Toán học và tuổi trẻ Số 138 (4/1984)

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Toán học và tuổi trẻ Số 138 (4/1984) sau đây giới thiệu tới các bạn những nội dung về vòng tròn nhiều tâm; nguyên lý khởi đầu cực trị; miền giá trị của hàm số; chứng minh phản chứng. Bên cạnh đó, tài liệu còn đưa ra một số đề thi Toán đại học - cao đẳng và vô địch Toán quốc tế.

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Nội dung Text: Toán học và tuổi trẻ Số 138 (4/1984)

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nhibu tilnr, thirn chl vd sd tArn, chring ta h$.'i: suv . '['a hfr.'r xet lem t{ Vdng lror: I tilr ,{ rli qua nglii thOm r-b n Khoing clch )). 1'r.ong diri sdng .8 li clrrcrlg eong gi'.) T:r thi_r. ngav r',irrgrlo lli. hirng ngiy, trr, ha1' n6r : ( K ho:rng c6ctr ) tu noi drrong thSng thdng thuung ,l'IJ ri i Lhi .,1 c6 nirv rldn noi kh;ic, ( Khoing circh l vE trinh el6, rllnh, B chav tr0n drrd.ng thing ,4'B rlri hai (Kho n${ cich r, vB tu' tirr}ng r'.r... trgav < Iitro&ng drr'&ng tiring A'd vi A' B. c6 dinh, do dri g(.rc cich ) tri noi n:\1, tl6n rrrri lihic"cfrng c6 thb c6 (dlnh'hu6'ng) giiia h*li dirring th&ng ndv klrdng nhiiiu n!\i dung: ( Iiiroing circh )) theo ilud.ng chinr ri6i, nghia li ( khoAng c{ch, lir it ddn I khclag bay, e Khr:&ng cich )) tireo tlrrorrg bd, ,r Kho6.ng .fAi. Nlru vA]' rtu&ng thing th6ng thudng ,4.'8 cich u theo rludng thrir v r-.'" Nlrrr .,.iy < Khoing li m.dt t6.nr.;{. ll*hung ngoii ti.nr A cich'r co ihb c6 rdi. nhiiru rrshiLr nhungdbu thdng I'a' ( ',,dng trdti > nil cr)n c6 r'0 sij tAm l rat la, plr&i lihOng c6.c b4n, nhuns ciilg di iribu lirm sao. Yi du ti€n dA1- Ii vi clr.r don giAn nli5t vE ur6t l6p lih0ng gian 16ng rii co nhitu tinh chdt rdt phong ptrrt tlo tric gil bli ni-r.ph6t. minh r.u :iA-v dung thdnh rn6t ld- thuydt r:6 thb goi li. {{ l-i' " /l irlr 3 thuydt r:./'Lc kh6ng gian Cri6h - Toin D hay a Ii' thr"rydt cAc kh$ng girn ui tuyQt ddi ilong )) (nhung d6.v lii dd lQch gcc gi&a phrrong :18' (hoirc BA') xin miEu giii thiclr ti],i sacr lai goi ii ( c6 tuvEt va phuong vudng q6c vrii &. tt6u ia dfnh hu6ng dii ddng tr) rniit ph&ng vir x6i grlc inh hrrung giila l:ai cludng 11 Uie u ddng niri r'0i c{c i-rqn ll t5.in qtrrn irong thing .,t';t vr\ '{'lJ rl,;a-v B';l vil 5'B) thi ta cri c'ria t(cicir nhin ): trOr-r nh&ng slr t'At cir, thtint kho&ng cic.h rlai sd AB (A truiic, B sarr) xirc chi rit cii, ndu c6 chch nhin nrrii vAn cc, ttrb ri!nlr i.rrlri /ifi. phit rnlnh nhiiu tliirtr rrrdi nrc r':r l.i'lhri. NGUY$N tf - KFisi 1^ ilAU eryE TRt o ) D0 n.i F,tt.\\(; c- @ St. noNr; rit)r {txt iotlli rr riltl lrudng 'chtng .ciurq nlr rr d6 1a irlrili r: I.t yrl,li: ( llrrllg. ululi.n lv t;ur rtap t aJ (::ic Ii-i'tiri lrgc sinh giiri, {r llirrring vi nE:-r",-0n 1i' Dir iq'iri0t (\'D rrgu.v0n li Dir.ichiOt n lri i lar: tlrt rrg S rrenr-iir l-i' chi'r'r:g i*inh: t:ti f trb xerl tl:0rri bii < Ngrr-r i-ri lic {-ri-r'ich-10 ,
t' i. +:-',.- i I [' '': I 2. LeTg Ltrc, l6p 12, trudiirg Qudc h9c FIu6, Gini t{trdng r6 nhiru hgc siiih dr3 thi I Binh Tri ?hi6n. 3. Dd Quan.el *qt, Iivp l2CT, tluoirg ts4i ]rpc 1.. Trudng PITH Trrrng.'Yuolrg. Quf Nhon, I Su pham lld n-rli 1. _ I ?. Tnrong PTTI'J t.0 llbtrg Pliong, Thdnh'ph6 i .ll IIb Chi llinh. I :-- ,- Giei xrrdt- eir: '' ' Giii khuydn khich: . l. Db. Drtg lihqnh, lup t0C l. trrung Ngrr.r ilrr 1. Ncug,;n Quang /)ing, lr;p 12C['. lru'ung Dai t \'f,n Tr'0i, \hrr f i':rrrq. 2. ) gugin ?iin Dfrnr1 .\, ir'rp 10CT. Dai hne 2' Trinh'Ydn ftifu, l(rp t1C'1, tnrong,Lam Son, l* I ong Il0p I la n,)r. r.\ I 3. 'Irdn Ddg 1Jini, lrip 11C, IrLro-ng 1'rune 3. Vd Eai ilr:Ai D\c, Ilrp 11CT, tnrring;,L6 I iibng Phong. TI). Ilb Clri l\Iiufi. te Giai nguni c6 nhibu ldi giii ha3 : 4. f rang Ldm'.Bdng, l6''p 11CT, trudng 1,0 Itrbng Phorig,'lP. Hb Chi Miqh. 1. Ngu,j€n TiAa Dfing 4, 16p I0C1', trrting Dai h.:e TBng hop IIi nt)i. 5. {6 Thanh Ifting, i6p liCT, truorrg Le tidng rln flrirfn. l6pg0 Cl"r Tlulng Phan Chu 2.-'f 6. Ngutlin i'iit flinh. .\in'j. Quarig, l,;p 10C1 , tluring Dqi 3. fld-ni herc Tbng holr i-Ii')rlgi,- ; Qttang Sai. 'l iip l2ii'I', trrrd,ng I)ai hoc 7. Iigugitt -tlinh \grin, i,.p lriC'f. Srr ph+m tta, npi l. {r'udng Giei d{c biQ.t: ye r'(tn phoJ, ttip 9E, rruong Ngul'6n Van Ti6i. Nha Trrn!:- 8. NgugEn Gia Bdel,16p l0CT, tru6.ri}-!{ Iii)ng Chu;i6p to$n" Qu.v-Nhop, irighla Binh.' ; Phong, TP. Hb Chi I,{inh. Giii truong dqrt t6ng s6 di6rn cao: 9. n-gugan lluu 't'tidn, 16p {}E, P]'CS I a Lc.}i 2l N/idI : Khdi C,lLrv€ulToirn truring })ai hqc 'l'bng Quy, Nhirn, Nghia Binh. 10. lVgrrg6n Binli Ngpc X&ud, lirp 98, tnriin'g Nii: Truring P?'lH PIan Chu ?rinh, Da, NAng. Iliri ?huong 2,.Pliycu, Gia lai':Corrturi-r Ba: Trri'dng PTTH. Ttrung Yuong, euv Nhon, "j.1. Pham XudrtTrunG, l6p 8A, trirdng PTCS l'ghia. Binh. - 'f rrrng Nhi, Hi N0i. *t,otr-zozol * alr4-..., (r,1,-ze2a,, + rl, ) I + ... * [ (r: , t ftr nnu It-I -]a ,,.o * ,,1li )1 - . #*r@ : - t -l-r dr f! ..: -it' or)".c. -,-I ttl(rr1*r.rt)'-n -l- -- 1o1 , ... -l- (.rt*rra)rl I l{oa*o3)2 .. + .", -i (,r2--irnl2l* ' BAi l/135. C&ri.ng minh'rdnq ndu sit N bdno tbng binhphu
\ \,- Lbl gtdt. cria\,frhn Qudn, t6p 10 CT, Di Ning. :ffi#:kl!ip sri dudng\ clt theo cQt chia bfing vu0ng rrongd6,l nz"Gii thAnh hai hinh elii nhAt nXk vi n X (n-t). Ta lgi cit ngang b'ing vu0ng thanh hai hinh chr] (l < 4 ( p). db'dugc e hir.lg rr.rinh. BB nhit ft X n vi. (.rr -k) X n. BAng vuong rlrroc chia DB giii bA.i totrn..ta chi chn tirn ldt ci chc s6 thsnh 4 minh A, B, C, D. nguy6n t6 p sao cfo : v6i mQi a, b ngui'61 mi tt * a'*b" i p thi,r j p r"a b t p. l
11h&ng ehi6c lilng lr) eic chri th6 < tl[ng tlong ta duqe I c{ch ndi rnri'i cd tbng d0 rtii cSi doqn ire,i l'II vir T'I s.i ;1). ilai bAo nAy nrLiSn gioi be hon. DiEu niy lr6i vdi c{ch chon ban dAu thi6u r,6'i c5c ban 1 nguy0n lj. ci:unu rninh thu crin chring ta, r'ily srUr rft dpem. tr.r.cri thb sfr drlng kh6 t6t cho ritt nhiEu L.jri toAn Vi'du 3. (Brii'toin Xir{-r'ex-te). thude c6.c dang hoitn toin khau nhau, il6 lu nguy0n l'r'6n rnirt phing cho n cliBnr (n ) gt. Biet r&ng li ( Iihli rihu cuc tr! u. rntri itudng thing tli qua 2"clibm bitt lij' dBu di NOi dung cfra ngrrycrn lj- phAt bibu nhrr sau I qua rn6t dibnr thu ba. Chri.ng minlr r*.ng cS n '[ronE m6t tdp hgp ttftu htrn (khdc ring) cd.e tli'irrr thang hirrg, sd thurc lrrrin'cd f,rrE cftcn drge s6 bi atidt ud s6 Giii : Gi6 sri' n clibm tihOng thing hdng. I,lag tdn nhd,t. v*i lrrOi di"bnr tt xit circ lihoilng c6ch tlucrng tri \ric6ch chirng rninh rdt drrl gi6.n n0ri nhrrirng dibru rli.r t0i t{t ci c6c duong thing di qua 2 eho,b4n dcc tn thgc hiOn Lir.. Tic giii rh1 xin tr"ong sd t'ii.c ilinh cdn lai. Vi s6 kho&ng c{ch d6 ph6p minh'liga r.iQe 5p dung nguy0n lf urav bing ii hii'u l.ran nii. s6 dibnr ctng iriru hq.n n€n ta cAc vi du cr.r thb: cri th'6 ciron l 1, 20 - 1 khdng chia hdt cho n" Yi da 2. Tr€n m5.t phing cho 2n alibin khOng GlAi: Gi& sfr c6 s6 n)I seo cho2'* l!n ed 3 dibrn nAo: th&ng hdng, trorg d6 ed n dihm rniu t16 r.{ n ilibm m[u. xanh. Chuug miqh ring Khi d6 n 16. Goi p li" udc srl nguyEn td bd tbn tai cich n6i t6t ci cAe ditsm d6 .rrOi .a" Oib* nhdt efra r. Do p etrlg l8 n€n (2, p) = I vir theo xanh brli n ito4n thfrng klr0ng c6 diBm chung. ilinh 11, Phic-ma ta c6 2P '- r, O. Ci6i: x6t ret ci e6c cd.ch n6i n cip dibm d6- Goi ,lr It sd trr,nhi€n bd nh$t eri tlnh chf,t ld xrr rrh bdi n tloan lhing. \'i cAch n6i 1i tdn tai 2k - Ii p. Ro ld.ng ft (p -t < p. Ta. chr?ng vi sd c6c c6.ch ndi l:\ h&u llan ta chon c$crh ndi nriuh khi d6 n j A'. ThAt vAv n6u c6 tbng dO dni e6c doan thing ln ngdn nh6,t" I'a s0 chirng minh d6 I'i cich n6i phii rinr. Thai n'= lig * r vcvi O ( {i,"tf,i, " r,f.5' ndu c6 2 doarr d,-Y vi 8I' ci't nhau iai rIiEnr q 2t' * r =(2t' ) .2I' = (m.p*t){ .2' L O ie, I} ruii-r ri6 cdn X, I' miu xanh). i
Yi.v n i 1t" Nhung- * d p n6n ta 5'r]' ril. n c6 rtdc rurinh vri nlrirng nqrrrii tlrur itlrung trqrrr'ri lhrtit sd ngul'6n td d;r 1r5i vdri c6cir elron F. l\{{u minir' (.hr'rhg nri:rh rirrg tblr tai it riil:lt rrrnt 8f u' tliuin ehirng Lil; iiii$ng to :t ttgu3'0rr -F 1 db clo tlr& tltrcrc goi tin tat cJ rnoi ngurri con l1i. !u*iin" Btii l$p 3: Trong iudt phlnr: irun,- bitlt r&ng Ban rlor il,ln irti;u. ti'iii rlirr cic bau .iir larn r.rEi ngudi r10u guen 'r'oi it nhAt 3 nguoi', Chung 'viri ngu56n i3. ( I{lr
*--.'-"--.-. ''{-r i. N6u O rrfrm 1r'in canlt cir:r I l:tnr fi6t: nAr: e'!,j 1) {:ku nS1 *iinft rdrlg I Ilo v55' Jt (i - I.,n, r n ,s ." 2 *,!tt, - a1.i,ri{or"/{p--rr0*r) *ai,s;las. {!to A,n, c td r=l uo$ n,fr.lo ,,,r,i -r- I oii'(F-.,ruo) ( * : : tan ldng radihn). Ild"g ch{rng mini r&aa: gtdc {aCI .1, at.o/oro 1 l1.cos /j * siad.eo; f,> o. t ,1 o I-*i gidi {oira nhibu ban). \'i .B < tt - C r.n 'r"qi.r 1 4 :,.,=fln= I ll:i rt6tr sin A {cas I -i- r:os C) > O. 'l,r -rr1l l.rai s=k2;{rri * n:1) t ki(j;? * Ir } n*2)-l {n-3}/ trrrirng lrrl;: ik",+- k -1- n-.3) + r/{g * n*t)* 1 lihi.k*+** n) Nriu B( r/z tlri cos pb O. Kh! 116 .4 cos g Y6'i at=.&, .oa=.1r2, ca 7 a4:...= dE=l thi ) 'in A cos B |iri vi A i sirr ^4. \.ir.1, s= k/(k * n*zy + kziut2 * * * n-2) .{ cos R {sin A cos C} sin,{ cos I f '= sin ,\ cos C 0. ) * (n -s)/(kz + k*n * 3) + rl(ez * n - r)-+l k--) + *. 'i.:hi I;) NCu B>n/2ihi ta ci: .{i6n cdt: Cdc i:9n .\.qrrr.;in I're, 0rldr9 (1il(,'f - cos B = r:os iat * C): crrs -! ro3 /l - DltTlt Ila ndi), ifguydn;\coc Yrin Kftoa. Irhn sirr d sin C ( cos .{ L,cs C. Qudn, Lam Tli'ng Giang (tOCT Phan Chflrr I'r'inh, i)o 116 cos B * cos d cos C. ) O. Di ning), 86 7'rpng Ytnlt, (tt CT' Lanr Sofi. Yi.,tg .{>d>o 'l'hanh ll6a) r'*. rnQt srj ban kh6c co ldi giii tuong r.ir cos B tg .i. cos .S d6i 16r. r,.i A cr:s B4-sin :{ co.s C )> tg ,tr cos fi I l['.8.M. .sir ,4, cos C = tg .4 (cos B * cos d cos C)} (}. ffiAi 7i 135. a) iifit1 tac rtinh did tri !*n nhdt Vi1'trong c[ huj tl'rrong lrop ta dbu c6 l,dt cdn ri $rro eho lbn tat sd b dt pluong.tiinh sau c ding tLr'rc phii r.h*ng minlr. lrn ngftifn; { rong ilaan [* l,t]t *2 + l,,r- "u : l/h{r rit: M0t sd ban drlng itao hlm dBr chungi -A - e." minh nhung khdng db .i ri.ng l, -B vA C lihdng D Ching mir;h rrtng khi d.:) cri hc nglii{rn d8u phiii ii bibu ddc ii p nEn cLring n:inh Iil.r6ng d*crq ,'hAt.!rF. t.;r! ,tiqi i,l,.r,r tlr,..,r ,.r, ,lr 1:i i rrl;: )r:.rr l"r'iirr J)..I}. h, ?)r:r1 1''ini-' lop I1 C"I I rurg Yrrrrr;g-- (,)rr',, l"lhirriJ Rai t',!35. ,,/r.r ri l,r sC :g t.lt:in ) j,: ,, l, o?,.... R] (li[ s& ;ihur;ng trinh rs - gz + ]r-a= 0 n cii 3 nghiQnr_ ri, .rz, #s (3j ( rz ( a:s) trong ao id ede rd duang,tdq, tr = X qt va S= dc+n [- 1, l]. Khr 116. theo inb li YiCt, ta col 11 t \r n I r1*.rsl-.r'3= 1 (t) -l ci'( F -trt+ )rot q tty adc rrn* 1 = d l. $ )lr'2 * rrr'3 *,r'gr1 = 6 (2| i -! {,1.'2.r'g=o (3)
l' ( rt rt,i rt * rr * rs 1) Ndrr .rg'< (t hoflc.r2 Goi rtt vi rt3 ttrtrng ring lr\ sd tirr cia Phi;r { rs ( I lhui,n r'{vi (1). nrtlrt eiri:i rtl tho Il vi l3' / 2) I'l6u.rr ( o- -( rr < rB, thi a = rlsz.rs < 0' Ta e6 dn : dl + (.,-l)llil = rr1 + (n--li x (11.19 f 1) = li:O * rit,* (n * 1). Yi (n*1) I 3) xgu 0 ( rr d cr :( rs thi rheo ildt diinq- thuc Cosi ta drrcc lfly cic gi6 tri- tri 1 rldn 8 nOn rll: 1, lirriLc r11 =: (3)' I = sr1 * 12 *'.r'3 > 3ii;1rg-'8. liIB an kirOng ciria hdt cbo 11.) trIit khSc I cn = at * ln -'1 ) (tr't{] -l- z) = Tr'r (3) suy ra 1) s V a ira" a '-( 1/27. Df,u 13.c * ri3 * !(n - !). Vi 2(n-1) lfiy c[c gii rling thftc x&y rd khi vi chi khi r i : c2 : r.: = I /3. trf 2,4, S,.... 16 nOn rig chi c6 thb nhAn c5c gi{ V{y nrax a= 112i" 'lh6 viro (2) ta dLioc -b:113. tr'! 2, l, 6, 3 ,lt anl rS) (t) t'- Iidt liqp c6c rIiEir kien (11, (a), (l) 'n {+t t^ $ Nguge lq,i irng voi a a 1 27, b= 1/3. ttri phuong. tri nh e6 ot chi cri th'6 nhfin circ gi5 tri 67, l9g' 27i' -- r? + tl3" ' tlm* a. N6u di : 6? =+ on i 6{}?; 17, ndu tl1 * 2il ": - c6 ira nghi$m $t = r? 5:.1r = l/3 Gi* 1' 1j' + o3=69? i 17. Yi]' *r clii cir thB rrhlin 1 gi6 b) tlibn nhi0n ilieo c&u a). tli du1- nhit lri fgS, ai do r:irng chi thn t:ri duy nhit c;ip sd c6ns thilrr mirn diu biri lit : I{;:dn ril : Cii+ baa .Ld. Tn,rnh ffd, Trtlcttg V&n t:ttbng, DJ 'J'rgng ylnl?, f , &n Xudn llinir -:- 199, 409, 6 t9, 829, I u39, I 219. 1{59, I rii''9, I879, '( i I CI' Lam snn, Thairh IIE5a) ; ,Vgugdn iVgoc IV/rfl,r *fl: Cic ban 7 ritn Dttg 'l;ttlt p rl'' 'l 1i' Irda J(rloo, 'Irbn Ori,tn. jiaq T'ung Sian.q (10 C'l 1'rung vlrong - Qui nhrrni; I gugdn l'i(t t2uanQ Piian Cli*-r Trith, Di-nEng); ,LC 'lnh .ldn (10 C f (10{l.fDr{rrI); La 7'ht€n'lhanh (Lop ittic 1r'uong Phan Bdi ChAu, NghQ Tinh) vA Di lltts Khd.nh trirng lroc An Ntron.I Nghia Binlr) ; A-gr:gi.n l httttlr (to cr Nguy'6n Yan 'lr5i - Nha tlang), cd ii,i tlti itrlp 1?C P TTII Lv 'l'huoug I( ri t) : 'l-rh n gili t6t. Qudn, r'i Ngugdn Arggc Ydn Khoa (10C1' Dr) N.r.'y. \tng),'.. cir loi qiIi tait. - P_,T. il . Bei 8;tiiS. a) clring minh rrtcg kh.6ng thb co lt) sd tgulidfl td ldp thdnh c&p'sd cQng ch*a trong Bni 9r t3 5. Cnn hot rlldng trdn hdnq nhie 1984 sd ttp tthlcn dilu lldn. (Cttud (Cz\ ttlp ttic u&i niort lai D' D$ng dri'tig b) Chirng minh rdng tbn lal duy nhdt 9 s6 l'inh }I.V cfia iludnq I rdn (C1.) uudnq gic ttdt ngugdn td IQp thdntt cdp sd c$t;g cht?a tronl 19eL d.trb-ng ndi tdm c&d'hai dYir!{, tron' Ch*n5 minh sd Iu nAien dirr liurn' rdng trcn n&a rlcdng tran if .l kltdng chtla D. I6n I-bi gidi. a) DE drng bur rn dtrrre tu c:1u b). lai di?m.t iao cho hui fie p tugln c&a dLr*ng thu iu {rdn (Czi di qtru A edr ilrdrng lrdn (r"'1) tai ,toi b) Ci5 sri'.&1, r,2, ".. crn {x6p tireo tflng d.i?rn B ia c {mri, A) tqo ninACAB can {.{: -4 = ( ll) din) Li m6t c5p sd cdng gburqcac sd nguy0n td ''C/ir!ng minh'rring kni-,{d qoc C}90o cl'rira.trona 1084 s6 tu nhiOn tlirr li6n vl c5 cOngsaid),t. Lbt giii, (igi 01, Oz lir tArrr ciirr cir,' iiii'rrtg 'Gr.ri P l{ s6 nsrrl'dn id nao- r1o nhtl hcn .9. tlon (L'1); (Cz) v{ bin kinir ciia cli[trig ln Il. Trdn nirg dudng trdn .rtf N kh0rrg chrlra , tr 1{v Ndu tr1:p ilri rp+r = a1+ Y: p(t * d) r-{: i-i' v i op+-1 nqur irr i ii. rridt diBnr.{sao cho 02!:2Rl':- Qiiiu n'i1','hIc +': var ur ) tl. clirr ldy rltrrrc biri vi R /r ( :n /t< 3F. I .r sE chri'ng tii rliEm A th6a m*n ili6u liif rt r:iLr r1B 'Xdi cire s6 clrr c&a circ sd ct. o2, ... ap Iih; .*l'r bir i (xern hinh r-i), ehia'cho p. Vi c{rc s6 n}1' 1il nhirng s6 ngul'0rr n t6 > p ndn cfrc sd du chi c6 th'b ln 1,2, .':,p- i. ,. Do fiO phni cir ! sij ai. {ri (1 < i ( / { p) cdr ', '' ':': ,il: .,, ,-._,.;-1* -,.-. -;__.-*,,-,1.,-.i;:*., ., cia.g sd du l;hi ciria clto p + gi - trl : (i*i) d .' '' .'. -.:.{ I p-+ ai p (vi 1{ i - t< p-r vrri p ld sd " 1.. . nguy'&n t6 bdt lij' nh;l hcrn U *+ d : 2.3.5 7 "+ d:2lt)ii (Jt(:,Vr. ' S{{ rir:ot + Sd < i9S4 '+.{ < 24S. \'a-t'd= 210' Ta c6 al ( iss+ * 8d l lgst - 8.210 : 301 (1) Ndu at : 1 thi &z = 121 i 13, cdn u€u ot ? 13 1 thi d? : 1273 j :il. vAy a? tz lzt g
\G '* Ta er.r'eos lOzQ = QLt2'AO2- i/:1'2 t+ tu(*)tacd cos r = 3,/-1 v:\trl dc tacri tos $=:. - Ji8 -- -l: { 0 non, C > 90o (d,}.c"rrr)' .r sin .10:0 = -/* ''l ll2 \ 2, Nhdn *lI: Ila s$ c:ic ilan. giAi l;ir rri.i dirrr . -ng Lsotoz ta co. d*i. Hi6ns ban Chd Qu,trig Quq?n (tr1 A' P1'1'FI Long Thnirlr, Ddng Nai) r:i; h)i giAi {6t hcn ca'" --- s A(tz + O()Z - AO\ 11 '7"\''r cos .{O2fr1 .BYT 2tl()z.O$z ! t- ,- tu dO cirng c6 :, sin :tO2O1 : V;i 8 Y2. Bii t$i'tr 35'Clrcr t,,dr ,;'.2 -,'- !j2-i :: { - '}7 }tdy . --a-:* r'-\ ,t\ tii giit trl lott nlt?it' utl gi i rr; ,,,1; n,ldt cir', dit Yiy: eos qAzOt: cos{;t020r + ,{Oz0} = 1/8 It .2
, :., a + fidi 3i 138. H. 3' tim nroi da thr'rc f(re) sao cho * Edi g tliti Iri.n rnit lrh;'ing cir n ,iilrrr ,E1 p(o) * o vd P2,..',Po \grriri la ndi trrll sd di'drrr r ui rrlrnu F{cr+i1 = [r(r)]2+r bring rn nidu khie nhau sro rho cdc eliEu kidn , i)ti Khang Od sair d rroc t h6li rrri n : , Bei 4l tr38. {lho duong conggl = *4+rr3+6."2 (t), U6l riibm P c6 sd ctiin -u(P) canh ru{t l-ec*d vri nrdl rludrrig thr'rng A eit dudng cong phit. tai 4 dibm lBn luot ld,ti, IJ, C, O. Chrtng minh ring neu AB = Ct) thi tbrr lni mdt dunng tlrang (Z). fri d'ibrn P turv _i. ta cd thb di lheo cAc c;lnh song song voi [, tidp rirc vtii duong cong tai dt rl6n mdt €liErn 0 triv v iiLie. 2 tl iarrr p''1 ri bi!'t (3). Voi nr6i dibni P Hodng Hoa Trai vi ruii rndr tirr a i1 { s ( m) gei us (/') lir s6 canh xtrrit phat tu P vl Bai 51138. Csi g it rhb rich hinh ch6p ram ginc S.4BC ehung rninh ring n6u tbn tqi mQt dibrn O sao cho OC = 1, O,l= OB =OC= 4 thic6 clugc tO"mlu {Lrl s thi us(P)< ** r(fl. b.{t dftng thri c }'< 9 ril tlodng tloa Trqr Chung liinlr rsug toin bd hg th6ng cliim Pl, EAi 6/138: Oiir sti .&{ ladibrn ninr troug tam ?2,.,., Ps vi cic canh dusc tO ir8i m miu nay- gi*c vudng c$n ABC (vudng 6;8) sao cho ill:{ : lAp thnnh l. trinh rna ta c6 thb r'0 b,]i m6t nct sao eho n6t r'.rro l'a' ndt ra tai I clinh cira hinh M.l]: M(;: I:2: 3, Tirr| gric .1.|y'-B iluoc 1$ $&i 2 nrr-\'u lih6c nhau. Fhuonll 7'hfra I'rt Dinh Lloa +3 SAi 7i 13S. Tinr i:[p hini s5 f(r) vag (o) th6r rnl.n diBu hiQn /(r)- lttr) = {"iyi. g (r-g) voi nroi :r' Y' * ),'guyJn Idn Mdu t0l!3It. ti
3: Ngu&i ta e[ia tAp hSp eAc dibm cria rn{t l _*+ qhing, dinh Irudng ni"q.''chibu t im Aan, i[] - D6 clo blng radiarr a(.L) cria. gdc (O,t, O,Yi vdi rij ( ct(.Y) < :l3r thAnh m6i sd hiru trgn nh*ng rSp hqp con k10og * Br 0, sao ctro tdrr iai irr$t diBm cirr:- C()- kh{c di}nr O, ta clro ,trrong r?ng I ct)trg rrrirr v rri I". /.. ?'hc;i gian litm b,)i: 1r girj S0 ph*t -' ql .-{ IIdi b/ii tlfim 7 diim , .Ngdg th* hai 5 thdng bdg 19Sti I 4. Gi5 si'r -4,BtB Ii rnot itt gilc liji sao cho Churig rniirh cuc L.rt ,!irrg lhirt : duu.ng rh:rng CD tir:p tuj i;n i 6i duong tr.on dudrng kinh AB. ' n-3 { 2a1p-( [n/:] lin+t)i 2].--2 Cbr'rng nrlnh clitu ki0n t.dn vA di"r db duong ([r] chi phhn ngu-v0n cira mi\t sd thr.rc r') 1!&:S.-18. tidp tu;.6n voi -rlrrong trpu dudng 6. Gia sr'r o, &, c,d.ld nhtrng sd ngu-y0n d.nong kinh C.0 id c{c during th&ng BC qn AD son-g song l6 ltr6a nrln cAc clibn liii,n: 5. Trong mirt phring ugucri ta cho rndt da gi{c i) o(D( c(a. 16i n eanh vdi * F 4. Ta gqi i' p li tbng eAc chrFrr ilAi r:iia tdt ci cAc canh cia ii) ad: bc, , .,^ tlc gi[c ; iii) q*d : 2" vit b*c _ :, ^fli A'\.-ii m !i_tuit$ilg rI lA thng c6.c chi6u'dAi cira t{t cfr. cac duong - sti nouvdn. ch6o cira da giAc. Chirng minh rX.ng a -= 1" 1 1'1rii tgian ldm bdi 4. gid ;i0 phut 1,16i bdi ehdm7 dibm & rEr quA cril DoAN flqC SINH YI&r' NA,EI lrldl nldt : Dim Thanh Soa (EltTtt IIA noi:) Gidi Do : - Yd Thu ?fing (pf,"r, Chu ,Irinh : , Quing Nal-r-Di N$ng) : 25 dibnr lrlol n/rl: 'D6 Quang D+i (OttSp t{a n6i l): ':- Nguy6a ?!t{re ,4.nk (tam Sor: . 27 dibm 'fhanh H6a) : 'J3 ditm - Nguy6n Ydn Hung (phan Clru Nguy6n Minh EEd {.Ciru yin An m& g*&H yffi$'guvhx gfrffiffi e&E [*$( w& '{,&g m&ws . : r. ,"1.Ej.NG 6-I9s4_ . pfrrif - khing kb thdt gtttn dgte ta chip db tkis i "(l&^0 .., . C6u 1: tiho phrnng tririh l) tlinh khi nt,= 114 Giai ;-r}iuong '{,:os6r * sin6r)/(r'ost* - sinzr);= rntgSn, m lA. 2) \'6i gi* tri nio cha .*r, phrtong tr.ifh c6 tham s6. nghiQm i, l.) r.)
j-L..*::-:ffi-: F_: I CduItr: Cho h6 tlrrerng trinh 2) Gie, sir b = c, Tinh thb tich cfia hinh t r'i \* s"r ,n, 1iI tlranr s'j. di+n lheo a, b, d '3 ,- YS = ;-11I-- :r -l- g = -l 3) Gi& sr"r D: ,: a'liiZ . inh goc e d? hinh 1) Giai hQ lihi rn=3. cir' (r) drrOng !iiirli I..f t;Fir rirr' i'iri tlrrdrig . 2) BiCn Ir :in hi (il r hi n.q CO. : "l'o. L&u lir: ..ac urnn l.ham s6 rn db cho dd thi C*u Y: Cho hai tioan thfrrg c6 rilnh AB, CD citl !titrr su theo thr'r trt ndrn lr'0n !rai rluirn.g thing giao y=2x3 -:l(m*2;r,2 1- u(m -l- 1r.r'* 3m *'6 . nharr dr r1'. 'Tirn qui' t!clr nh ['ng dii'rr M trong rnat phing (z,l sao cho lirrrg rliin iich ('5i' lam cdl titu, 1].r tri lrr diiirrr Llrri, nlrair. giic MA8,,LfCD Iihong dhi.. Ciu lY : Cho rnit irinh lrr diQn ;11?CD, trong Ghi chri : i dti BC:.r, .{S:.if =}, DE: DC:e, 6" lir goc phftng cria nirj di,,an cri cinh Ii Bf, {o ( cfl:). l) Ttli sinh tiirdi B ril cco tling iihOng phii t) Chirng'minh ri.ug AII I BC. Ylri diOu liiQn lirrr pirX.n.3) CIu tV ri c{u \'. nio d6i vtii b, :, c'lrrrrng thri ng rrdi di*rir giira I crta IIC v6l ilibm gitft ,/ cira AD li duons vudng 1) Ciring fa ndn hiEu &i md! chu g6c chung clia Sf) vi .{11. C,hitn.g rnin}r ri.ng khi . 2) Ban doe lhim r{3 n 1oo boi d ud d'D tchrt d6 iiinh clu (l t t'lirong kinh CJ] t1i rlua I vi J' lhirh cfia Tda soar.). 'frd : - Thue thl.1,ein rhrlrng rninh bing phuong ph{p phin chri'ng' a. ffifd,r fs.'$'fwedm &ffq 'Iiihy: - T6t l6n, em trith iiiv .li I 'Tr CAu c6 ]im duac khdng ? 53: phgi dni tlri ?0m chu khdng phli in 50m' K irActr nstrc lilli6n A : Euge ' a"' n"' Iibt'rrg"' i. : -*.vr NIIU ",VA\l - 6, aiA* dr-r cri gi iri quan trong diu mi ngii phii bin tlnr. Edn luqt nhi To[n hgc uggc, nhi0n : - 0og ndi sao ? Iih0rg quan trQng u ? N6u giXC, NHAY Sng mdi iii tr'On cilu dunc 50rn mir t1fl. rE sang tr{i thi 8ug sd rcri np;av xudng sdng..- trIQt t{m blng cd cliit tinh kr{' ia nhu sau I ndn ' ban vi6t di;' sd nguy0n ei; flz..,' 4r,... thi khi vi6t FEflAN AffiT}NG d€n s6 hq.ng ao, sd h4ng ititue trrrr;c no 3 buri'e tuc lir sd h4ng rio-e so tu ddhg't{ng th6m 1 doii tnat : Enr hIy chirng minh rlng sd vi. C6 circh nio 'r'i6t nbj d:r-v sd 1, 2,,'_.! 13 tr6n -10 -1- tdm Ling rrlrfr"v il6 lih0rrg 24 I ti sd nqtr-\'i rt Id' ? 1A E Ia ^
{", T'huc ch{t cfla nh&ng bii loai -cini1.- lA cic bq.1r Ddn dS1,i6t rtrrciug diri r:trcl.rr,i tlieo 1 ptrtroirg tim cdc cih tri crja ddi sci rlt tdt chc bibu thulc phip. Phuone phdp l:'l'im gih tri t.iia g rlB phuong J'rong trrrr)'ng trrrp hinr sri cho L6i clir: bjBu t}ric trilrlr drii vo i .r cd rrglri,"lrr : , dai sd thi cdn rn6t vdn dB nira mi. crLc tran cin 3 r.OS2.r .i- (.os.tr- 2-g = 0 quan tAnr d6 li rniEn ui6 tri cira'hirm uO. lti!., EAt l''.Ecrsr thi din diin l:ii to/rn tim giA tr-i cfra gin tr'l cfia hiin sd-*frc ti nhi'ng gi6 lrj c6 thb g db phuong tr.inh bqc 2 d6i v6i iin X cO,rsiribm nhAn iluoc cria lrarn sci y d? co -giA tri t.rla drji th6a mdn : -:l < X < l. 'f e co 1rtru.rrng lrinli: sd c lAp tu'ong ring voirio. 'lhi d; ham s6 q:x2 thi rrriFn gil ir.i h U2 0: vi r.di rrroi rid tri Jrz+r-(2+u)=o'' cria r'^derr lrrrrng ring r ui gif, lri cfra lrirrr sci . Xdt tarn tirire ):4e 2: f tX) * JXz + I'- (: + g; lt--a' )0 rn uuuoc lai rrii bf,t cu gil lr.i ndo DiBu kiQ.n db /(Xt cd nghiQm tlr6a n.rIn - i,< X < 1 .cria g ) O ae u tbn t4i gia tri cua ,uc, blru *2 g. le l{6t thi du niia ld heni s6 q =" sinr thi miEn = gi6 tri ln -I o ta c6 - 1 (y:sinc( 1 r.a r.6i lirLt cu gii tr.i [$rt-1)>o >o e {{*c}o *+ nAo cria -t =
r' [: t. fl '') * 3.q: -F "Y *'* chinh'la, q)-;, -*25i12 n€u x nhan I'i0n ph&ng eliEnr r:iil truc ltrng rna db tiri h*.rn ,il ,.i i rrroi giA. ti! tu1' ,- t':iosr nlln: sd kh0ng di qua r.iri mt.ri eift tri cia rn"ln;g)l itTi,,lAl Iro:rc A(._l,it. n ' 'Ia x6t di) thi gi,ii htir tr'€n tlonn * t(X ( t.'i'ri Va-v Ilir,tohn nr)1' thuc chit l:6t ta phiri tim :i ii - c16 ta thiy ngav khi * 1.
VE CHLTNG AAINH PTtAru CHUNG |,;o f hpc ud ttfiL tr-, ctrrlng ra dJ trAi vdri rn6i dib.u dt ng (dinh rrghia. titn dB Tntti, oc cui) tol.o.an nr(jI loai chr.rng dinh nf nh dC drlng rlE ,.lr,,rc . chr'rug minlr rnin]r. ;iriruoc d6 lA Joai ^1,{-, 't16iilr,rrrg ,rirfrt. gii,n {.i6p. toqi ehr.r;g :1_ ],lAc6 :"d :'.'.klri, l_i do h.ii diiu nr,J,, .'.rr.r,.., uting l]lil]1, s",:, ti6p,. co phrii r:h,.rng minti phA; rnudn t.l:iihg mirrlr 1rlii, clrr.rng, I'ong.s;ich giho khoa trrrln, ngucri ta tiI elri phi'i rljnlr ki:1 ILrin riii su_l r.rr ditu ,.-ll"l,g;truy. -l phcp chtrng nrinh ph*n chrlng girip ,1t,lJi.t.o rI'al \'6i giu iir;dt. :I:rntl cAc i-ran rrhri,ng minl.r mtil sdr nittulr t1b toin hol \.-Suyi. la thrrdlrg dr)rrg i irinh tlrrl,c,:lrr,,ric rgar {ri klti cic L:arr hoc torin 11 lr.rp , r;rjnir drioi. tii:i l)l)iln CnU]tt.slu rirl-r. : (x) c6e bfin rro.i herc hinh hoc ci&ng iran.. . \Jur,iri ddng rlrroc t. Fhn dinh lu.Qtt ai} rgj srl.;; rc rii#l lrrij rirj . Phdp ehrrrrg r::inli plrin uhrrng, €iA thidt. i'd[;-d: ;ii .Ti ;ii,,'iT,l,lii:'i,,i: {X rnr vi rl r.i i) Ii,.,* d "t}1* S,,. Chr-'i ;.,ritng rrr ( Sr' .tng ,rh,. 2. Ph; dinh luq.n itb rbi srt)r ra dilrr rrrii a&i ..=.r'li."trdi ,'l v{ii {I). Iic ro:i,-r,1,, aey:r,t,hu rnpt dibu d*ttg ndo d6. :,:"o"#hirrg,c. } r.:rr c, tr-ong dr.r s,,.is.r,:l &/ir, d,,, l. CLL'rng nri.nb ritng nt\Li a- ,o .l ffiinir aili,1.hLrr:. r:hdt I:\ l*dn 116 (nigrlr . .Yi thi ta cti i.re + Ad j { i. " L' = {r .j-Ctr - t :l',-iS :_.n:U" ytr"ilng rninh) sau d6.v: ,a N6u eriB rluing tnarig o, &, e ma allr. v;r blIe thi tsllb,. p1fi Giii: rllnh efia na Ii:6i,i 3 du6;; ;e,;;' 1] Gi6 sfr r.6i gii thidt nhu vfi), rrid ailc yd bllc nd c liho*g ,oug;o,,;;i, or. ", b,ctr,,t, lcr +idl}l" 2i.h$u tI{t *=cosg thi, vi oz + 6: ( a tih6rrg sorrg sons ,.*i a', ia phn dinh , =i nAn. i,:d:*^ tirll Irrirr tr aiib *, chrr kh6ng phili b = sinp" ln t.oin ho lr:in dE. if.4rr uqiy. miln pnfit ,h,l dint, N6u ,l5t c= coia lhi r.i c2 I dz i rriii i;nn t,i,a,, d=sinf,. - It1ar.?.nnnr trdn, n,to.t ta gnlp taf cd gia thtdt cia dir u6i phu atnn *At iuan cira na (i.l Ilo d6 lce 4- j= icosrp ,coss * sinrf sinal &d _! = I eos(rp - a) l)' t. t tidn hr\nh .theo B irroc : 3) Ei[Lr trdn Ii \.6 I- r.i _lcos rl ( t. vev OI"": t,: t,hri r!inh lu.rn ili. Btrtlc rrAy ,Juuc sir khdrrl- rh3 tb; rr; i.l r,, ,,0 ^^ {oi_14 brdc giit sti. q1. Sia ]1eu i*+#t .... - 'Rurir 2 : Rrit ra tliiu v6 I,i. grr6r nr\v dir?r: 3. Fnd iflnh !u&a alb rbi r$y rs, hai ggi iri btt** trug nguUen, -ilcu. ilibu ff6i I Bn&c ',: Btr&c l;{t tLtd,n. n Vi du 3. Chfrng ti; I ing phrrtrrig tr.inh : Chtrng rtrinlr 16rrg. nd;rr rrr6{ durirrg .u I, ., ,O', tnA .1. .onS.. v6j mirt drrrng ttring I,ui (rr - oXr-r) + (rr-o)f.r-r)*(r*c)(.rr_ai= 0 i lhx'{ li (rin rn{tt ^:ong bao .gid cirrg ci.r ngliifrn. ,{ m{t plring kh0ng e{rrra n6 thi n6 " !'$.t93S vdvi ctrinh mitt ph&nq dct. {ollg, }((p) .. jlnl r.^r) Gii sir phuong lrinh ir€n v6 nghi{m, i"t'.\ nnat trrdn, ta duoc phrrirng trirrh tuong diri:ng , bI l) Gia sfr v6i gifi thidt n!.ru viv nnA, o khOng vrii phrong triirh di cho I song song vdi (f), cir nghia in i X (p). 3-u2 *l(o * I, *ci r. .i- ta& *cc f DcJ = r;. 2.). Ct.,i (B) 111 rr,at lrhiing (a. b). Nhrr viv % . s€. lI giao tuydn cria (p) i.a (*). N€r.r a x'ipi,D *) u6i i thi. a phii eit giao tuvdrr b t"i fE o i pf .f.ti Odi cdi. bnn.d.d quen it nhila tdt ttgictr v6i giri thidt ollb. s'i{i ihidr ud, K td. tt€t tuln thi hinh tniellaa uan 3) So di r.6 ciiirr rr,:iu tiiuin tren ld tlti tu di. gil sr'r a X.{p). f.a.E oil(F). - {c l{i G *+ 8'. o, dti, phf ,ai* e&,a td Ia irdn d8i: ",i 6f; ; lludn ehrtng *inh ptrin ch&ngr ra phi di*h 0+K=GVK-(iK: -luAn'ilE rbi.lu dri suy ra su vri ly. S",rO frr'nJ *) -Yrm NqulJla thb Ie tlo dibu 'su.l ra irrli v6 j g;a tllAr ,-.0' ,ni Dbc Thuhn trong tt Slg utt ,:!,uttg mtni fiilsFll .v. 1g80. .' "*, /tSn r5
11. Chirng rrialr OI{ | tlac'} tit' fl ld lr{rc tiim 2) NArr ltlrtr"itii lli; l' r''5 nghii'rrr lhi cfii A arr;. - ut ; 1',,; { rl' (i) ',r) | = ,1o-, b'c;l - ll{sb *ift sir ring viri gii thi6t 0A I E{;, AB -li OC lihrrne l;i6r: tfi;i, ix rlrnl: ' vi oti l tlac) mI ll khong Ptrli lir truc tAm . ir. -. ,, z;r t A : ,.rl (r.- hl: , (o- *il l (1,-- c)t j ;- U. t?) cLia A c irC. 3) l,{&u thilIq I {1i vn tii} in phir dilh:.;a.nil-a.l: I gi6, sir hh6l{ {lib cir' co nghia 'iir :rhtrong V,3..' Cie " trinh dL t:hc c6 .tghi$rn , -';:'. I 4. ?hn dinh ,rdt &16r eds lu$n db' rbi s*i ra , .:.i,' t. \ I diEu'trdi a*i tfuide" gi$" , ,Z c"2 la r: t il , t,, *,| *' \i d+ 4. 0l.iLng rniiilr iir:'I ndu u-' &" - al,rtt. 1 L :ra -..:- -..-+ r* rri-:i l cria rrrr'rt cJp '\ ,i ,,,r 'I t bais6 hqnq t'i.n tiif ihcu thrii in d'v \.. -\, -\ i , ;.. . .''I I ;';;:;; i;il+"il rrti:-i-r:); tl(a*tr'I,cnrig liL ba ",* ,i iofi i,r" iidp rl"" il'it t''' d3' ':ia nr*t cip ' -