Toán học và tuổi trẻ Số 155 (3/1987)

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Toán học và tuổi trẻ Số 155 (3/1987) sau đây sẽ giúp cho các bạn hiểu rõ hơn về con đường từ hàm số logarit đến hàm số mũ. Bên cạnh đó, thông qua việc đưa ra những bài tập và phương pháp giải những bài tập sẽ giúp cho các bạn bổ sung thêm kiến thức về Toán học một cách tốt hơn.

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Nội dung Text: Toán học và tuổi trẻ Số 155 (3/1987)

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erJi mdg, rm,,{c *t,$.c F* f" i' Y* Ai,*s mdsa I3,{ qun t*a soqn b{o q-Ia{n }tqe v& tual CIuqc hlnh 1 b*og 8 durlag mgt aEi. B&ag ltp ir,6 3 nh$n duEe thu e&u t{p tht ehi€a ci lu{,a tuong tF te th5y hiBn n}ri€n:. horil lhu 68'- 4S{ (XgtS finu} h$i ring: Dinh li I : ilfu mQt tzlnh c6 In dinh ti thi 6il*,.bao nlrfu dudng rnlt n6t dE u€ dxge irlnft e&n &hdng .tt hon n dudng mrlt n{l dB uE ro. .tcar"j C&u h6i dugo iHt ra l& lifn cO ih} ,*€ ia6t hinh c6 tn Elsh l& b,[n6 {Ifrng ntlrr*ng m6t net kirSng'.) BiBu il6 dE d&ng cbuyfn vb e&u hdl tttoag duong rlng ligu raQt htnh kh$ug er iliah 16 n{o vA liin thdng (trfc tA t* rngt dinh ed rhe rrti r{t re eia ilinh kh&oi e6 thB v€ b&ng mgt ilu&ag mgt . a€t k6p kin kh6ng (bing ea';i: t}Orn vio gi*a rn$t e{p diqh 16 mgt canir n:r}i ntli ehung)? eau trl ldi trong trudng irqp nA.y I& cd : Dinh l! 2t Aat.hinh Jiin lt1ng ttti l;hdng cu di.nh 16 ndo d?u cd tht uE rltrJe birtg drdng ngt nil khtp kin ,{inlr j , Ta ohfag rainh dinh I-f nty b$ag e6eh sri dgag nguy6n lf eryc iI+|. 'ft} gl& thi{r li&a th6ng ra ruF re tlln tqt uh&ug du&ag m$t ndt khep ktn (kh0sg V& nguy&n t&e l&i Si&l bAi to&n nay hhOng uh{t thi6t p&&i ehfra t{t e& }*a .goi efia triabi. kh&o ldi giai bi,i to&",n'? ebite e{q & lhinh B&i vi n6u xu{t ph6.t tr} r*gr dinh ato d6 vi Ksaigsbrrg, m$t bf,i l*&* iluqc tlte gi&l quy6t di tbee e6o equL t6i c1[e dlulr kh6e thl nh{t & rhs k, r8" Sinh eB llie ta i6i rngt dlrh dE tii -qua (n{u 'lrirne bdt tn pl'r&a logl e{e d.lah e&a hioh ra kh6u-g tht dinh eu6i e*ng 16.i!u;;g i egnh xudt l&ra B loqi, tuqi din]l /d v& loei di.nh chdn. ph6t). Troag tdt c& rto dtuir;rg mr;t ndt hh€p kie Dleh t6 l& *Ae dieh *$ m6t o6 l€ rloan xuf,i ta r€t 6u&rg c6 ohiEu egah nh{r, phit tr! a6o vil cliqh ch&n ld'eae illah cdn lqi Bgn ilpc ed tliB eh&ng mlnLr d6 rliag riag khi Ndu v€ mgt dud*g mgt adt khdng kh€p kln d6 duSng rn$t ndt aiy ehria t{.t ei cAe eqnh e&a {tte ta dl+h iIXu va tllnh eudi ph&x b gt nhru) hinh. DB e0 tbb chrtag rninb li6p, e&e bge tham v:* rseb nt1 rs i(h$i iriuh, thl rioh cr,in !& erla Z khf,s th0m ldi gili bAi tong B/t3S d[ng tr$n tO dinh el&u mrlt e&a durrng thay dbi edn eria e$e . Ed' lll$65 bg,s ('IoAn hgc vA. tubi tr6 n, ilinh khie drqo gi& ugiy&u. T& do euy ra rodu T{t abi&n e6 baa s€ hiri ( 1rf,y v&i nh*rg hiah e6 th& r'E hiah 1 $"'&cg cAe du0ng m6 s6t thi ta e6 ?n * i dirih 16 rhi ssa ? i). Xia nhudng e ho &u li rbst td S dtr&rg rnQt n{t ahu v{y ;r[go&i *d,c bgrr eh*og nr:ir:h r&ng r*pi hi:rh aes rtl eo ':3*:i:.i:3-Y:-Yr *:: 3.:1" "Y' :: ju"*n==, ?) iru41 = i,*r, + u;), r,stl = \ 2 u1*us +*\, { S n = l,2,.. J r- -::rt :i_ Ch*ng ntinh riing ,u a'1= 3 F \l rrs** * Bn$* { ts /:o y(tt mpt ;r, ;'r i::.!,ti,,rj:i .J;;"ll l-Si giri;.' Yl u1 ). il, ur F S ruy ra uo p S, uu)0 Vn, !s t+t eO - l*S5, ut = 1985. * ve** = {lu, * eln-Fi u*i472is1,, -i' us} > S
Vr = ) rrg Yk = 2,3,,., 1, Ir..., su)- ra ug (*tlup(l ;*)*p{,rp). , V+y 0{(rit c us}13{u& trtl) { I, Vlr*?,3,,.. Trt.66 vI r& gi& thtdt "Ttl dd ) : : ru&+r'' ur&t sr {ar l l'1iu}af&{cg + r,'r} ( rls * rrg ' ?e Bil!: rt iiiBu Ph&i" cliirns,uiinh ' Air6r.r rdJ: EAi l*&u o&y chtr c6 h*i b4n grii r/198CI < ?. lel gi6i ddn u?iung ldi gi&i 6Eu khdng driag. M{t kh*.c d6 thsy 2n > 1 * n rio {16 D.H 1' tl tr 17 p 2"+t *J 22 1lo ;F 3. 6ii 3/l5g: Cio ddg s0 an, Un darrc sric dlrth . E&i vny .: theo qui luQl: Jn,= .ta*! * Igr- , u 1o!n2o lrs*t * un+t ( 2 < 3zl:lo $ll = I, ?... . Pn = Sa*1 f 216-1 cos'o R I{,T. uhi 'ru * 0". 9o ='co.9d: Chtirtg atinil r#ng; :1n E*i 2/t.5U. (ha cdt .rr..,., ,rr.. : c!!-t 0(.r,1 (cz('..{*o{1.' s?j thr.tc . - n;sys - = -:-, * 'rr:i 1:*) i=[ K$ hi}u 3:o = 0r ;rn+l = l. Giu sii rdc,sd nrig (C[ =' ri !l(n * r) ! r' !r thda mdn i ' " n+? T Ldi gtdi; (e*a -\gd Brio Chdu * Ael0 DHT[I). 1/h:1 * r.,) o j*0, j+i = 'I'a rc. { *o=f,n-t*2ge-1sinza t ,* = uo*r * 1:tt**1 cosza Ch*ng n**h r&ng ca4.1-i = I - s d6i udi Hhia 2 vd: cfis d&ng thrle tr€a v*i ess,ft, efie i = 7, 2,.,., &; df ag thftc dudi voi sin{r ta duqe Loi gidl: Gel P(.r,) s (a - c'6) (r * wt).., : + f to.ota la-!(o$i[ gs-1eind' sin2a'. I go ,ina = gr"-fsinfl * tit*'1*oto. cip2fl D+1 Pr(.d = x, '(.r ,'t) j roeose[ * gasiu* '- i*0, t+k * - 1ro-rro.o * Yn'raias)(l * sin2a) I u4l I coooto - Unrina - DE tLdy. F'(r) = E rr(c) vt t -- (*o-ruosc - yr*t sinaXl * rin2a) " k=0 ' Trty h8i dhs''t165 r,.o ta.ed . U+l [ *r eosa * Yi rlnt : j k:() o (*o.o*a * Yocina)(l + sin2a'o k+j I tu (lo | *o.oto - !/n cina = *'sin2a)o : [ = (rocoaa - ys oina)(1 n*I p"(.ri} = : pl(.r,;)l(ei -. rj) _ f tru eorer * Yrrsinatz * -i, k =o [-j_*io':o(r*rin2e]?o j +( I4 r=ts *.X - rr} * o k=0-Pi(clji{;u k+i | * lainttz(t - cfn2sj2' ' lr do dliEu kign eria dEu bii vri,i i = 1, !,..., r. * Trrl 3 {!*eg \hto trGn theo t€ tht cd - Do d6 *(o 1).p"{*) = (n + CXn+I}p(*) (l} {rjynoi$ft. sasP k .. SE deryg th6y r&rg clrl tba *gi duy :nh{t mgt ila tb*e b{e rl * ! v$i ,hS sd eao nh{.r blug I -- i r. aial2a[(t * ain:e]so -- (t * Bin:d)sal th&a aiEn (l). .,4 Mf,i kh*c tla thrir fre.i = (*tlo. p(l - ru) r ( rt t th6s rnilri phuong tinh (t). gi.r.) 16 6a thrlc b{.c n * 2-vCri hg *'{ c*o rhdt b&ng t. YSy , = +:l sinz2*i i (i=t .1i-t (rin 2oi2'*1i .J
, fn I l.o: gidt: D;&tr ki{r:,r'1*;:.il -r 1:r, s euyo a a* uio:" | : el'-l (sinicritrll .r +Eri+ l.n'2, .r*/.rr'3 Li--t :' = j"'*rgift:'r * ,t 'f;: r:d e ::lg ,i' a ! lgrtr + tgh.: I I cfli-'t istri :e roi I rd.p.,: rn) { Li=t J hay tga; tgl"i $ lgSc *olg$r * cotg3.r. tgrr.' : I ,\ ;tQn "rcl : e6* b6u g&i bil {Srr il&u xust pli6,t tr'f.iy iei s* tgEr: i- t6?3r * cotg33,t' = tg"r:tgg;rf iu r ig* iin:r h* sd i sa€ r:$.o : * tg*:r,ao-.g3* $ cnt;3.rt6.r it:n * 7,g,, = 1r(;r'* 1 1 l,yo.*r), ** ({g:r - tgt,r}e -{- (tg*S * cotg}.r}3 .f- "+ {cttg$.r -- ig.r}? '.' 6 .1{i:rrxg ii66r dd rn*l s6 bqu bi6n dbi c0n ri*;,. ,** tS.x :: tiqPr: ::* *ctgli.u. i{g phuong trirr}: **p S.F,K, vt', ughi$qi" Y6y phreng tritsIt *& chs v& nghigm. li&i 4i159 : fitirn ,a, l'; & *,n t;4da r:tr-;r /; i{o} : Nlt&tt c:\tj Ila e;5 **e. i:6n dFu 6i&i tlieo cA.eh "" s. 6gy vi *6 l&i gi&i drixg. Mgt sS bga di €6r: kdt ?.t lint f(.r) . A. l*i* eei: IE m rBng ; tg:r :i, r"gtr * cotgS.r = ij- x* ti tli6lig l;aa lta&ng Cirrg Bd,, /}rirm (triCT QuSe 3) * rr. "yr 6, n. i'(t) ,1ttt.2): 1r?..3 hqc ltud) rlang bdt d&ng tlr*c riE gi&i c6 bu.3eg Cfrii;rg *inh rdng fir) * 0, V(;') 6 n" ding atirrng thiln:-l&n. , 1$a ur.*,* t-di qidi: Trt li v*. t) suy ra Va}0 bd irlr j. 3E >0 sa.o eho c6 B*l 51152; sll16 [ aArr :180 cd cqnh bdna Brj. lftc)l{ a v*i icl { e. ' (u) 'I'r€tt crlnh AB ldtt itOt t!;itn E t:t[ch A ngt khoanq Ki hi&u T,l& a6 l0n nh5t thuge [9, +mj ttr6e ;:'.f ri E oE rt phiu tttluii:i lutrt giac nQt tla sttnyl mAB st;ng u6l ttti. ?'ia ndy ctlt cung chdn,buj -iB cfia r/rrcrri.q trdn ngttg.i tlilt {am gltic AItt ip n. Xa1
---.E &o (rr ilr) = ! r:f.n 16 {r'1. rt-l .r':iit* utt=t \,ii_l-x$r+-2*'2>4 _ (t) va rf i-,r'r,i + ui tl ss li'r>3, tri (t) co ilfrt knda (.i)'Z * 8i2;F0 sti (x? * ""rgt +'tfi ncr e6 '{s = rr} +e,1yr * gf a4n {xyz}s *l 16{xva}*o{>0 B&ng e&eh t!r& trqie- tidp e1 = I, t,3,"1.5.6 ta hay xzYsrabyutxyz *4)= 16(x"'l-1'+u +? d) c6 dlusr ? nghi$m.r,'t E I, gi xti vS it'l .=6.rII=1, Y&y xzl'sz: lta,a-x'$' I' + z - 2 (2) Thay vdo {3} tB. cS 2 nghlgn:r : T* {l) vi (2) $u}, ra SiBu'e5.n chring rr':i*h. $,,= t ia "-:* r.i I *r\ . Muds e(r d$u e ;s xth! ta ph&i e 6 -I*1' = Z A ? YE i g=t: t- { H=i": iirc l& a! p* s) - b/(p f)**c/(p-- c' x 2 *ay 'lrcxg e& hai tt'udeig.ljilp tri rl$u eS ij = 1? ?p : 3o :: 3& s 3c" 'fla e6 tam gi** dBu' NguEe :\lr{n rdt: C*.e ben ,l'girilin D*c Thinh, }}adrt Iai lihi d = & = c thi ts lqi eo*,{.,= !'* z:z At:th Trung,.s!ii ?Jlinl? r!fl, (10 CT BI{SP siah, do
Ui+v*-- ctsldri? )> ."{5'e * $1ips) o,r (4) esc S$r g;q,$So rs% "/-\a, l"rsi, C6d.l '* l&. e$* gde Jrir:r.€ o;&2111.1 ) r(SsB.1 * Snnc) {5} lqi itis5 .f.r erla tr? di$n LEtf)'* ed t&eg ir&:r{ cs.b3{/3,1.J } riSgc,l * .Ss6s)- ($l i8*s q411 B rli&,m -41, 'S, ,tg th&ng hAnS vfr tii (#') r'iflng rd y/ri vF 6 hdt 6frn3 ibritr tr€u ta dir,ge : 'euy r.*. /? id drlm giira ci.1 ln.t.l $I*t efiatrr tuoxg try, te. eiiru6 ed Av, D, .ls tbiiug irsug vA O iri J:rh (hl"f,?r) *.aa&x {he '+,rla) | esb6 {is t.d I lie> ) -tr (S*s* -t" $afis * $e ne + ,Srse) dlilr:* gi$'a *&n A* '{s; 6r. r), -1i i.}r;lng ii*ng va C l& d;&m gt&n iila "{c,tr.'I'iL cl*;r ei5 il;iag srrv ++ ri&t {frr *dr} * *a,}p t,ll*dri 't **irs{&s ?;!lloa 3e diBlr e*n ,: hrlug nrin!:' o'" l6 I' {:) r ,) SJilrr kl€n
a oudi iludog r{sy, OIr +bi sS cta agsl nbA durye Si t* ,{31 xan6 hai nh{ b0e eq.uh, NEu di ttrreo rdi v*ir.,{i, l* j2. Ci[ c& rs co ehi ed 7r r*t ibi ehibu t& dit theo 3 aEnh r'6f, to, ta d[r ltda /f+r lI ehi eS *fia egsi nh$. du+e ndi vdi ag8i mgt.nhi 4y'.nto 4td n,bgn Gi$tr. tr! 2 nh[ bSn nhi ,l;r*1 ' l{ *gdi rh& eB ehi ed eho shs.t clnh (trie ld 3 sdc to d{e d€u s6}. I}lEu n*iy m$u thulo vtti (e). (G*) t?cng t{t e& *te ng&l nh& & exsi Su&xg d*-v. Cu6i etiag la ph&t chl ra lA tr} nh& rr&y tlign rtiQa Hhu v$'y thl v6i m6i e6eh m*e *lign ta e6 ed tha t&i m6t $h-& ttty j,. Ch6 g' rdng hC rhdn6 &nqe rn6t bg {i* j2,..; Ja-ri v6i 1{ jr(n e&a ta ed n doaa d&y ddn. Ci&, sri r&ng ndu nd {a = i, ?,..-. n*t} duge rAe din}r mgt e&cb duy ehis tb&uh khu: vqe .r ye * ahd kh,&n$ rlr,rqe noi ah{t. ?* tb$y dE driag r*mg: . vdi nlr*u b&i elogn d&y aAo thi b*ug qu{ trieh + { il*nrio 1,, ,,,.., ni\l,ir" .r:,.^..Jn*r j 1 1 Sdm eAe ubd &. eu6i dlu*ng d&.y vA, lAs lrrdt r'[i b6 ie= t, n i\;1u, /a...", -iu*11 { chriug. tht ta thfly *& f ,lqug riAy & tr:nng m6i mrn I e, . , T u*) 6qrn
,, trang edc 96* A, B. 0 iue,a$ r1;tg uGt ccic cflnl: on r: rju -- .d 4a g ,i. LtC "i. C,, FlC, CA .t!+. Clti..tirt minh r'iiny. licu -{at: p& * :E * -1.111 + -P{. I C.JJ * .S/11 : Sf I l/ri laor giric ,{EC d}tt' .: il_iir +- 6 tI (*) Ltii. ylitti., {*&a Le yhi /{rrdng, }#p SA FTCS ,Nlr*xg d+ "SSr = &]t ::$a tr? t{n) ta t{l Se*g Yg, Thar'h H6x). Jl,U*.8:'r-{t=7r, {iqi .Y t& tlsp elibnr efi* v*r:g tr0:l h*rg ti$p irc*g 96r {-' v6i eqeh fJC. I{li d* ta o6: El,/ =CN {r'i eilng b}ng ,r}. Sr.r3 ri.: e',i r: CJ/' Nhrrng ;.'i.v'I-"-i- t'rir,3 .r:' * :rg -- o '-\"ic gJrr .r:'l:ih S/rc I f ,? + ,,9 -=.; &&i Sl! 5;i. {ilirj y,h**ug trirrh I JJ'lr'] I i(Ji Brt$I" ehe s+* :i;;;' I{:ri di}m s ehq.r. H&i Alxtxl - iX:].r1* 134 =' 1 ' 1/o (1) !r6n O.r, C alr6y trBn Oy s*a cho ehu vi AOA,: v*iS{r-'dl bfing ?rr eho tr,r6s. i',jr:rri:: lihit: Mi:ttt Xie Sinl: .8, C sa* r:ho diqe ti.:ir AOSC lSn uhf,t. &Ai 6,1155. rli*i phuang tri*lr . !)rirn I'dn Nhi iltg * t r - i =':-i'i.'l ?rrl/,t , C*e E&5r trYgffi &Ai ?/f 55.'liur t6,t ,'r.*. *Se h*m sS i, i}:8s m*u: f("')f(U) *{{rz * yz) v+l n:oi s,i thr,,c r. g tsdi 3/155. *qt So * 1 "{.' t/2 +" 1/"} *.,, + !,,'n. II0y cir*tg r!:;sl, rlng "i' r'd n X ttdn l)dnq 10
BAi &"r iSS" &'*i r's, hl4* ]r&rl h0;' n1198t ''i- oitlsg' '.d. clclog. l*- mh& nkdt {1-:eirr,r' * sinrr = sins o,+.eH,oc it uiasr: ..r *, ,elney '= .6ss3 Duo 1'rudn11 Ctung I i' sin'':t' sirr 4-g * eos?;( .'\"rutlAIt I L.r"'r :lJriu et] e*+ eanh birng nhau v& bEng -{C', 6$e tere B&i $,,!55" (:i* e& ta, 15, {u IA .e€,: duong di€n dinh ,4 Ifi d0u, phf,e gi6'e tr*ng'v& JI.la, rib, :::. l* ei.. dr&ng a) Tieh sS 6o'g,ir riae mtt sir* tarn dign tr".rxg tuy66. r r'& il tue*g r:ng I& b&il klair ciri d!*lt l. ' e6e ilu&ng tr&n uSi ti$p vi. egaqi tiFp, p lA airo b) l{gt il}t phiing e$"t eri* ,e&Eh dS, .,tS rA chu si s&e tem gi&e .CEC. Chtlng nrinlr r&ng: ,{.4'}Sn !,;;t tai tr,,\,,p t's cii .trl'& 0.Cl.rung t9 + rf + 13 { pa{J,a * ir;...i} (rrrg -1 n,[ .+ nrt urlah r$ n6 ,t liSQ " llAlf -!; 1l'1i\r {- 11,'1P vA dd,r ding tlirire u&y la khi rA chi lihi ,1 RC id , Phqm !)dng t:ong tam g'l{t r dlu. Ir. L*lienberger (ilo:ing Ilu.-" T*n s*ri lh::z) ffii,i X2ln55 'i'1'6r m.{i }&rr ;rgut}i ta 6du nrgt sei btnh trbn edt i:&* ki*h b&ug nhau sar che Bn; i0/155 Tln Olr !rcing ;4c rern ciif u d6rr !rh6ng eci bai hiqtr, rrlri nh*r gia
6t6,p. lL L6,p l$' l) GI&{ hQ 1,hu,eng triah: i) *lm v& lugn p&rtron$ trie}. .L iU*i "b{€a r*y iatlLd i 4x*{Bq+!}tx+oE*{E o, iu'"i-u+'r:6 {n':*c*:=1, g; Xee diuh e&* gi*" tr! eria a tl& phuwag trinh {"r[*l f (n * T)r * t = o ?? $i&i phunm6 lrinh e* drli*g ? *ghi$m c$. * nis*; .& d;;.". g "p {;"* y)* 3) T'im a$e khosng dBug bilia e6a hllrc s5: 3) flh*tl6 miah h&c6'd&ug {br}* ; * $} * Eloal * U * 3j{,.r * IS/iS,ein!.t * I,3l xia4r'. [e'c(a g $] - n:ncllsin{T lein{i +^ a) -'s:mfljlgll*(T - ai * ei*Fl' 4) Tfu& tte b eria *rgt l{ng trr tenl giAc €Bu 4) Ch@ t{r gf*a /rECo nQi tiep trotrg m$t SE&ng, b&rg 1'. FiEy xae d!*h d& &di c*s e4*h €{y l5ag lr*a"s& ngrgi tiep rn$t i!u&*g trirs khf,o. Ki' tru dB dign tteh to*n ptr&u efia i[ag trlr lt rho hi$u S l& diBa tleb e*a t* gi&o vA. ct. &, c, d l& uh{t" d$ dsl tee egnb erla t* Si*e {e. eh*e g mish: 5) Chs ha u[.t ehc €0i rn0t tisp rrie r&i alrsu 5 - lrahca vt erlng ti6p xue vrli mdt redt ph*.ng. Kh*iog 5) Cu* bixh lflp pbuang -4Bf,S.{r$efr101. L$y c&.h Sita c{e ti6p diBm c&a cAe m{t clu il6 i*i&m F tr6a ennir fl6t sao *he CIP x ?{.iP, XAe. vdi ra5t. plr&ug l& cr, b, c. I{*-y x5.c l$inir c&e b{n dinh s6* srua hai -,*r nl;&%"u:,r:r,iri;*. klnh ei* e6c r*$t r*u. $gugin Ehdc l{inh i1 jir:: i r.*. . li ,l ,.1: .$.,.:t.i lir ",,;i :l t!, ;lt . r: 'i.:,;;' :i -.; . ','.. 1i. tl i.* ,: *,:, , E*#& aS eir fiHAu Bdi I "' efto n PhL hlit rg I;'i nh k.ttortlrg * T.2*i - i ( i5 * :riix E {. ar &r* *lap*(a1 )tr)'-i.-to I*,uot or{ --l .fi : [1,, 1/9;5j, VAy trl hP tbue ti) te ed t*1,2,3,..,tn. G{ti l[i tddngthrcnt n, bat ,at :y c&a 0at ki o€c;{i.t phuong trinh*ng Ltati {ka lrunt:.sd ir. * ioGcr. *lltr \.t r winh trdng ns: Thco dinh lf d$o cria tam thrle bfle hal, n \,'t!J la ed: /n lx 11 q & ,-I \I v trir, .i? I+1 1t lr,!!*ari€o(i*ffi) a !r jt,IJfJf, n. il.r \t*' I + E rlt *4 X 'r;{o &ti't grfri{: isl i*! ffs q# | * / n Ii \, It', * (3ar)e *' +iai * 3&l '{' tri ',= 61.30, * !) > Q * { E .'i l/,
tsai 2: 6pi /i riti r !d, bdn klnh cric hinli e&u =' nlL' n- ftFl*aea=na/0eeror. a/i " eotg{r/n} ttgosi tiEB uri nSi titp hiuh chrip t9B6 giac db,u, * ncr?eotg(u I n) I lcaga. 1) Chtrpg minh rdng wrb | * licos{* liflil. S& duag dfrag thft* tLong {S) v& $eu rngr sS il Tinh rliQn lich :tung tluanh c&a hinh clzitp bi6s dbi ta iluqe : c6 cqnlz d(rg bdng a lihi & b&t d&ng thirc trdn = aaz{eer(*in) * l}i{etn(nlni. *riy ru ddu bdne. ,t"n Ldt gklt: Ta r€t bAi to{n , r"&i hirih th$p n SAi :t : Cho M{u) ttl n'tlt da tht?c bg.- n $do 916* dBu. Ktri n = 1$46 te tr& vil &&i to&n ph&i rho il($ * 2U u*i tJ : 1, 2,3,,.., n * L Tlm gta l. N(n*!), Bs hiah ehdp Sn cho !.r ddu nOn o6e {&r* O Lbt gidi: Nnqa 16[ r&ng: vi I efie hinh c&u ngoai ti€p v* n$i iidp ri&ul 1* -(r * i)'o , tr&u 6a&ng aeo SIJ e irs tria.h *it$p =.i, + *i. . "+"cX q&i m nguytr duong tr*trg d6 'Ch = nr(rn * 1)(rl *'t\.".kn: k * illlr! X€t de tir{re b6t n; ra P(y) = Z X {y*;)( 11 "rl ,.{y. i)iJ".l t,*-. D -.-U j :,*, -' I ., ,t , r(;i) = ?i ui*, + c|*r + ..-i- -f-li* rI Hhr* v{y hai €u th*e bge n: }I{g} va P(g) i", : -=;:li'.t" rrlug nhau tg.i n S i g*& trg ph{s t}i*t xr8r .J ,'i i 'i'\ '"' ;"''^r' i' P(g) re #(g) v8r ta c6: . '.i f '"'' ..as*_;*;i*=_"#"1 It: i: . .1t &:, ,lr(***) * *.{*i*, n *l*, +:..+e;+ii Ta cS r :r -tr{/ o {1.F', tg II|H = =.Ei! to+J * eiTii u-l-rl = an*t*:, 'd I' fr' Es i crtg..": !g _ (ti S&i 4.' f,to .&irtft t,udng ABCft tg.nh 2s. Trong 2nz rr'_*\ n$t phing t)udng g(x: r;6i ntfit ptr&ng ABCD qua voi $fff := lx v&" f"fl t{ phsa g.i&,"r gde }J.FS" lfi ta rlqng tam giac dbx ilMB. tr)t rlibm S chtlg ?a l4i e6: tr|n .48 cdch B inlt khaang SB * s. GOi P fu) ,j'--\ dt$q /dnh ch'i4u cita l{ lrdn,lt ud E, t} theo thir'ttr SIf e fi.{A . tg S.:-"ir ,- ld" cric rfiEnr yi8a cria i.B, CM. tsie(c'rn) {fas le b{a h.lnh drr&*g lr*r ngeql tigp n gi&c t-] Tiru qut tteh ctut P kht S c.h4y lr|n AE dDu) vA khi 0 niur irung hiah rhop lhi. 2) Tintr cric gi,i tri ctrt rlEL od. c{rr: Iila c&.a SA Sll: SO f Oli == i? * -V';.4? -;B = Lrii gkii.' 1) 'fheo diati If ba elu
X.) l& trung ttrydu etia liiin gi&.e d{Sfl *&a SU . ,90? = ltts;r? + ss?)'; Mur j/*. $!ung M,59 =.:r-a *da} *'46re og$Se ",s2,+4ot .-?G.L' ' sl? =.{o$+.s?: MC3 =. 1u1, 'i'he* b&*g. tr4e i*. th$.y r&*g iI gliar nigt 16 l. ,'v*.g So e6o sgi khi e =;,'-a,l*,; ,s$ gi$ lrj L;;. sd l, bu si 3,.. ii r.' .n i-& rr. = !n --./i1,, eg* d4i b&ug Fa. $Or'etge ttEu khi j&- = fi,";l i't': , Ta b5y ii;a &r. C&c ES hqilg e"3= ti €Bm ct6 gia tri cqe titu bA*g.r1i;,;1. *h&n6 *l:i:t_l e * gi& trj i6"p lei : B&i $: ?inr cric giti lri lr rrr'iiin etia n (n) l) {r), i: ri, i3 s *i j1:.:r, l* : U'i ). l.:*1 iEn lliin . n E*l Ta ph*i' tier eh*t:l i,- ,J ir,;ng nb6m c,J ho, i.du -tr--T thu * thi gie t,'i .fi* ri6 li. L i'g ,a.' r'4 i- .l I L.. 'J'a rnlr rlng pb&n ,. lal i*,i etfi* d$y lrnsle xhd-;rr * l&:ro$t sd i, l:i* s6 i,... j; -t e6 .t*1, r$i i + ?"j- I,i .. * L.*i;-,,* dtinyl utst mpi ,iri {i = l- 3, i},-',, rri = (l; - x tl'i sd hang . 'Ldt g;di: Y6i *rl w :r* *.:1..x; :ir*:3-r::: ! 1:$: .llh* vgy, ililu iA--t)irli :l*i < n i.Jia1; i i)//: I t re x; *. E{t **irg l&*e *h* ta: thi iril .- ,Ii. ?& gi$.i i,rq blit pb-r;+e< ariaL tyria ' ny*t*a,"F(n*ti.t €'b t;'rri .t ti;e* n *ghia ::.i tinb 11, iheo n. fi*y ra n,{ $. Ta vi6{.'bdt d&og. tiirr+ *l& rla ai ii;*:)Jeilr i I q n * ti* yr"*-;ll,e ',.1. { (r +" li il, ?il: Y€ 3r&i t$r m$t iq,rn th&e b$e,&ai #t{i i*l r.6. U) ii; i' + l),1 +' i ) 11 c* '* r * '!6Til1) ? 1= ir Tqm thi?e esy kh*rg 6rn vd"i i,Eel r& };hi v& ci,i (*, + yil::;rr < hod.u ,,,. khi bi$t sd A - (ri +,.. + ra-:)a * l{rl * . .trf;*u} €o Ftiru vil : t {"fan-;)l: < i < r1a {fr[];;3. -t ' hey.'a1*! + .,. + ***r) * (rr *. Suy ra i:: iif + frr,*z;lxi = r- {aEu i1..1 eUl ".+ r:,,*l;s (ti. phia n6u;;tn eia.r.). Apdqng i;{t r$&ng th*'e fl&oi * B*x}:i*rspxki.ehrs .r't! n2. .., r'n-tr bf,t ky vA t, j,.' . I(n I ltss! re:6 ; 1'9.5: : i:11 s ?:r * l(t + 1651-?,lii, n = i, 8,,, tn*rl(r? * :.. * ,f*r)-P (rt * ..- + *o*tlr ;.u i .Yiidt :r'if : Qi.ra vi&e eh4m edic bai th: eEa; eay vel 4'P n*1, t* {:} ra tAgy tuSt 6&*9 tla'e tt} ehrleg l0i th$y'tr:g! sS irAi ttnr r** bgc lg rr:gr Iu&n x$"y ra r&i txgi 11, rg,,,;r c**t. e,5 ttri€u e6t e*.B fih&e phge : Y$y c&e gi& tr! trg ahi€n erla ;i tl,,l: e:&a rl&u bait*n*?,3,d*5- 1. Xil si.i !r'uirt:l hc!t, inlng 'rgir: Khi gl&i blii o6 l,- e6 nhibu bei gi&i iih0xg xdl iru6ng hGlr Bdi 6: Tu ddil sd tlt rthiin 1, ?,3,".., ta tgSt i;,tr*: t; m$.i r'Bu rrgoai ti$p ninl uger&l birh etr+p. ddtT sd sou: .5d hqng thw tlk&t ld" sd lJ, cu th?,iti 3. XtLlrrg fit i;iiln ttioi hqn chcl qu! tic.[. f Lt sd 1; Hal sd hqng tiip theo ld cdc sd ehdn ! udt4 : du nliu & bAi 4;'x]:i&u li&i kh6xg gi'3i hq,u quf Ba sd t4ng ti!5t thea ld cdc sd ti 5, 7, 9' €tdn a*u d5:e dS:r. i*o;g, {*ng q*: sul l&rn trdiBg si h1ng tilp theo ld. cac.si, citfin 1(.), 12, 11, tti pf.rrhl An 'It}.s. .&drn sd hqng ti6p theo ti edc sd lA #t 19,2t.2&'; 3. 5& du*g e &ng li'ru* lih*tg this$s ph6*i 8i 25. T'im sd hang thft n tfic *'rTr7 sd /r'eri. e ut r.hxlrrg irislr toair pirB th$rg v& plr[, thfrng 1 si ,gdt.' eiruy&n le{an, re[ klr$*g eh**g raiab' Chimg ]r6n , Ta se x&e.h d&y eS ,t $& *ho srii ei$y" s$ "B v* 6*i khi gi&: bli 3 rn$l sd bAi gi&i Ss e& dr3ng t'{ing J? "* [ I ] fa OSy ad g&.tn hi{c nh&nn{ as hqr:g ru+ *6 thr"rr: I-:t*grF;r6 il-trgl'*ngti ln& i
ar;_:{-T tt r I A .tt f#u&,i\rff## ?A# ? ffi#H BAg Y&AF* flue Yffitr ffiffitr$ (,SfrAN KSryC Tffi"4,Nffi 1",fih6t fuibu bdi tsir$. * Q{o * t)} =p{4i,;1* ti':0{a)} (3) 'J(Qia) hgp: $i& sft eri 3 pirc fiSi 16p lhrrrm ebi$P {ehfrag Ta x{t trtrt}ng ? i:ga 2 ngu&i *h'nl rn$t tic, choi rr&o 66"' BSn r' 1/ Gi& *& p :5 q = li? {+:}:&r:S }::!in. kFai B bdru ,+ ,:d trrl luo'n,J c {,rg:;di, tiEt. f iB,.,.} B&e- h nglti.,a-, sfre) I(hi ht vti ttici ,t e6 16* lugng t. Si6 si'r & r,16: tr$n {usn' 1&rr e(a) * {1{1t* l}: Q{r: + 1) -- {{t} '"= a7 Silu, tr&n etaah,.".) kt:& usng tir$ng &.b&n .r tril p Kl:i {iS Q{a,} i4p tlr!:*h ;'fip sti r:irlg r;il e&sg e li il trl €p{ 1}, c*n k}r' ni:r6 thua iA E (p.-}" 'l *'1)' tihi tti*ag ir&* .{. d-sp* t$*g thi:ri,a (dwr7e liru&ng' eia) -:: Sr(ili ;- ar/ dreqe trb st,n; " "i lqrc luq*g el1a minh th&rr I d$r: Nh&(I)tac6rl'-fiaibl vi, khl thi:a b$l: A b! gi&in {5i phgt, bi mfit""} Y$,v : Q(rz) * tr * rrl(o :- 5) "= ,i {n'+ e} (4} lgc ltrp*g cila s',iah tr dsu i'i. 5*u rl{rt oi$ iB* 'ftcng iu iti dr&u kiq,n P(0) : *, P(tt'l- == 0.1 1 d$u eB t-lit xSr rn tinh hr:.Seg: hS& ,{ b! mflt tx tlrn drerp* k$t lge luarrg c e** txi*l:, irosr: h&a ;1 ttlu r3*g* P{c} :;: a,iicr * &) ttlAn b$ lq:c iugse a *'b vB phia urinti' 'linit hinh 6y duge gqi 1A sr thiit btLi deii v6i b6a .{ l'{hrr v{y ta iu0E *o .P(u} * {(*i = t hofle ibst bei d$i veri b$t B, Bii t*6n il[t' ra ti Gja sfr p {' q. #{t 1 '' ,i/p tr} ($) te 66 ln hfiy x&* 6iab kh& nEeg th{t b+i e&a }:E* '4 C(c * 1) * {{ai * rl(Q(r:) -* 8(c * li} 5 l& bao nbitu ? * =: .." E 7'a(Q(1) A{o}}" 2' M6 hinh te6n fugr: ciia b*i teiin: Lr{.r tbng 3 v€ thes a t& I ii6s tlx bgt k-j' ta e6 ${$t eh$t ei&m {het) ax6t pli{t t* g5': tpa 6* 'vil di d$ag ir€n m{t .ttho&ng h&u bqn' Klii dat 66n &iBm nn*t ciia kl'c&ng thi di d$tig eh$sa Qti) * r;{;:i -l li - ti.,il) ' tr(i}i't4} , ,*, -' i, dGt. Cir&ng irq,u gi& s'! hat di fifng ddn eS'e dudng rt,Ulg ,7 """- a h"y ,, =' b {o' b}0} tht fbtl ! a*rr(tj tri' (5i 1* e& .Jl A -- ':.--- 1,"t" -r! eh{rn dui. I}1 d{:eg .'l} n:& tdi & tr6n tus:eg *ng :- -,: - r*i di Si:;E€ ri':3 he.t x*6t 3"ihii tr rli&r:r ti = tr #iii l 1,","' r,'i c6 bi*rl 'tai e6e drEcr t,' = * r'*'.il = 'I + Ir' C*Si ,tr}n5; la *{r : Gei G(cr l*, gS &o irhi nlug aa$ xri{t th5i tt} r.3.* r,a*b diEm g e r:6at d6n du&ng ttrirr, g y '* G sctr l;
T- BA,ag eau &&y eho ta rnst sd gi& 'tri "g tbb * tSuy td. &{ag eu6i) lrhj eheii s$t €fi rhfi e ria Q'a) c& lue lugng manh (gi{u) ndu khfi ndng thdug & mSi tr{n v$a eeo thi klt* xsng tbdt bgi-v&a b€.. Cudi eri*g rq x€t trrrrlrg hf p gl6i hqa khi Lau ,L ehsi q*i ddi th* e giAu v6 han r (b qia) = ac] v* gi& ofr hhi {y ta vfli e6 p } C, (Len a *ir,ri :- cO ngh$ thu&t.hsnJ. 0,5 ,. E&ug cseh'gu'e gidri han khi lr *+ o" troat 0(a) v& ehri g r*ag h : qlp { i ta di'd6rr C(oi -* li. L', $ 50 50 6,5 ii,5 0,5 CA I Id 'I0 {i. v& k h& nlng thfi.ng cris. b$a .4 cinn d€r gi6t s,45 {},s5 4,6 CI.4 $r} 1S odl 8,ff0p hB aI /r", 0,CI1? Nhrr v$y btn .{ e6 }r.reiugag a lin cS khfl a[ug' th6*g kb6ng it tdrn nrg* aiu tai r.hrj *ri;r nr, g!tru r 6 .lrau, -#ruxg tesn boe cd nh&rg bAi ioea kb6 nli te *& rrh*.$ s6,t : ?tb b&xg tr6n t ti6ag he{,* ri a$ kh&rn phA dugc n:Qt quy hr{r -* ?rong tru$a6 hryp p = rl =- :i3 (hei bgn khLrh quen nAc d6 trrh& s&u o{e, hofle-vi a6 gi&i &Srag r$c) b$n uio eu lEe luqag y$n h*:t [I _ cJui'dt dugc nhiBu yAa dA e&a tbuc tiOn [b{ phoog phri. BAI to{n vE s sr; thdt b1l I trta {tfly e0 kh$. eBng th.tt b4i l6n hom. m6t ir$ng nhff.mg bSi lo{a tLu&c lo4i dS. C6 iha trinlr b*.-v erl irE i&i eil* u*e sgudi llnk bfiag lr€ng o*i : T'ri. cA* Si&i &rep bili Sdrig ,..i;s'qiflii a l:i;rr,#i F{i.. f *l: FEdi*; I,{6; dS; G&t LAe -b6xl.{gu&i rri r& }t&i agu&i lixh bdr kj, rri$t treeg r&u h6t **u ed-xS* djnh €uq* *rla n6rr*! itrb Suryc h6i clang g&* l* e &e gi: c&r . r-6u I : i{gu&i n6i th{t g&c r,i!e s6eg irh&i kh6rg ? Lac Ger C&u 3: Sgu&i B6i d6i g*e cr&a f h$r phai kh6ng i, C*u S.: ligtr*i e*i th$.r g*e e&a oh$i gh{i - kh6n6 ? tdi gl&t ir6n {&.y t& l&i gi&i th6t}g tl:u*ag. -, C{u 4: I$g*$i lt$i d$i g{re '*ire *dug ph&l Bgn ltcr:.uh !dinh {.Ae'S qugn ) tirSnh phd Hfi Chi kh&ng ? Mirh c$n dre I&i gi&i r&u kr.i hs;.r : usu&i tri tli (Treug d,i u&u I sd e&u ! tr.rong dw*lrg vrii nhqu, 'h6i mgt tror:g hei *g,:rri lisb e&u ir6i: s F{dt tdi s&,u 3 v*, *S* { turr::g S*o*g v*i nh-sti). h6i arrh iinh gA* c{ra ben kia eii'n elh 6,r e&e ld. DE rr& Idi c&.u t ho{c cltu ?. riguJi ilnh g** *ue edng & tiri x.rih 5v *6 ggt dBu ph€i khOng ? ) o&dr cdn6 bao gi& cfiug 6flt, agu*"i iiah c6e e r.r.a ho{c i rfi.u hfri tuseg tlf nbrr v*y blng e{eh efet.|ao.g.id e&og tic; vi vgy rh6y og,r?, rlni, thay tll < sdng ) b[r:g tri ( ehdt )] lrr]&€ ril fi gAt )] gflt *eu thl agusi rir tii y&n tt* di rI* l$i crla blng tfr E l6e )). {t6, tb6y l$.o ti&u tbi agu&i tir i,i prrni elo hga tr6 ruy lu{n s€ thf;y [St giAi e &a ban lg"qi_Iinhg6i ra l6i efra k1e, vili efi.u S h*4e r:{u { r}:l Lan *&ag ii' iiri giai du,g' rguqe lqi. .p.(1.G. .{ ' g&&trg ffii{$ t 8qi rfuu ti sl + nam 'lgii tro .JL gid ni6t :;6 btirs T'odrt hpc t)i tubL tti td ld Adiry I n l3S8J **du 1'4i Xi nghiSp {a 7$ $*Irlg t!& * *8& }i$i; t-kl *d: !t 884 SS in I lSls?, Iil xr:ng r& grii lr*u ohitsu tblinS ?=1587. Gi6, g,iltlO