Tạp chí Toán học và Tuổi trẻ: Số 225 (Tháng 3/1996)

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Tạp chí Toán học và Tuổi trẻ số 225 tháng 3/1996 gồm những bài viết như: Một cách tìm hiểu lời giải của một bài toán, họ đường thẳng luôn tiếp xúc với một đường tròn hay Cônic, một đẳng thức mới trong tam giác,... Mời các bạn cùng tham khảo nội dung tạp chí để nắm bắt thông tin chi tiết.

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 225 (Tháng 3/1996)

Nffi Bo GrAo DrJC vA DAo TAo * HQr roAN HQC VIFT NAM ,,*rortiiiffi iiil :: :. NAM .,:.1" 31rru; "" 1996 rrnl sa , tap cni RA NGAy 15 IIANG TI{ANG I MEr cdcH rim NHIEo r))^ LoI crfir cof, MgT B6t TO6N HQ DUONC ffidnC ^.? LUON TIEP XUC VOI MQT DTjONG TRON HAY CONIC a MQT oArrlo rHfc uol TRONG TAM CIAC * oE rrrr r{ec srNrr Gror LoP lz quAxc xcAr * TRO eHor Dorlr{ cflu Co gioo Lc lltr .\:gatr, tltot,gioo Dattg Qtrortg Hcri L'ict ct'trrg doi tu.vAil HS gioi lop I Chuvin toatt ci)a Ttl rd lTcli Drrong
TOAN HQC VA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trartg Ddnh cho cic hun Trung hoc ctr sti For louer seconclory sclrcol leuel. ftiends La Qu,ang Tntng =. X'IqI c6ch tim nhi6u loi giAi Tdng hiAn fiP : ctia mdt bii tr-ran. NGIJYtjN (]ANTI'|OAN Gini bdi ki trrdc Phd tdng biAn fiP : Solutiotts of problents in preuiott,s issue NGo I),\'l 'f(l C6c bdi cria s6 221. I lOAn-G (llltjn-G D'Aru ki ni1' Problertts iit t!ti.s issue Tu225, . ., T10225. Lll225, L21225 HOl oOttc atEN tAP : D&nh ,tlto cic hurt chudn hi thi vio Dui hoc For cotlcge ond uniuersity entrance danL preparers' Nguy6n CAnh Todn, Hodng Xudn Bang - Chfing minh rnQt ho dtibng Trd,,ru Chung. Ng6 D4t Tfi, L6 Kh5c th&ng ludn lu6n tiep xfc vdi mdt dttdng trdn BAo, Nguy6n HuY Doan, hay n+6.t conic cd dinh. 10 Nguy6n Vi6t Hai, Dinh Quang a DA thi tuydn sinh Dai hoc B6ch khoa He N6i 1995. 1,2 H.1o, NguSr6n Xudn HuY, Phan a Httc sinhlim titi Huy Khii, Vu Thanh Khi6t, L0 Young friends' search in Moths i{ni Khoi. Nguy6n Van Mau, I{guy6n Thd PhtonC - Mqt bit cl6ng thrtc rn6i Hodng L6 Minh, NguY6n Kh6c trong tam gi6r:. 15 Minh. Trdn Van Nhung, Nguy6n Dnng Phdt, Phan c Dd thi hoc sinh gi6i torin l6p 12 Quing Ngdi. Bia 3 Thanh Quang, Ta Hdng a i *ian bun drc. Bia 3 QuAng, Dang Hung Th6ng, Vu o Cidi trr tourt ltut'. Drrong Thpy, Trdn Thnnh Fun tri tlt Motlt,'ttto Ii < s Trai. LO Ba Kh6nh Trinh, Ngd Binh Pltttorzg - Giii drip bni Vui nam m6i Bia 4 Vi6t Trung. Dang Quan Vi6n. trtrgo HarL - Trd choi dodn cdu Bia 4 o Bun ci bidt ! Do you know ? 'ludn Thanh. Triliion id gr Ria 4 Trry s6 toa soan : DT: 213786 BiAru fipud. tri s4 : VU KIM THITY 458 Hhng Chu6i, He NOi 231 Nguy6n Ven Cil, TP Hd Chi Minh DT:356111 Trinh. bd.y : TRONG THIEP
I - S",, q cti\!ithdliqr tt dugc ldi giAi cria mdt bdi to6n, trong nhi6u tnrdng S G= Igq, (hoac khdc \dt'quA h6ac crich d., md suy Sihi rifr,r"g giaT hidu dtrocvi sao ccj cricidch giei khdc)."" Sau diy"a"f, m6t vdi vi du minh hoa. xin i6u GD -, tr- z I Blri todn l. Phdn tich d,a thtc 13 - 7x * 6 thimh nhdn ti. Gidi: f(x) = x3 -x -6r + 6 = x(x2 - 1) -6(r - 1) ru ES = (r - t)(xz + x- 6) = (x - t)(x - 2)(x + 8],. 4 G= 0, vindu 1-r < (4) 0trtcr 1thir2 jP +zz > l,trrii > r-{ _.Blrli:O + v6i (Z).Tuong tu : 1 -y > 0 vi L - z > 0. Do dti @) iayra khi vi chi khir2 (1 -r) = y2 (l -y) : (l - z) = 0. Suy ra m6i s6 x, !, z bing"2 0 ho{c 1, tr} dci c
K6t hopvdi (1 +r2) i d suy raZi d. Mad le s616v6yd:1. Vi ( L + r)( I + 12) la sd chinh phttong r do (2)) nrd (1 *x, I *x2) = 1 Viy 1 *r vA 1 +r2 d6u phdi lnsdchinhphrrong. Tathdyx2 .x2 + lla2 B,diTllzzl. Tint s6tq nhian n ldn. n hdt sao sd ttr nhi6n li6n tidp m?r d6u lis6 chinh phuong. cho s6 1995 bd.ng tdng cia n s6 a, 1a2, ...ran Suy rax = 0, trorug do cd.c sd a,(i = 1,2, ..., n) dbu ld. hqp s6. Khi dd theo (1) thiY = 0 hoacY : -1' Ldi giai : (cria ban La Minh Dic,l6p 7\ VAy ta c6 2 clp nghiQm cta phtrong trinh Ii (0, 0) vd (0, -1) Thudn Thinh, He Bdc) Ta cd a, * a, * ... * an = 1995' Trong c6c s6 NhQn x6t. M6t s6 ban cung giAi dugc bai niry nhung trong lAp ludn cdn dii ddng ho4c o; phii c 9. Vay Nam D!nh, Nam Hi ; L€ Thd.nh Cdng, 7T, Pham Huy Quang, D6ng Hung,'Ihai Binh 1995 = or *... * o,, 24(z - 1) +9+ TO NGLIYtTN Bei T3/221 . Clulng minh bd.t dang thitc : 1995> 4ru+5:+/r +n 4 497. b-bc+c c-ca+a a-ab+b' Y6in = 497 tathdy ab *bc *ca 1995= 4+... +4+6+9 '- il-a+b+c . 495 s6 trong d6 a, b, c lit. cd.c sd duong. Ldi gini : (d$atheo Mai Nggc Kha - 9 Torin Yqy n ldn nhdt ld 497. Trdn Dang Ninh TP Narn Dinh) Ta cci cric Nhan x6t : Cric ban sau dAy cci ldi gi6i t6t : m6u thrlc d vd trrii d6u dtrong, ching h4n, Nguyon Minh Phuong 9A Vi6t Tri, Hi Xuan Gi6p 6 Thanh Hda, Biii Vidt Ldc 8A BdVan Dan bz - bc # c2 = ? -;)' * r'i >o; hon nr1a, a, Ha NQi, Pham Vtr Toin, 6 chuy6n Tit Li6m Hd 6, c ddu duong n6n d4t A In bi6'u thtlc d vd triti NQi, Ddo Quang TiSn 7 Th6i Binh, NguySn vis = 1a2 +b2 *"2)2 ,taccithdti6nhinhbidn Trong HiSn 7 Hi Tfnh, Trdn Thi Thty 9T Hn ool sau oay : Tinh, Trdn Tdt Dat 8A Chu Van An Ha NQi, Trdn Tudn Anh 8 Torin L6 Quy D6n, Nha S: Trang, Khrinh Hoa. I Cd hai ban cci drlp sd sai. M6t s6 ban cho drip 63 sd dring nhrrng I4p ludn chua chat cho + {Qr:iaaffi + c'z - ca} 0z DANG FII,NG TTII,NG 'c3 Bni T2l22l. Tim nghi€m nguy€n cia wr-6rrry)' phuong trinh o'z -ab +b'1 x(l +x +x21 :4yO + l) Ti bidu thrlc niy, 6p dqng bdt ding thfc Ldi giai : o&a Cao Xud.n Hba,9Ar, Hdng Bu-nhi-a-kdp-xki, ta c sdl6. M[tkhectt (1 +r; i dsuy ru(l - x21z d. > ab3 + ac3 + bc3 + ba3 * ca3 + cb3 (3) 2
KhOng lim rndt tinh tdng qu6t, girtr st Nhgn x6t: a>b>-c)0,tacri: Giai tdt bdi ndy cci cdc ban : a2 (a - b)2 + (a - b12 (.a+bxb - c) + Nguy6n Nhrr Chudn, SNK Thudn Thinh, +c2(c-o,)(c-b)>0 IId B6c, LO Ldm, 9 To:in Chuy6n phri Tho, oa4 _2a3b +a2b2 +u?b _a3c +ct2b2 _ Nguy6n Minh Phuong, gA Minh Phtrong, Vi6t - a2bc - 2a2b2 + 2a2bc - 2ab3 * 2ab2c * Tri, Vinh Phu, HodngHii An, 9A Chuy6n U6ng Bi, QuAng Ninh, Pham Thu Hrrong 9A,, Hdng + b3a_ b2ot * ba_ b3c+.c4_- c3b_ c3a* czab > 0 Bdng, II&i Phdng Biri Vidt L6c, 8ABdVin Din, * abc(a + b * c) .==:o4 + b4 + cA >_ Nguy6n HoingLam, 8A, Chu Van An, Hd N6i, > ab3+ac3+ bc3+ ba3+caj+ ct3,trlc ld (B) dirng. Cao Thi Ly, 9 Nang khidu Vu Thu, Thrii Binh, Kdt hop (2) v1i (3), ta c6 A> a *b *c. vd Bni Thi Kh6nh ThuAn, 9 Chuy6n NK Y ycn, chi cbn phAi chrlng minh : Nguy6n Trong KiOn, 8T, Mai Ngoc Kha gT ob*bc*co . ?rdn Dang Ninh, Nam Hi, Mai Thi Thu S6nh, o *b *c > lt-;-,- (4). ThAt vAy, ta 8A Nga HAi, Nga Son, Thanh Hcia, V6 Si Nam, A-rO+{_. 9 Chuy6n NK Drl'c I'ho, Hi Tinh, Trdn Chi Hda, de bidt a2 + b2 + c2 > ab * bc *ca n6n 9CT DAo Duy Ttr, QuAng Binh, Nguy6n Hoing (a+b+c)2 > 3(ab+bc+ca) ; rnd o *b *c > 0 ThAnh, 9/1 Nguy6n Hu6, Dd Ning, QuAng Nam n6n ta cd (4), suy ra dpcm. Da Ning, Trdn Tudn Anh, 8?, L6 euy D6n, NhQn x6t. Nhi6u ban mic sai ldrn cho ring Nha Trang, Kh6nh Hba. s6 nao nhAn vdi s6 b6 hryn 1 crlng giirn, thAt ra di6u d x2 + 2dx = ZaZ -111 haY PB - MB:; (1) '+ (r * ci)z = 2s2 4" 4z Nhu vAy x + d ld canh huy6n cria tarn gi6c Trrong tu, ta cfing cd vu6ngcd canh gcic vu6nghang at[2 vdd. Tt dd suy ra cSch drrng doan r. Cach dgng : *.;u=I e)ui,#.#:; (3) DUng dtidng trdn tAm O dubng kinh BC (da cho tnr6c). Dr"rng LBOM vu0ng can crj OB : OM (= o). DUng LMBE vu6ng d B c6 BE : d. t{ U11S nrta dO ddi phdn gi6c da cho). Ldy / sao cho EI = d. MI chinh tn do dai r phAi d1rng. Tir {d .yy ra giao didm D cria ph6n giric vdi canh BC. Ndi M vdi D cdt drrdng trdn (O) 6 A, Ta duoc AA.BC cdn dting. Chtng minh : Dinh cho ban doc.
Co.ng (1), (2), (3) v6 vdi v6 rdi bidn dcii, drroc .'*+ t) ] ,l 1. ,t 1. ,l *cN)*\"*- I, Suy ra : 1 < 6n * < Vn € N- (1) (ap * BP)*(^, 2n@ Ztt Bil= 11 111 n-l*z :1+- L - -+-+- nLtLp T[d2't = (2.1.1...1)/, tLn ABACBC11lI- (n * l) sit - PA.PB ' NA.NC MB.N|C - nt n p N6n tr) (1) suyra 1 < b,, . I * :Vn € /V-. (2) 1 kh6ng ddi. Nh4n x6t. C6c ban sau diy ldi gi.{i t5t : cci :\ = I ndn trl t2t tadudc 1 Bni Vidt LOc (8A TFICS BdVan Din - He NQi), Dolinr = lirni\ t + nl 1 : tt'+q tt-+a Trlin Thi Thny (9 Torin Nzing khidu - I{d Tinh), t. Pham Tudn Anh (8A Luong ThdVinh - Ha NOi), ,r]*r: Mai Thri Au (9 Tori,n Trdn DAng Ninh - Narn Dinh - Nam Hi), Trdn Tudn Anh (8 Torin L6 Nh}n x6t : 1. GiAi trrong ttr nhu tr6n cdn * Quy Don - Nha Trang Khrinh Hba). c6 cdc ban : Hd Duy Hung (11 chuyOn Tin - PTI.{K Triin Ph(i, FIi.i Phbng) : Nguydn Quang t>ANc; vt lrN Nguyan (11CT Ha Tay) ; Nguydn Xudn Son. Bei T6/21 l. Cho ba s6 x, y, z nguy€n tlttang Duong Van Y€n, Nguydn Cd.nh Hdo (llCT than ntd.n hQ thtc PTTH Phan BOi ChAu, Nghc An) : Nguy4ru ,1 +y'I*za=1981 Minh. Tud.n (11T PTTH Dio Duy 'fr), QuAng Chung ntinh riing sdA : Zff + 1lv - 19961 Binh) vd Ld Anh\\7 (lzCT Qudc hoc Hud.). httOng thd lit. ticlr. cia 2 s6 t4 nhi€n lian ildp. 2. Ngoni cdc ban de n6u t6n d tr6n, cd.c ban Ldi giai : Sd A 16 ra phAi li sau diy crlng cci ldi giAi tdt: Trdn Hodng ViQt A:20x+l1Y-19932 (12A PTTfi Chi Linh, HAi Hrrng) ; Nguydn SI Vdi sd A nhu dd bei thi bni to:in li tdm Phong, Nguydn Qudc Thd.ng,VuViet Atul, (10A thttdng vi A khi dd hidn nhi6n ln sd 16 vi do dri PTCT DHSP Dai hoc qudc gia He NOi) ; L€Van kh6ng thd Ia tich cira hai s6 ttl nhi6n li6n tiSp Cudng, Nguydn Ngqc Hung (11T Larn Son, udi nqi x, y, z e, N*. Thanh H6a), Luu Trudrug Huy (LZA PTTH Ba Hon nttamQt s6ban (DuongDinh Hung 11A Dinh, Nga Son - Thanh Hda) ; Hb Huu Thq Nga Son, Thanh Hcia, V6 Thanh Ting, 11 Qudc (11A PTTH Nghia Ddn, NghO An) ; Nguydn. hoc Hud, D6 Bich Diep 128 chuy6n Hria DHTH, Minh Tudn, Trdn Thanh Tn (.llCT PTTH Ddo Nguy6n Nhrt Chudn 8NK He Bic, Trdn Htru Duy Tit, Quing Binh) ; Nguydru Nge,c Tud.n Lr-tc 10CT QuAng Binh, Trdn Vi6t Dong 10A (11CT tnlong chuy6n QuAng Tt\) ; Trd.n Anh trudng LA Quy D6n, Dd Ning, L6 Van M4nh, Lud.n, PirunAnhHuy (11A, PTTH LO Quy D6n, 11T Nghe An) da chfng minh rahg kh1ng tbn Da Ning) vd Nguydn NltQt Nanr (12At PTTH tai ba sd x, y, znguy€n duong th6a md.n h.4 thtc Vtng Tiu, Bn Ria - Vung Tdu). 14+y4tza=1g84 3. Ban Nguy1n NhQt Nant di chfng minh DANG rruNCI'H^NG dring khing dfnh sau : "Vdi {o,.,} ld day duoc xic B,niTll22l. Cho cac day {an}, n c. N' ; dinh trongbii dd ra, vd v6i & € N* cho trudc ta I { 4,.,} , ru €. N* thda ntd.n cd.c cing tlulc sau : n(l n\ -t nt'(l ntt\ * """'l"-t"o 'o")@i = ' a_=1+-- +... ++Vn€N' " l+flt l*n't 4. Dd giei bai da ra, kh6ng it ban dd cd nhi6u cd ging trongvi6c dua ra c6c drinh giri khri phfc 6n= ("+l),,(,r+l) Vn €N' tap vi vi thd ldrn cho ldi giai trd n6n nldm ri, ,rn c6ng k6nh. ,::::n", 5. MQt sd b4n dd pham phAi nhtng sai ldnr La,i giii kiaPhan Duy Hing,12CT THPT v6 ki6n thfc co bAn, ch&ng han : Dio Duy Ttt - QuAng Binh) : D6 thdy cd : o Tr) r, < lnYn € N' suy ra , = "1(t *ll < zYnG N'vi Yh dd =Tp.Do ,,Y'=,,l]}r,'{t')' l*nu trl cdng thfc x6c dlnh on ta cci : a,, 1 : nkhk+L\ n * 1 < an4 2n * t < 2(n* 1) Vn € N' r Tt ,a1=ra1 *irti*rXn* 4'
ct Vfi ViCt,{nh, Nguy6n Qu6c Th6ng (DHSP), Bni suy ra linr --l= n-t1 : Vi6t Ldc (Bd Van Dan). --, TP. Hb Ch; Minh: Nguy6n Ngoc Bi6n . : nktl+nkt IIdi l>hdrug; Hd Duy Hung, Ta Thi6n Todn, = lirrr:i- n'r t +) u, 1;n., --:-\'- ' ];. ( I ?). Nguy6n Phan Anh, Pham Drlc Anh, Nguy6n tt-e k= lt -*(n*r;(l+rlj.; Flrtu Tudn. N(,i:yirN Kri,t(.M'NIt Hdi IIung; Nguy6n Qu6c Khrinh, Nguy6n Bei T8/221. Ching ntinhbdrdatry tlutcsau ddy Van Luet (Phuc Thinh), Trdn Hoing Vi6t, tL?-+b) b2+c2f, c2+a2 ol+bz+c2 Nguydn Thanh Tnng (Chi Linh), L6 Dai -r__ ____0 c:t Trttdng l{u.y, L6 Xuin Dung. .. b *c Qudng ?ru : I{odng.Vi6t Phtrong (D6ng HA), oc(c - a\ bc(c - b\ Nguy6n Ngoc Tudn, Ngrry6n Vi0t Tidn (\,inh Lituh) ]_.,a a+b a*b ' Qudng Binh : Nguy5n Minh Turin, .Ird.n , ablb t-:--+ - o) 6c(6 - c) Mai Son Ha, Bui Ngoc Thanh, Phan HuyI{r)ng, + LC Chi I'ho, Pham Hdng Thrii. c +a c *a ab(o -b1 . ac(a -c\ Thita Tliin. - Hud : L6 Anh Vu, Doan Xudn T, ---, :-- b*c r -;-------b+o 2 u Vinh. V0 Thanh ?r)ng, Trdn Thanh euang. Dit Narug..Phan Anh Huy. ac(c - a)2 bc(c - b1z Nant lla; D6 Ngoc Anh, Vu \,Ianh HAi. (o +b)(b +c) (a+b)(a+c1 Vinh Phi: Nguy6n Minh Phtrong. ab(b - at) + (a+c)(6+c)'" ' )0 Htt, Tay; D6 Anh Tuan, Nguyen Anh Nguy6n. Thd.i Binh.. Trdn COng Cudng; Nguy6n NhAn x6t: Ngoc Khrinh. . 1. Nhi6u ban cbn cd nhi6u cich giii khtic bing crich srl dung c:ic trdt ding thrl-c co bin Phi Y€n: Trinh A quO". Khdnh }/oa: Nguy6n L6 Hodng Anh. Bunhiacopvski Dbng Nai; Pham Hoii Lang. . 2.llvIQt s6 ban dE rnt} r6ng vA" chrlng rninh dtng bdt ding thfc dang tdng qu6r I Tibn Giang; L6 Van Lai. Cho or, o,.t ...: an > 0 ; dilt Qud.ng Ninh: Ngr.ry6n Hoang C6ng. Hba Binh: Hd Khdnh Todn. Sl :0r 4ar*... +on; 52: a:r+a)+. +fi NGUYT]N VAN MAU Khi di B.di Tgl22l. Gid sr2 O ld, nt6t didnt nant " s'-o,- (2.-:. T+-* s2 trong tant giac ABC. Cdc duitng tharug OA, OB, OC cat cdc canh BC, CA, AB ldn laot d A,, 8,, i?'St-'i -'"'sl ' C'. Tint fip hqp nhttng di€nt O sao cho : 3. Tda soan d6 nhan duoc cric ldi giai dring s21oAC'1 + sz(oBA,) + s21ocB,1 : cta cric ban = s?(OBC,) + y2(OCA,) + s21OAB,) Ngsc {ol I Ngoy6n Tudn Ngoc, Trdn Nguy6n Ldi giei (dua theo Nguy6n Hiru Cdu, 10CT, _(DHKH?N), Ngrry6n XuAn Nfrry6n, trudng PTTH chuydn Ddo Duy Tir, QuAngBinh) Trdn Tdt Dat, L6 TrongThu6n, Nguy6n HoAng Lam (Chu Van An), Dinh Trung Hing (Mari Det : s(OA'B) = s, s(OA'C): s'l i Quiri), Nguy6n Htru Cr:dng, Nguy6n Si-phong, s(OB'C) = sr, s(OB'A) = s' 2 ;
s(OC'A) = s3, s(OC'B) : s'3 ' nghia li li mqt horin v1 cua (s t. s2, s) ' (s'1, s'2, s1.) ve : s, *s', : :x rsZ*s', :.1rs3 * t'3: r. (Dinh li Vi-6t v6 tinh chdt cdc nghiOrn cria phttong trinh bAc 3). Thd thi ta drtoc : X6L cdc ho6n vi khSc nhau cria hai t4p hqp sl ^^t 5l OA' s6 tr6n, ta di d6n kdt lu4n : Mu6n cho didrn O :-:-= nim trong tam gi6c ABC th6a m6n di6u ki6n I sr*s'r' sr-ls', OA (*), cdn vd dri ld O phAi ndm tr6n mQt trong ba s, * s', drldng trung tuydn cria tam gl1'c de cho (tril c6c dinh vd c5c trung didrn cria ba canh). NGUYF,N DANG PIIAI' ^zx,xy Duy ra: s t: E t: Blin rclz?L.Trong kh.1ng gian clw luti duimg ,n, ,4, thang chdo nhau a uit b. M uit. N lit hai didnt !) Tudng tU I : chuy'dn d.Qng ld.n luot ffei,ru a uir b sao cho ctudng xy,yzyz,2 th&ng MN hop udi a ud. b nhftng g6c bd'ng nhau. "t z*x'- t z+-x'-3 x*y'- s x+y Tint quy tich trung didm. P cia doqn th.ang MN. Tn gia thidt : s2, + s2, + s! = s'l + s'!: s'3 (*) Ldi gini I : Goi AB ld doan vuOng gcic chung cria o vi b, Thay cic gra tr1 cria s, vd s', (i = 1, 2, 3) vdo (*), A=AB La,B:AB tb, ra duoc : (*)
phingsong song cdch ddu hai dudng thang ch6o Nh{n x6t. Cdc bqn cci ldi giei tdt : Nguy6n nhau a vd b) vd drra bai toan tim qu! tich trong Dinh Thinh, 1iL, PTTH Phan B6i Chau, Vinh I kh6ng gian v6 m6t bAi todn tim quj tich trong rn6t phing don giAn hon, Crlng trong ldi giAi NghQ An) ; Ta Thi6n Todn, 12A,, PTTH Trdn Nguyon Han (Hai Phdng) ; Nguy6n Van Hidu, tr6n ddy, chiing ta kh6ng cdn tdch bach hai 11A2, PTTH chuy6n Lc Quy DOn (DA Ning) ; phdn chrlng rninh thuAn vd dAo ; hai phdn ndy Phan Van Binh, 11 Nguy6n Hu6, Tuy Hda, Phri Y6n ;Nguy6n Bi6n Thty 118, PTTH Bim Son, dE diroc trinh bdy gQp tr6n co sd v6n dung didu Thanh Hda; L6 Ven Lai 12D, PTTH C6i Be, ki6n cdn vA dtr cria di6u ki6n ddt ra cria d6 toan. Ti6n Giang ; L6 Quang Thdnh, 11CL, chuy6n 2") NgoAi crich giAi tr6n (phrrong phrip tdng QuAng Tri, LO Trdn Thd Duy, 10L, LO Khidt hgp) bdn ban sau dAy da giai bai to6n bing QuAngNgei, PhungDuy Hung, 8.118, chuydn phrrong ph6p v6cto, dd ld cric ban : Dinh Trung Li, DHTH Ha NOi ; Dodn Dinh'Trung, LlL, Hing, 12M Tnrdng dAn ldp Mari - Quyri Hd PTTH He NOi - Arnsterdam ; Vri Manh HAi, l' 12A, PTTH Duy Ti6n A, Nam Hi. I NOi, Pham Minh Hoing, L2A1, PTTH Kim Li6n, Hi NOi, Ltru Trtrdng FIuy 12A,'PTTII Ba BdiLzlzzt. cho nrqctt di4* #i) fif;Li rn Dinh, Nga Son, Thanh Hda vd L6 Chi Tho, bidt Et = 70V, tr: 12; Ez = 6V, Rt = 2rr. Khi"no LZCT, Dio Duy Tr), QuAng Binh. Kr.uit. Krd6ng, A, ctti 3,6A; K.hi Krd.6ng A, Ldi giii 2. Goi a*, a-"a c Ia cac v6ctd ddn vi cni glzA. Khi K3d6ng A, chi 2,sA, Ar chi chi phtrong qia cdc dtrong thing a, b vd tAB1, r6i bidu di6n MN rheo i,6'"it ri tlAtfri di6u ki6n 3A. Bo qua diAn bd ctta dd.y ndi uii antpe kd, hay ctra bdi to6n dat ra duoc bidu thi bdi df,neo thfc - : tinh Ro; Rr; uit. eui,ng ctQ ddng diQn qua Rrhlti lu?;[: lMJraJ Krd6ng. Tt dd, ta cring thu dtroc kdt qu6 nhti d6 chi ra trong ldi giAi 1. 3") MOt sd ban dA b6 s
B,niT71225: Cho him sd rp : R * R DAt A, : {x € R, p(r) : a} I Ar:\x€R,p@k):yy GiA st1A,\A, ld rn6t tap hgp htu han vd tdn tai hdrn sd/: R *R th6a man f(f(r)) : 9k') vdi moi r € R. Chfng minh ring sd phdn trl cria A2\Al [i rn6t sd nguy6n chia h6t cho 4. I cAc lop rHCS NGUYEN MINH DUC. BidiTLl225: Giii phuong trinh. @d N i). rr + 1ari5 - 7)i + s2 - 28,[s = Bai T8/225 : Cho dEY b, : O,b2= l4,br= -18, 6nn I = 7b n-t - 6b n-z' Chrlngndnh rdng, vdi moi : (84 - tz{i -3r2) \i7;7i6 s6 nguy6n t6 p cd 6,, chia hdt chop. rRArr rriiNr; sox (7hdi Biilh). oau vAN NHi (Thdi Btnh). B,di T21225 : Giii phtiong trinh nghi€rn nguy6n dtrong (dn x, y, z) BdiTglzzlo: Cho hai dttbng thingr, y ch6o nhau c6 dinh. C6c didm M, N thay ddi tr6n r, x2 +y2 _2077tqesk *t (10 _z)(h€N.) c6c didm P, Q thay ddi tr6n y sao cho MN - a, ucuv6N NGec Kr{oA PQ : b. trong dd a, b ld cac do dai cho trtl6c. (Qudng Ngdi). HAyxac dinh vi tri ctaM, N, P, Q dd cho bAn kinh Bdi TSl225 : Cho o,b,x ,y ld cic sd thgc cira hinh cdu n6i tidp tt? di6n MNPQ li ldn nhdt. {*' .vt I thoarndn. u: o+b I I()AN(; NG(J(l ('riNI I l;* l)) (l ld 'linlt). [x- +Y- = I Eni T10i225 : chrlng rninh ring,# .t#:d* Cho ngu giric d6u AA/.1A, canh bang o vd m0t drrdng thing (D) trong rnat phing chda TI
M vd B vi. ur* gitta B vi.l[ l6ch pha vdi nhau ngang, d0 dei mring Id L rndt gcic ld p= 90 d6. Qu! dao cta qud cdu c6t bao nhi6u lin drrdngsinhAB ndu N(;I tYF,N QI.]N N(i IIAI I (tra N0i). ban ddu tr)A quA cdu chuydn d6ng r,6i vdn tdc b6 theo Bin L21225: MQt qui cdu nh6 cd thd trlrdt phuongvu6ng gocv1iAB. kh6ng ma s5t theo mdt mOt rn6ng tru brin kinh ]'I{L,dNC; ViN.T{ DI E,N R mi truc nghiOng ru6t gcic a so vdi phttong narn {Qudttg Binh) has a unique real root, denoted byrn, and find PROBLEI\TS IN THIS ISSI.JE ,,:::, m1225. Let be given a function p : R.-R. Put For Lower Secondary Schools A, = {x € R, p(r) = r.} Ar= {x€R,9(.p@)) =r}. Tl1225. Solve the equation suppose that Ar\A, is a finite set and there rJ + lsrls - O*2 + s2 - 2Nb : exists a function f : R * R satifying f(f(xD = : (34 - l2\[5 - arz; {rZ+zffi. - tf(x)for all r € R. T21225. Find positive integral solutions Prove that the nuurber of elements ofAr\A, (x, y, z\ of the equation is divisible by 4. x2 +y2 - 291lee5k +1(10-z) (ft eN-). TAD25. The sequene (b,,),, :1, is given by: l, Bl225.I-nt a, b, x, ybe real numbers satis$ring : b I = O,b2: 14, b,t -18, br*r:7bn-r,-6b,r-r. l*' .v' 1 Prove that for every priure number p, the jo b a*b number b, is divisible byp. [xj +yr : -1 tl qgJ \)l9A1 9 Tgl225. Let be given two fixed noncoplanar eaet Ur.tv7 (a + b1eq7 lines r, y. Two points M, N move along r, two T41225. Prove that for every triangle ABC, points P, Q move along y so that MN : a, PQ : b where o, b are given lengths. Determine p:.-h2"+hi+h: the position of M, N, P, Q so that the radius of where p is its semi*perimeter. the inscribed sphere of the tetrahedron MNPQ .1 T5l225.LetABCD be a quadrilateral inscribed is greatest. I in a circle. Let M, N, P, Q be respectively the fr: nridpoints of the arcs BA, BC, CD, DA and T1Ol225.In a plane let be given a regular r-| {O,} = MC i AN, lO.} : BP O DN, pentagon AA/.$A, with sides a and a line {or} = QCoAP, {o*1 = BenDM. (D). Let M 1, M2, Ms, Mt, M 5 be respectively Prove that O,O.,O-rO, is a rectangle the orthogonal projections on (D) of A1 ,Ar,A1 ,A1 ,A7. For Upper Secondary Schools Prove that : TGl225. Prove that for every given positive, -) a,rul+ u ptzr+ M integer n, the equation rltli+ M /w2r+ M rlil1 = + ,Jn+l _r11
Dail, eln oahn tlodr litlt,no tqt ht a RAN XUAN SANC (Qtttitr.g Binh) 1. C6c bhi to6n md ddu : ("-l)r+(y-1)2 0) v6 nghi6rn, dinh thu6c rnAt phring O.r:v (116 D6 thi Tuydn ta thdy ngay tQp hgp crir: tlidm M(x,.y) cdn tiur sinh ctra Bd Girio dtrc vi Dno tao. D0 39 - th6a m6n brit phuong triuh : CArr IVat 10
t Ta quan tAm ddn 3) vd 4) vd chi n6u kdt quA : sttM0, nt).D6 ddng vidt drroc phrrong sd nr., giA Phtiong trinh hinh chidu D' c'0.a D l6n rnat trinh cta tia Mt : px - Znty + 2nt2 : 0 (*) *1: o I ph&ng oxy titl::':- vsina GiA st (x, y) ld toa d6 nhirng didm kh6ng l'-" thu6c tia Mt ndo. Khi dd pt (*) v0 nghi€m nz. Tap hqp nhringdidmM mdho drrdngthingD' e+A'-y2-2px
\A 94 DE TlXr TUYEN SrNH DAr HQC 1995 MO}{ TOAN I TRUONG DAIHQC BACH KHOA TIA NOI I i i (Thdi gian lam bdi : 1S0 phrrit) I PHhN CHUNG psiN Tu CHQN B : Cd.u I : Giai phrrong trinh 2sinh(4sinar - 1) : Cdu Bb: Gi6i bdt phuong trinh = cos2r(7cos2?-x + 3cos2r - 4) log"a* -ort Lr2ox-a) > l,olisddrrongkhric 1. Cd.u 2; GiAi hO phtrong trinh Cd.u 9b; 1) Chrlng minh ring trong mOt hinh tri di6n, 4 doan thing ndi dinh vdi trong t6m ctia {x+y+z-7 t" mat d5i di6n d6ng quy t?r 1 didrn, didm ndy chia 1*2+y2+22=zl m5i doan thing dy theo ti s6 3 :1 tinh ti dinh. I[*'-J .. - ",2 2) Chfngrninh rang trong moi hinh trl di6n, Cd.u 3; Tim tham sd nr dd h'6 bdt phuong ndu R vi r li b6n kinh ctra hinh cdu ngoai tidp trinh sau cci nghiOm va hinh cdu n6i tidp, ta d6u cri R >- 3r. Chti thiclt: Thi sinh phii lam phdn chung l*z+Q-JnP\x-Gntz 0 t{ chon b. Cd.u 4; Ti6p tuydn vdi dudng congy-a { 1x, D6p rin cit trucOx tai x : a, c6t truc Oy tai y : p . Vidt Cd.u 1. Phttong trinh de cho phrrong trinh cria tidp tuydn dy bidt dlJ = 8. *8( I -cos2r r , -cos2r; = Cdu 5.' Cho hdm sd f(x) : )e4 - 2ntx2 + 4, nt z )-(1 ld tham sd drrong. Tim gi6 tri nh6 nhdt ctraf(x) = 7cas3Zx. + 3cos22-x, - 4cosx vdi0
khi khoAng (nt +2,nt *3) kh6ngldp kin doan Cdrh 3. AB = f7 rt ,r crich 1. D* 6dil = 3 l-2 ,3m2f , di6u nay khi nAo cting xdy ra vi khoAng (n + 2 , nt * 3) cd dO dAi bang 1, cdn doan [-2 ,3nr2) cd do-dni > 2 Ctuh hhnc. He da cho cri nghi6.m khi (nz+2) > > -2 ho?g_ nt *3 < 3nJ1 * (nt 7 4 ho6c urr- 1-{37 Z m, 1+{Bz ltodc Z t,trlcli.Vnz. Cdu 4..Y : x * : x6c dinh Vr * 0. Phrrong Cach 1. Thuc hi6n ph6p quay tAm A gcic 1 "x *60t', didru Bc n thi f(x) dat g 0 ( 1) Ta cdn chrlng minh Trong LMAB cd AB2 rangvdlx*xtvdx*x2 = MAz + MB2/\ | 6ax*2b ,7 I _\-: - WA.MBcos AMB. 3ax2+hbx*c 2 \ 3or2 i2bx 4cl ^ nhung AMB = l2O{\ +.1\5, - r[7. truec ca 6a(tux2 + i.bx * c)'z < *f*, + 2UZ Mt=Bl =600 => a!fu2*2 * L2abx * fu,c < l*a2xz + l%,bx * AC2 - MAz + MC2 _ 2b2 *2b2 > fu,c - LMA.MC d 1.Vi limy con6nddthicdtiomcin = a + MBC = n - (x. Dinh li hdm cosin rip dung .y+l*e vio s6c LMAC, MBC cho drlng r = . Vi lim y = 1 oo nen d6 thi c
Q VAV phrrong trinh ctia Q ldy - z : O (2). Dd tirn phrrong trinh tham sd ctra (d), ta tim ntOt = lim -1 dicm M trdn (d), ching han M\2,-3,0). Vhy _r+ @ fr+1 phuong trinh tham sd cria (d) ld x : 2 (3), l,-, + I ! : -3 - t (4), z : t (5). Toa dO cria.I li nghiOrn Vdy dd thi cd ti6m cdn xi6n ldy = r * 1 cria h6 (2), (3), (4), (51 + I (2, -*, -Z) = Cd.u 9a. Ddi bidn u = rE=T * Lr2 : x2 - -l n* ee {E4 +OI= I q+ 4 =; I du +Zudu: Zxdx+ I : I I I *u2 2) Cnng cri thd dung c6ng thtlc dd tinh O1 T' CQu 8b. Bdt phuong trinh da cho ccj nghia I khiorx - (o - 2)oz - a > 0 (6). Gia str a > 1. Bdt Jf, 1t Jt phuong trinh da cho a - srcrTul 4 6 tz', (di6u ndy k6o theo (6)) I V6 0=cx > a+ r > 1. r = -+dx 1 dt - GiA sfi o < 1. Bdt phuong trinh da cho + HoAc a2* - \o - z)ax - a < a. K6t-hop vdi (6), ta cci t t2 0r+R>3r, TI
I llgc sirrh tim tdi WMrDfoffioffiffiTAffiffic I Sau khi doc bdi b6o "Md rQng h,6t qud. cila Ta thdy -l6ic : Toricelli" ("To6n hoc vi tudi tr6" th6ng 91L994), t6i nhAn thdy trong tam giric cci m6t bat ding l^ = 18P - fric ^ thrlc ddng chfl y. Il^CMA = 18ff - C'B'A' Dinh li I : Cho tam gid.c ABC. It/I ld m.Qt d.idnt. I uwn = 18op - A'c'B' t trong tam. giat d6. Khi aA udi ntpi didm N tac6 : V4y theo bd d6 tr€n ta cci : sin BMC.NA+ sin CMANB + sin AMB.NC > B' C'.NA + C'A"N B + A' B'.N C >- > ,in 6ic.MA*sin die.ma*rin ffa.uc. B',C',.MA + C',A'.MB + A'B'.MC (l). Dd.ng thnc xdy ra hhi ud. chi khi N trilng uoi M. D&ng thrlc xAy ra khi vd chi khi N trirngM. Dd ban doc ti6n theo doi, tru6c khi chrlng Ap dqng dinh li him sd sin cho tam giric minh dinh li 1, t6i xin nh6c l?i k6t quA co bAn md A'B'C'ta'cd : bai brio n6u tr€n da d4t drldc, xem nhu li bd d6. B,C' C'A' A'B' Bd d6 : Cho tam giac AEC va ba s6 drtongx, y, z. sin.6?c' Suy ra : sin dFA' sin .f?B' A - GiA srlr, y, z hhdng phei la dO ddi ba canh ctra mdt tam gi6c nio dci. Khi dri, ndu B,C' C'A' _ A'B' ,/t\ - . .1, , t. y * z < .r thi (xMA+yMB + zMC) nh6 nhdt sin BMC sin CMA sin AMB t khi M trungA. Tt vA suv ra : (1) (2) v^ 2.2*x ^ khi M trung B. sin BMC MA + sin CMAMB + sin AMB MC. 3.x+y < z thi (xMA+yMB+zMC) nh6 nhdt Ding thrlc xily ra khi vi chi t
>AN+BP+CM. BC, CA, AB. Tim vi tri criaX, Y, Z dd chuvi tam + 2ABC + co\.ntb + co*r.nlc) > g16c )(YZ nh6 nhdt. 5.(cosZ.nla Giei : Gsi (O1, R), (02,82), (O3, R1) la >(a.+b+c)12. drrdng trbn ngoqi tidp cl5.c tam gtil. ABC* co5.m,u * + cos'.nld cosVt, 2 AYZ,BZX, C'Xf. D5 thdy c6c drrdng trdn trrOn cirng di qua mOt didm. Ta ki hi€u didm dct ld 7 3 (h.3). Gqi I/ li truc tAm tam g,iilc ABC (h.4). ,4@+b+c). Nhd dinh li him s6 sin vA dlnh li 1, ta cti : Ding thfc x|y raa() - / IlA.sin BHC + "B.sin I/B.sin CHA + HC.sinAHB D6 ddng chlng minh rang: IIA.sin BHC + IIB.sin CHA + HC.sinAHB^ = 4R.sinA.sinB.sinC. Vdy : XY* YZ + ZX 2 4RsinA.sinB.sinC. [re = zn,
Al DG TffiE HOGSEHH-..lGE6EEOHHh6PM qeOr- cUe riuH gUANG NcAI NAItd HQC 1995 - 1996 Ngay thi : 08.12.1995 Bhi 1 : Tim tdt ci cac hnm s6/(r) cri tdp x6c dinE vd tQp gSa tri ld doan [0, 1] th6a man c6c Thdi gian ldm bii : 180 phrit (khOng kd thdi di6u ki6n : gian giao d6) Bni I : Hdm s6 f(x) xdc dinh vdi moi sd thttc a) f(x) * f(x) ndu x, * x, r vd Ia nghiOm nh6 nhdt cira phrrong trinh sau b\ Zr - f@) e [0,1] vdi Vr € [0,1] (dny) : c') f(2x - f(x)) : x y3 + Lxy2 - (, + l)2y = 2^x(x + 1)2 Blri 2 : GiAi thich tai sao tdng cira tdt ci cac 1 Hay tim him s6 f(r) vd v6 d6 thi ctia nd. Suy s6 cci dang ' " voi p, qliL c6c s6 ttt nhi6n vd p.q - ra gia tri ldn nhdt ciaf(x) 1< p
TRO CEdI DOAN ChU Gidi dip bdi Trong m6t trd choi cci thudng, ngudi ta quy Eui nEm mdi dinh nhu sau : Cci 6 qulcdu gdm 3 quA xanh vd 3 qui d6 Goia, b, c, dld" c6c s6 cdn phdn tich. drroc dd trong 3 hdm kin. M6i hbm 2 qud.O ngodi Theo ddu bdi ta cci m6i hdm cri ghi c6c chtl : XX d hbur thf nhdt, DD a*b*c+d:1996 (1) 6 hdm thtl hai vd XD 6 hdm thrl ba. Di6u d