Toán học và tuổi trẻ Số 201 (3/1994)

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Toán học và tuổi trẻ Số 201 (3/1994) bao gồm những nội dung về một phương pháp giải phương trình nghiệm nguyên; hệ thống bồi dưỡng học sinh năng khiếu Toán ở Liên Xô trước đây; bài toán về điền số; phép biến hình đồng dạng đặc biệt và ứng dụng vào việc giải toán hình học.

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Nội dung Text: Toán học và tuổi trẻ Số 201 (3/1994)

Be GrAo DUC vA DAo TAc * r{oi roAx Hoc vrET NAM , .... I |) ,; ,'r[,u. tap CHf na UAXC TUAXC it _ ! !;;,o A,.tA\- l*r* rxoilo no-r !)u6lrc Hoc slitH ilnug HHI€u , l'" / TONil 6 U€il xO rnu6c nnY Thi hoc sinh gi6i toin qua th r,r # VOLGOGRAD I99 I + :!r n 6):e i.E D"O, Z- 11 OI'o ;Jr*ll iJ-rH "tri, ?lJ^ XC zez A tJJ _J AJFI{ TJJ t-!| #,9'H. DQi tuydn tod.n ud dQi tuydn tin hqc z]l\ Trudng Dai lwc Su phant HiL Noi I dtt thi qudc gia Db ra ct}a cuQc thi dac bi6t chdo mirng 3{} ruAna rAP cruf ToAN{ Hec vA rribr rRE
rAP cHi roAN Hec vA rubl rRE MUC LUC Trang MOt phrrong phrip giAi phuong trinh nghi6m nguy6n 1 Tdng bi€n tfip : t'nNIt t o,rN Nr;r rYtiN H6 th6ng bdi dttdng hoc sinh nang khidu to6n d Li6n X0 trttoc dAy 2 Phi tdng hi|n fip : Nc';6 oa'l-t'rl Giai bei kj,tn-tdc rf HoANG crttLlNct L5 sinh nhdt mbo DOr6mon I HOt odNc atEN rAp : Bdi to:in v6 di6n sd I MOt bei torin hai cudn sdch ldm sai I Nguy\n Cdnh ToAn, Hodng Chlng, Ng6 Dat Tt, LO Khac Dd ra kj' ndy 10 Bdo, Nguy6n Huy Doan, - Nguydn Vi6t Hai, Dinh Quang Problems of this issue 11 Hdo, Nguyi:n Xudn Huy, Phan D6 ra cia cu6c thi dac bi6t t2 Huy Khai, Vu Thanh Khi6t, Lo Hei KhOi, Nguy1;n Vdn MAu, Ph6p bidn hinh ddng dang dac bi6t vd Hoitng LO Minh, Nguy€tn Khac rlng dung vd.o vi6c giAi torin hinh hoc 13 Minh, Trhn Vdn Nhung, Nguyi;n Dang Phdt, Phan Thanh Quang, Di cd kheo 15 Ta Hbng Quang, Dang Hilng Thang, V0 Duong Thuy, Trhn Thi. hqc sinh gi6i to6n qua thU Thdnh Trai, LO Bti Khiinh Trinh, d Volgograd 1991 16 Ngd Vi6t Trung, Ddng Quan . ., vten. ^ Tru sd tbo soan : 8l Trdn Hrrng Dao, Hh NQi DT: 260786 BiAn fip uir tri sq : VLI KIM THUY 23r Neuy6n Van Cn. TP Hd Chi Minh DT: 356111 Trinh bdy; DOAN I{ONG
n ua bdi vi6t niy t6i mu6n trao ddi ctng ban Bir,i 2 : Gi&i phuong trinh nghiOm nguyOn : \fdoc mQt phtrong phdp giAi phuong trinh F-y'-2y2-sy-1=o (2) nghiQm nguy6n. TOi nghi ring phtrong ph6p niy c0ng li mQt cOng eu t6t dd giAi m6t ldp Gidi : (2) + rt = f +2JP +3y +.1 (*) nhtng bii to6n vd phtrong trinh nghiOm Tac6y2>0;5y2 *2>0n6n nguy6n, T4m d{t cho nd c6i t6n lA Phrrong phrip "khir dn". 6f+ zy2+ sy+ 1)- 1sy2+ z1 < f+ 2y2+ sy+ 1 <
Dd iam quen v6i phrrong phrip grAi tr€n, cic * fyT - l) = (x: + 1)(x2 + 2) b4n hay giAi cic phrrong trinh nghiQrn nguy6n sau : Lyy-1)=(x2+2)(i+3) 1) "/ -r'r = 3r Kethop v6i (***)+l':=: ' 2) (x -2)a -xa = f Lxt=l x=2;X=-2;X=7ix=-7. 3)/:xJ+2x*1 4) xt' - 4yt - 4l = 2 + 3y + 6f . Tt dd tim dttgc nghiQm nguy6n eta phrrong Tin chic rang s6 cd nhi6u thri vi ddn v6i ' trinh (4). ban v6 Phrrong trinh nghiQrn nguy6n. -Al./^bavi Hc rHOilg 001 Duolfc t? Hoc t^^aA slllH lfnilo HHI€u ronil o ucil x6 rnu6c nnY NGo vIoT TRUNG Cu6c thi hgc sinh gi6i todn (HSGT) ddu L6ningrad, Kiev vA Erevan. Nhttng nha to6r ti6n dtrqc td chrtc d L6ningrad nam 1986 bdi hgc trd d cric thdnh ph6 tr6n da di khip Li6n ts.N. Delonu. 0rg crlng li ngudi td chilc cudc XO dd td chdc c6c cuOc thi HSGT vd t.uydn thi HSGT ddu ti6n d Maxkva ngay nam sau chon hgc sinh cho c6c trudng chuy€n nQi trti. dd. Hinh thfc cuQc thi do 6ng nghi ra hiQn nay Cung trong nhrJng n6m 60, c6c trddng vd l6p dtioc dirng d nhi6u nrJdc vi cA d nrrdc ta. Sau chuy6n to:in (kh6ng n6i tru) cflng drloc mOt sd dai chidn th6 gi6i ldn thrl 2, thi HSGT dttoc nhi todn hoc ndi tidng td chtic d cac thinh ph6 nhAn rQng ra toln Li6n X6. Cu6c thi v6 dich l6n. DAy cfing li thdi ki mi l6p nhi to6n hoc toAn n{6c c6ng hda Nga ddu tiOn drroc td chfc ddu ti€n trtrdng thdnh trong chd dd Xd viet cci ndm 1961 vh cu0c thi v6 dlch todn li6n bang con brldc vdo hoc c6p ba. ddu ti6n d{Qc td chrlc nam 1967. Cho ddn Bd sung cho cdc trtldng vi l6p chuydn cbn nhfing n6m 70 c6c cuQc thi toirn ii6n bang cci Tnrdng torin th6ng tdn todn li6n bang drr
Nh$n x6t a, b, c khOng ddng thdi bang 2 vi trrii v6i gie thi6t d +b2 +cr = 4{;5A Do drf : a(.a _ 2\ + btb _ Z) + ck _ Z) < O *a2+b2+c2 31[abc (2) Tt (1), (2) suy ra : v6il 2ct[1ab + 2 > c. Ttrong t{ : ,I",'-' ,o,--no- 2>-b;2>a lE, rnCiNc, NrrA'r
B}ri T3/197 l Cho duitng trbn tdm O, duimg I{ud i Nguydn Hfru IIQi A, 8T LA Khidt, QuAng hinh AB. Ggi C, D ld, hai didnt. theo thu ttt ckia Ngii,Wgzydn NhQtNam.,Bd Rla - Vrlng Tnu. trong, chia ngoiri dogn-AB theo ti s6 k > 1. Gqi chu ui hai tam gidc chung dd,y CD, ldn luot vO KIM THUY : c6 dinh M d ffAn duitng trbn (O), N d ffAn ddy cung AM, lit. Pruit Pr. Chtng minh rdng BAi T4lf 97 : Vdi m6i cQp sd nguyan duong Pt. Pz < 2.AD. nguyan fi cilng nh.au (p, q) dq.t : s =[9] + lU1+... + ;- -l)q1 (P 'Lbi giii : Lp) LpJ L p J MydidmD'thuQc trong d6 [x] lit s6 nguy€n ldru nhd.t hhdng uuot tia qud. x. Xd.c dinh'cd.c gid tri p, q dd S ld. m.Qt sd ^ BD CMB ^sao cho = BMD'. nguydn $. KhiddMB, MAliln Ldi gini kila nhibu ban) : Vdi mdi a e R lugt le phdn gi6c dat { o} = ct - [a/. Khi dd, v6i k € N e6 . ho, rk trong vi ph6n gir{c t;l = : , d diY ro la sd drt trong Ph6P chia ngoii cta g6c A kq cho p. Do vAy : CMD'. Ta cci : BD' AD' MD' 1+ (P-l)q- (2*2 '"' -'\ BC AC MC s =p ', * ...* P p \p p + ...+ p / TiI dd CA AD' BC = BD. suy tidp ," = y* o# : 2 -(t'v-"'* (p-i)q ,'t ,'2 .rp-t. p) , . CA: DA. \u' ca i,B ) 's", o D'A DA ep = 7g'YilY D' = D' Vp) = I n6n ro * 0 Yk = 1,2, ..., p- 1. (q, Hdn nfra, ta sd chfng minh r,, ,t2,...,tp_l Trlc ld MB, MA ldn lrrqrli phdn giric trong d6i mOt khric nhau. ThAt vAy-, gid st, ngdgc vi phAn gi6c ngoii rl c,0'a CMD. l?i, it,i e{1,2,...,'p - il ii
dinh S (theo (3)) n6n trong trudng hgp niy, ak + I ) 3 . Do vQy, theo nguydn tt kdt quA cria trudng hgp 2, ta drroc S ld sd ak + I eNn6n li quy nap, ta cct (1) Y n > 1 . nguy6n t6 khi vA chl khi : O TiSp theo, ta chrlng minh : ndu a,, > 3 thi f'l lp=z**1,udime(P r an+11a,,(2\ ^- ll't -' I tsi ?h6t v{y, tr) ar, > 3 + a,, ) 4 l[p=s (door,€N)+ aD '1q = n * 1, udi n e (Atdn #2 (mod3) [' 2r-2 * Vdy, tdm l4i, tdt cb c{.c cAp gui trip, g cdn tim li c6.e cilp p, g drroc x6c dinh theo (4), (5). - -Q rr- "dr, + q,+t= a,, - lZ) * 74 o,,- 7 'o,,' lZ)'2 Nh$n x6t : 1. Trongsd 112ldi giAi mn Tda o Tt (1) va (2) suy ra 3 i e N* sao cho soan nhQn drroc c 7. Thatvfy, v6i n = 7 c6 at ) 3 (theo gie thidt) .. tt'nl'an= [.r+{c] nduc>S Gi6 srl da cct (1) vdin = h (h > l) . Khi dd ,r+6 {z L +i"i nduc < S ak S (i ci t
BAi T6/197. Gid.i phuong trinh Suy ra : A3 83 : (3AB - 5X3AB - 4) 2518a- el - t+ + z5:*o - af + a, : hay (AB -3)' + / - Q*/B : 3 - V + Vay, { A3 = -4+3AB = 5 -3 ffi , dP"rrr. = 2g - 18. 3r4 -8t2 + 12 - 7xa -8t1 16 Liri giai : Cd 36 b4n dd ngh! srla l4i d6 ra Nhin x6t : C6c ban grli bdi v6 d6u cd ldi gi&i dring vd d6u sir dqng dinh liViet cho phrrong vn giAi dring theo d6 di sita : "GiAi phuong trinh : t*rinh baE ba. M6t s6 ban cbn sit dung cric hang l, ding thrlc dAi s6 dd chrlng minh trtrc !i6p bei si + rq - {rz + 8' : torin-. Tuy nhi6n, crich giAi niy quri dni dbng. 25(y4 - + 252'*" *18 x 3ra-8r2+ NcurvEN vAN MAtr = -7to -8r2+ 12 16 Zg D6 dnng drra phrrong trinh v6 dqng : Bai T8/f 9?. Xdt cd.c sd thtlc a. b, c (a * 0) gt+1q5t+2+V=2g sao chophuong trinh a(ax2 +bx*c)2 *b(ax2 *bx *c) *c:0 (1) v6it=1x2-ay2>O uO nghiQnt. Gei vd trrii ld f(t) thi f(t) ld hAm ddng bidn. Chtng ntinh rang ac > 0 Vir>ononf(t)>f(O)=2s Ding thtlc xey ra khi vi chi khi Ldi gini : Cach I (str dung tinh chdt li6n t-0 0 . Khi dd phuong tt\nh f(fk)1 : Q ehring minh phtrong trinh cd 4 nghiQm. v0 nghiQm af(f(x)) > 0 Vx e R. Tt dci, c=f(f(x,)>0vdac>0. l0 rH6Nc NHAr 2) a 0 (dpcm) ap +py +ya - -2;oFy = 7 Nhfln x6t. 1) Tba soqn nhQn drroc sd ltrong rdt l6n sic ldi giai grli v6. Nhi6u bqn nhdm l6n Stt dgng hing ding thdc trong lQp lian - aOi v6i phuong trinh (x + y * z)3 = x3 + y3 + z3 - Sxyz 4 aP +bt*c=0 (v6i t= ax2 +bx +c) coi di6u +2(x+y+z)(xy*yz+zx) kiQn v0 nghiQm
3) Tuy vQy, vi6c ard rQng trgc ti6p bii to6n Ldi Sai. tr6n drrgc d4t nhtr sau : Qua M kd cric drrdng thing Cho d.a thrtc Pk) ftQc > 1) c6 it n.hdt hai tudng rlng song nghiQnr ud. phuong trinh P(P(x))' 0 uO nghiQnt. songvdi c6.c Chung ntinh rd.ng cd.c nghiQnt. crta P@) citng canh cta d.d.u. Ivlong c6c b4n xem x6t lai crich giAi cria ABCD. Gqitip minh xem-cd thd 6p dung dd giai bei torin tdng hqp cAc glao qurit tr6n dttoc kh6ng. didm ctia chring T{CUYEN VAN MAU vdi c6c m4t cria ABCD ld E vdi BAi T9/f97. Cho hai didnt A, B cd dinh E = lAr,A"r,A^, C ffAn dudng trbn td.m O uit. ntQt didnr. C chuydn 8,.8".8",:,D:. dQng ftAn duitng trbn d6. Tint. quy tich trqng D',, f). Xot mat (BCD), ching h4n, ta c6 cac td.m tam gid.c ABC. arii,iid,, e-';:";incnr sao 3ho MA , tt AB t MA, ll'AC!; IilA^ ll AD. Suy ra hirih chr;p Liri giii. DatR MA',AA.. d6u (d5 ddng d4ng ph6i cAnh vdi li b6n kinh dttdng ABCtr, t6ur ddng d4ng ld giao didm cria dudng trdn (O). Goi M ld th6ng MA vdi (BCD) Kdt hop vdi MA' t trung didm cta G.rA./,r\ j4t c6 AU trongJ$m A A hay AB;Glitrgng MA,-+-MA,+MA.=SMA' 'A.A, !++J+€++ t6m tam gtrdrcABC, Suy ra 2 Mxi:3(MA'+MB'+MC'+MD'\:nMf Q,) ta c6 G nim giita .r€ t C, M sao cho GC = ZGM. Udy didm I (vi 1 ld trong t6m c.ua A'B'C;D'). MAt khric, nim giira O, M sao xdt dinh B chane han. ta c6 BA ll MA, v6i A, cho IO = ZIM, ta e (BCD\ ; gC liMO, v1i C, e"6BDi ; BD fl c6 IG ll OC (dinh li Ta-l6t dAo), suy ra MD, v6i D, e (ABCi. Ntrtt -u|y, M, B, A1, C 1, IG MI 1R IG D) fa 5 din1r cria nrOt hinh hOp trong dd MB IG'R = OC: m : ,h"y = ,viLc Ia-mOt dttdng ch6o vd 4A, M%, MD.l? ba ,R. canh. Ta c6 : MA, + MC) + MD) = MB. e(r;g) Suy ra L Mr,= MA + MB+MC+MD=4MG (2) -€4i-++ Goi ,41, B, li cric giao didm cfia AB v6i .t e Il (vi G ln trong tdm ctta ABCD). Tt (1) vi (.2), (, , 3). Dao l4i, rdy didrn . ; (, f;) "*, ta c6 MG = 1MI hay M, G, -I th&ng hdng. Md cho G'khdcA,, B, ; Goi C'ld"' giao didm crja tia G cd dinh n€n ta cd dpcm. MG' vli (O). Gqi G, li trgng tdm A ABC', ta c
Ldi giai : Tdng c6ng sudt ti6u thu d m4ch P = Pt + P2 + P3 + Pj *P-, = 700W Crrdng d0 dbng di6n chinh P 700 175 u 24 6' #-ara/' nluo *0=0*SntUu Tr)/ = y': * I.t--' 175 160 (J, 216U) -_-l- q 180 suy ra U_, (1) U 35U2 - 192 suy ra r,r8 . Ap dr,rng dinh luAt bAo toin Ti 1_, = I,, ' I-, ta cd 3 co neng: Ps P1 P2 L ,u3, = ntgho uz-u, d=4 u2 z"lg) (32U2 - 408)U., u2 suy ra [/, = 31U, (2) h, = - 384 Suy ra Ift (1) Rs R2 Trudng hW 2; Luc ddu ta xdt su va cham trong IfQC chuydn d6ng vdi vrln tdcf,r. Trong IfQC ndy vAt B drlng y6n cdn v6t A chuy6'n dOng vdi vAn t6c uo ddn va cham vio B, gidng nhtr d trudng hSp 1. Do d
Gidi ddp bdi L5 sinh nhat nnEo o0nEnnont Tr) nhfn x6t cria Ddr6mon suy ra crich m5i vi mBo c6c mOt vi thi crj mQt vi m6o cdc. Vdy eai to6n vb dibn sd cd d6y cQc (C) vi kh0ng cdc (K) nhu sau : Di6n cdc chtr sd tr] 1 ddn 9 vdo 9 6 cria bAng ...* c * c * K * c * c *K* c * c * K *... dtt6i dAy, sao cho : Nhu vay giua 333 d khSch ng6i c.ich mQt - sd cri ba cht s6 nim d sE c6 222 mdo c6c. Bdi vi da sd khrich ld c6c hdng ngang giua bing tdng du6i cho n6n d6y ctng s€ nhu thd ddi vdi s6 cfia hai s6 (cd ba chfi s6) ctia mbo cbn lai (ghi ddu *). Nghia ld cd cA thAy hai hdng cbn l4i. 444 mdo cQc du6i. Ddy dugc s6p xdp nhrl sau : - sd cci ba chir sd nim d cQt giua bing tdng cria hai .., CCCC KKCCC CKKCCCC KK.,,. sd (c
Bei T10/201. Trl diOn OABC vu6ng d O, cd dudng cao OH c6 dinh, cric dinh A, B, C chuydn dQng. K6 HA' t OA 6 A' t HB' r OB d B' ; HC' t OC 6 C'. Chfng minh rang 6 didm A, B, C, A', B', C'cung nim tr€n ur6t m{Lt cdu vir mat cdu niy luOn di qua hai didm c6 dinh. Cdc ldp PTCS t-t, QUii 2 t}l.i P (Q (r)) cong li mOt da thtlc. Kai P+ cho tia 11( di I Chting rninh rlng P lA m6t da thrlc. qua f,leu ti6u , *-t=- A{ n 't/v I F didm -F,I cta NGUYEN MIN[{ DUC kinh Klnn kinh va[. vAt. ( Bdi T71201' Tlm mQt hdm sd f(x), xdc dinh Tia IK sau ul vdi mgi 916 tri thgc ctia r, sao chb v6i r bdt ki khi qua kinh f llkl = -r vafr) nhin giri tri nguyon khir nguy€n. m6t so cho tia KF di qua F vh k6o ddi di qua NCUYE,N DONC I} ". Nhu vdy F Ia Anh thAt cira F, qua kinh dd quan s6t m6t vAt phing crrc nh6 AB dat tru6c BAi T8/20f . Ki hieu la, lb lc ld dQ dni cric liinh vit sao cho &nh ctraAlg md m5t nhin thdy drrdng phdn gi6e trong drroc k6 tdi cric canh 6 crich m6t m6t do4n l. Hay chrlng minh dQ tudng (tng a, b, c cria tam giric. Chring minh b6i giric thu drroc lir : .abc LD. a, *=f,f,(l*z) ^)4*r"*q;r+ q*4>''t3 (D la khoAng nhin 16 ng6n nhdt cria urit la Ib lc S_ b)6+c*"*o*o*o=7v3 ngudi quan s6t, A la dO dii quang hgc cria kinh hidn vi (khoAng c;ich tt ti6u didm Anh cria TO r{AI klnh vQt ddn ti6u didm vft cta kinh m6t), f, Bni Tgi201. Tr6n parabol I y = x2ldy 6 vd /, ldn ludt le ti6u cil cta kinh vit vi cta di6'mAr, Av A3, A1, A5, A6. AtAscAtAAta klnh-m6t. Tt dd suy ra c
TBl2Ol. Function P : R + fi possesses PROtsLEMS OF THIS ISSUE property : For every polynomial Q(r) with real coefficients with deg Q >- 2, P(QbD is a polynomial. Prove that P(x) is a polynor-nial. For lower secondary schools NGI.IYEN MINFI DLIC Tll20l: Find all integers r such that 25x * 46 T7l2O1. Construct a funtion'f(r) such that is a product of two consecutive integers f(f(x)) = - r for all r € fi, and f(x) is an integer NCUYEN DUC TAN for each integer r NGt.rYIrN DLING TZl2Ol: Find out digits x, !, 2,./ such that T8/201. are lengths of inbisectors lo , 16 ,1, Ty .yz = VE (x * y, z *"0) corresponding to sides a, b, c of triangle ABC, . LE HAI KTIOA TRUNG respectively. Prove that T3/201 : The lengths of 3 sides of triangle ABC arc.3 consecutive^inteqers. Determine AB, BC and CA when 3,4 +28 = 78V ilh.#,+#>,tg l,r lb lc NG UYEN KI.'IANII NGUYEN b)b+c*"*n*o*=av33- . I'0 I IAI For upper secondary schools Tgl20l.6 points At,A2,A1 ,44,45,A,, are lying on the parabol ! : x2. A,A, meets A.A* T4l2Ol,Letp (p * 5)bean odd positive integer at a ; ArA,, meets ArA,, at p and ArAa meets and d be a positive integer different from 2, AuArat y. Show that a, B, y are colinear. p, p 18. Show that from the set M : : {2, p, p + 8, d.! can be chosen at least 3 distinct pairs DAM VAN NIII (a, b) with a * b such that ab - 1 is not a Tll0l20l. Consider the tetrahedron OABC, square (pafu (a,6) and pair (b, a) are considered having three right pland angles at vertex O, as the same) such that O and its orthogonal projection I/ TRAN DUY HINTI on the plane ABC are fixed (the vertices A, B, C can be moving). Draw HA' t OA at A', T5l2Ol. Prove inequality HB' L OB at B', HC' r OC at C'. Show that (q'+q-")z>2(q2+q-2) the 6 points A, B, C, A', B', C' lie on a sphere f.ora2,{T,qr 0 passing through two fixed points. NGI.]YE,N VAN MAU t,E QUOC }INN D6 VUI MAY TixH Mriy tlnh b6 trii th6 so cci nft bdm can bdc H6y tim mQt s6 mdbinh phuong bitng nrdy hai, cho gi6 tri gdn dting thidu cira c6n bQc hai tinh d:(ung bling 10. vdi 8 cht s6. N6u binh phtrong c6n thidu ctia Drip : Hdy bdrn s6 N l€n, ta kh6ng drroc N. Thl du, bdm 8trtrtrtr trE Sd phAi tim Id 3,L622777 Xudt hi6n sau khi nam i-il ta drrqc cen thidu cria 5 li 2?360679 rdi dn ti6p vi sd 10 xudt hi6n sau khi bdrn tr X = ta dugc binh phuong thi6u ctla nti lir 4,9999996. NCUYEN OONC 1l
6.Eil A- CAC LOP PTCS: .a H Bai 6/PTCS : Cho a, b, c, d, e ld 5 sd nguy6n tty y. Chrlng minh rdng tich h P = k - a)(e - b)(e - c)(e - d)(d - a)(d - b)(d - c)(c - a)(c - b)(b - a) chia hdt FE NGLJYEN HUY DOAN re) r BAi 7/PTCS : Tdn t4i hay kh6ng c6c sd nguy6n drrong x, !, z th6a mdn xz + y3 + z5 : 19991098765432t t TA H6NG euANC; H B}ri &/PTCS : Tim tap hgp A l6n nhdt gdm hiru hqn cric s6 thrtc, sao cho ndu x € Athir + 1 - x2 GA vi x - 1 *x2 € A. ,G HA HUY VUI Bni 9/ PTCS : Cho dudng trbn (O) vd mQt dAy cung PQ. Tr6n ddy PQ ldy didnr M, qlua M v€ c5c ddy AC vit BD bdt ki (B vh C d cung ph{a so vdi PQ). (, Ndi AB, CD cdt day PQ tqi E vil F. Hay x6c dlnh v! tri cfra M trdn day PQ A MP ME vE dd cd *: MF E ucuvBN KHANH NGUYEN 2 BAi 10/ PTCS': Cho tam $6,c ABC vd cdc didm M N, P tr6n cdc canh tudng tng BC, CA, AB sao cho : 'BM_CN=AP=n rG MC_NA-PB_*' Chfng minh ring AM, BN, CP la do dii ba c4nh cira mQt tam giric. Goi S(H le, H - diQn tich cria tam gi6e niy. Tim A sao cho S(k) dqt gi6 tr! nh6 nhdt cd thd drroc. NAUYEN MINH HA \I Ed B - cAc l6p prts : -\ U Bai 6/PTTH : Tim chrr sd khrlc kh0ug cudi cirug trong bidu di6n thdp phdn G. E cria sd : (51e64 + 5lee4) ! oAxc HuNc ruANc _gg. Bei 7/PTTH : Cho 0 .oi < a2 < ,.. Chrlng minh rlng hai khlng dinh sau E- tudng dttong : d i) lim g, = +* E -l )G F ii)lim " ( . ru.\'' "\l -+-.
Tim hidu sdu thAm Todn h7c phd th6ng PHEP BIEN ninln o0ruc DANG DAC BIET VA UNG DUNG vAO VEC GtAl rOAN nirun Hoc xcO rud psrnr Cho hai cip didm A, A' vD, B, B' v6i A,B' = Blri to6n 2 : Clto LABC bdt hi. Ve phia !4U.trongdd k > 0lb, mOts6thucduong. Cd ngoiti tam gidc d6 cd.c tant giot BCM ud ACN = phdp ddngdangbisnA thinhA' vdB thirih 8,. hai sao cho : (^ |tQt nheq.Arr nguyOn hudng ctia mdt phing goi ^ - - ArVC ld ph6p_ dbng dq,ng. tttue_n Mot pnep aai hia-ng I BMC 90' cria m4t phing goi ln ph6p dbng dang nehicti. lcM = 1.CN 1 Sau ddy ln m6t phdp ddng dang thu8n aac UiOt guo Ar{ = z c
tga, * tga, A NMM'T-ANBC tg KMN = tq(dt + dz) = | * tga, tga, Da bidt LPAC - ANBC 11 _r_ Tn dd _3'2_., APMM' - ANMM', 1 ,6__ 1-' Hai tam gi6c niY cti p MM' chung vi cci Y4y fuN = +sn hu6ng nSl9c ndn bing nhau (d6i xtlng Ta lai cci cosza, = ---l- tgza, * t qua trqc MM).YAy ,1 FAw = fritt -- fei = 4bo + Fttt't = cos-G.z = --.:-- + I MP=MN tgJa. -tg22+l VQy APMN vu6ng cdn tai M. ,KM,z : I = (rr.r) nr1 *t= a Sau dAy li cric bai tqP tudng tU : KM 1. Cho LABC bdt ki. DUng tr€n cdc canh KN ctra tam gi6c vi phia ngodi cria tjrm glac cac tarn girict6u ABM, BCN, ACKrg-sf P, Q ldl Biri torin 3 : TrAn cd.c canh cia L ABC bdt luot ia tdm cta cdc tam gl6c BCN vd ACK. ki dUng ub phia ngoiti ctia tam. gid.c dd cho Chfng minh : cd,c tanr giac ABM, BCN uit. CAP sao cho : a)CM L PQ ^ -^ CAP= CBN=45" b) cM = PQ{5 ^ ACP: ^ BCN:3U, Cho trl gi6c l6i ABCD. TrOn c6c c4nh vd ^ ^ phia neodi iria tt gi6c ta drlng c6c tam gi6c ABM= BAM=l5o ia" eEu, GDN, Be I vd ADJ. Goi P, Q li tam Ching ntinh cia cdc tam gi6c BCI vd ADJ. chitng rninh LPMN uuing cdn tai M. MN t PQ vn MN = PQ{5 (Dd thi To6n 3. Cho LABC bdt ki. Drtng v6 phia ngoii cria tanr giric di cho cdc tarl gi6c EBA, FAC qudc td ldn thri 17 sao cho nam 1975) l^ GiAi : X6t a IAEB= AFC=9V ^ ^ ^ I F= e(r, ;, io5") . e(P, k,1o5a) . R(M, 15v) Itgane=ryFAC=2 tron ,B C ldy didm If sao cho BIf = Lf," lr" =sin4b" _u sin45" - --r. lPl- sin3oo 6iiva v6i h = = vz''' rinh ti sd ffi "tr,3o" 1r, _ sin3o, _ 1 thing nt, n cit nhau tai lLlC sin45" h 4. Cho hai dtrdng i S tao v6i nhau m0[ goc a . Qua S dr;ng hai Suy ra : drrdng thing bdt i(i Q, b. Cho M e-nt;N €:ru ;haMA,l- a vd MB t b; R(M. ls(P),2(P,k. lo-s") ,r (ru, |, ror; .lfa, r i ve Nfi, i b Tim g
TruyQn cu Khoai TYPE st = string [10] ; VAR L6P bA : Array tl..Bbl of st ; LdP 58: Array t1..201 of st; DI CA KHEO LOP 5 : Amay tl..55-| of sr ; xuAN TRUNG PROCEDURE Gh6p l6p; var i, j, ft : intrger Hai l6p 5A vA 58 tnrdng t6i cd it hoc sinh ; n6n thiy hiQu trtrdng d6 nghi sr{t nhAp thdnh m6t l6p goi lA l6p n6m. Vi6c l6p lai danh srich Begin tudng ld chuy6n don giAn hcia ra lai gay go. I:= 1i {Hoc sinh ddu ti6n cria 5A} _ Cri ban d6 nghi tflp hqp cA ldp lai, mdi ngudi j : = l; {HSc sinh ddu ti6n cria 58} viSt ho t6n cria minh ra tb gidy rdi n6p cho l6p k := 0; {Ltic ddu ldp gh6p chrra cci ai} trudng mdi. Ldp trudng theo nhirng td gidy dc; sdp xdp lai danh srich. Ching han ban An drlng While (j < = 35) ADN (j < = 20) do tntdc ban Binh... Cd.c ban giAi thich ring dci Begin In kidu s6p xdp ABC g)6ng nhu trong cu6n tt h: = k + 1 ; {s€ th6m 1 ngudi cho didn vAy. ldp ghdp) Cu Khoai nghe duoc chuy6n bin trin drn i if Ldp 5A t,l < Ldp bB fil then bbn bAo tdi : begin Ldp 5 {kl : : Ldp 5A tll ; - C$u bAo cric ban ldy sd didm cta hai l6p Dtra vio I6p gh6p tr6n vdo theo kidu di cd kheo li dtioc ngay th6i i: = i + 1 ;X6t ngudi tidp rud. theo cria 5A - Ci kheo la thd ndo ? Mdy dtng cri dr)a. end else - Le dd trlng btr6c nhrr trong rap xidc dy begin Ldp 5 [k) : = L6p bB[t] cAu a. ; Dua vdo ldp gh6p j:=j+1;X6tngudiridp theo ctia 58 1 End {While} ; 2 {Ndu 5A con ngudi thi dtra 3 vio l6p ghdp) 4 While (j < = 35) do 5 begin 6 7 k:=hll Lap 5[k): = L6p bA t,] ; i.'= i * 1 ; DuyQt hai danh sdch tr] tr6n xudng drr6i, end ; m6i ldn lAy mQt ngudi d m6i b6n. TCn ai drlng trudc thi dua vio danh s6ch m6i rdi ldy ngudi {Ndu 58 cbn ngtrdi thi drra ti6p theo 6 l6p dd. Sdm mu6n s6 cci rn6t ldp vdo l6p ghdp) h6t ngudi. Cu6i cting chi cdn ch6p ndt nhttng WhileU
Thi hqc sinh gi6i toen qua thu d Volgograd (6.H.L.B Nga) do h6i cAc nhA x*#i:r'"y;,I:B?;:l.JIItqa:*liiru":.'*dt.1ff ch{ tqo mAy pndi "niitli".Uilil33,i!&i"r,,S.i,lTd; [ii"'JA"'"p6ii"iiiiXidiir.irJ"!'rnJi n"" rn{t trdn.sar'*tri dii ttri ca=n.phAi. srri bAi siai trons thdi t"n.Oiiran"sau *niadlni i,lO" c6ng.bd.tren ta bAo Ephil (PhAt thanh) chuyan in cec€hudng hOp. -t F Fl i;1";';Hi;;;niira iirvEn"ni^rt-tii,"g iJan-d votsosraa..u's,.roi tnam sia cuOc thi kh6ns bat. buOc -r E Itli Siai'ta-ili'"a.'6aiioan Jang rTen bao. Nh0ig"nsudi iuoc giAi trong cac vdng thi qua thu, se b,lOchoi tham d\l vdng thi vidt tfp trung. d dOi dr/ tuydn dd gii tning Ngo?ri giAi thtldng. nnOng l',1q" .inn auOi" di€im cao sE dddc chQn vAo ai nqE Gp"ve dAo tlo tai da"-tr,.rjng dai nqc hAng ddu trong C6ng hda li6n bang Nga' C) DU6i dav lA c6c bAi thi coa vdnfl cu6c thi qua thu nam 1991..Cec bAi ttr c;""b;iiJ ldp 10 vA 1 ddn S dAnh cho cAc c) nocliiniii} i, a, s] 6 ddn rt aann cnb 1) C6 thd dtji mot tb bac 25 r0p thanh cAc td bac cAc hec 16 sinh lo4i 1 r0p, 11. 3 r[p, 5 r(p sao cho nh6n duoc (t) cA tnAy rc td Uac 16 cAc loai duoc kh6ng ? E Et . lrao nam 1991 d Lion bang Nga c6 c6c loai td bac 1 rtp,2r0p,3 r[p,5 rup,10 rip,25 r0P,50 r[p' vA 100 rtp). d hiy a g6c 19" thanh hai phdn tli l0 v6i .6 vA 13' !# 2) Bing thu6c vA compa chi 6 vu6ns ld 1 ddn vi dei' ngubi.ta.vE m6t du-qn9 d ii ita, m6l to gidy x! a.rrong voi.cannlocia inong i't hEn 250 dinh 6 vu6ng ndm b6n trong hinh trdn-v6i uin xi,.n ro"a';.j. c[,i"i-61]"i'-,an-b- trdn dA cho. chr't sd giting nhau C,E Fa 4) C6 thG! nhAn hai sd c6 hai ch0 sd Ad tlcfr nhan duoc lA mot sd c6 bfin duoc khong ? 5) Cho bidt menh dd sau dAY d0ng : Q.r ts ir*g .nAt phing, h)nh F bid; thAnh cninrr r trong ph5p quay tAm O' g6c quay 48o' F+. HAy x6t xem cAc mQnh dd sau dAy dring hay sai' a) hnn F bidn thdnh cninn F trong ph6p quay tam O' g6c quay 90o' ni H'infr F bi€in thAnh chinh F trong ph6p quay tam O' g6c quay 72o' ot cnino minh rinq trong mot t,t-giac ndu teing cac binh phr.rong c6c canh deii dien bing nhau o P)r th) cdc dLroig ch6o cria tu giAc vu6ng g6c vdi nhau' E i) fi* ba sd nguyan td-lnac nnau cno bidt tlcn c6a ba sd d6 gdp ba ldn t6'ng cga chrng' 8) Trongi
TRA Lot THUBAN Doc Ban doc thdn ntdn ! Tba soan da nhan drroc rdt nhi6u thtr Vd thdi gran ra tap chi thdt cria ban doc. Ndi dung thtr.ctia ban doc thudng tdp trung vdo mdy vdn dd sau dAy : - Thdi han giAi bAi cho m6i sd tap Chuydn tt hai th6ng ra rn6t sd sang ra chi. hdng th6ng tba soan dd cci nhi6u cd ging - Sd ngridi cri ldi giAi dtroc n6u t6n dd tap chi crj thd ra tai Hi NOi vio ngdy tr6n tap chi 15 hdng thring. Tuy vdy 12 sd ctia nam - Thdi gian ra tap chi. 1993 de ra tai He NOi dao d6ng tt 12 - Tim mua tap chi d dAu ? ddn 18 hdng thring. Ri6ng hai sd I vd 2 Tda soan xin trA ldi chung nhu sar.r : nem 1994 do cci trd ngai ctia nhd in vi vi6c nghi Tdt Am lich n6n mai tdi 22 vd. I/d thdi han gi6i bii cho m6i sd tap 28 moi ra duoc d Ha NOi. Tba soan xin chi thdnh that xin I5i ban doc vd nhirng chdm tr6 ndy. B6t ddu tt s6 4 nam lgg4 Vi tap chi ra hing thzing cho n6n c6c tba soan s6 phan ddu dd cri thd ra ddu ban cung chi ccj thoi gian Ii hon m6t ddn vdo ngdy 15 hdng thring. th6ng d6' giAi bdi cho m6t s6 th6i (Tt khi tap chi ra cho d6n 30 th6ng sau). Vd viec tim mua tap chi 6 dAu Khi nhAn duoc sd mdi cdc ban lai phAi bit tay vdo giAi c:ic bAi cho sd mdi. Vi vdy rdt nlong c6c ban cd g6ng giAi c6c Tap chi To6n hoc vd Tudi tr6 phrit bdi ctia nr6t s6 trong thdi gian hon m6t hdnh theo hai con dr-rdng : COng ty ph6t th6ng dy dd tao thcji quen, Kh6nC Crei hdnh brio chi vi cdc COng ty phdt hdnh dttoc bdi nio thi b6 d6'ta lai b6t tay vio s6ch vd thi6t bi trudng hoc. Vi vAy dd giAi cric bdi cho s6 moi. nghi ban doc tim mua tai : - C6c buu di6n thi trdn, thi xA thdnh Vd sd ngudi c
r. YEU cAu csuxc On cho hgc sinh l6p 12 chudn bi thi vdo c6c tnidng dai hoc vi cao ding. Tod.n hec uit tudi tr| dd tidn hdnh ddi m6i . glTin lzqc; Nham gi6i thi6u cdc khdi ni6m, nQi dung, hinh thtlc vd md r6ng d6i tttong phuc vu. Cdc bhi vidt n6n nhlm phqc vu ddi vdn d6, n6i dung, phuong phdp ctra tin hoc tuqng r6ng r6i bgn dge b ci cl6c trudng trong c6c tnrdng phd thOng ; m6i quan h6 qua Phd ;honttrung hgc l6n c6c trtrdng phd lai giira tin hoc vd to6n hoc... I I I th6ng co sd. Bii vidt cdn chri y xiy drrng cho lol Todn hgc uit' ddi sdng: Gioi thi6u nhtng "! { trr duy to6n hoc, rlng dung cta to6n hoc vdo c6c linh vuc kh6c I hqc sinh phong c6ch rbn luy6n ndng cao khA nang phdt hi6n vn giAi quydt vdn nhau cita ddi s6ng. aO. Cacn vidt cdn nhg nhAng, vui, di d6m' lll Hoc sinh tint /di ; NhrJng bai viSt ctra hoc sinh phd th6ng, tim tbi nhttng ldi giAi kh6c tr. YEU CAU CnO CAC CHUYEN I\,ruC CHfNH hay hon ldi giai c6c bdi to6n trong srich gi6o ll N6i chuyAn udi cd'c ban trd yAu tud,n : Chit khoa, trong b6o THVTT vd nhttng tdi li6u v6u nhim mric dich rbn luv€n ttr duy todn hoc tham khio kh6c" 6 aav kidn thrlc khdng phai ta muc dich urd 12l Ngodi ra cbn cci nhiing chuy6n muc ld c6irg cu, sAn phdm ng\i6-11 lhig". thu dttgc . nhrr : Lich sir vi nhirng mdu chuy6n v6 ti6'u sau klii giei quydt vdn d6. Kidn thrlc chi nou to6n hoc,'Tin tfc day hoc to6n kh6p vita dti sio ch-o bdi gon md dat dtloc muc ti6u stt c6c nhd xAy drrng phong criCh hoc to6n cho hoc sinh. ndi, Ban cci bidt, GiAi dap th6c m6c, v.v " 2l Tim h.idu sd,u thant. tod,n hoc phd thdng : 'l3l Tod.n hgc uit, tudi tuA cbn dang Anh phAn Nham muc dich ddo sAu, md r6ng vd hQ thdng 6nh cic hoat ddng day vd hoc to6n cung cdc hria nhrrng kidn thrlc trong trudng phd th6ng. ho4t d6ng khdc cira cdc tnrdng va dia phr:ong. 3l Budc d.d.u tint. hidu tod.n hqc hi1n d'ai : Girip md rOng tdur nhin cho ban doc. Ldm sao III. NHUNG DIiU CAX CHU Y dd v6i vdn kidn thrlc da cci hoc sinh cci thd nhQn li ilai viSt tr6n mOt rnAt gidy. Hinh ve ro thfc so b6 v6 nhilng kh6i ni6m, phuong ph6p ring, cci ghi ch6 dat hinh. M6i bai ddi kh6ng qua Iufln so ding cria toSn hoc hi6n d4i, gcip phdn hai nghin chrr (kh6ng qud 6 trang d6nh mrly) kich thich vd gAy hrlng thri tim hidu to6n hoc. 2lM6i d6 ra phAi cci kdm ldi giAi, viet ri€ng 4l Db ra ki nity; Nhirn httdng d6n nguoi trOn mQt rnrinh giay. CuAi m6i bai ddu ghi ddy giAi t6p duot vAn dung tdng hgp cdc kidn thrlc dir ho tOn, dia chi itra ngudi ra d6. D6 ra cdn da ccj dd giAi bdi torin bing phttong ph6p thOng ghi 16 ld sang tac hay sui tdnl (Ndu cd thd xin minh vd s6ng tao. Dd ra cci thd ld sdu tdm ndu [,ii kOm bAn dich d6 sang tidng Anh t bii todn cci ldi giAi hay. 3/ Bai dich hay sUu tdrn cdn ghi 16 t6n. nAnr vd noi xudt bAn ctia tii li6u dci. Sl 6ru5 kinh cdi cd.ch. d.ay uit' hgc tod.n : Phin . 6nh nhtrng y kiSn cita hoc sinh, phu huynh hgc 4l Bdi chi gui vd mOt ftong2 dia chi : l, ir3 sinh vd c.ic thdy, cO gi5o v6 c6eh hoc, c6ch d4y, 81 Trdn Htlng Dao, Hd N6i vd s6ch giSo khoa vd s6ch tham khAo m6n to6n. 231 Nguy6n Van Cr), T.P Hd Chi Minh 6l N\L cuiti tod,n.lzpc ; Nhtng bii vidt ndy 5/ Rieng bdi giAi cria hoc sinh chi gti vG 81 cdn rdt vui vi cci thd gdy tidng crrdi sAng kho5i Trdn Httng; Dao, Hd NOi. thu gi6n. 6i Anh ndn Id Anh mdu cd 9 x 12, 6nh rndi, 7l Ditnh cho cdc ban Trung hgc co sd ; NOi ccj chti thich n6i dung vd ho t6n, dia chi cira dung bii vidt d6 c{p c6c vdn d6 phuc vu ban ngrlbi chup. doc ti l6p 6 ddn l6p 9. 7/ Bdi.kh6ng dang kh6ng trA lai bAn th6o. Bl Ditnh cho cd.c ban chud.n bi thi udo dai hec ; Gdm cric bdi hrrdng ddn crich hoc, c6ch IOAN HOC VA TUdI TRE ISSN: 0866 - 8035. Chi sd 12884 Gi6 :1200d Md sd : 8BT03M4 In tqi Xrr6ng Chd bAn in Nhd xudt bAn Gi6o duc. In xong vn gfii luu chidu th6ng 2 11994