Toán học và tuổi trẻ Số 139 (5/1984)

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Toán học và tuổi trẻ Số 139 (5/1984) giới thiệu tới các bạn yêu Toán những nội dung về số và lệnh; tìm đa thức với nghiệm cho trước; vật lý toán trong khoa học và kỹ thuật hiện đại. Ngoài ra, tài liệu còn đưa ra một số đề bài và cách giải bài tập Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 139 (5/1984)

7 I YrEN IiH0A. EOC ffiffiffi yrEr NAlr EOI TOAN EQC YIET NAE # TBING * tong n:7- Dten top l Nguy6n C6nh Toin Trg s&: 70 'Trln Hung. Dao - Hn \ioi B Phd T'bns biAn tQp: NgO D4t Tir Df,.v n6i : .52825 x6r rsAun TicH zo TrIM EoaT oOnc Hd Nfi, ncdy 31 thdnrl I ndm 1984 H0r E0Nc NHA NrI0 c C{e ban thAn mOn, 'I'rong 2U nIm rJua, b5o ( Torin TaXC THrr0NG hoc r u Tubi tri, ) ill cr-r nhfrng dong 96p quan trong BLo roAN Eqc vA'rubr TRI . Hoc toirn vi lAm to/rn'hhOng chi ld rli tim c6.i hav, c6.i rlep cfia to6n hoc md. phli rlcm to6n hgc r?ng dqng c6 hiOu qui vio moi iinh v1rc ph{t tliEn kinh td vi xE h6i. Thi.nh tich cfia.ddi tu-vbn hoc sinh nuoc la trong r:Ae ki' thi toAn qu6c td chung t6 tu6i tri; nguoi Vi6t ft-arr ta c6 nhfrng kh& \,-iQt Nam. neng r6t Aeng cr"ri trqrrrg r i phrit huv vE to6n hgc. 'I'u duy va ki6n thric toArr hgc la cOng cu ngdy cAng ciin rhi6t tl6i voi nhrtng ngu'di ldnr khoa hoc k! thuAt, nhirng ngudi lim kirh t€ vd n6i chung ddi r,6i nroi ngrrdi trong linh r-uc hoaI tldng c6r urinh db tinlr'toArr dring, lua chon nliing r phuong 6n t6t trong cAc qu1'6t djnl t trong r.i$c'ldri-r. "a s Chric t.[c barr ire ;rhan cl{u db tien bo khdng ng*ng trong viQc hoc to6n, lirrn to6n THU C0A CHo T|CH i'a ung dqng toin, Ch0c b[o < To6n .hoc vi tubi,tr6> c6 th€m. nhi0u nguoi doc r:i HOr obNc B0 TRuoNG. ngi-v- cing c6 tic dung thrdt thr5c. PHAM VAN OONC ) Gr?i bgn ilgc nhen dip Bao To6n Hg. FhSm Yau Dang vA Tu'bi T16 trdn 20 tubi \-
*r.-...:--l-- I 't ,j HII ftIIT.OI NAM HAS YSAffi E{OC YA Tff&f, rRE sAng Iiidn eirr Ban !in rlong thirrir lrilr tAc. b6o T'oAn hoc vi Tubi iri rlir nhu i:hy nrlng B ToAn IIoc Viit Nam viL su frng ird tich i{6i f)O ctc vunn l6n giira mira ddng bio. r-urrt qua rnoi ]ilio , cria S;- ban I(hoa hc.rc Nh:'L n,,-c i,,gil u"o- Iiht'rn gian khb gA-r. nOn b6i nhr-rirq brriic leo thang li 'Ii.r ban Khoa l,oc vd K! ihuit Nhi rru'Lr:) []ho chi6n lranir ngA_r. cing irc iir.t cira qutn thir, 1'oip }roc vi TLibi tr6 ra rnit brn doc vio liriing Jriruc vu ban doc drroc llirrriltg ru.r-0n vi li6n [4udi nhrrr 1964 i'iri rririe tiich ( gd17 l;hdng ktti Iuc. Trong lric rn53' ba1' M1- nriil ilinr a'ieo r[c sdi n6i, ltito hftnq hac loan I ron q i lrunh tht(a nl€n, dau thtru^ng vii it6 nAt tr0n circ pher phrrirng lang rtQc hi(l ld .hoc sl.nh cric trtitng phtt lhing cdp i} mac, bAo Toi.n lioc..r.:\ Tu.Oi tri rr=in vrro.t qua (ngdg nttg Li cnc trtrd'ng pt{6 th|ng trung h1te) l bom iian d6n r,6i ban iloc l;0n miinq hiinr tlii Nh&nr rnuc dich di db I'a, ndi dung cfra biro dn hav du6i rn[i trrrdng so tirn, rrrili theo circ lip trung vio nhirng cl6 mr.rc: ^\6i ciru-r.0n viri hdrn thrt tha-v cl6i cria cAr ciridn sl pl:r)nc lihOng bg.n l.rt3 -1'0u toirr; 'I"im .hibu sirti toin iroc ph6 r.lijn b€n rnilrlr ph6o, canir trlnr ra clu, qrirritlr diu th6ng; DE ra lii na-r' va giiii b)i k\' trrr,rc (11i-r- [6n lirri hal iip fr trong Lra 16 ciir r-i, unh h0 la 1lflq ,'lr',r .viu ri Ilrrrrirrg xu.vi n cii:r rriui ,-;d aliri Cu I{I dang < x6 cloc 'I'rrtting sr-;u rl i cL'ru LrA'ot: Bir0,' rlSrr tiiii hitri lorin lro,. lrirrr dai: rt iruc u, f'oiln hoc vi rldi sdng; Liclr srl-. ijtr .", cAu Cudc lilriug ehi0n rircing II i gi:\nL tl,rng nirirt chu_vQn vi gili lhoai toin hoc ; Ciii rt6 p thic
w P:'6r ! --'L .l * C6 tAc rlr.rng s{tt xa vi. tft2irg ydu ddi vtri 3 todn od, Mm tod.n khdng chi td etl tint cdi trcg,cdl viec alito tao, bbi du0ng vA phit hiQn nhtng tii rlgp cila lod.n hgc md phdi ttem todn htlc ang ntrng trir 1:\nr nt)ng cdt cho vi0c nhanh c1.r6ng tlung cd hiQu quit udo mp! lTnh or;c phdt trltn A:t'tih tlho tao vt xirv dung {0i ngil cAn bd khoa hgc 'E uri .rd hli tt. ki thurit n6i chung vl to5.n hqc n6i riOng.' Db xr1ng ddng. vOi phhn thu&ng cao qu! crin Dinh- gi6 thi'nh lich cria bf,o TH v:\ TT trong Nhir nuoc, Ab lintr hQi s0.t sic vir thrlc hiQn sinh 20 ndm qua, tl6i dbng NhI nudc Nusc Cdng hoa l6ng nhfrng loi chi bio cira Chri tich tlQi rldng Xe bdi Chti nghia Vi0t Nam d[ t{ng thrr&ng b6o B0 tru6ng, db chdp t{nh uoc mo cho c[c b4rt TH vn TT I!uCrrt t:huong lao dQng hgrtg nli' tr6 r0u toin, girip hg nAng cao nhiQt tinh vi E)A-v ln phin thu&ng crto qul' vri vinh du i6n ph{t hu-r. 6c sing t;1o, n}ranir chtlng chi6irr linli lao mir Eing vi Nhi'L rruric tli\nh cho t:.1 p thb iOn nhftng dlnh cao cfra lri tr0 hay gdnh lric nhiiltg nhfrng ngudi'tii. g6p ct)ng 96p sirc rA1' dtlng vit trgng trSch trong c6ng cuQc dtrng xi1- r'i\ bio vQ lirm nAn th)nir tieh cho bAo i'll vi T'f . clf,t nr-r5'c, biro TH r-A'IT'nrong cluoc sq cdng t{c 'cOng tic, t-nong chX.t chd hcrn nii'a cira cAc ban Ngny 10 ttring 10 nx.rn 1967 Chri tich I16i dbng cluoc'sg gi(rp dO vd irng tr6 nhiEu hon nfra cria c6t: BO tru6ng Pharn Vfu'r Dbng cln c6 l.hu gii'i ban co quan criq Ding vi. NtrA nudrc, mong drqc str clgc cira bio'I.IJ ld T-l nrii l€n vai trd cfia b6o d6i hrrirng ung, a6p j', phe binh cfia ddng diio ban tlgc. vdi tubi tr6 r,i rldng vidn thanh thi0u qi0n ( n€a NlrAn ngir-r' ki' nif6 v6 vang nAv b5o 'II{ r'A'l'T cao chit nllhiu anh ltting rhch mqng ttong uiec .r.in bhy ti1 li,ng l.liirt on rldi v6i cAc co quan Ding tidn qud.n uao l;hort hltc, k! lhu!'l t. Nlrrin cl!p blo vh Nhi nttdc tli hudng d5n vi ginp dd' bdo Ttt vi TT t|dn 2tt tubi ( hfi tlch l?i vi6t thu cho hoat rIdng, xin chin tltinh ciim on cic c6 rihin, co ban ilqc eria biro r{c nhqrn r 7'rott11 2a ndm qua, .7-ttdt quan, ctodn thb aln cOng tic vtri biro vn dA gop bd"o ?r Torin hoc utl tri.tt dit cd nhftng ddttg phiin tirrn rr6n tlr:1nh tictr cia ]:r1o. gdp quan t rgng tI6i L,ot phonq lrdrt hr.tc toan ct)a tubi tfi ) vd An cin chi biro cho chtng ta c llpc Biio TOI,X HOC VA T{.TbI TEG I[6i xhuyArc t]bi c*c bqn tr& ydu to{rn, sO vA LfrNt-t NGUYEN CANH TOiN fv ,+C sd thdng ttrut;ng (s6 thrrc) cdn c6 gi rlirrg sd th{y ri.ng thri'tr.r nriy ld, quan trc.ing)'MQt c{ch su-v nghi nira nhi ? Nl.rrng chi viQc clbr c6.ch t6ng quAt, ta n6i ring tich ci'ra hai ( lenh ) (ldJ* nhin thi lai lim dibu mrii xuf,t hiE'n, it ra ciing theo m6t thu ttI nno d6) Id mQt ( ICnh ) duy- nhf,t li moi r16i voi nhitlg ngudi ehi c6 ki6n thr?c c(r tdc dgng 10n ucn m5i gi6ng 4hu tdc tohn hoc phb th0ng. Ta hlv ldy mQt ph6p nh{n drlng cria v-iQc thrlc lriQn liOn tidp hai hai s6 thuc, ch&ng han ll. 4. Ta c6 thB coi (il) ril cho, (theo tht' tu d[ cho) 16n ctrrg < v4t I c16. \i-' d Iil mdt lQnh vd. (4 ) lA vit chiu lQnh: llnh ra ngoiri pltam v{.t ( 12 ))..( Lenh ) r.ir
-.=i\{r BueG vA I' dEu IA clp sd thr.rc tht h'ai: 1.. Ta ctng cbu 1, ring, q€u trong (4) < ISnh > s&p thrt tg, cdn Q ln sd thqc' ta guy u6c r$ng ( Itnh I (p,g) tac rtQng l€n (f :) ., ,'u"* (61),,,, k€t quA ei v€ sau cfia dvft.r (*,U) ttri eho kdt qui ld s6 thuc pa*qb. (4) sE gi6ng nhu kdt qu& rr vS sau efra (3). Btr*c th& ba: i, 7, Q dEu ltr cIP sd thtc. T6nr lgi, m6 rQng nhu ir (1) le dA bao trrlinr Ta qu1- udc'rBng
tiin hge hai cria l. Ai chi bl6t *1 vd *l - Tlch hai lsnh adi ehung kh6ng giro boin' cl6u e6 i.iiuh phuong b[ng 1. Nhuug dAu di hdt. B4n cu thir tinh rliA x!ru. Trin ki-a tp=0 ahung pq +0. Ta lrlv xdt l6nh e= (?;) R6 rAng d6 khOng Rd rAng ta burvc vAo mrjt thd git''i nrCvi, rn& r6ng thd giiri mi & '116 cdc < lQnh > lA cdc sd phii ln lQnh < giit nguy0n r. Nhung ban tu minh thuc. Ct&c ban hlo huc nru6u bude vio thd giOi c6 tlrb itrir d'0 th*._v ring c" la. (( lQnh ) < gii! dti. Nhrrng di vlo de;,, phii r{t chn {hAn nhu ngrr,r,6n }). VaJ' 1, -1, e vir -e dEu cri binh phuong lihi burlc vflo mdt bei rnin. t)bHng hair vio dAy', I-ring I niu gir.p d&ng thitc ab: ac. Voi o *0, ch6 vqi L'&c cta khoug. 'Ia hd1' 6p dgng l6nh: rrirc lugc dF c6 b:c, Vi aD=ac cho ta oD-sc=0 hay c(b;c) - 0 nhung c(o-c)rO c6 thb x$y ra / t-z\ r'6i cr f 0 vl b * c+$ ttc D + c. Hoi.c ph&i rdt ,= ( -i iJ "*" c6p (rr, tr). rheo (1), ia c6 c:ip thfln trc2ng voi tht tu c6c thria sd trong rnqt ta *2h, - 1o -l-4b). Tn lai 6p dung lgnh : tich. Dir vAy, ldng h6.o hrlrc ri6i r,6i c6i rndi chfie sD thuc dhl' b+n sfim mOt nr6r der nrin dt di vtio s = (l l) "*" clp (a - 2r5, -.ro +lb)' rheb (4), bli min do. Cudi cirng xin n6i llng, t0i gr.ri " l$nii ' eho ta cltrgc cap (0, 0). t"1y: tlch qp = U m{c 4u n6 nOm na, cori trong to6nl hr.re ngrrdi ta ggi c6c p+0, C+0. q ra p gr.ri li ciLe uri'c c&a lthlng, < l$rh ) Lrlng nrot ilanh tti v5.n i'6 hryn l.ir : m$t ili6u eirng mdi lq. a to* n tr? >. ttm hibu sd,u thAm todn hqte phb thding ilr ol IHSc Yfi rcnltil cm nu0c Nco },IfiT TBUi\G Hit xet bii io[n: Iim mpt rla th{tc odi hi nhdt (r'ni he sd thr,rr:) nh+n /il+ V:f tao, sd ngugdn co bdt: nh6 nhdt nhdn irrrgt s$ a nghil:rn lA x -1it.- Vf nA lo1i c{n bg.c }ra ta chq lrtdc ldm nghilm; Eir-r' li rndt bli toin eb clibl c6 li6n quan clr[i cliE ddn r."iQc phit rrli*n a6 r'rri (x -lTlz + (x *t':=lVil+ ivf* uir uhAn tlrror. tlu thtrc' tribn sti lihii nidm co s6 cria toAu hqc. nrQt Ngirl' lay nguoi ta Lidt r.ir phii gi6,i quydt biri rr - r/ilt3-s : ^\3 -ty'e*x? + 6.I .* ztT *'t. toirn nay nhu thi nAo. Tir.r' "nhidn vdi lrinh dqr Db to4i cintqic hai ta lai uliin ti6p.r v0i phb thf.,ng cI6 vfrn ld rn6t bni to6n kh6. -\3 + G.y-: + y'z f rrz * l) r.n nhSn drrr.yc .r Cd ,t!ts r;r,rnfr hoa rlibu nity qua trudng hop . Pl,Y) = 1x3+sx*B)l *z(.:xe+e)? z = 11 2 + V3 . Dtv li rudt iIB bii cria ki' thi Sau lihi tinr tlttgc .P{X) nhibu thi sinh nh6"n € 'qua. hoc sinh girii toun iap tf ioan qudc r rr ret li: r.i 2 vd 3 ngult\n t6 t.ung lhau nOn 6: Hhrl h6t thi sinh cliiu tim thflv: 2.3 phii ln bic nh6 nh{t r'fra c6.c da ttruc veri P(.[):x6-6x1 *6x3 + nxz - 3ot + 1 hQ sd nguydn nhAn iy + Vf-lar,, nghiCm. Do ln mOt da tlruc r,0i h(! s6 nguyan fiT-+ Va d{t P(X) l[ da thuc chn tinr. K€t lrrin iibu nhtrug lirn nghiQrn uhung lihdng ai chring"fr,1nurinh cltrgc rrlt giin r\r th$t. Tai sao khdng ti6p luc n:i6t e5e}l 116 li ala thirc c6 bic nh6 nhf,t th6a rnln tliEu bigJr thurlng b$.ng r.ifc gii_si c6 nr6t ila lh-rk:r kifn tlEn. Q (x) ** *-q1{ * ..' * a5x} r'6i trqr sr5 rrgnyon ViQc tim r:r P{{) .hoirn tofn tlou gi&n vr}. dqa nh{n y'J- +Vr ttn, nghif nr ri)i chirng minh vd.o nhin x6t sau: Tich cfra ni0t da thirc c6 f,o =' . '=oS = 0'l nghignr li .r v6i rnirt tla thuc bflt lii vfin lir m6t Dibu ki4n duy nhft r.Ang bu6rr.: c4e sd nguy6sl da ttr*c co nghiQnr k\ a. Da thirc ci' h*c nhir e0,.. . , rr5 vdi nhau lir g tVt*+ Vill* 0. V{n dE )
It phii loai rluq-c e{c cin -bi.c hai vi ba' Db C[c phrrong trinli nA-v cho llri.v'b.r. ".'. D5 phii Inli vif c niy la hiy klrai,trih ia. lir1' thria cria lir cf,c io trlin' Dlt bs-lh;' " ' hs:2bs r'iri thal' y':-+ Vi iro,rg 0 t r'/z + V Tl riri lei viro hQ phuong trinh tr'Ori'vir gian rrdc sil chung" uhrruno trinh!: 0 'idt iVz +Vs-) :?, 0+ dutri d+ng 4 di ta lai ilunc nlot lr$ plttron{ trinh tuo'ng ttt l-3 t- viri hn lir' b:, ..., b;. Tt r1{1' lai suv ra t;' .'., b;, bo+ br\tz + bzV3 * ogV o + b4v 2 l/3 + ph6.i In c6c s6 chEn v.v..,'l'6rn 1ai b6' "', 15 chia + r,5/z-iio = o. i.gt .ho mgi lny tht'ta cria 2. Di.:u ui1" chi xh1' ra'khi b0:... =b5=0. c6 thb d€ tlnng thrl lai db tird.t: Nhu vdy ln tu di6u ki{n a (Y; + '\'fl = s i-r1n dr.rc bo=oo *Zaz* 3ds * 4aa * 5045 la dd srry ra tlugc 6 Phuong irinh sau: Dt:al * 2o3 * llat* 4as bz* a! * 6a3 * 3aa * 20aS oo'l Zaz* 3as i- 4aa * 5045 = g' QL + 2u 4- 12aa * 1o5 = 6, l.ra:3az * l2ar 1- los h4=tor-1-.5o*'1.,o: cl*6tr3*30+*!045=g' I';=llo3 +:rl(I5. saz* 12ie* 3os - 0, 2ut* 8aa*15os : o' I"oai cin hAc hai ta nl:in dttoc phuong trinh (nn- rzV; + I,,r't/u l2 - ztor * orVa + or\/o )' = 3o3*2045 -0' : Ban dqc c6 thb dE dnng gi&i h0 phuong trinh r,3 + 6ba&: - lbl - t2babs * nur- r'rt nhirn dugc ktit qui ag = "' : d5 = 0' l-lt'v gir-r ta c6 tlri' 1i6t lu&n 6 IA b{e, ntr6 r;1'4 *{2b.bz + 3b3'- 4bfta - oail Vs + cria c6c da thuc r'6i trQ sd tguv6n nh{nl 2 -r- \ 3 litrn rrglriFnt' + (:b,.r,s +ol * la*5 *lall i/3 = o ' Tlong trudng lrop tbng quit kh6ng phii liri Tiuh j sE thit ir.hrrong trinl [r0n cho ta m6t s6 a r,i-o ta cirng tim rluoc mQt da thirc vrii h€ da lh[Lc b3c hai Afz + BX +C r.oi hi s6 ngu,vOn s6 ngul'0n nhSn a lint nghiQm' Vi dq nhtr chc nluin V3 l:\m nghiQ.rn, Ditu ir1iy, dll d6n li6n s6 rf , e kh0ng thb l:\ nghiEnt cfia bdi ki mdl da thuc vdi hf Jl-s6 nguydn ni(', vi r'{,v r.rguoi trt goi lyg"S 3 ln bric*fih6,'nh$t'c&ai d6.e "ila t1i{r.9. ahqn- chung lr\ c{c sida uidt' Ngric-rc Iai ngurii ta 13 lirnr nghi0rr.r. vi do il6 ,1, Ii, C phii biing goi z."ta mdt s6 dai sd ne\u ca) mdt da thirc vdi Lh0ng. Chung urinh di6u nil lihbng kh6. 'tri gii a' laln nghiQnr' 56-io i c[ng 3/_. ti t6 "gu1'Cn "aAi nhrin llrii;t,4\'C + 1l\i *C=0lacr.r : II mot ;"0 so vi n6 lir nghiQlr cfia phrrong (,tils + eV3 + clrailr** E)=3,,12 - BC- trirh-\'t1=0. - (D- -AC) 1/i = 0' Cic d4i s6 thucrng c6 thb bibii di6n qLra ctrc s6 Iilru cnrr bic ba sd nhin dtroc cin b{i tq nhi0n cria circ s6 ngul:On vrlii cAc nhdp tinh thbng thrrdng Iih6ng ph6'i r i.\c rir: (3.* - BC)3 *3 (82-dc)3 : o iinii aa ituc vtri hi sd ngul'6n bAc ntrir rihiit : D[t m 3A2 - /JC vi n: 82 - AU. Ta c6 thb "a. nl:an chrir:g lir.rr ngtri6. rn lirc uio ci)ng 116 d:\ng th:il ngay' li.m, n ptrii cliia h6t cho 3 rrghia ld nhn truo-ng i]gp a: iz 'r ili' sot' rlar- li rTr(it nr:3m1, n :3rn1 r,di m1. nt I:'L nhitng s6 tlsurOn. clch xiLc tii"n cac da thirc nil dua thec cec kdt 'lu lai,',, rrrl 3nl =- rr, lrl dt-v Iai su1'r,,nt, r,1 qui cira loirn hoc cao c[p. - phli r:hia heit cho 3, r'.r'... Nhu vdl rn,.nrsE chia Gil s&' P(X) ln rndttla lhf c vcri hi sii rigtrf in h6t cho rnoi ily thri'a cfra i]. DiEu ndy chi xh_v ra lilri rn = n:0. Tr'r d-i6u -liidn 3Az - BC - 0, bic nh6 nhiit nhAn nr0t sd 7 ii\nr ng-tri('nt' N0u B2'*AC=0 ta lai c6 3c3-83='0 r,a clo d0 A, B bic c&a P(X) ti n thi nguoi ta c6 thb tirn thdl' phiri birrg lih6ng nl'ru 6 tr6n, Quav lai vcri gil n s6 crl, ..., c{11 kh6c nhau sao cho: P(X) : a(.X*a-r) ... (X-4"), thidt :ril 9-+ rVT+ C: 0 ta c
' Ndu a iri n:6i c6u bic ar cila rlOt sd n g u,r'irn ( - t,{? i- i vr 3,2Ji7ll e+n ilr;c co thB tirir lar dE o khing'cri ,1,,n* o = btc r.(ri I I:\. nr6t uric sd t lr ri,r' . cfrn nr tiri chc sd li0ir .hep cri:r a li' 2lrr / cos -_-_ 2k; \I tll /_. (.y-,r,/T-V lt rx + Vr- Vllro ' x (x * vt +YTn + i./, V ,rrl x - l5r!r -.__ o { m /V rn \ 1c = 1, .... m, v.i tr cri I'(X) t ;Ym * o lir nrdt >< (x * /T + V il2 - iVTV sc)x cin >< (x + !1 + \t vz+i r,is .V s/2J x rla thirc tiurl ! Ndu ?" cti tht i,iibu rli6n dudi ci:rng F' (p, T,...) >< (x + /t+ vui I li mdi rla tirrlc nhiEu bidn vtii hQ s6 nguy0n \'l ttz-iVT.V :iz)': vn p, T, ... li cic sd ci; dangfi a nhu ti'Or s:ro :x6 * 6.14 - oxs +',2x2 * gox + l. clrorrtirr F=i/D lriY =i9 tht b vi c phii Crin phii nhfin nranh. rXng flhuong prltip t3'€n * ngu1 0n td cr'rng nlrau, thi c{c s6 lien hcrp cirn er r ch'i girip ia xic clinh P(,Y) trudc viL ta vil ph&i sE c6 dang I"(p', T' ...) r,di p', T', ... LI nrtil. chung minir bing -c6c ki6n thuc so cip ring da cip c{c sri li6n hcrp tuong xirrg cirt I, T,... Cii thrc P({) tim iiucrrr thco cf.ch. nly chinh la da srl at, ..., a.n lri c5c' s6 kh.{c nhau c6 dang F(p', thuc voi iri) sd nguyOrr b4c nh6 nhfit nhin a ldm T', ...). Da thric (X--er) ... (X--:a.) sE c6 hQ s6 nglr ii rrr. lir. c6c sd hi}u 'ti. Qu1, dbng vir lo4i b6 rn6u srj ir i'ric lrt,sd ciin ctr thirc na1'ta sd cti rla thrrc /'i-\) Ban cltlc nlo quantirn d6n c6c kii,n tlritc [o6n cin Iint.. hgc cao clip din cldn phrrong ph{p tr0n cd thb tim doc qu1'bn ( Eai sd n cfia Serge [-ang (Phln lI- \Ii clrr nhrr lrirtrng hgp a '! z + V-T, chnong YII vir VIII) do Nhi xu6t bin d:Li hgc "a..0 Iion lrcrp c,ia'l/Th -]- V z la .i,* V s irV t] , \.ir trung hc,rc chul'6n nghiQp xuit bin nim 197'1, = {..,*+1 * il:r. + 1$6 1)'- i,t1-;, - 4,r-r * 196'{) : [tOS+(a-+r - ar+1) "i- (r-+r - rl+1,J - - I [19,\] (e- - er) + (r,, .-* rt)l : [(a,,+r - ar+1] - 1(am el)l (theo (I)) 1984 Su.v ra 51:ro-t - :r1-t); 198'1 va ao (5, tOe+t:t (r*-r - rr-t) : 1981. Bli 1.13(i : t;ho dd1 s,5 l.rr| rftu .scrr : .t:L = 20, uin 'f ti dtl' Ia t'i, : r6-t .I'2 - 10A u,i = rl--1 Tuorrg trr ta e [r rlg t'(t r1-n=/',,,-g vu eu tidp tuc .tnll - 4.r'n * ir',1-I * 106i, V tt 2 2. qLrri trinh tr€i ta nh13n iltroc : t:hfng ntinh rti;tilg. tbn lai m$t sd c{1tr dtiy chio i.z= rzr(.m*,) I rrrl Iril" tho 1984. ' tr - ri1(rrr*l) ) ' l,di gidi: l)irt .rn = 10E43'n * rr,, Sn >, 1 Ta s€ chiriig t6 ring nm-l: 198{. ., er1 r. i rs li circ sd ngu_r'Jn la 0 . _( .r ( tgA*5 T'hat li.v. ta co: ir Xr.t rlir,r' rr.r p sd (r1. rz) : (r:. rs) ; ... ; (ro' ' 5.r',r--t = .r'm-l+2 - {,t'n,-111 + 1964 In+t) : ... drn-l* 2+ il'm-t l? .ii Yi dn1' rii-v v6 han rhi sd cAc sii l'i hiru han - 1984 *, ',1 dm-I*i * r'*-i+r) + 1961 {1,984 rrirr plrii lrlrr tai hri s,I Ir.r. rrhidrr I, m (gia sri. rn ) 1) sao llro (r'r, i't+iJ = (r',n, 1m+t) tlreo irghir '= 1$84'(crn-r-1-z;- 4o*-t1 ) * (rz* (tneb 4t" 1) + 1964 ct)) rl="1 Tt}diilu kiQn dhu bii ta co t'1 = 20, 12:196 .,+, .= ;;\ ,i (r) vir dr:r viy : 'l'a s0 chr'rng tir ring, lihi dir rl*t s./'m-l 5;r*-l : 1gg4 (flr,i-r j-z * 4***r+r) + 1gS4. Tlrlt r ir , xel lriirr : Su'r' r'e 5.t*-r.. 198i1 vi clo (5, 19E4) = 1 n6n ;)lflt_1*;).r'lrl - 1r,-r 1 irsl (il .p.t.nr).
tfidn v61: {hc, ban D6 Duy K}rdn}r. (10 CT suy ra -uk+l uk-l. - ukak*z -' ui - t'l-t Nguy6n Ydn Tr6i - Nha Trang), LA Thaah Hit *) ur*r (ut+r * ux--r) = u1(u1 * u6-z) =+ uxl(ur+r * ur*t): rrt-*tl(ut I b'r Lam Son Thanh H6a), Ld Bii radng (r"z (2t ut-z) Luong Y{n Cir. - Long Xuv6n, Trdn Hung (12 Tuong tr; CT Ngo Si Li0n Hd B&c)' NgugEn {gsc Vdn : "' (hoa va Ldm Tino Giang (t0Cf Phan ChAu ur-ri (ur f us*z) : rtk*z/(utr*t * uk-c) Trinh - Da Ning) c6 ldi gi&i tdt. = urli.rlz * u") = 1/(2 + 1) = 1i3; NGUYtN tTHIC },IINH Do 116 url(ul+r * t* uL+1 = &ut - u6-t us-1) = 1/3 Bai 21136: Vdlm6t s6 tu nhiln 4: o1o2.'.ao '+ u1a1 nguy6n (vi uk, ut-l nguY6n) + (at*al ta kg hitu A k) sd on on*l.'. a1. Hdy 14Y s" nguYBn r'6i moi n' , ; tim tdt cd cdc s6 fg nlri€n k ed tinh cJrdf rau: lVlrdn.cdl: Clc b4n Lt Thanh Hd Q2 CT Lam N,u s6 tg nhian A Id b|i cia A lhi sd tv nhi?n Son, Thank H6a) ; Ldm ting Giang vir Nguy6a .:r: 7 ,Ang ld b|i ,'fru k. IVgpc Yitn f ioa ({0 C'I' Phan Ch,.u Trinh' DA" Nins). Trdn Hwng (12 CT NgO Si Li6n, HA B&c): Lbt gidi: D6 th{.v 1, 3, 9, ll, 33. 99 li cAc s6 rtg"li'" Quang rhai ttt CT DHSP Hn NQi 1) c6 tr nhi6n eo tinh chflt n6u tr'6n. Dd lti nhfrng u6c toi giai tdt. sd cfia 99. "l'a.s0 chitng nrinh mgi s6,lt c6 tinh D.B.K chdt'rr€rr lr0n ld u(rt' sd cria 9$' cil slr k 1r\ s6 c6 rn cht s6 do tI6 /i < l0m. Trong 10m - 1 s6 tq nhi8n tiOn tiip tu 10m + I Bii 4/136 : Cho 01a,b,c={ r' ciring zriai; tl6n 2.10m :Ig_fl_Lhdt rn6t s6 crhia hdt eho k' a!$ + c * 1) *ttl(c + a * 1) * clfu + b + 1)+ Sd d6 i.d 1apz..' o-. Khi i16 sd on"' ozoll cfing +O-a)U-b\O-c)r{, chia h€lt cho *. V{y k vii 10 nguy6n td ci)ng nhau' Ldr gidt: (crla b4n Ngug0n Hb Anh Ngugln' Cnng vqrl' trong 10m - 1 s6 tg nhi6n lidn ti6p tdp 11 CT Phan'Ch{u Trinh, Ed'Ning): tir b.i0'+l * 1 ddn 51.10m - 1 c6 mdt sd chia h€it Trudc h6t ta chrlng minh, n6u O
* Tu do suy ra: Ap dung rllnh 11' hi.rn s6 sin cho hai Iam a(t - a) u(1-aX1-i,Xr -c) gifc ABP vi CCP ta rlusc: \ (u DIIIAP-Bin(3/2)isina vil CPI AP * sin(I/2)/sinC (a*b*cXl*e*c) a*}*.c 6(l - D) ! b(r *ax I - bxr *c) Do.d6 sinz(x/2;=,*inB.sin(j tuong clrro,g v6i (2) BP,CP =^dP2. Do IJP.PC= A"P.PP, eho rr6n (a* b* c)(l * a *c) a*b*c c(t *c) c{r * a)(r - bXl -c) .€} (s; AP,PP, =AP2 (a*tt*ct(t+a+a) a *b*c 1+ PP'= r{p Cdng r'6 voi vd ba brit ding ttrric (1), (2), (B) la duerc bdt tlfing thu,. (rr). Vi nhu vAy bdt r+ !{P: pru (\.i?,+ i; ddng tlrut.crilr dE ln drroc churrg nrinh.. -Yh(ln rit : 'l'rong s6 trudc tlb ra c6 su. nhiim lin P.rI =PM thA.nh MH:MP. Tiit pi c6c bii Cltri y: i. Er'ing thitc xir, ra lihi r"i. chi khi giri rt dBu al5. chinh li vi giii tdt. Trr:ng sd cAe a=fr:6 = 1. bni gfrl vE bdi r,ira em ligigin |r'goc Yan Khoa . 2" Bing phLrong ph5lr crla loi gi&i tuy hoc lcrp th{p nhfit ntirrng c6 bdi gi8'i ng*n trdn co thA ctrirng rnintr dugc h{t il&ng thric gon, s{ng.sfra. nh{t. t0ng tlrrit hon: QU6C TO;.N Vdi o < ar { t, (i:G) tti : nn, I ni'(l +s - ai)* c[ {t - a1) d t Bai 6/136 z Titn tdt c& ctic rla t h{tc'p{iluot h€ i=l i-l rd ngugln, cd bQc khdng qud 4. uL'ti linh ch6t: n fDn tet 5 sd nguy|n khdc nhau x:1, .I2, cf,, ir{r o rlirv : S= I rr,. .r5 SrrO fhu r]r''' P(r1) e(r2) P(cs) p(ra) p(:r5) = LdmTingGiant (10 C'l'phan Nlr6ln,udr : CAc Lran -J. Chiu Trinh, D{ Ning),, Lt Bd 7-tdng (21 Luong Ldt gidt: Yi vai trd ciln ciic srj .rt. 12, s3, 14, Vdn Cn, Long Xu5i6n), Ngaqdn r\gpc lrt!n .(hoa :r5 nhrr nhau nAl1 ta e6 tht gii tlri6r lp{rr)}}p(ril; (tO Cf Phan ChAu Trinh, En NEng),... c6 loi j=2,3,4,5. C6 hai khi rrlng P(r1):=3 r'ir P(r1):-3. giai rdt. NGUYEN KHiC .uINH ;i) P(rr):9. Khi d6 P(r2)rqr3) p(ra) p(c5)= B:L S/Eg: Te m. sidc AtsC co d0 ton htai C5'n phin bi{t hai tl'rr}ng htrp uh6 : gdc R od. C kltat nhaa, AIl, AM, AP ltong ang .{r) P(rz) = P{rs) = P(,r+) =. -1; P(-t.;) = t. ld. dtrrlng c*a, trung tug|n, ph&n gtae dvng lt Khi d6 P(a.i=Q(rXo-.r.2)(a. *r3Xe, - .ra.)-1 (rl J, Clrring minh^ rdng PII: PM khi uri cfti -&hi r6i Q(.r) l{ nrdt da thric r,6i hQ sci ngu;.6r (c6 stn8.sln r : stn'(,1/2). bic khdng quri I ). ' Ldi gi6i: (C&a Ngugdn ,V90c Vdn dfroa i6p 10 Cl'. Phan ChAu Tri4h. Dn Ndng) Thar' .r'=.i't \ i r=.rJ vio (1) ta dugc Phln giSc -{P l
Az) P(rz) = - l: P(,r'3) = P(r'+) = P(.r6):1. Su-y ra fi1-:t:2, r1-.r' j. l'I *.r.1 , r-l ,r'i Ii chc tr1:c Khi dd P(:r') = Q(.r)(r -.u3) (.r'- r.r)(.r'- r5)+t {irng drin r:ira 4. Nhin stit rine ,rl-rz+=:L4; vdi 0(r) lA nrdt ila thric v(ri h1: s6 nqu-v€n- Cho :L,l; 2 vi khi'dd lihdng tlib i.hon dror: ,1 Lrric :t' 3 alrl vA lt: : J2, suy ra : &h{c nhau. Sur la : 0 (e'r) (e,r - :r:) (ert - r:+) (rt, - rb) : 2. 9L-rZ- -2;.r1-..jr3 * -1;.r'l -'lr-4 = li ' ,? (:tz\ (rz - .rr) (3't * :r+\ (tz - :r5) : -r. 't'17.r5:c c6 lhb coi. ring 15 ) ...r ).r's thi .rl - rs, 3l -.rJ. *.{r-_.r-2)X tri d6 i{:-.I. Vg.v p(r)= :u1 - rb vir 15 - cz, 14 - J'2, fiB - ;I'z lA cAc iroc X (e-e1-t) (r -.r'r * 1) (e-.u1*2) +-1 gilnr tlin cirn 2. Ndu .r1 ).rz thi r t - .r3, .r'L - .r'4, .1 1 *'xs; h a-1' rz-irb ii 4 uoc giim clin cria 2' tliEu ndy khdng xiv rr. -P(,r) = - (* -. rr)a + 5 (.u,-.r1)2 -- g; Ytii .r1 !a. nrdt sd nguy'6n ii1' 1.. N6rr 11{.r'z tlri :5 -.r1. !:s - x2. .r4 - rz. x:g - rz ld 4 uirc giinr diin cia 2, di6u ndl cung NIrQn .ril: CAc ba n I-e Bri 7'ttirtg (11 - Luurrg khdng th'E xl-r'rr. Yd.n Crh, Long Xuytln) Trdn, f/rrng.(12'Ct Ngir Si Li€n, Bic Giang, FIi flic) c6, tdi giii t6t. I{dt 8) P(;rr) =. -- ]. Khi do P (rzt P (es) P (ra} sri ban kh{c, tlo nhdrl lln trong su-r. luiin r,';i P (rs) * 1. l-a phin bift 3 irutrng hgp nh6: tinh to{n clI di rldn cAc k6t luAn: .Kh6ng tbn tai BliP(rr) = P(r3) = P(.r+): P(s) - - l. nghiQm hoQc t6n tai hai da thirr: litr{c nhau thila Khi dd. mln iliEu kiQn bAi rr. P(:z) = A(:r: - rz) (r - .r'a) (.r -. ,r-l) X N.V.t{. , x(.r: -.t:s)-l ({) (r iii .4 lA I s6 ngu.r dn) BAi 7/136 : t-ldg tim gia !ri nhi nhdt ctla Thav.r: = lr1 vAo (4) ta iluoc /:-;1r2,.fl - a-s. irdn sd: $l * 14, "rl - x'b IA 4 ucrc lihAc nhatr cria J, y : -6 - 10.15 * lgra 'F 62:13 -- rilr? g6:i i 2rr7 dibu ni5'' khbng thB x5-'r' ra" ,3 _ alltr=F 10 - trin rlt,rJn 1 - Vi i -1,/i I. Cd th8 giir th:i6t ,-3) rz; rs ) r+. I(hi dd P(;r') - Q(r) (r: * tz) (e - 13) - I '(5) t.oI gtoti ta c, { = J3 - 5,r'? - 3.r"f I.,: 'Ihay r : 11, .1r : .{, r:.I 5 ver, .1 (5) tt tluoc = .2(, - S) - 3(.r -. 5) = (,r2 - 3)(.r * 5) : g{,r'1) (rr - r'z} (rr - r-s) -2 I = (.r -'l/r )(,.+ /i) r., *ot (1) 0(,r.r) (rr-..r2) (.r'1-,r'3) = I llat lihic. him. sd tr0n t ir tht, Liiiu tliir;r rf urvc " Q(.r:5) (.r'5 :r'! ) (r5 - - )= :r's ! d uui dang : Y{1, r:g * ."2, x.t - x2, 14 13 ld c6.c udrc i/=,r3-5r2-*3,r - iu rl (.3- 5r3- l.r;l{i) (l} giim rlin crja 2. 5tr1' r'a: - tu (1) ta thd-l' ;1 ) tJ t'cti l"t ( /s-n,,i rll-5.r'2 - a) tloirc r'i -rxz : 2;rl--i...2 = 1::r4*r'3: *i lJ.r* 16:A+1 ndn tu (2) {p dung b{t ding {hri'c suy ra :ir5 =;13 vO lf. Cosi ta cc, y ) 2 r'i cliiu bing . clat cluor-' lihi li b) Ho{c :rr5 -. .r2 : 1: x4 - .r'2 : * 1; chi khi: f4 *:rB * *2 sttv ra rB = rg vd lf, ,3-5."2 - 3r* 1ti = j,/(.r'3--i,.2 -.s.r t- r6) Bs) P(cz) : P(rt) : P(od = P(r;) : t' i+ (r3 - 5,r2:3.r'.1* t6)? = 1 Khi d6 . a - , '- | ,. ,. I I r' rl- - lvoi i.rl-_( V3 . *cr) ("r-13) (c.-:t'5)+l +=+ .r"--5r.--3.r-Fl{i: P(r)=1 1, (r- :rr) (6) (vdi ,1 lA 1,sd ngu;'6n). Thal'o-: 11 \'io (6) ia t+ tls-5.,1 2-:.3.,#lJ;0. r,nl ,,,.i ',,,,,,, clugc r.1-t:2, $y-.L8, rl *,.1 4, x'l -ri trOn iloqn I-l;- t /5 ] ta c6 trai rrghidrn tir , ili'- .L- = -|. VJ le cec udc khAc nhau cria 4. Cti thA coi ! rzlrslallr5. CAc rrdc cria .4 lit: l'rlnr lai giA tri nhrj nhrit cIn lim f/min - 2 r i --.1, -2. *1. i, l. 4. clqt ilurlc tal r' :/f-.t,od*,.r,--*y'.I 10
- --.E:H,F*F+!.F ...., -\.hfl1 cril: Cac bLn }lri .4rtr I'n (10 Cl. El{Sp CAi.'LB;: {;A.(,f) :}ttz 1i = f ,2}. Do rlo C c6 I{a Ndi), Ldm Ti,ng Giang (phan Chriu Trirrh, cirue 1;lrucmg tich voi hai nrdt clu (Ot) I'i (02). EA Nt'rng), t,E Ihi€n T'hqnh, L! Vdn t'ndnh (pbb ha1' C ninr tr'0n nriit d&ng phucrng cfra hrii mit th6ng I'rung hoc An \hon, Nghir Binh) vir ban ciu d6. 'l'ri dti, goi 1' ln, ti6p,dibm cfra hai mdt Ngug|n-Thqc !&n (truong phb thOng trung.hoc ciri nh6 thi ta c6 CI' sE ld tiip tulen cira chring fhlng Long, FIA N6i) c6 10i g'iii t6t. vi t,'12: C/i.CBi - 2R'. \'+-r-f ihrd.. nr[t cli 'r 'T.\i. f: tirn.C birn liiuh fiy'f c6 t1!nh (rI p.ir.ni). BAi B/136: Goi t' i! tQp hfp tttt cri cd.c tant D thtl,c bQc hai P(r) = ,r:r'? + b.r * c sao clta 1p(r;)l'-(l odi mgl gi'a tri 0-(r^(1. lfdtl litu crpc ttlgi c&a IP'(i)l udi mr.ti P ntim trong V. ,f,,,'i "-\ A ,' 1,'- B' Ldi qtl.i: \'i P' (.t) = 2ac + r lf, milt. hil"n bAc 'nd;::ii- ji-*f.- Yr.l %"izi\ nh6t n6n trAn rloan l0,l] P' (c) nh{n cnc gi{ tr.! /--+ cuc ilai-vi cuc tiEu tai 2 iiiu mit 0 vi. 1, do dfi \:'{./ s{' !|n,.,,i cgc il4i cria. P'(.,r) clri c6 thh 6 t tronr: 2 dibrri :T._U llll\' .l': I. \ Y,\ :E*- Ta c6 1"il) = 2a*b=3(6*t,-pr;- (a* 2b*4d*c \ \ llll \ \rii it + lP'(1;; - l2a + bl < 3tr + r,-r cl + la *:b*acl t * lcl=3lp(t)l * +'p(ri:ll-r- Ip('o)l( ll[r' Iihac p'(01:b-(CI+26+,tc);{a*}*c) -3c c =lol(la * zt,+.1.1 * in *D =+lp'(oli .l-cl *3fcl= NltQn;r:it: ban li 7'hanh IId (i2 CT. l,anr Son, =4iP(1/:)/ + /p(r)i+ 3jP(0)l
ll{t khic, Gqi r lA trin kinh 'Jtr"rng trtin niri 1;t;p A''tS(i' Ta cir : a2 : bz 'f" ,;2 * 2lrc cos.l 7 : za, - *Zhc casA. * r: 5l p: ( I, - a )tg(.t t'l): { P - f )1q(ri2) = 2fic(1 cosl) :4bc.sin2 (rt/2) =(p * c)tg({)r12) (1) : .lirc.s sir7( A,'2\ il .l. --_._--- cotg,4: - )-- ) c2 -o:)21r,' (h2 *- : t: t:-o' 1r1. sin.A 25 r5c .1S ."= 1S.lg(.1/2). 'fri (1) r'ri {2) ttt sit,v la. l,ir dii ng 'tlr(rc cSn ? prhii chrjtng mitrlt tttrrtlg'itrorrg vr'li: Do'trr'-:-*l,ts.:ri d(r su_r' ra : (rs)B
F -" :=] r a *Sii S;t39. Cho p::eoeraz...ab (t):,fl ta ngoai tidp AeIlC. Chung minh ring tulRs)z (,113) cos(^+/r) cos(Al2) cog(C/2). rirgt 16 nguv€n t6. Ch(rng minh r*.i:g phrrong trinh ol rk + nl rk-.' -F n2 ,pk-? * ...*a6= 0 lihdrrg T a lt &n l'lt c6 nghigni hftti t*. {'Bii lO,/139. Cho hdm s6 f: (0.-) -+(o,r). Ngugin f iln 'Ydu Ch&ng rninh ring hai kh{ng dinh tuong clu,rng: :-, a\ [{.xf(u)) : cfk) vr'Y)0" 6.Eei 9i 139. Goi R1 lA b*n kinh ilrrdug tr6n b) /'(cr) = {(r) f(u) Y.r,,ylo. ngoai ti6p tanr giAc c6 3 eanb lA l! duiing trung . ft{(r}}- , Vr)>tt' turl-iil cfrs "{,lfl', rn fi Ii. bAn kinh tludng trdn Yguyin Minh Dtrc TOAN HOC v4r ri T0AN TR0NG KHoA HOC YA KY THUAT HITN EAI VL Ps+M EUYftN t. Y$t ly to6n ln gi? Tilrirc hdt h:i-v'l-rong 1d1- mbt irh[p tii:h phAn ric dinh citr dai lirong clic inrng li dg, vE plrttong ph[p v$t lf to{n. co ii tri rn6i r.f tri, ni6i thc,i cliBrn.r.ili ,r'6u.c&u sao hgc, db tirn chu-vbn d6ng c&a m(rt. r'dt trong khi-rirg cho,kri gi&i todr: hoc.lhu drqc phli pht'r hop v6i gian, ngudri ta.c6 thb m0 hiuh h6a r'{it d6 duiri dang cirr s6 li$u quan sirt thgc t6. Tu' rli-v cdrt c6 thB -Ot Ai'b* cir kh6i lrrong m vir gii thlr tqi thrii rrit ra cic kdt hiin lf thrrl"6t lihfc nhau cho cdng tlc tlrqc hlinh, thi dr,r rlhtr lrong co hqc, c6 iliBm I & vi trl vdi v6etcv bhu kinh r. rbimdti'r thE chi ra cic luc ciin tric dring db cfc chuyBn chuyBn di)ng bing hirm r = r(t)' t^d" bing ddng xlv r':r theo t'€u cirr t:ho trutlrc v r'.,. :1 hiim u = drlrlt. gia t6c blug =ru datidl" vit 6p Eb lnp chc phuong trinh r,i ph&l rnir tA hicn dgng dinh luiit Niutcrn m0 t8" quan hf giiia lucr tuong ta plr&i dun vio cl'rc gii thul'€t hirv cdn t{c dr.rng F vi chuyEn rlq)ng b&r:g phuor.rg trinh ggi li c6.c ti6n db hrl clc nguv0n l'i, li edc ioirn hoc' clrin I-i- ituor: rr"'rt ra tir qua tlinh qtran sft lAtr dz; dr\i vi srr tbng hop thr.rc tE s1u si.c qu6 trinh ,r_F = n. vit"l-i'. Tlong ro ltoc -d6 li c{c li0n 116 Nirrion. 2. YAi n€t vb srr phlt triEn c{rs v4t ly roAn' D6 li phuong trinh 'ri phAn m0 t[ chuyBn Nirrtcrn (tg+g - li2i t ld rrguoi cliit n6n rn6ng cho c16-ng cira vrj.t iliBnr trong kherng gial. Phuong phrrong phhp vit li- torin nhtr n'rOt lihoa hoc. V(ri cir hgc c6 thb klr{o trintr ni-r' cho pl.rtlp cf,c nhir'dinh bii to5n co hoc vii chul'brr ddng, Niuton dfi xi-v s&t chu,r'?lrr ddng biug cic hi to{n hQc, dung duoe c{tr: trr0 hinh vii c6c vit, cfic phuong tirc qua tich phAn c6.c phrtong trinh nAy c6 thb trinh vi phAn chuvbn -lfing trong co h
\ ..ffi! :-- (hirn tlic trrrrgl chi phr.r lhudc rudt bidn), plrr.r.rrng J lritrr Yrrung. l(lri qrrnp s;ii sno 'l lrirrn \-uong trinh tlao hinr ri0rrg (trirut dirc tlLrng.phr.r lliu6c r!ri lirii.ri i jrrr llu(. \ga Lec.ren lrhrrrr llii.r.qrri da,, nhiEu birln), phrlrrng- trirrir r-i tich phfrn y.y..: thuc kh6ng pliu Irop v
F l -\=,'#q- ..,. i f \ f V.'rr** d6c u*'j tr;qn &g c&ro rHEnf, mOT cnir r sSry r DINo yr6N . I ill T .i'i i.r7 (tr.. t) cO nou l;ii to6n I (i,ri b:lo ffHOSr; lo..rrr l), u:rl r-Ji Irri- gi&i sau d$v: ud Bii to6 n 2. Cho hinh thang 'DC. Chrng ntnh r&ng: ",LBCD udi rldy AB EAi to6n f . iii; tt:p -gia tr_! Ian nhdt t,d girj ACz +.ttD2 = ADz * Bc?+zdB . cD $ls2 r'HrTj. lrl b! nttdl r:rta bii e i,\rc 12 -r gz uoi dibu kiin; Trong ldi giiii cli bai b6o chi n€u trrrdng hop Q' - !l' ; : rl +{.r2rl2 - t,2 -112:g (1) cic hlnh chidu il vi 1/'cria A .r.i B xudrig C,I) Ltli girii. l-rii,r ^r. n (t)iriin dbi clu.gc vii clang: dbu & trong- doan [-'D. 'I'rong Lbi dri, cfr ci,-ih&v 6 trucing hcrp'libi"c nhau (cic h.uriug hop 1l ho[c (,1,2 1,':)l-:l i..2 f 92)*1*4;r2=0 (Zl 11' tlring r-ao C hole 1) tlbu droc coi' ]A ni.m ha-v (x2'l- jit,'-:j t...2 + y2) * 1- * 4r2" (rl trottg tlorn ( D). n{,thl:r In lui girii clri d,ing nC" I:)r-rt u : ., 1 - ,,1. Khi rlir, t* (3), ta c{:: xhy ra trucing hop trolg irn.q vcii hinh d[ vE. rrl -3rr- l'1'r Ul' BAv giri, ltim thd nio db gi:ii tron ver.r brii to{n ira3. (li - 1' 5) I ,1. r.r - (:: + lT lz. cho ci 5 trudng hop con lai nira,? Chic circ ban dbu thdv ngai ndu'plrii vO thOm cho rfti'5 trudng Giri tlj i,'r:r i:,..'i ril bitu.thri.c e-2 -l y2 li: lrop hinh vi cdn lai va tidn Linh giei ti6p tftng. {:l+t,5)i2. irrrong hcrp nl6t vi t1{-v qui. lt nt,.,t c6ng viQc Cin tr-i l:c r,irr.i , i:a bii\ lhirc cz * gz Ii, dlng ehan. Nllrng, th6 thi plrAi lim thd niro ? Ilan d[ tr'rng gip nt6t h{ tlrirc nlo rld bao cluit '3 - I 5 )13. lruong hop hinh ve chua ? Ban sip nh6 r,a 'fhr.rc rr, 1ui ::iiri ru,ij. clli iidt lu0n -moi "rdi ducrc li.ng rlfl-y... chinh ii hc thirc Sr-lo. BAr, {id, ban IIax (.r2 * ai:) = (:j - i/Jliz t.i min(r2+g2; =" iriv chring minh hQ thr'rc t6ns qu{t sau' di_y vE : (: - 1 5 ),2.c., rl2 +- 92 dat cturrr: circ gi{ tr! biirh phur;ng canlr ,{B trong tam gia. aaC (k-t Ong rir'. Du li di rlins tllii.v ring tliBu clil xAv ra cri irng r.r_ri 96c C nhc.rn, tri ha_r, l,u6ng) : khi r: = 0. s,rrjs lilrcrrg n0u tiibu dri ra r.in 'lA. rnriL thiirLr siii. 78, * x:'+Er:z +tFc.[n (rlong d6 rl ia hinrr iiii' giir. 1a l:i,i. chi6u cria dinh .r{ xudng canh BC| rbi 6p dung _ tliir xct khi nlrrg :ip clr.rng n(r vrio viQc giiri blii toin nii1.. ci'rl r:iich siiii dri r:[]ir tirlr.dlirt6i r:hiit di]u tie" (fI. ciriug han th:r_r'ir:ing di0u lii6n (1') sau t-Ifir,: Ta c6: (*' - y' i 1)2- t'.1.r'g' - 4r'- u'-:l = o (1.) tc2=TD2 + 02 +2cD . ul Bing ci.clr i;i6n rl6i tur;ng- 1u nhu trtrn, ta sd c6: ,r2 --3r, *1
r" Ltii gidti.l. Cqi ,.r: li t'+trh cfc hinh vu6rrg ptrli mirrh rrrfnh d€ ph.in t15rr eia n6 {tirc cirlg iling elt di, ta cd tlrh tich tr' cira ciri hQp li: gi6 r.di n6) sau dlir: