Toán học và tuổi trẻ Số 106 (1/1979)

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Toán học và tuổi trẻ Số 106 (1/1979) bao gồm những nội dung về một cách chứng minh bất đẳng thức Na-sơ-bit mở rộng với n = 4, 5, 6; Pôlyminô và các bài toán trên bàn cờ; lời giải các bài thi Toán quốc tế lần thứ 20. 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 106 (1/1979)

H(rr i'ciN I'I0{1 vr$r 1o* $ffi n .. :- .BnO P.A HAI THANG rr/1Q i Ki ()tit niiiirt, i';CUtr$':iq CAN{I TOAX t'ilr.i.g iri* soarr: HOA.rq] CHilt\C , .f,o:ll tl]_r],1'l li11e rr+o * iin I\iSi r,aj,.u6i: 6z62s ffipT cAept [Fr#rgg ffirHH p6r o&we gpg#e ff&-s{r-s!? :. t Li . ffi*' R0H6 Eo,i nE S, 5, .'6 a, ;: i---/ r v,{* NCO CiNrr ItUT(} va .THiir; QUANG F *.* t, dfB i. t,t\ I.\li I 1l f . iliic lran ihsn mdn ! . cbo rt=.1,5,6, chIl1r ttftt rrrSt.th]i giqn rlai bAi t' t t C6ch ddv hon sdu nftn. tr'€n sd liiro 66, thdy : h gir[o Phan Buc Chinh dd r.rtli rr r6u chuygn hhOng Sa u khi itrgc bi6t < c&u ch ri5.Sn kht,ng do,n ,. {:'= . do.n gi6n"vl udt hii toiu (na griuD. l)d lA c.{u giiini: mii nguoi kB rir;3r€n Igi chue cho bi6r chul''ga v6 bdt d&ug th&c iri-sr.r.-l;:t rrr& r*ng. TtJ i otcAch chung mi.nh c*;,r tr{ooc-dol. n}ru th6 nlo, ,: vA sau khi xem c{r:}: ii.rrtng i -n nrin}r bXt d&ng thuc noi tr:€n ct,o -tiuirng hg,p : 4 trong l-.ei vj6t : ifdrr at, a.],,.. ,_ i, ] O, ti; clra rrrh Li Qitdt /irin il[ng tr€n ,sri,bdo 88, chirng ' r lOi rifi lao viro chung minli b{t d&ng thrtc niy "*--!:1--- - *--ttl- 1- .. vdi. rn(rt ni6m s*y in6 kh0r;g bigt' htpt mtii. Sau . o2 las 43"|ir4. rqhiilu l*l tlirit bqi, ehrlng tbi dA tifir ra ldri gi{i ' dn-l Bq ,rl ' klrdr gon cho cic trudng hgp rr : :1, 5, .S. 'Sru dty lA cach chtng minh erla chirng t6i. a- anlnl . o1:l a2 2 'T'ir bSt thirc * J Sudt 5r) ndrn, nhiFtr ug,rrdi,Jd tirn c6cli t:hrlng .r18.ng ,Bu*nhia-e6p-Iki (r.?+.,'l | ,.. + *?l h:i+ irl . .. r:i) I ruinh ii{t dfi.ng thrlc n}1' vA mli .t5n rrlm lgi:[ A rL- IIo
trong'do eile trh Ui (i = 7, 2,,.. n) Ie sd thqrc tty 2. YOI n : S. Bnt d8ng thrtc cBu ehr,tng miril lir ' ' '(ar*a,+a3 lar+ar)?. . i. i. erl = u1i! a{az * as), 12 = azl'{ az(as * aa), ) -; ':. i > f[r,(o2.+ 0s) f ... f.a5(a1 I ax)] : ... , :ui : aalrf ar(q * az) ; s 6 .. (oi*... *a elz b :; , . [(ar+ ... + as]4 .* tu71- .',. + a?\i y?,1/-l-':-, ar(a2.1 as), yz=''l az(o31atl, . ,4 . t+ 5 (of-g..+afrr ) (,ri+...+astz ..., Un=1an(,,r*as) 2+ +l,t r r& r2i (n?+.., i rijr)16i 1.,..* us)2. i2 + 12) .' EAy chlnh l['b{t rIf,ng thirc Bu-nhia-cQp-xki. ' te duEc '-Vay (!) dring vl clo rld (1) Atrrg roi lt:5. 3. VOi n=6. Bdt dEng thirc cln chr.rirg minhJA |f _ ,,?. "r . +- "ult._-_1_...T--_.-.--.--__. ".r? lx 1 (ar * ... * ou)r' )> 3la{az* as) *... I a6(nr * az)l L oi(nz+ne) az(as1 a4) oo(nt+oz) J Sau khi .kl'rai ililn vir rirt ggn t* co b6t d&ng ' thuc Irrcrng tlrrong: . X lar(aa + os) + ,rz(as * or) 'f ... -{- aa(err 4" gal] 'o? + .,, af; * 2a1an * 2azas t 2rr3116 2 {u "Y e? + '., + ,,)2 hay' i .7 {+, [c1f ar * i*(a2 g as * irs t oolj2 + 2 olGl! +... 'r 3 a2*ag " a1* ny r- "--=(.ar : or -l.as - ae)z b rs. : (ar,* ... * au)a - B{t t1&ng thuc cuoi tng ney la hi6n nhi6n, e a1(oz{ as)*... * ao(at'l-os) , trudng hqpn*6. duEo clirng rninh xoug. -: I Nhu vfy &8 chilng rrrinh btr d&ng tbui: Na- in SE cirirng minh {1) ta chi c}n chring minh vd so-blt..cbi'cH.u chring minh trdt ding thuc (2). t' ph6i cfia, i;{rr d&ng thirc trtn 'khdng nh6 hon V{n diS d{t ra li b{r d&ng rhoi (2) cOn ating .: ". n>6 hay kh0ng ? i,{Oi "U_,.uF c.ic ban ti6p ruc sul"nghi nl2, t&e li e hirng minh b6t d&ng thuc (D sru dAy -: .:"vts vd'u de dd. Ch0c_cdrc Dqn dar ,iliEu i,nq>6 [ct- :,t ) g ,4 ni'i- -'- . ----rt6r.q"a.- : * :, r f) (41 *.'. +ooX;-'!* * ,.) ktri rrhAu il.uoe biri nI5., tda iorri ", ".I:S. -'lai .Sau nh{u ti6p duqc hai 'biLi nira c6 cirng u6i _J - X [ar(a2*as) +,.. + an(ar*ee)l (z) duug v&i bli niy: - MQt bii cria hai bgn hge sinh^& HuE kh6ng ghi hq ten. .. 1. VOf n.: 4, B{t ii&ng thue c!.n eh&Dg ' - I{Ol bAi cfra brn Ld ThdngAlftdf, c6n U6 Uno, minh li F - (Ngodi ra trr.Oc d6 m6t s6 bqn de grli bii vE (ar * c2 f . a3 -f ai? El*2r.i1a2 f atas * s*ag'+ vi0c chri'ng minh tifit dfing thrtc }ia-bo-bit rn& t .'t tllS aen T&a soq.n, song c{c cich chrtng minh. 4- 62a4 | alaa * a3o1 -l o1at * a4an) crla c{c bqn d6 dBu c6 ch6 sai). . .r6.19- []ua vio cich chrin'g minh_cria bJn L€. Thdng t) ai * ufi { ai + ai > 2aLu3 -f tu*t -l'ldt, tlong bhi nhy chrtng rOi da tnuy d6i chtl tt cdeh ch&ng miuh cho lruong hgp n ='6 criu c6.e tdc gii cho gqn h-on" +*Iar- ns)'* Gs* a)2 ) o. , " Ngoti c6ch chu*glririnh nhu tr€n, tr6ng trudng ' . . B{t itr+rng thuc cudi cr\ng Ii (2), dtug n& dq dO (1) dirng v6i n = hibrr nbi€n. vAy hqp n= 4 hai ban hoc.einh & tlu6 cdn d,ua ra 4. rf
p0rvr*tN0 vA cAc BAr roAN rn*rq BAN ccy r.6 rrrdNc NE,tr.' '\l [, lnirxc sd ljfo tor (sd -] Eiln] 1978) toi'itr giOi qua mdt trgc) duEc coi lir nhu nhau. n*hu rdy :'1 r thiQy yo,i cdc hgn mQt sd lini to{n rr6n bin clsu nrin6 c6 t hi&u, tt6minti cd I ki8u, 3-miub cti. U trAi bdo thr h*i ui-v tOi xin gi0i thiNu c6 2 hiEu, 4'* min6 ed 5 kilu. Cri1, h4u s6 tiriy. vOi e6c b4n m6t kbdi ni{m nroi lt p0lynln6 5*min6 cd 12 ki0u, 6-min,1 c6 35 ki8u, ?*min0 vA. rnst s6 tr.)i toaninir vi Ii€n quan gina pOty,6;n6 cd 108 ki6u. 8-mind cd 309 ki$u, I .. miu0 cd 1285 vi .llin ct} ChAu Au i kjbu, 10 -rnin0 e6 {460 ki6u. Yi0c xic dirili m$t ' Trrrdc hEt p0lynrin0 lil gt7 'fa hf,y ret mqt cOngthirc tluhs6kiEric&a n-miiro1A mgi lAnde biln c& e hiiu ,iu vr\ coi rn6i S vudng cfra birn Iih6: Cho tdi nny voi n)>10*tin chua drgc t10 nri"y et}li mgt 6 vu0ng rlryn r-i.. Pdlymind cd th0 hi8u gifii dip, ngav tltrrlng lrqp nr-9 1,i n,= 10:c[ug II m$t hinh gtr6p bii,i cde 6 vu0ng elon vi sao rnf i il*gc R.C. ltead tim r:a nh m 1962. {)d nhibu loqi cho mdi ii vu6ng kE it nhdt v6'; niQt 0 vu6rig ioan chung quanh ptilynrin0. Rdt nhi6u nhA tbdn khic .tlr*o rnSt c4nh chung. Pdiymins tqo thinh trgc tren th6 gi*i dd rrghi€* ciru vB nh&ng vin b&ng crich tltn t.lo ghdp n 0 r'uOng ilun v! iltrgc t18 uAyi ch8ng hqo nhu M. Gnrrlner, S.\Y. Golonrb. ggi li u - minO. Hinir ciri gbrn rngt 6 vuOng'dvn J,lLrtndersou, J,B, kelLy,. v.v.,. NhiEu lqp chi vi ggi li clsn nrind. .\gnii re ngilr;i ta thulng to{n hgc rIS c6ng b6 ciing r:hu (lE gi6i thi{a vb !9i 2*minfl iir d6nriu6. nh&ng v5n .16 nA1'. I(h$irg nheng th€, nhiBu bAi todu vE p0lynriirO tli iltrqc Igp chuong tlinh cho mdy tinh diOn -tr} gi&i. (,l bii bdo niry, t0i.chi f, dda aal gi&i thi(u nr$t s6 bni toin thugo loli -rlung c{c m tiilla6 l pirlyrxind phr'r klu bdn cd ch&u Au (irb6c adi thro cich iroAn toAn t,uong tluong iir ehir bAa Sc ITI 3 - nNtldig chdu lu thAntr nlriiug pdlS min6). Bii tof,n 3 & bii bdo tr.u&c de giai rluy{t v{n & rTi,Tl ,- tuitl*,6VyL +- ntai /&9 dE chin biin e& ehid.du thnrrh 21 hlnh 3*miu0 th&ng vi t hiuh don mir:0. Bfly giir ta d{t v{u dB lhay 3..mind th.ing b6i 3 - rnin0 vu6rrg gde tbi s€ nhu tbd nio? ffi +- r*h6t*1r 'I'ri c6 b&i to{u sau: Bii. toAa B. ll6t bdtt cis. chdu ii ci.i lft& chier ry +-,frilr? I daqc tltdnh 21 hin/r 3-rnind utloti{l goc ua ntgt # tlon nrfarl utyi ui lyi c.i.a dnn milld nda clio tr$&e Iny y, ,1, abi *iia Lai.gii.i. D& gi&i bii toln. ia lln I ust xgt : 1) Gri.sfr tn c0 'mQt irinh vudug kich lhr6e E:: l- miad: L ? x2, 116 rirug eiri don mild & vi tri niro thi phLn cdn lgi cdng li. nr$i,3 miriO vtr6ng goc (hinh 2a). 2) GiA srl la c6 hintr vu6rrg kich thirdc lX4, trr $inh X chia nd lirrn 4 plriin kleh thudc ?X? (hinh ?b). Db tinh ddi xrtug ta giri sir vi tri don mino & I rong .l .0 nu0r-rg & phhn gqrch soc, tlreo ptrhr 1, llinli 1 gisi thieu v6i cit: Lqn rngt s6 rtgng thi phiiu crin lai eria phEn ed gqch sgc lir ur$t pdlyrnind yi tCn gqi cfia ehring. CAc. p6l;'mind trinh 3 - minO ruOng g6q. eOn ha ph4ir [h0ng sai kh&c uhau m$t phlp d(ri (quay ho{c ddi r.(ru6 guch so; tr cho drvn nrinO & vi tri d{.rrh cl{u uhu
q/ . ., hinh 2b vir ba don minO n.by i4o thlrrh"mOt hinb B*i. toin 1O. linong ptrr'r AIn dtrg-c tiit ci ttic . 3 .-- min0,Yu6ng g6c. 6 ott|ng li6 mdt cqnh cfi.a bdn cit bdig cac hinlt '1 ,ffi 4 - nind riin. .x BSy gio ta i6t biri to'rn: X x ts*t toan 11. Khing thi- cltiu btitt .cis chdu ^ thritth Au fi hinh 4.'-tttindT ud n$t hirth 4.*min6 tlinh 2a re 9tto.nq ' H irth 2b Lbi gi&i: 1'r0n.b[i:'cd' ehhu Ar1, b[t cu inQt hinh -1 -, miri0 'f n5.o dBu chfta sd l3 ed 6 tten vii s6 16 sd 0 tring. GiA s& chia tlnqc bin,ed ch&u \ Au thinh 15 hinh 4- min6 f vA' nrQt \inh-'l- mirfd vudng thi v&i 15 hin} 4-mir:O T sE chidrn m$t s6 16 0 den vilmSt sd 16 o tr.6ng, edn mQt hinh 4 -min6 vu6ug ,-bao gid eflng chi6nr 2 6 den vi ? 0 tr$ng, vf y su;r ra'sd 6 ,ieu-cr1a trin cd I& 16 vi s6 6 tr&ng crln birn cd *rllg le, v6 If. Bty gicr circ 59n-h6y .nghi cAch td mAu l4i bIn co rtB dring l{p lu&n tuorlg tu l;tri toan 1t Cl&i bii joan sau:(v.{n t9-.tn{u aen S:.0), Hitrh 2.c Bdi t*6n L2. lfndng thb chla bdrt cis chdtt Au thdnh 15 ltinlt 4-rnind L od mdt liinh 4-ntit$ 3) I)6i vdi hdu ip Au (hicl, ihutvc 8XE) ta ehia ban co;thadh"t,it,4 phln kich thu6e'4 X 4 ' v&i,lAp !ui4*nhu hai bAi {r0n vi tO miu 14i (hinh 2c). Cae bau th{y nga,v l$p''lu{n tuong tr.r bdri co nhtl hinh 4 cAe bqn sE gi&i tlrtcrc biri phhp 2; fa sE di, ddn k6t qu&. , .Nhi '.'{y dr\ng phuong ph6p qui nqp ta cd th8 roal sag uu1: r,. Jt- chia h'ir:h vr4irg kich thudc 2o X 2'1h[nh (2" X 2"-1); 3 hiutr 3-minO vu6ng g6c vit 1 dcn mil0 ts.ii to6n L3. Xhdng th? chkt birn cb chd.u & vf tui ttry y. A'u tndih 7 hinh i - minA uudng'uti 15 hinh- 4 - ' E6i vdi i6c hinh 4-mind ta cd cic bii to{n: min6, tiortg dd gbltt i4$f 9d (c6,tnZ tdt cd) hinh ' EAt ioAn'9. .Rdn cd'chi* Au.co tlt| chia dtqc 4-miid th8.ng cr)t lqd lri ctic hinh 4 min6 riitt' thtirth $. hlnh 4 - tttinl cins ,ki?u, trit ki?u 4 - . fi1ina &i|n, C6.c b4n thdy ngay l0i gi&i nhd binh'.3, Cdu ,,/z/ /./ . ,ffi , .i , '1" rldi vdi kibu .l -mind xi6n ctc b+q h[y trl"etr{rng fi a ,il ,4. minh kEt qu&*manh hon: fr % ,*, ,4 #n w ffi n ,/;: ,4 7t fr ,fr /rt 'fr a w ,4 fr ,rt I I 7i, ft;: I I Elnft fl fJ tnfi 4 I 'v *,
Truoc kti x6t cf,*'tai Loan vE 5 - min6, xin gi6i thiQr vCri c6c L7n ti-t ci i2 kibu 5 = miud vA 1011. gsi chringr (1n'eo t,iutr &,trliin s. lU:U,. f--t rl . H--lJ Ii t_i Lmi$ 7 t&rta ,f.aatrtf ary)ffi. - w.'\./ in / S. n;ta! W . d, m*i )t i-"r-'l .--F-t 1l ii t^-i*.1 l_-,- ,-J 5-e'">ii : - 17.,2:; f &.,!7ilA0 I 'c I i--r--l r--il r- IJ -l- ^-rt t'-. / -/' ,f- r::nd y' ,r- E r'dLq Ft i-m)ti e Hirth 5 q* e BAi tof u 14. :l6t tan'c6 chd.u Au c6 th1 ehitt'd,nqc thd.nh-l2 hitjh 5-rnind uar4 hinh.dott 3tr mrn6; tr.on11 tfo cti.c hinh 5 mind yl'du khtic ki|u - nho rt. &. Lli g;fl1 LAi iorr u 14 rit dri dqng tbF hi€n & hlng r'5t nhilu phuong ph61i chia, chi xi'n gioi thi6u r.0i cic hgn 3 efich chia & hlnh g. E[c hi0t, 4 hirrh don min0 I4i cc thE nhflp' Iai thnnh rnQt hinh 4-mind vu6ug & vi tri trif j- I $ Bdl toin 15. UAt br)tt.co chdu A,tt c6a th|. chia dtoc tfidnh 72 hinh i--mirrci gbm cac kibg khdc nhan.od m.Q.t hinh 4-nin6 uu6ng1,. ucti ai tri ctiu 4 - tttitd vtu)rtry rtittl ld ttig g: . Lbi gilii, 'f a th6-i"c6 49 vi iri c6 th0 c,.r cfio tArr hinh ,l-minC, vu6ng (hinhT); nhung
1". -! ?., 5 * rnin6 Y. Cdn phln bOn rrgoai'hinh vu6rlg cuoi ci'ngxin gloj thigu itiri sic b4n bii to6n g>{g ta -ciria iharrh 11 hinh' 5 * minS gom c-ie vE sg nh{n gdp ba do giio su R. r\I. I{0binroa ki6u" kh{c . nhau vI ktr6.c kiEu 5 - min0 1'' nhrt (Mi) n0u rn : r! hinh 8. ' Rei to6n 17 . Itfrg cign n$t troi.cl t2 kipn .,5:':tndnd tlh u$i9 trong sd 11 ki:-tt 5 * mind cdn iri c6 th? -x? p tttdnh mfit hinh dbng dattg ubi " /,'ii,u 5 nih6-id chon. {Tltit nhiitt.hinh lSltip -b&i . "9 Ailrr 5 - ni.nd sE dbng'dqng uoi iitth i - *inA dd chpn theo tfi fd 3t l). Bhi to6n duqc gi&i'voi i{t ci il2 kibu 5 .- min6 ; xin gi6'i tbi$u l&i . gi&i cfia 2 kiBu (binh ' 9),, cbn c6i lti giii khqc mong c{c bau linr toi; rhtc cAc bqn gtp nh,ibu ljr thir" '# e flittt I Illnh I R6y gi&.e6e bair hey suy nghi.vilrr dE suudty e r 5 - miu$ hhsng g[n vdi bAn ccr c_lriu A u n&t : fr . Bii toAn 16. Trottll- ctic hinh r/ifr' nh{t /.'It:h tttunc lxfi, 5x12, 4;a15, ud 3x20 cd ftinh c}ft Yi tthilu .la tto ntn cu$c thi gt&t toa'a ilhit nds chla d.ttpc tltdp.h 12 iinh S - mind gbnt kfi niQm 75 .ndm *rrd{'q*dn bdo l'IlvTT . cric /ri8u khqc nhau ? ' khdng tidtt hinh dvtta rrfra, iMeng cae bqn' thdng cfun. C6c bgn cht g ket qufr tlugb tr&L loi li q${ loqi hint chii nbst irln dEu ehta {Lrqc. '..-" nl 6 ,
--l- - -*-i< LgI GI,A[ CAC Bff TOAN THI QU$C fE rANg TI{Tr 2S Tt ().\(j s6 liao i04 dA gi6.i thiQu r_6 kj, rhi G loiu Qu6c tJ ll"n thr'l 20 vi cAc bii to{n 6k*1 -l- 6tr-? 1 ...+ 1 = & (nrod 5), Vj ve;,- (*) la trong Lj thi. }iin gi0i thifu ti6p v&i c6o bqn il u o,ng*voi /. ; 5. Det /i= ltt, la co tt-.n.=2At. kli gi&i ciir cic bii todn d6. ^ 4lurrng - \:+]- IoJ"-* J == lu3tot- 1. Lli.giei iJii 1 & dfly cfia bqu Trhn Cao Son ' T;'.0 tOa2:- 10609 IUO (motl 125) (l&p 10 ,chuyOu toirr Tru0ng D+i hgc Su phgm : -" -15 (mod 125). Viuli) khtc chtt it vfi cich'gi&i trong dgi dg \'{y loi2o' - I : l2i++ 161o' -l : 1zF thi cfia la. Ld'i gi&i bni 2 dtng p}[u thuf,n cfra (+u6?)5t* 1 : 12i*+65i*1 : 125 ban Sun, cdn phhn tllo ilrrrp'e chfrng minh mgt cich gon nlr.6t. Lrii giii b$ri 3 efia h1n LiTttdng +.;,;776'" - I :. 125++26t -'tl- 125 *=i 26t--l "j- 2rjt-2 + ... i- I : b ' Nt?di (EHSP Vinh) srl dgtlg c6ng thuc rnl hai r.{:tl i b, v;iyn - m i 1oo. bqr Ngrry6n Tl'tanh Tring v6 Ngug\n Trung Hd trong dfi du ilii n0u ra. Bii .l vir bli 5 c6 nhiEu f ir cic diBu kiQn n-rn; i00'vi m)3 ta suy cich giiri, 6' tlay chi rtnh bny rnQt c6ch (cAeh ra v6i- n: 103, nr - 3 thl l.ilng n * rn cr) gi{ tri gi6i tlong hii criarhai bqn LE Thdng Nhdf vi nh6 nhf,t. ?'rhn Cao Son gfri Td:r $o;n). L&i gi&! bli 6 c&a BAi 2. Pltiitt tltudrt. Cii su hiuil r,&rr cli cjio ciia h4n Irfi Kim ?uffn (hoq sirh ctri ttrl). c6 tlm lb O. Ta tinh 0Q. . BAi I. Di€u ki{n dE 1g7BB vi 1978,,i.co I cht sii cuiii biing giSng nhau tuoug duong v6i ', 1978r*1978m i tOOo .++ 1g7g1,r 1,rrrr,_r', e tl : to.oo DiFu r:iy xi.r' r,a l rlucrng,r,dri PQ=Pn + PB +PC l03o-*i-1: 12b. vl vi PA, PB, PC d6i mQt vuQng g6c'n6n De l,lln-"'- 1 : l2J tlri: PQz -PAz + PBz + PCz Nhu vdy: 1) 103"-'* 1: 5r=)n-mi +.oxt p;'1n=4k. oQz:api + pQz +ufi.fr 2) 1,r3n "'* I i 25, tuc l0rak - 1 :'2S. ="apzlp{+rd*ry,' lllo 1l;l -=. 3 (nrotl 2I). + 2()P,(PA + PB a PC) ,n€n diiiu ir6n tuoug .: tluong v6ig4k- 1 i :r +=; s1r.-.i ; *,n9 f;- i'i ;; (p.,fz.+o pz +2ffi .iit + \P Bz + o p' +zfp.fnrl (vi 8l -- 6 {rrroti 2J) ) :
- I U cil) t? ,lo4,u/'L l'8, t'{) lilug nh.err. 'fr} rlti ta su1' ra ' r'O I /'"\ /rt;Ji '--Iop*-- \ rlngvoi mSt
-*/ i)o rld bb d€ dirrg r'6i r:rgiid t1r nhien n.)> 1: Do dd ta c6 Yi rrgy: g(r) = 1i7fii)] lF l:/tl) -i- n. O.tK :}' lI'B * tatl ah' a, tra3 1iltn)] -/(n) i n - 1. hly rj ilo 1(r) - i r'o.n .ri(t) :;|1(tlj -] I :2. OIA' = 1-'R ..1()11:lF (1) "
/- - {trt:o { .+,Gii sri khdng eo=nrlo,: ii:!o c6 hqri dd lh urdrc /J; 06 hiQu tld un!. voi ca.c sii thi vii*r nra gd tht. trt b$ng t6ng cic s6' thrt tu tu'bi, Dr,..., llr{Dz {... {Doo. Ta lai I,66 r'd ctta hai hQi vidrr :rrrrlc d6 hoic Lrdrrg Irai thn s6 xdf c6.c hiBu 6oo*&i, b6s - bt,..i, ixt--br, bz 'thu tu cfra mdt h6i ricn n,rnc dd. Ta rit 329 tht do gil sir ba4 ilhu, 65 hiirr riay klrOng-Li,, tht hiQu a2 -- tr1 48 -- o1 . ., o3go-at. .{,{c hiiu ni,r'- li sd thir ttr crla h6i vi€n nAo nirdc B,-crrng rISi m6t khdc nharr vd do gii si! vua nflu ri*rr nhu nu6e .{. \ {.t 6;r hi6rr ni5, Ii sd thu lr.r cira chring khdng phli ltr c6.c sd thrr tu,cira h6i viOn 65 h0i r,'if'n trong 4 rr,ii0c-cdn l4i; :v.y... Cft ti6i., nio tr nu'oc A. \]qy -chirng tri sO thu t+ ci"r+ l.uc nhrr trOn. cn6i tr'rng /2 - f1 kltdng, Il.thu tu 329 hdi, vi0n cta l-r nii6.c ..con l*,i. \Ifi.y tbn tri Eir:r l:oi vida-tlrdc na6. r.ri Ij vi hiri yii.n ciia IonhI :sij {t nhf,r l:rt\t- J -n r =66 hifu ti thrt io .iro 6 nu-oc rlrrgc d:inh .sd thir tr,r tu 1 riio 1978. l5l -f,icn + Miu thudu do lac b6 giii s,:r. J-rarr rlAri. yit5- ta c6c hQi thu6,, rir6t rrldc rrio d6,.chhrrg han c6 diFtr l,hli clrung rninh" a Ndu, s{ra iai. dlu- bili rtrrr s.,u thi bni to6n'trii ndn lih6ng qu6' tl6 : 'Chting minh r&ig uiti tn,)i sd tli tthiin n ihi sd &e nnu . I:ii' ,4nti ' o4,r: ' +2"-' .1 *i kiE {*er&e ludn chia hdt cho 22. ' I'a cd 24o+l:2..42o:2 (mod:r) vi 4--;- I(rnotli) ,\ - V{y Z4n+l :2 + 5ll vdi k li sd tq'rrtri0u. Eat .N:,32no*r+2 thi N - 3345k l* 2=9.213k + z lo BAi f/103. Chi'nq tninh rdngh( phtrctng trintt (rnod 11) vi 213-.1 (mod 11). Nhu r0l'N i1l. r' * uu.=- 151e64 Ta lql ed 34o+l = 3.g?n=3 (mod l0) .r-i g:- 1 ' d , J .t'u (1) (urod I0). \'ey 34n*1-=3.llrl idi I lA s6 tu nhi0n. I *'y :197915 (2) Dit Jf:2r^4n+1' +g thl. klt,6ng co.ntthiim nguydn..rioi ry,qi s?j lV nhtdn k. t JI = r3+t0t i.!: g,lZr! + 3 =j 0 (urod 1t ) Lbi,ei&i. 'tl0 phrrong trlnh I Isn kbdug c6 ng.hi€m ll 32::1 {mod 11). Nhir vey ff j t1'. 'ngu1-6n v'bi rnqi sd tU nhi€n /r. Th{t vdy,'gil Tir c6e kEt qui lr'0n ta s,ri'ra 7'.=.\-l-.U ph::i srl tdn iai c5p sii rigrrl'on (e-,9) th6a min_phuonp tr.inh (2) thi * vri y dEir l& gic sd 16,.,-k vi ryk chia h6t clro 11. Nhu_ng d6 th{y ring ,I Ii srl " chin, rio do T i 22. eftng ld cac s6 16, ril vqy c{p*(p, u) 116 knong tib ] r..r. l_a nghi{m cfib phut}ng trinh (1) ttuqc. Eni zlX.03 . Ch{rng mirth.r&ng o6i nroi sd t\t BSI 3/I03. Tim tdJ crt ctio s61 tir n/riril.p sao rriidn n //ri sri cho 13P-1i53. I+r*r ,,1.=Br. ,-*2{-_,.+b, "i,, +i I-ui g;tii ctia 7'r'.in JIrn/r, lr;p 7,\ irrrtlng c5p 2. luan lrt nqp so. 'I'rung-\'u il. nnng lai khOay' nh6 ph.fit dt t:P-J chia h0f clro ll.:'t'& tl
',/' 6 Ad k ti .6 rtgu\-Cn vd il(r{13. '[e ed: )littlt ud itii Mitttr Hdi (Ao9 t)lil'It), Drong Ei[c 1BP-1-131Sk.r- 1 = tsr(ls13k - 1)t (13' - 1) -Long (9E, Hung t'uorrg, \'inli irhri) c6 lSi giAi tdt Ta dd nisi rg13k - I i 5i; vd1' rnu6n 13P- I i53 L.H. ,* thi bit bufc {3r-t phii chia hdt cho.5li. Nhung- r{13 ri0n r:h. BAi 5/103. Clro rl?i-r si.r,1l ri2...,r.n theo qu*. 'li6lu-lgi.otic sd ngul'on 7r dF cho ilPi-" r igll Irtr,ll sau: - . phIi c6 d gng -;r:11]t (/t ngufin bfit hi). , . !*, ' r! =:sindl . t-.H. *2 - cosc{l sinag , :r3 = cosGl c()sd2 sinctg. BAf 41103, .\crn lfr'l S ctic ,tfiy sd fgam m6r sd lritu liau sd hqng) co titth ch?it sttu d,dy: T'hng . 3n.*1 c-os?!cosct2....cosC[s:2 -Sin""r--1 cttu 77 sd hgng tidn ti(p {fu i ld nrdt-sd dtang, ia rbh6 cria 7 sd hqng litn tiCp ti17 i1 ld n$t sd trn = cosal cosd,,., .sosan-2 cusa4-t. dm. ttdi m$t d&y sd ubi litttt chdt trtn-ttd' cldi Chtrug mintr r&ng *t+**,*...+ c* ) tln; ntfi{ gbm ltuo nhtCu sd hang? Ldi gidi.I)[y s6 phli li,n gbrn l0 s0 harrg. Ldi giiri (cria nhiiiu harr). hqng c0 ' Theo gii thi6t ta c6: .. , ' Qul v$y; dey s6 eau diy gbm .tr6sfl*13' . linh chtt'tlt n€u: 5, 5, *13, 5; 5' 5. 5;5, ,, fl, 5, 5, ._13, 5, 5. tBqn tlqc hdy phdt hi(.n -'e i t .1. [*1 =cos'41 cosza2,.;]couzr*,, 2..,,02c*o-11- -13, qu,v lu{t tlq'ng d5y sd niry dt glii triii to{n t6ng ''+.or2r1"or2*r....or'2r,,-zsinza,i--1 qu{t) ta chi eOn phii chr?rrg t6 rilng n0u mSt : aor2o1 .or2(*z...ao.2cxn--e ("oazan*l.+'siu2c[r1 J] ctl.v sd gbm tri 17 s6 hqng tr6 lOn kh6rrg lh6 co = ^os2?Ac's2 o2.,." or2 oo-2.. tinb chsl tlir n0u. .'fi + r',,'3-, -.o12*1.o.2qz...cos2s.n-3x Gil tlri6t trrii l4i lii ngdEv sd a1, n2,.,,, n16..1r'.' "3-l - X (eoszs"-2 4siuzo;-p) cd tinh ch{t dfl n6u..J'a co: : c0s2 cll c.ruzaz..,cu"Zar*-d S=(ari oz*.,. -l- o1e * au) * '. * (ai 4 at * ...1 ali'+ ota). * r*. ,'r,D "{--afl. *1 - *ri 1- + ct,.2-t I - r, ,. Lng rhtrlc: punl"riac6pski cho 2n s6 a TIheo bli JtrT d AI *(a7 1 aa * .... + {116 * ari).} 0 ltBrrg rih trc' air &2 &2,:, on vir btr, l|:,:,., bn thi cria 7 s6 hqng duoug)- I \ln {Ilrt bi,r) I.o! I I d( I -( o! \i.l:l5I i:-=I l: I .(a,z*ds+...*as)* .t d'1a,uudl Ith ri'c, \.,:ariyyrrar hhi vn chl khi oi:tbi (1alt gt ' + (au * are + '.. + otr) ( 0 (tQug eria 1l ' .i:l ., ,-,....11. Thh-r-'t' ql 1, 6r =-i,? vio b{t d&ng lhire. lri-ilurrc s{ hqng Am). .-". Do rld gEp miu 1hu6n. .2. (r 2 + -r 1+jt_z .,. trZ1 MSt c6ch thng" qu{t, eho nr vi n 1A hei sli tqt nhien nluyan l
-" $*i 6 .[03. Ctro hri ohurrns trinh OIIiI c$t du&rig trd.n t'tqi ^41 vI i\ thi.iM v4.4.1 .2r li hai ti€p tuyEn c,i, duo'og trdn ltln 1r€n'n:It +/rr,Utl:U {1) rrYa ph*ng do' Ta v6:"ri6rrg m{l ph&ng Ol/A n-hu h}uh ;iF: 1;Gai 1 li giao di0m cria MN vi Of/ thi / uirtr = l) f'.,1 'tr0n m5t phfrng chira cludlg trdn c. Ta ctri cln . fr-n) churrg nrith 1 cd dinh li xoug. I irit tliEu lei{n cua nt dt hoi phttotrty trin'h d6' o ttri'ttq dtt'onq. Lb'i.91iiri. t. Hai phriong trlnh (1) vi (2) tirong tttong ndn chtng,-co cing mQt t*p htrp nghidm rdng, tti'e lri phuorrg tliuh (1) vI phttrrrrg lritrh (2') sarr r-l A.v uung ud ngltiitt ( r') Phuung trinh (1') vfr n$hi6rn khi vd ehi khi Ar=m" -l(0 har i 2{m( z. (2 ) vO nghiQrn khi vi chl hhi . Phuong ir;inir Az.= 1.- lm ( tl .hay m) 114.' , ' Vey (1) va (2) v6 rrghigm khi vA chl ktri ' l/{(rrr
jr,_--. Il{ Il$c), DtLo.Xudtt Ofing (Hgc-Cheu, Son La) cd 'Goi kho&ng cieh tir Dr t6i ].ra m{t c&a t.t'r' diCn 1{}i giai t6t. cht?a 1);{6i dif n rt, Il, C lA ra, rB, ec, tac6" .1..s. ts * rc x'A = "l'I! :;l'f' = Bni S/lO3 , Cho lhdi tft d.1€n ABCfi; cae
Gi[ s& P, 0 ld lrai t1i6rn nflm lrollg -di€n khdi,t* _ Trong d8ng. thrtc (1') ta nh{n th6y y + I ( f. nBCD vd nr6i di6m dbu c6 khoir:g eAch t0i Nhung rI Y * I vd 2I: N, cho n6rr *it ncn hlng' n&a tbng cic liho&ng ia*t to ,tl+d=to+ T,' t' (6) ' aliBm d6 tdi ba rir{t cdn lai. Khi d6 mc.ri iliEm ' .thu6c do4n FQ d6u,cd tinh ehfl.t nhu P vi Q. V*I1 1:N. ' ' \7,) . Ii€p dd ta nhirn th{y- i:fi.ng- (nhu & cieh 1) They N = 2I vio (I) ta duod ,.. ' .idc ttitm 8", C' D' pd Linh ehfit cln phiri chrlng. ; minh. IlSy gin gii s1} ,U ln mol. ililm bdt I(6t h
,. BAi 6/L96. Ci.,rrng !ninh rfiug vdi nroi .s6 tU :, n'hi€n n, ta lu6*u c6 . *' I! /(, + r)! 2t+t1,7 t \-. Ngug?.y Httu Itann (DH Kinh t6, "l'.P. Ii6 Chi r,tirtr) . *- : i. i-:i{t 1 . IJ! ' Fei 7 / BSi X1106r TiLn c{cid tn Dlii6s co tinh cirdt : /1t,'8. Cirrtng-minh lang phrttng. :trinh bau df,y kliOng cd nghiqrri duong r.6i khi cbuy$n chit s4 tI[u tiE* ci:a nii v]o vi lrl nrgi sd tq uhi€tr rrr,. n "g.ryen sau cqng thi ta duoc m$! s6 liOm S lln s6 lran i!hrr. (9 +/6t )"+(g =lu )':r.Zx*n+1 Rtii Quang 'l'rtdng (Ha NOi) Ng,476rt Ngpe litrrid ({Ifr N0i), I n+i Lllfi{} (,ho pr, =-ttL },1'. Ilirl, ttm t{t c'i i:* circ gi.i tr i ciia n dB pn Li sti nguyen t6.' 'l'iti Rfu Ngoong (DI{ kI thu4t c0ng nghiAp Viot B[c) ri, a Bit 3/t05lttro.tru6'e ba s6 duong a, b, c. ,/ Bdi S/106" {]hitug rninh rfi.ug ndu hin}r hQp i",. :\guui la thlnh l{p cdc di1 sii llrr.',,n, rl,n nhtt satt: etr& nh{t c6 ba cqnh:l} d, b, c. th& ticii. lA Y, ')- ba gbc tqo b&i duqng cheei.vtr ba cgrili ,1,.b, c, tuotrg: ung li d,9,ry thi ta'co brit tl&n$ tlirc ! .' ;tlii ti)2 thi rin: /r"-r,"rJ, rro = /f-, 6 0-b-lc- ,6 r 6 ,r'n: y Ilr._I Un_I. --lr'. ' -*: ir----F---t,r;' ) 2!s71'' {f. ' cos'"Cl cos'-i, J c.os'" ii- rtm {4ror rqn cus cric dily sd dd. (Cb(r f :' ' .. . d1 . iir,r Ax '=-1 vr-ri .4 lA'tri.rrg s6 dirolg)' Ll oir.n g lt d-n P hn'o' g li{ ii. (1011 u i"' lt-t, Kim Sr-ru, [IA. Nanr i{inh) Liri U' ,l ' BA.i 9ll{t6. Ctrirng uninh ting G.: Bdi 41L86. Giii i;rhuo'ng trlnh \ 271oc6v'- 6.6{loc6t : Br2,- i lr:gloc6: .k .h ,,.1 - a . Edo Qiottg Eitn (Tludo{ BTH trurig uarig, iiii l{rrng) , 2!n 1/,t1 -?u ; : r-Oi n vir lr ln's6 tg ntti0n. 411i- n 1r inh Citi11 ,\'-g tSci nttsP vir,li)" .,liar -4 < ,I, ';; ( 1 a n. ' l' * l10u ..,Ii lso'r BAi f 0/106: C6 tilri ta.i.hn,y irhbrrg inf t til thuq*r./(') hs s6 nguy€n khong d;,ng nlrlt l;i*g / trol:g cl6 0 cri gi6 tr! tu-t'Qt rtdi cic h$ sd r:ho hrvn 8 inri i." . chia,h6t cho ilri l"Lrr'rc a i {, _-4 I { iu / l:t /s *-1-- tt-' \ t=l Ii = ,l-' *-''4t' V1 V2 o ,"',-1\{Wt g(r):4rs -- +r2 - idzp & l)vo tt.c1 7- tt 1' h r\ t h" it t .v-. /l.., l1 E * Li thdng I'lhdt (DttsP \:iriu) " (gCT llil'f LI Ha Nai) --r5 { i
F-, .'--"i .ry -.-F t,, ': , Y6i iliEu kisn dlfl cho: x.dt r,nSt.iret,.r atutrr,- rrdo d6 iht chi phi viu rhu.r.Fi nr.l; '. i,: hanr h,la liiit kr> theo chi6u lrAt kj li rrhr. ii.;r.u, rlr " db a6'rir tqtn, ra 11, il}u ,iti,i "ri'lttt',a"iai,,," hia :qlwqc udn chug?n'thto cltjln iii ithi mti,u';., 'l'a clrii. khu d.in cu llrauh h,ri l;lrrr. \ ri:c rrrr hinh'vE. - ;. .- .t itr' ,q ' t) Ctrung n;lii.tr ndu din.rli6.; -l,rt lttr r:-flrn l, G]AI DAP BAI ri". II.thi rrrl phit Li rlia rti8m 1,; :iiu f i;:J .tt XAY HHA. MAY l1r lqi nh6t nim o khu & SAU,? rr tli6m r,uc tthi ur, pl,fli :i*^ Thgl v'{y, rdng hing hda &'khu vr;c I I:\-1.rr 6t cirung d"6 giii bei toi.r: thuQc luqi uly ta tiu eQng v(}i hieg 6. L 20 tflI lhlnh 1gd tdn.' ' phii bi6t rluqrc chi phi vqin chu.r'En mQt tfln I6.n hryn nua tiing hing hou {:ii0 if; n). N liu vtiv . hang. hdr trtltr tuug do4n du0n$. "l'rdu cu ndu. nhi inil' d6t 11 r1L9 d;6ur l;hic, clrirng trir. sr! cld moi c.6 th6 tinh ts&n tai ltrlg ilia di6m, ilia iljBnr JI, {hi khi chur.€n nhrr:ria1. r.A i, s. "so sinh. rui rihau sd lirri tlrroc dia diEr4 lqi nhdt bot 'tlrrqc chi phi r.an'chu.r'frrr tsrJ tdn hi ng Irca .. (iliri di6nr iira -riy nhi rnty tqi do sE cri cl,i 1bi 1: 11 tlong t;hi Gd i[ng rhdnr cirrig: lin, ' lilA chi'f6n phi vAn chu.v6n 1{i{} liu he nF liotr tri LI 'v?u ciiuyEn'hnng t-hring {n it .nhAt). tsiii todm ni5' ll rnQt truo'ng hg'p ddc oiQt: trong.lihu vltc eIAn tl6n 1, (ndu hAng 6 Il vd U clrriyEn'Aii,, I t tLai,r cu c6 dog.t duo'ng PiL gtdng nltu rrr$t cai ciiu qua M'tdn- chi t;hi ir hon'lihi cirul.Fn q*.o.'.if,,l.. (tr6 doen clurlng EL di thi .khu vqic bi r{cli h&u thi con'-lcri rrhiBu irrrn nfta.). Ihu v;ij.O .AoS i - lA.rn hai), dilng thoi h{ng h6a rn6i }-rdt c}u x6p Iir din tlilm ltli nlrir cia khu vu,c II. xi'mgt n&a t6ng tiiidi lugng hirng hda' 'Ia,di (ihirng lnilh trrau6g irr td co Ii ln ,1i,, rliFrrr- vrlo vi0c giii bdi toin' Iqi nli?it cria khu vgc I 2) Cli cOn llhli so .enh II t,a L. " _: . ,-,. /40 \ i Ifa ntr,iir tlAy,riin; tlu rihi m[] d:ii ; lr truyti :, g Jli t{t ci hAug hoa cirng dEir phii qur hoi - rlia tlitim ndy. Yi v{y clri rrlri nra-r. dirt J, lI hay 'r &"-L thi vigc vln chul,En-ir,:ng hoa iren t::t er,r cAc iloqn ilLrdng.tru, dtoan HI, ll itorir i,,;,,, khOng thay rl6i..Nhrr vAy chi cdn i;bii rci ireu fa) . doqu iludng HZ ld so sdnh dir,.rt. ?n L ' N6u nhi. mdy d[t? fi ttri Li.6n itorn IIl, hang t tEi th[ng phli c6 t8r) tin hing hoa {cira khu vrrt lii ,.\ :H G \ .20 I ' chuyEn (un.' Ndrl rrhi 'nriy dil & I, thi tr6n- .l tloriu ,'rL hln$ than* c.:hi cc' 16C. ldii hring hr_,:, i t' \25 I \ --JI (ciin khu_r1u'c )) chul.En qul). \ KI \'{-v .t lA'dia {liiiin lrri nhit, lrtc ii. n6u xi .t* nhi rr:a5, tqi L thi l6rri1 ciri phl rin il:rr1.Bn !rarr,r : h6a hilng thllng I:\ it nhdi, B --o ; f ' 'l '-- --*- - ' ------ -* , Irr ' 1ti 000 e uJq,tqi.,\i in 7.'r - i-i llittitg tlli IIL. NOi. S5 in nghiQp :i0/7U. -\ong ngty ti,*-tgZti. ':GiriluuchiBut|{ng4-t979.0his6:12B84. .::l :