r :i JE lC tu vho (12), la dtrr,rt': 5= dl3 - .r -.' t' ,'ut' .. ( 13) lni=i1B +;'cltl I cBJ = IBA + BAJ Ta thay viQr: tirn .n.. ,X, cira S' bing riQc tim IIIA c uc tri ltra hd nr P (.r') sau tltiY , ./\-
: ve: Cring ehf n5 rninh tiiong tu, udu IA = c.J thi . +d 3t! IB= I(: =:"-u. Dc riri, crrr'ii cu ng ta di rI€n kdt P(.,'rr)=pnrin. - ' -Vlll -{.r^dl\ gU) l,ttAn nhrr sau l!+d : Tron.q sii nhrlrrg tani gi6c viia niri ti6p rlrrring Blil' giri, ti.rt: rrin (r) - rt:Ii-. d, n1lx i.1).= trrin (0. 1l) r,un ngoai tidp rlirr'rng .tr'--'rn (f .- r) -- rz = li i-r! : tlrd thi iri (ll). s.r;i i; jii Jriilr to:iri l;rrtr girir c,i dien.tich rrh,.i nir5.l. (lon trh;it) Ia r'!r rrit !'()n rrbtr lre tlrri,, (l). tir 11r i, ,lrrut.: tiinr gillc tril uhdn rlrrtrng tlrilg O1 lAm iruc d6i :i1l + {J xring. Di::lr crla tarn uiAc cin nd.v cat:h t.Atn f i {20) eiri dudng tlorr rrQi tidp mOt khr,ing nirr"r nlidi t Ii-rl (16l rrhdt) l;nng li - rt (biug n-i- {1). Gi:i tri nho vA: nhd.t vd l
'f ir (1) YA (2), it ilrrqe : *o (5) auy rr : I * sina -f ct:sct ={ Vl + f : 1}{{l _i ON ,,[.{,\r = 1J11/1--,r. / sin:t - ('Osctr \ (l) I d(] slr-\' ra ( r{&\ ,. -1t: --'-- i_-' I r..r \ tTr'osq I*sine,l *Bari CT Le l{irna -N'ryurTf'n'l'hanh Alft,i (1'l X6t hArrr sd: q.=I(.rt:sinrl(1 {cos.r) Phdng, lhilnh'phri !Ii1 Chi }f{t dua m cich * c,:e'.l,{l-}-"ir:s} gifii sarr : Drlrt ()lI:r, t)N=y, NO()z: fr, ,U{)Ql :F, (o('r: o e r + 2R- - .2r it:i:"> a (1) 1 * sit.tc *"eose tet> 2n (1/t- i), i,nr' 11-\'= t>2R (r/i - r DE
I -t"\F I Bdi 3/120. Cbrhig minh ring rr€!u n,b,c I:\ ba (.{B) vd ('ltl) fr P vrt clnlr ciLa rrr6l t;rm gii{e llli Q. Chung rnilh r.ring rr6u iI li trurrg dilln: cfia rlor,lt ,te thi ,lre *lmU, os + b3 + c3 - 3,altr) n (b - crZ f I, (c - a)' tron.q 116 M la irirng tlilni ciia ,llC. l *c(a*b)2 'l-rt Hdng t)uino (\ri6n 'Iroirr hoc) I(hi nAo -rir-r-r'a d:.rng {hric ? . - Rr'ri Quang Trring (Hn NSi) BrIi B/l2O- 'l'16n c6c'i,rrrrg IJ() , {).4. AB't:iia tlrdng irdn r'igoai tidp tam g.i.{c -4ll{..kh6ng - Bai 4i1!O. xac diob'cric sr-t ciuor;rj -1,.i-i,Csao r:hti'a -ciic ilirrh .{,R,(.-lrt Ii-r lrr,rrrg ring t.jrc rli6m 'cho tlr thuc / (.r,) =,1rc * g:.J - r.r. .,]t, 13t, L't. Ch11ng nrirrir ring diEu lriEl ciin vA :jl ,Si1 {r'i ngrrr0rr yrii nrr,ri :r- nqil\.L,ii vr', /rl), du d0 .{Ar, Bl}u CCt d6r1g qui tn f(21, f e') t.i' gifr Iri rr!ri, nhjit, .4,Ily BLt. {;:lt. =;{18. /jtl'. (ti,"t, 8di 5/f 20. a) Giii sri ioil ( r (i = r, , ,,.l ChLi'rrg rnirrlr lir;g. Bdt 9iL2O. 'I'r6ns rlrrt p6irr! ff c;o nrdt g6c (l-Frr1)t,1 tu,il.. (t +/.n, i.(i-il1)(l * {r z,}. .. cdy r1ua1. xiri:g'gur;ih .di6m ,l r-rii dir.:h. O lA ruQt (t - ao) {2". rli&rr. c6 dinir rriur rrgriiri rnirt pi,i,rig T[ sao cho t.iiir thri 0 {:.-i,
-;:ii?BE#a:ae-Gq: '.\ 'l'irrr gia tri ngul'Err cira rrr it& t6ri tgi rr6t da SAi 3. Cho b6n tliiim irong trhdrrg gian A.B,C,D, t lr ri r' lt (p r ,,',0 f t p) : qr 1'J. fi (p) r'(ri nl{)i gi:i tlirlng vu0rg g6c cl.ru ng .,UAr arirt ir*! doal C^B I ll ut' r:tirr p. v{ (10" r'ri 'tB luOrrg 96o r,"oi {.D. Tliilt ling "{.11 = J78 = a. r,.',\'= ,YD =. [, rly',\, : iri, lr'ir 1_i1l tr.on{ Bdi 3. {-hri hrii di0rrr,1l r'{ rV r'v n{i)ii IrlOt khitng gia n rn0l di6r:r S sir() clr() tbng t ic k lrtrlirie { plr;i:;3 ,rl. \a,' 6;,,!1 r i Iri riia rliFrn A {r-6n rrr:i c6ch ILi,S t]6n cic diSn: A,B.(:.b iii cuc iiiiu vA 1? sati cht.r ii srli ;t.l/idl{ l& r:rre tiiu" lim gii iri i:rrr tilu rlti tlttrti it. &. ttr. t. B;li 4. (iirii hd grhrrirrrg trirrh: *2 +- ,JY -F ,t + I!:50 ,'2^u2*:B-tz:.'-').1 rfr guA mt rut cmgl{ IrOC sr NII x.z : 11.1 cI0I roAN eudc r-y+z*l='0. Giat ci nhan. Bei 5. (lho n sd thuc ti. t2..... fn saocho0(p GiAi nhtil : )igugbn l'fi' Long,'1 11p,ing. D,li h.,c,f