Toán học và tuổi trẻ Số 217 (7/1995)

Chia sẻ: Physical Funny | Ngày: | Loại File: PDF | Số trang:20

Thêm vào BST

Báo xấu

84
lượt xem 7
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Sau đây là tài liệu Toán học và tuổi trẻ Số 217 (7/1995). Tài liệu này trình bày về sự chính xác trong giải Toán; hồi ức về giáo sư Tạ Quang Bửu; định lý Lagrange và các ứng dụng và một số nội dung khác. Mời các bạn tham khảo tài liệu để hiểu rõ hơn về những nội dung này.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Toán học và tuổi trẻ Số 217 (7/1995)

7 p1t1 rap cni na NcAv tr uAxc tsAi.tc fl Srg ctrintr xic trong giiiri Toin E *a> & ue- anc 7q Zaa*ry- €cza tl "oai- U6 C6C UN6 DUNG t( LAGRONGE OINH l Anh , Ci6o su Hoing Ch(tng c0ng thAy rrd trudng THCS thitr6n DAm Doi, tinh Minh HAi
ToAN HQC VA TUbI TRE MATHEMATICS AND YOTJTH MUC LUC 7'rnng Ddnk, cho cdc ban Trung hoc Co si For Lower Seeond.aryr Sclnal Leuel Friend Tdng bi€n tdp : Vu Httu Binh - Su chinh x;ic trong giAi tod.n NGUYF,N CANH TOAN 1 Nguydn Cd.nh Tod.n - H6i rlc v6 gifo str Phd tdng biAn fip : NGO DAT ].TJ Ta Quang Briu II()nN(i ('lltiN(, Gini hiii ki truric Solution of Problem.s in Preuiou.s lssru c Crlc bdi cira sd 213. n0l oOr,rc erEN rAp ; Db ra ki ndy Problents in This Issue 10 Nguy6n CAnh Todn. Hoang o Ddnh cho ccic ban chudn hi thi vtio Dui hoc Chung, Ng6 Dat Tr1. L6 Kh6c For Col,lege and Uniuer.sity Entrance Exo,nt Preparers Bio, Nguy5n Huy Doan. Ngul'en Ngd 7'h€- - Dinh li Lagrange Phi€t ViCt Hei, Dinh Quang HAo. 'r, vA c6c rlngdung 12 Nguy6n XuAn Huy, Phan Huy o l,,lguyin Viet Hd,i - D6i tuyen hoc sinh Vi6t f{am du thi Torin Qu6c td tai Canada 1995 Khai, Vt Thanh Khidt, LC Hai 14 e Cao Qud,n - Mdt c6ch giAi bai torin Hinh hoc Kh6i, Nguy6n Van MAu, Hoang kh6ng gian. 1b Ld Minh, Nguy6n Kh6c Minh, t Nguydn Van Mdu - Dd thi tuydn sinh ldp 10 Trdn Van Nhung, Nguy6n Dang chuy6n DHTH Ha N6i 1994 lb Phdt, Phan Thanh Quang, Ta s Hod.ng Dtc Tdn - Tiep nOi phr;ong trinh jriOlu Fermat. H6ng QuAng, Dang Hr)ng Th6ng. 16 o Gidi lri to(in hlc VO Duong ThUy, Trdn Thdnh rt-un uith Matlrcntatics Trai, LO 86 Khdnh Trinh. Ng6 Binh phuong - C* vd gh6p hinh Bia 3 Vi6t Trung, Dang Quan Vi6n. Ngd IIA zr - Trb choi vidt sd Bia 3 Hoang Clfing - MQt sd ki hi€u to6n hoc khd"c Bia 3 Tru sd toa soan : 45B Hhng Chu6i, He NQi DT: 213786 BiAn ffip ua tri sq; VU KIM THUY 231 Nguy6n Van Cil. TP Hd Chi Minh DT:356111 Trinh 6dy ; THANH LONG
-.- f Dirnh cho c6c ban Trurig hgc Co s6 S U,gflNIHXAO T&ONIE &[A[ T,0frN VUHUU,iNH (He Noi) :- {-r -g ,1. MQt sai ldm d6 m6c Chring ta dE quri quen thudc khi vidt 2 thinh {E-r =- (3) {, .r[2, vidt 3 thinh {5. {5. Nhrrng hdy cdn , CA hai kdt quA (2) vd, (3) cct thd vidt chung th6n : kh6ng phAi hlc nio cring vi6t drroc o thinh {; . ,1" . Chi cd di6u dcj v6i a > 0, nhrtng drroc dudi aur* vdia< 0thio=-{--o r[=o H. Tlong chrrong trinh Dai s6l6p 9, d6 cd hing 3. MQt ldi gi6i kh6c ding thrlc r[AT = lAlnhung nhi6u hoc sinh ldp 9 vdn m6c sai ldm khi giAi bii to6n sau : NhAn cA ttr ri m6u cria M v3i bidu thrlc li6n hop cria mdu : Rrit gon bie'u thrlc * : 12r*'^*,'E- *tlxz ? tri. ' ^, _ (* + s +b[7 - s) lz(x -a; - f;7 - o1 -6 -9 l2(x - 3) + {r"r - elt2(x - r) - {-iz--q Da sd c6c b4n bidn ddi nhrr drr6i diy : _2 (x2 -e) - (x $)t[?4 +a@ -qtfx 4 -z @z -g) rr \E +a . fr-TB + ztllx + B)(r - B) _ b[x - s. {t -E +{1r + a1r 4(x-B)2 - 1xz -9) - a; _(4x-12-x -3X,7r- _ s@-ril.[?=E ViTTtVx +T + zrlx - s) _ .fE,n @ -3)$x -12 -x -3) 3(r -5)(r -3) ffi=l*-stzl {?4 C6ch giAi tr6n hidn nhi6n kh6ng drlng. ThAt x-B vdy v6i x = -5 ching h4n thi bidu thrlc M cd Ldi giei nny khd ggn, tuy nhi6n ph6i n6u -5+3+2.4 nghia (U : 2(-5)-6+4 6 I di6u ki6n nit gon ld x * 5. -72 z 56 5 d ddu ra ? D
1l ;iji ,:,::'it - ffilo\, * rr rtrw -:r.-fi# &,' w $#, lii *''1r i:l :. :1 , i..!,:l\',f qffi&ffiqs &es Lli tba sa3ll. Gido su Ta Queng Btu sinh ngdy 23-7- d huyQn l{wng Nguy€n, tlnh Ngh€ An vd dd 191 0 vtnh biQt chungm cdch ddy
Ldi gi6i. T!u6c h6.t, rdt hoan ngh6nh c6c ban d6u phit hiQn so sudt cria dd ra kh6ng n6u 16 a, b, c ln dQ dii ba canh crla mdt tam gi6c vd d6 tri bd sung di6u d 0 vA x*Y 1z : a*6 *c = 25. Suy ra (x * y)(x * z) : 4ab ; (y + z)tg + x) b, c th6a md.n666 =ffi xfr x 7 = 4bc ; (z*x)\z *y) = 4ac. Do dri : Ldi gtui : (Cta c6c b4n Trd.n Thi Ngoc Hdi 9T Quing Ngdi, Luong Tlt'd.n AtLh 9T chuyOn (1) +=)as 4ab *4bc + 4ac > 8s Nguy6n Dlu, Mai Thinh. Hie.p 9B Thanh Hria, -c) qs -r) * - Mai Vinh Quang 9iJ Thanh H6a, Phg.m Thu +y)(x +z) +z)0 +x) +x)(z +v) Huong 8A H6ng Bing, HAi Phdng). o@ x,yz *U *(z 'Ib" co a6fr: a6 .ac .7 + 4(x+y+z) >- yz zI x,y 10&-6 46d-= 7@)@) xyz *a6(7ac-100;:67 Kh6ng ldm mdt tinh tdng qudt, ta coi nhrJ 6Z *Tae-100=ffi x > y > z (> o).rhdthi + * 1'u =+ xYzl Vi 0 < 6io6. lO
(1 - o)(1 - b)(1 - c) < ,3-(o.+6*c),: I \ e )-zt (2) I Tn (1) vit (2) ta 0 < (1 -a)(1-bX1 I* ab * bc c6 * ca -c) - (a < 1 n +b + c) - abc 4 M Or N (./ 128 < ZTol
Chrtng minh rd.ng, tbntai cdehd.n g sd duong s uit t saa cho < xn < filn udi mqi n > 1. "fi , Ldi gi6i (cria Tlinh Hrtu Tfung, 10T Lam Sg, - Thanh Hda) : vdi m6i z € N dnt tn = lmax {rn, yn, zn} vdun : min{ xn, !r,, zn} . Tt 6c gia thidt crla bdi td, d6 thdy : tn+7 4 ,, * 1 € -, N (1) vn ur+1 Z u, + 1 ,nVz ,Vn Trt dd LAIIF - ANDI'+ AFH = NFD --...-\ -^ AF HF eN (2). suy ra u,,D uovz u eN un+r= 3 3a Suy tidp ra AFN = HFD va pW : "u DF - un Mmr ^ LHFD (1) Vz€N.Dodd: Tuong tV LAEM - LKED (2) tn to , ( uo u* - Vz e N.Vivdy, tt (1)ve (2) tac6 : Me, LAEM .^ A.EAN n6n tr) (1) vn (2) suy ra AEDK - AFHD Nh4n x6t : 1. Ldi giai tr6n cria ban Hd Tt Vfl, 8q Le Khidt, QuAng Ngei. 1 = 1-,., Yn>L *. *= 1-,.r|;.h 2. Cdc b4n khdc c Hqnh, SM Mari Quyri, NgOVd.n Sd.ng,9A, Chu Van An, Pham Nguydn Thu T?ang, VO Quynh nl'"*4)-v"'r Anh, 9C"t, Nghia TEn, Nguydn Si Phong,9An , 1. PTCNN, Ha Ndi ; D6 Tlung Kian, Mai Hd.i An, + /, ( - vn > I (3). Biti Quang Hdi, Nguydn Thanh Nga, Dq.ng lr,*4)ln ViAt Cuatng 9T, D6 Qu6c Bd.o,8l Thdn Dang un Ninh, Nam Dinh, Nam Hi, Ngri Manh Hd.,9T, ^un-l + 1 > uf,t+2., ufi_r+Z. ^ Chu Van An, Hai Phbng, Philttg Thanh Tilng, ufi >- , r_ - o n_l t-"_l L n gCT TX Thrii Binh, ViAn Ngqc Quang, Nguydn Yn>-l Thanh Hodi 9T, Lam Son, Nguydn Mintt un uo Thud.n, 9T, NK Hoing Hda, Thanh Hda, * rfr> uzo +2n . r" . >I Nguydn Thi Thd.o,9l Phan B6i ChAu, Ngh6 %r ,vn An, Mai TD.ng Long, 9T, NK Hi Tinh, Tluong uo ur, Vinh Ld.n, 9Cl XuAn Ninh, QuAng Ninh, Qu6ng Blnh, Nguydn Hta Khd.nh Minh 81, Nguy6n Tbi Phrrong, Hu6, Thr)a Thi6n - Hu{ Le Phil Thdrlh, Sl Chuy6n LO Khidt, Qudng un) .r[i vn > 1@\. Ngai ; Nguydn Long Khd.nh, 8A, Chuy6n Lrrong Van Chrinh, T\ry Hba, Phn YOn ; Mai Do u, < rn 4 trYn 7 I n6n tr) (3) vn (4) ta Dilc Thanh, gCT Budn M6 Thu6t, D6c L6e ; Dito Duy Nam,8T Le Quy Ddn, Long Khdnh, Ddng Nai ; Phan Huy Vtr,,8l NgO Quydn, VZ dtrgc : khi chon s = vA r: 4+! thi Dic Phri, 9l Nguy6n Du, Gb Vdp, TP Hd < xn < tt[n Vn > !. Chi Minh. "fl[ vO rru rHLjY. Nhfln x6t : Ngoii ban T[ung, e5c b4n sau ddy cri ldi giai t6t: Pham Dinh Tbuimg (1lCT Bai T6/213 t Ba ddry s6 Tbdn lrhri - Hei Phdng) ; Wnh Thd Huynh {r,} X:,, {vnl tr:o , {i,} 7:" thw md.n - (11Ar La Hdng Phong - Nam }J.ii ; Phqm. Minh xo lo zo dbu duong uit, : (111 10T Lam Son - , , Tll.dn vi Soo D6 111 xn+l = ln * ; i !n+l : zn *; Zn*l = rn * Thanh Htia). "zn'"xn"Yn i - NGUYEN KHiC MINH udimoin20
Bldi T71273. Tim tdt cd cd.c hittt Bai T8/213. Ch,o tam giac nhqtu ABC. Ching minh rd.ng f : R -R tang th4c stl ud' th.6a man ,3fr f(f(x)+Y)=f(x+Y)+1 > (; ) , uoimgix,y €R (sinA)siM .1sinB)'inB . lsinC)sittc Ldi giei Cdch I (cta ban Nguydn La L4c Ldi giii (cta da sd c6c ban) THCS, Ddm Doi, Minh HAi) Srl dung c6c hG thrlc co b6n trong tam gsdc Cho "y : 0 ta drloc 0 < sinA *sin-B *sinC = f(f(x))=f(x)+l'(l) + f(x) :f(f(xt)-l- sin2A.+ sin2B + sin2C : 2 * 2cosAcosBcosC, f(f(x)) : ftf$@) - 1l = f(f(x) - 1) + 1 (2) Th cci : (theo gii thic"t vdi ! = -l vd x ld f(x)) sinA *sinB *sinC > sin2A +sin2B +sin2C Tr) (1) vn (2) suy ra Tn gia thidt tam gSdc ABC ld tam gi6c nhon, ta nhAn duoc ftxt =ftftx)-l(3) 2 < sinA *sin-B *sinC = Vi/ la t6ng thuc stl do dci nri oon rinh vAy + tr) (3) suyra * : f(x)- t hay f(x) : x * l. Xdt him s6 Thrl lai ta thdy hitm f(x) : x * t th6a mdn f(x) : xlnx Cd.ch 2 (Cira bin Nguydn Quang Ngqc LL 1 TbSn QuAng T[i r-i Tlin h. Hrtu Tiung Lam Sdn, Th thdy f'(x) =; , 0 Vn > 0 n6n : Thanh H (;) dpcrn' f(x) : r +/(0)" ThaY viro ta cci '(s)\ttt'1r'\iltl'tstn\ ' f(f(x) +y): f(x) +y +/(0) = x +y +Zf(a) Nh4n xdt. Cdc ban sau ddy c
Nguydn Ngqc Hung, LA Minh Hidu, Mai Drtc s (O rO rO rO ) : s (AO TBO zCO 3D O 4A) - Khoa, Phan LA Minh, Nguydn Minh Tlrud.n, - s(otB%) - s@pv)-s(o3Dor) - sloo+\1 Le Vidt Hdi, Din h Trudng Son, Mai Van Minh, ndu trl giric l6i ABCD cci hrr6ng drrong. Sao D6, Nguy1n Khuydn Ld.n, Phqm Minlt. Tfu.d.n, Dinh Thi Nhung (Thanh Hda). Pltan - DAt AB = a, BC = b, CD = c, AD = d, th6 thi OrA = OrB = ab OrU OzC : Anh Huy, Nguydn Tidn Dung, Nguydn Hbng = Nhl.n, L€ Quang Tfung, Nguydn NltQt Nant, W, nT , cd OqD : OA : Tir Mi nh Hd.i, N guy 6n Huyn lt H d.i (QuAng Nam OzC : OtQ = : nT, 1,vdtacri - Dn Ning), Nguydn Hoirng Cong (QuAng 1-++ NgSi), Nguydn N hQt Nam. (Ba Rla - Vtng Thu), s(OrBO) : iBOr.. BO, sin (8O2, BO) : Tit Minh HAi (Daklak), Nguydn Phong, 1+++++ Nguydn Duy Hirug, Nguydn Xud'n Hito, Tldn =7 o6sin[(B 02 , BCI + (BC , BA) + (BA , BO ) Hod.i Nanr (Nani HA), Ng6 Van Hbng DiQp; 1++l++ Tfd.n Van Phudc (Vinh Long), Nguydn Dinh =]absinlg0r + (BC, BA)l = nabcos(Bc, BA) Inh, Hod.ng Tlnnh Til.ng (Th5:i Binh), Nguydrt 1 La Luc, Huynh IfO (Minh H.ei), Le Quang Minh = nabcosB. (Thrii Binh), Nguydn Long Quynh (Lang Son), Oa Pham. Dinh Tluirng (HAi Phdng), Vo Hoitng Tlung (T[i Vinh), L€ Ngqc Tti (H?t l.by), Vu Dtc Son (Ninh Binh), Nguydn Dinh Tbitn (Ha B6c). NcuvEN vAN MAII Bei T9/213. Cho trl gidc lbi ABCD cd hai dudng chdo uuOng g6c AC ud. BD. Gqi 01, O,O,O+ ld.n luot td. td.m. cd.c hinh uuOng O2 durug ffan cd.c canh AB, BC, CD, DA d phia rugod.i til gid.c. Ch.ilng m.inh rd.ng diQn tich ti Chfng minh tudng tU, ta duoc : 1-1 giac ABCD khong uuot quaf;ore* ilch tti gidc s(OrCOr) = |bccosC, s(OrDO | :) cdcosD 1 ororoto+. uir.s(OnAOrl = ndacosA Ldi gi6i. (DUa theo Pham Dinh Tfudng, - Tr) dd suy ra : 11CT PTTHNK T[dn Phf, Hai Phdng). Th x6t s(OrOrOtOq) = s(ABCD) +s(AO,B) + s(BOrC) bdi to6n tdng qurit hcn ddi v6i trl giric ldi n
1 li gcic phingnh! diQn canhAD, M7t khac, 6 > $(ABCD) +;AC . BD > 2s(ABCD) li gdc phing nhi diQn canh AB, n6n theo giA 1 thidt thi : hay li < ,s(OrOrOzOq), dPcm. s(ABCD) ^ BKH= ^ DBC=a Ddu d&ng thdc xA1' ra kiii va chi khi vd do dci : BKH ^ DBC AC = BD vb s(ABCD) = *O". BD, nghia li Tt dd ta duoc : khi vi ctri ktri tit g76c l6i ABCD cd hai dridng .BHCDBDCD srns = ch6o bing nhau vi vu6ng gdc vdi nhau. BK= BD= BK: BH NhSn x6t : 10) Sd ban tham gia giai bei BD2 CDz toSn niy khri d6ng dio, cri tdi 740 b4n ; rdt tidc BIP BW c6 2b4n giii sai. 11^11 (--:= * ^:\ BDz (-j=+--:=\ ' \ ntz BD7- .t =CD2 \ Aaz BCz I 20) T\ry nhi6n, gdn fnOt nrla c'6c ban {651140) BDz CDz*--; CDz chrra chi ra drtgc khi nAo thi xAy ra ding thrlc, Hav' li : -----^ * 1 : ^ BCt + ho4c chi ra kh6ng d(ng (nhu nrii ring khi BAz ABt ABCD li hinh vu6ng). '!=-.L-CDz -BD2 CD2 Cd 4 b4n giai tdt hon ci : Ngoii ban - AB2 BCz 3C,) Pham Dinh Tbrrdng, cbn cci ba b4n nta cring Cudi cirng ta duoc : cho ldi giAi cria bii to6n tdng qu6t hon, kh6ng CD2 BCd ddi h6iAC t BD : Nltft Qu! Th.o, ILT Lam Son 1 'd'P'c'm BC2 AB2 - Thanh H6a, Tfd.n Nguyan Ngec, 10B, DI{TH Ha NQi vdNguydnVir Hung,11D, chuy6n ngit Nhfn x6t : 1) Khri d6ng cdc ban tham gia DHSPNN HA NOi giai bai to5.n ndy, ccj t6i 113 ban, tdt ca d6u giAi dfng (tru 1 ban giAi chua d6n noi ddn ch6n) ; NGUYEN DANC PHAT tuy nhi6n lbi giai cria nhi6u ban chrta gon. Bei T10/213. Cho trl diQn ABCD c6 AB t 2i CO mQt sd it ban sir dung ci dinh li sin (BCD), tam. gi(rc BCD uuOng d C ud. nhi dien d6i vdi gric tam di6n mi ldi giai nAy cflng kh6ng cqnh AB bd.ng nhi diQn cgnh AD. Chtng minh. don gi6n hon lA bao. CDz BCz NGUYEN DANG PTIAT rdng: BC"-npz=t B.di Lll?l3 Mdt crii hQp ftoac m6t toa xe) Ldi giii (cta nhi6u ban) truqt xudng ddc. Tbong h6p cci m6t con l5c. Vi ,4-B L (BCD) vd CD L CB (g/th) ncn Ngudi ta thdy con lSc tao vdi phttong thing CD t AC (dinh li ba drrdng vudng gdc) 'ri do drlng gdc kh6ng ddi p = 30o. Ddc tao vdi mat I I phingnimngang g6ca = 60". Anhhudngcria i I kh6ng khi khOng d6ng kd, tinh h€ sd ma sdt I giua h6p vi mit ddc. Hudng d6n gi6i Con l6c c6 gsatdc f1Uing gia tdc cta hQp) vi chlu tric dUgF cria hai lrtc : lrtc cang crilgfay t?o3 ry tgg / vd. trong lgc tao ra gia t6c g. Th cr|'a = / +9, Gia t6c a cl&'a vat trudt xudng ddc ln a= Mgsilna - kMgcosu M (M le khdi lugng cria hOp (kd ce con lilc), h dd CD t (ABC). Bdy gid tt B ha ln h6 sd ma srit). BHLAC=H vd BKTAD =K, thd thi : BH L mp (ACD) = fr vd do dd B.F/ "l;Q_. Suy YCya-g(sina-hcosa,) ra : mp (BHI{) L AD = K vi vi vQY BKH = a 8
- sina -alE tJnukh6ng ddi. 86 qua giri tri dien trd - ra k=- cosa dAy ndi, kh
CAC r-OP THCS Bldi Tll2l7 : Chop li m6t sd nguy6n t6 16. Tim r, y nguy6n th6a mdn : # +Y = P[@ - 1)!]P ra HbNc ouANG (He NQi) B,di T2l2l7 : Tim nghiQm nguY6n ctra Bdi T9l2l7 : Cho tam giac ABC co ABtef, ld nghiem ciax2 * arx *bl : 0, phrrong trinh : 3(x2 * xy + y2) : x * 8Y. te 4, NCUYEN OE 1Hl Phr)ng) BCte la nghiem cia x2 * ag * b2 : o, tS BdiTgl2l7 : Chrlng minh ring t, f : CA 111- tS la nghiQm c'iua x2 * arx 16: : 0. 1+=*.-_+...+._>147 - 3{Z ,\fa jr/ggs i,tS 7 Chrlng minh ring tam giric ABC d6rr ndu NGLIYEN or-lc rAN (TP H6 chi Minh) (l -\ +blX1 -oz*b)(l -ar*b3) = Bdi T4l2l7 : Cho tam gi:ic d6u ABC vi m6t = 5616 - 3240{5 drrdng th&ng d. Goi A', B', C' li chAn cric dudng DAM ,.-AN Nui lrnai si"r,; vu6ng gcic ldn luqt ha Lit A, B, C xu6ng d . Chf ng minh ring gle tri ctta bidu thfc Bei T10/217 : TttjlQn ABCD c
ffi.'IIIIII11ITII'ITI]]-J TBl217 . Find all polynomials P(ri satisfying I pFloBLEMS.,[N]T,|{,l,g,i,.1$:g:[J,f i P(xi.P$) = p'^,r*v, (.t) -P-^,x-Y, (; | , Vx,yGR. Tgl2l7. Let be given a triangie A-BC such that FOR LOWER SECONDARY SCHOOLS AB Tll2l7.Letp be an odd prime number. Find ,57, tyi ur"the roots of x2 + arx *b, : o, integers r, y satisfYing BC ur"the roots of x2 + a4 * b, : tr+f=Pl(P-1)lp tg "2, tS O, i T2l2l7. Find integral solutions of equation CA ur" the roots of x2 + aix + b3 : O te i , tS i . 3(g2+xY+Y?)=x*8Y TBl2l7. Prove that Pr,ove that the triangle ABC is equilateral 1 1 1 if 1+Jr+-T5+ .*-m> -.- r+t (l -ar+br) (1 -oz'1-b2) (1 -a, *b.,): T4l2l7 . Let be given an equilateral triangie : 5616 -3240{5 ABC andan line d.LetA', B', C' be respectively TlOl2lT-. The altitudes of a tetrahedron the orthogonal projections on d of A, B, C. ABCD are h,(i : L,4). The bisector-planes of Prove that the quantitY the dihedrals of the tetrahedron cut the A'B'2 =. B'C'2 + C'A'2 opposite sides at E1(i : t,Ol The distance does not depend on the position ofd. from E, to the face which does not pass through E, is 1. Plevg that T51277 . Let AA1, BB, CC l be the medians 4 orlg-triangle ABC, The irioisector of angle 384 AC p cuts A4, and AC respectively at P and , =!l*ll j ?,*i i?,ni Q. The inbisector of angle BCp cuts BB, and t}n?>' BC respectively at M and N. Prove that if i:l AP = AQ then BM : B-A/- When do equalities occur ? ' FOR UPPER SECONDARY SCHOOLS T61217. Prove that if r, y, z ate distinct real cAc nax cul na.I Du THI numbers then csfv: l, -vl ly -"1 {1 +7 {t +V ,E+7 {T1-" M6i bei giai vidt ri6ng tr€n m6t mAnh l,-rl gidy. Phia trdn b6n trai d6 sd cta bii, ' ,114 'tr7;-7 b6n phAi ghi ho t6u vA dia chi. T7ti-ll. The sequence {rn} is defined by M6i phong bi chi gui bai cira mdt sd 4, b5o, ngoii phong bi ghi ro bii cria sd 11 = 1, Kn*r=Tfu,**n forn > 1. b6o nio. Find Chi gui vd m6t dia chi : .rr x2 r,, \ TOAN HQC VA TUdI TRE 458 Hing Chu6i - He NQi ,,'l(,r*\+"'+\*,) Mv7'l
, Dirnh cho c6c ban chuAn bi thi vio Dai hgc D{nh ti Lagrange v&r, e6,e frng dUng NcO rsf; PHrSr (Qudng Nam - Dd. Nd.ng) T[ongchuongtrinh torin 6l6p L2 phd th6ng Voi rnqi xt * x2thuQc [a, b] tbn tqi xo e (xn x) cci dlnh li Lagrange vdi rdt nhi6u fng dgng. Ttry nhi6n dinh li da bi cac srich girio khoa b6 sao cho : f ,(x): o = r
2) Khing dlnh mQt phrrong trinh cri nghiQm X6t him s6 F(x) = (x - aXx - bXx - c)(x- d). Thi drr 3 z Cho m > 0 cbn a, b, c lit. 3 s(i F(x) ld him sd li6n tuc vi cci dao hdm tr6n thgc bd.t ki the md.n dibu hiAn : ca dudng thing thrrc .R. Yi F(a) = F(b) - F(c) abc- L-r-=rl - F(d) = 0 vd F'(x) lb. m6t hdm bAc ba. VAy theo dinh li Rolle tr6n tr)ng doan [o ; b\, [b ; c], m*2 m*l 7L lc ; dltdn tai o < lr < b < !2 4 c < J: < d Chrtng minh rd.ng khi d6 phuong uinh qG *bx * sao cho F'(J) : F0) : F(t) : O . VAy c = 0 c6 it nhdt m.Qt nghianr thuQc hhod.ng (0, 1). (Db thi 105 cdu III - BA dA thi tuydn sinh) FU) = 4(x -yr)@ y)(x -y) (Giii bing phtlc,ng phSp kh6c d6p 5n) . Sd hanghingciaF'(y)ld -4yryyr. Dri ctrng ld hd s6 cira sd hang chrla x c.&'a F(x) lit. : Bni giii: Dat r,(r) :#.#.# -(abc*abd*acd+bcd). ld hdm sd li6n tgc tr6n doan [0, 1] vi cci dao Y4yy{ztz- (abc*abd *acd *bcd) l4 (1) hdm tr6n (0, 1) Sd hang ch(ta x c'ia F'k) c 1 -.1r, (DE thi sd 72 cdu IVa - B0 dE thi tuydn sinh) l*x2 Bai giei z Dat F(x) : lnxliln tuc vi cd dao l-nndur
ffi$f, tuydm hqe simh Viet Nam dg thi To6,re qude td t+i Canada xha m3 r.9s5 Tbong hai ngdy 5 vd 6 thdng 5.nim 1995 vd ggi d.a , ds , d" theo thf tu li cdc truc ding tai Hi Ndi de di6n ra cu6c thi chon hoc sinh phuong cria c{p drrdng trdn di qua A, c4p dudng vio dQi tuydn Tbrin qu6c gia dd chudn bi drr tron di qua B, cip drrdng trbn di qua C. cuOc thi Tbdn qudc td ldn thd 36 td chrlc tai thdnh phd Tbronto Canada vio gita thing 7 Chrlng minh ring de ,dn , d, d6ng quy khi nam 1995. Tham drt cu6c thi tuydn cci 40 thi vn chi khi o.a * bp -r cy * O. sinh dat tt 27 didm trd l6n trong ki thi chon Bei 2 : Tim tdt cit cdc sd nguy6n ft sao cho hoc sinh gi6i to6n qu6c gia ndm 1995 thu6c c6c cci v6 sd giri tri nguy6n n > 3 dd da thrlc l6p chuy6n to6n cria Thudng Dai hoc tdng hop Pn(x) = *+r +kf -87ox2 +194tu + 1995 Ha N6i, Tludng Dai hoc srl pham Hd Ndi 1 vn cia t4 t{nh, thdnh ph6 sau : Bic Th6i, Vinh cri thd phdn tich drroc thdnh tich ctra hai da Phf, Narn Ha, Hai Phdng, Hi Thy, Ninh Binh, thrlc v6i h6 sd nguydn cci bAc l6n hon hay ThSi Binh, Thanh Hda, Ngh6 An, Hd Tinh, bing 1. QuAng T!!, Thrla Thi6n - Hud, S6ng 86, Bei 3 : Tim tdt cA cdc s6 nguy6n a, b, nl6n Vinh Long. hon 1 th6a min di6u ki6n : CS,cthi sinh thi trong trai budi, m6i budi ldm . (o3 +b3)":4(ab)laes 3 bii to6n trong 4 gid ; didm tdi da cria 6 bii Bei 4 : Tlong khOnggian cho z didm (.n >2) thi la 40 didm. Sdu hgc sinh cci t6n dtrdi dAy mi kh6ng cd 4 didm nio d6ng ph6ng vd cho dqt didm cao nhdt trong cu6c thi tuydn di drroc 1^ chon vdo d6i tuydn todn qu6c gia : ;:, (n' - 3n + 41 doan thing mh tdt c6 c6c ddu , 1. Ngo Dd.c Tltd.n, nam, ldp 11 Thridng mrit cria.chring nim trong sd n didm d6 cho. DHTH Ha Noi : 32 riidm tsidt ring c p. qudc gia : H6y tim tdt ch, c6c bd ba sd nguy6n kh6ng Bni 1 : Cho tam gsdc ABC vdi BC = a, dm (n, p, q) sao cho cdc c5p sd (n, p), (p, q) vit (n + p 1 q, n) ddu ld c1"c cdp sd dep. CA:b,AB=c, Bai 6 : Cho hdm *:-' . Ldy sriu didm phAn biat At , Az, Bt ,P.2, s6 thUc f(x\ = 3(xz l) C1 , Cz kh6ng tring vdi A, B, C sar:t cho c5c - didm A, , A, nim tren dudng thing BC ; cdc ring t6n tai him s6g(r7 li6n 1) Chrlng minh didm B, , B, nim tr6n drrdng thing CA ; c.6c tuc tr6n R vi thdi c6.c tinh chdt sau : cd ddng f@(x)) = rVx e R; g(x) > xVx e R. didm C, C, ni.m tr6n dudng th&ng A-8. Goi 2) Chrlng minh ring tdn tai sd thrrc a > 1 d , f , T ld c5c sd thuc duqc x6c dlnh b6i : dd day {on},n : 0,1,2,..., dudc xdc dinh + d + --+ R ---> ---, 1, + A, A^ = 2 BC . B, 8., = L/.A'tz CtCz- LAB tCA, . bdi : ao = a, an + I = f(an) Vn € N ld dey tudn hoin v6i chu ki duong nh6 nhdt X6t c5c dudng trbn ngoai ti6p c6c tam giSc bing 1995. AB1Cl,AB2C2' BCIAp BC{z'CApl CA282 NCUYENVIETHAI ' @aN1i) L4
a) Tin:r difu kiQn (cdn vi ,Jn) dE M,VPO la hinh lhang ; b) Tim di6u kieii (cAnve dt) tl|MNPQlihinh binhhanh; .-ffi;ffi,e- D, ,I c) Gqi / thing .Gei ,r thing hang. I li gizio didrn cia hnng. MQ vA NP. Chilng minh rlng C h giao tli6m ctra MNvi QP. Cinfrngminh ringl, 4 Giii: a) Ta cti ; Mr! = nPIMNPQ) n \P@DC1 PN:mp(MNPQ)nn:ip(.CBD} CAO QUAN Sit clqng tinh chdt giao hoanvi k6t hqp cira phdp giao hai t4p hqp r{5 n6i 6 trC;-r, chri i ring (TP Hb Ch; Minh) mp(MNPQ) n mp(MNPQ) : n'ry(MNPQ) v) np(.4DC) O np(CBD) : CD, ta c(:' : MQ n PN = tttp(MNPQ) n CD.(l ) Tiong hinh hoc kh6ng gian. tlrtltlng phii xdt giao cirzr-hai Suy ra : drlcrng thing (nrit phing) : lrong tnlctng hrlp grao tlti la rdng. ta cri hai rltlrlng thing lrrat pheng) song song. Do r1
- HOANG DIIC TAN @d NOi) Phuong trinh kiar Fermat : n = 72.73 : V6 nghiQm. .r,| +xl +. . . +rfl : rr.z. . *" 1r; n=14: 28.776.440.335.96',7 trong d6x1 > 0 vi xr ,xz t. . ,xn € {0. 1, 2, ...,9} n=15: V6 nghiQm. Tiong bAi nAy t6i xin trintr biy v6i c-ac bqn rrQt s6 k6t qui n=16: 4.338.281.7 69.391.37 0, 4.338.281.7 69.391.37 1 m6i tim ctldc sau ddy : n=77: 35.8'7 5.699.062.250.035 a) Chitng minh phrlctng trinh (1) vd nghiQm v6i nxr/l n > 61. 35.641.5 9 4.208.9 64.132 b) Nghiem ciia phudng trinh (1) v6i 12 < n < 25 tim dlldc 21.89'.7 .142.587 .612.07 5 bing MTDT. n=18: V6 nghiQm 'A - Tru6'c ti6n ta chring minh phuong trinh (1) v6 nghiQmv 61 tt=19: 4.49 8.128.7 9 1.164.624.869 3.289.5 82.98 4.443.187,032 Tii didr kiQn cira phudng trilrh (1) drJcrng nhi€n ta c6 : 1.517 .841.5 43.307.505.039 0 ( xi < 9 Vl : 1,2, ...,n (2) D6 chfing minh a) ta khio s:4t him sd sau : n:2O: 63.1 05.425.988.5 99.693,9 t6 y:lg+Llp-x+l(x>0) (3) n:21 : 499.17 7 .399.t46.038. 697 .307 128. 468.643.O 43.7 31.39 1.252 Tinh clao hAm b6c nhdt cfra hirm s6 (3) : Vo nghiQm. y'= 1l(xln10) + lB9- 1 -- (lge)lx+lg9 - 1 trong dii e : Iim(l + I lm) khi m -> @. n=23: 35.452.590.104.031.691.935.9 43 R5 rang ley s6 dc,n didr gi6m khi vA chi khi tfuo himy' < 0. 28.3 61.281.321.319.229.4 63.39 I Titc li v6i c:ic x thda min bdt phudng trinh sau : 27 .907 .865.009.97'7 .052.567 .814 (lse)lx+lg9-l 6l Lh\ co m chft s6 1rn > 1.1. Ching h3n v6i n : 2 ni c6 th6 chirng @l,x2,.. .rn) li mQt nghi6.m nAo dtl cira phrrdng trinh (1). Do nrinh dugc ring lviQc chilrg minh lA hoAn toan sO cdp, nhr:ng didu ki6.n (2) ta co : dAi ddng) tdn.t4i v6 s6 sd tu nhi€nm dC phuong lrinh (1) lA xtxz...xn : xI +x2 +... *xX < z.9n c6 nghi6m. Y6in : 3virm = 2 ching han. MTDT de xac dinh dtldc nghiQm sau d6y : Chd )' tdi (5) ta suY rz : 443+463+643=444664 rn - *" < lon -l 1o; vdn d6 li rhii nay r6i s6 rrdr lai v6i o4c bqn trong mor bai B6i vi xr > 0 cho n€n s6 xrrz . . . xn l5 mOt s6 tU nhi6n c6 b5o sau. Rdt molg nh4n ddOc k6t qui cira ban dgc v6 bii todn n chr'ts6. vi vAy ta ph6i c6 : rt r...r" > 10n -1 (1) va bei toSn t6ng qu5t htia cira n6. Didu d6 li mAu thuin v6i (6). MAu thuSn nAy chilng t6 ringv6i rngin > 67 thi phudng trinh (1) lA v6 nghi61n. I7S. Sau bAi "Tlao d6i ve bai roan phrrdng trinh kidu FERMAT' cria c6c bpn lidn O6 Hnng -'LC Sy Quang dbng VQy a) dA dtl{c chitng minh xong. trCn s6 6(204)/1994 Lda soan dA nhSn dudc cac bAi cria c.4c ban BCdn l4i li vdn tlb: 'V6i n < 60 thi phumrg - trinh Vfi Thi Thu Hsnh (Y94E, Dsi hgp Y DuQc, TP. Hd Chi Minh), (1) c6 nghiQm hay kh6ng ? Bni Thd Duy (11C! Dai hqc Tting hgp, He NOD drla ra c5c Tlong thdi gian qua lap chi To5n hqc vi TLr6i tr6 d5 th6ng nghiQnr cira phuong trinh v6i n 12 ddnn : 16. Ban Duy con : b6o kdt qui cria phudng trinh (1) v6i 2 < n < 11. Sau dAy li dlla ra mot chudng trinh PASCAL itng v6i thuQt toSn dA sir dung. B4n Tldn DSng Vi6n (chuy6n U, PTTH l,C Quli DOn, cac k6t qui ti6p theo mA tdi m6i tim dudc v6i sU trq gitip cira : : KhSnh Hda) drJa ra c6c nghiQm v6i n 12 ddnn 77. MTDT: 16
Ban c6 nidt ^.4 MOr sO ri HrEu roAN Hec xnA'c 'frong thrli clai ngAy ruv, khi chtng la n]d rQng giao IrIu vdi nhrdr-r nu6c tren thd gidi. chrJrng ta n6n bidt nhr-tng ki hi€r-r toin hoc nr.r nr6t s6 nuric dirng Jrh6c v6i chilng ta. !. Crich vi6t s6 tq nhi€n, s6 thip nhin Giiti ilip bdi a) D6i vcii sd rrz nhi€n, sAch gi6o khoa cta ta clirng c6ch vi6t : ch*u rrrit khodng nhd giua hai kip k€ nhau (l6p don vr v;r lcJp nghin. l6p nghin v) lilp trie'r-r. v.v...). 'fhi du : cXt vn cHEP uiNu 3-5t)0 (ba nghin nirnr trirm) ; 24000(X) (hai tri€Lr b6n trlnr nghin). Nhllng tren hAo chi (bio Ntr.1n dAn. I Ii n6i nf,i. Sii Gin grii Theo chi6u ddi, c6t midng v6n thdnh m6t phring. ...). nhidr khi cac chu sd dtrr.tc v'idt li1n nhau hoic d.dr ttit hinh vudng cci canh 0,3m (A) vd hai hinh chit ddu chdrn gira cdc t/tp.Ilai s6 trong thi du t€n cri thdradr : va 24(X)000 hoic 3.-500 v:r 2.400.000. 3-500 nhit cci mOt canh bing 0,6rn vA m6t canh bang I)6i vdi .ri thAp phdn. chfrng ta tl:'rt ddrr phi,y (,) giila phdn 0,3m. Tidp dd ta c|t m6i hinh chit nhAt thdnh ngrryen vi lrhiin thAp phAn (ddu phdy thQp phAn). Thi c1u : hai tam gi6c vu6ng bing nhau (8, C, D, E theo 0.25 vir 561i. ;158. drrdng ch6o) vd ta gh6p lai nhu hinh vc dttdi. tr) O Mi. Anh va nt6t s6 nuiic, cr:lng nhu trong miry tinh. ngrkrli ta lai khirng ciirng d:hr phiv th:ip phAn r-na tlingddu r:hdtn th1p phAn. Ilai s6 thap phAn trCn dav cllkJc vidt lir : 0,3 A IB o1 0.25 r,i 5(ri{..158. Sd ti6n -50 116- la tra)n ( kh6ng cri xrr ld) d Lldc vidt $50.(X). Nair phlin ng-ryOn Ii 0 thi cti thi bo sai tl dv cli. ching han s6 0.25 cti thd uCt la .25. 0,3 Citn diiu phdv rhi lai dtrrtc diotg d.€ d.dt r) gitta hai lip kt nhou cirlt nra)r sai trl r:hien. Ban co thd thiy tren bA() ttr6ng tin cira nr(rt c(ing tv li0n ltranh v6r nrlrJc ngoAi : V6n ciic.u le : 3.(X)0.(X)() USI) (ba trierr L.rSD). 2. l){rr nhin. dfu chi:r vr\ crich tliit ph6p tinh chia Ve rllirr nhin. cltrinc t,r viOt : 3 x b hoic a.h holic ah. () r,hi.-tr nrt6c. rl:iu chiinr. v, tj 1 nghr a l; r l)hcp nhan. cittctc vidt c:o le n n]ot chrit lgiita thiin chLt). 'l'hi tlr-r : 2.1.1 la sii thAP Phen (hrrr rlrln vi rntlli hrin ph;in triur) (,Vrrilin |'hi Kin Phutrn,q. (). 'l'hi trin I)itc I'hr]. Qu:lrnq Nsli l. 1l li tich ciur I rti ll ','t Dilt* L"irt ('rtitrt.q. ()..\. ChLlven \'--1. I)n I-tt,1nq. Nqh.: .\n) l..11 l:l tich c[ia 2 vtti s,i th;rp ph,in.l-l ((l.l] : .ll). Y i
oufrl{G c60 uEN cONe NGHF vl orEr.r r0 (IMET) rdnc DFI olEtt rcY rnuer ve pntrn PHOI sf,N Ptl6M Weornes Tfil UE.f NfiM O VAN PHONG GIAO DICH TAI HA NOT : Tel. : 84 4 250761 ;84 4 267645 Fax: 844267645 a vAN puOxt; GIAO DICH'l'AI TP nb cni MINH : 4 Dang 'l'dt QuAn 1, TP Hb Chi Minh Tel.: 84 8 419134 : 84 tt 431064 Fax : 848 431064 lg: Miiy tinh Wearnes li siLn phdm cria Wearnes Thakral (USA) - li mQt trong nim nhi cung cip miiy tinh l6n nhdt. lg: Mdy tinh Wearnes dat ti€u chudn ISO 9002 do Co quan Ti6u chudn chit lugng qudc tdcip. ]g- Miiy tinh wearnes dugc bio hinh 03 nim ls Mriy tinh Wearnes dugc B0 GiSLo dgc vi Dio t4o chgn li lo4i miiy sE trang bi cho cdc Truo'ng Phd th6ng trong nim 1995 freWearnes ljr ##w#z*T==*