Tạp chí Toán học và Tuổi trẻ: Số 231 (Tháng 9/1996)

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Nhằm giúp các bạn chuyên ngành Toán học có thêm tài liệu phục vụ nhu cầu học tập và nghiên cứu, mời các bạn cùng tham khảo nội dung tạp chí "Toán học và Tuổi trẻ - Số 231" tháng 9/1996 dưới đây. Nội dung tạp chí giới thiệu đến các bạn những nội dung về tính đồng thời với việc giải một số bài toán, phương trình dạng toán vi vòng quanh, hội thi Tin học trẻ không chuyên toàn quốc,...

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 231 (Tháng 9/1996)

-. {- (t 5C \cihiJ BO GrAO DUC VA DAO TAO * nOl roAN Hoc vrET NAM r4r cui RA HANG rnANc * zo NAru NGAy srNH crAo srr Ncuy6n cArun toAru * TINH odua THoI VoI VIEC cTAI MoT SCi LoAI ToAN * HOr so cfc s6l rofn rRolrc DE rHt royEn slnH Dqr I vft il * VE MOT UNG DUNG CUA SO GAUSS NGUYEN TO p ? * IIE THI GHOTI DOT TUYEII TitfH THAIIH HOA -^. ---- + . * ?€ ,onoiU niru* qt"rE ?oi?r -t/? -l/dry z?/i?r? * H1t rHt nN Hec rnE rcu1xc cHUyEN TzAN auoc DQi tuydn ViQt Nam da thi Tin hqc Qudc td 1996
cH0c MirNG zo NAM noAv stNH clAo su NGuYEtt cArun roAx. r6nc elEN rAP rep cniroAn Hec vn rucil rnE UA fadt fAf xin chtlc Nhdn dip 70 ndm ngity sinh gi6o su 2B-9-1926 - 28-9-1996 tqp chl.TO.N HQC v?t tong sq nghiQp' Ai6o sr dbi dito stlc khde,iuAn tuan vui vC, cb nhibu thdnh tlch ro l6n trong cbng tdc rAp cxl roAx xqc vA rudt rnt mNH sAcH cAcnilvffirrn0N rolrvrocviradt tp.E crta cu,o su Ncwfhr cAmr roAN t Sd.ao ngu ngdc hay th6ng minh, sd 1(1),1964, trang 2-3, €. Cildu sAu cua mot bai to5n si6n d&r,9ei r z lTQl'913'tz s6 2(2). 1964. tranq 1- 2. zo. fiirriirs tidn C6hs trons toin hqc, s6 tl-12 (81), P74, cnshi 4' z. ca, tning", sd {4), 1965, z- s, sd z(s), ,i'ca"- - il; ir6 veu io6n ighi gi Fm gl.dd d6n muns- '! 1965,"irvd.'ra:;a"udns bai noi oanqTdnlhrr lV., a6 (94)' 1977' 2-3. quan tigng c0a mlOt c6ch nhin, s6 3(102), 1978, t 3; 1- 2. s. Mof bn,Jonq oh6o suv nqhi s6nq tao, sd 7(10), 1965,.! 2. "r"-diJ':a zz.'iAi +. ruddrinAinlnh noc tr6nq"t(i tni toi sinh.si'6i'to6n l6p 10 s6 4(103), 1978, 1- 3. todn midn B{c ni6n kh6a 1964-196s. 56 9(1?), 1965, 1-.3' 5. LUc cdn lA hqc sinh t6i dA hqc toen nhu th6 niro ? 56 11 ;?. ?XiI3l ii 1f#'TbD5*9'1).8'rol,5o3; €'., 1-2 (14), 1965,.1- 2. zs. An sau tronq Pitaqo. lA dclit, sd s (131), 1983, 1- 2. o. MOi idg6ns cAi ti dn c6ch lAm ph6p nhAn, sei t 6l;, tsos, 26. That vA siA-Oclit, a61(13s),1984,1-3. 14- 15. 27. s6 va l6nh. s6 5 (139), 1884, 2- 5. z. NOn hsc tdp tidu si c6c nhA bAc hqc nhr-t th€i ndo ? 3(18), 28. s5. lenrr va hlnh, a6 3 (1431 1985, 2- 3' 1966. 1- 2. 2s. Sd Enh. hinh vA qi n0a s6 4(1441,1985,1-2. e. phai'uidi nhin mOt kh6i niQm toan hec theo nhidu c6ch 30 Na'o'aa'ndt dau i6i s6va lonh, s6 5 (14.5)' 1985, 1-2. khAc nhau, s6 6(21), 1966, 1- 2. ir. Moi iidu ad toan mil, s6 (14711986,1- 4. s6 2 (148), €86,1- 3' 9. Ngay tJ nAy gid iAc ban hay t6p duot seng tao trong 32. MOt cu6c tranh ludn li thrl, s6 1 (159), 1988,.1 2 - toAn hoc. s6 10(25). 1966. 1- 2. 55: TliJ a0is ii tdm Cio hon mot ihut ma nhh, s6 a (to8), ro. Nnln cic' sd tneo ('uan ai'dm "to6n tLi" c6 lQi nhrr thd nAo ? 1989. 1- 3. s6 1(?8), 1s67,2- 4. s+]"idr'ni'in vd sLic s6ng t4o, sd 2 072), 1990, t z ; sd 3 fl. Cai_si tA quan trqns, Sd 4 G1), 1967,1-2. (173), 1990, 1- ?. P.fi6o tuc mOt cAu chuyQn c0, s6 6(33), 1967,1-2. ss. xa ta md ssl xd'ta mA odn qri,qrj, s6 s (175), 1990, t12. 2. r5. runAn 6ai 'ninn hqp" tiohs t
tffi lffi* Tinh "dbng lhdi" 6 phAm vi bii viSt de c4p li sU tudng duong crla crie d6i tuqngtrong rn6i quan hQ to6n hgc ndo dd. Trong chrrong trinh tonn phd thOng lM trung hgc co sd, mQt 9d loai to6n li6n quair ddn rihung'vdn d6 sau dby sc giof lffi{= b4ndgcphdnniohinhdungrat6cdungcrjavi6cvf;ndgngtinh,,itbngthbF Itm'd llDd.ngthrlc:A=B |l-d6@2 $: . _ 1.1. N€'u4 B lduictfphgpthi ttA = Btacdchring ttbngthbi la taphqp hiru han hoQcilbngthli id tAp hop v6 han... I #.3 5 1.2. Ndu A vd B ln hai bidu thrlc bing sd : * A, B itbngthbi ld hai s6 * kh6ng Am ho6c kh6ng drrong. nguy6n, fr ng thbi li s6 nguyOn t6, fr ng thhi lA hqp lkffi lffi F _ _1t, B " dbytg thbi" ld hai sd s6, ddngthbi ld s(i chinh phuong... * A, B'il6ng thbi" chia hdt cho m (m lw id- -- kh6ng chia h6't cho nz ld s6 nguyOn khric 0) ho6c ddng thdi :n A, B dbng thbi cci sd drr nhrr nhau trong phdp chia cda chring cho m lffii lm A, B dbngthbi cci chrl sd tf;n cring gidng nhau.... 1.3. Ndu A, B ld cdc b.id1t tlnc nguy1n chia bidn thi tri sd nguy6n cria chring | *md I .oU c{ng cci ddy dri nhtng tinh chdt dbng thbi tudng t{ n6u tr6n lw A(r) = or{ * arft-r + a;{-z * ... * an l&d Cho l* l4c vd B(x) = bof' +bfn-r +. " . + br,-{ *bn I @\ . 2.1- A(x) = B({) v6i v r ++g6c h6 sti cirng bac tudng rlng cing ddng thdi bing nhau o, = bivdiV i = 1, 2....n ' l,w lw lEffi 2'2. A(x) = 0 0thi )a,= 0 c=rV Aid6uit6ng I M i:t Il@cEe 3.3. Di6u kiQn xay ra ddu b&ng cria mQt s6 bdt da;g ih,t" qrr"r, thuQc : ching han ddu bang ctia bdt ding thrlc C6si , + > {ab xAy ra aa, b ilbng thdi I 0
Bdi toa.n2 ; (d6 thi hsc sith gr6i to6n kh6i phd {e;=4Y+25=5 la=22 (dPcm) thongqu$nD6ngDa - I{n NOi nam hoc 1995 - 1996) {@=firT 16 : a '=l: Tint rngi x, y, z trong phtrong ttilt' ' [' " _= - x+y+z+4 = 2t[i4++fY-3 +6tlz-5 (1) -@ -212 +9:0 Vin dqng chinh xric tinh il6ng thli cbn c6 tdc Girii ; Di6u ki6n c6c dn phAi ddng thbi th6a dung:girni lih6c phuc mQt sd sai ldm trole gtAi m6nr 2 2, y >- 3, z > 5; (1) O, i = l, 2, 3, ". ft mi tinh ttbngthbi dat d,rq" ddu bing ctra uic sd h4ng .k kh6ng th6a m6n" q )o, = 0 thi suY rudbngthbi c6cA, = 0 r) ^ Ldi giai dring: Bidn ddi A =2(x - 4)'* i- I 1^ 1 1 1 A, iting thili bing0 ho4c frng Lkn 2 : Al uit. +O -|), *r, +; >;VayA,,in: f dattaigie thbi khdc 0 dd tt $;1 - !)2 = 0 suv ra dugc tricria c:ic bidn ldm cho tdt c|bdt ding.thfc cti {i-Z-1:o trong vd phAi r/Dng lhbi dat dugc ddu bang hay Nhd vAy ta da drla bii to6n kh hb kdt hop nguyen thi hai v6phAi dbngthbi chia-hdt cho 3 v6i viQe sit dung tin}l' ilbng thbi 6 di6tt ki€n xAy = y- = 3ft. Thay vio ta suy-ra tic'p 15y phr:ong ra ddu bing cria bdt ding thrlc niy ld o; b phAi trinh cO nghiQm nguy6n tli khong th6a m6n vi dbngthbi bang nhau mQt vdbing 3 cdnld kia chia hdt cho 9 ln di6u Taco (x2 * 10r * 8|2 , +(U + l)(xz + 8r + 7). kh6ngrhd ilbrtgthiri xAY ra. Ddu blngx6y ra *v2 18x +7 = ?.x * | Cacn Z; L4p luQn reii dua vd ding thrlc, : ay2 16n+6=0+rr=-3+{5 31p - kz - n2) = 1 ta thdyhai v6cta dingthrlc d6u kh6ng ddng thdi chia hdt cho 3. rz = -B -{5 b)(2r+1)2-A=4 NhQnrdf : Ndu kh6ng nhin nhQn vi rip dung Bd.i tod.n 7 : Chttngt6 tap hgp cdc s6 nguy6n linh hoat tinhilbngthdi 'k6t hgp vdi vi6c stl dgng td dang {6nt 15} ld v6 han bdt d&ng thrlc mir giAi bing ph6p bidn ddi trrong Aa{toan 8 r Tim tap hqp tdt cA cAc cQip (m, ru) duong th6ng thtrdng s6 din tdi phr.long trinh sao cho bi6'u thrlc : bAc 4 phfc tap hon, ldi giei sE dii hon. Khi nang m@,[xa+x - WrTE - n - - nO{F* + t[1) 1) lgc v6n drlng tiirh dbng lhhi dugc nAng cao se +r[f(T +4 -{7 ='0luon dringv6i msir' giup hoc sinh dang On thi vdo c6'cl6p 10 chuy6n Ggi y : giAn Adi (5) dua v6 dang shpn giii mOt s6 bdi torin thuQc mQt sd d4ng, f1(m,n) g(x) +fr(m,n) = 0 vdi msi x'+cac ching han : h@ s6 ddi vdi hhm sd gk) dbngthiti bang kh6ng. Bdi tW 4 : Chrlng minh ring tdn tai duy nhdt Bdi tod.n 9 : Tim Gtt'UN cria bidu thrlc : mQt bQ s6'o = b = c dd d&ng thrlc sau dring : ar=({7+1)r+r-6{;+10 {W=G;T4L +t1dr475 = -c2 +4c +b(B) Biti tod.n 10: GiAi Phtrong trinh : @4@={)r-+6 +gp4f+16 = -1c-212+e S{i=Z + 3{Z -rr : x2 -Gr + I + 8rl7 Ta nh6n x6t vd tr6i lu6n ldn hon hoqc bAng Bd.i tod.n J L' Chrlng minh ring h9 c6c drrdng 9 cbn vd phAi lu6n b6 hon hodc bing 9. Dd cd thlng :! = Zmx*3m * 1 d6u lu6n di qua mQt ddu "=" t}.\ dbng tlthi phAi c
Phri Thg, \dnh Phf), Nguydn Trd.n Ngqc Quang (9T LC Quf D6n, Long Khrinh, Ddng Nai), Dinh TrungiThd.nk (88 Trong Didm PT cIl Ha T",ong, Quing Ninh), Nguydn Minh Kian (6T, Chuy6n LC Khidt, Quing Ng6i), Lfi.m Quang Minh {84 Chuy0n Bqe Li6u, Minh Firi), Phqm Quang Huy (9A, THCS Trdn Phri., Bai Tli227. Gidi phxong trinh nghi€nt Dfc Trong, I-Om Ddng),Ngqydn DtcAnh (614 nguydrt.' ri -x:J+3x- 2t- 5 =O THCS Ddng Mi, Ddng H6i, Quing Binh), Bni LAri giei cua LA Hodng Anh,9T, NK. DOng Dang Quang (9 To6n Chuy6n Tam DAo, Vinh Son, Thanh Hcia. Phrl), Vuong Trsng Nghia (PTCS Mi Xuy€n, Stic Tr&ng), N guy dn Minh T hanh ( 10E PTTH Ta ccj : *3 - *2y*3r - 2y - 5 = 0 (1) Tk Lai ChOu, tinh [.ai Chdu), Nguydn Tilng +=ry: x+ x -5 (2) (8CT Chuy6n Tr) Li6m, He NQi), Hodng ,: +, Phuong Ddng (A THCS Cdc L6u, Lio Cai), Vir,y E Z hcn suyra : x - 5 i x2 +2 Nguydn Khuydn Ldn (9T PTTH, Larn Son, Huy(* -5Xr +5)i x.2+2 Thanh H.6a), Nguy5n Drtc Tidn (8Al Chu Van x2+z-27i x2+Z An, TAy Hd, HA NQi), NguydnVan Trung (9T2 Suy ra 27 ! xz +2 Phan Dinh P.htrng, Tp Dh Nd.ng), Chu Dic Anh (88 Chuy6n Ung Hba, HdTdy), P hun C tu r ng \6A f)o r e Z ; xz +2 > 2 n6nx2 +Z ed thd nhan' c6c giatri li rrdc c{n27 nhu sau : 3,9,27. VAn Ki6u, Quing TY'1), Ti€u Thi Mai Pkuorug (1042 THCB Trdn Qu6c Tudn, Qudmg Ngei), Ld -Ndu x2+2 = 3=> x: *1 Thanh 7h (.8Tr Nguy6nAn Khrrong, Ho
chrra ddy dtr, Ii lu{n chua chat chc m4c dtr cd Nehq An z Le Thi 70.m, 9Ao Hung Dtng, d6p sd dting. Yinh,IiXud.n LOc,8T Nghi LQe' Ldi giei tudng ddi tdt cria c6c b4n: LA Thd Hd Tinh : Dinh Thi Hba,8T NK Th! x6. Thang 8H Hh NQi, ?r&n Tud.n Aruh 8f Kh6nh Qonng Binh : Phitng Ngqc Chuong, qC Vo Hda,"Dqng Hd.i Th.anh 9T Thdi Binh, Vuong Nin[, Qu-aug Ninh ; Nguydn Hfrq. QuYEn, I Trsng Nghin 9T S6c Tr5ng, Ldm Th.d.nh Dq.t Chuy6n QuAng Tr4ch. 9A Bac Li6u, D&o Phuong Bd,c 8A B6Van Dan Tri : Nguydn Hilu Nghi, gTL Qunng Ha Noi' Biti vidt Chuy6n L0 Quf D6n. '%t*f;rTfrHANG Ttrita Thi6n - Hu6 : Tttn Nhu Quang,9 BdiT4l227 z Chuy6n Nguy6n Tri Phuong. Gei I, G tdn luot ld tfrm duimg trin nQi tidp uit Quing Nam - Db Ndng z La Trdn Tlun g, ho.ng td.rncrtam$ttam giac c6 dQ dd.i cu canh lhn 8T Nguy6n Khuydn. luot Ih 2, A,4. Hd,y tinh d0 ddi dogn IG. Qu*ng Ng6i : Va Chi Thitnh,gJ Lq 4hi6t' Ldi giai so h/gc : La iloilng Dtlc Khd.r.h, 9T Nguy6n Nghi6m, Xdt tam gi6c Drlc Phd. ABC 6 AB :'2, ginh Oinh z Nguydn Xud.n Tli, Vo Thanh BC=3,C4=4. VieLgAQudc hgc, Quy Nhon. GoiAD vd.AMlir Khdnh Hda : NgO NguYan DuY,8T Le Quf phAn gtdc vi D6n. trung tuy6n cria Lem Ddng z Phqm Quang IfuY, 9Ar, Trdn Ta eo LABC. Phri, Dtlc Trgng. DBABl Ddng Nai :.Fl&Mi nh Ngqc 9/15/ Trdn Hrrng DC AC 2' D4o, Bi6n Hba. MaBC=Bn6n Blr Ria - Vfing Tlru : Nguydn. Quang Long' BD = \,DC -- 2. Vrlng Tdu. IDCDlGM tP ffA Chi Minh z Phsm Minh Hitng,9T Lai cd Nguy6n Du, Gb Vdp. d= CA: ,: GA -Ti6n Giang z Nguy€n Ngqc Binh Phuong, + IG ll DM IGAG2 . Do dri vir 9T NK Cai L$y. DL. = AI,I = g Minh Hlhi z Luong Thd Nhdn, TCT ChuYGn 2 2 2 ,3 ,. -1 76=-rDM=i@M-BD)= s \r-r) = , B4c Li6u. VU KIMTHTJY Vay dQ ddi doan IG\d|. BldiT5l227. Cho hai duimg trbn (O ; R) uit' (O', R') udi R' > R cdt nhau tq.i hai di6m. A, B. Nh4n x6t: Tia OAcdt dudng trbn. (O') tqi di6m. tht hai C, niy cti c6c b4n : Giai tdt bdi tia O'A cdt dudng trbn (O) tq.i di€rn thfi hai D. Qunng Ninh : Nguydn Dtc Thq,9A Trqng TiaBD cAt dudrlg trdnngoqiti6p tam giorADC didm Cdm PhA. tqi di€m thi hai E. Hd.y so sd'nh BC udi BE. IIai Phdng : N guydn Kim Tlfing,Sl, tnin Phf . Ldi giei. N6i cac dudng kinh AM, -A-l/ cira Hii Hung: Nguy6n Dinh Son, 9A Kim (O), (O').Tac6 cdctiaBM, BN cirng vu6ng g
n6i tidp (vi p, C qnng nlih MIII d{di mpr gric *9"9139"Mr = Nr.M|ML = Br; N1=€, f"Oi r 2'2 , 2, 2 (" -;)- -5 ( * -=) *4 = 0.Dety - y --- try.(q hse" fq), cirng chdn cung Aiioac AC): V4y Br = Bz (D€qi p-lA€r"o didm cria -y2 -by+4 - 0-!t+ l,yr_= 4 cdctiaMD,NC. Do PDA = pCA = g0o n6n tri Ttdd *-?=1*r=-1ho6c2 gpdc PDAC ngi pA li dtrdng kinh dtrdng x ligp vir trdn ngo4i ti6p PDAC, do dd cung li dudng kinfr 2 dudng trdn ngoai tidp tam gSacADC. Hon ntra, :t,--=4-x=Zf.r[6 x, trong.tarn gpdc AMN, cric drrdng cao AB, MD, Dtp ; Phucrng hlnh c6 4 nghiCm - L,2, 2 + {@ . p, n6n tia BA di qrra p, vd do "d {C $dng quy tai Cdch2: Phdn tlch thinh nh6n trl dci chrla tdm O" cria dtrdng tr6n ngbai ti6p tam # - # + hM, + 4= (r* l)(x - 2)lx2 - 4t - 2\ = 0 g?.4?.9.YQy BA li truc d6i xtlng crta dtrdng Tt dd tim drroc 4 nghiQm nhrr tr6n. t1yt.(!"), vd kdt hop v6i (1), ta ei c6c tia'&Cl, Nhan x6t : Bii niy drroc rdt nhi6u ban gui ldi B^E_ dqixfng vdi nhau qua.BA. Suy ra D hodc E giAi vn giAi dring. Trong s6 cic ldi giai tdt cd nhi6u ddi xt?ngv6i C 1ua BA.Suy raD ho4c,E d6i xr?ng barr d PTCS nhu : Br)i Vidt LQc gA Bdven Dan Ha v6i p Uua BA. Ma R' * R n6n kh6ng phei la.D NQi, Nguydn fidn Ttung 9T NghQ An, Tfr.n Thl nrd ld.E ddi xrlng v6i C qua Ae,lta{tiC BE. MlAn,gAPhti YEn, Mai NhiNgx, gT Thanh = Nhfn x6t. C
kh6ne thuOc c5c c6ng thr?c bidn ddi lugng gitic o N6u a > Zthi oo = 1 + |fG1"l' + (ry)21 Jrai ( tti. Khong iiban cho ldi giai hoqc qu6 tat noac qu5 itrdm ri. C6c bqn sau dAy c 2) thtu nwn Buon Ma ThuOt) ; Ddng Th6p z I{b LQc ThuQn '1 \-P)dt Att rHcg 'itri.*a Cao Lanh) ; Trh Vinh: tort*qt2+1-YB (t) iia" ru"ynh ritd Nhanh (11A Trudng TH Phti Ft a* tn"ai nuang) ; Ninh Tttott .I'€ z Ldi giai (etra da sd cdc ban) i"i tttn. truurre Chuyon Ninh ThuAn) ; Bi '["4 .-;t|-4at ni" --Vr.fig TbuI I'{guy6n Qu.an-g-long.(IOA, Trudng Ving Tdu) ; TP Hp-qhi ftIinh : ^-,rT4t =- r-*thi dx = --=- Det * "='1+t2 (l+t)" t inuu{n Ns& nbn (11A, PTTH Lo Hdng Khi dd Pf,o"sl , ffi,a*f, Hba : ird'n Tudn Arth' (8 ? (t - P)dt To6n-Tnrdng L6 Quf DOn - Nha Trang) ; r Quing Nam - Db N6ng z P-!y'a-n Aalt. Huy t4 +aP+1 rira. ?mn Lo Quf D6n) ; Thira Thi6n - ifo6" Caa Thd Anh-, Dodn Xud.nVinh( ( lf CT 1 -)- l=a' (1+f')'r \t;=z Qu6c hgc HuO ; Quing Trl z L€ Huy 12 Todn Tludng Chuy6n Lc QuJ, -Truung, Oon) ;--NBhj. An : -l;4r t + a-2 ri ', N euv 6i Dtlc Duong Vd.n Yan (9T, 1 1T pftU Phan BQi Chau) ; Thanh H6a : Truong u' 1rd'x 'zY't' Cao Dfine (L6b C, PTTH BiTN SON), I,A Vd,N =WJ7*z- Hitne f ris, pttlt L6 VEn Huu' Dong Son) ; Ninf, Binh' : Pham Trung Kien (9CT D6ng = 1 (arctgr *c) Cto - Tam DieP) ; Nam Hd't Phan Ttudn {A Giang, Vu DuY r/d, (10CT, 11CT, ?{TH Lo Suy'ra: Hdn{i'hong) ; Th6iBin}l' zLuon-g-ft!'ttng 9f Trtrd:ng Chuyen Kidn Xuong) ;--Ha. T&y : Bnl ', O-F)at tol*aF+l :Lorrrn@1= tla-2- '-' 7+t' lo Xuaru Ean ( 1bA PTTH Nguv6n HuQ) ; Hb NOi : D6 Ngqc Dttc (6H PTCS Trtrn-g--Vrrong), 1 {;4 Nshilit Thanh Titng (lOB FTTH Thang =Warctt- 2- 6o*), Nguydn Vlnh Chi (11-A PTTH Minh Kh#'- fi"yO" tt Li6m) ; Vinh Phl&: Biti Do viy ltl oor"rsS = $'84 r"> Dd.ng Quarug (9 To6n Trrrdng-Chuy6n Tam OAoi; Hii Hung z DQrt:g Thd.nh Trung (l0T 1- PTNK HAi Hung) ; tlh B6e z Phq.m'ViQt Ng-as D$;,{;4 =r thi x > 0 vd (*) 0' 2. Tt Bei de ra, bqn La Hd.i Td.n (Gia Lai) 1. '?;r de d6 xudt Bli todn khr{i qu6t sau ; "Cho sd . x - arctgr. thqc khong d.m a uit. cho day sd { o,.,} duqc xdt 7 *x' g(x) vli dinh nhu sant : ao = a, an+l = f(x)= x- ') , = an(4af; - L0an + 42 Vz e N. Hd.v xd.c dinh s6 hq'ne tdng qud't a-n ' g(x)=--.x ,-arctgx I +X,- Cnfuang Ac ki6n thrlc dtrgc dt c4p t6i tiong S-GK PTIH- Ghdne phan ban) hiQn hirnh, ban L'H' Tdn : (l +x\ -2x2 Ta cd : g'(x) dagiAi dfngvaWrangfn go.n Bdi torlrr n
Caa Thd Anh, Dodn Xudn Vinh, Qudc hoc Ldi giAi. Ggi H, O, I vd, a ldn ltrgt ld. truc tArn, HtiS;Ngzydn Minh Tudn, Triin Chi Hda Diro tAm cric drrdngtron ngoai tidp, nQi ti6p vi dudng Duy Tit, Quing Binh ; Nguydn Phxoc Hiep, tronOlecriatam gSdcABC )a,b, cvi.R, rlidQ Phi Anh. trIilng, Qudng Nam Dit Nang Vd Tdt dii c.ic canh vi brin kinh cric dudng trdn ngo4r vA Tlrung Amsterdam I{e NOi ; Hodng Tidn nQi ti6p tam gidc dri (chu y ring ro li trung didm lkang Bac Li6u, Minh tlhi; Nguydn Si Huy o&aOII). Tt cric hO thrrc v6ctoquen thu6c DHSP HA NOi ; Phan Anh Dic Hii Hung; ++++ Nguydn Vu Hung DHNN, Hh NQi ; Nguydn OH:OA+OB+OC Thanh PBC, NghQ An ; Trh.n Nguy1n Ngqc oIA + ort + rtd =d DHKHTN, I{i N6i. taduocl- + € + NGUYEN VAN MAU 2POf: z-€++. aO!+bOE+cOC-, (l) BidiTgl227. Cho trl diaru OABC uu6ng 6 O, ZpOa = pOA + paB * pOC, (2) c6 chibu cao OH - H ud. dO diti cdc canh cia vidodd i tanr diQn uu6ng : OA : a, OB = b, OC = c. --+ ---) (n -n) --+ Chtng minh rd.rug : Oat-OI=Iut= OA + - 2p a.cotgA + bcotgB + ccotgC > Slt (o-bl + (o-c\ + (trang d6 L B oit C ld. cdr g6c crta tant gidr ABC). +]-oB+v"_-oc; zp zp (3) Ldi giai. Det BC = a' , CA: b' , AB - c', (trong d6 2p : a +b +c). thd thi .. a,2 : b2 + c2, b,2 _ p2 1r2, Binh phrrong vd hri6ng hai v6 cria (3), ta drroc : c') : +F. a2 @ - a)2 + (p - !): + (p -c)2 Ta ccf (dinh ly him s6 cosin) : Sabc; (4) Trl (4) vA (5) ta thu drrgc : Tn (3) vi (4) ta duoc bdt ding thrlc cdn tim. Ddu ding thtlc dat dugc khi vd chi khi o2 ^ ,R .) -* - 4 _-Itr*t'=(Z.r)- al2=d,2=!: a : b : c, trlc li trl di6n vu6ng cdn. Dd ringE >- 2r, n6n ta duoc 1i : Nh&n x6t : Khri d6ngc6c b4n tham gla g;\Ai R bAi torin nAy vd ncii chung, d6u dua ra tai [iai d.,:utl:r_r; (6) nhu_tr6n. Cd m6t sd ban thidt lfp truc ti6p-cric hd thrlc : hayli:d=aI=p-r, (6') R dtgA: Ftgts : c2tgc {= zS) (trong d6 P = 7 la ben kinh drrdng trbn Ole) (rut tt rn6t bii to6.n trong 86 d6 thi T.S m6n Di6u dri chfng t6 rang drrdng trdn Ole To6n) rdi tr) dci thu drroc h6 thrlc (1) nhu cach giAi de n6u. ,R, trcuyEN DANG pHAr \, , P = Z) tiSp xrlc trong vdi drrdng trbn (I, r) n6i tidp tam giac. /. Bni T10/227 : Sil dung udcto, hdy chilng Nh{n x6t : Tdt eAcdcban tham gia giei bai m.inh rdng, dudng trbn O-le (duOni trbn 9 torin niy d6u cho ldi giei dring. Tuy nhi6n, cd didm.) cila mQt tam gid,c thi tidp xilc irong u6i mQt sd ban d6 str dung nhtng h6 thrtc kh6ng duimg trdn nQi tidp tant giac d6. quen thu6c l6rn, ching han.
gIC?=p2-16Rr+5P phAi cri Rrcosa = Rrcosfr (2) v6i (trong dd G la trong tAm tam gi6c) 1^atL HQ thfc ndy dring, nhtrng cdn ph6.i chrlng casrr=@,cosP= minh cdn thSn. NGUYEN OANC IHA:[ Tt (1) vn (2) suy ra Rt = Rz = 100Q ; B,niLu227 t1 Cho mq.h di.& mfic tha s db uc ffin tinh 1 uoi aL= *,nitra L=iH' UAB = ti[2siruot NhAn x6t. Cdc em cri ldi giei t6t: Nguydn Thanh Lon* 11L, PTTH Dio DuY Tr), Ddng "{1 H6i, Qutnibinh ; Vil DuY HAi, 11 CLTPTTH Le iT5ng Pihong, Nam Dinh, Nam -Hn ; I/?) Cd.nh Liem, 11 Li Hda. PTTH L6 Quf D6n' Qulng TYi; Pttilng DuY Hung .,T! Quang in;n/,tt^^ PTTH chuyon Th6i Binh ; Phan Thimh lii.g lZL, PTNK Hii Httng, -Trdz Thanh Xudn LtG, PTTH chuy6n HtngVuong Viet Trt Vinh Phri 1 Nguydn Thd.nh Hung l0L, chuydn Lo Khi6t Quing NgAi. MAI AN]] 1. Dd cho hQ s6 cing sudt cria toit'n m.qch Bi}ri L21227 : Ngudi pht4. trd.ch thang rnd'y bd.ng l.thi R1 , R2, L , C uir.at ph&i thfu md,n hQ cia mQt tba nhd. cao td.ng ld' mQt ngutti ldm. uiAS thrtc nhu thl nir.o ? dilng gid. Ong ta da treo #An tudng thang ntd'y 2. Ctw Rt : 100Q, C :# PF ud' tk'n sd mEt-ibng hb qud. ld.c chay drtng d6.biet hhi nd.o hdt giit ld.rn uiQc trong rugay. H6i 6ng ta f = 50H2. Hay tinh cor gid tri cila Rrud. L dd hQ c6 kdt titic cOng uiQc drtng giit hay hhOng ? sd c\ng sudt cila todn m4clr. bd.ng 1 dbng thbi hi€u Cho bidt thiti gian chuydn dQng (theo dbng hb dien thluAM uit. t) *o m cit ng nt Qt gid. fi hiQu &&g. dilng yAd udi uicto gia tdc hudng lan ud' Hudng dln gi6i. 1) Dnng phuong phrip hudng xudng ld. nhu nhau, m.6dun cita cd.c gia gian dd v6ctct, x6t doan m4ch AM vd MB t'a c6 t6c citng nhu nhau" cac gian dd tuong tlng sau dd ghdp 2 giAn d6 Hrrdng d6n giii. Goi f le thdi gian than S.av dd- -dtns v6cto f- lant truc) sao cho di l6n ho4c di xu6ng, ? Id chu ki dao dOng cria ddng fl.an =di* *O*ucingPhrrong v6i f hd dfngy6n so v6imflt ddt. thi 3d daodQng todn phdn crla ddng h6 niy fiong2t giay sO ld Tt c6c giAn dd suy ra I tr = Icosa, Iz = Icosfr, 2t 2t t ^lE Ic Nt=T=;W_:Tvl tga :7=RroC; ,1 r.(: g LA. Ggi Q vDr f, li chu ki dao dQng criaddng h6 Ep = t=i,, trong thang mAy khi v6ctd gia t6c o hr'r6ng l6n " vd htl6ne xudns. ta cd U.e.M= Rrlcosa; t f UMB = RrIcosP vi ngoii ra ,l u,qn Tt=2n ll fi,r::r"\ , _" U*usinp - re uaa (tinh theo ddng hd dring y6n) 56 Irtn dao d6ng tdng cdng khi thang mdy di N-T-a li = Ualasina Tt dd rrit ra t\ I \ --T- i l6n vi di xudng ttt- + o + tis - o) N2= T.4= rfo ({g r h€ thrlc Ir t_____\ R2tc ---D vlG +; +r[g -; < 2,[E nonN, < Nr. uua R?, Suy ra ngudi dy lim viQc ldu hon rnfc binh thrtdng. | *R2ruP* ---E> NhAn x6t Cdc em cd ldi giei dring.: 7d -- (I) R)+oltz ''' r - "48 Quang Chinh,l1A PTTH chuy6n Thdi Binh ; NeuyEn Thi Kim Oanh, 11I, PTTH Y6n Lac 2) Dd VIr.i. pf.ta iehqm $bng Thdi llArPTTlI Dho UrqM = Uuo Duy T\I, QuAng Binh. MA
rssS o=,Ir(-1),*, (T".#) HOANG HOA TRAI (Qututg Ngdi) BAi T9/231 z Trong rnQt phang cho duimg thd.ng d uir hai didm, A, B klidng thuqc d. Tim trdn dudng thd.ng d mQt didm M sao cho : cAc lop rHcs I MA- MBI = m (nald.rnQt dQ dd.i cho tru1c) BAi T1/231 z Gid.i phuong tinh sau : vrayldrgdD#Nc I f2 +y2 +&r -2y - 1 | ++ = 2*x -l xz -gx +21 =z-xy ncuy6N otJc rdN BAi T10/231 z Goi diQn tich cor mqt ddi diQn (TP Hb Cht Minh) udi cat didm 4 B, C, D cila m|t trt diA.n ABCD lhn Bhi T2l231 : luot lirS4 Sr, S.5S, Chrtng minh hQ thtlc : Cho ,, r.y, +3(x2 +yzl +4(r +y) *4 = 0, Sa + S, ..fB + Sc . IC * So.ID = O .IA uit. xy > O. Tim gid. tri 16ry nhdt cfi,a bidu thrtc : lit tdm crta hinh ch.u nQi tidp fi a. - "..fr9yg^Q6I diQn ABCD. ^ f'1 I .' NGUY(Hv*,fn:,"oaN M--+- x\l cu yEN n rura"r-r H (Hdi Phdns) r.r c u\dN .. Bni LU231 cAc oE vAr il Bei T3/231 : , a Chtne minh rdng ndu cdc sd a. b. c uit Trong mq,ch.u€ b€n c6 2 r,7 + rio-1 ,[i ]a nuu"ti, thi cdc s6 r[i, riE, rE udn h€ cil.ng diQn trd cung ld' hrtu ti' RrrRl=r,Rz=R3= 10f2, NcuyfN o6 ddn bda Ro rndi fiAu tht4 114 R/ c\ng sudt dinh mtc. udiPhdns) Bai r4l231-: a) Nnh r, Rr, Rrbidt riing Tanr giat.ABC c6 BC = 8 cm, dudng cao mudn d4n sd.ng binh thuimg AH = 6cm. O phia ngod.i tam gid.c, d4ng hai thi phd.i bdbdt di 2 trong cdc hinh uudng ABDE, ACFG. Goi M ld. trune diAn ffd, khi d6 sd chi cilaV, d.i6m doan ndi tdnt cila hai hinh uu,Ong. T[nh hh6ng thay ddi uit.bd.ng 5 ldn khod.ng cdth til M ddn BC. ovchi ctiaV". sd vreo Le@f 2. vu rruu eiNu b) Cho bidt cOng sudt dinh mrtc crta din l?t 8W, tinh E. Bhi rb/2ar : rran or*{'r;:;l (o), cho duimg hinh AB ud, dd.y CD (C, D hh6ng trilng udi A, B). Gqi M lir. giaD didm cila cac tidp tuydn BdiLzlz3l z Mit con ldc don ']'"#" c6 chibu dd.i tq.i C, D crta duitng trbn (O), N ld. gio.o didm cia p cd.c ddy gung AC_, BD. Chfing minh rd.ng MN ddy treo ld I = duqc treo ud,o bubng 16 Qn) uuong g6c udi AB' thang may ding yAn. PHAra nt,hrc Gia tdc trgng lqc lit. g (Hd N/)i) (mlsz). Kio lQch con I t qo ld.c ra hhdi ui tri cd.n CAC LOP THCB bdng 1 g6c ao rdi thd Bni T6/231 : Gid silphuong trinh luqng grdr sau c6 nghiQm. xo cho dan dQng hhdng ufun t6c dh.u. Khi con It *__!__ ' sin2r *kcoszx= j* +' L -zL ld,c uila qua ui tri cd.n A18 - sin2r cos2r bdng B thi cha thang Hay dnt nhttttg gid fi thttc @ tM c6 duac crta k. mdy roi tu do. I NcO vAN Hrf,r 1) Vdi a = 20- HdJ (Hd Nr,i) tinh thiti giaru dd con BAi T7l231 z Ilm. tdt cd, cdt hdrn i6 f : I .*E l6,c di til A ddn C (hinh u€). thm, mdn f((x L)f(y)) = y((f(x) + ll udi msi x, + .D Chnng fi rd.ng udi ao c6 d.Q l6n thtch hop yeR thi khi con ld.c chuydn dlng tit A ddn C s0 c6 1 NGUYEN DOC HUY ui tri mir. uQn tdc con ld.c ddi udi dd.t bd.ng hh6ng. (Long An) iinh o' Bhi T8/23f z Cho a, b ld cdr s6 thuc duong. TRANrnenc H,NG Tim gi6 tri nh6 nhdt cia bidu thrtc : (Quhrg Ngdi) I
+ !, , I tr For trpper secondary schools f;.t*.;l. $*i.Si,iii$$UiH TBl23l. Suppose, that the following trigonometric equation Fon lower secondary schools sin2r * hcoslx: -+ + *]; - z Tll23l. Solve the eouation sln'r cos'r l*2 + y2 - Lry + 3x - 2y- 1l + 4 - I *2 - : ?-x has o rootx,.. Find the possible real values of A. 3x+21 T7l23l. Find all functions /: R - R TZl23l. Find the greatest value of the satisfying : 1l f((x + t)f(y)) : y(f(x) + t) expression M : ; * -, wherer andy satisfy for all r, y € R. the conditions : x3+ y3+ 3(x2+ yz)+ 4(x+ y)+ 4 : 0 andry > 0 TBl23l. Find the least value of the expression *'l3l23l P1:ove that if the nurnbers a, b, c and 995 {o +ilb +t/c are rational then so are the I o'I hn numbers{;, {b, {7. T4l23l. The mea-sures of the side BC and the . , lt=l .t A:Sr-1 . zJ\^ll-'(***) altitudeAll of triangleABCare respectively 8 cm .where a, b are positive nurnbdgg. "' and 6 cm. Construlct the two squares ABDE,' Tgl23l.In the plane, let bp given a lipe d and ACFG at the outside of triangle AB C. Let M be two pointsAp not on d. Find a' point M oi"r dsuch the midpoint of the segment joining the two that : MA MBI : m (m is a given leghth) centers of these squares. Calculate the (listanc'e' I - " front M to BC. T10i231. Let Sr, 56, S., S, be the areas T51231. Let be given a diameter AB and a ofthe faces opposite respectively to the vertices chord CD (four points A, B, C, D are distinct) A, B, C, D of a tetrahedron ABCD prove the on the circle (O). Let M be the point of relation++€+€ intersedion of the tangents to the cirde (O) at SJA+S1IB+SJC+SalD=O C and D and let N be the point ofintersection of the chords AC, BD prove that MN is where I is the center ofthe inscribed sphere of pcrpendicular Lo AB. the tetrahedronABCD. Ddnh cho cric ban chuAn bi thi vio Dai hoc *a(\/Lt? MMOT SCI Mgg TEAN YffiffiFdffi M$ TMB TUYITN SBNH *ST fi VA EE NA$V$ ffiffiC T9S6 - Tg97 T4 NGQC TRi Trong hai dot thi vdo dai hoc ddu ti€n ula qua, - Ti6m cQn : * TiQm cAn dfng : x : 1 vi nhi6u hoc sinh dn rdt lung tfng khi gap phAi cac bAi (r + 1)l todn trong c6c d6 thi. Thric chdt c6c bdi torin dri dbi ,. llIT) ---------= = 6 h6i c6c ban phAi kh6o leo vdn dung kiSn thtic dA hoc r*1(r-1;z vA ld k6't quA cia c6c bAi to5.n quen thuOc khric. T6i * ti6m cAn xi6n : xin din chrlng ra dAy m6t sd bdi torin kidu dci, mong (r*1)3 rang qua dAy grtip eic ban ttr tin hon khi lim cric (cdY:-=x*5+ lk-4 bai thi trong mua thi nim sau. " (x-112 (x-l), -) Bii todn 1 : (cdu I - Dd thi tuydn sinh DHQG Dtrdng thingy = x * 5ld ti4m cdn xi6n vi He Noi) / -(r*1\3 1) Khd.o sat s1t bidn thi€n uir ua db thi hdnt lim []:---+ _(x* b)'JI = 0 (r + 1)3 ..-.--------^. ***L(x-l)z SO.'Y: " - BAng bidn thi6n : @-112 ? Biei luAi th.eo k s6 nghiQnt. cilaplutong ninh : (r+1)3 -k(x- 1)2=0 Tont td.t liti gidi : 1) Tap x6c dinh , Dy = /l{ 1 J (x+l\2(r.-51 +x- -1 : I=" -D6thf: [0
l-- Tl.nt gia. tri t6n nhcit cia *r, s6 y = cod'r sin4r (0 < , n q tir cdc sd tu nhien ldn hon $l n, 1 Tom ftt li;i gid,i : Ta di tim g.t.l.n criahdm s6y2 : cos4x sin'tx D4t t: coszx ta di khAo srit hdm s6 g(t) = P . (t - l)q v6it € [0, 1]. D6 ddng tim dtroc g'(t) : p . fl -, . (i - t)q - q . P . (t - t)q - | : Dua phrrong trinh dA cho v6 phuong trinh : fl - | (t - t1,r-t tp - @ + q'ltlvdlAp bAngbien (r * 1'1 thi6n cria g(t) nhrr sau : tudng dttong , * : fr. Tr) d6 thi cira hinr (r - 1)" so cia cAu 1) ket luan dO dnng: 27 U, : phtrongtrinh cci 3 nghiOm Z 27 k : phuong trinh cri 2 nghiOm i: * . i:27 phrrong trinh ccj 1 nghi6m. g(il datg.t.l Tt ld ( -4- Nhdn rel.' Nhi6u hgc sinh nrSi ring ddy ld dd n \p +q I)' I\p-{-+q t)' bdi to6n "I?" ,ri chtta hoc loai hdm s6 ndy. MOt : p;{;'t'q t,f" ld him sd y dat g1|trildn s6 khd.c kh6ng vE drroc d6 thi vi thdy drJdng cong khi f nhdt c6t tiem cdn xi€n. C6c ban lu6n nh6 bii torln r P rP/2r Q : khio srit IuOn lu6n duoe giAi quydt theo ttng budc trong sdch d5- n6u. Ngoii ra ti6m cdn xi6n (;i)'' *rl4'' khi'r arc *" 1 ,t ( r(t/2 vdi dd thi hdm sd dci ln ti6m cAn tai vO tAn. Nhhn xet : Mdu ch6t cria ldi giAi la di x6t hdmy2. Biri todn 2 : (c6,u IiI (2) D6 thi tuydn sintidai ' Biti tod.n 4 (CduIII.z - Dd thi tuydn sinh Hoc hoc Ngoai thuong) viQn quan h6 qu6c t6) :"ir,f : 5cos3r "in| tai trinh { # Ol Ching nrinh rang nla . nlb . nL, 4 Gid.i phuong udi nto ; nlb; nl.c lit cac trung tuydn tng uoi cdc Sxxx Girii ; (3)
ffiffiffireroreffiffiffiry@"ffi (Phil YOn) H&n cric b4n d6u thdy rang trong to6n hgc Ddn dAy h&n nhi6u b4n s6 dat vdn d6 : m6i ccinhi6u vdn d6 hdp ddn li6n quan ddn cric s6 sd nguydn Gauss cd mdi li6n quan gl vdi cric s6 nguyon t6. Trong bd1 vidt ndy t6i xin dd cAp ddn nguy6n t6 Gauss hay kh6ng ? C6c ban d6u bidt khai niom sd Gauss nguydn td, d6ng thdi nou dAc di6'm c&a cdc-s6nguy6n li : mpi-sd nguY6l rnOt t?ng dung khri thf vi ctra'kh6i niQm ndy l6nhon l d6ubiduthf dugcmQtcrichduynhdt trong vi6c giAlquySt m6t vdn d6 trong pham vi (kh6ng kd ho6n "i) dudi dang cta Jb frgc-ptrd [tr-org. nL = p\t . p\z ...pf;r vli pild cde s6 nguy6n td z, I. Sd Gauss nguy6n td l}r gi ? _ lir cric drrong. Trtr6c hdt xin ndi qua v6 s6 phfc. D4i-dd Oai"O "g"yO" vaiZat sd nguy[n Gauss ta crlng cd tinh rang : t6p sd thrrc duoe md rQng bing cdch drra chdt gdn gi6ng nhi;ay : th6mkhriiniQmsdAo,ddlis6imii2 = -1.Cdc Dlnnti l.NaukhOngXds{horinvf vhkh6ng s6 hang dang o t ib v1i a, b e R goi ln s6 phrlc. phAn bi6t 2 s6 nguy6n Gauss li6n kdt thi moi s6 Tr6n t6p C c6c sd phrlc ngudi ta xAy drrng 2 nguyOn Gauss a * ib kh6ng li6n [r6t vdi 1 d6u ph6p todn md r6ng tri ph6p c6ng vd nhAn tr6n cd sg bidu thi duy nhdt : a * ib = (ut * iur),tr (u2 + iu2)n2 ... (uu * iu1)nr " ilf,ij + (c + id,) : (a+ c) + i(b + d) v6iu,*iu,ldcdcs6 nguyen t6 lauss (a +ib). (c +id) = (ac -bd,) +i(ad +bc) Cdc sd phrlc o * ib vdi a, b e Z goild. cdc s6 n,ld. cdc sd nguy@n drrong li = 1, n) nguvon Gauss' c.hl"g ta lray tim hidu *1T:.9 ' nguy€n Gauss cd^nhtng tinh chdt gi tddng tu *"i$ viq" chting minh dlnh li ndy xin dinh cho nhrr s6 nguy6_n hay kh6ng ? _ "6";;;."^^-^' ai"f, H tr6n ta c6 : Di!'l'n'shia.Ir4?isdnguvonQausq-,.+11-va itiiir,,Zis.a)v6i a*iblds6nguyenGauss : kh;;;"1'ie;i;iloi r tfri i"o" tO" tai s6 Gauss c + id drroc goi ld li6n kdt vdi nhau (ki hiQu 'a +ib; c *id.)ndua2 +b2: c2 +d2 nguy6n t6u,1 lusaochoo + jbphAichiah6tcho Vidu: 2+3i^3-2i u*iu. . i- | II. MOt tturg dlrng cfia sd Gauss nguy6n td Di.nh 2. a * ib vd c + id, liL hai s6 -nghia a * ib dtroc goi ld chia hdt cho ,MOt sd ngu]ren nf,rr thdnno thi bidu-t[i atroc nguy6n Gauss ; thAnh tdng 2 s6 chinh phuong ? ". Sau mQt thdi c * id (ki higu : a -r ib i c + id) n6u tdn tqi m6t gian suy ,r"gtri toi thdy rang ccilhd trA ldi cau h6i sd nguy6n Gauss il * iu sao cho : nay bing c6ch stt dqng sd nguy6ri td Gauss. T6i a +ib = (c * id)(u + iu) xin trinh bay vdn d6 nnydd c6c ban tham kh6o, Chfng ta c
Cluhryminh. Nhudanddtr€nndup = q.2 *b2, nguy€n t6q dlng4k + 3 phAi cci sd m[ chin, vdy p ngu)€n t6 lon hon 2 thip ui dqxg 4h + l. n p!e! cci dang nhrr d6 n6u trong hQ quA tr6n. Ngugc lai n6up nguyOn td crj dang 4A + 1 thi Pd, ddy rndi cdc ban tidp tuc suy nghi, ch6c theo bd d6, tdn tai s6 trl nhi6n J sao cho : chin cdn nhi6u fng dung thri vi khric crla Gauss acpvia2+li p. nguy6n td, girip cdc ban cci th6m nhttng c6ng cu Theo nh-{n x6t 3 cd 1 sd Gauss nguyOn td m6i trongvi6c giAi cac bdi to6n sd hoc. Ap dung u -t iu sao cho : nhirng kidn thrlc dE trinh bdy d tr6n, mai cric pi u*iusuyrap2i u2+u2 l*) ban hay tht giei cric bdi todn sau dAy : D.o az + 1 p n6n a2+ 1 = (o +i)(a Bdi todn 1. a, b, c, d ,e N, gid si ! -i) i u * iu ctng theo nh{n x6t 3, o * i idec a - i a2 +b2i c2 +d2, chtng ntinh ritng , "1*u1 ph6i chia hdt cho u *iu, kh6ng mdt tinh tdng qu5.t, gia stl a*ii u*iu suy ra : bidu didn dusgthimh tiog z ,a rniin rfr:":,# a2+li u2+u2, vi Bdi tdn 2. (Dd thi hqc stuih gi6i Hunggari - 193[]) a
oE rmr cHoN Bol ruvdru roAlt t-oB -s mqM rlCe tsss - 1ee6 rirun rhlANH HoA $80 phffi khilng kd thdi sian Phdt db) Bhi 1 : a)'GiAi phriong trinh trong tap hSp c) Vdi m6i biSn sdr, y, z, t chicd thd ldy mQt cdc sd tu nhi6n trong c6c giri tri sau 1"010, 1945, 1975, 1995. f -JS = 1995 ' Haylim gi6 tri c:iuax, y, z, t sao cho tdng : b) 'Iim s6 tu nhi6n n nh6 nhdt md c5c rr6c s6 A = (x - y)2 +(y -")2 +(z-t)2 +(t-x)2 trt nhi6n cria ncj Ie dl, d2, ..., th6a m6n : c6 gla tri nh6 nhdt. it d, < d:.... BAi 3: Cho AABC vdiAB * AC, AB * BC . Goi M, N tttong rlng li trung didm cria c6c c4nh ii) dJ : 4d.r - 5d, BC, AC vi G ld giao didm c:iua AM vdi BN. K6 iii) d,s : zd.d pl - 1 c6c drrdng phAn giSc tro ng AD vit BE ctn LABC' Chrlng nrlntr ring 4 didrnD, E, M, N nim tr6n rBlri 2 : a) H6y tim bd s6 thuc (x, y) th6a mot drrdng trbn khi va chi khi tf gtae CMGN ln' mAn: ttt gtdc ngo4i tidp mOt drrdng trdn. ['*' = 2 Bei 4 : TrOn td gidy k6 cric 6 vu6ng c4nh lrt *xy *f:3 bing 1, v6 drldng trbn bdn kinh R = 10 khdng di qua dinh ndo cria cdc 6 urOng, kh6ng ti6p xric dd bidu thtic M - *2 + y2 - xy d,4t gia tri ldn nhdt, nh6 nhat ? vdi canh ndo cria c6c 6 vuOng. H6i cci bao nhi6u u bf drtdng *u" b) Giai h6 phrrong trinh vtii tham sd o : ""u:* "tlJStrualliNu [x+Y+xY:a]*2a (Thanh Hcia) v: 2o1 uaio > o 7x1 + pB rffi chtoN oOlrtsvdrq rrQc SNii olo[roAry u6p re nrAnn'Hoc less - leeb rirun rHANIl HoA 'Vdng I- 180 phfit khdng kd chdp db Vdng 2' 180 phrtt khilns kd chdP db Blri 1 : Cho brit glac ABCDEFGH c6 tdt aA BAi 4 : Cho da thrlc : cric gdcbdng nhau vd d6 ddi c:lc canh ddu Id c5c /(x)= (1 . x * 2. x2 +... + nl')2 - s6 nguy6n. Chrlng minh ring cric drrdng ch6o AE, BF, CG vd DII ddng quy. - o*2 * o*t + ... + o2,,rv' Bai 2 : Chring minh ring hQ phuong trinh Chrlng minh rnng: sau khOng cri nghiQm nguy6n drlong : 5n2+5n+2 (^ dn+l * an+2 * ... * o2r, = Cr?r*t. )*'*f - 22 -P 12 t*^' zt l*t - - BAi 5 : Dey sd {r.,} cho bdi c6ng thfc + t = dWT 1li, xt ='[5 | {n + 3 " v6i gi5 tr! BAi 3 : a) GiA srl P, vd P,ld citc da thfc h6 X,, sd thltc cta c6c bi6n rr,' r r, *', . Hdy chtlng minh ndo cria o thi d6y s6 f , = o^ cti gidi han ? ring kh6ng thd xAy ra ddng nhdt thrlc BAi 6 : Cho tam giric d6u ABC, D ld didm el+12,=xl+x)+x!. bdt ki nim ngodi mat phing cta tam gi5c. b) GiasrlPT, PJ, PJ,..., P^ldc{cdathdch6 Chrlng minh rnng : s6thtlc cria cdc bidn r, , x) , ... , xotJnla m6n ddng a) tr) ta do+nbA, DB, DC cd thd drrng duoc mQt tam $5c T nio dci. nhrit thdcpj + P4 + ... + P?,: x2, + x2, + ... + xi . b) Haitam g;6c T vd' T' c6 diQn tich bing H6y chrlngminhrdngn > k. nhau, trong d
riu utEu sAu ruEu roAN nqc eu6 ruOna @ mwuww sffiM ffi$NG mmmm ruq vgmm qrraruffi ?.rcuyEN o0c suy (I.ong An) . Chring ta dd ttng quen bi6t v6i hQ phtrong trinh ddi xrlng Kidu 2. M6t dang md rOng ctia = f(*,,)> f(xt) * g(x) > g(x) = xtz x, he ddi xrlng Kidu 2 le hO phuong trinh dang VQY xt = n2 vd tt) dd suy ra : "hoein vi vd g quanh". xr = x) = "' : x,, f(x) : g(x) ii) Trudng hgp.r ch6n : xt 4 x3* f@t).- f(x) + g@2) > s@a) f(xr) = g(xj) +x.t7xj (r) * f(x) -= /(r+) + g@3) < g(r-s) ii,, -,1 : s(x,,) +r, (.r"s .......... f@): s(xr) * f(xn -,) < f(x,,) + E(xn *l) < g(*r) - Cung nhrr hQ ddi xrlng Kidu 2, dac didm chung cria hO "ho6,n v! vdng quanh' Ii n6u h6 cd *xr_i f(x) + g@) > f(x) + x,,) x2 quanh ctia b6 s6 ay cung li nghi6m cria h6. Vi the viQc giei he phuonglrinhll) trong trridng VAy :r, 0,voimoif + 8@) < g(rr) + x3 4 xl =3tz* tt-t+l €-R . Vay f(t) vir g(t) cittg d6ng bidn tr6n rB, n6n = f(xn - t) < f@) + B(x,,) < g(ri) + x, ( 11 theo 'cria tinh chdt r thir :,/ = z : t,v1itldnghiOm Ydy:xr
Giai. f* _ x.J : = xr)g ltt - Vi vd trr{i cira c6c phuong trinh d6u dtrong i^-_ .: lt2 - tJ-"'-'loo nOnt, !,2 > Q GiAi ra ta drroc : 1)11 =r::2 (loai) X6t hdm sd/(r) = +t2 ,' Glzl 2){,:x2=-l - {2 (loai) PG)=tv6itE(0,+o; " Tacri: 3)xr=x2=-1 +{, f(t) = -(21n4)(3t2 " q . (;7) 7,2f +P 0 l*, : o (0, +-;. 4)1 ' l*:= -t SuyrafgiAm ffin (0, *oo ) adoetanr va dog tangtrGn tinh chdt 2 thir : v : z = t. vdi t (. = N6n theo ti: NOn lxt -t Id nghiQm..cta nghiQm.^cria phrr.ong g trinh f(tl-= 6(f) va ya ro 5)1,r=, rdngf tren ( 0L +oo ) phrrd'ng trinh/(t;-= ino trinh'ridy 6(tr) ndv ch-i'ccj 19 chi cci mOt nlOt n8hi6n t : l rrrarus hop 3 : r,
Him s6 g(t) c6g'(t) : SP - St * a > 0, v6i + xn e fifa,a] * fB) e f{t[a,aI) moi/€R +B(xn_)eS(r[i,"1) (vi A' :-lG - Ba< o do o r'j, . rg + xn - 1 e F[a,a] + ... => xz e F[i,a] biSn trenaB vA do g(o) = 0, ^ lyv> 0 6:.{d1S ningft) khi vA chi khi f > 0. VAyr, e t{s-al, vdi moi j = 1. 2. .... n o Tr) vd trrii ctra cric phrrong trinh kh6ng dm T\.6n doanfl a,aJ hitm 6 f gShmc6n g tang nOn : ta suy ra xi >2 0, vdi mgi i : 1,2, .... n. . - TrOn klioAng [0, +-; cric hnm sd f vi e d6u tang, nen.x/ : 12 : ... - x,n = t , vdi /-ln nShi6m *Khi z chin thi , )*' - &3 - "' t' - 1 lra ho ctia p.huong trinh 4t2 - dt * a) = 0 vd do lx2=x4=...=xn z5 tf ) o, Z n6n phuong trinh ndy cci nghi6m duy lzrl = *, +\ nhdt f : 0, tfc h€ cci nghi6m duy nhdt : dtroc thu gon thdnh : ] *', Il=X2=...=Xn:0 25 lr*', = *, *T r1.ef,ruan:a>2. VidU_-5 : Cho z li s6 nguy6n ldn hon 1 va ."-" b"),=x2z*a o f .O.Chrlng minh h6 pf,rr-ong trinh sau co : xl -r a nghiOm duy nhdt. lzx,x) - .a2 Thlvdtaduoc: (rj -x)(\ *xr*2xrxr) =0 2xl:ar1, - +xl xr(vixr1-xr*xrxr>O) +Xl 12=...=xn a2 Zxi-x-*- zJxl o Vfly trong mqi trudng hSp ndu (xt , x2, ... ,xr) Ie nghiQm cta tiq thi xl = x2: ... : fr, = t, vdi / ld nghiQm > ,[a "Aa ^. Zxi_l:xn*; a2 *n phrrongtrinh :2t2 = t +{hay2ts - t2 - a2 =0 Hdm s6 p(t) : 2F - t2 - a2 .a2 xfi=x,*; *l c6 9'Q) = 6P - 2t vd cd bAng bi6n thi6n : Giai o Tacrithdgiesir:o>0 Xdt hai hem s6f(/) : , *+ vit g(t) : lsz _o-__------> a'.*.-.-.-- r *o NhAn x6t th6m ld e\[a\ = -a1[a - h2 < 0 Hems6f(t) c6 f g)= t-lvaabAngbidnthi6n n6n tt bAngbi6n thi6n trbn ta suy raphudnitrinh p\t) : 0chi cri m6tnghiOm t e [rlo, +-] . Do d
crrt oip nAr r6t que trfn ddu BAngui 4 d6i, m5i d6i d5u3 @n. VQy tdngsd @n 4,3 ddu cria Mng Ie ]5a : 6 t@n. M6i tran "thing - Dofn nS&2 sint) thua'Id 3 didm, m6i tran hda la 2 didm. Ndu 6 tQn Trong gid ra chdi sau tidt h99, b^al T!6ng - thua tlii tdne sd didm ctra bAng ld ae" fa it a"e"Nhrrng ogai aoai, ilgdy, th5ng, nam sinh cho 3 ban Lan, 6 x 3 = 18. tdng sd didm cria bAng-Ii 16 do vAv bAns cd 4 tran th6ng - thua vi 2 tr€n hba' Hdng, Ngg". '"Gie g;n dh6ng bAo cdc ban dci Idn lugt thgc hiQn ;t 4 doi ctra uane ld A, B, C, D vd A ld cac phep todi sau dAy, ri6ng cira m6i ngudi vd d6i ddu bAne &uoc 6 didm. Vav A th6ng 2 ttQn ai bi6t : ',ra ttrrr" 1 tr-An. Gia st A thing C, D vd' thua B' -- chorgay khdng sinh cria minh nh6n v6i 2 r6i c6ng Tt dd kdt quA cira 3 trdn cdn l4i (1 thing - thua, - fay 2 hda) cd thd ld : th6m -- 11. Trutttug hop 1 : Bhba C, D vi C th6ngD (hoac - Lay kdt quA vrJa tim drrgc nhin vdi 5 r6i D thing C) cQngvdi22. ----1Ldu Nhu-vAvB drroc 3 + 1 + 1 = 5 didm, C dtroc k6t quA vrla tim drtoc nhdn v6i 10 rdi fl c6ns vdi th6ni sinh cria minh. ' 1 Ldu 3 = 4'didm, D duoc 1 didm (ho6cD drroc 4 kdt q-uA tim drroc nhAn vdi 100 rdi cQng didm,Cdrrocldidm) .rr6i na# minh vi c6ng th6m 33' liut*g nop2: BthuaC (hoacp)v-dh0,aQ Qoag - "i"frtri" C) con Ctioa,O. Nhu v4yB dugc 3 *7 : 4 didq,Q fvfOi ta" cho bidt kdt quA cu6ictrng pal Thing drroc 3 * I = 4didm (hoacD drroc 3 + 1 = 4 didm) s6 nrii dring ngny th6ng n-em sinh-crla - ttryF" P auq" z aiCm (hoad C aqiz didm). ga, Lr:r, Jtto UiAt kdt quA cu6icing ld 3-39514 -ti CA hai trtrdng hop ta ddu thdy kdt- quA cria gan Tha"S dd n