Toán học và tuổi trẻ Số 211 (1/1995)

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Dưới đây là tạp chí Toán học và tuổi trẻ Số 211 (1/1995). Mời các bạn tham khảo để nắm bắt những nội dung về việc sử dụng yếu tố phụ trong việc giải bài toán đại số; thêm tính chất cho tích phân xác định; cách nhìn hàm số lỗi; phương pháp diện tích.

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Nội dung Text: Toán học và tuổi trẻ Số 211 (1/1995)

Bo GrAo DUC vA DAo rAo * noI roAN HQC vlFr NAM NIC ui_ rap cHi na xcAv tr uAuc tnAxc rd
TAP CHI \ It TOAN HQC VA TUOI TRE ,,:.::,,:,::::::::::::::,:::,:.:,:,,.'.,,: ,::: :::::::::::::, Ia.fie.:bla:n 4ip ; :,:: , ,, ,,Ncu*Erv 65fuH,,roAN ,,::.i1i11.t;,,1:.1.,'.;.:.,,"p75',:5ng,,:b.iinlirtp: .', ,' N:C6 DATTU ,,,,, tiiiiiiiiiiiiiiiiiiiiiiiiiiiiii".i"-.,,,'"".,"i ?H*rta, ............ ...'...lllll l.ll.l lus'....**0,.*,',*u....o#*.lll',.,ll ll ll. NgAyEn,,,,,,,,,,C*ffiiiii'i....To il;....;.'lUoane, Nna'.,,.oat.,,..t-*;......r4.......tm*a ,,,.,chdsgr B8o, Nguy6n Huy DoanjNguy6n IIaa, ,.,...v16t.........fJai;,,,,,,,rDinh,'Quang Xu&n Huy, Phan Huy ......NsurAn Vri ?ha.nh.Khi€t, f,c ffai i.....Kh*i, ......I(hoi, Nguydn Van ffiu, ;:Hoirng ,',,.'I,' M'-h Nguy6h Khdc Minh; .t .:.:::::. TrIin. |'V:an Nhungl,Nguy6n Deng I :::::::::::::::::::::::::::| :::::::: ::::: : ::::::::::::::::::::: :::::::::::: P,het, Phan,:.,.. franfr Quaug;,,, f+ Ueng q uaug;.ih6 ThAn g, v,r,,,,,,,Drlong tUilv, ::trah "e.Ht,ng Thdnh j ::::::::::,::: ,:.:.:.:.:., ,...,. *i1......t6 ,,,.Tf .8..s.'..Kilamn.....trtiilhi .NgB ....lfig.ti.i.trru.ng;,,,,Dpn$,,,,Qu4n,,,V.i6n,,l Tru sd tda soan : Bian ffip uit tri'su: VO XtM THIfy 458 Hhng Chudi, He NOi DT: 213786 Trinh b&y; NGLIYPN frfN DUNG 231 Nguy5n Vtrn CrI - VU KIM THUY TP Hd Ctri ulintr DT: 356111 "Khai bit" : Tranh bia cria PHAM NGOC TOI
Danh dro c6c ban Trulrg lqc Co ?- /< x SIJ DTJNG YEU TO PFIU TRONG VIEC GIAI BAI TOAN DAI SO LE QUdC HAN Trong t6c phdm "GiAi bii to6n nhu thd nao', P6- li- a cho NSh€ An reng , y€iu td phu nhrr mdt- nhip cha dd n6i bdi toin chn tim ra cdch gidi v6i bdi todn da bidt cdch gidi. abcd Bdifip4,:Cho7:E=a: D Ta h5y x6t thi dq sau dAy Thi d1t 1: Ddn giiin Uiar tnirc Chring minh ,[aA +,[FE + r/Ze + 'lTD : :,t1a-+--b-+Trq(f TB-{T+__D)- 1a+ b+ c)3 + (a- b- 9)3 + (h- c- a)3 + (c- a* b)3. * D6i v6i nhrlng bAi toen phic tap, nhi€u khi ra kh6ng Giai bei to6n nAy bing cach khai triiSn va u6c ludc cac s6 chi phAi dUa vio c6,c bidn phu mA cdn phAi drla vdo c1c hitm hang lA m6t phudng phSp thri c6ng va k6m hdp d5n. Tuy nhi€n, phu. ndi ad y t
Danh cho s5c ban Chudn bi thi vho Dai hgc - A' { THEM TINH CHAT CFIO TICH PHAN XAC DINH SSch GiAi tich 12 de gi6i thiQu khrii ni6m n12 tich phdn xdc dinh, cac phrrong phrlp tinh, + (n - \ !siff-zxcoszxd.x : nhirng rlng dgng d6' tinh di6n tich, thd tich. T6i mudn gi6i thiQu th6m mdt s6 tinh chdt vd nl2 rlng dgng dd cdc ban tham khAo ;'v6i hi vong : (n - t) [ si*-2x(l - + sinzx)dx nt*Z*r+... *+1 1 (ft + 1)! Thay/ bdid c6 Tt (4) ta cbn cci : .. x,2 lk+t d > I ** * 2,.+... + (ft + = dpcm. z _ ri- 2,2.4.4...2n.2n ry "",' Thi dg 2 : Chrlng minh ring 2 n++rc 1.3.3.5.5...(2n-l)(Zn-t)(2n+t) r (Zn)tt -tz L n , (2n\tl -rz I L €,r-l)tL] z"+t':' t
Ldi giai : Ta cd :4h = -l(modp), 4h -l: : -2(modp), ..., % + I : -2k (modp). Suy ra : (4k) | : IQH 112 (modp). Mat khric, theo Dinh li Wilson tac6 (4k) !t 1 : 0 (modp).Do dri : IQk!) +1:O(modp). (1) X6t n = 4h.(2k) !. Theo dinh li Fecma nh6, Bdi Tll2O7. Tim ttit cd cdr sd httu ti x d.d ta c6 : 2n - | : fzQk)t1ak - 1 = 0 (modp). (2) sd m : tlxa-+-Tgu +Eg ld. s6 huu ti. Tt (1) vi (2) suy ra : Ldi giai. GiAsrrr vdm eQ. Khi d6m -x e Q. n2 : l4h.(z\ll2 a 2ak'(x)t: o (modp) +2n VQv Sre st {rz + 19r + 93 -r = y e Q. Hdn n[Ia, d6 thdy nz + T : 0 (mod2) vn (2, p) = 1. Suy ra n2 + ?l : 0 (modp 2p). Ta cri xz +lgx *93: x2 +2xy +y2 (1) + r(19-2y):y2-93 VAy tcim lai, cdu trA ldi cria bdi todn Id cci vd s6 n, th6a di6u ki6n cria bii to5n, ching D6 thdy 19 - 2y * 0 (vi ndu 19 - 2y : han li n. = 4k.(2k) t. O suy ray : 1912. Thay grd t4 ndy criay vio (1) Chon ft : 249, tt kdt qu6 cria bdi toSn tdng 'Ta ccj gg : 19214. V6 li. Vdy c
x5 -x x3 -x T
Thanh H6a, Trd.n Anh Son (8M - Mari Quyri H6a, Pham Huy Timg 9A, BdVen Ddn, Hd N6i, Ha NOi, Nguydn Thi Thanh 7 Torin - PTCS Mai Thanh Binh SM,PTDL Marie Curie, Hi N6i. Chuy6n - thi x5, Mai Dinh Hoitng 9T, Tnrdng THCB : Tluong ThuQn PTTH Cd Maq Minh ndng khi6u Nga Son - Thanh }{da, Trd.n Hd.i }JA1 Trd.n Van Dqt PTTH Phan Rang Vo fhi Son 9A, Gi6ng V0 II Ha Noi, y, Trd.n. Cuitng Ld.n, VA Thi Ly PTTH Luong VEn Chdnh, Tuy Hba, Phri Y6n ; D6 Kh.oa Hiq, Phan Hodrug Vi€t, 7 Todn - PTCS Trdn DEng Ninh - Nam Dinh Nguydn Minh Thq Qudc hoc, Quy Nhon, Binh - Nanr I{d, Vil Drlc Phrtc 9 To6n, Tnrdng D!nh. Nguydn Thi Hd.i Ydn Qu6c hoc Hud ; NguySn Du - QuAn Gd Vdp, TP. Hd Chi Minh. Nguydn AnhVaru (B) Ii Quy D6n, Nha Trang; oANc vIEN Vo Qudc Hing, Nguydn Ch; Liruh, Pham. C6rug Thietl T'rndng Chuy6n LO Khidt, Quang Np, ; Bdi T8l2O7 : Ddry {xn} cho bdi *, = LZ Ddn Thi Thiin Huong, ItOVan ThaaP"ITHDOng Ha, Quang Tfi ; Nguydn Thi Quynh Hoa, La Anh : xln + x-Vn x-n+r -, n > 1. Tlm phnn nguyan cfi.a sd : Tud.n, Thd.i Anh Vu (PTNK Hd Ttnh ; Nguydn 11 Xud.n Tltong, Kibu Van.Ty PTCTDHSP Vinh ; A=rr-r.t+xz+l+ xrrn, * l' Nguydn ViQt BdL, L€ Huy Khanh PTTH Phan B6i ChAu, NghQ An ; Tfd.n Thi KhuA, Dinh Liti gid.i krta nhibu bgn) : V6i cric gi6. thidt Truimg Son, Trd.nVinh Thu, Tri,nh Htu Tfung, cria bdi ra ta sE tim phdn nguyOn cira s6 : NhtZ Qui, Tha, Phqm Minh Tl,td.n PTTH Lam Son, Thanh }J6a;Vu Dtlc Son, DinhVanTd,m i11 PTTH Lrrong V[n Tuy, Ninh Binh ; Phqm A(m) = rr + 1 * r; r+ ... + ._ + | Thhnh Nom PTTH Nam Ninh, Nam Hi, ?rtiz Dtc Quybn PTTH L6 Hdng Phong Nam Hd ;Zi 6 dAry m li s6 nguy6n duong cho tnl6c. Van Mgnh Hodng Van Thtr. Hda Binh ; Nguydn Y6im: l tacci COng HiQu PTTH Nguy6n Du, Thanh Oai, Ha '12 Tdy ; Trd.n NguyAn Ngu, Nguy&t Nggc Td.n A(1) : ,,* 1 = E= [A(1)] : O PTCT DHTH He NOi ; Nguydn Hdi Anh PTTH Y6n Hda HaNOi ;Nguydn SiDang,DQng Thanh I{A PTCT DHSPi Ha NQi ; Tfd.n Thhi Hodng, Xetm > 2. Tteiethidt ddthdyxn > 0Vz €Af. Vu Huy Phuong, Dd Thd fdi PTNK HAi Hring ; Do d 1 (1) N guydn Din h Todn PTNK He B 6c N gO Dtlc Duy, Phnm Dinh Truimg PTTH Trdn Phri, HAi rr* 4= E Phdng ; Tq Hd.i Anh, Tlinh X.rd.ru Manh, L€ Mat khric, do Quang Minh P"INK B6c Thdi vit Nguydn Lorug xn*t: 4 * *, 111 Quynh PTTTI Chuy6n Lang Son. -, n +L-;-,-n ntt 3. Bai torin dd drroc nhi6u ban hoc sinh (Vir,>0)n> lndn: THCS tham gia giAi vd hdu hdt c5c ban cho Idi giai dring. A(m\: 111111 Xl +-+- xZ xZ ^*...* I* x3 Im*l NCUYEN KHAC MINH BniT7l2O7. Gidi phuong tinh 11 _ : __ 1 < 2 (Z) (do r_*, >0) gx8+84+126xa+s&2+1 Z + Xl Xm*l Xm*l x8 +s6F + 126# +84* +g Tt + lA(m)l = l. (1) va (2) ga8 +84a6 + 126a4 + s6a2 + 1 lo ndu nL -1 a8 +s6a6 *726aa +84a2 +g -0 Tdm lai : lA(m\l:1. [, neu na >z', Ldi gi6i : (crla ban DQng Thanh ffd 11A Tt dd, [A]= [,4(1993)] = 1 DHSP HA NOi). Nh4n x6t : 1. C6 $t nhi6u ban gti ldi giei gx8 + 84x6 +tz&4 +9tu2 + | 'cho Bii todn. Chi c
Phui;ng trinh da cho cri dang ta c6 a, b, c 2 0, Ma a, b, c * 0 (dd vd tri.i c6 afqr) +'xf{a) : Q nghia),n6no, b,c) 0,vi0 I =: -a Son 11. Toan - PTTH Lttong Van Tuy - Ninh DApsdix:-a. Rinh), Nguydn Long Quynh 12 C"l PTTH NhQn x6t : 1) Nhi6u ban c
(mo * mu * m,)2 < 3(*Z + mf, + *7) : C6 ch crich giAi chi sir dung kidri thr.lc hinh q hoc bdc PTCS. (5) - Tnrdc hdt, chdng minh =1(o2+bz+c2\. 4' : Theo c6ng thrlc L6pnit, ta cri h6 thrlc : hi-Z(& +a,2 +b2 +b,2 +cz +c,2) (3) lo < u.u...til dri suy ra : 2Rcost , la+la *la < R (3 + cosA + cosB 1'cosC) vit Nhtrng, trong tam gi6c ABC ta c 4sot[5 @) r'
Trong cdc mat khric cria trl di6n ABCD ta gsp xe (3) sau 17 gid chay, dudi kip xe (4) sau drloc c6c bdt d&ng thrlc trrong trl, cdng vd ddi 18 gid chqy. Hoi xe (4) gep x.e (S) khi nd,o ? v6 4 bdt ding thrlc cci dang (4) ta thu duoc : .. Yo9rg dqn. Cd thd giAi hingphrrongph6p 2(a2 + a'2 + b2 + b'2 + c2 + c,2) > 4S\iB (b) {9 *rt hodc blng cdch dp dung cOng thrfc auong di dga vio qu6ng drrdng mi hai xe di duoc khi T'I (3) vd (5) suy ra : dudi klp nhau, chon g6cthdi gran Ia hicxe (l) dudi 8R2 > S{3 (6) kip xe (3). Dirng phrrong phap thrl hai s6 thu6n tiQn hon v6 mat tinh torin. I(dt quA tinh todn cho 3V thdy : xe dap gap xe md.y hic 1b gid 20 ph. Thay S = vdo (6), ta drroc bdt ding thrlc - NhQn x6t. Em V6 Thanh Tilng,10 chuyOn cdn tim : 8R2r > 3\iBY Torin PTTH Qudc hoc Hu6 dE c
Trong bai 'Bdt ding thrtc Jenxen" thdy girio Trdn Van Ving dd gi6i thi6u vdi -6 c6c ban s6 brio 21199-l vd cdch nhin c6c bdt ding thrlc Cdsi, Bunhiac6pxki, ... theo quan didm hdm ldi. Trong bdi brlo ndy chring t6i crlng sE ding crich nhin tr6n vdo m6t s6 bdt ding thfc khdc nta ctng kh6 thri vi. I. Dinh rughia hitn s6lbi ; Him s6 flr) x6c dinh tr6n (o, 6) gqi ld l6i tr6n khoAng d
dn tt: 24"2l2t[6 af, + ltun- r/0 V n - > l. nnr, *'+y =, Xdc dinh cOng thric tdng quSt o, }{UNG CUONG \ RA ldpKI Bei T8/211. Cho ti giscLdiABCD. Chrlng Cdc I{AY TIICS minh rdng ABC : D BAri Tl/211. Giai phrtong trinh : tsT+ts4+ts4 t-tS4+ -.2 a i6
dnti = zhe-n{6 af,* 154,n *'[6,vn > I iiFR OEUEnn$,r.;,lN.;.;lf Hil$.iii;il$$UE Give the general formula for on Fo; towCi Seconaari schoois [IT]N-G CT]ONG '1'8211. Prove the follorning inequality for Tll2l1. Solve the equation : a convex quadrilateral ABCD : vz3 *'7 : ABCD *' *; - rr'* 2x * 4y tg i+tgz+tg 4 tS 4-+ :,[w+r7T6i+w , -l---- A B C -- D-. - 16 I NGUYEN DI,'C TAN 4,+tgi+tSi+tS Z+tg, '121211. The sum of three given positive numbers is 1. Prove that the sum of two of T gl2l1. Let be given a pyramid SABCD, them is not less than 16 times of the product the base ABCD of rvhich is a parallelogram. of these three numbers. From a rnoving point M on SA, dlaw the line PlIAM FIUNG parallel to AD ; it crrts SD at.N. Take the point v -l CO SM Tgl2l1. Let *: wherep is an odd Q or' CD such that = Find the position Ci; So prime number. Prove-, that nt is an odd of i'i.f on SA such that triangle jl4NQ tras composite number, not divisible by 3 and greatest value. gn *r : 1 (mod nz) NGTJYI1N VAN I-OC DANG HUNG THANG T1Ol2l1. Let be given a circle (8) with center O, radius R and a fixed chord BC. A T4l2ll. Construct in the outside of a point. A moves on an arc BC of (3). Let -Ff be triangle ABC the equilateral triangles BCA', the orthocenter of trian gle ABC. Find the locus CAB', ABC'. Let M, N, P be respectively the of the orthogonal projections M of H on the midpoints of the segments CA', AB', AC'. inbisector of the angle BAC. Prove that MN = CP and the angle betwen the Iines MN and CP is equal to 600. DAO'IAM llIAN XUAN DANG T5l2l1. The aititudes AA,BB, CC, of an ,,,.' .MOrr.,.EiNH GtAt,fffuoig,6,,'.,.'.,.,., acute triangle ABC meet at H. The circur:rcircle of quadrilateral CA.HB 1 cuts the gtait qi'on ...E buae, rnedian of triangleABC. at 7'.'Ihe rnedian CM, ......,..,,,Tat'.'c4...cac...U4n,,,,00pt of triangle CAtBr cuts the circumcircle of thi i h-Am:1993 vn cu6c thi d.?.c biet d6u i cci .g!A! thuang, ::- triangle ABC aL 7,. Prove that 7 and 7, are ,: ,,, ,, :,,:::,:,,:: ,, :,,j, symmetric through the line AB. :'.,., 111,:':ifiong:.:diU.:'!ii: ni4m l.S,0.,,nxui.l,vi..ttAu. ki6n. NGI-TYEN I)i] khongiCho phup, bin td ihte chi moi ceie bs;:d Ht xoi:;'Ha Tey, Thanh IIc,B til bdc For Llpper Secondary Schools dia,:phttong d gdn ho4cicci nhidu:.gidi vd du 16.., i' I : TGl2ll. Prove the following relation between Tda io4n mdi cAi ban cbn l+i chua linh ra distinct positive integers ay a2t...t an I gi&i,vd 45Ii f,lang ,CfruOi [5,115qn. 1g5u n khOng.,dl drroc cd thd riy nhi6nr cho ngddi 2 (2 / khAci..ddhi..,lihh, k:r "?, ).:r"o\' :lrhOng,thd d{n,drroc vi 6 xa,bsn NC-u When does equality occur ? hay egi Aia ihi c.ti thd hiOn nay vd cld NGTJYBN [.,tr DI.]NG chfng ibi gfi giAi thrrdng tdi D+c,bigt T7l2l1. The sequence (an) is defined by : H nirIAS 9 V a.,*,r, UnU,, dA,bhiuy€n t nfi bn g eACr , can tiu y ad gihi thuong khdi thdt I4c o r) *l = -- (7, \,6 11
Dinh cho cdc )An chuhn ban bi thi vao vao Trumrg PTTH Chuy6n ,^^., PHUONg PHRP DICN a TICH Cdc ban d6u rdt quen thuOc cdng thrlc tinh NcO rHf; PHIEr di6n tich ctra m6t tam giric : Dit. Nd.ng S : (U2)a.h^: (U2)b.h,o = (ll2)c.h, NhAn cdc vdcGa (1), (2) vn (3) ta suy ra : S = (U?)absinC : (Ll2)acsirB : (L1?)bcsinA ACt BAt CB, :1 crB '- AtC-X" Bl.. II. Chrlng minh f,ai dai lugng s = 1aryEl4 ,: hinh hgc bEng nhau Tt c6c c6ng thrlc'i Biri to6n 2 : Cho LABC co canh BC : a, rdt don gran 'dd AC : b ud. AB = c, cd.c dudng cao tuong ilng chring ta cd thd giAi ld ha, hbuir. hr. Ggi S ld. diQn t{ch ud bdn kinh mdt sd bni todn kh
III. Chfrng minh bdt ding thfc hinh hgc Ddu ding thrlc xAy ra khi : Bhi torin 3 z Cho LABC ud' tnQt didm O b€n dalb =,{bla oa : 6 trong tarn gidc. Goi cor khodng coth tit O ddn vd: W:{ih+b:c tam gid.c lit. do, d6, d" utt. khod.ng ca'ch til O ddn cd.c d.inh A, B, C ld Ra, R1r, Rr. Ching minh Sr.ryra ddu d&ng thfc xAy ra khi AABC d6u 2 c.d, + b.db a) o,.Ro D6n ddy c6c ban d5 ldm quen v6i phuong ph6p di6n tich. C6c tran h6y bdt tay vdo vi6c b) Ro+ Rb+ Rc > Z(do+ db+ dc) gihi cdc bdi toSn sau ddy. Chric cac ban thanh (Bdt dang thtlc ERDOS) c6ng trong ki thi s6p t6i. (Dd 48 cAu II - Bo dd thi TuYdn sinh) Biri gini Bni tQp tU gif,i 1) Ha BK vd CL vu6ng g $la)d, * (cla)do (9) tit A, B, C ldn irrot la A1, ,B1, Cr. Chring minh Tudng tu : rdng : Ro > @lb)d,+ (clb)du (10) AH BH CH Arn+ ar1,+ crg'-a R, > (alc)do+ (blc)du (11) Vdi tam gi6c nao thi xiy ra ddu ding thrlc ? CQng crlc vd cria (9), (10) vn (11) ta dudc : (Dd 144 cdu V - 86 d6 thi tuydn sinh) -Ra+J?b*R.2 > (alb + bla)d., + (alc + clddo + (blc + clb)du Cu6i cing do: ('{alb -rtb-/aV > 0, sau khi khai tridn ta dudc : &i':::':'.cAC alb+bla-2>-0 Suy ra : alb -lbla >- 2 Trrong tt '. alc* cla > 2 blc+clb>2 Til dd : R^+ Rb +,Rc > 2(du+ db+ dc) t3
rHEM riruu cn{r... Chfrng minh : (l -x)h = : czn - c)nx + cf,rz - c'r{' * ... - (Tidp theo trang 2) Ttri au 3 : Chrlng minh ring ...-cfi-Va-r + %h -".)""cosr 20Ur 1 I I r =Jrr rdoo;dx = zoor' -x)hd,x: I Ch(rng minh : 2OAr zOOn , = [ {4, - cL* + 4n*2 - crr,t, + ... - r: I^ COSX : J __;- : e CLSII)I ,d*-d., ' ro0z ... - Cfi-4n-r a gPzr*ca)dx . SlnX r20bt 20An - (l-*\h+l ., 1 r I #l: SlD.f, : r I l00z I + I rdr*x, = (rcw-*"i*,+!cz*z- - -d.x. 11 2OOtt r dX I -*c!-r *a .#n"**r"*) =,J*7 r2o0n l,o* = * l) 111 -#n= r cL-i"5,**"L- 100n zOUr 200n' tintr ctrdt 2 : Ndu flx) -1"'.+ - *o*-.#nr*, = g(x) vdi Vr e (a, bl bb Tinh chdt 3 : thilf(4d.x = ls(x)dx. b Thi ar1 4 : Chrlng minh ring IftOs@*'>*a* = yg(n) -f g@-L) +f 'g@-2) - fl 't -fz)r(n-z) +... + (-tYl,,>Slll:+ onll rd - L 1 \ ' -L' .u^i+l"n - n*l I=U b + (-1.),?+r Chrtug minh : If*rror, ,a (1 + x)n = q, + C)x + A#, + ... + Cfixn 6 d6 f(x), g(x) vdtdt ch c5c dao him cria chring 1 ln li6n tuc tr6n [a, b]. "' J {t *x)n d,x = Vdn dung tinh chdt niy, ta dtra ra c6ng o thrlc khai tridn Taylor cria m6t hdm sd ! 1 Trong tinh chdt 3, khi ta ldy g : (b - x)n = c)x + A#, + ... + cf;*)d,x+ thi g' = - rL (b - x)nl, g" : n(n - 1Xb - x)o2 [fc|+ ,...,g(t) : (-l)n(ru - 1) ...2.1,g(r+t) = o.I{hif(x) (1 +r)u +1 1r c6 f'(x),1"&), .... f"*1)k), n-rl lo- O = ff(r) (-17" nt -f (*) . (-l)n-rnt (b -x) + : .1 1. (c9. +2c)x2 +... + ci*.) l: ^nl- f '(x).(-l)"-''^ (-l)"(-f)n-r fn-r) 61 3 1 2n+t _1" - n! {O-=)2...+ huy Z iyrc; = n+L . (" --i)I (b -x)n-t a tnion E : Chrlng minh ring b 1 +1 1- + .. + (-1)' r@@)fiO -O") l+ au - ,ci" Bci" +crb, - - b 7 n>r-t n2n + (-L)'*t I l"+t)1r;1b - x)d.x - - 2n"2n ' 2n * 1"2n - - 2n+l' a t4
: (-1)" I nlf(u) - ntf(a) - nlf (x)(b - a) - *,f'(o) ,. .y'')(o) n! .L 2l x'+ "'* nl t'+"' - ,.f 'fOfo -a)2 ...-7^)1aYo -"f)* Thi dq 6. Chop(1) : I, p{x) : p(l)p(r - 1) b + p(2)p(x - 2) + ... + p(n - 1)p(1), V n > 2. + (-1)n+1 tf a:Ri hi6um:*ir[1"*t)(r)], 1! Tn f@ = ; - ZQ - 4x)tt2 suy ra : M : mal.{f" *,)(r)] f (x) : (1 - 4s1-rrz, a
vii cAcu NrriN gAtr ... (Tidp theo trang g)' Ndu ta sit dung f@) rlnr ld hnm l6i tr6n moi khoAng cria nrla truc thrJc drrong vd bdt ding thrlc (6) thi ta crj thd tdng = qudt hria bdt ding thrlc d vi du 1 nhrl sau : Tdng qud.t h6q. uf du 7 : Cho o; i bi (i : 1, -a) la c5.c sd drrong. Ta cd bdt ding thrlc sau : \ 2", lr ,a-_I t" ( , (7) ,i,o"'i" l>, ",) Tr. /-t "t ) i:1 Ddu bing chi xAy ra khi : a1 aZ a, Thrrc vAy, v\f(x) : E:4:"':E rlnr ld ldi tr6n moi kho5ng cria nira truc thttc duong. Tt dd 6p dung (6) ta u; : (mpy* m*z* ... * mnx.n)ln(mp1* ... +mnxr) =< ( nelrllnrt * ...* m.p,n.lnxn (mi .O ;2 mi = l) yur taLrr cach i:1 dat t. l*;: : .l bit} b,1i :\i) VOr "*u , < i_ I a, (i : \iJ . lb,r, = Ta cci ngay bdt'ding thdc (7) Vi dLL 2 : Cho a, b, c }} cric sd duong. Chrlng minh ring abc offi.b,**,."ffirlrr+b+c) Ldi gidi: Bdt d&ng thdc dd cho tudng duong vdi (--+ t c \a+b +c \ *1--2 i"1 ' r t,,, lnb+ ("*6n6')'lnc> )'"'*' \ a*b ,a*btc, = ," ( B . Bdt ding thrtc nay tudng duong vQi ) alna*blnb *clnc,a*b *c,,a-lb *c, *--- B=-=\ g (8) /t"\ B / ." St,dy"q:.f!* rlnr ii hnm l6i tr6n mgi khoAng cria nfta truc thr,rc duong vA srl dung bdt ding thrlc (6) ta se = chrlng minh duoc bdt d&ng thfc G) -lqi Di6u dac bi6t thf vi !a.tr:.yi dp 2, nhir bdt ding thrlc (6) vd k6't qu6.f(x) : xlnx ldi v6i r > 0, ta tdng qurit h- \ill)rllM, n6u xi, xz> l. Tr] dci tdngqurit hcjabii to6n. 2ol Chrtng minh bdt d&ng thrlc : ua+bt2 * * r"' +eb1 @; b > o) Tt dd hay tdng qudt h
DAr HOr DAI BIEU roAI\ eudc t Ax rntr rrr HOI TOAN HOC VIET NAM Ngdy 3-12-1994 tai He NOi, HOi Todn Hoc Vi6t Nam dd tidn hdnh Dai h6i dai bidu todn qu6c ldn thf III vdi 103 dai bidu chinh thrtc. D6n du v6i tu c6ch ld khrich cria Dai h6i cci : Gs Hd Hoc Trac, Chri tich Li6n hi6p cdc h6i KHVKT Vi6t Nam ; Bi thu thrl nhdt Trung rrong doin TNCS Hd Chi Minh Hd Drlc Vi6t ; Gs Nguy6n Vdn Dao, Chir tich II6i Co hoc Vi6t Nam ; Gs Phan Dinh Di6u, Cht tich HQi Tin hoc Vi6t Nam; c6c Nhd gi6o nhdn dAn : Nguy6n Thric Hdo, Ng6 Thric Lanh, Nguy6n CAnh Todn,... Gi6o srr Nguy6n Dinh Tri, Chrl tich HOi, da doc b5o crib tdng kdt hoat d6ng cria H6i trong nhi6m kj' qua. Cho d6n nay dd cd 624 hdi vi6n deng. ky chinh thrlc. HOi da gcip phdn ddy manh crit Lioat dong cia c6ng d6ng to6n hoc. H6i de td chrlc thdnh c6ng nhidu hOi nghi To6n hoc trong nU6c vd qu6c t5, dec bi6t la HOi nghi qu6c td v6 giAi tich rlng dung (TCAA -93) v6i 60 t
Gidi ddp bdi THAY CHU BANG Sd HAM LUOI NHAC - HOC + HOC HOC KHEN CHE CHAN a) b) c) Trrrdc hdt ta thdy K: L: dd cho HOC cri nghia thi 11 * 0. (1) 1 vd ! vi CHEti m6t s6 chia hdt cho 5 n6n E : 0 t,oirc E = 5. Nhrlng ti b) ta thdy n6u E : 0 , thiIl:0.VayE:5(2) Trl a) vdc) ta ccj : c +c : N (hoac 1N) suy ra M : C (3). Tt dri vd tr) A+ O = A ; M+c A+O:ZsuyraA:E:5(4) Tir a), c) vd (4) suy ra o : }hodc o : 9. Nhung ndu o : 0 thi H + H kh6ng thd bang 1I1 duoc (xem a). Viy O : I (5). Va tr) FI+ H + I : lfl suy ra 11 : 9 (6) Ti c)suyraN* I : CvdC*C : 1N. Viytac