Toán học và tuổi trẻ Số 146 (6/1985)

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Toán học và tuổi trẻ Số 146 (6/1985) sau đây bao gồm những nội dung về ứng dụng của tích ngoài hai véctơ trong hình học phẳng. Bên cạnh đó, tài liệu còn nêu lên cách giải một số bài tập trong kỳ thi Toán quốc tế. 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 146 (6/1985)

rF- J"' it:,:i:,t:j-;i::.,:r t, =i-i.t- ' l -ttl' YI$il rsoa' Egc:, ::1;j rrfr Ht"tr( .:.: nS.l roAr., x$e,... vtBr .N"lll , t" s T 1385 i .at. - .a . .'.'...:t :. : 5d 146 S,{O NA HAI 'rfiAfis u0r fi.t f8ng iiCn ldp.: ITguy€n CAnh Toia Tru : 70 'l'r$n l{ung Sao Phb tbng btin tgp: s& * Ha }{Sl IYgO D*r Tfr Dd1' u6i: 52825 ... HGUYfrT TfiT{ flAO fHOXt; I I * igEI. toi di gidi thigri r.cri bau 16 DB cho gur: hcrn n$a, ta qui u6c Arfi c{ _, dgc tich ngoli crir ,hai vccts;r:, y ahu lir 't?ifiC rt{p ldrt v& d{t rngt h&rn tuy{n tilh, ptrin x*ag.,i*.l,ung. * #_ FliJ {t} D{c h,iQt rnQt tlrrh ch6i quan trgng lA: ydi r v$ 8c_vgir) *ho S dilin b6r ky g la hsi reets khlo kh0ng, thi chdng in drlng {: B. C._^Chriag sE ph*ong ndu' vi ehl a6u tf&ne unug nfu vil cr,i n€u re i6 "e ff aang p1uong, tur li tich ngoAi eia chdng btng 0: L Ie' i/Jori' ' 1.+a, irJes (t) 'frong bAi n*y, sin gidri thifu m0t s6 r.6s dS I ho{e b*.i to6n li nh&ng rtng dung e ira tich ng**i. Att-()lJ -{r.{e E -A I. Oibre ki6n rh6ng hritg eho 3 tlllrn. + .+ Trong mAt prh&ng, ta h6.y elcrr grSt dilnr O J{{-'=OC * rl.}j ry i; *,{ viry: Ir)_v y. !rt6i diEm .S cria nr{,t phfrug sa sic dinh l.4Ij, "{f'- I = [B - 1. {._ dl !* mdt veclo r :'{)-\!, Ngugc [gi, ndu cho af,n di$m 0 vd gcclr r , tbi tlilm d{ l& ho&n tr:&n x&c . = [8, {:}+{{:, }] + {4, fiJ dinh. Yccts 6?. ,* grlc O rJinh ain, g.;i tt ,*rto 8{r qu[ iA,: Bc dt]nr A, g, f id tlrdrig hdng nlu ud s/ri fi8a: hdn i,inh rrla ilitrn # \6 lI urf t rrc, lLr &,rrc. Eib ' [.r, ai + {t, r-:1 i- [r., .+j =p " dou c{ch viar, j .t: {s) ,gi{p T'dt ri-hidn, khi 6p tlqog (ii), t* Srrgc ehgn t*y * U. f gde O er.la 'a$e ve rto b{n klnh ci'ra d,'8, r:., "0i! ,-}
g. Hinh 1tr gi&e boAa toirn. Cho 4 duoug "tfing llnr nhu v{y 66i vdi IJ'r'A Cl, cu6i ctng th$ug b$t kf. Chirng gioo nhau trtng e[p tli 6 ta duqe di&nr ,4. B, C, D; f. l' {xem irin}r i). MQt irnlh , B-o,c ry-',: C-9,r A-18 hhu Lhd elugc gci lA n:St r/i F ffi , *, C' & J---i::- tG). It? ,gtdc hod* lorirr. Nii i*d l-p 1-l e6 { c+nft li. 4 dtrdn;1. Ettsu ki$n clln l'& && clio "{'; ii', C' thi ng th&ug di cho vi ti dinh 1{ eic giao diBrn ct'ra { [.t', n'] + [B', c'] i- [c', 3'] = o. eanlr liEt hqp tung ddi rnQt.'Hai dinh gqi lA ddi Thay c6c giA tri {6) vAo, ta s€ duoe: difn n6u chtng tiru0e {[r, B] + fB, cl + [c', .4]] (t - crPT) = 6 hai c{p euntr lihrlc nliau' Ntrung A,, ]), O kii0ng th&ng hing, eho ndn: ?rong hinl.i, *lrc c[p {1, la, s}+' [,B, {;j + [e ' ,'1] + o D), iB, r), (t, f') ln V{y diEu kif n c{n vlt dd cho' A', B', C' thing nht?ngciip tlinhddi di{n. a hfing t$ JL, CAc doq.n thf,ng n6i c{e cAp dinir ddi r1i$n gqi l& drring ehto.'l'a cd dinh . a.p,T:l (7) ly : 7'rong m$! tir gtdc lrodn tadn, trung di0m cria 4. Ty sd iIo"n, ti a6 k6p, lring rA ch&m cde dtrbng ch{o tci.3 dlaln {ir?ng lldnq di&u !rda. Chtrng.';rirrlr. a) f! sA don cia lla tiiBm A, R, (: ti:irng i:lrig, theo thr'r tu, ld mtrt s6 thgc p sao eho: Te ggi P, Q, It l&t tugt ld trung dihn erju = IIA (S) An, RE, /.';-.' thi: PCa P = 1,,2. (A -+" r)" Q: ii?, (B I E), Ta kf lii$u p bing c{ch vi6t : r? = 1 2. (C + r), v^-t- rl . R,C) (g) f'heo i3), diBu kiga thiug hdng clro P, Q,B la: T& (8) suf ra : ta* n, B + E1+ [B + E, c + r'1+ .P(B-C)-(l-C). Vdi gdc O khlhg ndmtr|nduing thdng A.1),i. . ic+F,4-f /limg. ta hfly l$y ticb agodt cia C ttdi hat uE' sE dugc Kliai tritn c{c tieh ngoli, te e6: la, r] + [-{, El + [r, Bi + [CI, ]?l + l.s, cl + (10) ' + [a, r'j + lr, c] + [E, r] + [C, ,1] + [c, ol (4) b) Tfi sd /rCp clia 4 dibrn ihffng hAng ^1, B, {:' + iri, -4j, [F, "0)] = o Il, theo thri tqr, hj' hi$u lla (A, B, C, Di. dinh Nhung.r[* b* ba diBm sau ilsy lA tirftng h&ng: nghia li. m$t s6 thuc x6e dintr b6i hg thric : (,1, B, r-). {A, E, c}, (r, #, r}, U), B, c) i(A, B, c, D) = (,,1, B. ct'.{A, B, D} Cho n0l ia c6: t [rc i&. : [a, sJ + ['8, F] * [F, ,ql = o i . (A, B, c, D} = [c,,{ll[c,IJ]:[r, d)llo,n] rtll' [a, r]* oi + J:' 4l= 9 Ir. r'l {rrl Gdc O ph&i & agodi dudng th&ng' In,El + lE' + []'' ol: o ( Bdn diEm A, R, C. D th8ng hdng, drrgc goi li lo.al -t- [s' c] + [c, tri = o I rngt idnE di\m di\u irda ndu 'theo diBu }:ifn (1) vn ( e6e diog thitc {5) k*r: 12) nhu thd iA chirng rninh cho dinh U. 3. Binh ly Meaelaus. Cho tsm giitc ABC. (rs) T'r'€n ur6i canh (k*c ii'l,i) {JC, C-4, ^46 lln luot cl Ti sd ktp a&a m$l cfirirn dtdng thc.ru gbrtt cho *6.c diErn .{',.R', Cl' Ta hii.y vi€t diBu kiSn 4 tla O,l, OB, {}{:, OD, k! hi$u 1[ {OA, AB,0C, thrlng hhng cho 3 drbin,'1', lJ', C'. Ta gqi a, p, T Ofrl, lt sd th ue '" cAe tj' sd nrtr A:. al', ,:,' l&n lrrol chia e6e dloqn BC, CA, AP I {c)A, AR, oi:, afi}, = [c, al/lc, r]: [D,A]/[D.8] (r4). - {f _; **- iE !r*,{*i,* t - * /-ryi,- l. -* i-+ .r f ._;r" f l, \-,+'i'-'J \8"4i ' \(r B,I, t Di&u *&n chirng minir r:gay rlt cho ilinh nghia r0n drlngv frn g lA n6 ft?tdng phtl t hrt$c rcclt chon 'l'a suy ru cfufrng hgn cdc rlilar A; B, C, -0 lren rn6i tia {tla & aat' te f,(C *;t') *3 "*.i.{' E+ {l '- qid' = {J * ut: itrror:g thilng). Thye vfly' ndtr, ta ihn luot thay o$ a's t1(t,* e) " (s * fiC). c{e yecto, A, l}, C, D bing *A, Fl}, Tf,', E0 (voi ?-
# 1, T, E lA nhirng sd thuc tuy ) tbi bi$u thric ilEu gi& nguysn. Vay ta he;, thay iioan ,{B hArig (tl)F,v6n giir nguv€n gin tri crin .fn'd. Uan itge e6 tlog.o OP tr6n r1, thay doan CIr harg rli,rili q;p trU:; lhb !u chirng 'minh di6u d6. d', sao qho 0?= A8, AQ ^ CD. l'a sS rirri \ldt nlran rdl quan lrong li ndrr ta cllon A, R, quj tlch trong rfl,ph lrr.tag gbc ffi. 'i',, chqn {i C, D tr€n nrgt dudrig tLirg cAl tuy{rr cira rhr\m, thi ta thd-v ring: l$ s6 kdp efia 4 drtlng thd*g 1&m gSc. Ydi l:ai rIi!-m."bnr ki :!J vi ,ii' rirr:dr: ddng qui citrg l& lg sd kip crio 4 r{iEnr groo sd 11u_t tich Lrong 96c pOe, ta plirii "ce : ehing rac dinh b|n m1l cal tuudn brll ki, t,'2,.IP.,tr]+1is. [:i. g1= tll. [e, ru' j+ i,? i,:],",01. t) Ch0m I Od, AB, Oi:, OD I ducrc gr,i ld rrr0t 'i n su"v ra clrrirn dilu .hda n6u t.i' sd hdp' cria *6 [[ng * i : ["u"0* rj*={,u,,e * F: iF (o,t, aB, oc, (JD) = --1 +[,u" Q..ri -[as, o*tsl="6 Theo nirAn xel 1r0n, nELr m$t chrirn 4 dud;ug tric Ii fll/'-tut,e*fl '=o tiring lA diBu hCra, thi n6 giao vdri mot c{t tuy€n [.tf,U'. Prjl= rr {fS; thinh 4 ilibnr crla nr0t hing diEir hrir" NgirEc ,EiEu ki0n nAy chung t6 rirri; y,}J'i,f't vsi i{ Iq.i' ndu ta udi dit'irr O ltri { ,li}nr cria nrot hi\ng vil ,"r1'li nrf,i r:6p dibm bfrt ir-i lhu6+ q*i tieh.. diBu hde tr0n rndt dudng ti:r1ng khtng riira O. thi ta ducrc mOt chi'rnr di0u hdl. Vgy {ri;1 tich }i :,n4i iirrirng thirrg song s,;ng I,bi PQ, tdt nlii0n I{ hal ch$ !rong grlc ffF*o utA, 5. Ou&ng.thing rong son&. qu1: tir:,h i,i rlasn lii$ng.ll1.tr'z {Xcrri hir;h r,( ti. Hai dt1)ng thiing p.idl &aft, .{A v* Cl}, lir song soog khi v& chi khi t(ch ngoli: v6i Sl,ttcfi =J.ttlop - ,5. ' hli [.{8, co] = 6 con iai crja bii ic*n'$u.t r.* tlti ding. [:)hiirr Ch&ng han ctro tam gi*c 0,1l?, v[ h:ri dibrn .{." 8. ,?gdt b&i toar rhi r0 dieh q*tic r$ l$g4 tr&n O.f, B'trEil (JlJ. Vdi () lt Bd,r rlir cd Dt btit. Cii sir ABCD lA rnit tr? gidc ilri. s*r; thb dar: cho drrilng th&ng CO ti6p xric vt)ng trdn iiurlrrig d' = cr:{, a' = pB kith ,117. tlhirng, arinh r$.ng dutnrgi tliiog _tR li{p . s * C'8'i vist rlrse l&; xirc vdng trin du$ng iii*h CIl. ndu r,& ehi nf u _'teb_agoAi IAr, BCi i AD. lar, I'a'j==lB - d, {}B=aal = (a-Ft f,+, e]. Cfit. Do [.s,, si*o ctro n€n rich ngoar ffi ..fF,';-g ndrr vA ciri n6u q.:9.l}|i chinh lA rllnir i-f Tha' lcs drrrli dang thu gon. '6. ShoAng e6eh tdr no6t diBrn ddn ergt + rtuung thang. Ching hqn. mudrr co liho&;rg cielr 8 tu djEm ? il rtr6n dudng thing .{B thi chi.16 r'i0e eh!a diQn tlch tanr gi6e .tI,lr3 cho ilB riii nhin 2 r 'lSuan rll 8: l*tt, iial (r5) llav iA 1B 6, trong t!6 tich ngoii plrfii l{-r. theo chiBu qrrar 'ftreo gii thi€t, trung di8m O c*a..tB ;rliirl r*r:h ittun3 (ngtroc chilrr kim ddng lro). dudng th&ng CD blng 1/!. .4f1. Ys;.: 7. Uqt bdi toin thi vAo SAi bqc. l9&4 , ++ Ilil brii, 1'rong m{t phl ug ctro lrai d-.an th:ing lo D,' .*; r-ir: !lce-* il?.,4 It cd dinh AIj, CD, theo thu trr, udrn tr€n hui ttrr*ng t+ |rri,, tt{:l: l.')...7lJ.t.il {I:}. thing giao nhau rJ vA rJ' (giao nhau tai O). 'l'in; s 11u-i lich dibm llJ trong nr{t phring sao c.ho tbug Ci:* rnLiSn clrcl vdng l.r'd;r rlLrrr;rg i'iItlr C{.} ii+p rli0n tit:li irai rarrr giac M^{fi; .VCD lil,dsg dbl :rit udri 8 lhi,ciln v{ dri li.: ."1 (bhng S). fliril. 'i's cd ltr& clio iloar.i r{B trrrr,rl ti"iii ,J. Cil Io'tj.0'rl l;,sfi = 112.L-.{J trrrgl {r0n r/'r:nA riif n ti.i ,U.{ll r.rt diF:r lich &fr-'l) -*ffis. ii]1 = 1i2.ct) a.L-.
, V{y, theo (l;), ta e6: .-&, _* Cdrt diEu kign song song cho d^0'vA IIC ldt 4 [0'r, o',t] *[at, au] i sel' tfr,r, Fcl = c r+ le -' s, c* /il = s L{vJ*u*+O l}.m sdc c$r' verto brr0c, ta sl c6: *+ [,1 * t-], {: + al x c, 8= -1, O'B= --d.*t]', o'rl*3. *{-i', o'*ii3. (c+ l). C::di criug tra ei-ing e6: Do dd I iCTt,6.*qj:i-a*o',4-o'l * [-4, ? 6'l - [.], c t, tj]. ^l . [a, c + CI] = lr, cj 'cria bir tr,ln ill duEe $iEu ki*n (tg) tr& neu : ditng ih {lgi. Alerrt dB [,t, L' + J]l - [o, cl i isl. e [':rg rninh. I H*H h",t*l vri'[ viHtr -- - i' !i il"t ir" i;i,:i' i iI'i{ jii-r I li i. t$i* si A.N.Kdnm0gorsp e!flh nEin 19{J't, m5m bQ mdn khee.,. I)s dd eg b6 minh trong nhirog nay 81 tubi, nlm i8 tubi d[ li sinh'vi€n iinh vlrc hqp sd llrn thui ctrQt ci nginh khoa tru&ng Oqi hgc Tbng trqp L8m0n6xdp tr{ax- hoc l{.rn. D6i i,6i toAn htlc, ttnf,- t,lnf, tlon giin eova {1920), ld giSo stt, ch,i nhiQm kho.a vd hi$n hsn, vl c5c e0ng trlnh toAn hgc c6 tlnh iip thE: ld. chi nhi$m b0 m6n toAa lfigie cira Khoe to{;r- c6 m$i sd niii to&u hqc nghiOn ciiu nhtng vin co tru&ng SHTII L0m6n&x6;: Maxecrva' dE r6t hep, nhung ctng ed nhirng ngrroi nghiEn Vi$n si'K0nm6g*r8p iA m$t trong sd cde nhl cuu e6.e v6n dE liBn ngd.nh vr1 tu,\' lrghi€n cuu trinh eira Cng hep ntrrrng vfie hibu c*ng lrinh tc:in lir-,c li c0ng toin hlrc dfr'u rI{n hi$n dai, cdng trinh t{p th&. thuqe ti& loqi-tu c&e cSng trlnh lf thuy{t rat truu tugng ddn e6c eOng trinh ung dqng to&n bgc ' H*|. Yiqu si slio bi6t nr6i quaa hf 5itls tc:in cq thb trong eo hqc th&-v ktri, ccr hoc dtrP hli" hqc u.ng dilng v| ts6n hcc lf thr:r 6l' Trong thdi giag ehi6n tranh eh6ng ph6t xlt Euc tlsp. Trurlc h6i cln luu f rLng viie phln ehia ang nghiOit or"u iy thu-viit dudng dan. Nhung phHn ra toi{n ung dung r-i to{n 1! tnurt:t chi lA sU l6n cic c6ug trinh c&a 6ng ld vE if thuydt s6'c qui udc hbAn to'In. Clic r'6n dE tu&ng nhir r{t ruflt - & aey Orig d[ x&y dgng cA'c tien dE cs lj' thu5€1. r{r. xa vdi ung dung nbiEu- !':hi lgi b6.n m6 r$ng nhiEu ttinh I,f , ph6t hi$n nhiEu qu* b5t i:ga tr6 thinh cOng eu guan trong cira ohiEu t rin h... {.ing Crrng thue 16, m{t kh6.e khi ugh:&l cTu cAc ViQn si gi'trng ti:.iy & tru&ng dai hpc tbr:g hqp v5n tlts ung rii:ng nhi tsdn hoe lai thuoas rtrvEn vtr cir & cAe trudng phb thOng, s{ng IAp trudng phiri gili quydt edc hii toAir r{t. Cep nhung phb th0ng ehuy0n toAu & {ru*ng Eai h-c-c T}ng ch&ng c6 rlng dgng tnge ii6p nAo. Do d6 ngb!€n irgp &faxcova, ltr chir tich H$i rlbng cii cSch crlu irng duhg c&ng ph&i nghien ciru r,)ng. I-0 phuong phAp gi&ng dr11- td{n phi'r th6ng' vj5t nhiEu dllL'ng nfuien gi&i qu}'€t c&c ddi h6i cirs thuc t6 ra.f, giao khoa toirn lioe cho hoc sinh'.' YiQa Ie nghia vti c&a c6e nhir toSn hQc, r:birtlg si l[ igtr]i shng l$p b6o Lug-ng tir {i;Ao toiin ci: np c&n n chi&n c riu cA e&c r'5.n dB co thb lrgc vA-v&t. l-f hpc d&nh eho tubi tr6) v{r lA P}rd ctrui e6 rltn! clung'nga-v. nhung hSp din vA Tbng bi&n t{p e'3a toa l-rito.'" c}n che sg ph&t triBn cfra to6'n hoc. Du&i diy li n$i dirug ctr$c mq"n dam ciia ft6l. Yine vi€t r&ng 6rig ta bB nghi6n ciru ViQn ei vfi ttla soq* b*o c Lugng tI?; n6i trln, giti tiuh h*.rn vi vudng Kilnm0gSr0p' -\ia Yif n E6!. Ngu&i ta thir&ng n6i vE or3 ehu;0n ::rdn si:ho bi$t f ki$n vE sryganh dtra irong toAn hoe' h6a treng khoa hqc, trong irhi dd Vi*n si iai Dap' TOi kh&:rg hibu l{rn vE 't cri:r Yir:e' T6i nghi0n eiru nhirng vdn dE riit xa nhau - tu If ktr6n6 nghiin c*u nhibu vE gifii tic! h&rn. C']ng thuydt sae xu$t, t&po Sqi sd, t6i lSgic to6n' If trlnir !f thri nh(l cria t6i irong linh v*e niy thuyEt c6c h$ d$ng lge '" V{y, iheo vi$n sl tuong ll (Dgr'rng'roi: :l \iine'vA mQt s6 du&ng cong lJ lai c&a khoa hpc l& gi * van nd::11 hriy *hrr1'*c tiru kh{t trorrg iihdng gian, Iiinbe }. mdn hda ? :i YE ganh 8ua -- n6r.r lA tlLra l$i htu nghi tht Tr& let, ToAn hgc r5t rQng kln, m$t rrguSi *d kirdng ktr*e gtr sr; *q)ng t*'c, Su c$ng tic giira khOng thB nghiOn eilu dirr;c t{t e& cdc bd nr0n hsi nh} tof.n hr:c crlng ilirng thoi stiy- nghi vE tofin 1pe. Yb m!.t ri*y e [n e6" sirehuv&n rn$n ciicg rnQt v6u dts nhiEu khi rfrt e6 hiQu qul" h6a. Nhung dirrg thdi tri6'n hoc lil kho'r i:Qc I'{huag cd kiri xly ra tru&ng trgp !n srg tham gia th6ng nhst ng*y e*ng e6 uirjBu mdi ii*n hf giita efrr r:r$t ngrrdi nA.o dd thuc su l& thira, thi ngu&i cee tr'+ mOn' Lg mOu tlar tr& ,thflrh e &ng aq riie 6v n6n vui vi: tlAuh sarrg b0n'
_ iL6l. fl{l ph&i iru; u&q: tc*n hoc eiing llr d6i f ur'#t * r[{a X6 gi&. Sau dd ngfr kho&ng n&a gid. tLroug ir{r ri{n riia YiBx si kh*ni; / Viga ,ri ehr-rn An li1i vdo t7 git}. -(au bile tdi llm c&c vigc phg .ughB 1*m toAo tri khi nA'-r ? r:i:ir chdp lrii l-rlu rhder. Bubi t$i - rgqc s*cn, nitre Bsp. Khfrug phii nhu thr!, ceu ilir&ug v.tu t?:ieh nh4e, ti6p khAch, tr'rw&* klii di ngir cbdng tOi loin hor cira tdi r6t ngo&n ng{lgo. lliii l;d t8i l}n: ri4o choi, ng& v$o tue ?3 gi&. tinh si6i \.i thich.. c&c bii leAn sr5 h.oe, tr-1i hoc 'i"dt nhi&n l:\ kiri .tim rluoe eAch gi&i mat biri dai sd kh{ sdm. 0 c6"e lCrp tr.i.rng hoc tdi thiclr to.{a nAo d6 thi moi eh6 r1g lh&i gian tr& n0n cAc rn6n'l
:r ilffi's #$eE flefr BAI T${[ TSAN QEJstr ?tr WAffi NAY Doilx QUtxE S&i * tA;rhi: t..ira :llSl drrorg trdn idm ri lren ,^ tLAi thi DtiA = {tt - r ),2 = t'. 1 c6: nOn tu ctrn,lr 48 c.ir* u:q! lri grric lbt AIICI] ud lilp ric giAc Oii{i"D n$i ti6p duEc; ttt d6. urri 3 conrl t:r)n !(Ii. Chtrag minlr rdng, ntu ltt $i{re ABC\ riqi, ,idp' ta ed Al)+ RC-AB" ',rt :G ie, =k* Tii tiz = i:i -iii : + #rip ,.-ir: fir,ri O lA t{lm dudng trdn tidp'ruo v*i ba canh e&a ABCI). t' I& bAn kinh eia n6 Bei Z (lJe):6id sri n lii ni2i sd'.jt: n,t.rr, l; vi {i, J l{ ch&n eirc
DiBu hierr l) eho ii I{ci'} = ffrl vdl rnpi illr()c 4 #t!&n lri pAda 6i6l ciia M sno rlto tich aGZ' crta chilng td tily thiru b$e 4. cfia rr$l sS ngugia. , * fidp rl* i nr6i s6 *j trong jU cd deng ' ntuu rrign t) v* i) ehe ii) I(; = 1tffi orrl rnr:i a fi Z. Vqy v*i rnoi sd ngul'6n.m. do ii). e 6 : * ail *'&- r{ pi' ot, ngLry{in kh&ag Sm, trong d6 pt, /((m + lli) = f{(.m* 1)& * &} = f,{mk) "l*- , --li-% i*l vi (&, n) = r, i+p l;}*l rn e zl : zn; vil' p2,,.., pg ld eAc sd nguy6n tS bd hsn ?0' Tich' s{l 13:r'6 JA mdt sd chinh phuong khi vA. cli khi ft?.n) ld t{p chi cd rr$t phhn tr'r. aq *.a1* = 0 mod 2 vdi mQi i : 1; 2,,"., 9, Yl' BAi 3. (ttl t.an): .lYdrr P /,i rls thL?c a|t lii sd e$e b0 I scl nroduln'l IA 29 n&n mqi t$p con sd ngugPtt. cria lf g6rn it nh{t 2e * 1 = 5i3 sd ph&i t}r*a ' o" -i^ als * oziz * ... + ou*k hai s6 rn} tlch ld" rn6t s6 eltinh'phuong. VAy ta ktJ hl(a bdn6 }V(P) sd c&c ,ri$ sd aj sor.' rrlo a,; ru{t phet i* Id ve b&ng eieh nl]xt d5n tr}ng ciip sd nhu tird, ta dlirr.rc qt085r61;J)/2 * 736 ) f tS ld myt s6 lf. ydi txgt t npuu|n duotig huq b&ng 0 ta dd.t Qt: {t + r)x. chlrng mlnh rting, aoi c{p sd nhd vAy. Xdt e&c cEtt }:ie 2 cia h.rn 513 tl*h cAc c{p sd v&a nhSt ra d6 v$ l{p lgi argi lrp hDu itrr nhftny1 sfi nguqeft i1, i2,..., Is l-i' tu.lri tr0ir, su'1,' ra d.iBu cEn e hirng minh. ru(Q;, 'f Qiri...+ ) Lu();r) Qio) BAi 5 {t,;er xo): aha't.arn gidc ABC. tutlt * r:oi;g lion ld;ln O Ct r1ua cdc dil,t A uri. C vri lqi #,1 p 16: N€u &=2n ihl (l**)k=l *r(."-)**h, cdt cdc doqn AB oC 8$ lhea tkir tq tq.i hatdibnt trong rid r td nrQt da tlr(rc v6i hf, sd chin. lAv pkfra biQt /f vi r . Ta gid s* cde udng t-rdn ngoqi nCu bi.c efra da thirc P bd hon ,A' = 2m-thi ti|p cia cic tarn 11idc AB{) uri -Ii8fd cd.l nhau tat ru (Q.o . P) : 2w tl') (*) dting haLdibtn ith&n biit. ti uu 5I . Chftng r:ninh Bny gitr ta chrlng nrinh bdt dfr.ng thuc trong gdc 0tL R ltil)1,tJ. dE lj&l bAng'qui nap theo in. * frdp dn: Gqri F lil 6ia* dilm c&a e6e Cuoug Ndu 11 = 0.lray 1 thi b[t iling thr'rc ii t[nr tlring ,{N va AC (e&* iluocg th&rg nAy khOag th rrdng. thB song song vi cic du0ng tr$n ngo4.i ti€p dAC Gi&,.sir bdt tl.elng thire dirng vdi mci dny vd fiai, cit ahau tqi hai rlibra ph6n bi$t) ; c6 g--( ir ( re {... { io mtr to ( 2n (m } r)' nay' thB gi$ sri c thuQc doln PC. gio x6r dly 0 -=={ rr ( iz ( .. ( ro. m[ 2* € ln ( Gii s& BP cit tluong tron ngoqi ti6p I{.S;t: (2n'+t. xpi hai tlutlng :lrop : tqi B vi M'._BS.v gid_CrV,4f 'P l{.m6t tu gi"6c nAi i) ir>.3-. B6y gi* rJnng (*) u,t gii thi6t'qul tidp vt IVII'B = r9Ki{ ACB. Tn"il6'g-fh*i r+p, sny rn nga\: bf,t dSng tlrrro cEn chung = r:.jlr'c e EuiY * nri n h. !hu6. c__ttudng_Idn *!3' ii) i! < 2-. BS.v giu, d$l k : l* vi vi6t ttfiu'c : 8/iJtr -i Npc -: bAC. Yty M': ll. F{r): c" * u1r + "" + ak-t uk*l \i" + D{r l0,{l= ailOal = b, lOPl = p thi e6: * (r + * + &o*r,.t*:) l,r.- + ... IBMl.ta,Pl = lEJf l. lscl = be * o2 k-1 ")k(bo !r-1 -k-1 .li,p!- lrNl.lPrl : pz * 0,2 lparl \ .r* 116, cro tepiz: t#\,,ir::,tf;p]Bpi = - E ni.r' + I &r.ri + ,k X ri e' + ri (-r) , i=0 i=tl i=0 I tr.r:ng +16 mqi hQ sd efia da thrlc ,t l* chiri.) suy ra : lBr!12 : (b2 -' oz)''l{t' + p2 * 2*2i Iihi tirh ru(P) m6i at l6 it.rong tbng thft nh{i.) ipulz : (r,2 - i')'l{a' + pz * zoz) nrA cd bi tu
T-- P t.i- \L j',. : -- .{. ,i' \, 1 'r E 1 -#i- t { , * ..i!. *i.a ?-' Of trung irrrc erla rloqn ,{C r:rtn ^SJ song rong YOi nqri.rr ngus0n ) l, ea aa ) 0 du.r nhdr voi.1{O/: x6t ,Bl l& tidp tu,r.6ir cirr T rai f tli ty mi fo (aoi = t - l1a v* cd 0o) 0 du-i nh{t, luda tuong tu chting t6 AI s()ng song v6i OJ. ml /5 {br) E I, vi r5 rlng fn } oo.' Vgy O/BJ 1A rnot hinh binh h{nh; giao diErn I- efie cdc iltrsng ehdo BO vA IJ lA irung ditm Ilo fn+r (au) = fr, (co) Ifo(.o) t l,,nl = eia doqu 8O. So f./ ti trung truc cfia dogn Eff : (l - I/ni.I = Iotaa) nen f,.8 = [,rU. Tr] tI$ ram gitc (/.UB vu0ng tq.i ]/. ci'n, Io*i {oo+i) EE J - I,'(n * I) } 1-ll.n= . "8da $ (tt,ri-u Diln): L,&i m,]l sd thrt x1, tifry (.r'r,in -' f" lao) nsn ar.pt ) rr;. ) 1. d*ga :xtic dinh b&r cottg I h*c {iur nqF : fir*l = * 7 t ni tttl nb. t. .r'a(.r'n "chir:,ng' u-,tnh ,:d Do falt (hu) ,= t+l/n ),t+r (bn+t) aAn t*fi! ud rhi mgt sd irlr,rc i:! i-ijo ,:/to ,Jrti {;yr)uX bn+r { bu. I lhda'rn{l*: 0{rr:o {.t-c+.t {'t ;,ril.;niii a} I - *ap rjn: Dat rp ::,1'ri;r:1) ldi rlrroc hiur yJ lvllrt khAr, tr d r, * an ( {o {t,o\ * fn(cu} = 1 'n, f/:r) {r*i m6i sd rTguy0n n}tl. Khiing rtjnh nen dSe {ie}. leh I \:i e. ] e6 gi,ii han ehung I crir d8. b&i trrong drong voi kb.{ng diih: c* c = lim dx = lim &s,.dn(a{bo v6i mgi n rngt vA chi nrgt n rjt I -* i,,n { /o(fl) 1. 'l"ri d6 suv rr tlnh duy nh{.t ngu.r'en ) l). cfra a t !r6a mf n ddi h6,. : Liri girti : (prja nhipu bB*) :' Blt r = Vr.+r, r'=ii;;*, phuohg trinh .t,.-. ..leili d& cho luong {irrcrng vdi hS ' l11 + oZo*eu=Voe.- Bti l/143. Gtdl phtroag trinh I u"*u"=lo Vo--;:;r-"* VG;;IE- + V;:?,- V;f X6l ha i trrrtlng hgp:
li ai a * 0 Chia I v$ I phuan6 trinh cho nhau B&i Sll{$ I rinr gid t ri l&n nhdt erta ta ed btbu I}ric : rr*o=2Vc / ,-*u iz = ,,,=* !r* .+l-:-, lab-rU L 2 ; Vue .- rq I Lr:, y, ;) = l,;- I * aa uE * ut .j- uu r* 3(s * r)?_ = 0 ,* r Gs *u, la&*r: . er--_ Srr-v r.a Vr*u = *Y*-a 1*.r: = 0 I -r laD-:.r'l . --- {y *;) rc ,(z*r:)g, fr) a = 0. Tr) u? * u\'+ u.t, = {.u*ul2t2"r3ugi?*p li'il a{.r,#,2(}; a}0. Suy re u=i)=0*r-(t. Lnt gidt: (eria cle ban .Vgrryin Tdt g1111,.11 1, \'S.-v trong nroi trucrog hqp, phrrrng trlnh r6 Tty Son, tli N$i ; Y0 Kirn Crroufi ltk t,im m{l oghifmr:0. Lien Ht liOi : ..)., II,B.K Tru&e h6t, te chftng mlnh, b{t tt[ng thrlc : Bei 2/t43 : t;idi phr:ang lrintt I tr6 *,rg I i(.r: + S) { $ * a)12 (l} * t {tega.r)2*.rtog?{3+ ::) = logzrlr/t* -]lc97(E 4".a)J. hr-v : { (ab *.rg}2((c + gl2 - a)z (t, l*l gidt : Si&u ki{n tlfii e6i s : e } 0, Gpi i.) r+ [lob - 2:r.g * (e; * g] {"b - q}l F?ct, - 2ry f li phuorg trlnh ri* cho ta c6 : *(,r+u)(t) *CI)j
7 ihiln trr ,:r'i rriin lt + rfa !=* *" ,Jg=:-l":> '.-fi/ ' Iheo b&ng LiPn 't'rpa. .;; - n \ Y uzn3r4 "i/l;r_) v aln2trsl Jo'f; .V = min lf d-, f I nll-r,/l*;, 1l . i-r Ltii gi&i: lrdi or, I s il; lA cd p sd egng tc c6 l *1+n/(tl*lrl)i' \ ut I az*... * gn = nt} . (rtt + oa), r'r€n b$t deng Xrxt khnc ta lu0n ehon drrtrve sd tu nhi*n o thfte 'ci.n ctrung mlnh tuorg dluong vdi b5t sro cho d&ag thirr: ' {a i. r)z } n },2 h*., lll*l = ,, n! Xdt hoi lriri\n6 hgrp: \ aib V'-r*,, {V ol!uza3a+ -L J- ai Neu ,2 + u )nlu2 io ,uy.u i=l O(n):min l a * nl*, o + I * ni(a +l) I = tt + ttlu 'Iheo b{t d*ng thire Cdgi e ho n s6 ta c0; ,- VrE.,,-, ) ,,, n$u [b (,i)l | a * nlc l: ,r ]- li:/u I - la. r = tr{{t kh&c k hi dd t }/"4r, 1l- I= 2n. 2 i-* b) \eu (o r fiz 5 rt) a2 r a rlri \f ow2...n, ' \f ,ltnzue,a I.,(n) * min lc * *ic, 6 +.. I t n/(a + r) I . ni(o+li : V,- ; --( ({or * a5 * .. t oo)i 71 =Lr*l,f nsrr [b(n)l - lo r- I ] nf(,t + l) l=2, * ,. I{{t kh*c khi dd tV1ll+r l=2d+r. i/ , 1s, 7*@'+ VAr- v6i moi g!6 lr'! cIa n lrr nhjtn ta
Chung nrinL t rrong ir.r la cfing cd : ti (l * b)'l to&n qur-n iri6t. ({,StIr) lr tieg, tir.r€n -vd tr} S +tl(p * c)74,'a vi 1,(tr * al + t/(p * e?+lO' vCri vdng tldn tArr 1l b5n kinli Ot = ! tiz'dz14 * COng v{ vrii vd 3 ir;it d}ng ttruc tr&n la c6: = 112'IgtJt1g R l1 b.r1 linh (o))' (:udi 2,1{p - al * 2i i ti - 51 -f tlt r * r'tb 2{2! a +21 btzl t) cing vi Srt . SB = SC . SD : SE . SF nOn til gi&tr nhAn 2 rd '. oi (Sill t:r ,li ddn: ABEF ndi ridp rA ctrr$ng trr)n (CBEF) chinh ltr sl(p * o) + Si(i; * i,i + sl{p *. e) }:s/o * drdng trt\n {,() ph&i 'dtrng, thi.ia nrin mqri tlibrr kien cia bai toan. Ra) r'ang l* ndu 0 (d(2R : + lslb + 2s;c. v$3' ihi trong trrr&ng hop 116 b{i tc'iln lu&n c6 nghiQrn ln * rp i rc)2 fto *. Jrr * Jr" Ddu b*ng x&-v ra {dne bi€t ndu d:2R thi chi e6 I nghiQmi. khi gr\ cbi hhi a = l) = c trie l;i lihi tarn gif c cii sir Oe(.{Ai nr& OA+Ott thi ta vln ,{ /J/ ddu rfc'11inh duoc 5 vI bdi toirn trirng lu0rr c6 i{gugin Td.! Hiqu it tg," pf fU li'hdn rd{ : C$c bqn .Nlrrriiri ngirifm vdi diErr ki$n 0 < iJ < ?lt. Ctrtr OA= AB ff,1'Sono Hn Ni,i): Tndn tihantr vd C&rr tHi S i-, v0 tAn (vi CDri,tBI do d6 EF cirng dr,rirg .{,n /ihoc (1" lf n Trlir Piri. H;, Nrli): l"rdn Dinh 11,-yqe lf At), v&i c1iDu kiQn 0
! lai bA.i toirn, ta gAi t* rlign mdi tqo Ih6,n,h i* B&i lOi 143. Cfro hitlttattdng ALICD', cotdmO -l'D'C D : Sncn, Sacn, Seno, S6sq l[n lurTi lA ririnr {ror.q nrqil pttittg 't*rl ,j nui ni, duoiyl c6c diga tleh eiia cAe tam giitc IICD,-,.1C!). lB{, thdng llr, Dg audng g6c adi tnqt plidng {pl ua i .ABC: lrsgp. fiacn, Itnap, heso I[n lust li c{e oZ eing rngl ph{a c*a md! phdnE il.ll. H(ji. dt\m kho[tg cAe h gi&a e d,e m{t phri ug qua e6c. tarn J1' N cftugBn d$nE tr€,n Br lri Dg" Tirn quS ilch gi{e BCD vi I'C,D', vl .,1'il'S',. ABO vA ACD e(rc hinh chlbu I ur) I c*t 0 trin cac mri,l phdng A'B'D'.,48C vtr..l'B'C'. Ta niian itrf,rr trai tu diln:I.IJC.O vA d.'tt'C'IJ' ld dDng aang jUOi cintr [,tvrf f pA lCy,vl trong !rtdng hqp BIII*DiV=rrlg. theo tl s6 ir nAo d5y (c6 eAc mAr tirong rtng sc,ng, song), Viy Vz: ksvt (1r lp a,JrS .oog ti,r. (r ) t" .o t'ncoB,c,D, = Sg6niiggp (*2 + * ;,),rB . I'epe.c,p, = S46sh6on {ke + r{ | l} ;1 s ..: i 74ss4,s,s, = SanohA.sil (ts + ,* + t)/S I'4p6itr.s,, = 5A86 nlne(tr? + ft + l)lS. Lb! Utdt: {,:ia "lfoi Yiin l'lt-ltnli,16p it,\, P'i'TH liung Vurrng, Quy Nbun). Ha 0.rilndlv, ndi, d[. t'i lfir, oL]_i-ii,ea. 'r.A .a0 | trn non d{? _L [rr. oyi =+ -11.V I hr;rii. H4 ilt -l- *tr rUt 1 l* htnh rhitiu cria O lan [t&{Nl Ldy II ln tli6m gi*q eria ,&f.V ra c6 Oti = (BA +. + ,N)/? : M.t','2 = I{!{r'? = "ttH -^r E+ rl{A = C}tl! =- }ioti . ll inh Vi ():l 1r' drriing tr ung hl,:i; cira hioh thang ? il-tlNJ ngn '+ l'u dd la ti rilr diriyc ; l BMA=HOti =t)lti,,. i,-2 r i't tl'i:cnn,o,n, 'i Frru.r,6rpu -i ti-.rgpa"a,d, * Y*"r'' hai tam giAc vudng lfg(i yd. iI6r) b&ng 'iveuca uc, nhau =| 01i : US (tfrOng dbi). Tt 11( su;r ra li nir.a drrrlng trdn dulrng kinh .I3.D lrong rrfi.rn tr0_n -e Ft * (,S$eohs6o * 5.16plass r S.lpplr4ss I r, t.tyl, trtr h:r i drbm B +^ D. rrrrrl phi ng I tl + 56.rslr,1sg){/t2 -l k + I; .1. Tam.gidc ,1Ol{ c6 OJi : f}ll: ()A ngn ditm I KCt hop rdi {?i ta cd: ld . lrung dibrn e ril .4 Ii. VA,v I l5.rrr tr€n nria ts1/i3: Ir1* { Ssg etisg s *S&q pft4[ p ,i- S4g sft46p ] dudng lrdn li &nh cLla rr"ra tlrrong trt\n gu-i * *S,1sq/raec){"? * 1r + t)li ilch cria 1( qua phep vi lrr t$m.rE ti sd I/9. Nfra Hay r, 11.1' - 1): (Ss5p/i666 f Sa6p,ha6p * duong trdn niy c6 du
( & (Bll'.: M'&', DN'= ('Ar'). M{t kh&e O[t' : oB * iliBrn B, O) lr&n lBt, Dyl qua phep vj tq tfrn d : Oi{ vi tlo I' iA' &nh eria f(l qua ph6p v! tp li sd 1/1. t&m d, ti s6 1i'! cho n€n ,{J'=,1 '.rt'. Tr} dd th[-r M&t cAch trong tr,r, qul' tlch crja i tf o,1" ax' LAI
L()t orAr oE TOAN YHI T UYEN " SINH EAI HQC c vA CAO DANG h{AM t9B5 ffc$:*iH ainu 'rni ' l: t)cial i"+T *'r4*-l Tdm lai hg d5. cl:o ehi cl'ar6t nghigm Ctu = I idm . ti 7'in m d.? cho hl phto*p lrinh .TA U tbudrg lihi m )zsll. ' : ' cli cd m\l ngfir{ri €6u trI: Cdc goe erta larn giic .1 RC lh6a md.n Iti t huc \ !'2=.,u-4"-"'' (siuB * (sir,2B -:- I ,, : yt - 4yz + ,,,y. sirrC) sia2(l) = Loi g;dt: - sin I sin2.1 1) Fhuong trinh er1 nghia. khi * 1 .-( tt(4. 1) Cht?ng minh raaE ros8 *.os(, = l. Vidt n0 duiri dsng 2) finh cdc goc B, C biet li sd b c = k, trcng -V'.'+1 =t+'t/o-r. dd b, e tdn tvlrt l(i d0 ddt ttk cq.ntt d'|t di(n udt E, C-: C6. hai v6 dBu khnng flnr, .nBn binh phuong hai vd ts ilroe phroug trinh tuong riuong t.dt gidt: I) Vi A = rr '' (B . r ) nan 2A -.a = Lrt * t. (B + C). I{+ thil-c d: .r'ho rrrong Vl-.r' =,r'*2c+ OU(rDg vOt ( 4 -,r : {.r' - 2i2 t r? s- 3.r'= c -B+(-'cos .-Ij-c 2sin# 2sin {.B it.) cos (f t e2 -r-.; l.t-220 I r72. B+C BtC Kdt hqp v6i di€u kiQn n€u tr8n, tn th{y 2sin 2sin (A:r.) cos (B+C) phuong. tlinh dir eho cd nghiQm l.A c = 3. 2',) 2) He aa cho luon Iudn c6 nghiQrn r:g = 0 11 -' !)-'. 2cus." t_ COE I rn{ ta gr2i ld nghiQm t&m thud*g, vi vAy ta ph&i 2 2 - tim rn itb cho hQ kh&ng c6 nghlQm nio khAc B+(: , E-,. nghtQnr thm tbuong. - cos--__ 2tos' - - 1 2 rlg da cho ktr0ng dbi irhi la hoAn ri r vA -ry, 2 do iltl ndu h{ c6 nrbt nghifm lti {r. 17) thi (r7,.i) I R+(. 4 1'gs--".
trorrg do I = cosC. Do (l) ta co 0
d6 cf,c hinh:hilu cria 4, B, J, K, L tlrlng nhau. Ldt E$i: ciA g& dg 'chllu xu6ng rhtnh > l. C{c cnn}rcria hai-ram $!yu0ng C/J goc vuOng gtLc ACD. BCf,] dt,r < l. r{6'lF L Ca, er-I Ct, C't,'J .-V-i CI{ = Dl n0n. C,K'q D't.'. Tam Sif; Alt L_\ec!)' oai ei = *,- e,i i ["'d;;re cfui a'C'n' cd dir&ng. eao. .4'I' lrrlng,vdi cludag eirs Cll. Troog ranr giac aZi,,, sirie f cO lrung turdn, n$n C'l' I l'!)'. d,a d6 CI = ltt. AC? r ene = l/:. (: Dz +2AHz " , CAu V z Churt1 mtnh rang n€a f rong $t\t hlnh tu d?n cti cd mpt cqnh ;i dd dai L*i nan I Yl dCK t,.{, < I nea rzr,2 + za}/< i * tfti ,ft} ilch crtu nd khdng t6n tiontlB. .+ j.{.ff,,-( r -r2l* * Ly< /F;EE 'l-& cic lam gi6c vu0eg .{F-.1/ , -1F lt , tu e6 i * : i . .{lr(rr(a&r O ,Ci '* ' o.-(r ( t, v*-r. h(x) dirng'bi6n-r6i 0(r:( t, 1, -t* ds d6 a (r)(h(t) = 3i24=r t. Do d$ y