Toán học và tuổi trẻ Số 123 (1/1982)

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Toán học và tuổi trẻ Số 123 (1/1982) trình bày về số nguyên tố Mec-Xen-Nơ; kỳ thi vô địch về Toán của các trường phổ thông Liên Xô lần thứ 14; giải đại số bằng công cụ hình học. Đặc biệt, với những bài tập và phương pháp giải được đưa ra ở trong tài liệu sẽ giúp các bạn bổ sung thêm kiến thức về Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 123 (1/1982)

iNry H0r ,r0AN tt()c VIfT NAM i:i!::i!:!i "i+rrtur#liflii[Hiir:iliitliqf BAC RA HAt IHANG MQT K'i tthiQnt. NGUY&N C,iXU T0An- Chtli 't'ha kg {de *oqn; E0AnG C}tfiNG I'ru s&: 70 'l'rdn liung Dao * Ha s*di llav u0i i s262$ t \6i ehrryQn viri oAc biln tr6 yeu ro6rr I I 16 MEC.XEN.NC} i: t'I i, [- s6" NGUYEN I PHAN O[t( CUilVt-l ED.io rHr"l'T. sd l,'1981 ccr bai thong h'io vdi kdt qui nho r;h6 : dr)ng phtrcrng phip ( sArrg ) ! lD cic lrrr ir r e s6 ttcttten t6 trrn ntrdt duoc CA.c ban dA bidt ki6u < sdng E-ra"t6-xten >: vidt bidt. d.r I *', 22tz0t - l, do Nikel vi Noll c'irc sd tg nhi€rr tu 2 tr6 di thinh dfl-v, sd ? 1A tim ra n[m 1979. Nhidu .ban 1'6u toin c6 r'i6t -efra t6; ta sing ltot thu nhdt : gqch t1i c6c nguv6n thu h6i t6i vE j'nghia cira k6t rJuii ni1-' l6i 2. sd dhu tien con lai (sd $) lt nguy€n C'.ie ban ili bi6t vai trd quan trcng ofrn ciitt t6; ta sAng luot thri hai: gach di eAc bQi crla s6 nguyda td' Ilrn quan treng cang t6 rO hon 3, s6 ilhu ti6n cdn tai (srl 5) lir nguy0n t6; ta r,oi sn phit tri6n cria If thul'dt s6 v& sg xAm siug luot thri ba: gach dr cac bbi cta 5, s6 ilhu nhip cfra I,v lhuydt s6 vAo c6c ngi.nh to6n hoc ti6n cdn lai (e6 7) lA ngu-r-0n t6, vi cfr th6 m5i.,. khAc. Vi viv tr)' xua i16n n-ay, circ nh{ toAn hoc, Xirr gi6i thi(u vOi cac bAn m6t ki6u silg 1r1' th€ h6 ni1, sang th€ hQ khnc, de b6 nhi€u kh6c; coi nhu bii tAp: cdng sric r1b' nghi€n cd"u r--Ac s6 ngul'tin t6. RAI TOAN. T'roug tldg ltt nhian, ta b6 di: CAeh diy hrvn hai nghirr ninr, nhA todn hoc cb dai O-clit d{ chung nrinh rr'rng c6 vd sd -' IAt cd ctic lfiir th*a c&.a 2 : s6 ngu1.6t t6. CAc Lran dlir hi6t chung minh cfia .,o r t< o1I ,,9,13 . ,n 0-cli!, d6 lir mOt cAch chr?ng minh kiFtr mltr u(u - ---z - 1) v0 st tbn ttri cira mdt d6i tuong toin hoc, tuv tdt. cd cdi s6 cd clqtg r.u , trong hh6ng cho ph6p chi ra cu thB d6i tuong {y. V{n dE rliu titn dugc tIXt ra iA tirn clch do u, u ld htri sd lV nhi6n thda ntttn iti|u l:i€n : < toin gon lt 1{t'cfl r:6c s6 nguy{p t6. 0 dAy c6 ,(r.
1r \ Chl"n..t! minh rdnq sou Alj trtoi lri:r (nc r ho n vi6c phiit Ir i0 n lic si Fec-rna ngriydn Crar: nhi torlin boc thdi trudg rti titng co f td: cho d€n nav, ngud'i la iih ng bi6t cdc s6 kv Fec-tna r;gu-r.6rr t6 nio. ngoii 5 -.ii
II0NII ub, 2" .r'r,, if p iri sd ttgugin t6, thi p Baker-Grunberger (b&ng i6n nhdt hi{u nay phfit t* sd rtgltt:yirt tt\. khoirg th6 i n thAnh sdch. e hl cd du
I s6 Mec-xcn*n() m{ khdng iinr ciic s6 dr.rng khAr:'i hoin r:hi nh lii. thi it plrii ,'t, v6 sd sd horln D6 ki i'i ba rrguv6n nhilrr : ,'hinh li'l 1) DE chirng t6 rang sd N l* nguyBn t6. vii xi rr rroi th6rn vdi cric bq.n vd tlnh Bi-r- gio-. rnit nguydn t6e, ngriri ta plrfri clting c6ch sirrg (don giln> cira dli sd l{er-xen-nr-r. YA m4t 't-ra-t0-xlen, aoi cfrch khAc, ptri i k ibrn rghiem 1f thuy6t thuiii: ttil;. drrorrg rrhien con c6 rrhi€u rirng N khOng co u6c nguydn td p f/ H xt O"g dIv -s6 ilon gif,rr iro'ir, Chri ng ha rr diir' nr,ri cl6n sn t6n hi:rn, nhtr;rg viQc IA,ni-{ ni-r- khOng n2 -1 1 (6:: l), 1, ?,...) thB thqc hi€n tlLrgc khi .\ c6 hon hai chuc ch* s6 : c\o cldrr na1-, ngttoi ta chua bi6t dtroc tdl i)il nly chria vo so nirirng hrrp s0. ch,rng cd c:lc s6 ngtiyirii td c,r tri rnrrdi chu s6 trri' han cAc sd hnng rlrng vtri l iAn cirng birirg 1.2, d xu6r'rg. r ii,5,7,8, 9'(trri trLtdng liup n = i, 2), huic T =2mz Cudi tlrd kj; 18, rrhe lti6 ri lro,: l.cibrriz ili thu rrg vdi rrr ) l. biir r i kfri tld n" -F 1: {rrr" * I : rninlr : Ln,2 I i)'- lnr?= (irirs - 2tn + l):*.t2m2 - 2m +l). +; 'r!- Nhirrrg dir.t :1-v citrg chtia Irhi0rr sd rlguv€rr 16, ,/ DINrt t.{ LEiBNIZ : #! s0 tv rthitrtp> 2 l{r chine hau: nguyen tb, ttibu ?.'i{n cI.rr ud dil Id sd (p - 2)l *' I chirt hil t'ho p 12+1:2,'12 l=i,42 +l:17.t,2+l=37, luz1l=t01.t42+1=1u7, iri?r I=r)57,')(P+7 Dinh lf Leibniz khdng pliii Id mdt c6ng i:u c10", ild Ap dung d.ilrh li't.ei- rrbi I Ngndi t.r cfing r:hra bi6t ur6,t diEn kiQn bniz, la phii .tinh (lg * 2) l, c(r tii} coi r"ing eln r.A dri, cd Itil:u lrrr', chr, bidt ;r' + I lA mQt A'- 2: n l lu" vi [;, t., s6 nguvdn t6. tlq,c biQt eho, ph6p phat hi€rr nhiing si rrguli rr t6 Lru dang n' -l I. / \lu ->ru 7.108 ",= (t)", l;/ ro8 Con e i; m6t ioai dir,v s6 riit don gifin, rni cae ban rlBu bidt: cAc c{p si cQng . n6i c6ch khne ; (tr - 2)l cO hon 7.1(t8 ch-ir sd, co a -t nd. (n : 0. I. 2....). lE co n liu lt'irrr r6c m61; tinh tli$n tu m6i c6 khi nlng l[rn vi€c vtii cic s6 niy ! C[c dly s6 tr1 nhi€n, rtl-r s6 chirl, cliy s6 ii',. ., Nhu vA.r- chi c6 viQc ki6nr nghidm c6c sd d6u ln cf,p s6 cQng. C6 thB chung rninh d6 dri,rrg Mec-xen-no, nhd dinh I,v Lucas-Lehmer, rndi ririrg bdt ctl cdp sti cdng n{o (v6i a vA d }i thuc hi6n ilur:c ther: khi n:rng cfia mAv tinh uhir'ng s6 tr,r nhi6n) i16u chirrt v0 s6 nhfr'ng hqp s6. Nguoi h khdng bi€t rn6t diEu ki$rr ciin vd di{n tir, may cAng ( tn+trh > thi cAng co rliEu kiBn tim ra s6 ngur-,0n id Mee-xcn-no ltrn hrrrr dir, cd iri6u itrc, cho phdp xiic dinh rnQt s6 hatrg a* nd, clu cria mOt ctip s6 cqng cq th6, le s6 2) 56 l{ec-xen-rru c6 dang khdr clon gi&n : nir nguy6n 16. Thd nhutrg nhl-r 1837, nhir to{n hr-rc Ii nrQt liiy thira cira 2 trri di 1' Eidu rrAv r'{t Di-rieh-trd ilir chirng mitih l thuAn tiQn cho cAc ntiy tinh didn t&: trhtlirg linr vigc vdi hd d6nr theo co sri 2 (\'i rnach di6n tri I)lNIi i-lrt Di-HIO-Lll. NdIl a ud cl la hai sd /u hai lrang thiri: hr>[c. ttdlg, lroiic mir.;. nhiin nguytln tii cing nlruu, t]1i \ cdP sd cQng 3)'l'rong l-r: thtrydt s6, s6 tr'iec-ren-ncr cd Iidu a'l- ttrl (n:0. 1,2,..-) ehna o6 sdsd ngtgAn t6. quan vdi c6,c s6 hodn chinlt. Mdt s6 tlt nhi0r: I' Ii€t quii nd,v ilunc ituoc goi Id hoAn chinh n6u n6 bing tbng t{t crir ruc ro cfrl 1i,' thuv6l sd. Ngudi ta cang th6m eftc u6c tu nlri€n nh6 hnn P cria sd P' C.Ac srj thia voi kdt qui 5y hon, vi bi1' gid iiI gin a*\ hoirr chinh dl['dugc guan tAm ngal' tu thr-ri h6t thii k(, 20, ngoei cric c5p s6 c0ng n6i tri'n, ;flf, 0-c;lit i' theo ng6n ngir hi0n dqri. 0-rrlit il1, chi' trong c6c ctiy sd don gi:in thuong e-[p, nguoi !4d ra ring n6u i"Ip 1e mf;i s6 NJec-ren-ntr, thi 2P-1 \lp ta chua cirri'ng rninh n6i c6 r'r'rdt diy sd niro il 6 ld m6t s6 hodn chinh (ch:rn). 'Ihd h-i 18,. 0-lr: chita rd s6 sdi nguv6n tri I t'tI cliitng rninh chi,it chO: t{.t ci cic sd hcrdn c}rintr chin ld trlt ci chc sd trOn.'IhAnh thri'bi6t dutvc Trry kdt quii ph6t bi6ri rlorr giir n, nhung dB bao nhi6u s6 trguyAn td Mec-:,:en-l
. (& = 0, 1, 2,..1) h:ry 6ft *; (li = 0, l. 2,...) chira kii:n thtrc ioiru hgc & ba hay- bdn n{tn cllu 6 vO sd stj nguvdn 1fl. vi vfry Irirm 19.18, nhir loan clai, hoc. hoc Selberg chung nri nh lai din h l-'i" I)i-rich-16 l'orn lai, cric brrrr tlris lridrr girr chi ddi .vdi blrrg phuorry phfLp srr cflp: chch chrrng minlr [v c6c sd Mec-xett-tto. ta m
i1-[ duEc dl6ng g6i trong cAc eOngteno (thing {j. Ki hi6u S (n) la t&ng cAe chir'sd cira s6 ttt tlung), Sil cAe c0nglencr kti0ng it iron 35, khdi rrhi6n n. lur.rng cfia c{e hdrng v$n chuyBn Ia 18 tdn. C6 7 'c) CO ha-r trhfrng s6 tu nlri*n ,! th6a rnfin n ] con l[u vAn chul'8n riane t$n Ti6n bQ, m6i con s (n) : 1980 ? thu cd thb dua l6n qui tlqo 3 t{rr htrng. Bi6t b) Chung minh r[ng mQt trong irai s6 lrr n]ti6n ring rn6i t&u vin e huj6n, c6 thE tldng th&i dua 1i6n ti6p thi6t lAp drrdi dang n t S(n) r:hr: tn0[ l*n qui ilao mtrt trohg 35 edngteno bdt kli. s6 ttt nhi6n thu ba n nAo dd-i'. (7 rlibm) Chtng minh rdng r:t)ng mgt hie cac hdng v0n chuybn tt6u dur;c tlra l6n qui dlao. (9 tti6m) 7. Trdn rndt td gidy c6 k6 vir lran cdrc 0 vudng 4, C6,e dihrn 3{ vd F li trung cliEm eria cAc vA iluo. c tO bing cdc rnhu ito ho[r: xrnh, lh6a nriin: ctl m6i hinh ch& nhXt rich thuoc 2 X 3 t- cqrrlr BC vb. CD cira tu'gi6c i6i AB{:D. Bi6t r{ng AM + AP: a. Chunq rninir rring diQn tich cfia thi c6 ddng hai 6 rndrr d6. H6i ring mQt hinh ttt giae rrh6 tron;rzr3. (9 di6m) chir nhAt. cri kich thudc 9 X' 11 c6 bao nhiOti 0 i1 -L' mAu t16 ? (7 di6m) Lep 9 8. Nirtcg ngudi ltn & thanh phd Hoa b6ng I, C6 hay khdrg c{c sd n(uv6n td r, g, z th6a nhi€n bi b6nir ctln. Trong mQt ng'iy neo dd co mhrr plruong, llinh. rnOt vAi nguoi lun bi cirnr. nrhirng nguni .khdi + ry3 = 't'2 (6 di6m) khi tdi thim ban 6m sE bi bdnh ngay ngal- hdm '4 2. Che rIi$rn D trdn dlrrcrng hinh ",lC' cira mQt sau. Bi6t ring rn6i nELroi lilr bi crim chtng mQt ngAy vi sau khi bi benh sE e 6 rn6t'hotrc rn6t s6 tltrnng trdn. il[1,k6 qr:;r,L'dir1'currg Ill) sao cho ngir- nao cl6 {luoc rnien dieh. }Inc dr.r trong thdi rli6n l-ich tti giric AB(:D 1* lCrn nh{t. (E rli6m) gian c6 dich erim, t{t c& nhirrig ngudi hin dEu 3. il&i s6 li cht, !6p 8 (8 rTiEru) di the,r, c[c ban 6rn cria mlnh. Khi biit dlu c6 4" I'r8n urdi r:{i hO tror ngrroi ta chon mOt dich, t{t cl circ ngudi lun ddu khdng rlrrgc ti6m sir (lia rIi6m ndo ddlr. {iiril nirfrng ,lia tlihni n&5r eh&ng. Chung minir ring drroc n6i vdi nhau brii irhfr'ng chuy€i,i tdu thriy. a) NEu tru6c dich crim mQt !!:l;" mi nhtng Bi6t r{ng hai die rii6m :l vA B clLrcvc n6i vdi ngurli lt\n nAo i16 ti€m chfrrrg r;i cb miEln dich nhau b&rrg ci;u_r.dn t6u khi vi chi khi tri phia trong ngiy iI&u titn. lhl dlch critr c6 ',hE keo ben phii theo bd hb hai dia diErn -{ vi B khdng dii bao IArr tirl -r'. drr?c ndi vdi nh;ru }:ing rilrfrng chrly6n thu ti€p ttrei:.. Chung rnir:ir rling l.ri in6t dia di6m b{t b) N6u klrOng rgurri lrln rtdo c6 nri6n dich kj't6i rn$t dia di6m bdt !
r'- Lcrp tO ddu phh_v cria nd 1a ilrrcyc sd th5.p plrAn vd han 5. Gt.ri ,!1 Ii tAp Jrrrpr cac sd ngu-v€n. :i6' :. lA mrli. ki hi6u ltl- ;1r1. K hi lroAn vi cic chil sd kE s6 nli6 nhdl vA si l0C lrl s6 iri:t irhiit cria ,U. fr,tai 1ri sd thrl 2 ndm trong 6 ch& s6. sau d{u phiy mOt s6 trong,"f/. tru s6 l, tlB.rr b&ng tiSlg ha.i s6 cfia,rL tr drro-c s6 .r:2. Noi churg, khi lo6n vi eAc cht s6 h6. tt sd thr1. k * I ninr trong k + 5 trong M ihai sd co thb nhu nhau),'l'rong chc tAp hop ilf rihrr ve'\,, hi)- chi r:r t:{p hop c6 s6 chri s6 sau ddu phhv cria s6 ek, ts iluoc s6 rk+t.. phhn tri l.i lt;rhdt. . {ti rliErn) a) Chrlrng minh r[ng cAc si-r rg nhArr tluoc i-r '/ ' 6. Chri'ns rninh rrinq tb'h tai v6 hair c6c sS o 1r'dn tao thanh d5;. sO lrr.l Iu$rr cir gi6i tran, I{i hi0Lr gi6i han nir. ld g. mA pirur-rrrg t:.inh v r3,?,,r3,2t li) fli-v chi ra thuc ch{to co thb ldn dr.ing hoAe l't l-rly'l=a klrt)ng r,&r riung i:ic bLroc lirn li< 6 tren, tIB tirn ] cd' i't nhdt 1950 nghigm tu nhi6n "r. 17. (6 chbm) mQt s6 rO t? * I'i'm6t s6 hii:u ii r n'io r16y. f 7.'l'rong tu dien ABCD e:6 AC LBC va rA_L A/1. Chirng minh ring e 6sin cria goe hgp hui hr i c) Ii6;,- tim nrdt phAn s6 ir vi nOil cdc bu
DB j, raug 44s : I (mocl lg81), ,Io 4,{3 I= Ldi gidi. Do d,8. C d*ong vn /(r) cd gi{ tri &4 ' x(14" 1) + = 48.198t. Vd.y co th6- vi6r -l' 4-l l) nguv6n khi r nguy6n, n€n f(t), f(2), f(3) phfii d = rles;A + t)44,-t- (teslr ;riE+2lrl li nhftng s6 nguydn drrong. 'Ia co = 198lq-i-412++*4-t /(1)=A+8+.' = 7981q + t98l = 1981 (q + 1) :32A + sB + 2C, l-Q) (& tIOy 1t vi I lA hai s6 nguyOri rhlch hgp ,rd /(3) = 243A + 278 +ZC:.: q=k*l). Ctl th6 coi ddy li hA phuong trinh dcji vdi cAc v{Y ,: tgtt. hn A, B, C.'Iri hai phtroug trinh d[u, suy ra "*: Nhd.n rit: \rdi e Aeh gi8.i co b&n nhu tr0n, d6 30A+68 =ftu_21(D, 1* bing gi&i dubc hAi roAn tbng qu6r: sau d6.ti6p tue giAi I.r!, ta ctunc ,( Chung miuh r{ng rng' + m' * I chia.h€t cho '/L m" * m + 1, rdi m vA n lA cAc s6 nguydn duong n - il/r?u)l fa - vQ) + 5t(t)l Q) vA n khdng chia hdt cho 3 )). R= (tl,.ill - rtel + Er(2) - l3/( i )l (3) (lic ban T'rhn Tidn Dqt (8A Chu vdll An, Hi * NQi) vA Ngugi.n tlodnh Cti)'ng (10B Tru,ng Yucrng, c = (1/30)l/(3) s/(2) + 45/(r)1. A) Qui Nholl) da girii bili lo6n tbna guAt, nhung \rl /(1) nguy6n duong, ndn gi&i bi rig drong tdi Iiirac. l(1) > 1 (*l P.Q.(;. 'lLi (1), do .{ r,d .B duong, /tZ) va /(1) nguy6n duorrg, suy ra f@ -21
Nhdn rit: D0y li mgt bii toan li6t hrrp s jfi2Bdi 6/120. Cho da thtc h( s6 thryc f(d = ( Sd hoc > .r'i < L)r.ri sri >. Sai Iiirn cita nhi6u + c.r * d L,d mlt sd a->g. Bid,l riiug uoi ar? 1- b:r'? ban lA: 1) Chu:r t.li,im rO di6u lii6n cria bAi toarr . (c6 irl-( i c6 bdt d&nq thirc l/f..tlq d, tint uia tri tri n nhdt cfin lal, lo r, i, j, ftii. rht'ng bair. sau hdi IAp luA rr phuc tap, di ddn , h6t tu{n rl 1 Ii : (. : 0 l:l). Ldi gfii. Thco giri thidt t,r cir Nnieu b4n c9 i ha-v trong viOc chring n:inh -.2) Iri0n chn (tliln t,(.!-l) ( a. /{_ili2) < d. (1) tiiiiu hinh ; ban 7-rhrt Itdn Tin ilou llA Thudn An.'lhanh II6a), nhu.rg thi6u quan- rrita * tAnr tI6n di6u kidn drl. .{ = l,- l) : _ o a b - c 1 d 3) Il:rn NllugFn ltan l{inh(t Ii I phan ChAu Trinh, B: [( - ll2t = -,ri$ r hlt - r 2* rt L)i Nrl.nq) di tlua rrr ICri giii ho.p li. uh{t trong C = f(|iD = n/8 t bl4 * cl2 * cl f c{e brii giii dii,i rldn tda coan.'Ba,l Trkn Tudit H.i€p (11OT DHSPt I{A Nai) .te gii,i uror io.i gii.i D-l'(l).-,trh+c+,r1. 1ft.ctlilt ch€, nhung qua ciiu h j, (sfr du,,g. it6n , 'l'i bii:r hd thri.c tr.drr, rrit r.a dinh l-i Fccrna nh6). a:(-2'3) t + (ll3ttJ+( {/3)C + (2,8\t) Bni 5/12O. a) Gid thir lol( r ti .= 1, 2...., n). c: (l'6),1 + ( -s/(i)B + (8i6)/.' r ( -tlilD. Chftng minh rinq Kit hop voi (1) tu 6rrsc (1+-or) rl * az)... ( -i on) * i,l .-( r:';ri.rl +( I 3)lBl +(Jil)lr-.1 +(2i3)iDl),0. du dri tiirru 'fa rrir I ra chch ldr- Ai == 1 * oi, lii - ai (i = t. 1..... l), : b:(112)E+(t/zlu - F, rt:F. thi tu (l) suv, ra b{t dirrg=:,1 thric cAr chrlrrg mirr}r. Kdt hop vrii (5) ra duo-c b) C6 {hb coi ring in( n, tirc ld n:nt-l l: v6r' -lr ngr-rr,0l ) O. Oet Iol ( t t;2llEi + t ltDlttl ,, 'Fl x{ 2a rc) ..li : (1 +ei) (1 - yr) -- 1*r1-*yi_e1yi, lLtl=,t;li1z' (7) Iel :Zaxz.* a, vdi b:Zo , d=-q. tla thf'c/(.r) dj = I +.r:1. lJi= 1-.r'j, (j = rn+ 1,..,,nr *.lr n). Ta thdl' trong khoi'ng l.-1, 1ll_ux :/(=1) = a, = \/{y rrong liho{ng l.l( t tli *a thi tri (l) su.r.rir {'lt"; q/(o): (l * rr) ... il .L .r.a) il ut) .,. (l y. ) [/(rt1 cr. iict hop (6). (7) se cttrsc - i- + (l - .r1)... (1 - r.,) (t + !fi) ... (1 + u,,) iD].u, = 2o, ]dl,rn*: *. < 2-.: (1 - .rrqr) (t - t2s)... f r -- ...f*l n- hdn tdt: IIiu h6t r:Ac ban gri.i bAi iim ra q dap s6 dring. song 11' luAn ihid,r. ctr{f. = 2n = 2max(n!m)' ,ur-1-lr "o,., chC. .\'ltdn .tet: Bai toAn n:iv chi cri v6 phtrc tap N.I(.M. vd hinh thiLc. C6 nhirng han qu6 .er'f.l;, pnai vin drlng rldn bf,t ding- thric Cosi, Buuhiaiidpxki. Bai 7/190. MOt rlabnq thttng di qtta trqrc tdm.II tam gidc ABe, c&t cit: dridnq th&ng Phan Dt?c Chinh .cfia (,.1.B), (.4C) tai p, e, Chttng ntinh rcing nd'u il la
trung cli?nt cda d.ourt thiing PQ. thi PQ L *tU, trong do lL lit lrui'tg di&,';r r:clrl: ijL'. Lbi gidi the bal g11'i i.ini du thi dI giiri theo irhrBu cdch l.:hAc nhari. C6cir gi8,i sau clal' cuzr ban 1-riLn 7'udn lli€p {1i CT trnr}ng EIISPt. Hd N6i) la mdt cich giAi gon vA dnngf htnh vc: dor giirr, Ta thay vi€c cir6"ug rninh f'Q.l;ltA t,if ta trung di0m cira lJCi bing i iOr clrirng rninh: ndu di.rdug v'-r6ng g6e v6i PQ kA tri'l-I eit 8C tai .l/. llri l,t tA tlung tlrtnr cua Dr.. ,s 'lam giSc .,lF/P dbng dalg vci tam giac C'if ff (ede g,6c tuong r'rng la catr got: co cAc canh ld tuong irng r.rr0rg gric nFn bHrg nhau), 1"iv ta c6 ,\talilti = afilPq (D NhAn cac d&ng thuc (1), (2), (3) r'6 v6i r.€!, ra d rroc r ABt.BCr.C AtlAtI).RtC.CtA : r ha.r, AIll.BeYaAy': diB.Btrl.rlr,{. (") cia BBr vbi CCt, ta ehirng riri nlr,431 rli qua f. Ndu trii lai thi ,41 cit ciuolg trorr ngoqi tidp tam giac .1BC tai .42 khl,: ,-tt. l'heo ph)n 1), ta c6 ABt'B{.t.CAZ = AzR'BtC'(-'tA' ('ei*) Ti ('s) va (**) su). ra CAtIC ,tz .: AtR lAzB Tanr giac zlllQ ri,r,;g tla:rg vfi tarrr i{c i.il}'/}I, -e Nhu vAv Lhi hai tam gi{c ArBC vd ,{2.8C dbng r.'q1,- ttr cir dang vdi nhau, suy la AA :'41, di d€a m6u )tBli[:tt -,.ti)iQti (2) thuSn' VS.v .,1dr di qua 1. lrLi (t) vn (2), r'i P5,-=Q11, lin .i/C--.IJB, tric Nhf.n xit: NhiErr ban khAc cr)ne giii tuong li .14 li lmng ililm carrh Bt.. tu nhrr cdch gi&i .tr0n,1 song phin 2) e ri c1',6, UnU.. Eal A7lU0. Tr€n t:ac carrgr ,5{.', CA, AI} ki$ng Bei 9/120. Trong'rnQt ph&na x cio mQt 96c cltti'ci cric dinh A, D, C c&a ihtbng iron rt.gor.ri rAll quag l'ung quanh mQt rli?n A cd dinh. O ld tit.p torit gtidc Afi{, ta. ldil ttvng an$ cac dibnt m(t dftm 0 ngodt rl sao eho OA kh6ng ou6nq At, Bt, Ct, Chu'lg minh rittg rli'iu ki€n rhn pri dli gdc utt rC. Cdc dltrdnq th&nq ki tir O ou6ng goc ct} Atlt, R Bt, CCt dbng qug ld ua'i ed.c mQt ph&ng @Ar), QAg) cdf tt tqi M.N. 4l31''9C1'f;''t1 : At'B' BtC'C t A Ch{rny1 minh rdnq .l{iV Id mdt dtroaE th&ng cd Liti gidi (cria Plhari qtrde Th&ng ' ilCT ThAi dinh. (Xem hinh 6 trang sau) . Ldt gi&i. RO rdng mAt'ph6ng (fi.llN) vudng Lai; Ii&i Phdng).-' goc vrli Od. Nhu v{y M vi r\ lu6a ndrn tr'6n 1) l?iau .L'ffa chn. D5 thfly tam gii.c ,1181 dbng nrdt ph&rg chi'ra O vu6ng goc vdi Orl vA mit * d4ng voi tirrn gi6c -BJat, tu riri c6 phing 5t. Vqi' .11 r,A N rr&m trdn giao tuy6n ABrlArB*teltA. (1) r:ria hai mht phing cd dinh ilo. Do dd ,U.\ lA Tuong trt, tam qiAc S11-'t ttdrrq fl56g vdi lant rnt"rtilu'rlng thf ng cd dinh. giAc C/r,1. tr c6 NhQn xdt: Ddi vdi cAc Lran cl h tlrroc lros hinS tlC yt' fliC == I B ll(; . (2) hgc kh6ng gian thi bAi n{v ld nc6t bni rdt d6. Tarn gifro til.{1 dbag dqng v
n . auo (+.x ..:..j ., % (ntO+OA)'!*ntRz € ,'t i:] - tI vol -c6 Il .-*D_ n,ut02 tf rr r: + 2ildl ifi I=l I=l J= il €+ n,t!()z + M0 z r]ii=o (qi O.{? r -*1 ill/ n6l '*€ Z oAr- nctG (2) \t: -j'--- i .l r i=1 |i i; \ -:\\ Klri do ta co o .q {1) +-i 0 -- n.rto' + ]l (t 1- I O.lr ^= : n(l|o? + MO.)G) i'*,_- Bei ! Ci 120. 't r€n mQt mQt chu cho n di?m At At....ln (n)> tl. r im qull tich ctia rirPnr 11 \-i 0 cij rlinh,..li c6 rIinh. nOl ti ctng c6 dinh. sao cho Do cac ph6p bi6n clbi tldn lh tuong (lu.ung, It0'rr c6 thij k6t luAir rilq tiu! tich ce.l cIi&nr M ll ^dudng x rJJtrlrai) = ,, th6a nrirn (i) li mat cirr co liinh OG. il) l:I Nhdn :t(t: - Di&m rl thira min (2) crhrrrh Id lrang rto ,\i tti ,qiuo ili?m c:in ilf Ai :,iri mdl cdu trong tflin cria h6 di6in .4 1.1 rl2. ...Aa, tri.c lA dilnr (i = t, 2,...,n). tlr6r rrrirn lr0 thric Lbt gitii . (ioi f) ld tam r i B li ban hirrl-r cira mit cAu. \'i di, .ai (i = I, 2,....ri) thudt: mirt vri ru,.,i .li,li dbng ,1u i tai J[, r:0rr r-:irr It'-ecai *o i=1 uMz - Hj *,W.rr.lf 4 - LlAc ban 'lrdn T'udn tti(p tlt C'I. DIISpt l{n IJo d6 (1) trrong drrong vr>.i N0i)".rai Qaang Dai r.a pltam 'l,hunh phuongl (11) C'f DflSl 1 FI:i ,n N0i) ririrg 1ir.n ra qu! tich, X ,ll .ril = n tli2 - u.ttzt song c/rch gi&i'quA cirng k0nh. . l:l N K.N.I QANI',I SACH cAc 14ni c&r lrvr orAr cAc rAr roAN RA TRoNG s6 sAo seo tk' nAO PHLrC R,irr fHnng V",rcr,,g, \iTnh phrj); (Ngu1-6n 'LiAi, T.p. Hb Chi Minh): Ngd Dinrr I A-f il6 l'ie:rr [)rt, pham Xr1i,n ilrii {Chu lrdn Long (1.€^Fliing phong, T.p. Hb Chi IUinh); E{ng An,-Ha \di): Nguien ChjSoii. plrarrr Tnrong Son (chu.y0n toArr Eil.ffl HA f{$i); 'l-lring errde (Thei phia,r. Hai phong): I,harn Khuong F tri{rr ll5n.g C:, i. euirrg *Ninh) ,l.rh.r, Ngul,6rr Qu*ng Dirrg, Ngur.6n .t'i€n Ban, D6 I'in (Di Arr. I'iruin an, fhaih .H6a) ;; .tr rh.n Ntrm :.lrl'.C1, V[n Quang _Eqi, Pham Thanh phuong, .l.rin Tu6n DIng, Nguydn VAn i\fjnh, Nguydn ].hanh Duong, IIi€p.'Ir'5n Thantr'l'r(rc (chuy6n to6n EIISp HA \6i); Ngul-6n t-inh Vir (ilpS Trung iuolg,.) 1 Triin IIi-r.u IIuarr. Ngu.i ir -D,i" ftinn, \grr.r.i,rr ( 6c bqn sau il6.y hi6n kh6ng ptrii \i5n Nlo (phan Chiiu 'Iririh, Da Nirg); .frEn Kt . la hgc sinh Plruc. Dang Virrlr Hrrng reuang Trung.'eui \hon) pho thong ciing gLli iAi giAi il6n tda soarr: ]gu-\ rr IIr.drrlr . urlng ( frrrng \:uorrg, i ^ Dodn(SonQuang Manh (H&i phdng). Nguy6n vi6i. eu-v \hryn ): Cucrng La), ]Jai VEn .\uy6n iftrarin Hira), (PIeik11). \gu_r.,-rr l-rong ,,3nS..1,L,iu..Nghi Tu{n DoAn Th6 Vinh (Binh Tri Thi6n), pham Xuarr Son (An Kh6, Gia l.ai - Kon l.um): DIug Trung I.ui (Thdnh ph6 Hb Chi Minh), 'l'rhn IIii (Cri.u Long). 11
tlri r clung Tim tAp hup tdl cfr cdc gi{ tri c/-r th8 c6 atu'oe lur.v crla biFrr thric sau dAy tto obc th. ' 1+b--a 1*c-b 1*u*c Tq'Iibng Qudng rYidn 1'o6n hgc) tldi 71L23. I{iy tirn eAc tir diQn c6 thE tich Bdi I/[23. Cho tgp trcrp ,4 gdm 992 sd tu Ion nhdt trIm lrorr Lrdn trong nr6t khdi liip phuong nhi0n ph:in 'biQl, rnSi s6 chua.vrrnt qui lc. Tim cho trttdrl. gia tri I6n nh{t c{ra k sao cho trong .{ eo it Phsm Hodng (He \Qi) nhdt nQt sd ld bdi t:ria sd khhc cffng thu6c A. N6ugZ n Nam Son (Quing Ninh), Bet B/123. Trolg khong gian cho b6n tli6m phin biQt,hi-,i b6n cti6m d6 phli thda min di6u kiQn B&i S/123" Tirn tdt c6. cAc nghi€m nsuydn gi dE bdn ilrrdng trdn, rndi eluong trdn d.i clua ba dunng (;u, y, .z) cria hd phrrcrng tlinh trong b6n di6rn tI6, bing nhau'i I *' * ,J' * =2 :7glz Ngay6n Ddng Phdt (lld NOi) I r * g *: -- 68 Bei 9/'123. cho binh chdp .tau S. ABCDEF- Phon Dnc Clrinh (HA NOi) Qua rnQt t1i6m O trdn dudng cao SH eria hlnJr Bni 3/123 (iiii lrd plrucrr,g tlinh ' t:h6p,.ta dung mot mlL ph&ng, cit S.r, SB, SC, i ,' : i..i -., ifl=t=o SD, SE, SF tuong ring tai At, Bt. Ct, Dt, Et, Ft i ,'.= lyr*,r t"r"Slo Chung mirrh cic hd thtc 1) 1/SAr*l/SDr = l/SBr + ilSHi: llscl+l/Sfl. Fhan Dftc Chinh Bei 4/123. -I'irn t:it ei cc c{p s6 tu nhidn 2t llsAr+1lscti1lSE1* l/sat+1isDr + llsFl. (m, .,r) t16 (p -1- l)?tn - p"i ,dat gia tri nhr'r nh{t, . I-A Qudr flcin (DIISP vinh) trong 6ri 7; li sci ngu.r:Qn 16j. Ngd Day Ninft (Quy Nhor:) H9c srNH Tiff, TOI BAi 5i123. (iiAi vd biin luin theo k bdt BAi lOlL'23 Giai phrrnng trinh phuong tlinh t'r-l 'ryor-rXr-, . rlfir.,-itp - lX.r+ l) .. + 1'1, + rf-< t ='vl2cS Phan fitrc Chinh Ngd Dtrc Hodng Bei 6llh3. cii th& a, b, c li ba sd kh6ng (lap ti, truring 1'hanh Da. Am. Ii€n h6 v0i nhau bing bi6u t.h(rr: o*b*c=1. thinh ph6 H6 Chl [{inh) GIAI DAI SO, gANC CONG cU NiNrI HQC ll AU nay trong que trirrh hnc tAp chring ta rlit 'rninh Nhu vAy vE trAi cria bf,t d&ng thuc cAn chtng lL h;rl dirng cong cu dar sd (le giAt hirrh lroe, ohiuh bing hai IAn tbng tli6n tlch cia hai nhrrng vidc giii dai sd biing cdng cr.r hinh tam gi6c AflB vfr AllC, luc lA bing .hai 'lln hoc thi cdn it. IIoc- tdp & truring chriug t6i liit diQn tich tanr giirc ABC. l,I5t kha7. hai l6n di€n thich rlirrrg c6ng cu hinh.hoo dF giii dqi s5. b6'i tich tam gi6c ABC kh6ng l6n hun tich r{B.AC=y'ib vl nhi6u bAi to{n t14i sd ml chi don thu[n di)ng Bdt tling thric dugc chung minh. c6ng cu dai s6 dE gi6i thi r{t kh6, nhung khi dH churbn n6 sang dang hinh hoc thi rdt clon giin. Trong biri b6o niv t6i mudn gi6'i thiQu vcvi cic ban vli bAi todn dai sd vd dring hinlr lroe cle grar. Eei f . Chung minh b{t d&ng thuc lr
Bai 2. Tirn gi6 1ri b6 nh6t crja him,s6 Qui tieh nhirug di6m . rr. ri) th6a p.0u nay g:'/iz--..+ r + 1/.;3+ * + r. tludng thri ng y = -r f B. i dang Ta biirn d6i L{It hhac- Lrr{ t tiarrg thric dA .h,, f o,r,,o1Y}?0 : vdi rllg s:\,1(;Tw;ot27 + /GT il2p; llD? tzo - rt + or/ilr+ ) ac +Dri +. cr1+} '11 'lrong h{ il6 .r ()1, ta l{v c{c ili6nr * tr.r.rc toa 10 2 (ac + bd 1- cd))> (lt * at/ntz. (3) A(--112, !iz's. nrtlz, tlin), ,{/(r, 0), thi ita cd 'fha-r' * ,r2 b2 = | g-BI1 +J.l/. 9 - 2cd:.c2 + d2 vAr_, (B) la cir : o2 + bz + cz + dz - 2at _2bdltl*lz flz t+ 1a* d21(D'- alr lt: *lnziz aj l/ra - c)t + (b .* ct)2 ) rg _ ':t,t/: f \'6 tlai cria b{t tltuc la don,,.r)N *Oi,,t/ durrg thudc rlrro.r:g trr)n, N thuQc dutrng tlri ng. V6 phii l?r iloan d.B. Ta lu0ir c6 f{N > ag, .,,, ra rli6u phiri churig ruinh. Bdi toan eria ta tr& thanh bai toAn hinh hoc: Bni 4. Giai hd phrror:g tlinh Cho tam giAc ABJ/ co cAc dinh ,l va B cd di rrh, cdn dinh M char. tr6n dtong- thing song song 6 r:1 y2 - o2 + 1/-z--) r.6i canh ,.18. Tlm vi tri cira ,U db chu vi tam ) _ /:t-1 giAc SBIV nh6 nhdt. Bii torln niLr. rdt ilon g iin, iy=\,:._b" _1..r_12 chu vi tarn giic -{MB tiat nh6 uhit khi tam gi6c, 1z:v;z-:7-tl,tz_r., MAB ch.n + .., : 0 * 376ia :2. Cho a, D, c l&n hon khdng vn 1la 1_ ]lb ) 1lc, Bai 3. Clto a, b, c, d ltr c6c s6 rhuc sao cho llb + tlc}. 1la, 1lc + tla)'izo. o,.,rg t",n giac t, ABC co AC:g, IIC = z, clutrng cao a +D:1, (1) ha tu. C bring a. Theo pbrrcrng tr.inh (1), ta c6 AIJ *_ e. {: + d :3'. Dung (2) tanr gi{c AtBtCt co ,{tBl : e, CtBt : cao ..lrq. tu,- 8r bing .0. 'theo prlu"rg z, duc>ng Chung minh biit d6ng thrlc i.inl, (2), aclbd+cd( {s+al;)t,t . ta _c6 .41C1 = y. I)nng tam giric A2B2Cz c6 o,!.?: t, = y, ilutrng cao hr.r tL'r ;ti biLng -czAz Ddy 1A bAi toAir .ndu giii bin6i dai s6 thi kh6, c. 'l'heo phuong trinh (2) rhi Bztiz birng :. biri vi vd trri b.it rling thri.c kh0ng d6i xr.rng Clrc tarn giirc ABC, AtB(,t. ,,lzBy,/,s brlng nhau va vd phi:ri Ii s6 phric, !?p.BAy gid trr girli blng ndri chLlng cd cli€n tich br.i np nh,,u - hinh hoc. aq = bll: c. = 2,\ (") (S ltr di€n tir:h m6i tarn gidc). Ilu tarn gi6c n6i l16n ttuoc ghep lrri th*nh tam gi{c ABC c6 cic c+nh Lt !J, : r,a c6c rlrrcrng cao tunng ring vri.i c6c r:alth 116 I.i a, b, c. .fa .tinh cAc cqnlt ciil t:im uilr.theo a, b, c. F l'" Qui tich nhiing rliEm M(a. D) th6a rrtn (l) l* d.udng tr.r)rr tiru 0 bArr kinh l. Theo cdirg thfc 116rdng, ta cd 13
sd M, D6 rj=l{tp- *p*OO:i i) MA p = (r *,g * )12:S(lin + tlt + tlc do (*), nen r ':"1/(* * *+) (+ .: - *) (+. * - ;) (+ *+ -.-t :F l\ * cl f I v(+ 1I ' A 1 r__r-lt* t\/r ./[a .*-*)(:.+;) (*-+-i) : Ta tlm tlrgc: r = 2Si a, y = 2S/.b', z: ?S/c, S 'l'a cin tim gia lri ltrn Irhit vi gil tri nh,lr drroe tirth bang cdng thu'c tr6n. nhit ..fr1 .P =:' :3ri ,vdr (.u-( ; 'I'a :rE dir thi tfta hdrn sri P = u'- 3ri. ^0 tr€n do*rr [o.s]. Bdi 5. Cno p. q lir circ sd thuc sao cho phuong trluh rz * p.u * g = 0 co hai ngiri6;rr thuc c6 tri tuy{t d6i khdng vu'gt qrtA 1. Tim giA tri l6tt nh{t vl gii tr! nh6 nh{t cria bi6u thric p - p2(p2 *zql - 3) + qz(qz - 3) Gqri .rt, r2 lir hai .nghi6rn erla phucrlg trlnh .z + p., * q:0. 'I'her.r grri- thi6t, ta co fcrl-( 1, lrzl( t. vav -'5 ';,1,!lll;. ,,'j f vi phnong trinh c6 h'ai nghiQm thuc, n0n [:>O H0,.i iErr kifrr duoc vi6t lai l? li,r:3, 1 t-p+qlo Ta bi6u di6rr mi6n nghiQm cria h0 tr'6n m[t philng pOq. PhAn rnftt ph&ng bi g:rch kB ci blen li rnidn nghi6n.r cria h$ 'l'a c6 thB vidt Nhin vAo dD thr, ta thdy gii tri lOn r:h{t cria P = (pil * q't1t * 3( pz + q?l. P bnng 10 r'A gil tri rrh6 nh{t cria P bing 2'25' DdLu:pz*q2. DAy ld phrronglrirrh iluong trdn Do khudn khb bni b{o' t6i chi rin trinh bdrl' c6 trirr: ln O vA b6n kinhbidn thi6n tr6n'miErr m{-y bAi toAn nh6 vAy thdi. Mong c6e ban trong ngtridrn cia hQ trOn. Nhiit tr6n iI6 thi ta lh{y qu6 trinh hqc hiy lAm guen v6i phucrng phep umax lthi dutirrg trdn tli gua hai di6ra (-2, 1) nAr', ch5.c ehin ring cAc bnn sE tiru duoc nhiEu di6u tL(r vi vd bb ich. vA (2, 1).+y'J:f/f =+ {:5. va u-in lihi ir=0. BAy gid xin tlng eAc ban hai bii todn l. Cbo a' b, c, d la .ic s6 sao .'ho io-ol *14-rbl =z' c*d-3 - rninh Chung r':rng t€ L1'- ioz - tll"- 2kdt ac * b(t> 1i2. t *i 2. Khio sat s6 nghi€rn cria hQ plruirng trlnh theo a I t,'' -- r',:u ,,t2 + a2 = I | NguYtln PhLrong Mal (l6p ll chul6u tcian, Dli\P Virrh) t4
7 Giirp r.6e ban chuhn bi thi vio e6e trulng Dai hgc vi Cao d8ng \i rHr vAo cac rqtro[rro D+! Hge vA cAo o&ruo pF{iA BA€ '{'hrintl 7-tgBl DD tlti - mOn T'ctan 'I Cdu 1. Cho h&nr sri f,,' - (3sinn - cosa).r -- 1 * : , ,tcos2{r 0 (m*1) Gz-Jr)+2(2in+ r) lihi ilr:rrrr so n lrien lhiiitr. . g= ml'- nt 4. Cho hiirh ch6p S.,{EL,I, trcng r1(r rlir, Cdu tronp tlti m li tham sd. ABCD l,; rn6t hinh chli. nhAt, cri.rh b0nS.,r vLr0ng g6c_ r-6i rn5.t ph&ng da.r,. ,\iat ph&ng qua d vd 1) Iihio sit su' l.lidn thi0n vi Ye Ll0 tlll crtll vu6ng gtic voi canh,SC cdt .B o A', rnt SC i,r hdrn s6 ung r,di rn : 1. C', c*at S, rl .D'. 'l im cac didnr Il.en rlii thi tlir co toa dO hi 1) Chirng rninh rine lrinlr I ri si{c .tB,O,D, c(t nlrirng sd nguv€n. ha i rirji r,li0n vu6ng. g(rc 2) Chring minh rH.ng. r,o.i mcii gie tri l(hec 0 2) Chung minh r&ng ngI S rii chur.brr tr6n - r:11a m, tI6 tni cfia htrnr sd i1:1 ctro lutlrr t,6 rnt)t duong thing r.u6ng gr," r.,'ri urat phang ABCD ttudng tiQm cAn cd clinh, con dur).ng tic:.r,r cal l_ai -{. thi- nrit lihing ,,trj"C'_l}r }uOn Al qua mdt thir hai cria n6 luOn tli qua rrrdt clib-rn ccj riinli. rlrrur:e tlririe co dilh r,n cic dibm ,l, i. g,, C, Cda 2. Ggi a, b, c liin lrrot id c6c cantr tlui C', D, D'cung nlrI1. ti'€n nrdt rilil cii.il cd alinh. diOn vr'ri c5c gilc A, B. (. c.ira Ianr giic Altc.. Clirng nrinh r:'rng ntlu liui tjiEu l
r. h. nd.tt, r; 0) xern cli6m rLI di chuyiin tr6n dudlg r'l d) Chirng t6'rlng rlao hArn /'(.r') lrrirr' ludn t6n lei tt\C) tAm O birn kinh 2. Goi II la hirth t:tri6u tai, tir d6 linr ra bi6u tlrirc cria hiin [(rr)' Nirg g6c cira If ldn Ou' Db thu irai tli n) tinn cric toa tlir cira gilir tliErn P *triitr € circ Cdu '3b. a) Gi&i phrrong tririh Irrnrrg qi{c rlnt-rng tlrirrg OM vI llJ theo g6c a: (OA,OM). ,lcosr -1- 2cos2.r * cosl-r = - [. b) XAc tllnh vi'vE qu.r'ticlr ciirr P lihi ,rlf clra.r' lr0n ((l). (lho bi6t cric cluc tliirrr iirrr 11n$ lic:h b) Chr't:rg'nrinh rirrg phirwrr! lrilth n[]' .4ctlSe: * Zcos)r f cos'lr: - -7 t Cau in. Cho hi.rn sd /.(er) xAc dinh vrlri nroi sd khong t'u nehicrn" i i Ihuc :r, vt]r ihiin ruin cAc fliii'u Iii0n : Ctiu tlt. -['ronq trit phang P, cho hai tlut'rrrg ' (1) / (r) )> r + e' t'r'ri moi r" th&r,g di dl cirt nhau'iai c1i6m A r'?r tao vdi nhau (?) f (r * g) )-t f (r').f (y) r-,;i tntri .r. y. mor- g(:c n(0 ( a={0Ou). 'I'rdtr tlrrting thr'Lug 1.r'' i{',t a) (',lrttrg trtinh rirng vtii rrroi sti ttrrrc r vir vudng 96c vdi mEt pht'rng P tai -{, ld1' tliiln 'B rnoi so ttr trhiirn n, ta cti vdi ;{B : lt: ti:en duerng thing d2 t1i qua .IJ vA solg sorrg vdi di, lriv di6m r. 'tbi BC: 't. Cc.ri f tu) 2 f1t.'12")l?" . ll ld hinh chit:u r-ur)tiq iirc cirr C l6ir rt'i' i: b) 1'r'r (l) hair- chi re rnr-rt khoing lri) chitr () tl6 / (.r') )> 0 tltln lihoirng d6. a) TInh clo rtdi '{D Yi kholng irach tir (i d6n 'l'ir cfrc lit!t tlui cliL tirii tlrroc, hiil'sttr t'a ritng rn!t phing ABC.'1 il 116 su-s la tLl iit'ir tiil tr'r d icn -4,8(.,1). /(.r) >0 r'tri tlloi e;. c) (ihirng rr-rirth rirl( rrii ntoi.t' vi\ rngi h b) X[c dinlr tirrn mdt ciu rli tlira bdu t1i6m rlrrl.-t), Irr t',i A, R, C, D. * lrl t.r') l't thi phtrrrng trir:h lthdrig co righiQnr. Cdu 3e. 'tr'rdir nrit phnng v6i htl trurr toa cl0 l;) Gi&i phrrtrng triirh khi lml = 1/r. tlni'clrnln ().r',{)g, xetri circ c6rlie (Cn) ccr phuirrrq (ltltt -Itt'l'r'orig rnit ph:i r:g /'cho rrrjri tludng Irinir (r,i '- 4)..,2 - {U3 :'4(,r? '- il) trd, rr cl rrdrrg kinh ,1B = 2Il. Gil thii r11 lA rnOt rli6m tirl' f tldn nrrra itut)'lg tro:r, khAc v6i 'd' tlorrg d6 In ln lltartr sd. vA B; Goi H lA hirrh ehi6u vu6ng gdc cia ll a) Tily lheo c:ic tli sd cfrrr rn, chi 1|l [{p ghtit ldn ;18, cli.rt r : ,tB. Xeur irira duln61 thing 'lfru .! cria (C.,), v:i'clro bigt terrr, dirrlt, ti6u rlilnr vl'\ vudng gdc v0i l) tai ,l{, giii thidt J/a ludn lu.On '.' ti6n'r cAr ndu c6. ir vd-ctrng InOt phia cliii v0i l' hhi M thay dbi i h) VE (t o) vi t{ir-ri;) tlCr, cr\nq rrrdt hinh. tren ri1il ldv rliiim S vdi .!l{ - XIH" r a)'Iinh dO dAi c6c canh cia trir tlidrr 'tdB'}I' :;F; Cdu 4a' Vdi rniii sd nguvCrr d.Lrong n' ilht b) tihirn'g t6 rirng g6c nhi di€n carrh .48 tao I ' il' nOn brii ciic mlt pt,i,ng P vit 5'{B c6 gia tli i 'tn-l+I khong clili Iihi M thaY dbi. 1" - = J['-=---* .r"+"t r1r. c) Xic rlinh vi tri tA;n hlnh r:hu ngoai ii6p n*1 tu di$n S;lBrtf. Tlnh ba:r kinh hinh chu Av theo a) Chr'rng nriuh ring dav (In) n:1, 2,'..) bi 1l vir .r' v0i gin tr'i nio cfia r thi lrAn kinh {y i cltitn. c6 giA tli 16n nh:{t 'l lntttix nghiipinT5*77 Hdng llb f{ri,Vfi L)i sd 1?81J4 Gi6: 1,O0tl Sd tn 35is2. In rong ud 'iiri tuu chiEu thang 4-1982