Toán học và tuổi trẻ Số 199 (1/1994)

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Toán học và tuổi trẻ Số 199 (1/1994) giới thiệu tới các bạn những nội dung về nguyên tắc cực hạn trong hình học; cuộc thi quốc tế phát hiện tài năng Toán học. Đặc biệt, những bài toán và phương pháp giải bài toán được đưa ra trong kỳ trước sẽ giúp cho các bạn nắm bắt kiến thức Toán học một cách tốt hơn.

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Nội dung Text: Toán học và tuổi trẻ Số 199 (1/1994)

BO GIAO DUC VA DAO TAO't UOI TOAN HOC VIET NAM 1(rsq) rAp csi na sAxc rHANc ,4 't l I \_ 'I Xu* ulr {#;
\ TAP CHI TOAN HQC VA TUOI TRE q^&r't- il4^'w wh'Y"* C{-ir* Tdng biAn tAP : N(,trYl N t .iNtl I( ),\N Phd tdng hiAn tip : N(r() I).\l lt' II() \N(; ('llt N(i nOr o0ttc elEN tAP : Nguy€tn Canh Todn, Hodng Chung, Ngd DAt Ti. Ld Khac Bao, NguyAn Huy Doan, Nguy6n Viet Hei, Dinh Quang Hao, Nguyt;n XuAn Huy, Phan Huy Khei, V0 Thanh Kltidt, LA Hei Kh6i, Nguy\n Van Mdu, Hodng LA Minh, Nguy€:n Khdc Minh, Tru sd toa soort : TrAn Vdn Nhung, Nguyi:n Dang Hi Phdt, Phan Thanh Quang, Ta 8l Trdn Htrng Dao, Noi, DT :260786 Hbng Quang, DAng Hing Thang, 231 Nguy6n Varl Cu, TT' HCM, DT : 3561 11 Vu Duong Thuy, Trdn Thdnh Trai, Bi€n tdp uit tri srt : v0 Kl M THUY LO Bd Khdnh Tirnh, Ngd Viet vi bio 1 ud ninh hgo .' DOAN HoNG Trung, Dang Quan Vi6n In tai Xudng in Nhi xu5t bAn Gid:o duc In xong vn grli lrru chidu thang 1. 1994
Tro.ng qu:i trinh tim kidm ldi giai nhi6u bDi glac ABC cci ba gcic nhon n6n cdc didm A,, todn hlnh hgc, sE rdt cci loi ndu &ring ta xem P u C_, ludng ring nim tiong-.-APC, doqo BC.CA va x6t phdn tir bi6n, phdn ttt gidi h4n nao dd, trlc AB-Ndi PAJB, PCJa c6 + C,pB + ln phdn tir mi tai dr; m6i-dai luong hinh'hoc BPA, + A,PC + CPB, + B,PA'= gOdo. cci thd n-han Cui tr! l6n nhdt ho6c gia tri ntrO nhdt, ching han nhtr canh l6n nirdt] canh nh6 - ^Su_y ra gbc l6n nhdt trong 8 gic nny kh6ng thd nh6 hdn 600. nhdt cta m6t tam gSdc; g6c ldn nhdt ho6c gcic nh6 nhdt cria m6t da giric v.v... -Ktr6rq lim mdt tinh chdt tdng qurit ta cd thd gie st ring g6c APC. ld l6n nfrat, ttrg thi Nhfingtinh chdt cta phdn tir bi6n, phdn ttr APCl > 600. gidi han nhi6u khi girip chring ta tim drroc ldi giAi thu ggn crla bdi todn. X6t tam giric vu6ng APC t, ta c6 PCt . -.Phttong phrip tidp cAn nhrr v6y t6i ldi giei ^ < r bdi torin drroc goi ld nguy6n tdc crrc han. Ap = cosAPC, "o"60" = i. Danh cho cdc ban phd thOng c0 sd r--- l-ai L:, *- *Gi NGUYBN vAN viNH j Vi dq 1 loc tang tt 0o d6n 90o thi jo" . M6t nu6c cri B0 sAn bay mi kho&ng c6ch d BE. GoiB, vi. Z au -1qm C^1 txarrs r't1S le qic didm d6i xfng cira B vA .. Tr^o1rg i\t giric ABd cdba // \'.i C qua tdm E, ta cci tam gSac CrEB, nim trong gdc nhon, Idy '\ tam g15,c AED. trot aiOm p Ua"t lil:#+i:'ii[ ,11 i\ \"iar I AD GiA st doan kho,s;;l; ''N tr * ;n' -" oi"fi fOn--"fra? ;r\-'.- \ vdi doan DF,. trong cdc / -- - ' ' ll --." Khi dd drrbng /,' '',. ,.,' \ uroa""g cach"Li / t$n ? noi tidi ,l ; {\ didm F ddn c6'c I ::1 '**-i l ,, l.---= tam gidc cPr;t A E'-." thn tary gracf i-,-_ ,r nam bcn trons II ./"/.r,/--* " :-,j, t, gya H,o"s ' drrdng trbn noi -- J I I .: - ' - \i ... \i "hO'hd" -hocqe qhi chon hoc sinh gi6i torin ldfg nam dang phdi cAnh 4 l.) 1992 - 1993. Beng B) v6i drrdng trbn tdm ddng d'4ns E, ho s6 ddng dang ""Sl1o"?E?r",!,i"u331o?#:ifi *H ilY'#
Nhu v4y ,4ro rcrEB, = rcEB (rnuo lit ,-A rtri ' GIOI THIEU CUOC THI QUOC TE b6n kinh drrdng tron nQi tidp tam $ c AED). Vo ll vi tr6i v6i gie thdt rtcn = r-,o. Di6u v6 .,PHAT HIEN TAI NANG ll chrlng t6 ring"A : c, C{"D : fr'suy ra trl $6c ABCD ln hinh bin[ hanh. T'AN Hqc" Trong hinh blnh hnnh ABCD ta cci : pr.r = LE HAr KHoi = SAEB = SBEC -- pz. r vdi pr, p2 ttldng fng Th6ng 1l n5m 1992 trong cuSc g{p v6i lAnh dao Doi li ntra chu vi c6c tam g76c AEB vd BEC. tuydn thi Todn qudc td c0a Singapore mdy ndm gdn dAy, Suy ra : pt = p2, do dd.43 = BC hay ABCD c6c gi6o srl Koh K.M vA Chen C.C. c0ng nhr,t v& 6ng Yong li hlnh thoi. W.giAm ddc Nhd xudt ban Khoa hec - Vdn h6a - C6ng viaqa ngh6, m$t trong nh0ng ndi thr.tdng in cAc tAi liQu vd bdi Chrlng minh ring bdn hinh trdn drrdng kinh dL/6ng hac sinh gi6i To6n - Li - Tin hec, chring t6i drl9c li b6n c4nh crla mQt trl gi6c ldi phrl kin fi $6c bidt hiCn nay dang c6 cuQc thi qu& td Phat hien tai di cho. ndng to6n hqc". Md thdng tin vd cuOc thi nAy ddu do Gi6i : GiA sit M ld mQt didm tirf,y nirylQr NhA xudt bAn cOa 6ng Yong dAm nhiOm. troggltl g16cl6iABCD. Ta c
Bei T1/195 : Chg ba s^5 x, y, z nguyAn duong thu. mdn hQ thtlc f + t' * zz = 1992. Hdi sd A = Bh + g2i a 1ggg2z ci thd la s6 chini piuong kh6ng, tqi sao ? Ldi giii : Xdt sd A = Bk + 32Y + 1ggg2z ta thdy : 8h = (9-I)b = I (mod3). Bei T2l195 z Cho n s6 duong ay a2t ... an 34 :0 (mod 3). 1g%4 = (rc92 * 922 : I (mod 3). a! a) Yay A: 2 (mod 3).(1) a_ -L Mqt kh6c ta thdy mQt s6 nguy6n duong z (dt + a2)@1+ a3) @z+ a3)@3+ aZ) bao gid etng bidu di6n dtroc du6i dang : hodc n -= 3k ho6c n = 3k + ^1. Nhtt viy ta cci hodc + '+ a! r: n2 = o (mod 3) hoec n2 = r (mod g) (2) Tt (1) vd (2) ta thdy A kh6ng thd lh sd (q;qq-;$'4,1,o' chinh phrrong. Ldi giei : rdt nhi6u b4n) (cria Nhin x6t : 1. Rdt nhi6u ban cri ldi giAi nhu Goi vd tr6i cria bdt d&ng thrlc ld Q vd tr6n. al a! 2. Mdt s6 b4n chrlng t6 ring A = 2 (mod 4)-vd nh6n x6t ring": m6t sd chinh phrrong bao (a.t+a)(a?+aZ) (a2+a3)@3+a3) gid c0ng cci hodc nz = 0 (mod 4) hac nz : 7 (mod 4). Tt dd rtit ra A kh6ng thd la sd chlnh a! phrrong. +... + thiQ-P=(at- (a,t+a)(4+a?) g. MOt s6 ban dga vdo di6u ki6n x2 + f + + z2 = 1992 ciitng t6 ring ;t tar, "trrg - r) + (ar- a) *...1 (an* a) = 0 + Q = P Do dri ctta A li 8 vi di ddn k6t luin"ht : A kh6ng thd la sd chinh phrrong. 4. Cde ban sau dAy cri ldi giei t6t : Nguydn !4ng, -B}{; Trdn Minh Anh, Nguy€n Bd Hil,ng, o = jre+P)= (Ki hi6u a,r* *,i,C*#*A Nguydn Hodng Duong, 9H, Trtrng Vrrong ; , = ar). Dodn TM Dttng, 6M, Nguydn Ngqc Tdn, 9M ; Nhrrng : 2@| + o?*) > (a, * a,*r)z Marie Crie ; Trd.n Phuong, 7A,, GiAng v6 2 ; Duong,Minh Dtc, 9An Trung'Nhi ; Nguydn "pA"e + 2(a! + ala) > @i + a,*ry2ta! + ol*) (1) P_hrt Binh, 9A, Bd Van Dan; Ha N9i. Huy, Trdn Mat kh6c : Q1lang Phti Son, Luorug Dtc Vinh, BT, Vit Thu Hudng, Hit. Thu Thdo, Luortg Ld a@f + o.? * t)2 > 2@? + o! * 1)2 (2) Qryang, Ddo Xudn Duong, Triln Pham Ngiy1n, Tt (l) vn (2) ta c6 : 9T, Nangkhidu, Hf,i Hung. Pham.ThimhHi€.p, Vfr Van Hoan, Phgm Tudn Anlr., Nguydn Cdih a@! + o!*)2 > (ai+ai+)z(& +"!*r) Hiro, 9T. Phan Bdi Chdu ; Trh,n Thi Tharuh, 9T, PTCS Qu6n Hinh, Nghe An ; Ngo Vu Thu a! + a!*, Hdng, 749 NK Ti6n Son ; Nguydn Dinh Qud,n ; 2-41 (a, * an). 8Au chuyOn tstn Y6n ; Nguydn Hbng Minh ; 1a,+a,n)@! + a!*) 9T, NK Thuin Thinh ; Nguydn Quang Ninh, 9A" Nang khi6u, Hd Bic. Vuong Minh Thu, 88, chuy6n Vi6t Tri, HtL Anh Tudn ; Phq.nt .n Vi rQy: Q = qZr 2 1i a1 ttfl + a!,. Trdn Thg, 9T, Chuy6n Phti Tho, Vinh phri. r?r@,+a,*r)(af +"fu,) Phqm. Viruh Hba, 9E chuy6n Bim Son;?rlzrh ! " 1l Hftu Trung ; Ngl Xudn Anh, 9T, Lam Son, Thanh H6a. Phgm ViQt Thd.ng ; Trd.n Nguyan > ;" i >= I (oi*oi*) = i2 = t: ", r (dpcm) U-So" , 9Il PTTH L6 Lgi, Hd D6ng, Hd ?ay. Ddu ding thrlc x6y tz en sd bing nhau. Nguydn Thi Thanh, 9A,, Hdng Bang, Hai Nh$n x6t : Hdu hdt cdc ban grli ldi giai Ehbng. Trd.n Cdng Cuitng 9T, Pham Huy v6 d6u giAi drlng. Nhrtng bqn c
a! hidn nhi6n c6 (2). GiA sr} da c6 (2) vdi n. = k, > , oi- 4 11 ai+r h>1. I
Nhfln x6t : 1. C6 rdt nhi6u b4n gui ldi giai lint [(2x)tt . r2o -rr- t . 11x2'1] = 0 (x) vi da s6 cci ldi giAi dring. Cric b4n cri ldi giAi n+ *o tdt hon cA.: Nguydn Phuong Lan, Trb,n Quang Huy, Kibu II{tu Durtg, Nguydn Van IIQi, Hod,tzg Do vdy, tr) (1), (2) sc cti f(x) > 0 Vinh HQi (Tnrdng chuyOn Htng Vrrong - Vinh Vrc t0,;).(3) 1 Phf) ; Btti Thd Long (PTTH Chu Vdn An, Hd NQi), Nguydn Thd.i Hd) (10M Mori Quiri, Hi Metkh6c,v6i0 0 tuc f : [0, 1] ---> R th6a man f(x) > 2xf(x2) Suy ra g(x) ld hdm khOig gi6m trong [0, 1]. Yxe[0,1]. Ydy g(x)= 0 Vr € [0, 1] (dog(1) = S:(0)). Ini gtrei : Tt giA thidt, thay r = 0 vi r = 1 V{y Fk) = F(x2) =... = F(l') ta cd Vdi 0 < r < 1 thi lirri F(l') = F(0) (do F(x) f(0) > 0, 0. (1) tt' +@ 1 ^1)< ld hdm li6n tuc). VQy f'(r) = f(0) Vr e (0, 1) v6i 0 < x < * thi bing phrrong ph6p quy nqp, n1n f(x) = O V r € (0, 1). Da f(x) li6n tuc z suyra : f(x) > 2xf(x2) > (2*)".r2" *n - I trong [0, 1] n6n fk) : 0 V r c [0, 1]. 1x2"7 Yn>L(2) ncuv8N vAN uAu vi0
BAi T7l195. Tam gid.c ABC c6 BC = a, Long (Qti Nhon) cbn md rQng bdi todn cho cA=buita2 + b, = tsir2tgA + b244D.0) di6u ki6n z ttl nhi6n tiy y : Chilng minh rd.ng tam gid.c ABC cdn. ail +bn = teg@'tsd+bntgB) Ldi gi6i. Tt giA thidt suy ra A + 90o, NcuyEN vAN uAu B* 90o. Bei T8/195 : Trong hinh trbn cho 1956 didm hh6ng c6 3 didm ndo th&ng hd.ng. Chtng minh (L)+ a2 (, - u}uo) * u, (t t$ua) * =o rdng c6 thd chia hinh trdn d6 bdng 2 duanlg thd.ng thitnh ba phdn sao cho trong'nt6i phd.n - sin$sinA tuotlg tlng c6 2, 9, 1945 didm.. *a2 "o"fiore Ldi giii : C cos 5 . cosA Giua 2 didm bdt ki trong 1956 didm dE cho CC ta n6i bing m6t dudng thing, D6 thdy s6 drrdng cosTosB - sinVinB thing niy ld htu han..Nhu vAy tr6n dtrdng +b2 -0 trbn tdn t4i didm A sao cho kh6ng thu6c c6c C dudng thing tr6n. K6 tia Ax tidp xric vdi drrdng cos cosB 2 trbn. Cho tia Ax quay xung quanh A. ,+ a2 ,c * 62 ,c M6i dudng thing de ndi tr6n chi chrla dring -cos \Z*B) = 0 (2) \Z*A) 2 didm trong cric didm da cho (vi khong cri 3 vi o = 2RsinA, b = ZHsinB vit -^.cos didm nAo thing hnng). Do A kh6ng thuQc c.ic ,C * ,C +B) = 18e dttdng d
Md khdu d0 com-pa sao cho eA2 ddu com-pa P t, Qt, Rt, S1 uit' P2, Q2, R2, S, bd.ng nhau cham dtroc viro m{t qui bi-a, Dgng do4n thing (htong dd.ng) cd dQ dni bing kho6ng crich 2 mrii com-pa d tr6n mQt td gi-dy. Chia d6i doan thing ndy dd Ldi giei. Theo d6 ra, ta c6 cilc !{ giag drrgc do?n thang r.Ldy mOt di61m bdt klA tr6n PIQtR;St , PzQzBrS, trong adc hinh 1 m{t quA bi-a iim t6m vE dudng trbn (S,). Tr6n vd2.Illc6c hinh vfingAB'CD', BC'DA'0nnh 3). ts,l *lai khdu dQ com-pa ld r ldy 3 didm B, C, D'sao cho cci thd dnng com-pa do drrgc d0 dai c6c canh cta tam g;\ec BCD. Tr6n mQt td gidy ta dyng tam gitic B'C'D'sao cho B'C' = BC ; C'D" = CD ; D'B' = DB vit dttdng trtn (Sr) ngoai tidp tam guic nay. VE drrdng kinh MN (Sz). I.dy Hrdu dQ com-pa voi 2 ddu cidr "t nhau mQt khoAne il,{ f, dAt 1 ddu vio B vn quay mQt cung t"rcn dt (Sr) t?r E tlfi, BE ld dtrong kinh cfia (S1). ,A Do dd : 4 B, E thudc dudng trbn ldn ctia i bi-a. Tr6n grdy dltng tam gi6c MNP sao cho MN = AB, NP = BE vitMP = AE thi dudng trbn ngo4i ti6p tam gq6cMNP cd b6n kinh dfng Htnh 1 ld b6n kinh qu& bi-a. ViQc x6c dlnh tdm-O cria Do BD llB'D' (vl cirng r AC) non hai dudng trdn ngoai tiSp tarn g76c MNP ln d6 ddng, ttr dci d6an thing cdn dgng chinh ln OM. Nh{n x6t : Cac@n Nguydn Qu6c Thhi (12S - Amsterdam - Ha NOi), Nguydn Xudn Thd.ng (10T - D6ng Ha - QuAng Tri) drtng tam giSc vu6ng )(YZ c6 ){Y = AB ; XZ =Y uU ffi = gf , titX k6 dudng thing vu6ng gdc v6i )(Y cat YZ k6o ddi tai u thi )Itl chinh ld drrdng kinh quA bi-a, chia d6i y1l ta cd do4n thing v6i dQ dni li b6n kinh quA bi-a. C6c ban Phitng Son. LO.m. (12 - Phan BQi Chnu - Nghe An), Kibu Hitu Ditng (12A - Hirng Vrrong - Iltnh 2 Phri Thq VP) thiSt l6p trl dilnABCD c
Do O nim tr€n AA' n6n nci cach d6u cric dtrdng thang DF, BK. Ta cd phdp quay R (0, -J€0): !].ai {u.ng1 Nery6" Treng Nghia 11Ar PTTH Vi6t Tri, Vinh Phr1. . $0t cacn tubng [1r, ta cring cci 2l -- B^K:AII- E(0,r90p) x - EC;BJ ----GD; CL ---IA ; MT Pr(= DF CL) - Pz(= BK x IA) ; BAd L2l195. MQt cuQn ddy c6 d0 tu cdnt bang Qr.* Q2 ; R, - Rz; Sr - Sz. L, dien bd thuiln r, khtng dd.ng kd ud. nfit di€n Viy Pl Q r,R, S, - Pz Qz8, S, vi ta cri dpcm. ffi R m.dr sang song uda mQt ngubn diQn mQt I NhQn x6t. Cric ban d6u giAi dring. Crtc ban chibu c6 sudt dien dQng t ud mQt didn tro trcng sau dAy cci ldi giei tdt : Phgnt LA Soru (IOT. r. Lic dd.u hhia K md uit. trcng nmch kh6ng cd Lam Son, TH) ; Nguydn Xudn Thd.ng QAT. dbng di€.n. H6i sau hhi d6ng hhba K thi dien D6ng Hn - QuAng Trl .; Nguydn Tudn Hdi luong ch4y qua R bdng bao nhi€u ? (11M - Marie Curie - He NOi) Ldi giAi. Sau khi ddng kh
cuOc rHr cIAI ToAN oAc nt0r I . a: .. : ..r-:., CHAO MUNG 30 NAM TAP ChI[ TOAN I.IOC I Gidi ddp bdi vA ru6l rhE DA TIONG CTt{I Nhdn ki niQm 30 ndm ngay ra so Qud trinh udng bia vd "ddi v6 chai ldy bia" d'.rgc mO tA nhrl sau : b6o Toin hgc vi tudi tr6 dhu tiOn, | .-'-- 1) Mua 20 chai + u6ng 20 chai Tap chi THVTT td chuc cuQc thi gi6i tI 2) Ddi 18 v6 chai dugc 3 chai -*udng 3 chai. tt 3) Ddi 5 v6 chai * 1 v6 chai (di mtlgn) to6n dac bict. duoc 1 chai *udng 1 chai Db thi sO c0ng b6 trdn 2 s6 b6o 4) TrA v6 chai mugn. VQy l6p 12A u6ng cA M6i s6 se c6 10 bai : 211994 vit 311994. thtY 24 cha. 5 bai dirnh cho PTCS va 5 bai danh BiNH PHuoNG cho PTTH. Tdt ci hoc sinh dang hoc trl litp 6 LTi SIh{H NHAT ddn 16p 12 dEu dugc tham gia. Hoc nnEo nOnpu0x sinh PTCS c6 thd gi6i bai cira PTTH cdn hoc sinh PTTH khong dttgc giAi MQt sd chri mbo tr6n thd gidi dugc cr} vd NhAt BAn drt l0 sinh nhflt chri mdo mdy bhi cira PTCS. D6rOmOn. Quanh bdn trdn ddm drroc 666 vi kh6ch, da sd cQc dudi. Ta goi 2 mbo ngdi c4nh M6i bai gini vi6t riOng tr€n 1 td 2 b6n mQt vi khrlch ld ld.ng gibng c&a vi khrich gidy. Sd cua biri ra ghi 6 g6c tr€n b€n d6 ; 2 m6o ng6i crich mOt theo 2 phia ld cric trdi, ho tOn vir dia chi (16p, tntitng, ld.ng gibng thi hai cta 6ng ta v.v... D6rdm6n nhAn thdy : ru5i meo c6c du6i cci 9uQn (huy0n), thanh ph6 (tinh)) ghi it dung m6t v! Ling gi6ng thrl hai vd m6t vi lSng g6c tr0n ben phAi. Ngoai phong bi ghi gi6ng thf trt cing c6c du6i. H6i cd bao nhi6u ro : Thi gidi tocin dctc bi€t. Chi gtti vb vi khrich c6c du6i dU LE SINH NHAT. mQt dia chi TC THVTT 81 Trhn Hung NcuvF-N oiNs rUNc Dao, Ha NQi. Gi6i thuitng se dugc cdng b6 trOn tap chi vir sd trao vao nghy l€ ki niOm al) 30 nam TAP CHi TOAN HOC VA a:,Gi rudr rRE. ffi 6 Mong c6c ban tich cgc huong ung cuoc thi. TOAI\I I-{oC VA TUdI TRE
t BAri T71199 : Cho o, b, c ld. do dei 3 c4nh cta mQt tam gi6c eho trt16c. X6t oic sd x, y, z th6a m6n : RA Ki xAy I JL x+.Ynr=2, Tim gi6 tri ldn nhdt cta bidu thrlc I cAc l6p PTCS sinx sinv sinz Bid T1/199 : cho vdi mqi sd d Sbo Tim cric chft s6 a, b, c, td nhi6n n ta c6 ; F(x,y,z)= o + U + " UAU NGUYEN VAN I aa * a^-6f..-^cc --g+ 1 = pd -.:.::A + 43 Bai T8/199 : GiA sii H, Ao, Bo, Co ldrt --r- ---rr I ns6a ns6D nsti c -ns6d ludt lil trtlc t6m, trung didm c6c do4nBC, AC, TRAN DUY }{INH -/ BAi T2l199 : Cho c6c da thrlc P(r) vd Q@) AB cria tarn giric khdng cdn ABC. Cec dudng th6a m6n P(x) = Q&) + Q(1 -x) Vr € .B thing lain lrrot qua A, B, C vd vu6ng g6c v6i Bidt ring cac hQ sd cria P(x) la nhtng s6 HAo, HBo, HCo 6t Acdudng t}rhng BC, AC, nguy6n kh6ng dm vi P(0) - 0,. tinh AB ldn ldgt tai A1, 81, Cr. Chitng minh ring TRAN XUAN DANG feq) At, Bt, C, th&ng hdng. Bei T3/199 : Cho AA-BC. X6t didm M nim TA HbNG OUANC trong tam giric. Ndi AM, BM, CM c6t eic canh ddi di6n tqt At, Bt , Ct. Tim giri tri nh6 nhdt Bid T9/199 : Cho MBC c zs{s M,\ \ *r, *c, Pz*Ps "l HA HUY VT'I NcuYEN ruANH NcuvEN Bei T10/199 : Gqi G lA trgng t6m cria trl cAc T,6p PTTH di6n SABC. M ld mOt didm thuOc LABC. Cac tu nhi6n z ld sd cd tinh BAd T4lf 99 : Ta nrii s6 dtrdng th&ng qua M song song vdi GA, GB, GC chdt P ndu khi nld alc e,f.a dt - I v6i s6 nguyen theo thrl tu c6t c6rc mQit (GBC) ; (GCA) ; (GAB) t4i A1, 81, Ct. Chring minh ring GM di qua drrong o ndo dci thi n2 crlng ld ddc cita att - 7. a) Chrlng minh ring moi sd nguy6n t6 d6u trong tdm c:&'a LArBrCr. cd tfnh chdt P. NGUYEN MINH HA b) Chi ra ring t6n tai v6 han hop sd n cci cAc nb vAt t i \ tinh chdt P. (O0 arr tuydn thi to6n qudc td n6m 1993) Bai L1l199 : MQt binh c 0 dtqcx6c dinh nhu sau : H6i phAi bdm hft t5i thidu bao nhi6u ldn dd rntt ,=d,vrII *Blx | ,l 6p sudt d trong binh thdp hon 5 mm Hg. Cho trt+l = ax, * Fbr, Vn = 0, 1r2,.., 5p sudt ban ddu cria binh li 760 mm Hg "iog vd nhi6t dQ kh0ng thay ddi trong qu6 trinh bom Tim di6u ki6n cdn vn dt ddi vdi a, B dd ta LAI THE HIfiN c6 xn - *a ud. ln + *a khi n -*o v6i Bhi L2l199: mgirr,lo)0 NGUYEN MINII DUC Cho mach diQn A Bai T6/199 : Cho sd nguy6n duong ld ft vd nhrt hinh 1. sd nguy6n duong n th6a m6n,i < n. GiA stt ccj Trong dri ampe b6 sd cci thf tU (x, , x, , ... , xn) r,6i r, € {- 1,* 1} k6 A chi 0,1 A, Vi = T , n. Cho ph6p thgc hi6n i,irep to6n sau : vdn kd V chi ldv k nim d-ft vi til khdc n-hau. ctia b6 s6 s6. lzv. Ndu ddi (it , *z i ... , x,,) r6i thay mdi sd Mi s6 ddi cua nci. ch6 v6n k6 vd Cho hai bQ sd cd thrl ttl : ampe kd thi khi dci ampe kd ehi A = (al , a2, ... , arr) vdi a, € {-1, + 1} 0,02A. Tinli Si5 B = (bt,b2,...,b,r)vOibl e{-1,+ l) Yi = T,fr, tri cria di6n trd ft. Bidt Uru kh6ng ddi Chrlng minh rang nhd vi6c thUc hi6n li6n BUI VAN PHUC tido m6t s6 htu han ldn nh6o thav s6 ncii tr€n ddi' vdi'b6 sd A ta'cd thd nh'4n_drioc bQ sd B. NGUY6,N KHAC TVIINH - -- Chri v : M6i bii eiAi vidt nene tr€n mOt minh ei6v. Chi s6 c-ria bni ,1 goitren b€n tr6i, ho.-t€h va dia chi d g6c tren bdn phii. 10
dE---./ PROBLEMS.OF THIS ISSUE FOR LOWER SECONDARY SCHOOLS T1./tgg" Find all digits a d c, d such that for any natural number rt, the following equality holds aa=._ a D5.:6-cc-r + t = (m1.3 + 1\3 . ndiSita D n n.i --...- I - TRAN DUY HINH TZll99. Polynomials P(x) and Q(x.) satis$ the conditions i)P(x)-Q@)+ QQ-OvxeR Coefficients ot P(x)are non-negalive in tegers and P(o) ii) : g. Determine P(P(3)) TRAN XUAN DANG T3ll99.lrt M be inside the tnangleABC.AM, BM and CN intersect opposite sides of /BC atAr, B, and. Cr,respectively. w. Determine the mininum value of NGUYEN KHANH NGUYEN FOR UPPER SECONDARY SCHOOLS T4llgg. A naturat number n is said to have the property P if whenever l divides ao -I for some integer a, n2 also necessarily divides an - / a) Show that every prime number r. has property P b) Shov that there are infinitely many oompcite numbers z that pGsess property P IND.5 (rMO lee3) T5/199. For 4 positive real numbers ro , ! o, d ,B, construct the sequences { ,o } o o ura tY" ] o as foilows B " ", x,a=dv +L nf I Jn B lr*l = *, + . ;rt = 0, 1,2,... Findanecessaryand sufficient condition'f:ra, psuch thatxn +*o andlo *fowhennj+e Vxo, !, > O NGUYEN MINH DUC "161199. [-et/< be odd integer and a he a positive integer that k < ,r For weryorderedz-tuples (Xr, X2,..., Xn) witnX, e t-f, l) V, = 1,2,...,n; the fotlowing operations are caried out : For /c numbers from i< arbitrary different places we change them by other their sign (it goes from I to - I or from -1 to 1). Given two arbitrary ordered n - tuples : , A--.(at,-a?'__''i");B=(br,bz,"'bo) :: a., b. € {-1, l} ; i : 1.2,...,n: Prove that after a finite number of the above operations we can obtain B from l. NGUYEN KHAC MINH let a,4 c be the lengths of the sides of T7t1Lgg z a certain given triangle. Consider all real numbelrs x, y, z satis$ing Jt condrtron:x+y+z=z s! sZ ' Determine the maxinum value of F(x , y , z1 = l'uo'b'c + + NGUYEN VAN MAU TSllgg: kt 1/ be orthocenter of non-isosceles trianglelBC an d letA o , B o , C ,be rnidpoints of BC, AC andA-B respectively. ThelinesarepbrpendiculartoHAo, HBo, HCrfromA,B,C intersect"BC )C:ai|Z,gatAr, Brand C, respectively. Prove that A, , Br, C, are .oiin"u.. TA HONC OUANG T9rllgg: lrt a, , a2, a3be the lengths of sides of LABC and.$ be its area. Frove that for all positive numberspl , pz, pt the folloning inequality holds o] o?* ,, Pl a?>z,lj.s ol*Pr*Pz Pz*Pt ' Pt*Pt ' t HA 'HuY vul T10/199 r l-et G be a center of gravity of tetrahedron sABC, andM be.a poinr in LABC. The lines through M aud parallel to GA, GB, GC in tersect the faces (GBC), (GCA), (GAB) ar A, , B, , C rfespectively. Frove that GM through the center of gravity of L A rB rC r. NGUYEN MIN H HA 1l
\---- MQt s6 loai hQ phtrong trinh thudng gap v6 nghi6m, hO ban ddu cd nghi6m duy nhdt r trong edc ki thi vio Dai hoc vA Cao ding chrra :J:0. duoc trinh bdy chinh thrlc d s6ch gi6o khoa. Trim lai : HQ cd nghiQm duy nhdt ea > MQt trong nhirng loai dy la he ddi xrlng hai dn 2514. kidu II. Dinh nghia : H6 phrrong trinh hai dn goi _ Thidu2:Timnrddhe lo* t| ,l^, l-& l-l - tt -tro li he ddi xrlng kidu II ndu : ddi vi tri hai dn { ^ -i lzy+\x,-l=nL . cci nghi6m. tronghQ thi phrrong trinh niy trd thinh phrrong trinh kia. ' (D6 34 - 86 d6 thi Torin Bo GD - DT) Dac didnt I ; Ndu h6 cci nghiOm (xo , !o) Gid.i:Ddtu'-ldx-1> 0thi x=12 *l vd thi hC crlng cci nghiGm (!n , xn1. Do dri ndu hQ u: rly - 1> Qthi y =u2 + 7 ( cri nghiQm duy nhdt thi nghiQm cri dang lqu!+2*u:nt H6 tr6 thdnh : ]'' (xo , xo)' l2u2+2*u=nt t Danh cho cdc ban chuhn bi thi vio dai hoc Ht ncit xrrxc HAr Ax rrhu tr ^-THONG NHAT LE DQc didnt 2; Ndu trrf ttng vdcta hai phrrong Do u, u ) 0 n6n di6u kiun cdn dd hQ cd trinh trong he thi ta duoc phrrong trinh th6a nghiOm ld nr >- 2. Ydi nt >- 2 x6t hQ tr6n ta cci m6n khi giri tri hai dn bing nhau. Tr) dd ddn (u - u)(2u * 2u - 1) = O. (nhdtriltrlngvd tdi viQc cci thd drra phuong trinh ndy v6 dqng cira hai phrrong trinh). H6 trrong drrong vdi tich. tuydn mi trong dd cri h6 Dga vio qic dac didm tr6n, odc ban x6t drroc nhirng hO loqi niy. cci thd Ilu-u=0 l2uz+2*u=nt (r) Thidul:Timaddh6: t- l^.2--s-s*2t^ Dou=un6n: 2u2 + u * 2 - m, = A.Ta = - nf, * )'-, -, | "a nghi6m duy nhdt, c 2-nt ccjP= = ; (0n6nphrtongtrinhludn lx-=y-4y-+ay ' (DO 133 - B0 dd thi To6n BQ GD-DT) a2 cd nghiQm u 2 0. Suy ra h6 ban ddu cci nghi6m. Gid.i : TrrI tr)ng vd cria hai phrrong trinh ta Trjm lai : HQ c 2. (d ldi cci : gif,i tr6n ta khOng cdn x6t hQ cbn lai cria tuy6n. (x-y)tx2*xy*y2 -3(r+y) *al = Q Vi sao ?) He de cho tttong driong vdi tuydn [- zt lx-v=0 : - t l, " . .) (I) ThidqS:Gieihe:] 'r*t-^,2 lx':t'-4y'*ay' t" lY= t- -,' , xy.+ y-,- 3(* +y) + o:0 I )r'^* \II) Gidi : Ta cri x * +1 viy * +LD4tr : l*"=t'-4yJlay X6t(I):dor=yn6n = tsu vdy = tsu v6i u, u e ( - !, : !1 "a", x2 = f - 4x2 * atc==> D * + 4 H6 trd thinh !"u '.s2^' * tL-. x(x2 - 5x * a) = 0. HC (I) lu6n cri nghiQm ltgu = tgzu x, -- y : O. Ndu phrrong trinh 12 - 5x I a : 0 fu:2u-lkt Ilu:Zu+nil :- vdi k, n e Z. Suy ra: u - -(2n c ff. I ") f. / ?V llr 25 (-; ,g) nen k-r ru = 0 vd k, n a (-312, V6i o > 7 xdt he (II) : phtiong trinh 312). Do d6 (k, n) : {(0, 0), (1, -1),(-1, 1)} x2 + xy * y2 - 3(x + y) * a = 0 hay (u, u) = {(0, 0), (- nl1, nll), (n13, -rl3). tuong dtrong v6i : Tt dd h6 cci ba r,ghi6m ; (x, y) - {(0, 0), *3) + y2 - 3y * a = 0 cd x2 +x1y ( -{s , '[5; , (rl5-, -r/s). A, = -3(y - 42 + D - 4a 6m non hC (II) Chn i; HQ tr6n cti thd gi6i trltc ti6p khdng cdn lrrgng giric hria. t2
Thi du 4 : Chrlng t6 ring : vdi a * 0 thi vd ly. Vny x = !.Thay vdo h6 ta cd : 1 * sinr = (t 3n - .-, = ;, h6 cd nghiOm duy nhdt trong 0 lz*' = "v +L hC .l t, cci nghiom duy nhdt. (0,?n) ld x : J : 3n lzr; = x +L ?. S", ra a = 0 th6a mdn. t" x *sinx=2 (Dd 106 - B0 d6 thi Torin B0 GD - DT). Gid.i : Thdy ngay x, ! ) 0. KhiI m6u, ta c 3c > 0 *r * t): o cung khdng th6a m6n, vdy x : y. Thay vdo ..*Jf, ^-y)(2!y h6 vA giAi h6 v6i r, y thu6c (0,2n) ta cd nghi6m -1rr"-Y2+o2 duy nhdt x =.t: f t-ng l},?n). Do dcj e = 2 Do x, y > 0 n6n Zxy* x+ y > 0. HO tr6n z th6a nrdn. Tcim lai : a : 0 hodc a : 2. tudns duong '[x-J=o Cric b4n hay thri ren luy6n qua c6c bni tAp lr; i a) = o _ "u, _ dd nghi drrdi dny : X6t f(x) = 2f - x2 - a2 c5 f'(x) = 1) Gi6i ho : = 6x2 - 2x n1n bf,ng bi6n thi6n f(r) la : [^, 1 u)ll"-=Y*; "l l2v! = -n +- t" x . . [r r/l-:-F : 114 o) frV-l- -; : 1t4 2) Tim nr. dd hQ cci nghi€m : .[r+,[y-3:nr \h a * O ndnf(O) - - a2 < 0. Suy ra phuong =nL ttlnh f(x) = 0 chi cri m6t nghidm x > UB t:dLc [Y*lx-J .. {sin2x * nttsy = nt. hC de cho cci nghiQm duy nhdt. b){ ' ltgzy * ntsinx : nt f!*r,or=o t.) -y- Thi dq 5 : Tirn o dd hc lJ' '- 1*v2 lv 3) Chfng rninh hO : . " ., cci d[ng l-+sLnY:a .t- a -L cci duy nhdt mQt nghi6m th6a nrhn 0 < x < Zn, I *x- 0 1 +r/y *sinr > 0, vO ly. T\ronghl: ndu 0 < r < ytbtylx *siny > Q l3
To6n hga vi dtri sOng M6i kdt quA quay 3 brinh xe ld m6t bO 3 sd (a, b, c), trong dri a, b, c ld sd md mrli t6n cria brinh xe thtl nhdt, nhi, ba chi. CA thAy cci 63 = 216 kdt quA cti thd. Ta chia chfng l5m ba nhcirn Nhdm 1 gdm cric kdt quA (o, b, c) trong d6 a, b, c kh6c nhau d6i m6t. Nh6m niy cri 6.5.4 = 120 phdn tit. Nhdm hai gdm cac kdt quh (a,6, c) trong roAN rrec TRoNG vdN oii d6 a = b = c. Nh
TruyQn Cu Khoai - V6.y thi em chua Mc vQi. Hay thay 8 ldn 30 vi6n bing 4 ldn 60 vi6n r6i l4r thay bing 2 ldn 120 vi6n hay I ldn 240 vi6n. DAy c{u a ! Cu Khoai v&a n
N cAr cAcH DAY tilffiNE VA Hgc roAN truc xv cia AD . Goi F h pnep 6al xrJng truc.ry, ra coF:A-I);B-C hay P? = fi. Ap dung phAn h) dinh li tr6n, la co F li m6t phdp rinh ridn (l). Nhuns ta cfins co F 4,' b;C -? nhuns AC * DB 1vi cit nha[ : TRINH nhy ruHt, VAY tai I) ndn dp dung phAn c6 rsoA DANc crrANG ? a) dinh li rren, ta co F khcrng phAi Ii m6t_phdp tinh ti€n, mAu thuan vcii DANG KY PHoNG (r) (!) Sau dAy lA m6t g0i 1i vd c:ich trinh biy khAc: "Dinh li : Trong.SCK HI I - l0- t990 (ddng rr{c gii Nguy6n Cia Cdc - Cho hai di6m hdr ki A vi A Trinlr Th6 Vinh) treng 49 co n€u dinh ti: "Cho hai diem hdr ki A vd M a) V6i mqi cIp didm M, M'n€uA, M-theo rh( ru le enh cia A, M trong m6r ph6p tinh ridn thi Af,l" = ,{fu Ndu ,4', A' 4 trglr tA a{r crja trai didnr A. M tron1 mnt- a). phep tinh tidn nti AM : A'l9l' b) Dio lai, ph€p bi6n hinh F-bi6n di6m,4 thanh didm A' . , b) Dio lai, phdp bGn hinhlidn hai didm,4, M thAnh hai vi m6i di6m M thirnh mQt didm M' rudng t.rng sao cho di€,m A', M'sao cho AM : A'M' In mQr ph6p tinh tidn,,. A'M' = AM li mdt phdp rinh ridn. C6ch dien dAt nhu tren lA chua ranh mach-.ThAt v6y. vi A, Chitng minh : a) (ban cloc tu ridn hAnh) M li hai di6m da cho n6n x6c dinh. Nr,r'r rav,-Jd ifra. ,ii, OiC,i . b) Vi cap ditim 21, ,4' da c!g. inh z1' cta ,4 trong ph6p ki€n 46u _trong phAn b) chi qin chon hai di6m l, M' niro tlo la hidn hinh li.duy nhir nCn vecrd.4,4' lA m6t vecrd xic dinh. Hon co th0,kCt luAn F lA fQr p!..p rinh ridn_. D0 hdc bo di6u nAy. ra ntta. moi diCnL4/ lrengAr phiry rldu co inh M' rtrctng irng sro xet phdn vi du sau diy. "Cho hinh chil nhAt ABCD. X6c 6inh cltou(, '' AA'(doA'['l'-:,4(). negtheo dinh nghia phdp q$p t,_d*.. hinh hidn.y' rhinh 1). hi6n B rtrinh C hay trnh {ren. ra co /. = 1r.vdi y = AA'(dpcm),, D(- = AB". Goi giao di6m cia AC, BD li I eua / k6 trun! nr6t didm y € S nio dci nim tr€n dudns thans BAI TOAN J.XILVEXTH khoAne c:ich ti P 6i / sG nh6 hon khoAns tdi l. Didu dci # dan tdi mAu thudn do ta chon didm P vd dubnE HOANG OUC tATrT malg lI cci thing thipg ui khoAng co k-hoang cach bang r). Do A G dune Mne crich dung dung d-SsS va do xa eia thi6t reh I cd it nhAt IAB didm.cug ag eiA didm ctia t6n t4p -C_gij thd ki 19, nh) todn hoc Anh ndi tidng -li J.Xilvextr S, Sr.n6n.ta n6n-ta suy ra ring ffin Fgy-ra thing I y6 d.tqng thine h€n dubns v6 mtjt m$t dA dat ra bei todn sau ddy. phia nio dcitria di6"fi A so c
oUANG CAO a^? D6n dsc VAN I{OC VA TLIOI TRE, ban thAn thidt cua.hoc pinh vd nhd giSo cing d6ng ddro ban -vdn trong ^NgLIdi y6u va ngoAi nhA iruong. -TAp.MOt, ra vao th5ng I na1n 1994, .ddy 6.a.tranB, vgi nhidu bAi hdp dan cua cac nha van, nha grao... va noc stnn vtet ve: o CAc 6ng vin hay o Cac nhA vdn trong cAm nghi cua mgi ngLldi o Chuy6n n6n oidt - S6ch cdn dQc o TtJ li6u moi vd van hQc... Gie 22o0dfi bAn. _) Tim mua tai c5c C6ng ti S6ch ve Thidt bi trutong hoc, c6c dqi li var br-tu di6n. U GICX THIEU SACH. . . (Tidp theo bia 4) Thttc hanh ki thudt hcia ndng (1.250d), Thuc III. S:ich ngo4i ngU - Tir didn hdnh bii tdp hda hfiu co (10.800d), Gizio trinh BBC Beginner's English (Li thuydt) T1 dung hcia sinh hqc (7.700d), Thtrc hdnh hcja hitu co cho sinh vi6n, hs (15.000d), BBC Beginner's (D+i hoc srr pham) (3. 100d), Co sd Ii thuydt English (Li thuydt) T2 dirng cho sinh vi6n, hs hrja T1, T2 (4.000d, 8 000d), Gi6o trinh co sd (15.000d), BBC beginer's (Bdi tap) T1 dungcho li thuydt h6a T2 (4.000d), Truy6n kd c6c nhd sinh vi6n, hs (6.500d), BBC Beginner's English b6c hoc xudt s5c (3.400d), Sd tay Torin Li Hda (6.500ciu, Pra"ctice Tests T1 (15.000d),,Practice (3.000d), Sinh thrii dai crlong (3.000d), Di truydn Tests T2 (17.000d), Tidng Anh - cric l0i thudng dai crtong T2 (1.280d). Bdi tap di tmy6n g4p (20.000d), Hr.t6ng din thtic hdnh giao tiSp (1,988d), Girio trinh sinh li cAy trdng (14.000d), Vi6t (9.000d), TruyQn cd tich Anh (song ngrr) b5o d6ng m6i sinh (3.000d), Truy6n k6' cric nhi (4.000d), Sd tay ngudi dich tiSng Anh (17.000d), b6c hoc Sinh hoc xudt sac (3.400d). Ceic loai hinh thi vi kidm tra tidng Anh (15.000d), Ngrt ph6p tidng Ph6p, T3, TiSng Anh II- Sdch ki thuat Trung hgc chuy6n - kiSn thiic cdp 2, T2 (3.350d), Tuydn tAp cdc nghiQp vh d4y nghd bdi Test ti6ng Phdp (3.000d), Luydn tiSng Phap, T1 (1.620d), Gidi trJ tidng Anh (4.000d), C6c Vo ki thudt co khi (Trung hoc chuy6n bdi Test tidng Anh (1.800d), TiI didn b6ch nghiQp) (7.500d), Ki thuAt di6n (Trung hsc khoa - hcia hoc tr6 (4.000d), Tit didn thuAt ngu chuy6n nghiQp) (9.500d), Ki thuAt di6n tit (1.500d), Co ki thudt, T1 tTrung hoc Chuy6n Van hoc (bia ctlng) (12.000d), Tit didn girio khoa tieng Vi6t (9.300d), Tt didn giao khoa tidng nghi6p) (7.500d) Cdu tao sita chlta di6n 6t6 Vi6t (bia cfng) (11.900d), Trl didn thuAt ngtt (10.000d) Ki thuAt c6t may (15.CC0d, C6ch tin hoc Anh - Vi6t (30.000d), Thudt ngtt Anh - stt dung sila ch{ta tir lanh. Di6n ttl th6ng dung T1, T2 (3.50d), (4.00d), Sd tay Tho sira chiia Vi6t (gi6o duc) (3.600d), Tt didn Anh-Vi6t hi6n dai (40.000d), Tit di6'n M, - Vi6t (2.000d). co khi (10.000d), Ki thudt rbn (5.800d). Hrt6ng din day nghd ngudi (2.000d), Cac bai thuc hanh Tt didn H6n-Vi6t hi6n dai (50000d), Tir clidn v6 ngh6 n6 (3.500d), Hdi d6p v6 ngh6 m6c Vi6t-Hrin (55.000d), Tt didn Nga-Vi6t (3.000d), Tt didn \lhat - Anh - Vi6t ngoai (4.000d), Ttt hoc d6nh m6y chtt (6.500d), Sd thuong (40.000d), Til didn Anh - Vi6t (khd tay hudng din v6 kI thudt (3.300d) VO tuydn nh6) (12.000d), Tt dien Ph6p-Vi6t (30.000d). di6n ttl hi6n dai (2.300d)
OUANG CAO TRIJOI\TG MARIE, CIJRIE, Hil IYOi Dia chi: Khrrong Dinh - Thanh fri - ftar NOi. DiCn thoai : 584844 o Thdnh lAp ndm 1992 dttdi str bAo tro cria HQi Vat li Vi6t Nam. oHi6n c