Toán học và tuổi trẻ Số 125 (3/1982)

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Toán học và tuổi trẻ Số 125 (3/1982) bao gồm những nội dung về một bất đẳng thức của số nguyên tố; công thức tổng quát; bất đẳng thức; đa thức bất khả qui; một số tính chất của dãy số. Bài giảng phục vụ cho các bạn yêu thích môn Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 125 (3/1982)

h: /L_ HOl 'roAN nr.rc vrit na,na I t I ; ffi ,t',t} f ,,,iii$ii,"i'$Sr,{i,fii'pi3fl rij,t$Atm$ffi I " BAO EA l''lAI Ihr\r-rG M+ I v') t\! cttit nhignt. nCUytX C,llCU fO^[n ItEr *g r{id {srrn : ffi{lAXg CHf}$G Tru s&: ?tl I'r,In Hung IJA, _. H.r NOi lley q6i; 5ggzS YH tu*qlr B"4T *",q,ryfi Y}*,ri'il f,flis. ${} ruffi,L}}:frri Ys ( i \- - .-l '-[t * \u0 ( CItr)' :it:iti pfrt*a C{c tren th*n nisn I UOt h6m uhAn iloa bii bto ( MSt tinh eh{r nhtng "l:"e minh h€t srlc _itsn gii*, d{c d6o, hhdng kh6ng dring {Sn cu'erle to{n hpt crja rd 3t)) trong sO fao 6l (sd 718 rram'1971;, . c,io c{p mi 'Ir&llrrsep Oa "0ng dnng, m& th&rn ch* 'ro-i. mOi n&y sinh f ttinh vi6t bai n&y il6 trao hh6ng dring tl€n kidn ,,r , d6i v&i c6.e Lras rn$t b{t d&ng thrie vE c6c s6 ra ngoii ki{n thr,rc ilruc to{i, h,le oAn ,*.ro s6 }rgc 6 I6p i l Tf,t lrhi$n, r:guv0n r6. ehring rninh r{t tton giir rriy eirng pfrai mOi ,_'. Trr,'ng 36 tr&o d[ n&u e6 dd ra trii toSn : vii trang gi{-r. C6c b?n 1,,
-@rL.j,.- I L{y n = pk+t -1 thi ta duoe b{t tt&ng lhrira: Shrl Y r&ng lrl ( r, ndn ta suY 14 I< 6 oi (5) pr41.( e i::l .,r , oll$i-l) eri =.: fripll { = nl(Pi*l)' 'i-1 j.X.tn/pl) =I (1) n6u ,t-' Ta nhin th{}j, tt (5) suy ra duoc Yi vAy, tri bb dE ta suy ra duqc mQt bdt ehung minh tluoc: rling thuc I - r)a L.p1 {rt} r). ,., 'fjJ-l )' (n * r). \e/ - * ,,=/ ,]/tri+,1)n I ,u, pr+r ) n> or.o' Ta chi cAn chirng r,rinh: \i:1 I M[t khic, theo b6t df,ng thric C0si ta c6: (i*)'r'n'r > {-"rt )'*' . - l1+2*...+t\o :- /q+t1", hay I c)lt+-l I \'+t ''
.\ \ -Nhu vAy, la dh tini ra mdt LSt dftng ihrtc kepr Cf,c ban thin m€n I B[t e$ng thiic (7) rrdy thq mQt sd ngurdn t6 bdt kj': kho co thE suy ra t& b{t dSng thri'c (2} ciru lt'rcbu- k _,. lj (. tli *l,t sep (vi0c lAm nAy rifii .vdi tOi thi dinh chiu C JL . l:l -,1 D. "} pt+l b6 lav). Mong e&c har: he.r suF ngLi, rl&o t. sfiu thdm cho bf,t d*n*g thuc (21, d0ng thrli tlnt Jz rt ,,1t(pl r) 'l ra nhiiug ung dung hay c&a n$. trong to*n i-l hgc' t: I * vE MqT cONG THuc rbmc euAT l{GLlYfrr\ }Ii}AN$ TRONG dai s6. oli(rng t,r i.[rudrig glp tri&u Ip dung- cong thrtr rrlli tirtr.. Ntu-tsu i16r vr]l r tbuc ngut3n: {7 + I)k+1. t,, cO r(n, ;): ik + 2L + 3k *...* lk (t) ' v6i n yi lc li nhtug sd tir ntri6n. Circ hi6u (i,r-rlk" thr'rc dang (l) nhiBu khi lAm cho lr!) giii ctra =i 1-"i'r: L .-kr.l*i -,,k+']rsl E"'i l' rn6t s6 bli toAn lr& ndn cbng k€nh r{ n6u lhuc 1=6 ;-U hign cne ph6p tr.rAu r,6i n kha iryn tiii nr$t nhiBu lltv lii; thoi gian, dri cCr lam tr'6n m[1' lirrh iti6n tir di rr 1k+1 nira. B6i v[y, tguoi ta dI lim cAc bi]u'rhric rni-r) .."Kil\rl\i^l giii tich khAc tircrng duong. thay th6 eho =r]o[,',tt*' .,* ",*i {*i J dang (t). ehing han, ehung ta d6. quen bi6t c6c bii-lu thre vdi & = 1, 2, 3, 4 nhu sau: 0 r'6 phii cfra (6i, chuy8n ddrr xich*ma lr'r n(n l-' l)'- lgodi va-o ihu dunc d&rg thrtc (3). trong sO l"(n, 1.)=- Bay gio, chring ta sE d&n tdri c6ng th{rc tbng 2 quAI mong mu6u. . r,(n + I)(:tnrI) F(n, 2) :- '-------':---- -- Khai tri&n vd ph&i *fra (3), ta 16 : f) ' ,,t(r, + l)t (2) ,-i , , n' l:tn.3i=-.=----- . {n*t)raI=ai*,,loio. 1 n(n f 1)(2n * i) [an(n+t) - l.l k+l rr i-r F(n, 1) = +I cf x ,k+ r-, j=o t?) Try Y{y, vif c tim ki6m e6c bi$u thuc 30 i-Z ^,, dang {2) t
/ \i.ai-i il 'rr2(, + t)llrnr, + I) - 1l Tuy nhi6n, dQc giri iln luu j- th0m, viQc tim F(1,5r l2 ra c{c hi6u thuc dang (2) nhd c6ng thu.c (8) se r(n* t)(2r*'l)fsnzrn1t]t..Sn(n11) + ti khdng rlon gi&n trt-rng thuc hlnh khi k khA ldq' riIl,1,, -* ,,, 42 Vi viy, e6 thE ding c{c cdng thuc tbng qu&t dur,ri d&y dB xic lnp tlirng ddn clc biEu thirc ir2, -4n(n + l) r 2l a \n + -r9.. ' r,, t3rr'(n I l)a ,'tn. -/l: "*.. dang (2) rnQt cAch nhanh ch6ng hon: 24 Ndu k.'1 hi€ui (e) l'{n. t} = lrorrg il6 ni'{; = i1, 1,...,1) ld c6.e s6 nguy6n dn {6i phuong trinh nd5', ta d6 dnng tinl tluqc I bi6u * thfc bi6u di6n I &n s6'qua nrdt hn s6 cdn lai' d- gi$.n, thi vrii 1i chirr hodc 16 s0 c6'e6eedng th&c thng quAt trong ilng nhu eau: Chi viQc 'chon giA tri nguy6n nh6 nhSt cho hn F(n. 2D : r:( n - I )(2n -{- I }f(n, l) (/c = 2l) (10a) sd il6 sao cho tflt c& c6c hn s6 alusc biBu di6n ,F'(rr, ll * 1):,,21n + t\1 1(n, tt (k=21+l) (lOb) gua nd clEir nguyGn ld thu dusc f(n, D cu th6. Klri c?in 'tim biBu thuc dang (2) cho .F(n' /r) Cu6i cing ll 'r,iQc gh6p /(n, I) vAo (t0a) hofc l-rdt kn, ta r:hi viyae thiit lAp I phrrong llinh (10b) triy theo /r =21 hay k =21 -|- 1. Dde gii c6 b*c nll$l vdi I * I hn s6 lrl ao.'rr1, dl. Tu he thE ki6rn tral4i phuong phnp nAy mdtcAch dddir:g. Vi v{1' mle dr} chrirrg ta khOng bi6t s6 S nrA &u nnt thAv giAo d[ cho .{, nhung do nhAn x€t vua rbi, ta th{-v ring s6 S ctng IA tbng blnh phuorrg fp-sd## cria hai s6 tt nhi0n a vi 6, v0l d.f D< 12. Nhu r,{v niot nrlit ta c6 S : *z +y2, v6i e vi g IA hai s6 tu nhidn, r + g -) t:. Tri"d&y suy ra s=i'2 *y8 = (r/2)G + s)2 * (1lD k - ilz Bni 11122. ffpc sin/r A rdt gi6i todn. C6 thn > 0!D(.- + ylt ) too/r. tttdg gida tt6 .A ilodn hat s6 nguuen dtong, nd fhdg dd uii.t btt trong phong bi dan kin' nhung v0v .s > s5. chi cho A bt|i t&ng binh phuonEl'hat sd d'g. A Mdt khfic ta ctng eo S = o2 -l br,T,6i a. r,A b frd {&r: < Em chiu, liidng doan dtrgcD. Thdg lA hai sd tr5 nhi6n, a * b ( tz. Oo aO giao cho hilt thd.m. Hai sd dq c6 ting l&n hon S = o2 r DN : (llzl(ar b)? + \tilh * t:12 12D Kht.da A ndii < En do'in ra rbi' ud A dua ( zz + Ql2)ta * b)z ra kdt qud ding ! 1'hi nh thri Bsn hdy lim hai qd mri thdg giao dd A' chn gtdi thich chdl che iflp lrrQn c&a ban. 'rt dd ( tlz\k - ilz + 72> suy.ra,":,_t,fj?6t a)>b) sb f, {t Ldi gtdi. Ggi liai s6 phii tim li r va y. Luc d[u thAy giAo tho / biCt tBIrg S -.l-f y". Co C{rngv6i a+0
Vdi Ir = 3, la - th{y chi b6 thF chon a : 9 Nhtrng sd g0: gz + ?r2 kh6ng c6 cach pt an tictr 2) ltat ol =aI....,rn*t = flu_1, kbac thdnh t8ng erla hai s6.ch{nh phuorrg. 'l hAnh thfr hai s6 phii rim ld 6 vi 7. vA ld,r' o,, ) n. sao clro &Ai 2/122. CIto 2n sd d.aorto 2rrrrl o1, ,2,.".,.on, u;*...*,,i_l +rr* = a'r* + oi+..+r; bt, b2,,..b, thia mdn cac di|u ki€i i) bt < az :( ... ii) udi rr7;ti k :1,2,...,n, ta :( t,,, - u? * t!+...+ a2n, (r) cd ,! + of,+...+oi ( r,! + uzr+ ...t lf,. thi iheo nhAn x6t trdn, ta co Chang minh t dng: af + ,r; -r-...+ai,'-{ }r * bz t...1.bn. 1) ar + o3 *...* ao ( + bz +...* bn; * ' 2) X oioj( Dr E bibr. Hai r'6 1;f,t rliug thtic hni ld la drlrrc tlEu duoug, binh phucrng 1(i(i(n t(i(j(n n.n Ldi gidi' 1) Ta chilng minh bdng quv n4p ).a'.tt2 : eioj.-(Zt?+z : bibi tlrco i< ri ng + az*...*ar { 0r + bz*...ibu (k ( n) A) dt \"Ay do (3), ta suy r.a I o, rt'i ( d- li:1, d'o ar, ir ) 0 re ni
a, l;. 'fa rrii t,ir ci..:ri,: i".'irrniq hgp t?ng r'6i gi{ tri c ccto #Jto fif phrerngr trin/r ' crla (0, d) \'A (-, r,l : t, \c"u-*bc*c=0 (1) t) (,, 6) r'{ {2, 3) ' {bt-'"4"t.r:*a=26 (2) 2) (1, {i) vr (2, 8i ( .rr +" q.t. -l- 6 -2G = (3) 3) (r. G) vn (4, 9) r:o,rrl}idnz ficrrgiit, 4) (1, 6) vn i7, 8) 5) (2. :il vi (4. 9) l,&t gi&i : 6) (2, 3) vA (7, 8) C.0ng vd vrvi vd c,ira c{. 3 phLr,rng trlnh (t),'(2) 7) (2, 8) vA (4. 9) (3] iu {) T.. -'- ? n\tn Ggz X,rr', 59: :l {r'') .rAZ /\ uo ,g vtii'raoi c n6n ti'r suv ra o*b*c=0 (x), Iihi ) zlas (tlclt ton nhrlt) +' {y r0 rAng {l) c6 hai nghigm li 1 r'i c1o. Nhung ' ho'p thri hai kh6ng thb cii vi iitri 116 I kirOng phli ]A nghiEm cira hd vi tih6ng th6a -'i'ntdng -- ."lr X i" ) mrn (392 X 4t'. 4gz X 3a) ) 7-118 mlin (2) vd (3J. VA-v dt hQ c6 rrghigrn nguyon thi clo phii la nghiorn nguvdn "i,a"nO, Thaly i = el a - Trutrng hop thri b;r kh0ng thb c6 r'i hhi c{6 vAo (2) ta c6: ryz X ur, ) :mitr (356 x 2t, 35i x 26' 250 X 31' b {cldz * c(c/o) * n s2ti. : Kdthop v6i a { i-i.c 0, 251X36)>7448. ti.t ilucrc: ' - Chrtng minh trrong tu, sE th5y trudng hop nt -'2snt = .' (4). thrr ttr khdng thl c6. Db c =- ar nAn eB + o3r3, thav vAr: (-1) ta rJutrc - X6t(2truong hqp (b' il' ('z' r') trh{n c:ic c{p rr3 -- 36n?: o3rr3 +) as (t gia. tri 3). i4, 9)' - "3) . = 2sa? *) o = 26/(i rB) (5) Vi lGz X lu, tSz X 6u, 652 X lp cliu: kiii 'lh& hon nJn ehl c6 thb e: 5. II : 1' li 6' Yi a v{ r ld ngur,6n n6n 26 ; t - ,1.3. VA;- ZaaS, = *2: trrc ti6p tn thAy (*) khOng th6a min' , -"^"t ! l - t : t: 2: -11; 13: -iti: 2ti l. - 'l'rudng hqp (b, d), (;' u) nl.rAl cAc ... \:iJ- r" clti ci'p gi6 cri thE nh&n e6c giir tri 0, -1, 27 .+ *; qrhin c[c gi{ tr j ngusen ld lr 0, ii. Ihuv giA tri tri (2. 3), ty,'gf. ia, c; phii nhrin ci1: (9' 6)' r'i ndu - :' rla, vA li6t hop vrii tliEu nfr4n ceP n1.r6 hon thi ' ctia .r' vuo (5), r.A r: liitn (*) la nhrin duo-c.c[.c }ir (o, ], r,) 1uolg rrng obXcd{ 6'oot't viri giri tr! ciia r ld (13, Li, - t:l) ; (26, -26, 0) >.?,> min {152 X 43' 15i X {2) >6'000' (*1, 4, -lll. 'l'hav r:{c gi6 iri trrong ring cira a, rne rp &, c tinr duac vA.o hf ta th{-r- th6a nrln, x phii bf,ng l, vi ndu khOng thi .r Yz X- uo > i4'{8' YAr- cdc Lrd 3 s6 nguvdr (a, b, c) th6a rnhn vdn chu dlu hii l* (i3,0, *-lgl;26, -.26,0) v[ (.-1, Cnn 14i (g, n) nhAn c4P (5' a)' C6,cirrtdng hcrp co thB cfia rrl; X cd 1i L,,I.H. !i2 X fi3 : 5?9tr' 93 X ti? = 57S6 97 x 68 * 659S, s8 x 67-: 6566 * . B*i 51tEP. trilg I chr? sS ,. t,. .., I utdt titdtrh Cic trrrdrrg hqrp c6 thb cfia tgz Y uu lh #'. b6x sd ea 2 ho[e.3 c}0: sd s*o c!t* chting ldp thanh cric r{p sd eri tich hdnq nhau: 158 X {? = i356, 152 X 43 =' 6533: *' .: IEB X ,12 = 6120. 1,{? X 52 = 7436. ahYcd:rqzXttu (") Nhn vflv kh6ng co trlro'ng hgp nAo, thira rnln {ko l:ng udi gia tri c$a flclri hie.t e ri.t:h'oii-t ,fiX.a =ir**. mdi ir3i{*} ldn nhfrl td criu Tru
--r clp (5. 6), r =: l. $, u) nhfln cflp (3, 2). Ph6p BAi 7/122. t.no siiCts = 9$o. t[ot hinh nhii' thir tnrc titlp cho ta hai nghiQu : nhi.t OAISC ,,i }hdng tlbt, lhag v1'6i 5a1s chrt c6 ehtL 54 X 69 = 138 X 27 (: 3726); c(tc cqnh' OA. OC crio n{i luln n&rn tr[n cdt lio Ox ud. Og. e hri'ng minh r&ng dtrdrtg th&n* tfi qua 58 X 67 :.134 X 29 (= 3886). B Ltd pl1[,rn goc uoi AO luan di qua ntlt ,ti?m cd 'I'6rn,lai f6 ba cdch vi6t db tich m6i vd d&ng diith' ttruc (*) r:6 chft s6 hA.ng ilon vi bdng 1;. Nhdn xet: Circ b4n Phr.rm'I'hanh Phuong vb' Dd Quang Dr.ti, Tr'dn Thunh T'rrtc, Biti Hodnl Qudn (chrrydn toen EtIiiP I Hi Nc.,i) ; Laong Anh Tudn 11 P2 Yd 'fhi S{u, FICN{) ; T6 T'hdnh Tudn Y (7,\ 4 -279, IlCilI) ; Ngugin AnhTudn A (1; Thang Long, IJi NQi) c6 ldi gi&i tuong c16i t6t. * Bei 6/112. Tren tladng lrdn rltdng kinh JB ta ldy hai dibttt M od N phdrt bi|t. Goi B, F lheo tttti'l4t lci hintr cltidu cfis M ud N lrdn'A.Y ud, Alt. Cltting minh EF I A^B . Ldi 11i{ti. Ta loai tru c6c tnrong hcrp lArn cho rliiu bAi todn v0 nghia; dir Id truong hqp M Lh gtfr.i, Dung hinh vudng {)PQ8 c
disn $(dDC). (U6 thitl rhrrg quay Soc t)._1UnS qrranh duo"ng phflu gif,c 8p cfra gdc -l8D thi gdc tam di$n ,8(,D,{r) sE d6n.1 I'rlng vdi g6c tarn dirin 8t 3DC). {+ k ,.,\o Ylr ,? !a c(, ; .lE: = AB.AD..* AC' -* Tanr gi{e #d\ /\ r"--\ /\\ r--'1 -{EsC c*n *'ACI}$ Dt.El* AE*A=EBIC'i- E{8. * .EIBC n*x -/\ /',---- Nlrnr,,g AC,D p1i1i2 = E2CE hay Sau ilir tr0n tla 8.r: ta l$y diErl E 'sao cho' BE = BD: th6 tht ttDE ld mdt trrrn gilr: tllu vA do iI6 : DE = DB, dbng thli la cltroc LBAE - Ii.,Sn toLn trrunq tp la cri lhE ch(rng .."."\\ rninh LBCD (cgc) vA tlc, t[6: E&= DC. Iam giic DAlj AC.61 : Et{)I irn3_ iA iam giic (T) c5.n tim, c6 ba canh bing Dl, * _ _ DB, DC. C{ch gili nhy ln cria Biii Hodng QuAn A{:Er= Bl2 {Zl Ldp llCT DHSP He NOi 1. a.r'*\ 'I.e. cct: tCA=ACEI+EtdI (S) Sau dfly li hai c{ch gili kht?' Cdctt 2 (DQng Tnrbng So,'r - ilCT Di'I'fll IIn r''"\r Nh*rrg lTrC/ = "",*\- g&o * CIEtlT = 90o - c*aryt : N6i): Tr0n eirc tr:{ [DA). [1.rBr 'a Int;t- ta l{v cic r:* {ri. fro{=s6'- Al: r -ei (4) d.ilrn A', B' vi (." s3{r chr,l : D.q'DA *DB,DB'- DC.DC'=l . D6 Lh6y r:ing cac ci p tflrn giAc sarr Ilr [2) rA (4) r:sn (3) eho ta dAy la
th€ thi ,}rQ::: FE, Qrt : PC vr}. P,40 l& i{m qiik: c6 Eei l0i I 22. Lh*ng inini rir'ng Jr' (.;n nqtiri c{c eanh blng PA, PB,.PC). S*n dd, chi vi6c 6p dgng tlinh lJ Pi*ta-go vic) c6c tam gilc vutrng ttig $ bao gib:tu cfing cd thl chgn ra d.usc n\l PDA, PDB va POC ta c6 ngay il.p.e.m (m6i iloan nguot ho(t: dAi m6t qucn lrtdl nhau. hollc ddt m$t khdng qucn bidt nhuu. thr'rng OA. DB, DC c6 d0 dAi nh.i hon thng,eac dd d*i cria hai doau kia). (Trong dB bni in trong sd b{o 122, c6 il6 bilj 2) {}Oi 1l' ln di8m t16i xi'rng cira dibm O qua ,n al,,thenhC[: c6 th& ltr clo rtrng chft trong t*.m (,) eria tam giie dEu ADC ra dlt : O.t = s. llF=b, DC:c;11'C: a', D,B=b', D'C* a'. khi in V$v bii nny kh0ng tfnlt l*m b*i thi giii toAn). Ap dung dinb lf (thr? nhfit) vl fiuring trung * tu,vln vlo c€c tain gidrc D.lt', DBB' vn DCC', I.bL gidi. V6i a, ir la hai s6 tg nhiOn lon han 1 ti'ry f. ia dli Nia, b) lA rd tg nhiCrr nh6 nh&t trr ed ngey *Ac hB thuc sau : (i)ar + a'* : lrz + b'2 $ ,", tir N{rr, }) agrrdi tiry f beo gid cirng c6 : c2 + ( = 2ayz +-z?z), trolg al6 /? li b{n thE "hochon duoc hodc a nludi dOi mOt quen "'z trirn ngoai li6p trm gidc dilrr ,tBL')" kinh dtrc,ng brFt nharr. holl b ngrrdi ddi mQi klr0ng quen l{r.t kh6o, ggt .1' ln dibm c16i rring cira diBnr ;l bidt nhau. 'I'a e h{rng minh bflt dfrng thitc gua tAnr 0 *fra lam giiLc illu ,4BC; thd thi A' dflng in t[6i xr'rng eira O qua cgnh BC. Ap dr,rng .v(o.6) ( ,r,(a - 1, b) f .\'tn,l - l) (1) tlinh If vB tluong trung luv€n r"{o ba tam gidc voi a. lr lA irai sd tr nhitn lon hou 2. DAA'. DBC yA D0.1', sau khi linh torifi v') rril ]Iudn vAr ta chung minh: tri m$t t{p hop S gqn ta ducrc h6 thL'rc: g6m ^\'ic - 1,[) + N(a,b- 1) nguiri tuy y bao gicr ,.rz +b2 +.2 = :',t{1yf agl'1. cirng chgn ilugc ho[e o ngudi tldi n$t quen I:i6t nhau, hoflc b ngrr&i il6i m6t kh6ng quen {lting co thb ru1' rLr h6 thuc tr€n nl:anh r:hring bi6t nharr. Ta chqn mdt ngudri ,4 bf t k1i 11619 t{p hap S vit chia s6 ngrr'ri cdn lgi th&nh hai .2 t c-2 = -til-l'+ ?-r rr-z * I,- ii, fill' + DC'. nh6ni: + r+- +^ "4^ ."+c ,tr - l.r G.S, .r' quen bi$i / i, ,f),t4 + ffB'J -\ DCz =3but = (i.l' + Oil'+ oa" + { + Sa: I r (: S, r: khOng quen .4 l. 'l -9* 2])0{AA + OB + OL'} = t( 11q'' -1- R"'}, vl ..4 -+ + + vi isa I i {s+ i :lf(a * l,b)+,\(a.}-l)-1, n6n : 0a+oB+oc,=o. C{irrg [irrh trrong tu nhrr lr€n. ta drrac kdt quii hole I s1 i - v (a -.1,0),lrolc iSl> r(o. b-.1). a) i{6u t,sA i>,\(o - 1,Ir) thi tlong nh6m so (l;).az + b? -lcB'= c'2 *. I,'l -r- c'2 (.:ioDz +snz). 'I'r) crlc h$ !tr-rri'c (i) v{ (ii}. dd dtng su.v ra lio[c tirn dtuqrc 6 nguoi d6i mOt kh6ng quen edt' hd thuc sau: bi6t. nhau, hofc tirn tlrrge a - i ngtoi ddi mQt quen bi6t nharr; trong trudng hSp thrl hai ta n4qya4 c4=n'4+D'4 1"'4 hb xung:1 vio nh6m n - 1 ngu&i qi.ren bi6t nhau +trgrqrn.ttr.rtTt) bof.o ;. c'a' 4 a'ls' = b "r' * a *a' * tt "b -.. trong Sn sE drtop o ngu0i quen triSt nhau trong S. Tr'r d6 su-v ra : b) Ndu I Io ) N(a.b - 1) thi trong nhdm so 162 a .?-,r?ro' + (r'-, u2 -bz)hr+(n2'+ b2-.rl)*' - hohc tlm iluoc o nguoi rl0i mQt quen bidt nhau, ho,Ic tirn iluoc b - I nguoi tl6i nrQt lCh6ng quen : (r,'2 + .'? - o 2)o'2 +{c'2 + o'3 - ir'2)b'2+ billt nhau ; trong truirng hqp thu hai ta bb sung (a'2+b'! *"'2)r''. .{ r,do-nh6m b - 1 ngudi khOng quen bi6t nhau flay Ia. vi6i cho gon hon thl ahu aau: lrong SO sE clu
=- Trude h8t ta crr rrh$n xet l 'i heo gifi thi$i qui nap t'r c6 2 I'io-l,i;(C' f i2. b): c,lD = b; a'i- o a ', ' },i'i,l' * ir' . Coa-rI 1 ^ ,\'ro, I) * n, Co a -ti ^ - a. b -l " i ,l'lit-r 1 =| ,\{n, l,t { t:"a--t)*.1 , }- -J Nhu vAv cir th6 vi{t : ..a - '-l-r]r -2 1 ,r'i::. b) ( Ctl trlc li (2) rlirng r#i a, D -,,' li va Q 1- b":z k. \'-t!i -" a = 2 hoclc D = 2 thi i2) dung nirrr ilft nLi:'r x6t {" t\ .\(.r, :) ( 1r I ir 1r6n. YAv (2) dirng v&i tnoi'r, I' rnA o -l- b = A' t_ ,rl Brli toan ctla la l{ l.rudng hop dac bit"1 *n' * tir* li dfing thric (2,t riirng vdi nroi c, h tnit trr:1t d = l) : n-l 1. khi dd (2) cho ta mdt tlong hai s6 c16 bnng 2. 'l-rl nhAn rtlI t16rr ta c6 ugav vdi o*I;=1'ri N(n r i, rt 't) :( r:1r,. a g l.i: 5 thi i2) ludn dring. Vt tu N(n + 1, n * 1) nguoi triy { }rLro gio eing chon ra I ngucri hotic ti6i m$t quen dtrcrc n * Be1' gie gii sfr b5t dfrng thr'rc (2) dI drir:g r i)l bi,-!t nhau, ho[c il0i mdt kh0ng qLren bi6t nhau. moi a, b md a -1- b: /c - 1. {a chr'rng minh (2 nirr rdi rn$t tip hop tiry 1- s6 ngudi kh0ng it e ilng dirng vrii nroi nn 6 mi o * b : k. hon, I(ng 1, n F i) gOn Cln ngu,ii. ctrng phii ThAt vir', voi a va l; i* hai sii irr nhiin ma ciron ra cluge it nil6t n* I nguui hrilc d6i rn6t a*lr:k vA a,b)) 3 thi ta tld cd bdl diug quen hidt nhau, holc cldi nr$l kirring quen bi€t thirc (1,': nhau, -'"(a. l) { .t(n -' 1.0) -r .\'(a.b - 1r. P.Q,G. DANH SACH HQC S|NH OUr LOr GrAr CAC BAI TOAN RA TRONG 56 BAO 122 Yinh Pht: Ld Chibu Duong, Nguyin Phtic Vdn L{httgdn, ilQnq Vinh I!cnq (11I, QuanB 5r (10D, Hirng Vuong). He B6e : Nguyd n Qudc Trnng); Di6m ?hiu 02C, Quang Trung). Gia 'I'hdng (12T. Ngd Si Li0n). QuAog Ninh: Piram Lai - Kon Tum: Dd ttiru fidn (10.\, PIeiku 1). {ihlronq (ttCl' Hbng Gai). I{i NQi: Truong Thd Phri Klr6uh : Tlrdu Trong Il r)i(l t P4, Nha Trarg). E{ing(7A, 86 YrEn EAn); .vgagEnThanh Hd$G, Dbng l{ai: lfguy6n Nggc T'hiiu {1r;8, 't6n Phir). 'Irung Vuong) I ?'rhn ilIinh tbttc, rtb Ti€a ilqt Song B0: TqYdn ?'tn (ltA, Di An, 't'hudn An), # (10A, Chu \rln An) ; 7'rdn Menh C*dnq (10A, Ean Thdnh ph6 116 Chi Minh: Vd Eqi IlodiDr?c ..* bi Phugrrg) ;I,lgug7nThi Hou flbng (10B, \'i6t Drlc); (9A1, fl{ Iluy Tdp); .\rgurlin Qubc Ldn'(10A2, l{gug\n Anh Tudn A (l lH, Thdng Long). Ee Trung Vuong) i Llrong Anh I'udn (11P2, \'6 'fhi 'U*: Son Binh;'Phom-Ngoc Co (10A, Ivlai ChAu); Sau). Tien Giang : Phqm'fdn Qugbn(11, Truong .\lgugin Hodi A.nh, Ddo Anh Binh ( lltr, NgLryin Dinh, Gd C6ng). tsin fre: Ngugdn Yttn Kitm Hue). Hii Fhbng : Ilgug6.nThi Son (10CT, Th{i (l1A I{6 Cd-y) tiQu Giang: Ngugin ThanhQuong Phien) ; Phqm Qttang '{hrtV (118, NgO Qu.vEn). (11Pt, HoAng Digu, S6c Trlng). Llai hoc Tbng ThAi Binh: Dodn l'udn Minh (l2C'1, Ngu-ytln hqp Ed Nqi: E{ag lrudng So'n (l6p 11 c}ruydu Trii, VIr'I'hrr). Thonh H6a : NguglnTodn eugbn toan). Eai hgc Srr pham IIa NOi I: Ph(m (12T, IInm Rbng). Quing Narn * DA NEng: 'l'hanh Phtong vA fd Quang flqi (l6p 10 chuyen Nguy6.n 'f haflh Drong (1li l, Phan Chiu 'tr'rinh); toan); Bfii tlodng Qudn, l'rdn 'l'ufin HiQSt. 'I'rhn Qu6c Htng (128, Thdi Phi6r). Nghia Binh: Thanh 't.ric |(.6p 1l chuy6n toen). llai ban dii tYllug\n Hodnh ()udng (IUB, Trung Yuorrg) :7'rhn ilia clii klrOng dit: Lt. Ngapdn Hd,t (llC, -l'u Kg Phit: (11A, Quarg 'l'rung, Qui Nhun) ; Phqm Nghia); Li Vunl'hanh (12, Nrii Thanh. Ar:'I'dn). 10
Sei 5l125. Dung iam giio .,tBC. ctro bi6 ,BC = 1,. ga)* .1 l"ir ctri$u dAi d cila pli jir.r gi6c Ii lrll l.i,: .{. .\'1lu IJag.\i',:h (slrrr t11 ntt Bdi 7i f 25" Cho rn6t rlrldng trr)n l6nr 0 r:6 dutln: kiuh rl(,'=2.R. I{ai riibrn fJ vd O di cbuyiln Bei I1125. (ihn s6 nguvdn ducrns m e6 Met phr'Lng d6i .rrrng v6i mdt trung cli6n r16i v6i {!a*n: *ia-b!i (l t+ot, j+ jf -oO1r. nrit phin giAc trorig cira nhi di6n ph6t xuiit tr} Trr:ngt!'r:rrn{hop nio lhiriilra ddu il{irgthtre? cting mdt cq.nh cria mdt tu diSn drroc goi lI nrdt phan D,tc Ahinh tleii rung (tli qua canh 116) cfra tir diqn. " .t Chring miuh rlng s{u mit t16i trung cria mOt lSAi 4125. a) Chring minh rirng rr6u ira si1 tr'r diel ddng qui t4i m6t t1i0m. trgrry0;r t6 ldrr hon 3 l6p th*rrh rr*r*t cl{p sd cdng 'frlrnq qu./rt h6a. thi cdng sai chi:r h6t chci 6. ! ,\.,7u1;itt Btinq ph-$t b) 'l'i:n 9 sd nguv6:r if, lre ir:n 19I'l iirp liri lh tn0t ,'d'I sa\ c0rS, firt Trr&ng Gl;rriq {\'in1i Phil) lli'){, St\lt I 1-\l I 0l BAi 51125. nulg tri. giAr AD{:D. hi6r brirr Iid'i t"OitP$. GiiriphLrcng trirh nghi6m nguv6n di6ni .1t, Bt, Ct. llt liin lur.:t ld c6c rliErn drii xitng drr
Yi binh phuryng (bc * prt)2 hh0ng dm, ndn ndu d$ 0 (o, o, o). Khi do cosirr cfra ga,c gita cic b6 di, thi trl ding thirc tr0n, ta c6 i"l{t dllng duong ihnng OP 1,1i !Q se duoc xAc dinh b6i tini'c (11. I)du bing x5.y ra khi v{ chi khi ding tht'rc: Dc * 9d: 0, lghia ld hai cXp s6 (n, b) r.A (c, d) IA. ti lc r,
Ddu bing x61. ra khi r[ c]ri khi (o1lc * 6t) b&ng Nhu r.&y : . nharr c&, nglria IA. iihi ai/6i = 1i (Di + il). + sinz (rl2t ll4 + (rl2) Sau r1iv. ta gi&i mdt vii l-ir'ri tap \-in dung 3/4 cosr -: 1/2 cosz bdt d&ng tftrlc Cdsl*5 uarr:o. s(/r + 2kfi qp ngul'ou). :r = :1: Bdi tiip /. I'lm gin tri be nii:it i i'ra S - o'+g'+r? nn ' r-tri P : ar * ba + c:: r:onst. (a, b, c + 0). Bdi t1ip3. Cho i ci : l. 'lim pin f ,i:.: i I Theo b{t cl&ng thirc (.c1si - S vtrr,ro t}ri : i:1 i=1 trong d6 ci' di l{ nhilng sd drrr'rng, {lz+yl*:!,'(ar * b2 = ,2t )\a.- * bg * c:)z (Bdi todn frdn ki nrft brii todn trong li1 thulltt s: .2*y3 + .2> P'l(oz + bz+c2) t:tic sudt thdng kA,,kfii chn e hinh l$ klt Erd do I(hi df,u b:1ng ri.'ra iiri r'o min.S, tuc lt khi mQi dai luqng n lkn) lt},eo bai ifrng thuc COsi :t': Ao, g: /''6. : = ilc. irri nghia,lA-: -Svarxo, ta cr.r t' " n: aPlhz+b2="2), :'= bPl{az + ra + c2) \,i z: cp toz + bz + r1l) (,i,,:,t)(,i,,,,i )r(,:,,, ., -;)L, Bdi l0p 2. Ciai phtrong irinh Ndu d[u birng xEy ra riusc ,,,,, jrn?r? dqt gia 1/ 37a all"t i. l - i/ ri+-i.o*, frlzl= z 6 ddy a1 li si, irtrng thu nhf,t cria v€i tr6i, tri he rihdt. a2 Id s6 hang thu hai cfra v6 tr{i, bi:1 vi. n ^- ln b2: l. \rAv thec birt tling thuc C0si - Sr.ar-xo, trrin !a?,,i =r l: *L ta t'() : i:ilr Ji=t,rl if+ L\ + r sj*r l.)- ') / (-i+;*-f}]1,*,i 0 dtv, cldu'bHng xf,y.ra khi r/i ,, = * # , - ."a.2 * 1.1-i- . tric li ci =: ,k '(1/+ Va + .ou2 z a. I Dod6tac6: n 'ta phiri tim X ci = l. z> I 3h T;J Gtz) + I 17 .+ "-;Jl*/z) sd A sao cho i-1 N6u driu }riug ri1'ra ilrrtrc thi pinrong- trinh tii cho rr\
MQt ila thrjrr: vrf i h€. s6 h&u ti khbng phfli id Chri'ng minh dinh Ij niy rhi chn dring nhtng cta thuc khl qui du'qc ggi lA rla thuc i:{t khi ki6n lhiie df, ndu & bei ( Scr lrrsc lf tLuydL vB t1ui. chia hdt r16i vdi dla thircl vrl lf thur'6t ehia hdt Thi dr.t i: Chinig rni:rh rPrng dl thric efia clc s6 n'guyOn, nhung hhir dAi" ilE ;rglri cdc * * irqn tq tinr I{y. 1(r) = (r a1)tr: aa)...fr - ani r l, troitg riri {rl {rZ,...,rrn Id uhr}*g sfi ngu5dn, ln mQt da ih ut: t hi rtat J: Churrg n:inh {!x ih*c bf,t klr[ qui. /p{;r) = rcF-l + r,P-z -1...*r: * I, Chung minh blng ph&.n chung: tii& sir /(r) tri.rril tl6 p ln srj ngu-vdn td, l{ da lh*c bdt khi khdng ph&i tn da thuc b{i kha qui. khi do cd qui thB gi6 thi6t r'*ng I(.x) : ,l.ft(r-).12(;r:) vdi A lA Chtrng minft: Diit r=g r- 1. Iihi tlo sd nguy6n, fi(.r) I:i cac ila thue r di h0 s6 nguydn : (U*1tl'*t * (yr-1 .t;'-'z i- ....'(gi I1 -|-l ip(r)=g(r7) vn degf(r ) deg/i(*:) ) 0, ; : 1, 2. Vi irQ c6 cria (y + l)P I. s6 hang bAc cao nh{t eria f(r) }i i ndn ta c6 .-- = -l-l(y 1 lrrr . ii th6 gi& thi6t' d = 1. R0 rdng /(o1) e - 1' vdi (y*1)-! U i = 1, 2,...,n ndn hr-r5c /r(a;): 1, /z(ar) : - I ho*c 1. u ,r-I p(p- 1l .,-I t ,t /r(ai) x *1, /z(ci) * 1 =--lu'Tpg'' g PUI ' 2t do d6 :9. o-1,-rpA,n-t, t---;,1'tp*1) r-3 1..+p. Ii(ai) + fe(ai)= 0'viri I e 1, 2,...,n. , Vi ) 0 deg/i(:r,) V6i rnoi trt sao cho I - -( /r:( p* l, do p ngul'6r rin deg(fi(r) + /z(r)) ( des-/(r) = n td ntn (k, p)cel, viy k ! vi p ngri i-0n t5 ci ng rni /t(r) * /'.r3) nirirr gil tri {) 1r:i n grA tri nhau,d* dd tri (19:F(p*1)...(p --d-r-t)llr I ngu1,0n ** /r(e): * fz(.r) :.:: $ ,i1, a2,...,oo,* suy ta Tr : i1, * f).."(p '. I'+ 1),k I rrguldn. itr) /tl).f z{rs = -{/rl.'112. Do t[6 : ** u".,.+ p Dttu ndy mAu thufln ri h6 s6 cita s6 hang ]:{c fp(*) ** g(g) = gP-l + pTiUP'a :- p^12yt' cao nhilt ciia /(r) birg I lA sd duong vd v6i Tt, T2,... nguy€n. ' dcg/1(a r )> O. Ap dqfg ti0u chuhl Aidensiains, ia c6 g(y) ') lti dtt 2: 'l'im CliEu ki"n It ,'6 ve ,li db de L|6t khii qui, VSy /(*) b5t khA qui. lhuc /(r):..n + pxz + q la khi .r;ui, tror:g do BAy gicr, c{c ban hdy dtng nhtng ki6rr thr.lrc p, q li. nh&ng sd htru ti. dd n6u & tren db gif,i cdc bai tAp sau: Clri ddn: Nh&n x6t: Neu f(:ri) = 0 I) Chrlng minh ring da thuc thi /(-acu : 0. (r - at)g(c - a2)r,,.(r - or)2 + 1 Do d6 udu /(;r:) kh& qui thi n6 ph&.i duqc phrln voi at, e2r,..taa nguy0n li bdt kh& gu.; tleh tlutyc du'di dqng 2) I lrn crc gid tri nguydn a. &. c dE rho tarx f(*): (r2 + a1c * b1) (,rz t- ozr + ba). thue bfle hai ar' * br * c ph0n tlch duoc thAnh So s{nh hQ s6 hai vE, ta 6i ddn: EiBu kiQn tich eira irai nhi thuc v0i h€ s6 r:!u1.6n. 6t c6 r,d r1r1 tlt /(r) = *4 + .rr'2 + q khrl qui in 3) Cho /(r) : ,loo'+ airo-l -.{ ...+an la da thric cdc hQ sd p, 9 ph&i th6a nrEu nrOt trong hai v0i h€ sd rrgu.v6n vd c6 rn6t oi5 rrguy6n t6 p sao rliEu kiOn : ciro: hd s6 cria s6 hang cao nhdt kh6ng chia hdt t) pz - {q : rns; cho p, e6n cAc trQ sd ai.1t, oxf2,",.,an tl6u chia 2)q= ne:2n*p=/r?; h€i cho p vA s6 hang tu do khOng chia hdt cho trong do m, n, k iA nh&r:g s6.hitu ti nno 116. p'. Chung minh lirng f(.r) c6 nrdt nhAn t& b6t Cho trudc met da tlit?c /(r) voi h€ sd h*u ti. Di€u kiQn 6t c6 vi di dA l(r) kh6. qui la gi ? 4) Chr?ng nlrilh ring v6i moi s5 ngu3'drr td p CAu h6i 66 ddn nay r'6n ch';a clugc giii dap. vA nrois6 trr uhitn lc, da thirc nhfln clu{rc til'i;i8u 'l'uy niii6n, ugudi ta cirng dlfr tim ilusc mSt diEri k L-l dign dri t18 /(r) ]i ila thuc khi qui. Cu thE thirc (rp * r)i(rp * l) ltl bf,t khfr qui, ta c6: 5) Cho da thuc /(c) b{c n (n:2m lio{c 'l'iPu chuiin Aideilstaino: Gii sir n-2m * 1) vdi h{ rd nguyOn vA nhAn c{c giri n , n-l tri _[ I tai quA 2nr gii tri nguy0a cria r. Ohirng f(r) : oor" * alc"-'*'"..-.1^an r-ryi hg srl nguy0n ' vA c6 mSt s5 nguy8n t6 p sao cho p kh0ng ph&i rninh /(r:) bf,t khi qui" ld uOc sd ci"ra 4,, nlrung p ltr uoc sd cr?ra e{c h0 Ii) Chr.'rirg minh ring t6u da thrtc f{r) vdi h{ s6 ct-rn lai vA p" )ih6ng ph$i 1A udrc c&a ,:o.'Ihd rd nguyEn r:hgn gi{ iri tr tai qu{i bn gi* tr!rnguy0n thl i(r) ln mOt da thrlc b{t khri qui. eri.a c. thi l(r) kh6ng ehira *hen tu r * 1 14
,tll dge tim tol M+T sd TINH cHAr c0a nfi,y s6 loot : ir +. li2 -.t- .". + tinl fEL tltn hibu nhrtng tinh ch{t cria m6t ddl DB chung minir tiuh chdt 2, ta i.ring hai b6 s6 lA. mdt it6 tdi h{p din r16i v6i cdc bas clE.sau: tr-6 yOu to6n. Sau dd-v, t6i rin trinh biy 1. lin f mdt sd iinh chdt cfla dly s6 iro I = lt + tlz l]?) db an - e. n-)E r +... * I/rrl. ThAt vAy, vi cr ( rr (..- (r* (..., | 1.inh chdJ t, rrn (n )> I) Iiirdng ph&i In rn6t 16 ueu linr rrn hodc li *, ho[c eho ncuyorr. li s6 iriru tran a L d fi"+oo i Chunq ininlr: NhAn ret r[ng moi s6 nguvdn ndo tl6, Gii s& linr un = c[ (e:t *). 'l.a cO 1 n-+oo I l> 0). t;oi A ld tAp hop il6u c6 dang 2k.u (u 16. t6t I "a "{" s6 c6 iiang : ttrong vucr[ qu6 n r= * + l/5 +... + I/2k*1 +,..) 1 1i3 tn)>2). Bao gio cnrf c6 2lo la sd I&n nh{[ rrong |il+ 1i6 + . .+ 1l2k + ...) + (1i2 r. chc sd thuSc ,{, I-a sE chu.ng miuh rnoi s6 ( I )zttn + 1 4 + 1i6 +... + l/2k +...) ithi qui r'6 dang !p. g ig lir) m& 21t. r=t-.21a . =1+ 1,2+1j3+ ...+.tln*".. =a tlri p ( I". di ddn e -> a, 1dr diELr v0 li I V{y lim iln x &. 1'h{t vf,y; gii sLi p} to. n:- * N6u q = I t|i 2p -)- 2lo (r1iiu thu&u vdi 2lu Bb ,ti 1 2. : tl2 + ... + tt" {21tr. lun nhAt). D6 th5s IrAt tli ng thrlc tr6n d6ng r6t n:1,2. - sri bdt d&ng thuc dA dtng "vOl n=/c, Gii - Ndu q ) : ttri n)> 2?.q ), zp*1)>"21o. (nrip cirrlng ninh n6 cfing ihirg vdi n = k * l. ta thuin tdi 2lo 161 :rhdt). ThAt v{y, tr} gi& thi6t qui nap, suy ra: thi--u^.21o - I - i6i girin. 1+ ta.+ ... + llk+ 1/(t,+ 1) . Gi'i si'r, r.rx li sr) rrgui-in 21o 4& chtrag eCi dang 2k. q (q 16) r'd k ( 1"' Do phAn . * 's6 ndy blng phf,n sd tdi giin tr6n. suy ra 2k.g,21,) zl7. y{v ta c6 17/s.a r F,ffi hay 2k i 2lo suy la h , lo, ili ddii rn6u thudn. 21'* + 1r'1'6 41 < ff't-TT. (*o) t VA1' un (n , Z) khOirg thb ii, s6 nguy€n. ?t (*) ve (**) suy ra ztli + j/(tc + 1)
Vi lim uB sr oo (theo bb de 1), n8n cuY ra alai, ngr.rdi dtr cir d6n 544 cdng trinh lcin hoc ll --+ oo Ion nh6, ngrrcri Iirn to{n dl€n hoi thi cudi cung. mlft. ctng kh0ng IAp duqc biBu thr'rc cho uo. niA ehi nr&i ctrfrng rninh dlugc ring linl (rtn/n): it46 litrt (rrr., - Itr rt) -- c ic Irir h:ing s6) m m-1, r'... n+€ aon'" *am-ln'"-' *aln*o" : lim I{&ng s'i itd duqrc ggi lri hlng sd O-le, D6 li n-+ € &lnk*l +ru-rnk +..'+ 61n2 *borr nr0t s6. r'0 ti l e = 0,577... {ln rr trA lOgarit ef s0 e lim (ru/rr) { tim (2/n /ri) (do bb its 1)" cria n, e li s5 v0 ti bnng 2,?fi... ). Diiiu ndy chi n-*€ n4 00 noi lOn ritg ir nhirng sd n IOrr thi lim aofn) );0 (u" )'.0) 'In:ln n+C I 1l*€ Xung qurnh dny s6 la'i .on ct'r lli€t bao bAii cdn lim (2/n /n) --, u, l€u t:L to loin Iy thir khAc nta, chiir;g h4r : n+rc + Vdi p ola sd rrguy6n tri, p ,,.,?', tlri rrp = rlt '(noi vrl r - si p". Iinr n) - n'-l€ m m-1, +1,l2