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l' +r 't tn$ H*r 'roAfi fItlC ylEr xru 8ACI ni:ifni, nOUYtX cimm Trr,r r&: 7{) 'Irlrr IIung tlgo Chri ,H# B$n;G RA ,1, rn,{Nc MOr xy roln yho kg tdta loan Hra N6i HoANc cn0nc [)5y rror : 52825 8.t-{Ar NIFM OUGNG TRON CLE YA SLTSNS THANC OLI CHO DA GIAC NOI rfiP l I t I I i Cle ban thil mdn I ' f inh chflt .1 : C6c ilulng trdn co tArn lin lugt Kh{i nignr du0ng trr)n C_Ile vA.itudng lhf,ng l*- tl t, llz, H3, vi bin ktnlr b{lg bin kinh - dA etugc xAy drng d6i Ote cfia itudog trr\n ngo4i ti6p cirer tunr-giec..{rA.l,le 'roi tarn giue. Droi sE gdc tlQ cfr* verto, & bAi ndy t6i xin gi&i thi6u clt rrhau lai m6t fii$m chung H chi.h IA m0t cdeh nr,l rdng cA,: khAi ni0m niy d6i v6i tr-gc tflm cria ta nr gilc AtAzAi. da gi{c nOi ti6p. Tru6c h6t ta hiy nhlu lri (Ctc tran hfly ttr chring mintr ba tiull ch{t trCn) lrudng hgp iam gi6c ir g6c ilQ ndy. Biy gio ggi E tAdilm Bt0a cua OH, thd tbi D6i vdi tarn gi{c -lr^4e.{s, t.{c barr d& dang shung minh iluocrngay hai tinh ch{t sau tliyl 0- = (tl?) (Otr+ 0Az + OAil (t) 'linh ch?it 1; EiE:r ki€n c&n vd drl dB r', lf CAc bq.n de bi6t rludng trdn tim E bin kinh trong tAm eria tam giAc :ltdzCs li : t hfrng ltia bAn kinh cria clu&ng trdn ngoai ti6p +-.-4 trm gi6c rl1.{rds di qua cAc trur,g di}m Mr, }Is, rlitl = (l/3i(o.{r + oAz * ii-13) il} 'i113 cria cic eanh .,12.43, ,.11.131 At.4z ; cAc tr,rng di6m vdi S la {6m d*dng trdn sgoai ti6p tam gi&e (.r, Cz, /i3 crle cAc doao t'hing l/.tr, uaz) tt .lt; 1tt.{?As. c{c ehla ilrrdng vu6og g6c Br, Bz, BJ n+ tu Tinh ehdt 2: $i6u"kien ciin v{ tlrl i[€ J{ tt ele dinb dr, .1t, ,{s xudng cic larih a6i A;en. During trdn ni.r. duoc gpi Id drbng trin g dilm, trr:e lAm cria tam gi{c .{t;{rrlt l$ : .< lray cdn ggi li dubng trdn Ote Ifra tarn gi{c + " I Olt*OAr*0Az*0.{g, r&i 0 l*m tAm iludng trdn ngoli lilp eia gi{c.41r13d3. (2) tam ra ndu gcri //r. ltt, Ita l6n lugt l& c{e -r\.oAi tli0m il6i xrlng cria t) qLra rlrAl, .ls.{.r, .4r,{3 thi ta s€ c6 tinh ehdt: .{til2.l s. Ngo&i ra, do (r), (Z); (B) n€n 4 (li0m o, M, H, E th&ng hlng vA ilrrdng th&ng ili gua 4 rtigrn nny ggt lA iludng thing Ole crla taur giitc AtAzAs. Btroe ([Au ta thii m& rOng cjte khAr ni{nr tr6n In cho m$t tu girc nQi ti{p. - --/ r Gi& sr} I a cd ttt gifc nQi i i 6p '41 A2 As At' tludng trdn S c6 t6.rn O, li6n kinir rT lA dutrng tr&n ngoai ti6P tti' glrio niY' Biug nr$t cielt rri-'in tuorrg tLr circ tliir:g th(c (i), (2), (3)' ta dinh nghia: 't ngl,io : Tr u c t fr nr tt c irr iri gidrc {t Ar .13 Aa nOi ti6p troug iltrong trt'rti S. tem Lt, bun kinir Ii 1A diBm lhila mdn:d 4 4 € @ (2') O lt :0Ar * OAz + OAs i- tlA+ Oinh nghia 2: Trong t0m '\y' cirrr..tl- Cie" S' tAr;r O' Ar,qf es.la"noi ti6p tlong tlutrng tron mdn: ttr6a di6m ld ll Lin kinh € 4 s 4 € (1') + OL4i (oa\+ ij|, tiat + ail:(tl4) Dinh nghia na-v hohn loln phri lro'p v6i khti niQm"trqng t6'ar nra ta bi6t tir trutlc ddn rr*',' (c{c bqn trr ehirpg rntnh)' gif,c Bitth ngltLtt 3; Ndu ff Ii trrc t6rn eira tii O' Irezasan"oOi ti6p trong ilulng tron-'S'.tini li irung tim E v6i bln kr,rli /? thi duong trdn ililrn ctia OH, Lrh4 kinh tl,'2 iluoc goi lb' dudng Nhu trdn Ole cria tu gi6e nOi ti6p '4t rlz ;1g ;1+' mrjt^fluong trdn riinr v iv 4 diErl O, /,l: E, fJ sE -Ei rth ;il?r;,;;;,,g',r.,ing tln:" sqi li drrdrns th8.ns ak c&a tu giiic n6i tidP 1r '4xA3 Aa' . BAy gid ta hd5', I gihi0n cilu mOt viri tinh clrdt ,'i,, clc khai niQm vira ni"'r r$ng dcii r'#i tir gihc n0i ti6P .At.4zA3Aa. ln Tinh chdt 3': Gr-,i tt1, t{2. tt3, Ha lin luqt Ag'{a'4i' AzAlAq' tam c6c citn $iic tarn rrt," - tiiur lA .lnir.fr. ,1142Al; th6 thi cSc duong tlr)n (bin it,-Hz, H3, ll 4 co cr\ng bin kinh fi 1:rihintr nr{'1' J,roni t.an nsoni ti6p) sE t'il lha.u gifrc' tr1 rira diim tr/ cirinh it iruc t$m mitth:. 'la c6 + + ,..@ - Chtl'nq +r 1,.+ t+ {.}'1 ? + {i-i3 ' t)14\ l(O,Itl Otla\= fIfi4:lotl. : -- ffi''r-fffi: & + r ,+ rr*)i-ltl,lai -R. : li i?1 - llLlz * IiIJs n Ii' V{y ,t d ,{ = u2 * b*c +d, B:o* bz + c -l d, L -r.r* b.yc2 +d, D=:6* }i- c*d2 li c{c s6 chlnh phrrtrrrg, u)> .Tr! cliBu kign (*) ta c6 ,2 ( .r ( u2 +'ro ( (a t z)2. v4y r{ :ot + t * c * d:(a * "t)z ha.v d-(0*e+d-1112. Tir (1) vi {'t) suy ra a (30/:, va tr { g < ia,/2+ bz + ztt : bz + ?blz<.rD + \,*y I : a *b2 * c*ri ={} + l)2. rnnp444, Suy ra "4. = lOa - 2 hodr: .4 * 1{.}n : Z. Xdt hai tnrong hqp: l) .,1 =-10n J* I. I{hi ci6: ,{2 = 100a2 -1-{0a + 4, Suv ra .lLla *,tr ==- {,1 (p1od I00) .+ {0 (a * t):-(l (rrrod I00) "+ 2 to - 1):= g (rnr.rd5) =* c :5b -+-1. VAy : d : 10a * 2 = 50b -l 12. Suy ra ,{P:250062+ 12000 + 144 j= 4i14 {mod 1€00) + r00b (So +Z): 300 (mod 10{.}0) + b r5b 12) ,= B (mod 10). l)o clti D phrii ln sn 16. Da t b = 21,. + I, klri tlo (1.['+l) (10k+7):.3 (rnod l0)+4/rf Z:3 (mod I0) ".) 4k.= 6 (mod l0) -+ t : 4 (mod 10) ho{c 1t:- I (nrod l0). Ci hai lruring hop d6u din d6n b:2k* t =9 (mod 10), tri'c b cri dang D: l0c'- I. Suy ra d, = 60b * 12 : 500c * 38. Ta c6 t05(a2 = (500c - 38)2 ( to6 + t00VIo . {soOc - 3s < 103 "+ c = I ho+c c::,t.. { Khi do dr - JitB' = vn Trrrdc h6t ta cd v:ii nhin xdt: t) Sd chinh phuong c6 ttn'eung li i,$ s0 cd chri sd hrlng chuc la s6 chAn. ThAt v$y. sd c:hinh irlruulg 5v si' cri dang: (10a -l- t )2 : t00a2,-! 20a -t I , (lkz ;[3)z =100a2 :i:6Oa +{} ; 6- 0, i,2,... '1'* hai d&ng thuc nA-r suv'ra dpcrn 2) D6 'drtng lhf,y ri.ng s6 chinh phuong c6 t{n eing lir o sE c6 clrir sei hAng chuc li 2. 3) 56 chinh phunng c6 tAn er)ng li 6, sE cd ehir sd hing chuc tn sd 16. ThAt v{y, s6 chinh phunrrg dv e6 dang: (ttla:L 4)2 : l00al -}' 30a * td: a:0, 1,2.... 'l'u dSr tld dAng syy ra rlpcnl. 'l'r& Iai bli toln : Gi6 sfr sd ph&i rim, vj6t du6i dangrlhAp phfin la ,{z =;[};r o,. n,},..r t ) tr, t. 2.....9 I vir rn r * {t. \ri Az li s6 chinh phuor;g vi do r =7- 0 nen r ehi cir th0 nhan cic gi5. .tri, I,4. b.6,0. Lai do 3 .rrhAn -rdt tr6n rrdrr {:l chi c6 r - {. I)o vflr' sr1 irhii lilir cc drrro. .42 'la cd 105 (.42 = (S00c * 3s)z < 106 .+ loo/;l { iooc + 3s < to3 ,+ c ,,- 1. (lt 2'l2. (:i iray b: (u * c * d -- 1),2 lru (l) clrtr (2) r'E voi vd ta tlLrqu (3) o=0vAa=c*d--l .Vdn dB dlt ra ld tinr c.,.{c sd tqr nhidrr a, e, d l,t)?c >d) dA cac s6 o:2o *cz* d. D:2u* . +l rtl ia "ninn plruong. X6t hai lrudng hop: l)d=1. Ttt (r) ta c6 a=c vi do dl6 C= * t)2 lA chinh phtrong. -Db In sd chinh plruong l-hi D==lto-{- t - (3kt l)2. Khi ilo o:b:c:If'2:LZ. 2).i > 1. Tri (3r ta c6 c ( a (2c. r'rr (c * 1)? *zc +-cz + l( () < 4c + c2 +c ( rc + t)2. yA]' c = {c1 2)2,trlc ln2a } ca I d =(c 4 2)2 lrrry (a 2a*d=4e*4' ("1 ) Gifii he(iB), (4) ta ' ctugc e:itdl"L - i, ' Iu day la thfly ;=;:'; o = 5m - { } 1=td =.-!m. Suv mb 1. "-+3> D : 2o * c -]- d?-=4mz +Jnt -- 11>> (2m'a' 2)! tdo m > rt, vL. 4mz * l3m - tt ( (2m + +.r2. -it Ve.v chi c6 tlrF: a) I]= 4*2 + IBm-- tt:(z'm +2)?(+m =8. 4 = 6=5m- 4 : II, c=Sm * S =6, rJ = 2m:6. h) D - (2nr -t- 3)2 +* m = 20. a = D = 5m - 4:96, c = l}n - B = 17. rt - Zm i Vidc linh toarr sd tr& n6n nhe nhang vi ilon giin hon rrhi6u. n6u dE { rdng. [(r + rtz)Bl - |* * r/:)3J : *'- { 3lr2 n6u {.- i (mod 4) I i=t .2,8 I r*' * r n6u r.: o (mod 4) Khi d6 ta c6 ngn-t Ir s - X. (g*?rr.) + t/{ -F llti -+- rlt,J + 2g/tl a-r r -i 1t E k+tl4+ =$ k:1 t/s+l/l?+2&/13 =,10. 'l'6m lni cAc sfi tu nhi6n tl.nr rlucrc lA : rl=l)=c:3Jt1 +2k (t. I{ s6 tu,,tien), rl:1; 11 = g: i = 3*1 - 2A (& I* sd trl nhi6n), rl = I I a:b= II. c:d=6: (1 Lb!-!idi_ (cria Ngd Dug Ninh Qu-v Nhorr). - b? Trtn phri, Do Vn li nrgt sd tr nhiCrr ho6c mOt s6 r,6 tj. n6n crieh x{e dinh oo c{ri bAi ra ll dur nhdt.l Ta nhAn x6t moin6 tu nhi&n n th6erndn,** ';,;;.'ilu t!z)3 ae"u so.(k f t!z)" ve & _ rlzt" t.6"Jri-r. ci ta (t o,u, + i/zrg (L' n,)(r. - ll2\' sd ttr nhi€n. r'a lai..cd (tS + l/il3 ) rlisi ) ,.. ) lgr4 > . tr!)ts:12 ( 13 + l/:)3) t95:t, rrJn ,$rr ,+ lrt + r,zts] _ ltt .* trys ttlul' t_. * L - "_ i:I k:l A. + (rsrr t, * - 3 + l2l2 | 2713 + 4gt4 * .-r 195{-t- 1)it3 ji,i + t0st-6 + 14717 + lgriS + :43i9 + 300110 + 363irl + 43$/t2 + 28/13 r IL * {ir{) (1+ 1 ! + tl3) + =J --E arl: ,nr I 28/tB - 236-l!L 312 rcdt,:-Cteh --.S.ed" Ninh g{p phli giii tr€n cria ban Ng6 Duy rinh to{n }h{ c6ui tfntr, mOr sg 312 N.K.t. .Chnng mtnh rdng u6i moi ctich tV nhittr ddtt tient thgnh ra hai tQp hgp cirt lu6n lu6n c6 mgt tSp hyp coh chnq ba sd llp thanh mgt cdp s6 cgng. l.di gilti (cria .Vgd Duu Ninil. Gii ugrrqc lqi : t6n tai m6t cAch ehia tip - chosr"r di thanh hai t{p con L vA B sao cho trong { cfrng nhu trong B kh6ng c6 3 o6 ndc, Iep thtrrh mQt c{p s6 c6ng. xct 3 c{p s6 .c6ng (1, s,trt; {3, +.5) va (3,5, 7), 1'a th{-v cdp sd i3,b) klrd,ng thE nAm tror)g I tap hqp, vi n6u ngugc lai thi cde s6 1,4,7 iap thAnh rnOt c(p s6 c6ng vi crlng thuOc m6t, t$p hop, m0l lhu6n vdi gii,thi6t. Trrrrng tu, .ret edc cdp s6 cgng (S. S, Z) ; (5,6,7) vA (5, 7.9), Q,4.6) I (4, S, 6) vA (4, 6,8) ta c6 : "c{p (q,7) cting nhu crp (+,ot ttrong th} nim chialQp hop 1,2. ...9 (gbm g sd 3 tl.2),3.> I9,1 Bet 4lll-9. =b=96, c=51. ri=40. BAi 3iIl9. fry hiQu ao tds6 nguq€n san li nhdt. ildg tinh tbng S -- tlst * tlaz + ... + t/arsar. _ = :ao , trorrg mdt tAp hop. .Do ctrp (3,5) ctng rrhu ctrp (5, ?) khOng th& ndrn trong mOt ttp hop ndn B,T crtng thu6e nr6t tAp hgp vA 5 thudc tAp hop kia. Kh6ng miit t6rrg qu6t, gii s,i I,7 G i{ va b G R. Xet hai trudng hcp: 1)4t,.{. Vi 3, 4 G C fi6n 2 G A. Vi 4.7 C/ n6n 1 C B. \ri 2,8GB n0n S €,4. Vi Z,S t ri ndr. I G B. Nhung khi d6 1,5,9 (:E vd IripthAnh mgt cf,p s6 cdng mAu thudrr v0i gii rhi6t. 2) 4 c a, khi .td 6 c d. vi J;6 G rl r,ofi g L" E, Vi 5,0 € A,ir€n t c r4. \ri l,g q A rr€n t € B; vA vi 2,5 G B l$n I G .4. Nhrrdg khi d6 ti,7, 8 C A vA IAp th{nh rnQt c{p s6 eQng, mAu thudn vcyi giA thtdt. C[c mtu thu6u tr*n chAng t6 kdi lu{n crla bai toin Ii t*ing: N.K.M. _ ={ L Bdi 5/1L9. 'trong ddq tdc sd ch[nh,phaong r?. :2. :12,...,rr2.... hd.u ltiu ra tiit cit cdc h0 ba s6. sno eho rndi ttS ba .sd tiu Idp thrintt ntQt cdp s?i c|nE. Lot gr&i. Br'ri to{rr trrorrg llrrrrng vcri vi€c tinr t{t ci cAcnglii€rn rgrrvt\n dutrng (r: g; :, (-* ( U (:) cria lrhu'on;,. trirrh = 21Jz Di hni6n ta clri vi6c tim c{c nghiQm nguyCn drrorrg (:r: i/ i. :) th6a rrrirr th6.m didrr kiQn phrr : r, y, : ngu-v0n t6 cilng nhau, 'tuc li (2) (r, r7. ;; = i. rri (i) suv ra rlng r2 * z2 phii li mot sd chin, nhu viy :r.: phiii d6nrr thoi ch.in hay 16. lihurrg nd,r r',: abng tloi clr[n, thi ,l + nz 2 ln h6t cho 4, do dd theo (l), y ch&n, khi d6 "hio mQt udc chu ng crla c, g, z trai vf i (ll. l'hAnh thti vdi gi'i tbi{t (Z), ta co : r,: IA nhfrng s6 16. Khi d6 ndrr p ld rn$t s6 ngur:6n 16, ur:c crjr c va : t.hi p 16, vA lheo (l). p la u0c cfia g'. r"&v .p Ii nr6t urlc cira g., do d6 r,(i ld mQt udc chr rig cria r. v, :. 'f irirh thfr voi giA lhi€t (!), ta srry ra ring : r, z lir nh&ng sd 16, ngtry6rr l6 eLing nhau. Tu ket qui n*1' ta thd-r. r[ng r * z vl r - z It nlrring s6 chEn, do d6 cd thE vi€t (1) du0i (1) ,r? + ,2 d+n H [t* + ,llzl' + l(z *a;]lzJt har' cdn n6u chou u = i - q2, u'=2p,4" thi di zpq-,r' ' -' \**P, z = *utt t" $ l2:1,ri2pq_ qz D& -;,' (:)- r{ng ctic bibrr thftg cria .r trong (,1) va chi hh6c rrhau b6i ddu, mA tn phii trong (5) ) 0, thauh thri ta di ddn kEt lus.n : nghiSrn ngul'6n duong tbng quAl cria ptruong trirh (t), th6a mrlri diFu ki0n (2), dugc cho b&i chr,rrr :u l.=l(q+q)2-zpzt I o= r'* q' {.-(p+s)2-2q2. voi p, g lA hai s6 drrong triy j', th6a rniri di€rr kiQir: ip, g) = I rd p) C. 'l rong sd cic l.rqn giri hAi giii. kh6rg bai: r:ao gi&i t{ring bAi n}v. }-, K. \I, BAi 6r'119. o, b. c, rt ki bin sd d,tonr; * b = c * d. &dt phaong lrinh nhau, odl a l;;r /;17 + 1'. -J + l;rbt: hav =s', .t t (3) rf*tn=!l' *vdi u: (z+r)12, o = \z r)r2 li nhtng sd nguJ€n tirrong. N6u a lA u0c chung l6n nhd.t cira tr vi u, thi 2d ln m6t uoc chuttg cia .: * r va ; - .r. drr tt0 sau cIAv : cii t.hi c,-ri o(c(rj{b + b? * hk + til + cd),0 + cd) b(c-i-rJ-D), t*c cd) bo. Bir'lt pi111',,t;'1 i,r,i vd t:iiirrl). chi; r' a2,-2 to -- tc2,.t, f d l:etcd * abi di itdq phur:lg Irir,lr ed - ab+ ) : VG+.4 (.'+;t l/fl+lx 'l'idp tric }:inir pirurrng- hai r'6 vd udc luoc sd hqrrg *6rg d p.1:6 -s0 cli d€rr {rr -- .1 c6c : f ill[fl,:irFi P)ltiir'::1 iri:;lr chi e6 irghi6rrr rr ( ab; binh piiucrnl Ir:.i ti 1ae(roc ltror'sd di d6n (a--blz.r':0 +)l':0' SAi ? li}-S Cjro n sd drro'ng 3:t, t:2,..., Chax1l minh bltt rLl.ng :tL. th{rc BAi 8/119. Cho m|t dudng ti'dn. Gei Sn tti|n tich da 11iac tlbu, m canh n$i tilp tiudng ti,(to Chnng minlL rdng ta ludn cit 2Sz, s*: n*i r;13l- 'rt:t {.:,. "r:; nla*::n*i ' rr ',,., i 1; 1 .r'l .r'l i-1i"',1';r; r+1, i:i'i'i) , ,,,j,1, _,.-,.-_* -ir,r'il nr-t , ' Xel- IrArn f $ ? (ml2) 5in (t iltlm) s6 (d={r/Z} sin i2:rl r)' voi r "} )f {2.ri } /r.r t 'r- /i !' (l) t' Ta c6: (3(/2r)' /(2r) = 3c sin $t12il cos \ f{4r) = 2;r sin (s(l2r)' rninh tro thinh (1) t'irlrrg tlrr thirc BAt dilrg (3[12't) 4,r: sin (ttlZr) cos g65 (r{/r) } 2a'sin &l\d cos (fl/2'r) * 2r sin (XlZ*l ) ct':s {fl12:t:} cc's (3(/r) * (3t/2r) X *=) 2 cos (stl2.rr ) cos Ir - 2si;r2 (sti2r)] +1 (i cos 'tt( l2.x) > 1 - 2si n2 (rrl2r; s63 (3{/2n) (3il2c) r+ I - 2sinz (rtl1il> 1 - sin (c(/s) sin (3rl2.r). t I ) €) 2siI12 (tt llx) ( sin (3rlr) sin i+ 2 1 cos (rc/2r) \Ii sin (:[l2r) = 2sitt $t i4J, cos tgt/{r) y'll.sintrtl+') = ,]^,,., (:r/arl. {I"n* (fi1ail> ('6i .r') :r tlrl /i"n' (g(/1r') )-l)' vd sin (.riir) }--sin (3{l2r)}1/f sin txlj.r}' (1) tltl:t: n0n (1') duoc. rrghie nr t1{urg vs tlo <16 chting minh. nT'l ' nf} '" ItIn ' Ir;a-I 7 t"+ 1)n r1.:,3 ... crr'nn r1"rz .,. .tr, == (1 -l- !i 11)n # {1*:rrlri'*i {1"i-r2lna:s) ..' (1 * ro-1r713,o) (1 *rolrcli ? (t + tl nl". $' (:t/r) cos i;rl:d / (.r'): (,r/2) si,, (2fi/el) = r sitt :2.r sin {:f,/2o-) r:os {3(/2r) *us i:(/'r)' 'I'ha-y-. .u &.rl-ri-irr t* '10 'i'a,'h.riirg minll ___ ;- \.it'); i) 1'r'j r, Nirirr iril cir r:hn l-:g.t tl.r,'i 1!rric 11.;n" r-€! v6i r,6, r,:i ehir rir L;ri r'6 cira tliLrg ihir,: nlrAn dutlc cho l'l l'i;1:?.'. a!-e, L& dir(.rc (ri.i:.;-j-:u) {ri13 i-;r2} .." (;ra.. *:rn-tI (n:r r -l,ro) s& (bd.rrg t6ng rtiQn tich m tltn giArr cirn co c4nh ir* iarg-r vir gor: & dinlr i:;'rrg llt/ml' I f rr f ne,zi i l -! rz i nrr3)...( I i- so-tll:ro)( tr * r*"/nrr ) ) rt t tlnt", Ldi gtiti (cri.r i;hi6u i;aii)' Cfic sd rt:i li\ a:r'i (i:1. 2..... a) {Iu la t6<: sd dr-ro'ng,'6p rlur;g lrdt ,Jarrg ihuc (.dsi, t:r c6: ' l Liii (tiitt. I(hdng nr{t tbng quSt' ' ta gii' duong.irdn d[ cho co hin kinh b&ng- 1'.Khi aien lich da giSc clEu m canh nOi tidp ln ( IIr'1 -r ,.'l } S,-t'1 Sartz(n2 4)' : :tszo ha a} (1) ta dtrac' "'6i )> 51 i- 54n l'? '"-ac 4: I' S2r- .n->5+n* Szn 'f ilh i!ao hirm bAc hai cira f(.u) ta dury* ( f" {rj: ': t2ttrl,t3) sin (2ltir) ". Y{S /(r) lt him sd 16m' suy ra i ,(ot*' )=*l/t,* l^i *! l .-. ,'\( '':+)-11,,, )stt.,r-11Jj},p \ (') )-rr " (Xenr bii , sAng . tao b:ii t ,-l inOt ;rhuong phAp chuug ntinh vt rlirs ttrrir,) -'ttH1rit s6 Linr lggl. g..dtr li-1- q1 :.r, = l/2, d.du brit dfrng lhrrc p11g vdi hin'r Idm trii chi6u vrii cif,u ring r"ii hnm lili). Bdr- a gio ta chrlug nrinlr ) Slr+a * ,!*o (o*) vi Sp:7"(m,r n€rr trr (2) suy ra Sn * S,r-t )> Sr+r S, > Sn*r * .!o+r ) ... ) >S?" - Szr--l =+ ,!-,r - Sn-l ) {,Sin ,- Sn),/n. (*) v* ' S4o+z* Sao (.\^cn - .S+,-z ( .}+n-g - Sro-t < "' So _i * .S*n+2. B{i ding tirii,c n6u trong tl}u i:Ai duoc chtng QI'. . l'ini qpS t{ch di?m gi{Iu cfio pe. 'I'ir .Ldi gidi. gii thidi Xp = ey {:r'dn trrrng .lr:uc ci'ra },Q. Nhrr gi:ic c:in. /'\ rMO t16 bdn di8nr It. L p,(l rrinr tr6n n:0t rfudng trolr. Stry ra OI LPQ, vey 1 Ii. ditm giiia eira /,e. .T'e dn chring nrinh dltroe ilibnr giila 1 eria pe n&nr tr6n &t.V. De ddng thdy .ing 7 ehi ninr tr6n tt3trl -iis trdrr f/rV gicri hrrn b&i,1*i ilrrdng ch6o cri:r lr in lr t lr i rr g (lr . 2r. . Pl:-dl drio: Lri.r 1' tr6n IlS. Ndu I' ln diBni gita eriir ltS thi -{Al. .yy chria ,IN, p rrung r,6i }/, e trring voi N, sd c6 l-p = qy. N6u I' k16ng phii Id di6rn gi&a r,iia /1.!, ta drng O1.. a",,g =if .l aI'cii"t AD b p vi c6t Bc b Q. ca" t,i giai Net,o vn $Q/'O n6i ti6p duoc, suy ra ?"..._ ^.+ ---\\ N NO * ep{), pe0 = }J.\ u 0P{) vl rlo &+ = f # |, A Ope cin *t Xp = ey. tJO .. :1]U, r[ng trong trLrcrrrg hcrp rlrrdng rrung Lrinh ,&f .Y c0a hinh thang khdng eit .XI."tirc le 9lll9. CIto hinh lhanrs ed.n ABCD(,1tr= tidp trong rtrong trdn ldm A. IIAt jau Xf ?-C\ "0: thla d'6i c&t dD tat p, BC tsi e, sao cio Xp + *| -\\ QPO = pe} minlr. BAi Hei tam gidc vu6ne O.Up \.A {i.tIQ b{ng nha vi 0M=(),V, lp:Oe, \,ar i&=6X. .**\ T* dd : P0t1 = tr49y. . . Hai tanr giae cdn OM,l'ri t)pe t6 goe 6 diirh blng nhaq n6n dbng clang. yav tr,rorrg hqp XyllMN thi trung- dr6n.r cia Be nirn trln truc d6i xung IiI crlalinh thang. , . = su-v ra O rrim tarn vay fl.eq lA ."D1rng-dudng trung binh Miy cria hinh thang, g.ii sri -&Ilf cflt X]. tai ^I. Tarn gi6c 0.]fiy efing lii. ta rn giAc c{ rr. \ . r.r i '1 d \l q,, \i I i/ ri, : iii: Tirm lai quf richt'::'gi&a ciia pe ra hai iloan RS vh ItL r) hinh 2. Ban trfgrigdrt Duu 'I'hdi Snn (Hrrd) e6 l6i gii i tttt. Hinh 1 Bii l0/llg, t:ha mgr titi giat T ro diqn $ch S. ?'o chia m6i canh cr?n nd ,idnft .l phhn btng nlrau rdi ndi cdc cQp itilm. chia tuong *ng1 & cde eqnh d6i d.rcn. Ld.m nhr uQg, t* gtdc T dagc clrfa {lronA I 6 htnh I* gidc. trong tl6 $ mdt 6 f f iF khdng kE uoi canh ndo cfia T md lq get td. 6 trung tfrm. Ndi rinq diln tjr:h cria d trung tdm b&ng Sl9 co dung khdag J ,r ? \ l Lil gidi. \:6i nh&rrg kt hif u nhu hinh.? ta ed SDAE = (l/3) .SDAB, .lBCc: U/g) !'BCD. .Tri rtd c6 Sp6g + SSCC sEecr, i I I - : (l/3){S11,1B * 2513 = St.BGD12 has SBp611 = SC*,U I I t/ t/ ti d SBCD, = SiB M+t kh{e Sgpg : SEHG.'SCUF : SEr"-Glt I o g_ I I I II \l *.* _ S.3. BA5r giCr ta chring rnintr rl ng pS vu ()/l 6.{1 EII tai nh&ng clitm d. t chld doan EIi thnnh i phtn b*ng nhau. Thlt vi-v (xom hlnh 3) ta ccr PE = il813, f/S = (2r3) ]]B $, pE = HS/Z *+ fiE = KH12*EN=EHi3. Tunng t* H,!.:.EHi3. Tric le Ii, ,t e.hia doan E'Il thanh S phEn hrlng nhau. Tuong tg .lI,ry eltia iloan F6 thdoh :i phhn b*.ng nhau. Theo phlr trdn ta c6 ffintr 3 l: t: SttLlt,r- = t ll3)S gpg11 =.S/9 diQn rieh d trung rim dring-b&qg tiS ai€u .. Vriy tich cria rti giic di cho. Htnh 4 t) rt{oz(...(&o, ii) vcri mgi /{= I, 2...., n, ta e6 a? + dl +...+ai { Chri,ng minh rhng: f,&{ t) at * DE ra kdt gu6. drirrg + ... roA${ Yt{l stAt ToA}t r(i(i(,, t?s$ : Igg? BAi f lLZt" Hse sinh A r{r gi6i ro{n. Cri ldn th{y gi*o dd A doirr hai sd nguyen duong, mA tbEy tte vidt b& rrbng phong il Jan kin, nhurrg A hi6t 16ng binh phooog hai s6 dy. :hj :b: tr& ldi; ( Em cliu, kh6ng- iln6.n rluge >. 'IhirvA s giAo cho bii{t th6m : < }lai s6 {y c6 t}ng l6n hcrn ,* I?) Khi i16 A r:iri :

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