Toán học và tuổi trẻ Số 200 (2/1994)

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Toán học và tuổi trẻ Số 200 (2/1994) được biên soạn nhằm phục vụ cho các độc giả yêu thích toán học. Tài liệu giúp các bạn biết cách giải một số bài tập toán học. Mời các bạn tham khảo tạp chí để bổ sung thêm kiến thức về lĩnh vực này.

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Nội dung Text: Toán học và tuổi trẻ Số 200 (2/1994)

I I - ' i:"'li . GIAO DUC VA DAO TAO * HOI #sD tg: :, l:llij TAP CHi RA I{ANG THANG Db ra cia cLl(ic thi cl:ic bi6t clrio ntilng 30 NAM TAP CHi ToAN HQC VA TUbI TRE .DUA QI.,[Y NAP TOAN HQC VAO IdP NAO +, '>G E tr o CDJ ,>(tt .ctPe,Y tr 6' '=, !, \o !''- L - E, Lf) --Ez E€8 oa D6i tuydn Khdi Chuy\n Todn DHTH Hi) NOi du thi qudc gia C6ch giii phurrng trinh vh bfft phuong trinh bac 2 chria tham bi6n vh th6a min dibu ki€n phU
l- I 91 t TAP CHI TOAN HQC VA TUOI TRE TIN TI-IEM VE I]AI TOAN F'ERMAT [,]in clAyti6n si Anclrew Wiles cti grti den t',tic tlung tam Toan hoc tren the gidi nrot thrl ng6n, vdi iinh thiin sau : Vi ccj tin ddn v6 vi€,,c 6ng chting nrinh ch/r.rc gii thtryet Tanivrrnr:r - Shimurzr (Va ttlr dci suy ril chi'ing nrinh cria dinh li Ferruat). nOn dri trrlnh sti hidu l:inr clzing tiic,6ng nrrron tLi nrinh ndi l6n tinh 1r'ar.rg hi6n n:r-1, cira r,5r"r d6 nr\y. Trong rltra trinh hoin tl'ri6n chfng ntinh. r:rj nhi6,.r bai toritr tri.r, sinh v:i 6ng de girii qul,it clLroc phiin ldn ciic irii to:ir.r cld. Ttiv nhien rld cli clen gitii qu1'e-t gii thtr-1,et t'rrii tren tnot r::ich tlr.r11 1'6'11 r',in , ort ruot vin cld chtl;r gi;ii qu.r'i,1 xlirg' Tii,n si Anclrew \\Iilt,s tin ri\ng 6ng r:cj thd girii qrryet cltioc van cld clcj trong nrot trrdng lai grin clilr,'. lx1nS c::ic y tttilng nra erng clii trinh lriry trong tr;io crio t:ri hoi nghi d Canrbrige. Tdng bi1n fip : N(iI IYEN C'ANI t TON N Phd tdng biAn fip : NU(i I)Al 't'l'r IION NG CI II.JNG HOr DONc erEru rfn : Nguy\n Cdnh Todn, Hodng Chlng, Ngd Dat 70, Lb Khac Bdo, Nguyiin Huy Doan, NguyOn Viet Hei, Dinh Quang Hao, Nguydn Xudn Huy, Phan Huy Khei, V0 Thanh Khi6t, Le Hai Kh6i, Nguy€in Van Mdu, Hodng Le Minh, Nguy6n Khdc Minh, Trhn Vdn Nhung, Nguy\n Ddng Phet, Phan Thanh Quang, Ta Tru sr] tba soan Hbng QuanS, DAng Hitng Thdng, Vu Duong Thuy, Trhn Thdnh Trai, 8l Trdn Hung Dao, Hi NOi, DT : 260766 Le BA Khdnh Tinh, NsO Viet 231 Nguy6n \5n CiI, TPHCM, DT : 356111 Trung, Ddng Quan Vi6n. Biin trip t'it lri srr: VU KIM THtrY Trinh boy : DOAN IIC)NG
Sring tao todn hoc - dd ld mOt trong nhttng nhi6m vq hAng ddu cira ngudi lini torin. Nhung sdng tao todn hgc lh m6t vi6c laur kh6ng d6 va cci nhidu con dr-tdng khric nhau dd thuc hi6n didu drj. O Aay tOi mudn trao ddi v6i c6c ban trd m6t c6ch nhin dd brr6c cldu t{p duot sdng ta.o. TOi xin bit ddu btrng m6t b:ii to6,n qr,ren thufic. sfiE tsiri to6n I : Tr6n 2 canh gcic vuOng AB r.d AC cria tam gid.c vudng ci.n ABC ldy uic didur D vi .E trrong t?ng sao cho A-D = AE. 0x t\ t" Ndi BE, trl A vi D ve cdc dudng G. vu6ng gric vai BE cit c?nh BC theo thf td tai 11 vi K. Chtlng minh t) tt. i} ! Gh ratrg C'iI = I{K. f i - MQt trong nhitng cach giAi l'{' rs don giin cia hdi to6n ld : K6o r;' dei Dk cnt ducrng thing AC tai A & F, ta chting minh durrc (h.1) : 1t.l t0 5 ABE = 5 AFD {s.c.!"} =r AF = AB : AC, rnd AI{ ll p'K n6n C-Fl = HK (dpcru). - Bay gid ta hey dd y ddn quan h6 ctia cdc doan thiing trong bdi toan : ffi fAg = AC AB AD j* = AE* HC = HK lru;- * = t* = I + V*= t HC tt a,; ta niy ra mdt suy nghi &* AB AD : HL' q)ffi - Neu AC = AE = k thi Hl( bang bao nhi6u ? a) : . ,\B AE k thi-HC E hoAc ^" Neu N_, - Do dd ta cci cr1c bai to6n sau = ,rra bang b,rc.r nhi0u ? (b) a : Bai todtt lo .' TrOn 2 canh gcic vu6ng AB vd AC cia tam giSc vu6ng ABC ldy 0 td fr ni [::. i..,t- l _.:-" .) . cdc didrn D vd E tLidng itng sao cho AD -AE - ,\B = h Ut > 0). Ndi BE, tl. A AC ui ! vi D v6 c6c drtdng vu6ng gdc v6i BE cat g '1 I a" .r! -- canh BC ldn luot tai HK I/ va K. Tiuh tj, s6 a (theo la). ,F.Af( ," (t t.2) - Tuong ru c6ch giei bni to6n 1, ta cfrng k6o dii DK c6t dudng thing AC ^b al dXffi tai f,. D6 th.ry : (h.2) ol ui fi* AB ' HK o- U E L.A,\D^LABE*M=M=h*M AB= AE= it +,4c= '4-F AB'Ae:/t- haY rn = r, E o- z Biti toan 16 : Nhu bAi totin la, chi thay gii thidt br)l Ci 5l E.E /md AE:AB:, AD_ AC_,, bang Fl EZ - cl tE O bai ,Ay (cdc ban ttl v6 hinh) giAi ttiong tu bai la, ra cri,g cri : A AE AD AC rd. .ol U rr! i ffi7 Al'D ^ LABE* :HChayHKlHC=1, en = *\E = AU+A_F - AC., miAJ{ ll FKn€nHK U o olp .X . * Nhtt vny bang cach thay cldi ur6t chut suy nghi : coi hari cloan trring bang nhau ld 2 doan thing cri ti sd bnng 1, chirng ta da tu tiur ctiroc i -c U qrI bai torin thri vi nri bii to6rr 1 chi ld mOt tnidng hop dac bi€t. tC rEt fi o h. >, Bay gid ta tidp tuc c6ch suy nghi dci vdo bdi to6n sau :
Biri todn 2 : Tr6n hai cqnh AB vit BC cira trong tim crla vdn dd : Coi hai hinh vuoug li) hinh vuOng ABCD ldy 2 didnt P va Q tudng hai hinh chtt nhit cldng dang co ti lo 2 l
Bai T1/196 ; Lic riti nhit. di, ban Tit,ng xent gid thdy d'6ng hb chi lton 1 gia, uit hhi ddn truitttg thi tlldy hai hint dbng lfi da d.di ui tri cho nhau (trong thiti gian niry lrui kint lthdng chQp uoi nhau l.i1n ruao). . Tinh thdi gio.n Timg di tit nhit d€n truintg, lic Tirng riti nhd uit lilc Titng d.dn trttitng. Hd,i, 7A., Chuv€n Phong Chiu, V'rrorLg l[htlr Ldi gini : cria c6c ban ld Xudrt Minh,7E,, Tltu, 88, Chuy6n Viot Tri. Vinh Phfr. Ba Dinh, Tx Thanh Hcin vir Trdn Viil Dfing, 'l r't Nt,t l\'l 812, C}r.u Vdn An, Tp. Dn Ning. N GiA srl hic Tung rdi nhri kim gid,6 vi tri A, Ilni T21196 : Co 5 diint nant trong ltinlt kim phft d vi tri B (h.1). Hai kiur kh6ng cci vi uttAng co t:nltlt ld 36,7" Clulttg nittlt ritttg c6 tri nguoc lai vi trong su6t thdi gian Tirng ddn thd tint dttoc trong linh uudttg rud.y 1 tliint tno, trUdng, kim phtt kh6ng chiip voi kiur gid. Chia hhodng caclt cia no ddn 5 cl.iittr, ltia d,bu lan nint ddng hd thnnh lrcnt 10. 12 vach chi gid. Kim Ldi gini : Goi c:icr dinh hinh vuOng Li A, B, grd quay tr) A ddn B Cl D. Ndi tmug didm l( cria AB vai tmg cli€inr duoc y vach vi kim I cua DC. Tr'6n doan thing 1(1 ldy M vi N sao phrit quay ti B ddn cha KM = N/ = 8. Ta c
"Cho 5 didm nim trong hinh vu6ng.canh I thi NhQn xet: luOn tdn tai it nhdt 1 didnt tlong hinh yu6ng 1. Da s6 cdc ban mac nhi6n chi x6t bai torin mi khodng cich tir dcj t6i 5 didm d6u l6n hon ft, tronElinh v0 rnir minh cci duoc, trlc li lam l r2 tronfi 2 tnldng hop I vi 2 ncii tr6n. Phdn ldn v6i ft th6a rndn 3h,2 + hl -.2l . 0". Kdt qua la giAi dni dong. - 2. Ban Doan Dinh Trung, Trtfng Vrrong, Ha drins nhung chi tiec li tuan cria ban d phdn cudi N6i ldi siei t6t nhdt. lai lioi lung tring n6n khdng mach lac' '3.cciC6c bin giei tudng d6i t6t bii torin : t-E 'lltt"iNc; NuAl- T(! Thi Bich Hanh,lryd-ng I-.'!qi, Hd TAv ; Bai T3/196 : Wng trbn n1i tidP (O) cia D6 Truns Chdu, Viet Tri, Wnh Phrl ; Nguydn L ABC tidp xic udi cdc cqnh AB ud. BC d cd.c Thone ,Soru. trudns'Hdns Bdng, H&i Phbng ; didm E ua F. Phan gtd.c cila4ic A cdt EF d NeuvZn Phi BinllTrudnfBdVan Den, Ha N6i' didnt K. Chung ntinh rdng CKA = 90". tE ia Lonp. Lam Son, Thar'h Hda ;'Ngo Thi Moi. Nehia"D b,n. P ho nt T u d rt An h, trrrong Phan Ldi giAi : Boi Ch"au. Nehe An t Ii Phuoc Danlt, tnldng Dd bei 16 ra phAi ld "phdn gi6c cta g6c A Nluven tri pl.rbne, Thila Thicn - Hue ; Nguydn c6t tia EIt...r. X6t c.ic trudng hgp sau : Viih"Ngnienr, trudilg L€ Khict, QuAng Ngai. l)ACr_N, ^ ^ = ^ : BFE vO rtu 'ritilY B9! = BYL* 4E! Ta cci BAri T4i196 : Clrc n tn 2 2) day sd cluong t8v - EBF B_49 ij9! Mat khac phdn biil A,. A....., A... Ki lti1u N,kt lit s(j cac =^z^=-z 's(i hanp khdne-urtot dua x c-ia ddy A, ; N,{xt EKA = pAF - _EAK. Dsy_ra : td sd c\c so llang kttng uuot quti-x ctto plldn ^ BAC + ACB BAC EKA=ff-i=--= ACB ocF ^ " A, A A,. giao Riiiuoi !eN:1< i1. K n,, =j+--lV"i 2) AC < AB : Trrong tu nhu tr6n ta ccj : DO A,, A., ..., A., la cdc d6y sci thdv c6c dril'-Hon drrong phin bi6t. nrla, v6i"m6i i = l, 2, '1.1 rt vd v6i r € 7 tn\N;(x) =, , 1. T.rY ..., nhi6n. do A, o A, : A Vi * i (d6 thdv) non N,!x)'- u ovi *.i" .A Vr e Z+ .vA bdi vQy,-kh6ng n-t(n-ll'-'(!) ton tai Ai, Ajdd N,j,*) , -;: Nhin x6t 'lz)"' : Vi6c dd bei to6n tr6n xudt hi6n trons tiruc "D6 ra ki ndv" Id so xudt dring tidc. Tba'soan thdnh thttc c6o I6i cung ban doc. e C6c ban LA Anh Cudng vit NguYdn LA aL- -' Minh (Lant-Son, Thanh Hdaid6 nghj : Dd bili ^ fie - /\BC ro. co thd gidi dttoc cdn : BFE = a) Sita gid ttidt theo nqt tn>ng lni hudng sou : Met kh4q-ma = @ + 5k = i) Thay di6u ki6n "A,, A,..., A,, ld n daY s6 duong" b6i di6u ki6n '4,, 4., ...,'4, ld n day =%W=Y.r6clh 6FE= sd ngnyan duong". ii) Thav didu ki6n "N,r-r; 1A... i N,tx) Ia . . = KOC--Do d6 tVgiac OKFC n6i ti6p dttoc. Ai n AJSi aidu kiap'Nrtrt l^n so-9oc't,s(.nguv0n Suy ra CKA = CPO = 90o. khdns buot oua x cfio dhv 4 ; N,,(r) lit cac sO rlguy:en kt;nnb wbt qud r iila phan"giaa A, f\ A,. 3) AB = AC. Khi dd F = K. D6 ddng suy ra Dj.QM. VAy trong mgi tntdng hop ta d6u b) SonE sons v6i vi6c sila sia thidt nhrr treri, sita phdn-kdt lu4n thinh "tdn tai hai day A,, c6 CKA = 90o. A. soo cho : I
N:,,{*) > 'gT"l Cac birn cci loi giii tdt lir : Trin Thanlt Hoa.rt pOng_Ha, Quzing Ttil, pltan Artlt T,rtcitt {!JI., 2( il,' (1 1A, Nguyen.Hu0._ Hi 1'eyr, Trott etto ttg l-l tt.v (11, Ciruy6n Tr:6.n Vinh Phu), Tfin-pan? ltito .. B+1 .lrqlg vi ban Minh da cho ldi giAi dung d6i voi bii to:in d6 dtioc srla ddi" Hon !!4.-Dl]j.H_He N6it. TlfrL T'hi Honh r"l tCT, nqa, bgn Minh cbn chi ra dung rang v6i gin PTNK_HJi Hungl Tang iVant Long (l2C,f', "kgt l-uan cira Phan Boi Chriu. Ngh6 Ant, D6 ertine Mintt l,hidt de_ duoc sira nhrr tr6n, t-f-2A' !]ltt.v6n Tuycn Qulngt, A'grivf,z bit' Ki bAi ra vin sai chi it la trong cric trudng hJf (Irrirr_Phu._ilA Tinh t, NguS,611.Atllt '!udn tI0A,, n= 21 3,4. _ .3.- I,bl giAi cira Drip rin lA k)i girii cho bai to6n lpng LQ11, Bar-- Lieu. l,tirti Uait {)t.rin Ngrryen vdi giA thiit cAn duor: sia nhu dIi n6u d tr.eri, vA g,fudl_g'tl0A,, !o Qu- Dirn. D,r Nrirrg;. i,tin,,, Vic;l N*gu r lU, Trrin Pht, Il:ii l,hr,.,risr. lrirrl/ tfT ldj grai...niI chi cci thd cci ctrtoc khing Truttg Httttg, Ngtr.virt 'l'ltii Hd (10[,[, fLu.i. drnh rang bn toi9."ulg ltoi doy A,, A,.sao clto : NQjl ; Gio Cot Vo Hilng (i0A. Lo nr(n -c-) $uyli,_Hi Hcir"rg l)hung, Narn IIlr), ... I{ tl txt > | n,ft'. l I l 'i ll()N(; Nil,\'l " \z)n' BAi TGi196. Clrc ha.i rta thic bac bdn pkt ud Qr.rt co lie sd ttuu t.i,. L'lting ntin'lt riing tr,ttr P(x) uir Qd) co ntOt rtgltiint ciiune u6 ti iit lroi . Bei Tsi te6 : Ctw a,r, Clfing ninh rong : )tl';:)r:;f i:Y: rtgltiint cltrtrtg hin $. tlti prx, ult 9,11 citrt co tltirrt tttdl rtgltitttt clittrtg hltoc rttlo. .sina* sinp+ sirty* sll.(rr* + p) + siru({} * y)* sin(y*
Bai T71196. Tint tdt c'ri cd'c da tlt'ic Pk) Ki hi6u tdt ci ctic didm ctra tirp M li A t, cell tlitiru A,' A' irt''r I udi h.€ sd tltttc tlrca, nrunt dibu ki€rt : ';A, ..., A,, Noi tiit cri c:ir'th:'rng A-, I = i."'iit * *i"-Jo,.r, niii cltrJt: sd ch.r Pk)+ P(1) = | rrr, + 1) + Pb - 1)l Yt ta rir6t s0 httu han cac d$t)ng gllp khirc cldll, uii ur6i iuong g5p khuc'c16u cci tiit cri c'rc clinh ir) Ldi giii (cria da s6 cric b?n) ctic didm lt X6t clrrong gfp khfc dcrn (K) Dat P(1) = cl va x6t da thfc "O",lf. c 3. Khi dci, ti tl) vr)Aal*: I s''ry t':r tiil c;i r:ic ciinh A, cira (/r) voi I le lon hon 3 phrii I : II tQtr+tr + a(.t+1)l + Qr..r-l) + o(r-l)rl vr naur trc,ug t:irng m6t tri{lr luilt ph urg.mr} t:ci br'l Ia cit/ilng inirng arar. cbn trl r:ri titt: clil'r}'r A, r:ti:r *Q&) : ,lQ\x + 1) + Q(r - 1)l Vr I (l{i voi i chin phtii uaur tlr-rng r:ittlg trtot ntl:i nrit. phing (urd) d6i cira nilit nirtt phirtgl ncji tr'0u' +Qk) - Q(;r - 11 : Q(r + 1) - Q(x) Yx I H @+ Q(q+ r)(r+ P) lrri xAy ra : b) *'ytr'> Qr+ q)')(W+ rz)(rz+ Pr) 1) Truitng hop 1: A, B, C, D ln 4 dinh cira rn6t Ltli giiri (Dua theo ?r'dn Quottg LIu.t' - 17 tri girlc ldi.l{hi d6 AC + BD > AB * cD = 2. chur-en Toarr \'irrh l'hLit St lira hoac AC > t ho6c BD > 1, miiu'thuAn a) Ta cij tl-) + (J)(g + ,')(/'* ;,l) = z(.i:cls'n"'[** v6i giA thid.t ctia btii toan. cosM.r)rtcos,\{.+ cos.1/ ,) z(t:osl'/,,* cosi'/.t < il) Tntdng lrcP 2 : A, B, C, D tthong Phdi td, 4 ctinh cio nt\t tti giac lbl. Khi dcj se co mdt didur, ching han D bi chr'tn b8i AABC' -EnQ nhjF-n.-D * [CB), &-IAC] D dd Eidt -Dohotrc + ADC > 180" hoac tsDC >- 90" ADC >- 90", vi bdi viY ho6c BC > CD = L ho{c AC > CD = 1, luAu thu6n vcii gii thiet cria bai todn. Vdy hai cloan thing dO dai I c6 cac ddu mirt thu6c M phAi hoac cd chung rrrot d{u mtit-horJc c6t nhau-tai didrn tlong cria nr6i dorln (1)
, c. ) *"rrr, \= Z) ; B'K = A'H xfng qua tAur hinh 1rd6i / I r-l chu nhAt OA'MB') n6n h.ri taltgi6c vrreng < u,z ( s )" (bdth C6-sir < NB'K, PA'll birngnhau. hny l,lKB' = PHA'i Mt +Mz+ 'M,'), Kdt hop v6i (1), Q) ta cci ciic nhi di0n canh < xyz ( ,1 . t ('ri ha,','t OM ldn luot chtla NB', PA' bang nhau, suy g.6cos ra m6i uhi dien ndy bing cosx ldi trdn J( It1 -) -i;r)t=l-yz. . 180" -a d Z (= 90" - ,). b) dpcm * (i.*) ( i.il (i .?) = , Ta co : A'P) = A'H: tgl g0 - 1\ '\ / 2l = c.+ (cosM. * cosM,)(cosMr - + cosM,,)(cosM, * + cosM.,tf I A'H: (, = _,7 t3). Hon n[ra, tr.or]g tnm gi6c vu6ng tC-, )"o"Utf, c=+vT < (-g)t (bdt C6si) < A'OM ti' = 90"), ta lai cd : 111 Mt+Mz + ..+M ----- - * ----- = A'O: _ (4) Ti (3) vi (4) thay ,1 < ( g.6cos t) '-1 A'H* A'A4 6 cab A'P = Ddu "=" xdy ra khi AABC d6u. 2: A'O = i t A'M = ,, ta cci c6ng rhric Nh{n x6t. Cci 108 ban giii, tdt cA d6u giii n6u tr6n. Vdi rr = 90') ; ta cci dpcm, va bai todn da drroc giAi xong. dung. Cac ban sau ddy cci ldi giei t6t : Tron Nhirn x6t : 1. Ldi girii n6u tr.Gn cdn cri the Quang Huy, 11 chuy€n to6n Vinh Phu) ; dung dd girii bni tod.n diio (ban doc tu ph6t bi6iu Nguydn Hodng Cong (.107, thi trdn Son Tinh, vA giAi). QuAng Ngai) ; La Thanh Titng (llCT, Hung Vuong, Vinh Phir) ; La NguyOn Ctldt (L1T, Lam 2. Cd 96 ban gii.i, tit cri cl6u giii dirng. Cric Son, Thanh Hcia) ban s:ru diy c
ucoscr , t kdV, clti 1lV uit uon hd V, chi 1V. T'intt LJr.,, [,' = ud. d.iAn tud udn lai. I Ir = usinet - Zgt" Ldi gini : Ta nhdn x6t cilong do dbng diQn t', qua AC bing cudng dO d
ffi ' ad + 6tj a ad', (Bai 12 trang 57 Dai s6 10, ?riirz Van [:Inrt (chtr bi6n)). Day lA bai toan khd da,i vrli hoir 6&ffiKffiffiffi sinh di6n dai trA ndu chua drloc trang bi quy n4p torin hoc (Xem ldi gini trang ?6 srich bai cAr cAcn DAY VA HQc roAN tap dai sd 10 Trdn Van Hao...r. Sau diy lA ldi gini Lrirng qu;, irirl:i lo:itt h
Bii 'I'9/200 : Cho tam glnc A-BC cti dtrdng xn+ yn= 7 trbn ngoai tiep k, tAm O-, bdn l
2) For every (f7, tr..., tt) e T it exists PROBLEMS OF THIS ISSUE a finite number of operations after that it can be obtained n-tuPle (iA-J) For Lower secondarY schools n Tl/200. Does it exist a positive integer Find # ?. number with 4 last digits that are 1994 and NGI.-IYBN KT{AC MINTI that number is divisible bY 1993. T8/200. Show that for 3 arbitrary vectors of lengths not greater than 1, it can be chosen DANG HUNC 1'IIANG 2 vectirs such that either vector of theor sum or vector of their difference will have length TZ/2}A. Find all polynomials P(r) with non not greater than 1. - negative integer coefficients (all smaller than NGUYEN I{IJY DOAN 6) such that P(6) = 1994 T9/200. Consider the triangle ABC, its NGT.JYF,N DUC'I'AN circumcircle & of center O and radius B and its incirle of center -I and radius r. Another circle T3/200. Let .I be a incenter of triangle ABC fto is tangent to the sides CA, CB, at D, E, and BC = a ; CA = b ; AB = c' Prove that rJspectively and it is internal tangent to ft a trA}+ b .IB2* c.ICz = abc Show that the incenter I is the midpoint of PFTAM NGO(] QLJANG DE. For Upper secondary schools ESPI (lM() - lee-1) T10/200. Given tetrahedron ABCD with T4l200. Show that for every natural number BC = DA, CA - DB, AB = DC artd given a N > 1, the following inequalitY poind M. Prove that the square of the distance N fron M to one of 4 vertexes of ABCD is not saI \ < l-ln2 greater ihan sum of square of distances froln ,?t n(n + L) ztt M to other 3 vertexes. holds. NGUYL]N DANG PI{A'I - DAM VAN NI{I T5/200. Find all functions f : [0,1] [0, 1] such that coN s6 zs ri lA (TidP theo trang 9) l) f(xr) * f(xr) for x, * x2, x,r, € [0,1) 2) 2x - f@) e [0,1] V.r € [0,1] Hay ldy mQt bQi sd bdt ki ctra 73' 3)f(zx -fk)) = x Vr € [0,1] Ching han 18834 = 73 x 258. T6ch }IA I{UY VUI s6 niy thdnh hai Phdn TB]2OO. For a positive integer ru, find all - sd tao thdnh bdi hai chrl s6 cudi : oolvnomials P(x) of degree 2n * l.such that 34; Luery graf y = p(zk) (rJ (where PltK) (x\ - 2k times Ierivate of P(x) V k e N) has centers of - sd tao thdnh bdi c6c chrr sd cbn symetry lai : 188. Binh phrrons hai sd m6i niY : . NGUYEN VAN MA(J 842 = ttBO, tgS) = g1g44. Tdng cria Let ft and n be posotive integers hai binh phrrong ndy lpi la m6t b6i sr5 T7I2OO. ctia 73. ThAt vflY and ft < z. The following operation is caried : 1156+ 35.344 = 36500 = 73 x 500' For each time, k consecutive coordinates Di nhi6n b4n cci thd giei tri bang (xi* t, xi* 2, ...., xi+ I of ordeved n-tuple uich chrlng t6 ring cric s6 khric khOng (i,, x,,'...x,r) are'changed by oposite sign cci tinh chdt ndy. 'Iuy nhi6n 73 kh6ng coordinatLs e xj+1, - xi+2' ..., - xp). phAi ln con sd duY nhdt cci tinh chdt A,;. Can cri m6t so n[Ia, gdm 3 chit sd' ? is a set of alln - tuYles (tr, t, ... t,,) B4n cd thd tim ra s6 niY kh6ng. with the following ProPerties : 1) If (ri, t2, ..., tn) € T then t, € {-1, 1} ue,, ttiltQlInN(i V i =T,rt 11
DE RA cuA cu0c rur DAC tsrET Cfrin rmri'ng 30 nlrn Tap chf Todn hoc va tudi tr6 A - c,AC t_Op ptcs BAi g/P'I*fI{ : Cho x, !, z, t € [], p]. Tirn Bhi I/PTCS : Cho B sd duong n, b, c thria giri {,r'i l6n nhdt cria bidu thrlc : miin o * 6 * c = 1. Chr?ng rninh ring 5{a:t + b3 +-c.i .t ab *6c * co),3 + 3ubc > 7^ r*t-l 1_Ll_ tv, r*-r + l*r t H,ilv xtiAtt t>Axt, ), + z-, :Laj Rai 2/PTCSi : Xdt t4p hqp tdr cA cd,c sd x+.y .y+t nguy6n duong a, b, c sao cho tam thfc b$c hai ivcriyCn vnN n,tArr ar2 - br * c cci hai nghiGm phAn bi6t nam trong khoAng (0, 1). Tim giri tri nh6 nhdt cria Rei 3/PTTH : Gi,tri phuong trinh hlnr o, cira & vi. cira c 1 ,x, I>AN(, IltrNtr EAi 3/PTCS : Tim rrghi6m nguy6n '1-1 f citrong .i,-'-, /(t) - ,t \r) = cfra phuong trinh !1 -:trdrl tAp cdc hAm li6n tuc tr6n -t i * y2 + S9S0x * i-; , 3,j S9B9.y = 120464 NGIJYEN XLJAN ruuYNi r N
Ph6p giAi nhi6u phudng trinh thuing dr.ta diin viOc Ditu ki€n dt cd vb drt dd cho cdc nghitnt giAi nhiing phfdng trjnh b6c hai chia tham bi6n vA /.1 (r1 < tr) cita rant th*c ftt) fin rai vb , ,3 thda mdn didu kiOn phu. Tht du t. Gidi pttrrong trlrlr I- Cd hai dau nhd hon sd rhgc lit (tgsinr)2 - 2nlgsirx -- n,2 +2: 0 {r. > I o lat(.rt) > 0 Dd giAi phudng trinh nAy trrr6c hdt ta ch0 f ring { dd cho phudng trlnh c6 nghia, ta phAi c6 0 < sinx. Vdi IJ di du ki0n d6, dAt t = lgsinr, ta ddde, phtldng tr)nh la- r>t) t- f - 2n,t - n,2 + ?: r. Vi 0 < sinx < / khi va chi ll - ( d ltai d|u ldn hon sd rhLrc a lit khi lgsinr < 1), n6n phudng trinh dA cho tudng dudng voi nC n6n hdp a>o [r = lgsirx (2.kn < x < (2* + t]t\1 l^ af(a) > 0 jF-:.r-nt?+2=6 Ir o mOt didm chung v6i lruc hoAnh. 1121 - - Dinh cho cdc ban chuhn bi thi vio dai hoc C6ch giei phudng trinh va brit philCIng trinh bac hai chua tham bien ve thoa miin clib u kiOn phu NGO THTJC LANH od gidi bdt pnuong tr)nh nAy ta dat zsinx = ,. v6i 2) Ddu c0a h0 sd a x6c dinh chidu c6c nh6nh c0a .t dudng parabol. ch( f ring ; r :''n' < 2, ta di tdi hC tddng dudng 3) Ddu cOa (a) xAc dinh difim (a, f (a)) c0a dt-tdng t^ parabol nim tr6n hay nim dtt6i truc hoAnh. if + -z1m - t)t -,r,2 > o 4) Ddu cfia hiQu cr - r- x6c dinh vi tri c0a hoirnh It d0 coa dinh dudng parabol ddi v6i didm a. l;"= -i2a ' / H. I \ f(');o ' 13
Trubng hop ll d (r, v) 4t1 = 5 n6n a > O. Trong trddng h{p 3), 1 nim trong khoAng hai nghi6m. vfly ta phdi c6 a{1) < 0 hay a < 0. Trubng hqp lll 2 a < o e ,r < - - Trong trr;dng hOp ndy (3)) vi a(1) < 0 nOn chic chdn pht/dng trinh c6 2 nghiQm vd 1 nim gir.ta hai nghi6m. NhrJ vAy trong trrldng hOp niry chi nghiQm xr thbh hop. DoAn nh6n hjnh hoc Z a>0ent>-- Dd phudng tr'inh c6 nghiOm ta phAi c6 6'=(rr-1)(4n+ 1 +1)>Oerr l. It. 7 V6l aidu ki6n nAy, ta c6 ) Truitrg hrtp I : 1 , (-l) . o e 3n + 1 < 0 € ttt < - ; ll.6 .11 SGK Oai sd 10 c0a nh6m tdc giA DHSP HN di dua 34 nOn trong trLtOng ra bAi toAn : "So sanh cdc sd -2 v?t I v,'ti c6c nghitn cila hQp ndy ta phAi c6 phuong rrlnh 21 (3n + 2l x" - 21n, - 1\x - \m - r) = 0. -- 0 e rr > - e - 1): o v6i didu ki6n -2 < x < 1 ching han. Mu6n vdy chi viQc r0t ra til bAng c6c kdt quA trang 134, Phdi hop v6i c6c didu ki6n rr < 1 --vitnt>1 nhrlng nghiOm nAm trong khoAng dd cho. 4 Song cdch lim nAy, tuy chdc chdn, nhung ddi. Hdn ta duoc hai nh6m didu kien n0a vi cAbh lfp bAng c6 phdn mAy m6c n6n ta kh6 11 thAy drlqc ;i nghia hlnh hqc c0a m6i br.t6c. Qua thf du -i
Thl du I : GiAi phudng trinh '= lgsinr (lgsinr)2 - 2rllgsinx - n,2 +):o ta crLloc sinx .= 1ot (;iei. D|l r = lgsinx (2kt < x < (2k + 1) :r), nhL.l dA Tit do suy ra trinh bAy d tron, ta di t6i hC h6n hdp r = (-1lnarc sin 1Ol + l-r (^ lf -2rrr-,,,2 r2:o Sau clAy ld bang t6'ng kdt c6c kdt qua di dat duqc lr ,t < o qua sU bi6n luan cJ trdn. ,-_ X6t tam thuc bAc hai nt < -'12- x = (-1)n arcsirfl0'"-\zl^' - 1l + rn f(t\:t2 - 2nrr- ''2 +2 ta c6 -{2 < rrt < -1+ x =(-1)arcsinl0m t"['t'"11 o,,= a =1 sl2=m -1 < nt . y'Z - v6 nghiQm " A':2(nz - 1) -'l f\o\ = -n2 + 2 ttt > ,{Q > x = (-t1n arcsinlfl -fiI" V6i aidu ki6n c6c nghiQm r. vA r" tdn tai (r. < r-). Iltr tls I Gidi oAt phudng trlnh ?' ta chi cdn x6t c5c trtlo1'g'hOp' ? 4t'n* + (ri - 1)21+tin'- m2 t o "", (ii;i. Ddl t = 2'in* vd chu 1i ,anO I = 2''n^ < 2. t, ta duoc hA iudng dudng 2) t., 0 t.. t, Itt,1= 12 + 2trt - t1t - tnz > o Vl a = 1 > 0 n6n dr,ldng parabol y, = t'(tl quay bd l6m vd phfa tr6n. Do d6 11 l2 =, = , .Tradng hop 1. Fa phAi c6 Ta c6 a'>0 a=l J 6' : (nt - 1)2 + ,r,2 = 2,r,2 - 2nt 1- I ,'o 1r2 (0)>0 ,' ,J : - + ttt - I - ttt' - -(rtt' - trt + O) - L' = 2(n2 - 1) > O€/ll ( -llodr nr 2 1 _ 1 ^ 1_ s^ = -l!,,, - -.'+ -] .0 -:nt 0 e-€ < rr < y'Z .1 Tri d6 suy ra Vl a/( ,) < 0, n6n phudng trinh /(r) = // c6 hai -€
tAAl/\3 - l/H MOT HAI TOAN I-,IX[\&I I-IOC PHANG TRAN THANH THIBN I)u(ii dey li m61 bii to.in thi v6 dlch l.i€n X6 (c[), cich K div khoing hdn I0 nlrn : "Cho tam giAc AllC. 'lr€n ba canh cia tam giSc ABC, vd ,/\, phia ngoii tam gi{c ta dung ha hinh vu6ng. X5c dlnh hinh d:rng cila tam giriclll('ndu 6 dinh cria [ra hinh vu6ng (kh6ng r< ptrtii li ttinh cia tam giSc) nim tren m6t dtldng trbn" Xin gitii thiiu vtii han dt.rc nl6l s6 loi Siiri. l.iti grdi thi nhdt (llinh l) r , \u D .i r . '(np"v Llirth 2 f, . '.'-'i,' I-di gtAi tht ba : (Hinh 3) ' >-.1-{ /-*1 . ic I CSi P, Q li cdc trung didm cia lC. va l/G. llien IrlriCn 0. l'. Q thing hirng vi OP L AC, OQ L IlG. 'fa cltt.rn clticiLt ll,' l drJdng ctia truc OP theo hu6ng cta vecttt lQ. KIri d6 : Ol' -- Ilcosl) i I l (fi lA ban kinh tlrldng trbn (lBC). t'Q : X: = l/ilrrrli )_b V?g OQ? : (OI' + PQ)z : R2(cos/l + ?ri26): = R2(crsr8 + lcosBsinB + 1sitt1B!. Ta lai c6 : IIQ2 = (tuiira)2 : RzsittzB Iltnh 1 VlyOII = 112 (cos28 + lcosDsirtB + .ssln2/l; 'frudc het. han doc c6 thd ching minh dudc kdt qui quen = n2Qtirn - 2cos2B + 3) thu{ic sau : Voi hai hinh vu6ng,4-BI{1 vitACGH n
rEr oud pnlEu rnAnn oo Trong s6 tap chi th6ng 7.1993 Toan hoc t:it tudi trd c6 gii ddn ban doc Phidu th6m db. Ngdv 24.7.1993 phi6u trA ldi ddu ti6n vd den tda soan vz) ngdy 24.10.1993 phidu thf 115 va la phidu cudi cung de d-:n tba soan. Trd ldi cdtt 1 c6 231( s6 phieu drinh cho sd bao 191 (5.1993) ;dci ld s6 ddoc nhi6u ngliri thich nhdt. 56 b:io 181 (1.1992) dtroc it phieu nhat (.117,). Trci ldi cdtr 2, chuy6n nruc Di rn nhAn drtoc 807. s6 phi6u, tidp d6n ld, Ditsth cho coc bon chudn bi thi uao dai hoc, Db thi, Tint hidu sdu thant bdtt hoc phd thong o\q).Muc dng kinh cdi caclt day uit hoc too.n nh6.n dt{oc it phiel nh?it @71t. Trd loi cdrt 3, cd 35|4 cho rang D6 ra ld kh6, 62q cho la vtra, 37. cho ld da,3.l cho li hoi khcj std, cd 2'/r, cho d6 Vat li le d0 Trd ldi cdu 1, c6 3'i( cho ltirlng dd ra li nhi6u, 40',( cho iA it vA 55% cho ld vila, cci lX, cho d6 vAt Ii lA ir. Trd liti cott 5, co 6l',4 d6ng i; dua D6 tin hoc vrio vA ;igfl phAn cldi. Trd. loi co.u 6, crj 47(.4 cho hinh th(tc tap chi ld dcp, 49el cho lit tnrng binh vd 3./,, cho li rar. Trci loi cdu 7, co 65',4 cho biet nhdn dtidc tap chi vio crrriT thdrtg do, lge,(, nhAn cluoc vao diiu thdtry sou, 15./,, nhAn drroc rttttort lton rzr7o. (D6ng tidc Li trong s6 nAy lai cd nt6t s6 thu giii tii c:ic thinh ph6 Hd Chi N1inh. Nanr l)inh chit khOng phAi til c6c vung ndng th6n hdo lzinh) Trd. l.cti cotr 8, cd 5 y kien .ld nghi khong th6nr muc nio, 11 d6 nghi th6m mr.rc Danh nhAn to:in hoc, 10 d6 nghi thenr lld thi hoc sinh giiri cac ntiirr:, thdnt Giai thoai torin hoc (5 y kien.), th6m bdi r.'iet cho PTCS t5,r1 l
oudt'tc cAo sAcn ru6r cOn NxB clAo DUc l. Giui thoui trxin huc rtap I 162 irang, gia 3200'' ). Neii trong val) chucln g cd "vAn"ch r-id n o i i,,liii r.[n;r'ihi' ;;'iF trohsTodn hoccungcd nhtrfie catt chu"Yqn "Todri hoc truVen rhA'u" ket hop tai 1inh, ddv sdns tab c6i "tu duv iosi'c" vdi chat "hdi'. ch'Ct "lAng rrran" bans rn6t nshd thuat aoc Eao Tron-e fAp sdch nA1'. tdc giA .dX E'p hqp nhuns cau chtlven toall hoc co ltolt qtLan oen cac trirven thui'0i. stl lich lich" str, phong ltl.c, 8r1r thoar ve cuoc dol cac danh nh6n,.. dttoc lttu truy6n lllonR. d.an gian. tl'Ong sacn vo.,, Cu6n srich se deur den cho ban doc nhrtng ph[t gini tri ho ich] hhf,lf phdt .hien thtr vl. bAt-:',gct Ye r,'...' !llau si. ph