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Toán học và tuổi trẻ Số 129 (1/1983)

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Dưới dây là tài liệu Toán học và tuổi trẻ Số 129 (1/1983). Mời các bạn tham khảo tài liệu để hiểu rõ hơn về một phương trình hàm; giải các bài toán thi quốc tế năm 1980 tại Phần Lan; khái niệm phần nguyên của một số. Với các bạn yêu thích Toán học thì đây là tài liệu hữu ích.

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Nội dung Text: Toán học và tuổi trẻ Số 129 (1/1983)

  1. Ef": i. --F+- .t u h,{t"1 it' Ii n r\r,J\ - )"{ " ', rt i YII:IN Kl{O,\ ti$C i , vtET f{A}t Fl t BAo RA HAr THANG' MoT rY Chii :,-II:$M; HSUYf,N CANH TOIII '{hx,ki* rri€ roe$: SSANe CE{Ui\G : t. Tru, r.i: 'ii;'1'1trr, Huag Oao - Hf HOi : S&y *di : S?li?$ I t! ! 1 , N6i *har;,Sn vgi q&-s b,an tr6 .y€u to&a E i Wffi ffi**r. FKE#ffNG TftiNH HAM r . PH.AN }T,C CHINH' 't I & soan bi,o 'foAn huc tubi trE e6 nhii.n ilugc hAm ,E. 5 ft'r,t - 1r(r) xAc iliuh vlyi r hf,"u t-i, lidu c.i rni. tt ,rrdt sti lt:ri gi{i bAi td&.n s6 1 tiong ki thi rpng tluqr cho ugi sd thrro a. ching- ? ' . 'l +irrr & L{rc-xern'-bua : , B&i toAn 1- tbu6c loai c phtrrrng trlcli hArr'r e, F Bii ioin i l lffy rtnr mgi /rAm sd f(o) crtc {rlc-In &n l{ hAnr sd vd ph{i tim rdr cir e{,c h*i* rI dinlt rr*i prr{ e hitu tij, Itirja, nrfin cdc drtu &ign s6 ngiriQm bAi toflrr. ilh6p giii m6t phuang trinii thOng thudng ndi chung dt lA viQo klrSrig rla* lil) * ?, /(rg) = f(.r) Ttgt * f(r f g) + I gt&n, lO di nhi€n ph;p giii rngt phu,rr-ig trinh i u&l tt,p! " y triin fij. r'. , .CAc trii gr-aigoi tlClir h& ra ciirrg dip t'rrrg dugc him i lai eAng phril: tap hoa" . . 'Irong lrgt bni toAn vE phrrong tlinh IiE:m. 1,0u cAu dr)i hiii, vl bil.-i to6n n6i ring: htrs tlri. h{rn 16.phli tir\ buOc p}r&i th6a uran rngt rhrrl- irr0t hAn,.,. 5o ddr chhng tril d&i df,ne c6 th! nhiiirr) hf tbrlc dgi sdi co bin. \'ir t nui chung, ,rlrra ngay;,:: hi.i:n *6 f(r)c*c*- l, rec dinh r6i ndu khdng burlc thdm.,urgt v$i di0u kigr: phir, Irloi si lrirt, tt, thl c6 v0 s6 hdm sd 116 da:;g rSt kir6e l,hau, Nhrrng rl,1 khC.'ng ph&i it ib6m .? ctia bg1 lo*n nghigm bAi to&n. Ch$ng lran :- ! Thuc,.clrrrl lrii ..of,n ilui h6i tirn t{f cI eic hAm tfi tlroa nrftn uac diErr ki6n de nQu. i'a ifrtr.-il Bil torfln 2. T!.m tdt ett ,:ric hdrri frr)" ;i.iic l* /{"1 = r I I (:r irti{ ti): te ng}i9*';"".'nlrf- nhh u(ti- rnpi sd il*tc x, pd lJirie arfip diEr l:fl; cir-a b*i lo{x. flet ti6o trr,irs-{.d. r- ,T0Ag. f(.. rgi=/r.r)+/(9) khdng e.ti Larr nAo c nhd t rS1-" , ., ., " dur s6 nghi6ro h;t -l;. i -. iliirrg chi Fhqrn i so4rr ird hdi !Si : : \F r.t
  2. t [ --:... \-?.*--k\\,**.*-** ,- : - *-*'*- ""*" - | i -->'?-'' .,/ l' , I trong do cIe rn0t lrlngo6:{tr1lryie6 ttiair. }rihung. i,'..,.,kh6rrgthak{tluln,r:l1'g.:".. i) t'rong (ll): cl,n rr e g -: {r lIt rl6r,c , q(0) =9r({)) ji:er(0)- :. ' -,, , _./(r) = C.r vui nrc,i sd flipc..- l'', (.1) t {_r g(0) - g. ivrrJ}, t.trLi rrg hrn. rdi 4 vA C La hri lrlng ' :c6 ' ^"fi- tir.vlJ, 2) T'rtng (3), ,'1,u ., = L t:i drr,;t ) diith, h{rtl '-/'i'- ...''. -. ';)r'' . g(r) = 9(.t') g(1) "+ g(r, * p(r) - gi y + !1, I t- \ .{r ! Cs pdu r cd rlang ..,'= rV2 -r:ti's, \'0 r . /t-u): ( ! vtYi i' r h&'u it nf,rr s kh0ri.g biilrr diin drdi dang tg(**11:.g({)I.}frr+11. t1) I il !'r N{u g{ t)=r, thl g(.rr t ii=r} vii.i r.,l,,i .t.. tue' li,' i.j': '1i m$l nghi{m cira lili loAn. ' ilE irti tp{u 2 chi cCr nghigm (i), ehu,:Ait th*,:, i3. S(, t dbng nirit birrg 'flrnSh r), ,rr r r v,',i qii rlii6t. thir 9(1) g (1. 'lirplg (4), rko .r = *. lr ' di&u kien php. llirdng thrtirng_, dl,li toi C* sd t hi dluoc l :.. cac.phunrig trinhlh&nr, dO ld?{itu hi*rri-fdt) it=g(0):=g(- l* t):.gr i)Lgi .. trr :i, niA g(l) * u, nen s(-li - .. i' i '. - K-hiri tritrn lrdn tuc htt s,tiE rlu a p tl'ong irr;ng , .iltrirc i tqln.h,..rc. lihung khAi ni{m. Oy' rtii linh-.vi,,ntl 13) Trong (3).ehn g,= * 1 tlli phii.dr.ra ti6ir quan nif,nt cl,it che vii sd thrra. .\ gi* r):= 9{* t) g{a) +ig{j t) .1 gr,.) - ,(.s-- 1) Vi t'{y, & Irlnir {d, ph6 .th0ng., neu cO thl cf,r:g cbi cd tha dE edp c16n kh&i nigrn-', lian tu\- rlrgt " : --. 1 ^- g(.r: '.''i). cdch. str lugc. Nhan cA hai v6 ;ri' virj *r.gil), rhi ttiir: e) toAn lr chi u6i ddn hArn /(r) xrilc dirrh vrli r h&u - g(l)g(- *) - 6 tr):l.i '+ S( * * r]i : rjtr; (5) 'tt. Dt sno eAu hiri cfia ttAng chi'Giirr,r df tlrOi ireng lrg ihrlc rrrr r', citrl _ .r.. = r- i, lhi'drrrrc a _ .' .- th6t' 1$i; mir r.6ng bAir tr.rAn .1 clia eie .s6 th.rte,- -g2,1)=gr*l)=.-1. r kh6ng phii r&'ding khti li{m tiCn trrc. N6i r cich khte, co the li hf t^hrlc dai s6 trorrg bil r'4I gtr tl = t. N6rr 9(rt * . r. tiri (4) ir& rlrsnh I ta{n. 1 dfl rgxnr bao giinr kh*i iiem liea trio ? . g(.r':"lJ=- I .lir), . - Itghi.sat hon, tOi ih{}' r$ag r{ring l& nhrr r.ly, d r clil Dilng thdi c6 thb giri nr nhq gii tlr i€q, r,ri ia :i 'glb t : Biil to6n 3.Gi,i sri /(.u) ld m!.t hdm s6 stic. .t1flctii$t:g121-":g{a)*{t.'}ihtrilg,lrhidd l :dfnft r.rdi rnpi *6.{ftirc tr; ttd.'th6a rr&n diYa, *{dn ,,,, * 1 * g(t) * g\Z.tia,\ =.-g(:;gltil) i f(: g) =;(.r.)/tgt - f(r + tJ) * t (2) + 9{i) i1 gtl,rfi ,- s(2 1 1121 - str':l) - gil/31 i: g, ?l titi{ mpi: e: ud s. Tht ihi f(r") ld rl,lt'irsng rnAu:.thuf,u. \'{_t ta ph&i i:6 gil) li. 1-& i4), {S) .llal hdm sutt d.dg : "p}rrii = t' t ',t' 1,, -fioec/('r:);=1,&t"mqiri \ g(.r.r.li- = f(r) .. .l -.5 i; gi* r,) -:, ',r1,{r). .f," I . - lti{c'f(:r) = it t- i, ucri nrpt c. -t: ''^ . _{ ;':-: ,4) l;ai, ttreo. (3), tts. ,lheo irtl. 9{,r * :ili := ' t? , .' ild' tilfrc (2) tuy cS.r"6 da{ giirn. nirrog hoi E g($ + l),+ l rer g{trI l- X vAtrt(.z} =. r: g, klr6 lAm vi{e. lr[r 9(r) * 1'G) * I, {hi !ai ro($ f" : ' 0{2sE}"* g(ligt-'i 4., g{?.i 1 S(,r'} + g{i: *r ?r \ , Bai.to6ra 3'.'{iiei,,sr? rr(r)r ld nt6t'hdm rd *la, .-, = 2g(r', + ? r- s,(,r: * y(r't " 2=9d(r). ' dlch uot tnp.i sd liqrc a:, ui lhf.a rr:dn rlt'Ia A1{n 5) Ul-j' r'&lg c) +g(,r)+.g(g),:- g(r- 1 ., ,u) *9(r'g)=g(-'rg) 17(r:g) =,gt 9(q) **i : ' ., , I ;;(rr)g{,.i.vri,-hatiJ.}-.q( --,gi,:9(r-.-1;J rnQi..r'r,d Y. ?'fid l-lri :" . ' , '.1619'l -i* $i;r:) -- .liyl * 9i:. -,Ui.', '',: 9(a) dbig nhfu I'irtg $; ;i (3), arr.r !.:i ,r.\ = .r t'6'l met t. , :*.-t - - 2g(.r') r= g{lir) ^ ' '' i;- I thi ta e&. t * ;1
  3. clilng vdi' rnpi r,'a, hay vi€.t lgi rbnnh thr! u6u * ft t1 tl'.i . ' g(x) * g(fi r= g(c + g). i, ' o< g(.lr'* y) * pinr).1- s(-s) * s(;s) =g(yr, ha1'g (a) ? sU). ${cc) -=ig{,)9Q) (iiA fhn r Ii rndl sd tlrrrc, trly y. Ndu r hitu - Trim lai, hdm g(.r) c6 ci,c tlrrh cii$t: tf, ta c6 r,gev q(,r)'J .r- N{u r,vd t!, ,:l.rrerng l.ran i) s(1) - 1., ' brirrg e{cli xiip xi tiiisrr vA ihri4 sd .r', t*'-tim dlucrc lrai dRy sd h&tr tjr ii) g(.r * L,) : C(.r) * giy) v6i mqi-a, g, iii) 9(::y) ",, g(.r) g(g) v0i msi s, g, * , rt>r,rl)..'}r" }.. 'fr'r ii) bLng plrip euy nrp, ta durvc g{rr) = st{sz
  4. .'. .thb t)N{u BC i-q XY, thi iu (11 vf rri dinh lf h*m nghi€rn trgnj (do thtdu cfr& ( hoic rt; t:hi cr1 sd sir, suY rn. - -: li8u -- o6eh . r'i6t tIO ll hai ding thr'rc phii ddng ihdi. th6r men. . {.) _= . B&rl Z. l]&i 6or El,.(r1r,.., aB . d.ar,st tat rllnh ecrs^ll casC tLht ruu: sin 8er,sB -i- sinCcosC , ' '6u+1/!; . r: a r:r Ee r) s{,' . : sirr2F * slrrlo -^"-_'-_'.-- (:h*qg mtnh.rring 1,- l/n ( a'
  5. .r'.,i' I * do'fhr;y p;!6 rlE nghi ltm.{B'rhi ,, : mqt:bAi-to6n Jr.4i+t//d"+1 di-p1 1-t f tfi f - n -- I ; ta chring J, 2;i,.1 troog ki. thi ToAn th5 gidl Ifii 16 (1974) nhuns rrrinh .4 1.41o, /Ao -.to+1. (ii.ri tlcr ,lAi.currg 31.{i41 ld l&n hy, IlQi {t6rig, gi{rp .k}i&o cho, r*.rig"bAi n'1- n; ,y(1i i * 1,2,..., n * 1 rA rli dni cung .+rr.A[ tf, guri khd I) : . ." ,. ,ailo. Try .l,tA.zl lAn+t4n+,1. suy r.a' -'- ' , r Cho ct ld mdt td thtr.i, 0 1 a { I,'t,d.'n.ld mdt rrz -i ag14... + aD = rln .l:2 r ttn -i *...* 6ga (l) ..rd tr nhtin tiy {}. Vdi mdi n lip ddy s8 'l rrong trr. lw.ltAi4t '.{n-}-r rl,,;i r 1, i = 2,.".. n*1, Aoi (II, A2,,,,, dn , .:3 t a4 + .... t- {r4g1 ,a on*E * ...,*,112o,* a1 Q) -.i{ = o. rrk+1= tts + t1f,17(0 < k ( n)' 0o + o5'+..' * rrs+i = oa44*..;*o:n+41}43 (3) tldq elrittg'tntnh rdng lbn /ci ntf t s6. thr.rr ,l * o'u-1 i..,*a2n.l:.o2n * o1 *.,,{an-2 (o-r} ca (phttthugc ud..a a) grro cha (/(1(A*rr.)
  6. .l ,:.!. ..,' ." !,dt gl'dl; s*t}ng itifrrig'!, en e&t dudng cong B*i 6, 'Trqig dqcg .ttidp ptan er{s sd (Vf+ /r)1s80. ftfiy llm cJrfr sd ttdng rlott ,f ua u=*n+P'"*g"r*t'r*s cffii' rHliu fiAn sau cld,u Phhg. ' : tai 4 dibrn A, B, C,, lJ r:d'hei-*nli tl6 tuorrrg rirlg la a, b, c, rl thi q, b, e''.d ld'nghi$mcrla'phtrong ' Lbt gt&i. ta e6 bb ,IIi : triuh . :-,i ' aA ie. Ydri a vA. rr 1& c{c eS tu rrlrisn, thi tBn .4'j- pr3 * gc? + rs * s - in =.0 tgl *4 tr; 'iihi6n H db cho :, ___, ' . !i 5 rtt,c6 tiuh'ta*r gi*.c ndir -4IJ + AC> A'D (o+/rtTi)l =\ FVP=i '{*}e * d{b+ c (1i (! U*rn khOng c6 ilnh [anr gi5'c ndu CE +*ff
  7. B vAy phrrolg trlnlr. d* elto ciing, lih6rg ed nghiim ngir,v6gr (,r, g)" yiri U F 0. i t. 'i'om lai, plrrrr;ne 'trirtlr drr r:ho chi cd n:4t nghii,rn rguvel .r: y -s 0. .\ l.r{n .r'il : CAc 'biirr tlor ThF Dut. .\'gru\rt Tri Tt*i'tt11 rl.lA. Itlilh Lu,.. Iia Narrr \inLrt ed loi gi&i {61. Cri* bau Ld \-ttn Bill ( U ts. \ri i 'i'hanh, Qtrlrr6 \*in - Rrl Na ng), lrllr_trn T hrttrit !,ltttong, B*r.i ir]*f.i. Ilrr $i) dLdtrth{ni c&e I)d Qutrtii Dtti '(1'1 (,f. DfiStl t)llA l,i6i) cti liyi ltll ce{ran giii lrr,,n4 dni tdt. hibt rd:,g l# do ., trrn1l iltT:bu ttlii. ' t',t 1' a.n T t.r t i'." '" -. q ''1. L' (: ) t'a lBrtg rri,. clri!' si, , ri,z sd ,l 4 "nug bt). BAi 3 126. f im.ngi'i4nt nttuAin rtio phtronlr l.ii irrtt "'fia I ,I hlr,' llrn1,' lhyt l6 r'T lrrrdng irt rth ' 3 * r-il l'hnrr i-h6r, ['r itrl,. lt] r\irrgr. 'r -t", . o.'1 :U - E' l:" I lieo :tii Ilrii.t {ri r'r, Lbt qiAi (r:iia uhiEu trari). . + " I,r * ?i -i c : hlt -.llb Ch rnot {ro,rg. .!;s &:,r lrdi k} gi6 tri. b$ng 0, *+ :)C , C :.: il/) ta chi i[rr.rc nghilra. (lr, 0. 0). 't'a chr'rlg n:inh plrrurng tlinir kh6ng c11 rrrlhi,:rn ngti-v0u khicr g. \i.4.;v ?n -i. r l:liil Ir, tc'li,, g. lr a ( e riln?n 7 e
  8. , )iEu ,$i = 3&, v#i k nguydn hh6*g Sm thi & h'ai 6 c kli6e nhau- ;r : ljk (g,t- - ;l5.rl .'(Vi ndu tr6.t lai thl- sd \I'iv n ! I c6; hinh bhI' nhAt tao \i:u S1=$1*Lt thi rt=(ift*1) (9k!.-!':6lf:l t -SSz). fili rr { 2b -1 ijt' trdnl)' \ril .n kh6ng c ia irft cho 3. -Xg,, c* hai 6 a trdng kb nhau (cdirlr nhau 1 6 T6m lai ndu n biDrr di6n &ugi:'dudi deng tr0n c) lhi cll lih,ing .{, irbng thl n' lih6ng'cltia hdt eho 3 ho{c ni 9. 6 c & gJria dd Phii c6 b) l)iEu kiln dii : it. uhst rn0t gi6n bi' (Vt 'la ct: chtinE minlr hhi n kbOng clria hCl cho 3 a6u tr6.i lei thi hlnh hoirtr n chia htii. cho I thi n biBu tli6n duryc d,rr6i chr-t nlrAt tqo hiri ha i 6 dqng tren. Ttrrre vi-r : a vA 0 c d6 tr'6rrg)' ' NEu n:31c:F1, r.6i & ngu.r.An kh6ng Am thi : Tr'r nhilir rr-'t trsn su-t . n = 1,.3 + .t3+tt + tis :- sk.,ft (k+ 1) ra : &6 kh0:rg -{ thl tbng .33n lra5 n =.r; + ri + ,!*Srlrzrs v6i rt = &z e sd'bi tr0n cic 0 b, c phii l6n hon s6 6 c irdng. {n * l)r3 vd Gqlnlie66o trdug. Sd bi tr€n c6c 6 a c6 1, - {n + 2)13 i{ c{c s6 r:guy6n khOng Anr. bl It r thl Ndu n = 3I'+:. vdi f irgu-r€n khOng Am thi )36-n n = /eB + {A + t)s + (&+ 1)$ - g& (k + 1)(& + 1} (vt t6t ci 36 6 p). o6 Isu n = 0k, ,di k Id s6 tu nhidn tirt: '56 tri tren.c{c 6. b, c l} y thi dE khdng A: ,\ ' n = k3 + (k +i)3 r. {A*ti3-:ic (A * 1)(k* 1) Ab n"+ t ' lf&4ir: rdt':, CL,e b1n NEuyfn 'Fhanh Duong (1211, Nhu v6.y x*y},37.' f try,&'rg Phan ChAu Trinh, DA N{ng), I'it euilnh (12ts, NgO QuyBn, Hii Phdng) e6 tot giii r6t; t{c Te ehi c6 35vi6n bi n8n t{t phii'eo ,{ (d.p'c.m). ban lrlrr Nam'l)dng-(l0p 11 CT, phari-Ch6u Trinh, Nft4n rct: C6 thE thay sd bi b_tng 36, i6t{h6m il{ Niug), NgugOn Vdn Mintt (llll phan Chtu trudng hgp kh6ng co 0 o tr6ng. T1inh,,Da i{{r:g} co Ioi g.lli tucrng d6i tdt. ,' Mlt a6 bAn kh6c ion gili theo clc ench kh{c' Tq Y'dn Ts Bei 6lt26.r) Chrlngmtn h rang udt ni1i tam gida Bdi 5/126. .\'gubi ta rdl JS uftn bt trln lnll ABC ta ludn cri: -. mdi sdn hitth rul ng cd canh b&ng iLm. Chtlng minh {tliir ooc .d.-{- (1/4) sos F + (1/5) cos C { 5,/12 rting l;hl cdc hdn bl dfr dftng yin thi crj th| tim fl$t v*t d.qng lam gtd.c A,EC nhr fJiE'ndo fii cd du',;e tritt mdt sdn it nhdt mQt hinh ch? nhl, c6 dfiu tlting tArlc ? drfn /icJ'r ton.han 3mB khdng chti.a ui€n bi nd.o {tur: lal kh&hg e6 dibm neo Jfia hinh ehft nh{t ti -b\ Ch*ng miaft r&ng- v&t m1t tant"iide AE{; t* rliBni tidp xric eria mgt-vi6"n bi v.6i mit eg.u). Iadn cri i t.t,i ttiiti (eia Ngaydrt Thanh" Duong- l2/l phan 'cos .3. * (1/?) cor .B + (113i cog C.{7 $ :t ifErodi rai u&l mdt sd e > Q tfiA ! lu4n tbn t4i it i {.hia rrrdt stn thinir. e6o o vu6ng o, D. eAo 0 nh6l m|t dqng tunt giae- ABC i12 cho : el:ri nhit c nhu hirrb v6. coo I * (l/3) cos B + UiSt cor (Ji 7i6' E 'i'a chu rrg minh c6 it nhdt rnot hinh cht nhtt ec eligi: rich lon hou 3rn!, .lt htnh chrl nhit tao Liil $il ieta #{ng Trtr&Ag San * 12 CT Dai b(ri a "F lb -f Bc ho{e tao b&i 2a * c. kir6rg.U,i" Ui, ' $pi rinh ehgt db ed'hinh eLt nh{rnhu th6 ta Trude hdt ta chri'ng ininh .hb dB reu: ..t, ngry!,lc lai Id &/r6n|7 A. V6i mqi AABC vA o, g;'aCJ?; .r$;)0. ta ' id n,t.r{n xdt : "l'a : lu6r c6;: . (tic) o6s:t * (t/s) qds I * (ric) cos c -Ndrr nrdt 6 'ruOng e udo il6 tr6ng (kh6ng r:6hi) thi ale A,h6ag A, c6c 6 b. c *ung qu"of, 6 o .iidr plr&ri,c6 tbng a6 it ntr{t 2 vi8n bi v& nlu D{u b[ng rly ra ktri sA clrl khi sin.4; sin I eti c6'hai vi6rr thi hai vi6n bi 66,,ph&i n&B sinC a.:gii. E
  9. Chttn.t mlnh: Nguildn Thanh Droag (1:ll, Phs$.Ch0rr Trinh T&' (georA * *cos8 ^ zJz + (yairr,{rrsing)? } 0, De Ni.rg), Pfrlrn Tharih lthi$ott vA'Bd Quang ,D11f khni trr$n vA rrit gQrl t;t dtroc: (11 CT, PH$PtIN) edloi 'giiri tutlr:g dt5i t{lt. ,? -l- ul * =2 - 2g:cor.l - 2rjeos fi * 2;r:y (e os.4 eosB '- gin A sin B) . BAi ?/116. Cha tliLy r0 a1, Jr:r,..', .r,,Jsr rr,i'i Iray go:r hcn: tlnh clttit: ItD s.z + uz i :z - l.g;cosA -- 2.r:;cos/J - 2.tycor'C ).0 l) l'l = nig8, = l, uri 'l'a'r iSt hdt ding thric cu6i eung nir duui ' . .,.1 2) 11= (rl i + rr+l.)/[i (19s3 -- t)l -1- t I u0t I , ,lang: s 2.:t...., 1961. 2g;oosA l- 2r:cosB * 2rgaoso ( *o + u" +-l '$) r2: f lgSlr E:= fl980,'..,.r99t = .r99Bi ,ja chia c& 2 r-6 eho ?rg: {>0) thi'eo ngey di&u. . ':3 pFi6i chting urilh. Diu bi.rrg x&y ru.khi rS ehi lil,i: b)I (chi cEn x6b v6i eAc 16 e ) O , 11993 ah6 tty G -i'),-ta xet d+ng A.{&C c&n 6[rh 6 sa-o ' 2.v +i cos(l[ "- A) = - cosl{ !E eit*1 ' Do v{y v&i dqng- A.4a{'&6 thl; (d; t. 19s1'< 3' 1e8o ( :.. dsel^e92); vty H( ff/i'Jal + f *+ Y'< tsrj,'1g80s I + cosi{ * (1/2) cos8 + (iill) corC} - r080. 1 / (t - slz) + $|il{l - 3/2).= (1/3)(e/:-* 1) = T6ar lai t ( .n ( r + 1i ls80 r'dii mqi I:E i16' {t trD e } : 6 - t: (tl.p.c.m !' t,2, ..., 1982. , .lYJr,6rl .tii i C6"c b+n Vfi Qar)nlr (128* NB0 Cl,c.ban: d!6 Quangt $qi 't'il .Pfirtrn -IJran& Qrr;Bn,', H&i Pir0ng), tr3ug0n l/da ' ffinh, fiurn6 {l0p 11 {1'I tludng DSiiP }{e }i0i) . I ;-
  10. I I Ngugln .'l'hanh Euang'va .,1''grrgPri Ydrr t$firih N,lidrr ral, 'Sfi cfic: b? n th:r.rn gia =giii hAt nAy (lop 11 i truijrrg I'ira;r t.liflu Trinlr, tl'a llarrg)" tt. {.1{e bau. sau di1. err lii giii ci; 84rg gr2u lrrrn S,crn (1: Cl"l'lll'it'l-: iiir l{ii), lrl4irtT'.hsnh. I'rrr&ng #(ng f rrrcrng drn (ldp i2. CT trtrdng Bil 'liirrg Plwlng r n -Ild t)t.itirtrl Dri r lrr ' i)llrt) lla Ndi). ' i, I. hop tln NQi) eirng c* liii gi&r tdi' FIai l.lan .iYqrt'1Sitt-Iirtrnrl /irlo'tg r *r- "-'.'11uqiD '!'nri blinh, \12i1' f luiriig' Pita;i ilirau 'i'rirlir' .I)A Ni:rg) Bai 8, I26. Trottq mrtl ph.fino tho drtijttg ti'ih .cirng gi&i tItng, ".rri' drinot i'ir ,1ir,i: lf I{d'rd-la- (u) r,ri hrdl 1]dg;-c*ng AB tig ll . G+i I ld trang d,l|nt u-vt vi rli:rh 1i liArr s(l'ccsil tlon{ tarn giAc. erin'."'is, ff rli N td hai diiim niluntrin dB ud ddi Tuv nhido kh{ng *ri.La;r titlt: cii lir6.n r6t th€rn :urrnq r*i nfiau girrr i. La$ di?m {.) tittt li tre 'r'dir&rrg rlirrg bAi ir.i6n eirr r-lr lhl :rirriL i,iirr dudi tl;rn4 175v2.; rw r{rtrng thirttq.t..i{ zrri CN fhr:o lfrr? {1r 'ttriing tbu! quAt ho,ra., ' , t:&t, L11i ilvb'nq. !'ta" trl' & F\ ui Q. 'dit'b'trg ?Q
  11. suy ra , lP := |1!tlt ri S erla eic Coarr iharrg 4{i. 8I) q1 r,i rflv, dA (xf,e dinht' tu ,i; dirtrg sul lu rirri ticir giar.d;Frr, rr t,ila cire tru diErn I Jrrqc iro![ir d6' (tAnr c{r.a c&c lrlu.lr chir'nhnt) lA, rnrit rlir;ring. , toi a x,1c ilinh trEn trdrn, d6 lA drdng !rdn drdng A.rrrlr ftS. Alr. Suv ra ci.r clr . (r.rrig sru. irh(n tit. Chri j riing h.flt grri rria phir, tri6n Iu6n n6* & trdr: el
  12. !U 't , D[t BC y a,'CA = 6" ..tE -= 6, DA-= tr, bff = g, I{hi gi&i: h't nd-v ea" lari tt,ly he . ludn . c6 DC ::, HA: ,t'', HB: g', HC * :' nghiEin duang dul"nhit ;. ., = v aFTrT *7n . fi a ='v l.,c\ + zar - D?; :l , :=V(inl+2b'i-_cE,,J .vA 6ghi{m nd.r tiri:ra rrxn b5t {Iring thuo tam gi{c (c{e ban e 6 thE ch{rug urinlt del d'Ang }rir:g cach ehii j- 7. !l.t z la rrghignr cfia (3) n€r1 phAi c6 (e * y + :)(c * z * r\!z-p r* s)(.r' * y - :) ) 0). ; i' . ' Nhrr v01. nhiing rli$rn O'ti'0n A.thda nren hl(*) A Vr* -7=-iZ 'l{rrhirBg, tti'6ur .e&rr tirir vit nju c6. lhi c.lri i6 l; --,1-- rl hai tiiln tL$i ltng nhan qrra ,ll vd xia dfnh etri"k r) q - o , t- f)H'- ro -I .!''! i fi' * (lfi" n") = (2h" 't 2c" * rr-) i; - a- + 1/i' lqrtr, ' -'''- I = 2(at +' bz '-'"21 ;t - IEz .''.. 'u'2 y'r *'br a t, o? == + := 4R3 (1) Ta iI6 ct v0i-rngi D tre.n A tqi Bi|a. Iarin: "'2 "1 fq *O cl? etr*rrg minh tlutyc vdi mpi D trtn r! + o'= y2; b2 - :t 4 e2. * -". A rht ,r + az o.f? ian = ,2,+ ,zi.i , e) ' nln dilu kier: rlb c6 rrghidir,lA tr€n A co diErn D khOng trrir:g vdi 1I tli6a nr6n N6ri .tri :t)1 !J, i c6 thB dr5ng,duoc, mOt ,tam 7 ' +:-:a-*h-*c' 2 2 ,2 t giAe tht'di{n tlch J crla tqm giAe $6 th6a mdn {+at (l) EI ng th['c. ,q tt * N6u.r'." + g'- -:'D a12 i| b",2 ,!- ,q r-c'2.,.thi hhflng 165" = * g * z) (g * : -.-r,) X (.r. thD co (.1) bao gid crih6 cd x(;t.u-y)(.r*g*:f *"\drr r'2 *.,/'E + .'2 ( o? = b2 .,:t ihi hao' 'gin cung. oQ hql tiihm l} d6i xirrg trhrr: qrra-11 = (g2 * z? =,1;21i' -F (.2 + *z - gz)yt -i ,1, , t + g2 :2,2 il'oa nrirr' (1), vi th,,E.r'? + y? + .l ,-6ng dEn fi! t (.u - )z ., 'r + g'3 +'.'t d6,, o". khi l) t'ha.r trtrr -\, t ri H - M{t khhc,di$n tich So c&a.tam giAo. lBC th6a vN xe jtrr $, phii e5 vi tri . cIE i6og dd biing, nln iling thtic : : o2+h2+i.' gibu. kiQn ,'' '. oot +.g.'' + r'2 .--12 + b2 + rz t6s3; (b2 + c2 - nt,n2+(c2 + o2-b2,|',+ _ tuong rlsong v&i . 6 ,2 -t 2:2 -;,/ + Q" + b' - c-)c-. + (lRs - ai) + i+n2 '- u\ | Gnl'- ,z) ill hai.tanr gitc c6 difn tleh blng.nhau c .,2 i,. " thiv6y cEn vA d[ lt : d q- . b" 'lc- ll 5 * - "2)rz + ,(;2 + "2 gl)y' + (gz + :2 '3,,'). + (.cz * oz - ,2)rz = (bz * .! - o!)o2 + '\ '' ; ' I C{c bqrr f}drq T'iudng Son '(l?C'l , DHTH }lt . * (cr + at - bztbz + (oz + b2 - (B) I;01) vil Nguudrr Y'dtt Minh {12ll, Phnn.' chiu "2)"1. (2) Tr:inh Ddr'NEag) c6 ldi "gili ttrcrllg:ddi. tdt; ]I+t :, $au khl bi{n dbi thi hf hri phuong trinh td (3) turrng itucrng vdi h6 s6 barr c6 thidu s6i,'ld kh0rrg tl6 cAp ddr.r bf,t ( 12+ '0, +bt=12+"2 "'=u' (+e) i * *"r, =, o, * u, o ,, ", leua€n' Ddrtg Ph6.t 1 LA'r- :
  13. ffi \ diu {lfirrgthrir obl riy ra khi at: Dl : cl = a2: bt: cz = eS: !i: r.3 ) UaJl ar& .rQng k{t qu& ticn. : f , I - . .. phan Dir
  14. /' vH xuAl mr*mr u prthrq r*cuyrm e[Je ffi{}B' sG o xeuyfiu lur:,'c t'nui.r- ' f-1ft phuc vr,r cho-vi-ge it&i p*" irdi to6l- cS li6n vi ** + 1i'J > 0 ngn i ,' + t,',.t1 --, i-,, L,c ,16 ,U i16 r,ii khAi iniirn ( phiin ugtiytrr mOt s6 r, [-..o * 3r.l = [.r-" + tlll*ft > 0 (!),;-l.c-irg d6 : ,xln n€u so lrqc & cIAy kh&i nJ0m 1:l- m0! n ld sd ngu-riu. sd tinh ch{t.cion gi&n eira pfthn ngrnydn niSt rd, Bidu tbrie *" + 112 cd rl:?r ia.r giii rlr i.i.rr irao diing t&tii giii, rn6t 'i'Ai iii ioAn giup ban ilgc nli6u iirng ctlrqre. i{h,.ng, i,i6rr rl,ri,. - -r'? + llr li,rn quen" Ph[n cutii euug crie ir]"i uA1', s6l ntli thi kh6ng r,hu vfl;-. rl hat t.l;ii, - .r.a -r 5r : q,,* i'it vi0c r-E ai; ti,i tiai dang hArir sd c6 ch t'r piriir rtquvSn',lirrrlng itav g:rp nlrdl. - i."'o- 3iu): - (k * AIZ)'$. ." 9t,\= Sl.i - (r -slz\" (sll . .. ' ]Yr&n ngullin c&a m|t s6 a &iqs6 ngugdn ldn I'u d6 sur' r& r:ing , gi& iri ! ln6 riilSt cria rrl!lil &htil:{' ulitll qud at, }!i-iidu Ia.[r]. Ching han -,r2 + sx la 9/,1,rD(, dtl- ;2 r- glt
  15. 6;- ..1 \ .Gtrii". *4t .r l&''s6 ag&r' i,i:&-. lrpn' i:Anh truiirig eho, p t* I;/e], ctri* hdt, t:ho p2 ia is/px.l, chai 'l'rrigg:qu6* ri!nh ciiir-r,irir qir$t. Nhu r:iy, dudng h(t, che ,p3 le {ni'ps], r'.r'.:; Di0u rn,iy chr}fr6 i6 chuvbrrlqn{n '*fra'n-'iiir:rg. s15i ..u inm. Nhrrrrg, tr'€n llnq s6'lria ind sij nguy0n t6 p c6 mAt trorrg su, lhyc t6.,6 'rgd.r r!,u.'.tt qr:3rr lA r'ii;. truirg ph&n tielr r, I ra thri+ so :igur6n t6 Ia l,ang i6ng t16 o6^:igBy n6hi 1*i t,: l{"r' + li.7i:11 })Lr.Jo 66,,*6.r* ia,pl I ln!{,zl-i ..,+. lnip'iI r'*i p"'
  16. . { \ -. i{6 duong..thirng :r E, n, n = o, *1, 1) Ki:,, +3,-.l.vA ta, e6'c{c }rflng.tao b&; eti tlir$rrg tlingS'=rJl:rtnf!. 2) Lf,y nhiLng glao di6.rn cfie e{c drtrrrg rhfrng t c='n:"t " * " +'t'r6f :f*A tdi ll&rn g,{, - f(*). lihil'ng tti[m n{y phAi thudc'r'Ao db th! h&rn d ' f * t7:*/ , = ltl.r l) cl c6 lroirh fl0 ln nh&ng sd r,grry8ir (trong htrrlr dudi li c$e (iFrr: 4,.6). ' 3) Mudn pd cfc {i0m khAe cfi,r rlb thi hArn sd y =-/(l,x'l),.ta vd cde hinh r:hi€lu vuOng gdc cita phBl t10 thi-iludng t,irn rE g = f(:re) trorrg n:6i hftig tao 'biri c{e ihing ;:.: n vt .r:i n * 1 tr0n duong tlrfirrg g * f(nr.l{inh tiriirr tlc tlruOc - I :. .' ,vAo tlO lhi htm 16. g a It[.r,]) r'i rnst dihm ,N ,-:3) trIudn eo clc'diErn-kh{c ofir db itri Uam sO b.{t kj,'tr6n pir[n aB tli hlm s6-y - f(.r) c6 ,'= [11*)i ta .vi c{e hinh chiiu vuOng- g6c efie )rpAnh d6'lt r.,, mA n'{ro{.n + I thi [.rol : a, t:fc phin db thi hAm s6 g s f(.u) trong mdi bf,ng Bqn de.rc hey,th& '5rlr: vtii c{c bilti.p sau : tao lrtii cAcdrrcrng thiing 31 : n vA.UEn * I tren 'dndnu' th&ng y = r, Hinh chi€lui tl6 .thtiQc vi.o l. GiAi phrrotrg tritrh . dt thi ha.m i6 o * [1tr)J vi mot ilitrn -M ]r{t k} tr{u-pirin itb thi' hAm s6 g : f(*) c6 tt.ng dQ i)L=, f srt-rgI l-ie(r+1)'l j=L ln y", mA n {yo{ n + 1 tht [yo]+n. ,,[+#J " .I l t. ', Ye ilb.thi ham 16 y =f([x]).'cti sir, dn * ed rt0 thi irim sd U, f(r). ]{uCin tE dB thi irir =[-+J i 1n hirn cd g = i([.tl) ra tarn nhu ssu-: s.r tr12 - lrl =:. 2, Chirrrg midh nl kh6ng-ehia h6t cho 2o ,t r ,. 3. Ohring 1 minh .r ' ,. \ ' lrl + [r * lln] *..; * [r' "r (rr. - 1)l,l= J,r.r'L {. VE dA thl eAc br\rn s6 I = [r ll ; fl.= [a*l r'6i o ) I : y + [sirr,r'l;'g -- llgr l: y = y-o["'l (o
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