Tạp chí Toán học và Tuổi trẻ: Số 229 (Tháng 7/1996)

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Phương pháp nội suy và các bài toán, xung quanh một công thức tính thể tích, tìm hiểu thêm về bất đẳng thức lượng giác,... là những bài viết trong tạp chí "Toán học và Tuổi trẻ - Số 229" ra tháng 7/1996. Mời các bạn cùng tham khảo.

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 229 (Tháng 7/1996)

Nr.'; T k*jts eo crAo DUC vA DAo rao * nor roAN Hoc vrpr NAM 7
ToAI{ trQc vA TubI TR.E MAT'HE.MATICS AND YOI.JTH MUC LUC Trang o Ddnh cho cdc ban Trunghoc Co sb For Lower Second,ary School Leuel Friettds"- Van Vinh - Phtiong ph6p nOi suy va '.i.iii t ii Nsuv\tu 'o "a" ainn aa tntc' 1 c Phan Tudn CQng - Dd thi t-qyd' sinh nhd ' Tdng biin tdP : th6ng nang khi6u ti"f, ffai Hring 2 NICUYEN CANH TOAN c Gidi bdi ki tudc Phd tdng biin ihP : Solutions of Probtems in Preuious Issue NGO DAT TTJ ba"uii cias6225 3 cnfNc HoANG c Db r0 ki ndY Problems for this issue r{hi0,,..., Tlol2lg,L1l22s,Lzl229 I o Nguydn Dung - D6y Fibonacci vd mach di6n 10 ttOl oOllc elEH r4e : o Tim hidu sAu thAm tudn hoc Phd thing Nguy6n CAnh Todn, Hoing To HetP Young Friends Gain' Better Chung, NgO Dat Tri, L6 Khic tlnd.eritand.ing itt School Maths Trinh Vinh Nggc - Xung quanh-mQt BAo, NguY6n HuY Doan, i. .o;J\t ti"fi irra ticrr Jria trl di6n 11 Nguy6n ViOt Hai, Dinh Quang H6o, Nguy6n Xu6n HuY, Phan o Hoc sinh tim tbi Young Friends' Search in' Maths Huy Khfri, Vrl Thanh Khidt, Lo Hei Kh6i, NguY6n Ven Mau, so - Mdt phrrong PhaP chrlng minh c6c 13 Khic ilai a'b"g tnlrlt aai Hodng LO Minh, NguY6n o Ddnh cho ctic ban chudn bi vdo Dai hoc Minh, Trdn Van Nhung, For College and Uruiuersity Entrance Nguy6n Dang Phdt, Phan Exam PreParess Thanh Quang, Ta Hdng Le Thdng Nhdt - Tim hidu th6m v6 Quirng, D4ng Htng Th5ng, Vfr fat ia"S"tlrrlc lrrong gi5c trong talar' g;t'itc L4 Drrong ThuY, Trdn Thdnh o Nsd Dat Tu - Gi6o sti LO Van Thi6m, Trai, LO 86 Kh6nh Trinh, Ng6 ,"rfia toa" hoc cci c6ng ddu trong vi6c Vi6t Trung, DAng Quan Vi6n' ;; ddt "e ph6t tridn n6n tosn hoc nddc ta o Thi chqn dQi tuydn ho.c s-inh Vi6t Nam D>^ 51a oo du thi lo6n Qudc td 1996 o Gitii tri todn hoc Fun with Mathematics Binh Phuong - GiLi ddP bni Di6n s6 vdo hinh vu6ng Bia 4 o Nguydn HuY Doan - Kdt quA trAn ddu' 458 Hirng chudi, Ha Noi "* ud' tri su: vu KIM IIbNG DT: 8213?86 BiAn tQp THUY 2J1 Nguy6n V6n Cil, Tp IId Chi Minh Ot, ggSOf l 1 iri,n bav ; QU6C
ffi €rE Trong cric ki thi hpc sinh gi6i, thddng c6 cS.cbii todn v6 x"c dinh r46t =h EEI da thrlc. Dd xdc dinh c6c hO sd ctia mQt da thrlc, ta thudng dtng ph6p chia da thfc ho5c ding phrrong phrlp hC sd bdt dinh, phrrong ph6p 9;6 tri ri6ng kdt hop Ehi vi6c giAi rn6t hQ phrrong trinh. Bii vidt ndy xin gi6i thi6u v6i c5c ban m6t --1 phuong phrip n6i suy cta Niuton (Newton) cho ph6p tim nhanh cric hQ sd cria m6t da thrlc. ffiH It t-: >,= -2 1 Kidn thrlc cdn thiSt g6m : Dinh li B€zout (BOau) : Phdn drr cria ph6p chia da thrlc P(r) cho nhi e> thrlc1. bAc >o nhdt r - o bing gi6 tri cria da thtlc tai didm o, t(tcld P(a). ffi ru, t----E{ Zlo *g.i tr 2. Dd Phuong tirn phd.p nQi suy Newton oi o. + 1) didm : C thrlc da P(x) bdc kh6ng quri z khi bidt girl tri ctra da th(rc tai (n trH Z 1, C2,.... Cn+ , ta cci thd bidu di6n Prxtdti6i dang : Pk) : bo*br(x - Cl)+ br\x- Cr)tr - C2) +... +bn(r - C,)... (r - Cn)
2. Suy rit gid. tri, crta tdng sau ddy (n ld. sd BAi 5: , nguy€n duong) Cho bidt da thrtc bqr hai P(x) c6 3 nghiQm s6 S = 7.2.3 + 2.3.5 +... + n(n + 1)(2n + 1). phd.nbiQt a, P, y. Ching minh rd'ng P(x) = 0 udi @6 thihqesinh gi6i Tb Hd ChiMinhlop 9 : L992). mei x. Giii : Cho r : 0, suy ra P(0) - P(-1) = 0 Gid.i: mn P(-1) : 0, v4y P(0) = 0 Ta cd P(a) = P(p) = P(y) = O. Chor l2in lrrot crie giritr!r -1 ; x = | ; x : Deyt P(x) = c I b(x - o) * a(x - a)(x - il. tanh{n duocP(-2) = 0, P(1)= = 6 ;P(2) = 36. Chor = a, P(a) : c,v4yc : 0. 2, D4tP(x) = e*d,(x+2) +c(x+2)(x + 1) +b(r + 2)(x + t)x + a(x + 2)(x + 1)r(r - 1). Chor= p,P@)-b@-a):0 Chor = -2,P(-2): e iF*asuyrab=0 suyrae=0. Chor = y, P(y)' a(y - o)(y - p) = O Chor = -1 ta suy rad = O uiy*q,T*Psuyra Chor:0tasuyrac=0. a= O.YQty P(x) = 0 vdi moir. vSy P(x) = lh I a(x + 2)(r + th(r - 1). b(x + 2)(x + Dd luyQn tQp, cdc bpn h6y glhi cdc bii torin Chor = 1. P(1) = 6b. vAv b : l, sau diy : Cho r = 2', P(2) = 24'+ X4a 36 = 1 1. Tinh eic tdng sau dAy ,v4ya= (x-a)(x4) , (x-b)(x--c) , (x--c)(x-a) 2 A^ = vdy P(x) : 1 + \2@ +2) @-e)(c4)* (a4)(a-c)-@=7fr=d i,*@ (x--a)@a)@-c), (x -c)(x - d) - P(x - 2. P(x) L) = x(x + 1)(2r + 1). b = (d-a)(.d4)ld,-c) - -b)1, Chor: lt2;3:n tac6: 1a= o11a 41a - a1 :'1.2.3 d)(x a)(x - -. (c -4)@=q@- P(1) - P(O\ (x b) P(2)-P(l):2.3.5 4) P(") - P(n - l) : n(n + 1)(2n + t) DapsS:A=l;B=1. suy ra 'P(n) - P(0) : 1.2.3+2.3.5+ ...+tu(n 2. Tim mQt da thrlc bQc 3 cho bidt + 1)(22 + 1) P(0) = 2, P(l) = 9, P(2) = 19, P(3) = 95. I Do dci : s = P(n) = + L)2(n +2). )n(n nii rsr uuvdrq srNrr pHd TITONG NAxc KHICu roAN r,6p ro v0nc z xAvt Hec lees - 1ee6 TINH nz(r nuNc . Thdi gian: 180 phfi (hhOng kd giao db) Cd.u 1.' Cho phrrong trinh x2 - (a - \)x - a2 Cd.u 4: Cho o, b, c ld sd do 3 cqnh cria m6t +a-2:0 (1) tam gi5c. Chrlng minh a) Chrlng minh phtrong trinh lu6n cd 2 ,t 1 *o*"*"*")* 1 nghiOm trdi ddu. (a+b +") (o+a b) Ki hiQu m, n ld nghiQ^m crla phriong trinh ,L Sobc (1). Tim grttd ciaa d6. rnz +nt dat gi6 tri nh6 ' (a + b)(b-----+ c)(c + a) - E nhdt. Cdu 2: Tim nghiQm nguy6n dtrong ctra 5; Cho tam gS.6c ABC. Cd.u phrrong trinh a) Ldy 2 didm X, Y tr6n AB, AC. Chrlng 1 1 _f,_r r_ t t[44++ minh: L2- 2^3 -1- "''rr(, + 1) = 644 dtAXY AXAY Cd.u 3: P li tQp hgp nhtng s6 t1t nhi6n cd dtABC = ABAC tinh chdt : ndt 2 sd thuQc tflp P thi tdng ctia b) Gqi M, N, P ld chAn cric drrdng phAn gi6c chring cring thuQc P. Gi6 srl a - b ld. s6 nh6 nhdt trong crla tam giric. Chfng minh ndu dtABC = trong cdc sd dangr - y v6i x.y li nhitng s6 thuQc 4 dtfuNP thi tam $6c ABC d6u vi ngugc 14i. Pvd.x>y.Ddtd.=a-b. Cd.u 6 : ABC lBr tam gi6c d6u c4nh bing 3. a) Chtlng minh b chia hdt cho d. Ldy 2 didm E, F tr}n AB, AC. Chrlng-minh ,Etr' qut tim tam grdc ndu c6 hC thtlc b) Ki hieu K : 62 Chrlng minh v6i s6 tg 11 -2. d nn + Ae = 1vi ngUqc lai. nhi6n A bdt ki md k > K thi sd hd thuQctap P. PHAN TUAN CONG 2
vdi 2ollnxn, y = 201lnyn. x= Bing cach thii trgc tidp ta thdy nghiQm nguy6n drrong crla (1) ld.xn, ln, z) = (1, 1, 8), (1, 2, 5), (2, 1, 5) vd. (2, 2, 2) V4y nghiQm nguy6n duong cria phuong trinh dE cho li z}tln , 2OlLn, 8), (2011n , 2.2}t7n , 5), B,ni'I11225. Gidi phuong trinh: (2.20L1n, 20ltn, 5), (2.20L1n, 2 . 201tn, 2) - 7y2 + bz - 28t[b = ra + 1srl5 199* + : (34 - t?/t[S - 3r21x2 + arIS itd6n = 1 Ldi giai. Nhfn x6t : Nhi6u b4n tham gia gi6i bdi ndy Dato : x2 + 4t[5,tac6a>ovd"a2-7a- vi t6t ntrrt : Trdru Tud.n Anh (8, Nha cri ldl giai 23: (34 - 3,o)lG- Trang) Biti Mq.nh Hitng (9H Trtlng Vrrong) e(a2 - 7a - 28)2 = (34 - 3a)2a Phan Chi (9A chuy6n ngrl Hd nQr) Ngd Kien a(a4 - 23a3 + lg7a2 - 764a * 784 : O Cudng (9, QuAng Ngai), Nguydn Minh NguyQt e(a2 - lZa + 1&)(a2 - lla+49) o (9, Hei hrrng), Nguydn Trung Ki€n (7 Hi n6i), T^ = NguydnVd.n Quang (9, Thanh h
La Phuong Thd.o, Trb.n ThiVietAnh 9A, H6ng p(p- q, hZ(2) (ddu':'xdyrakhivnchi Bdng, HAi Phdng ; Nguydn Quan'g Bdng, ST ; khic:o) Nguy€n Qwynh Dia.p, Hoir.ng Minh Son, Trd.n Dinh Ngoc, 9T, NK HAi Httng ; L€ Thitnh p@ - c) > h7(3)(ddu' "xdYrakhivdchi Cong, ?T, Pham Huy Quang, D6ng Hung ; l
NO2. go'dd NO1 * NO2, trlc ld L NO,O, cdn (taiN) (2) *r: h*lr[*rr+l -'# Lai c
Nhfln x6t : Cdc ban sau ddy c6ldi giAi tfit : uit. hai didm P, Q thay ddi tr€n y sao cho MN : Nguydn Tidn Dung, Huynh Ki Anh (Lo Quf a ud. PQ = b, trong d6 a, b lir cd'c d0 diti cho Ddn, QNDN), Nguydn Anh Hoa (L6 Hdng trudc. Phong N.Ha), Nguydn L€ Lrtc (DHQG Tp Hd.y xd.c dinh ui tri crta M, N, P, Q sao cho HCM), Phan Duy Hilng (CT, Qudng Binh), bd.n kinh hinh cd.u nQi tidp fi diQn MNPQ ld Nguydn Vi€t Dung, DQng Dic Hanh, Le Vart ldn nhd.t. An, Trdn Nam D{rng (PBC, NghO An), Dlnh Ldr grar. Trung Hd.ng (Mary-Quyri, HN), Nguydn Tri Gsi y, S vi r ldn lrrot li thd tich, di6n tich Dung (CT, DHSP Vinh), Luong Xud,n Thily toin phdn vd b6n kinh hinh cdu nOi tidp trl di6n (Bdn ?re), Triin NguyAn Nggc (11, DHKHTN, MNPQ ; XY ld do4n vu6ng gdc chung cta hai HN), yl; TdlThd.ng (Amstecdam, HN), Nguydn dudng thbngx viy, Xtr6n r vi Ytr6n y, XY = Si Phong (DHSP, HN), Ngzydn Nggc Htng d, (v"y) = p. (LS, Thanh Hcia). , NGUYENVANMAU B,diT9lzzi : Cho dd.y sd {b,} ff = l duoc xdc dinh bditbt - 0, b2 = 74, b3: -78 uit.bor: 7br.r- 6Wr- 6bn_rVn > 3. Chtng minh rdng, u6i mgi sd nguy€n td p ta ludn c6 bo chia hdt cho p. Ldi giai : Tt hO thrlc xric dinh diy {b,.,} ta cd phtrongtrinh : x3 - 7x* 6 = 0 (1) li phrrong trinh d6c trung cira ddy sd d 1. Ldn luot cho n : 1,2,3, t; (2) vA giA thidt ctia bdi ra ta duoc : 3V Tacri:r= s la +2b -Sc:0 la+4b*9c:14 *a-b:c=7. I 1 V= . PQ . XYsin(x, y) [o+Bb-27c=-18 ,MN Vay 6n : 1*2n + (-B)n Yn > I (3) 1 hay ld t V : Ydi p li s5 nguy6n t6, theo dinh li nh6 i abdsing kh6ng ddi. Tt dd suy Phecma, ta c6 : 2P = Z(modp) vn (-g)p : ra: (-3)(modp). Vi v{y, tr} (3) ta drtoc : b- (l +2 = r: max.os - min - 3)(modp) Vp nguyOn td, hay boi p fp nguyOn HTMM, L y = M1, NN, r./ : N1 vdPP, t t6. x, QQt L x = Q1, tac6 : 25 = MN(PP.+ QQr) Nhfln x6t : 1. Tba soan d5 nhAn drioc ldi giei + PQ(MM.+NN,) : 25 r*25r, trongdd : 25 r- ctia 59 b4n. Trong s6 niy, chi cj 2 ban cho ldi a(PPr+ Qqt) vd2Sr: b(MMr+NN1). gi'Ai sai do da giAi sai phtrong trinh (1). Tdt ce Ta chrlne minh rane : S, = min vb,S- : min eic ban c
Chrlng minh trlong t{, Sz : min 5). Ldv [rol didm O ndo dti tr6n PTCT Dai hqc Stt pham Vinh ; Nguydn Quang d,trds th"ane I (aa drroc'dinh hrr6ng) rdi ldn Nguyan.,11CT, Nguy6n HuQ, Hi Tdy;Nguydn IttgtdUng cds,vecto Khd,nh Quynh,11Ao, PTTH Phan Deng Lrru, oiSi J AAi*, (i = l, 2, ... , n, v6i quy u6c YOn Thdnh, NghQ An ; Phg.m Hoiti Lang, A^,, = A,). GgiB',lihinh chidu criaB'tr6n A ; 11CT2, tnidng Lrrong ThdVinh, Bi6n Hba tinh ttidthi :8,. a.. ..'.. B- li oic dinh cta motn- DOngNai ; Nguydn Httu Chu,10T, Lam Son, gidc d6u ti6i arianii trbn tam o, b5n kinh R Thanh Hda; Trd.n Nam. Dtrng, 10CT, Phan BQi : a, vd ta"didrrgc : Chdu, NghQ An. €+2n tr6n (OBi,OBi*)=f=; 2") Y tudng cd bAn cria ldi giai bai to6n li chuydn viQc x6t b6n kinh hinh cdu ndi tidp ttl Goi r li g6c tqo bdi A vi A,A 2,, cung tl?c la : diQn sang diQn tich todn phdn.cria ncj (nhd h6 r: (4, oBrl,tadugc: 6,AA;+J: (A, oB,) : thfc rs : 3V : const) vd tdch S : Sr *Sr, rdi x+(i- l)fl(l < i < n)vb': tim giri tr! nh6 nhdt cria ttng tdng S, crlng nhu MNi * t - OB'i -- acos[x. + (i - 1)P] S, bang c.ich srl dung ph6p chi6u vu6n9 g6c, nnn'l c6uydn bdi torin hinh hgc khdng gian v6 bii oe'f :"2 to6n cira hinh hgc phing d6 gi6i hon ; cq thd li : 2 MNi*r=\ | "o&6+nP):o2q i:l i:l k=0 AA trong dri : Sr : s(MNP) +s(MNQ) -ttin
Nh4n x6t : Nhi6u ban tham gia giai bei Lol t89t : to6n tr6n vA cho liri gi6i dung; tuy nhi6n phdn d6ng c6c ban cho ldi giai kh6ng dtroc gon l5m. R ,d tSgz: RC\, . Theo d6 bni p, Ngodi bdn ban n6u trdn, cd"c b4n sau d6y cd ldi * g2 = 90o, suy ra tgg, : cotgg, : 1 rrit ra giAi dfng. tg?:' Ha NOi : tsili Manh Hilng, g}J, Vfi Tdt L: CRz = 27m.H. -. Thang, Nguydn Vinh Chi. Nhdn x6t. C6c em cci ldi giei dring vi gon : Hn Bic : DQng Hod.ng Viet hil; Vinh phri : ' Dodn Dinh Trung 11L. PTtH Hanoi Dqo Monh Tharug. Amsterdam ; Pham Anh Dung, 11 Li Hda. Ha Tdy : Nguydn Quang NguyAn, Hodng PTNK HAi Hung ; T0 Quang Chinh, lt}. -- Ngoc L\c; PTTH chuy6n Thrii Binh ; Nguyfiru eudc Nguyan, 11 Li, PTTH Phan B6i Chdu, Vinh _ Hiillrtng_.Deng ThitnhTrung, phil.ng Dtlc Tudn, Triln Hoitng ViQt, Hodng inann ftdi lNgne An) ; Nguydn Trsng Nguy€n,11A., Lo Hda Binh : Trdn Quang Huy, phitng Minh Quy DOn, DA Neng (QuAng Nam - De Nilng) ; . Duc ; Hd. Khanh Todn Triin Quang Vinh,11 Todn, chuy6n cdp II-I[ Nanr Ha : Nguydn Anh Hoa, !qKhi6't"QuAng NgEi ; Vu Qudc Khdi, ttBt, Thanh H6a:Vi€n Ngqc Quang, NguydnVan PTTH Bim Son, Thanh Hda ; Nguydn Ngdc T14.ry,lyy !1udn-g Huy,Nguydi Ngqc Hung, Le Thi Miti. Nguydn Anh Duoig Ti.rng, 11 Li, PTTH Dao Duy Tt, Quing Binh : L€ Quang Thd.nh, 11CL, chuydn 16 Quy D6n, _ _ Ng-he $y : Le Vd,n An, Nguydn Hbng Thung, QuAng Tfi; Ddo Manh Thdng, 10A, chuy6n {_Swy Khanh Quyryh, Nguyd.n fiinn, fiO Hring Vrrong, Vinh Phri ; Nguydn Van phuong. Huu Tho- Nguydry Tri Dung-, iqng Dic lianh, 11 Torin 2, PTTH LO H6ng Phong, Nam Dinh \_guydn Thi Minh Huitng Ea TIn[ : phan'Thi . (Narn Hi). NhQt MA , -. Q"1g_Birth: Itod,ng Manh Cudng, Nguydn Minh Tudru, Nguydn Httu CAu B.di LZl225. MQt qud ciiu ruh6 c6 thd truot Quing Tri: Nguydn Ngoc Tud;., khdrtg ma sd.t theo mOt md,ng truc bd.n kfnh R -- Qupng_Nam Di Ning : Iluynh Ki Anh, md, truc nghi€ng m1t g6c apo u6i phuong ndnt Nguydn Tiai Dung, Nsilyfu dhuone Trini rugang ; dO dili cila ntd.ng lit l. Qu! dao cia qua Khdnh l{da: Trdru-fidruAnh, NguyEn Minh ciiu cdtbao nhiAu ldrt dubng sinh AB a).a ntdng Hoarug ndu ban diiu til A qud ciiu chuydn d6ng u6i uan Bu6n Ma Thu6t : L€ Hod.rug Su t6c bi theo phuong uu6ng g6c u6i AB" Ddng Nai : Ph.am Hoiri Laitg, tlu!4rg d6n gi6i Ph6n tich trong tuc Ldm DO^ug : Phan Thanh Hd.i - = nlg tric dungl6n quA cdu ldm 2 thAnh phdn : P Thdnh phdH6 Chi Minh :Nguy4ruVi€t Cudng Bdn Tre :Luong XudnThrty - Thnnh phdn dgc : theo truc m6ng mgsina, ldm cho qu6 cdu chuydn d6ng doc theo mdng NGUYEN oANc pHAT vdi gia t6c a : gsina ; B.ni L11225. Cho ntach di€n nhu so. db tr€n - Thdnh phdn vu6ng gdc vdi mdng, nim cho bidt.R: 3Q ; C = 30004 F. Hai dd.u M, N tr6n m5't phing vu6ng gdc vdi m6ng, mgcosd, cia doon vi ban ddu quA cdu chuydn dQng v6i van tdc nh6 mqch duoc theo phrrong vuOnggric vdi AB nOn thdnh phdn dat m1t hi€u ndy ldm cho quA cdu thuc hi6n dao dOngv6i chu di€n thd xoay chibu. Coi ki ? : ,"\E vdi g' - scosa.. di€n trd hoat d/ng cia cudn cdnt ud, cia day ddy ldn rudi rdt 56ldn quf d4o quA cdu c6t dudng sinh AB. nh6 khOng ddng kd. Hd,y xdc d.iruh h€ s6 t4 cdm L cila cu6n cdm bidt rdn g cdc hi€u di€n thd u*u N:2. LL f2t @_ina M ua B ud. ur* (gitta B ud. N) tenh $i& Ttt \ E"o's" nhau mQt goc A : 90o. Nhdn x6t. Cric em cri ldi giei t6t : ruguydn Thanh Chinh, 11 Torin Li, PTNK HAi Hdng; Hudng d6n gini. V6 gi6n d6 vecto (hoec Nguydn Quang Tuitng llCL, PTTH chuy6n drta vdo c6ngthtlc tinh d6l6ch pha) s6 thdyz*o Phan B6i ChAo, Vinh (Nghc Ar') ; Phan Anh nhanh pha hon i m6t gcic p1, cdn ur* ch4m piil Dung,11 Li, PTNK HAi Hrrng; hon i mdt g6c gz, cd d6 l6nx6c din['bdi MAI ANI-{ tt L-_ _
BdiT8t229: Tim tdt cd cdc sd thuc d,uong a > 2 sao cho : L f \1 - t:)dt 1L ",,tr*ai+l I RA r 0, z > 0 : Gqi AAr,BBl ld cdc duitng cao trong LABC Ching ntinh: nhqn trung tuydn CM cia n6 giao udi dudng uit. (t r .l\ x z v (xyz I 1) i trbn n goai tidp A,A.B rC tai T, cdn trung tuydn ' y zlI +-{-+a 'l*-+-+- z y x D x * y -l z * 6 CM, cfia tam giac AP p I giao u6i dudrug trisn Dang thlic xdy rd khi ndo ng.oqi tidp AABC tqi T, bntng minh ring T, ? ddi nlng udi T qua axtyts thdng gB. uc6 rn6 purEr (Qudng Nom - Dd Ndng). r-rju xuAN rirun BdiT2t229: (rhanh H6a)' r Tim. nghiQm nguyAn cia phuong trinh : Bni rro/22g : x2 +x3 +)c4 +xs :27144.0. Cho ffi diQn ABCD, trong tdm G, nQi ti6p !: NUI QUENC TRIJONG mQt cd.u (O). AG, BG, CG, DG tlrro it u'tu l1 (HdN1i) (O) tqi A1, B1, Ct, D Cltting minh rdng : "it BdiT}lz2g. t. Gidi phuong trinh: GAIGB iGC fGD 1 > GA+GB+GC +GD (x2 - Sr + 2)(x2 * lsx * s6) + I: a - NCUYEN MINH I-IA PI'IAM HUNG (Hdi phbng)" (Ha N6i). BdiT4t229: cAc oii vAr ri. a - S, b : Cho t_ary3igc ABC u6i cd.c canh ^ c = 7. Tinh khodng cach gita tdm tsdiLuZ?9: 6, ctudng MQt con lac lb xo c6 kh6i lttong m : 200g uit oAxc rY pHoNG dao d6ng u6i chu ki T = 2s. Tai thdi didm t : (Hd NAi). 7s con ldc c6 uQn tdc u = -25 ,{2n . L}-3ntls ud BdiT5t229: c6 thd ndng tlt * 12.5. ,r2" 10-6J. Hay uidt I:1"_r-rgt phang cho Elc xoy cd d,inh. f *Ol -_60"). M1tJsrugiac cd,i MAB (MA: pta :"o phuong trinh dao dQng cila con l6c uit. tinh ln9nga9i. AMB = 1200) thay ddi ui tri sao clto hai nang luong cila ruo. *nh A, B.cl.tay tr1n caciia tuong Oy. Tim qujt tich cilo d.idnt M. ing Ox, DO VAN TOAN (NghQ An1. Ttl uOnkd chi 24V. Chtng minh raig h6i tu han ctta n6 d,dy xn ud. tint gi6i 1) Tiruh di€n trd g crta rnAN xuANr oANc (Nam Hd). diQn kd uir. R, ft3= /'foJ? BdiT7t229: crta u1n hd . Clrc hdm sd f(il lien tuc tu€n t0, 11, co dao 2) Tinh diQn trd x trong 2 truitng ho.p chuydn .hom trong (0, 1) udf(0) = f(l) : 0. Chtins minlt x sang llAB thi: ld!^S-!.on tai mlt sd c e (0, 11 sao chilc1 = 1996f (c). a) Di€n trd toitn mach ngoiti dat cdng sudt .H6i kdt luQru cfia bdi tod,n c6 thay ddi kh6ng ndu f(o) = f(1) : m, uoi m lit sit thui hhac 0 cho b) Di€ru trd x dqt c6ng sudt cuc dqi p*_r . tru6c ? RAN vAN vrNu LE HOANH PFI0 (Hd N6i) (Dd Ndng).
Prove that the sequence {*rJ ha" a finite PROBLEMS FOR THIS ISSUE. limit and find this limit. For lower secondary schools. T71229. Let be given a function f(r), '111229. Let be given three positive numbers continuous on [0, 1], having derivative on (0, 1) r) 0,./ > 0, z > O. and such thatf(0) = f(1) = 0' Prove that there Prove that exists a number c € (0, 1) so that f(c) : (xvz*7, f 1//i +L +1 +L > x *v * z * 6. 1996f'(c). Is the conclusion still true if in the '\x Y z/ z Y x hypothesis, : f(7) : ttl, trl is a given When does equality occur ? number, nx *^0) O? T2l22g. Find integer-solutions of the equation T8l2zg.Find all real number o > 2 such that L- x2 +f +xa +f :271440, I G-P) toro*aP+l d,t=!. TBl229. Solve the'equation : a (x2 - 3x + 2)(x2 + 15x * 56) + I = 0. Tgl22g. AAr,BBr are the altitudes of the T41229. Let be given a triangle ABC, the sides of which are a = 5,b = 6, c : 7. Calculate acuteangled triangle ABC ; the median CM of the distance between the incenter and the the triangl e ABC cuts the circumcircle of center of gravity of ABC. triangle APp at T ; the median CM, of the '1512-?.9.In a plane, let be given a fixed ang-le- triangleArBrC cuts the circumcircle of triangle xOv. xOv = 600. Find the locus of the vertexM of "variaile isosceles triangle MAB, where A is ABC atTr. a moving point on the ray Ox, B is a_moving point on ttre+ay Oy so that MA = MB = a is Prove that ?, and ?are symmetric through fixed and AMB : l20o. the line AB. T1O1229. Let be given a tetrahedron ABCD, For upper secondary schools. inscribed in a sphere (O), the center of gravity T6,1229. Let be given a sequence of numbers of ABCD in G. AG, BG, CG, DG ett the sphere { r,} satisfying : (O) respectively at Ay, B-1, C t, D l ' Prove that l l. DAY FIBONACCI VA MACH DIEN Ddu thdng thil nhdt co m6t d6i tho con. Ddu thSng thil 2 ta co d6i lhd tonlDdu thdne thil 3 tho dB, ta co mQt d6i tho l6n vir Rro R6 Re mOt d6i thocon. Ddu thfng thttn ta co bao nhidr doi thd ? Gidi : Goi s6 th6 ddu th6ne th[z la an, ddu th6ng thitn + I sd th6 dv vdn sdns nhuns co ihem nhune con mdi dE. Sd d6i rho mr,i r; ddi dAu ihAns tfi'iln + t Uhne sei-Ooi th6 c6 mit tnjdc do hai thdng. trlc la b5rig an - r. Ta cdcdng.thilc cd bin un*l-:un+un-r(1) DrJa vAo cdng thilc niry ta ldn lUOt vidt ra dudc sd d6i thd d6u cdc th6ng li6n ti6p: Gidi : Hiqt dien Ih6 cta di€n trd R: bing t
@dnnh) Khi siei torin hinh hoc kh6ns eian. dd tinh ZSeac. Sordsina 2 oosinn thd tich-cria m6t trl di6n hodc cricidEi lrrbns hinh t7 :;.L(*) hoc ta hay dirig cric c'6ng thrlc quein biiitEau : "ARCD - 6d, 1) y: I Bh (B ld di6n tich d6y, h ld chi6u cao) Do thidt di6n di qua t6m hinh cdu n6i tidp n6n : B 2SouCS.sin| 25ouo.S.sin$ 2) V : itt *."o r*rina (AB,CD td c6p canh t7 ,ABCD - (**) 3a 3a ddi, MN le d6 dni drrdng vu6ng gric ch ung c:iuaAB Tt (*) vd (**) ta cd : vd. CD,a ld gcic giiia hai canh A-B vd CD). ad, 1 y = j Sr.r (S,, ld di6n tfch todn ?Pq'i,,o 2P5sm 2QSstn, 3) phdn cta trl 3a __ 3 J_ a ' '3 di6n, r ld brin kinh hinh cdu n6i tie'p tri diOn) V.v... ^d Zpqcos , Tuy vQy, cci nh{rng bii toSn ndu dirng c6c c6ng thfc tr6n dd tinh thd tich Le thrlc hoic tinh tie.h ho{c t.inh c6c aAc o *o dai lrrong li6n quan nhu g'
Bdi ndvd6duoc sidi Goi S,, S, l5 di6n tich cdc m4t chung canh o bans cach drine firdt S-. S.'Ia di6n tich cric mirt chung canh 6.. ph+ilg dl qy'l:A ya ;:'*,%-f"Aditai"riCca-nticriamat"ccjdi6ntich Wong g
Hoc sinh tlm t6i tlg ftfl$T PIIII$Hfi PHAP OIII,INO MM OAO IAT ilAHO TxIIICI ilAI $O lE Hoaxc aNH, BUr DUc HoANG, rnAN viNn lrNrr, oANc I tuu MrNH, NcrrveN eueic luAn- Lop 9At, Tradng PTCS Gidng Vo, Ba Dinh, IIa Ndi C6c ban thAn rndn. chdc cdc ban d5. ouen 4\@n *r< -yt -k1 +(br< -t< -4r-ky >A srr dinethf c C6-si cho 2 s6 tioac nhi6u dunebdt = @k -r!1 16n s{_qd HAi mOt's6 bdi rozin vd bdt^diire thfc b) Ndu ak>&>-bk>-o vi IBDT)U d6v'chunE t6i xin trao ddi cirfiE cdc on*k 2 Cr-k > bn-k > 0. Khi dd ban nr6t phdone ph?n don eiAn dd chr-lns firinh ni6t s6"bdt dinithrlidai sd ld phtrons pftirl srl Ar= (bk -ak)(bn-k ---cn-k1+pk -ak11C,-tc -on-k1>0 dgng hang din"g thrlc ia cd,c b5t dinfthrf'c co ban sau : Trfds s_u.y ra di6u phAi chrlng minh. " a2 > O dins th'Jc xdv ra khi vd chi khi o : 0 B.D.T.6 ab > 0 khil,a chi klii o, b cirns ddu Gia thidt ring ba sd a, b, c khdng dm vA o. : nlax(a. o. c). L* ,2,f i:r i:r ,?r{, B.D.T 1 (Bdt dirrg thrlc 1) trong d/ai+t>O (.i: 1,2,...n), az +bz +c2 2 ab *bc *ca drr+l = al,nl,ft € N, 0 < ft < nz. * chu{s *gi. Kh6nB mdt tinh ,dfg,o"u,ng}i Ching,.minh. Theo giA thiSt ta cci A, = (a2 + b2 * 1ab *bc * ca) = + c21 o\ >ai> ,.. > afi : a(a 'b) + b(b - c.) -r(q - b + b - c) o,{-k>df-k>...>a7-rc : (a -b't(a -d +f6 -c\(b -c) ) 0 xdt hi6u T']'dd suv ra di6u ihai chfns lnintr. B.D.T.2 a3 + tt + it > o62 q"6r2 1"rz Chine minh. a) Ndu a D b > c >- 0 vit Ao:@\t+a!+.+aii\- az > bz J ,2 > o. X6t hi6u * - k +a5"T * o *... *4 _ ray - k +afidi - k; {a\a! A. : (o3 +b3 + c31 - pbz * bcz + ca2l : t4 @i-o - oT-\ + 4 @T-k - o\'-k)+... :a(ol - O]) + b(bz -c21 - c1a24z+b2-c27 : (a2 - b2)(o -iy + 1Oz - c2)@ - q > O + o,(,_, (ai_{ - aii,-k1 b) Neu a > c > b > O vb,az > b2 > d > 0. rt k Khi dc, = 11a\-af) @r - orit - + (4-4) @,), -k -4 -k) + k1 . Ar: (b - a)(b2 - c\ + (c - a)(c2 - o2) -- O Tt dd suv ra di6u phAi chrlns minh. -4{"T-o -ayt-k +a!-tc -...+dl_-f -"?,-k)l B.D.T.3 a+ +bq +;4 > o63 q"6"3 q 163 + @f,-t - o,;,-k1l > 0 Chine ntinh. a) NdU a > b 2 c z 0 vi. "f,lt"T_-f - a3 > b3 J c3 > o. X6t hi6u TiI dd suy ra di6u phAi chrlng minh" A, = (a4 + b4 + ca,l - 1ab3 + bc3 + ca3) Bing phrrong ph6p hoin toAn tudng tu nhd :a@l-$3) + 61b3-r3)-c(o3^- b3 + b3-c3) tr6n, chring ta cci thd chring minh drroC cdcbdt = (e' - b')(a - c) + (6r - cs)(b - c) > 0 ddng thrlc sau cho c.Ic s6 kh6ng dm : b) Ndu a > c > b >- 0 vA o3 > c3 > b3 > O. BD.T.7 a2 + b2 + cz D t[Ft + t[Fi + {Va Khi dd BD.T.8 a3+b3+c3>a2{ao +b2{bc + c2{ca A, : (b - a)(b3 -"3) + @ - a)(c3 - or), 0 Tt dd suv ra di6u nhAi chllns minh. BD.T.9 o4+:64+."4 > a3,{ab +b3rlbc +c3,[ca BD.T-10 B.D.T.4 Lq + Oq +;4 > o262 i 62"2 + c2q.2 nn Chtingntinh. Gih.sita2 > bz > c2 > 0.X6thicu .- \["pi * t, Ao: (a4 + b4 + ca1 - 1az6z + b2c2 + czaz) 2 i:r "f 2 i--1 "f-, = o2 (a2 -b2) +b2 1b2 -c21 -c2 @? -bz +bz -cz1 6 ddy ar*r: a r. : (o2-02)(", - "2) + (b2 - czylOz - > O Chring t6i duoc bidt tdt ch cdc bdt ding thrlc Tt dd suy ra di6u phAi chrSng minh. "\ tr6n cring dring cho tnldng hgp s6 mi c'(ta a, b, Mgt crich tdng qud.t, ta cd thd chrlng minh c (hoac ctle' a) ld cric sd thtrc bdt ki. Do khdi bdt ding thrlc sau ni6m "hiy thfra vdi sd mfl bdt ki' chrra drroc d6 B.D.T. 5 an +bn +* > akbn-k q6krn -k q"kon -k cAp d6n trong chrtong trinh phd th6ng cd sd, Ching minh.. a) Ndu ak >- bk > ck vd n6n chring t6i ddnh dtng srr tim tdi cria minh d Qr-k > bn-k ->, cn-k x6t hi6u dAy, Qua myc Hgc sinh tint tbi oia tap chi Ar=(o! + bn + e) - pt'6"-t' *6*rn-k4*on-\. THvTT chring t6i rdt mong c
Da^h cho cdc ,hudn ffiufiffir mom0awmo$fu-Wfloffiffifir (HdN1i) Chu didm-bat a.ine thrlc hlons ri:ic trong Bay gid tt 1b, c6c b4n suy ra ngay: trong tam siSc vta hav. vira"eav nhi6u'lol6irs cho cac moi tam gS:6c ta c6 ban dhudn bi thi vao D'ai hoc. Mira thi-vAo Dai SilS hirc nam 1995. nhi6u bdn di: thi da ohAi bri tav sinA*sinB*sinC
Tt sang3\ tudngtu nhu tt 1a, sang 1b. 3a, chuydn Tt dd tac6 : trong mgi tam gi6c nhgn *:*lB-3[>cositril >0dodtita ABC thi : co: tgA +tgB + tgC > ?t[i 1 A_B B_C, 2\cos 2 +cos , )= iGz) -ABC + cotS, + cotS, (13) tgA+tgB+tgC>cotg, A_C A+C_28 = cos n cos -*--v- = +-r[tscz A_C 3 ,O :cos':V:-cos) (, -r) = tr'*r* * (Gqi {*n$,,n, chrlng minh (14) : chrSng minh :cosA-C 3 , jL r O 4iB-;l tgA. tgB D cotgz^C tgA + tgB > bttgAtgB ,+ =","j lr-3 | :cos * tr-51 Tuons tu cflns c6 2bdt dine thfc nta vd > 2eotgf; .r, ", OItsA +{tgB12 > atcot$ c6ngttnEv6ctia Sbdt dinsthfcEG cri didu nhAi chrffie minh. DSns thfc chf xAv ra khi vd chiA l-a : B Z C. Cd thd Ihdv kdt qudtdns ou6t hon : =tftgt +tsn>-z\l cots] ). NFq A, B, C ld cdc cria rir6t tam"giric va p ta g6"c sd thubc[0 ;3/4] thi Ti 3b, ta c6 : trong moi tam gi6e thi : A_B cos 2 *cosB_C lcosC_A ABT U ^ 4 ts;+tc;+E;rrJFrrrl 4cosk ("*) +cosk (r*) +cosk (cS) Osl Bey gid cdc ban hey ddy : trongmgi tam giric thi cosA * cosB > 0 n6n sinC(coiA + cosBJ < * Ndu bing con drrdnE nhrj tr6n. cdc ban cd thd sd.ne tao ra hhi6u b6lt tl6ns cosA * cosB, s6 e6 2bdt d&ng thrlc tudng tu va thrlc nta. - Ch6ne han x6t : c6ng trlng vd ctia 3 bdt d&ng thrlc ta cd : A *cos,B A+g A':B cos , = 2cos O-cos- vn dd y sin(A+ B) * sin(B + C) + sin(C+A) < 2(cosA n- * cosB * cosC) - A+B
K,: "rz* 5 ,can, a2aq,adt QS e ?,Za ?ka,a. 1 crrio su Ln vAN THrtM, NHA rb.riN iqc c6 cONG o.Lu rRoNG vrBc xAx DgNG vA pHr{t rnrdx xbx torix Hec Nu6c rA. Girio srr L6 Van Thi6m sinh ngdy 29 thdngS Dupna, Chri nhi6m vd Tdng bi6n tAp ddu ti6n n6rn 1918 tai xa Dtlc Trung, huyQn Dtlc Tho, cric tap chi To6n hoc, Acta Mathematica tinh Hd Tinh. Nam 1929, khi 6ng chrra trdn 11 Vietnarnica v.v... tudi thi cha, mg ldn ltrot qua ddi 6ng ph6i theo GiSo srr L6 Vdn Thi6m dd cci nhi6u d
m$r yUyE}{ HSS SrryH VrEr NAM AgAA rE{E C${QN &ry T',E{E T&AN Q{J&C TE 1996 Trong hai ngity 17 ud 1B-5-1996 B0 Gid.o Bei 2 : Vdi m6i s6 nguyan duong n goi f(n; duc ud. Ddo tao da td chic cu)c thi chen hec sinh .n-L- uiio dOi Todn qu6c gia dd tham du ki thi Toan tdn nhdt t;rC,'*t .3' qudc td td.n thrt 37 td chtlc tai Bom Oay - An OO ld. sd nguyan d.d s6 \ ch,ia j:() til 5-7 ddn 17-7-1996" hdt cho 2{(") 35 hoc sinh dat gidi trong hi thi chon hqc . sinh gi6i Tod.n qudc gia (3- 1996) tit 23 didm trd Tim tdt cd, cd,c sd nguy€n duong n md. f(n) : l6n d bd.ng A uir til 28 didm trd l€n d bdng B da 1996. ub dU tht tuydn. M6i ngd.y thi, th; sinh phdi litnt 3 bdi toan trong 210 phnt. (Sd didm tdi da lit Bei 3 : Xet cd,c sd thutc a, b, c. Tim gtd tri 40) K€'t qua ld 6 h.oc sinh dat sd didm cao nhdt f(a,b, c) : (a + b)1 + (b + c)a * (c + a)a - dxdi ddy duoc udo ddi tuydn Todn qu6c gia +t -1,"'+b4 +cl1. 1996 : 1 - NgO Dac Tudn. HS l6p 12 Trtritng DH BAi 4 : Cho ba didnt A, B, C kh6ng thang Khoa hoc tu nhiAn (DHQG HiL In)i) duac 27 hd.ng. Vdi nt6i didnr M cia nrQt phang (ABC) dtdnt goi Mt ld didnr d6i xtlng cio M qua dttdng 2 - Pham LA Hitng, HS l6p 11 Trudng DH th&ng AB, Bai Mzlir didnr ddi xing cila M1qun KHTN (DHQG Har Nli) duoc 21 didnt duitng th&ng BC uit. gei M' ld didm ddi xing 3 - Ngri Dtc Duy, HS ldp 12 Tntdng PTTH cia Mzqua duitng thdng CA. Hay xdc dinh tdt Triin Plri - Hdi Phdng, duoc 19,5 didnt cd cd.c didnt M cia ntat phang (ABC) mit 4 - Trinh Thd Huynh, HS ldp 12 Trudng khod.ng cd,clt MM' be nhdt. Gqi khodng cdch bd PTTH LA Hbng Phong - Nant. Dinh, Nant Hd. nhdt d6 lir d. duoc 18 didnt Ching nt in h rdng : udi m6i didnt M cita mQt 5 - Nguy4n Thoi Hd, HS ldp 12 Truitng DH phang (ABC) hhi ta thuc hi€n li€n tidp bo, phep KHTIV (DHQG Hd N2i), duoc 17,5 didnt ddi xtrug qua ba duitng thang chiaba canh cia 6 - D6 Qudc Anlt, HS lop 11 Trudng DH tam. giac ABC theo thu tu khd.c (so uoi thi tu KHTI{ (DHQG Hat Noi), duoc 16,5 didm tr€n) dd duoc didnt tu[" tli khodng cdch be nhdt cfta MM" cung bang d. Dudi ddy h dA thi chon dbi tuydn Tod.n qudc gia 1996 : Beri 5 : Nguiti ta mudn miti mQt s6 ent hec Blri 1 : Trong m.Qt phang cho 3n didm (n > sinh tdi dtt m|t budi gqp m.Qt, md trong sd d6 : 1) ntd. kh6ng co 3 didm ndo th&ng hitng uit m6i em chua queru udi ;t nhdt 56 em khdc, ud kltodng cach gitta hai didnt bd.t ki kh6ng uuot udi m6i cQp hai em chua quen n.hau thi dAu c6 qud.1. it nhdt mQt ent quen udi cd hai em d6. Ching minh rdng c6 thd dung duoc n tam H6i s6 hoc sinh duoc mdi tham d.4 budi gqp gid.c ddi mOt riti nhau uit th6a md.n dbng thiti mat n6i bAn c6 thd ld 65 em hay kh1ng ? cdc dibu ki€n sau : 1) M6i, didm trong 3n didm. dd cho ld. d,inh BAi 6 ; Hay tint tdt cd cdc sd thuc a sao cho cia dirug m,6t tam gidc. day s6 { *"} , o : 0, 1, 2 ..., xdc dinh bdi : 2) Tdng diln tich cia n tam gidc nhd rr: fi996, 1 hon;Z . xrti-l: . udi n : 0, 7, 2, ... I +xft (Hai tanr gidc duoc gqi ld rdi nhau ndu ching kh6ng c6 didnt niro chung ndm. b€n trong co gidi han h{lu han khi n + @. cung nhu ndm tr€n canh. tam, gid,c). NGt]YBN VIE'I'HAI
Giii ddp bai DrF,N sd vao HiNs vuoNc Co rdt nhibu cdch dibn cdc s6 til 1 ddn 16 ud,o linh uuOng c6 76 6, th6a mdn yAu cd.u ctta biti ra. C6 13 baru gtii liti gid.i ddn thi co 12 ban nt6i ban gtti nrQt liti gidi, tdt cd, dbu ding ud. khd.c I(ET QUA TRAN I'AU ? nhau. RiAng bqn Hod.ng Nant, 70, chuyAn Li, Dito Duy Tit, Dbng Hdi, Qudng Binhc6 gti ddn Kil thnc giai doqn 1 gidi b6ng dd EURO'96 16 cdch kh,d.c nhau, trong do co 15 cd.ch ding. (nghia ld. sau hhi mdi dii dbu dd ddudi mdi ddi Chnng t6i khdng dang hdt duoc cac liti gidi cia hhac citng bd,rug ding 1 trdnl, ngui,i ta c6 nhdn cdc banr, ntit, chi n€u ra dd.y 4 liti gidi dd cdc ban xet ub hdt qud d bdng X (c6 4 dQi tham. gia) nhu thant khdo. sau 1) Tdng sd didnr cila cd.4 dQi trong bd.ng X 1 14 15 4 o 16 10 5 ld. 16 (Saum6i trdn, dQi thang duoc 3 didnt., d6i hda duoc 1 didnt, d1i thua kh6ng duoc didnt 8 11 10 5 6 9 t5 4 nao) 2) D/i nhd.t bd.ng li, d1i duy nltd.t duoc 6 t2 7 6 9 13 2 8 11 didm 13 2 D 16 t2 n I T4 Hdi hdt qud. trQn d.d.u gitta dQi nli bd.ng ua dOi cu6i bdng ? Bidt rdng ngdi thtl cia hai dQi c6 sd didnt biing nhau duoc quydt dinh theo hi€u sd birn thang tril sd bitn thua. Ndu hi€u sd nity curug 4 15 6 9 15 4 I t4 bdng nhau tli theo sd bd.n thang ud. ndu ud.n kh1ng xong thi b6c thant 5 10 3 6 10 5 8 11 NG[JYITN tlUY DOAN 11 8 13 2 6 I L2 I L4 1 I2 7 tf 16 13 2 A 4 hinh uuOng trdt tdng cdc s6 d nt6i hhng ngatug, hirng dqc uit m6i dudng ch€o dbu bang 34. I]iNII PHTJONG t)x\ CO THE TIM MUA TC TOAN HoC vA TUbI TRE O DAU ? Tap chi Todn hoe vA tudi trd ddn tay ban doc theo 2 con drrdng : Brru di6n vA cdc C6ng ty phrit hinh sdch vd thidt bi trubng hgc sau ngAy 15 hdng thdng. Ban doc Hi NOi cd thd tlm mua TIfVTT d 81 Trdn Hung Dao, 5? Gif,ngV6, 25 Hnn Thuy6n,.. Ban dqc 6 cric dia phuong khdc c