Toán học và tuổi trẻ Số 206 (8/1994)

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Mời các bạn tham khảo tạp chí Toán học và tuổi trẻ Số 206 (8/1994) sau đây để nắm bắt những nội dung về kỳ thi Olympic Toán quốc tế lần thứ 35; một số bài toán chứng minh bất đẳng thức bằng phương pháp hình học; một vài tính chất của nghiệm phương trình bậc cao; định lý Pitago trong các nền văn minh cổ đại.

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Nội dung Text: Toán học và tuổi trẻ Số 206 (8/1994)

.li. ffit 9t 7alt'a. BO GIAo DUC VA DAO TAo "1" * uor roAN nod'vmr NAM a rep cni na xcAv rs uAxc rsAxc s!,nI!rf*..2. ftI$Y Sffi ffifiI T[Ifiru EHtrHG ftIIH}I BfiT IIfiT{{i TFII''S I "a_ ffi&FIII PFIITI}?{[I P}If,P HIruH H{IG uQtvru flrlH cHnr cAc ltcHltu crin f,!NH r[ Prrncr(o .} rNC}HGi CITC NCN v^? t'lOT PHUONG TRINH BAC CAO l,NN TUIINH CC} I'TIT I H tr E . I U tr z Cd.c thiry giao ud. hqc sinh ldp 12, ldp I dtt ttti hoc sinh gi6i, Qudc gia ci,a truong PTTH Lant. Son, Thanh Hoa Ki TFIr ol-rwrPrq roArv er.rdc rff (IIVIO} T ,r\, LAT{ TI{ 35
',\ta-- TOAN HQC VA TUOI TRE MATHE,MATICS AND YOUTH MTJC LUC Trang o Phan Drlc Chinh, DQng Hilng Thang Ki thi Ohmpic Todn qudc td ldn thf 35 i o Ddnh cho cric ban Trung hoc co sd Tdng bi€n tdp : For Lower Secondary School Leuel Friends NGUYEN CANH I'OAN . Nguy€n Dic Tdn - MQt s6 bdi to6n chdng minh Phd tdng bi)n tip : bdt d&ng thrlc bang phrrong ph6p hinh hoc 2 NGO DA'f 'ft-I o Gitti bdi ki trwtic HOANG CIIUNG Solutions of problems in preuious issue C6c bdi c:"ua s6 202 4 c Db ra ki ndy nOr oOruc BIEN rAP : ProbLems in this issue Crlc bdi tt T1/206 ddn T10i206,Lt1206,L212A6 rc Nguy6n CAnh Todn, Hodng o Tim hidu sAu th?m totin hoc phi thing Chring, Ng6 Dat Tr1. L€ Khic Further study of school Maths BAo, Nguy6n Hry Doan, Phan Huy Khd.i - MOt vei tinh chdt c6c nghiOm Nguy6n ViCt Hai, Dinh Quang cira m6t phuong trinh bAc cao 12 HAo, Nguy6n XuAn Huy, Phan c Lich sft Todn hoc Huy KhAi, Vu Thanh Khidt, Lo History of Mathematics Hei KhOi, Nguy6n Van Mau, Ngo Vi€t Tntng - Dinh li Pitago trong cdc n6n Hoirng L6 Minh, Nguy€n Kh6c vdn minh cd dai 14 Minh, Trdn Van Nhung, o Butic dhu lim hidu todn hoc hiin dui Nguy6n Dang Phdt, Phan Your first steps in Mod.ern Mathematics Thanh Quang, Ta Hdng Nguydn Tidn Dung - Bhi to6n con lac di 16 QuAng, Ddng Hirng Th5ng, Vu o Gidi tri todn hoc Drlong Thqy, Trdn Thdnh Fun with Mathematics Trai, LO Ba Kh6nh Trinh, NgO Binh phuong - Giei dap bai Thay chit bang s6 Vi6t Trung, D?ng Quan Vi6n. Nguydn Hilu Du - Chay nhtr thd ndo Bia 4 o Kdt quA cria dodn hoc sinh Vi6t Nam tai cu6c thi Tin hoc qudc td ldn thrl s6u Tru sd tita soan : 81 Trdn Hrrng D4o, Hh NQi DT: 260786 Btan ffip ud tri su; VU KtVt THUY 231 Nguy6r, vln Cir. TP rrO Crri Minh DT: 356111 'rrinh bay : DOAN Hbxc
ii-: " 1,; *iri;l i" .:';f 1 1: f I i'. .ii. i i, !: PHAN O6C CUINH - DANG HUNG THANG ^iuOc thi Todn qudc te ldn thrl 35 drroc td Ban Td chrfu: khOng x6t giii d6ng d6i. Tuy f"rchtlc tai H6ng KOng tr) ngdv 8 ddn 20 thring nhi6n khi xdp thrl trr (khOng chinh thrlc) cria 7 n[m 1994 - Tham drl cuOc thi ldn ndy ccj cac dOi c
Ddruh cho cdc ban Phd thing Trung hgc Co sd sfl sfir mflu cufruc mtHH sflr ofinc rn BIHG PHUOIItr PHIP HIHH HOG - Ncw6* #" ta* sflc tii li6u srich b5o viSt vd cric phrrong Gid.i : =1eng I ph6p giai bei todn chrlng minh bdt ding Y\x, y, z, t d:;ong n6n ludn ludn t6n taitl6 thfc d P.T.TH.C.S hdu nhrr it dd cap ddn g75.c ABCD c6 AC t BD tai O vli OA = x. i phrrong ph6p srt dung kidn thric hinh hoc dd OC=!;OB:z,OD=t giai. V6i v6n lidng hinh hqc P.T.TH.C.S trong D6 thdy m6t s6 tnrdng hop ta cd thd glhi c1rc bii todn AB : r[irT7 chrtng minh bdt ding thrlc m6t cSch ngin gon pg : {fTZ vd c
lu+y-z>-o 4u-y -2 < o lr, -* -4 < o 4 Chrlng minh ring *2 + y2 > ; Gidi: Goi I (x, / la didm tr6n mp Oxy trongd6 r, y th6a mdn ddu bdi. T{p hqp c5c didm I (x ; y) ld thu6c mi6n m5t phing gr6i han bli LABC. Tuong t! 4h? s (o + e2 -62 4h1J "5 Sau dAy li cric bdi torin tu luy6n : 1) Cho x, y lh. cdc s6 th6a man di6u ki€n 2r |y)2,2x-y2y A-,,1 g ;:6,.,: ,1."','* Tim gi6 tri l6n nhdt, gil, tri nh6 nhdt cria TrGn Oy ldn lrrot ldy cdc didm D, ,8, .t'sao cho x2 +y2 . OD = d, DE - e, EF = /Qua A, B, C k6 cric 2) Cho a,, b, c > 0, a >.c, b > c drrdng thhngd, , d.2, d3song song v6i Oy ; qua 1. Chrlng minh ring : D, E, F k6 c6c drrdng thing ar,e2,03 song {c(o -c) +{Z@ -c) < \E song vdi Ox ; arcit drtqi M, arcirt drt4i N ; 3. Cho a1to,2,. . . tan vd b1,b2,. . .,bn\e. cric s6 dttong. arc6t d.tai]>. D6 thdy : Chrlng minh ring : oM = vLVW, MN = WW i Np = {7+V ,[W, +ttazTq +. . . + {a,+q > oP:w. ME OP < OM +MN +NP 4.Choa,b,c>0 Yay:@< Chrlng minh ring : - *b)c Blri todn 5 : Cho x, y lh. hai sd th6a m6n 5. Tim 916tri nh6 nhdt cria cric di6u ki6n sau : {wx +T +{xa=Tfr
- Ndu abc > 0 + a * b * c > 0. Lai c6 ab 4 ac* bc > 0 ta se suy a a, b, c cr)ng l6n hon 0. ThAt vAy, giA sir LrdJlai, khOng ldm mdt tcing qurlt ta c6 a < 0. Suy ra bc < 0. Nhrrng ob + ac *bc > 0 n6n ab * ac > 0 "+ a(b -l c) > O Do dd b*c < 0. TI] dd a+b *c < 0. Mdu Bdi Tll2O2. Tim cd.c sd nguy€n. td dang thu6n, ta cri dpcm. 2te()4n117 (n. e N), bidu d.i6n duoc drt6i dang - N6u abc < 0 thi o * b 4 c < 0. Lai cd ab hi€u ldp phuong cia hai sd tu nhien. * ac *bc > 0. Ta se suy ra a, b, c cung nh6 Ldi gini : Theo Vt Quang Hda, 8T, PTCS hon 0. ThAt vdy, giA srl trdi lai, ching han chuy6n thi x5 Th6i Binh o. > 0 + b *c < 0 vd.bc < 0. Ma ob * or *bc > 0 n6n ab + ac > 0 hay a (b + c) > 0 Do drj Dat7l _ 2tee4n +17 > 1T (ru e N)(1) b*c>0 - Ydi n >- 7 : 7994n cri dang 2k (k e N+) *a+b*c>0.Mduthu6n. +P:22k+17:4k+77. Viy ta lu6n cri a, b, c cung ddu. Ta thdy 4k : (3 + l)k : I (mod 3) Nhfln x6t : 17=2(mod3) Bdi niy nhi6u ban lam drroc nhrrng li luAn dai. Giai t6t bai ndy Id c6c ban Phant Huy Yay P : 0 (mod 3). Thbo (1) P le hop s6 Ting, SA Bd Van 5an, Nguyen H'6ng HitA, Nguydn Dong, Trdn Vi€t Binh, 8H Tnrng - V6i ft : 0, p - 21ee4" *17:19 ld sd Vrtong, Ha NOi ; D6 DQng Tao, 98 Chuy6n nguydn td'vd 19 - 33 - 2'\. Ung Hda, Hd Tdy ; Pham Thi ThanttVdn.6H Vdy s6 nguy6n td phAi tim ld 19 (v6i n = 0), Minh Khai. Hii Phbng ; Bni Nanr Phuong, NhAn x6t : Hdu hdt c5c ldi giai ddu dring. 8A, Chuy6n Phong Chau, Phan Ngoc Lan.9CT C6c ban sau ddy cci ldi gini t6t : Kim Ngoc Chiry6n Vi6t Tri, Vinh Phu . Trd.n Phu Son, 8T PTNK Hii Hrrng ; Nguydn Dinh Quan. SA Minh,7M, Marie Curie ; Phan-r Huy Tilng, 8A, Chuy6n TAn YOn, Hd B5c ; L€ Thanh Tinlt. Bd Ven Ddn ; Trd.n Vi€t Binh, 8}J, Trung L€Van Cudng, LeVidtHdi,9T Lam San. Phant Vuong ; Vu Td.t Thd.ng, 9CT, Nghia TAn, Hd Tidn. Dftrtg,9T NK Nga Son, L€ Ngoc Giap, NOi. l/gzzydn Khdc Tufin, L'A Nggc Gid.p,7A. NK Dong San; Phanr Anh Tudn, SH chuy6n 7A NK Ddng Son, Luu Van Thinh. ST NK Binr Son ; LuuVan Thinh,8T, NK Thi6u YOn, Thi6u Y6n, Thanh Hda .: Phant Tudn Anh, Vo L0 Von. Cuong., 9T, Lam Son, Thanh Hcia. Nhu Quynl4 Nguy€n Xuaru Son.9T Phan BOi Pham Hbng Linlr, Nguydn Xud.n Tuong,9T, Chdu, Nghe An ; Thai Ar'h Vu.9NK Can LOc, Phan B6i Chnu ; Nguydn Khd.nh Quynh,9T, Hn Tinh ; Trdn Thanh Qu-ang 9T Qu5c Hoc NK YOn Thdnh, Ngh6 An. Dinh Tiei Hoitng, Hud, Thta Thi6n - Hud ; Nguydn Minh Qudc, 9 PTTH Hoii An, Binh DinhNguydn Huu Hdi, 9T, NK Ti6n HAi. Dito L€ Dung,9T, NK Vfl Thu, Th6i Binh. Vu Thu Hudng, Nguydn 8T Chuy6n Le Khidt, Qu6ng Ngei ; Luong Thd.n h Nam, 9T, NK Hai Hring. Nguydn Minh Tud.n Anh,8T1 nguySn Du, TP H6 Chi Minh ; Vu, 7, NK Qud V6, Ha Bdc. Le Quang Nd.m., Trd.n Thanh Ting, I T Le Quy D6n, Long 8T, Chuy6n Drlc Pho', QuAng Ngdi. Nguydn Khdnh, Ddng Nai. Minh Qudc, 9, Hoai An, Binh Dir.h, LA Tudn VO KIM.I.F{OY Anh, 9K, L0 Ldi, Hd DOng, Hn Tdy. BiLi TBl202. Cho tam gidc ABC nQi tidp 16 NcuyEN duirng trdn (0). Dudng phd.n gidc trong cia Bd1TZ12O2. Chung minh rang dibu ki|n goc A cat dudng trbn (0) tai didm thu hai D. dn dd cac sd a, b, c cirng ddu ld, cd.n ud, MOt dua,ng trbru thay ddi lu\n lu6ru di qua A, D cat cdc dudng tharug AB, AC tai cac didnt ab*ac*bc> ouaL+a+1ro ab ac bc tuong ing M, N. Tinr tQp hqp tntng didm I crta MN. Ldi giii : Ldi giii :Do cdcttS gSS.cAMDN, ABDC nQj . Di6u ki6n cdn : Hidn nhi6n tidp ndn dat a : MDN,tacci a :^180"^. MAC . Di6u ki6n dtr : :_ :BDA n4at 9(", do At: Az n6n DB DC ) DM = DNhay DB : DC ; DM = 11i Tr]db , + -,+; ac bc > 0 tacr obc (a+b +c) > 0 DN. Ta cd ph6p quay thuAn chi6u kim d6ng hd (hinh v6 b6n) i 4
ax2+Zbxy*cy2:n (1) Ching minh rd,ng M(n) lit httu han ud M(n) = M@k*) udi moi k > 0 Ldi gi6i : (cria c5c ban Pham Huy Tung, 8A BdVen Dirn, Hd N6i ; 7r&z Trung Thdnh, 8CT Nghia D6, Tt Li6m ; Nguydn Xud.n Thd.ng, 10CT (QuAng Tri) "fa c6 ax2 + Zbxy * cyz - n (ax + by)2 * py2 -- na (bx+cy)2*pxz:nc Roa : M -Ni B C ; do d- 0 BC ta chi cdn chrlng minh M(n) = M(pn) Z huy IK ll : HC.Ydy HCKI ln hinh binh Ta hdy ldp moi ph6p tudng rlng giua tdp A hdnh, vd HI ll CK (1). Trong L cdn CNM' (CN cd.c cd.p sd nguy6n (r, y) th6a man (1) vd t6p = BM - CM'), mung tuy€h CK ddng thdi le B c1,c cdp s6 np;uy6n (r, y) th6a min phAn gi6c g6c ACM'. Hon ntra, tt hinh_binh h?+MBCM', ta c6 AM ll CMi hay MAC + axz *2bxy * cy2 : pn Q) CAlr!^ : q0" ; do dd A2 + ACR : MAC --t ----'.\ + CAM X6t ph6p t,rl':ng fng = ---- Z = 90o' YQY CK L AD (2). 6=-bx-cy Kdt hop (2) vdi (1), ta c6 HI L AD, trlc ld .t rl=ax*by (3) R6 rdng ttt d&ng thrlc nim tr6n dubrtg thd,ng IIx uu6ng g6c u6i AD kd qua trung didm H cila BC (B). DAo lai, ldy oez+zbxy*c.y2=n---' I, e Hx, vi A, ddi xfng v6i A qua I, ; k6 a(cy * b$z - 2@y.+ bx)(ax + by)+c(ae * by)2 : pn AtM t ll AC ; ArNt ll AB (M, N t thuOc c6c suy ra ndu (x, y) e A thi (6, ,D e B. Ph6p tuong dr.rdng thing tuong :(:ng AB, AC, ta cd hinh rlng tr6n 16 r.dng IA rnOt - mQt. Th6t vAy tr) !l-nlr_ h3_nh AM / jN ) vi 1, Id trung didm cria M rN r. K6 dudng tr:dn qua A , Nr , D c6t AB tai (3) suy ra di6'm thrl hai M r. Theo phdn thudn, ta cci trung af + br1 @) u,_d-jr_ thu6c Hr. Gid srl I, * I, ta b€+crl=px "ii -tfN, cci IJ2llMrMz)(drrdng trung bfth cria Cudi ctng ta. cdn chrlng minh vdi moi cap LNIMN) gsx{a AB-LID (vi AB ll Hx, AD (€,r) e B t6n tai cdp (r, y) e A sao cho chring t Hx),hay BAC : 2 BAD :.79}o,mdu thuAn. li6n h6 v6i nhau bdi (3) (hoac bdi (4)) Ta cci Yqy Iz = Ir; Mr: M7, vi phdn dAo dA duoc (€, r7) e B suy ra chrlng minh. Vdy tflp hdp rrung didm 1 ln drrdng thing n6u d (3). (a[+br)z*pq2:rpn NhQn x6t : C
Bdi T5t202. Giei h0 phrrong trinh Phant Diruh Truitng (10CT Trdn Phri, Hii d5 Phdng, TO Duc Thd.ng (10E PTTH chuY6n -f,:---:--COSrTf. rqL Thei Binh) PhanrManh Quang (10T Lam Son, {5 cos/ru3 Thanh Hda) ; Nguydn Minlt. Thq (11A, Qudc x2= g hoc Quy Nhtrn) \r5 NhAn x6t rdng x3= q+COSJT-r , - tr {5 - irl2 < at'csinzr = r' 0 < arccosu ( z n6n +qt , cos/TI -r.: ilu Ldi gi6i : ria cdc ban Tran Quong Huy ( - nl2 { arcsinz * arccosu V tt, t: € -t1, 11. 11CT Vinh Phri ; Vd Hod.ng Trung, 11CT Trd "t Vinh ; Ph.am. D inh I'ruiing, 10C Trdn Phri ; [r B L€ Anh. Dung, I 1A PTTH B5c Thrii ; T0 DOng Vu, LLCT DHTH He N6i) VAv:a)Ndu l- ls", ll, -.t a tr -, lril'{5 < q=.2Vi = 1,4-cos,'rr, > 0 1 Tacci l8 ihi phrrong trinh\6 nghiem 1 +Qqrr., I 3'7 -nlz < *, = Z n-4t < -t < 12:r(*) b) X6t Neu r, ) x,+9651;2 > cos,'rJ+, Do 'lrr., rxo € (O, ,) JT Mnt kh6c, vdi u e [-1, 1] thi arcsinrz * atccosu : nl2 - frI2 { ZfO + X24 X4 --+ COStrtrIr > COSZr3, dO vi arcsin(-u) : - arcsinzr. ; arccos(-u.) = - tt. : tf, ' arccosu. trxtlnx.\ € (0,;t .+nrl (r rt-rl (Jl. Suy ra .i n arccoEslnSc:2-* V4y rr : x3-x2: ro. VAy r, : f"o",rr-. voi 0
3n1 r]1e.7 -2\tt4.t - ='-2ktt =-r€ 2. Bang phuong ph6p cria Cd.ch 2, c6c b4n ZA Nguy6n Quang Hdi (l0CT, PTTH Chuy6n, '- 32n * 32h,n Vinh Phri). Nguydn Vi€t Linh (10CT PTTH ll^: v6i k e {0, 1. ..., 5} Trdn Phir, HAi Phdng), Nguydn Tudn Hdi t7 (11M, Marie Curie, Ha NOi), Vu Duc Son (10 CT, PTTH Luong Van Tuy, Ninh Binh) Nh?n x6t : lldu hdt cac loi giAi grli ddn Nguy€nVifi Kian (Thanh Hria), Nguy6nXudn d6u gili sai. khdng chu y ddn tinh chdt crja Son (9T PTTII Phan Bdi Chdu. NghQ An) vd hinr -sci f(t) : a,-c sint, g(t) = arc cos/ cd c6c Nguy€n Quang Tuy€'n (10A PTTH Sao Nam, tAp xac dinh [*1, 1] vd tnp gi6 tri ldn lrrot iA Duy Xuyen, Quang Nam- Da Neng) dd chr1ng nrinh dung khAng dinh sau [-tr 2' t 2] vd L0''"' : "Cho da thrlc bAc n (n e Nrt ftxt : a,pn + *,,rryr.N vnN M,\rr o,.rn--l a ... + o-- ,x * a,, voi hd sO duong.'Khi db, V,rl G N'.'hr'> Z,'La cd o[n-t f,r, ,- Bni T7l2OZ : Cho tant thtc bd.c hai. fk) = orj +1.,r *cudi coc h0 s6 rl,uonguit o.*b +c: 1. [f ("fi)]"' Y x ) 0 (o, : au:- ar* ... *nn. Cltt?rtg rttinh rdng u 0. todn deu nhAlr xet durrg rarrg "RDT r{a niu tron.g bit.i ro r:lti ditng uii, i 0". Xin c6o l6i arf gul].!dS'ban.daleu t'6n d tr6!, c6c ban ::"g^r::.i:: t'a. va nnl/xrn n,_1,.:: Do sungdleu::j..'T.i:"-: bei dd bii torin dttoc hoi.n chinL. j: *: klenr'>!.1u \-ao de Fir+1'1".'J:i,-ffii'flrfl,;;il',';!:;fl#i PTCS Be van Dani Ha Noir. ?r.ri,rz Trung i3i, Thdntt eCT, pllcs Ngfri" ia", Ha N6i) ; Vo Ldi gini. Cac:lt I .' Vdi r > 0, x6t day l{hu Quynh (9T. PTTH Phan B6i ChAu, Ngh6 t,,,,:,V.!r fl tlr 0. 1.2. ... Voi nroi n € \t Ant, Aguv&t Lt; Lttc (8A' PTCS Danl Doi' 1, : rh.o bar dang thLrc Bunhiacopxki ta co : H,li,tdlX; ,o,'fi.li'rff;:f;;#,::::r';r";i ,..-, +b:"o1{; *c)'''*l: (11A.-PTT"H'Gang Th6p, B6E'T]n6it ; Trii,i rt ' = toznt '16z -H'ong Quong (t2A PTDL Chdu Phong. Vinh ttfo fi .'"*'\[7 +{t {t.'"*'fi+{J fi;21:"iih;i ,"fr;:;:s cd,m (tZT, PTTH Ha Long, < (o. * b + c12"(a2"G +b2'{i * c)2" = u,,, flil1}-.Uitltn^?fr,X\:rtY:f|::rfi;ffiIn Nhri viy. d5v {22,,} li dav kh6ng tang. Suy ra : PTNK Hni Htlng) ; fioartg euA- Vuong \llA', ,u{,)1.- nlT ,,\ > ,kvlr € N. 0, Nang, Nguydn Coo Son t10T, PTTH Luong Van Tuy. Ninh Binht ; Plton Quong Vinlt. Li Cach 2.' V r > 0, theo bdt d6ng thric Thoipliong,NguyAniriphuong (1-0T,pTNK BunhiacOpxki suy r6ng. ta cd : Ha Tinh) ; Van Ditlt Qudn ( 11A. PTTI{ trl rofi)r:* = ( ,*fi 2*{; ,o,[op + z*v6 *[l;F"lf:;"tr1,r,:;:!i,ryf3{:# rl:;,,td?!4, 2o{1, 'orfa, + 2o}E 'o{r)20 r,:\!i,fl"6ig?t6}:?"1,"d.1*',K #',#n, Phant Qua,ng Huu, NguydnVan Hibn (10 CT, < (a * b + cy2r -1. t..ax] * bx * nc) = f(x). PTTH Ng6 Quy6n, Ddng Nai) ;NBd Dang Hir Cctctt 3: Vi ham g@l : r2o lA hern l6rn tr6n An (IIAPTTH LO Q.uy D6n, Long An) Nguydn [0, +-) va do o, b, c ) 0, a * b * c = 1 n6n I{hQt No,nr (1OA PTTI{ Vung Tdu, B} Ria - theo bat d&ng thfc h}"m lorn ta cci Vung Tiu) ; v.v... : .k- ,k- . N(illyl:N KI IAc MINtI 0. g( - Vr') * b.g(z,tlr ) +c.g (1) Bei T8/202 t Ki hieu N* lit td,p cd.c sd nguy€n > E@ z"li .k- 'k- +b 2"{c +c) Vx > 0 d.uong. Tint tdt cd. cd.c hitnt f .' N* * N* th6o * fdt .- lf (20\[i\2r v* > o. nnn ctbng thiti cdc dieu kien sau : Nhenx6t:1. Crirdtlhiglb.a.qgrlildi,giai lf(n+ cho bii to6n vd da s6 da giai bei todn theo - / "- 1), f(n.) Vz €.ty'* cd,ch t hoac cach 2. D:uy nh51 ban trlguydn ViQt 2) f(f (n)) = tL * 1994 Vn € N* Ki1n (71A PTTH Nga Son II, Thanh Hria) da y !.: _-!2: 6ung ld ngudi duy nhdt da gi&i bai theo cA 3 HN 1) : Yi f(n) e N. czich.
V ru € N* n6n tt gif, thi6t 1) suy ra f(n + t1 a) Tinh dian ich ti gidc MNPQ theo >f(n)+l a:ABuitg. Vz € N*. Tt dd, kdt hop vdi giA thi6t 2), ta cd : - b) Vai goc A nito thi chu ui phd.n chung hai hinh uudng be nhd.t ? n + 1995 :, f(f(n + 1)) > f(f(n) + 1) > f(f(n)) + 1 : Ldi gini : a) Goi K Id didm chinh giua cung nh6 AB', ta cd A, A'ldn luo_lddi xttngvdi B'B =n+1995VnCN- qla-trUc oK DaL-a - -AQK, ta cd 2a : Suy ra : f(n + 1) = f(n)*7 V z € Nx AOK + KOB' -r AOB + BOB' = 90o * p hay * f(n) : f(1) * n - 1 V n eN. (i) a=45"*92 +f(n) -n=f(1) -1VzcN* Trong L AOM, ta c f(n)*a- 1 Vru €N. thing A'B' 2) fi@): ak' .n +b vz € N* vdi BC, B'C' v6i CD ; C'D' r'6i DA ; D'A' vdi AB, tudng tu d dey fu(n) == f(f(...f(n)....)) nhrl tr6n, hinh tao bdi hai hinh vu6ng ABCD, A'B'C'D'cdn cci c5.c truc d6i xfng EG, NQ, ktdnf FIf. Suy ra circ r:anh crla bat giac MENFPGQH : n *I, bing nhau. Det z : HM, ta cdn tim di6u ki6n (D6p s6 :f(n) nd,ua : 7 dd z dat urin. Dat tiep r : A'H, y : A'M, til A vu6ng A'HM, ta c6'x2 + y2 :'22. Mat khac, b(a l\ f(n) : an * ), --!, ndu o > l). x *y * z : A'l + HM +A'M = AH + HM + a* -l MB:AB:a.Dodd t6t NguydnViAt Liruh, 4. C1.c ban cci ldi giAi ) ). 1 (:r*y12: (a-z)2 *22 + 2az _ a2 Pham Diruh Truitng (10CT, PTTH Trdn Phu, zt = xt * y'z >J- +1_ 7 Hli Phdng) ; H6ng Cdm (12T, PTTH Ha Long, > 0 hay z > a(,'[5 - 1) va z^in: o(1/$ - 1) khi QuAng Ninh) ;76 D6ngVit (llCT, DHTH, [la r =y hay p = ,L{>o.:D6p s6':-i} : 45o. NOi) ; Dirzh Tidn llod.ng (9T, PTNK Tidn HAi, Nhfln x6t : C
phuong cdc khodng cdch til d6 ddn cd.c mat Cdng v6 ddi vd (i), (2), (3) va (4), ta drtoc : cila ti clien bang k/ cho trudc. dtr(M, ABC) + d?Gtr, DBC) + d\(M, DCA) Ldi giii : Cd thd giAi bdi to6n ndy b6ng 4 hai phttong ph6p : phriong phap vecto vd phrrong phrip toa d6. Sau ddy ld ldi giAi cua T6 MSW,DAB)=>d? = DOng Vu (11 chuy6n To6n, DHTH He NQi) i:l bang phrrong ph6p toa d6. 4 Drrng hinh ldp phuong AB'CD'. C'DA'B = L*3 + yZ + zf, - O1x" * !o * 2,,) + b2f ngoai tidp trl di6n ddu ABCD dd cho. Chon 4., b,t , b,-t , b,t 1 - * * dinh D', ddi xrtng v6i dinh D qua tAm O cria hinh l6p phrrong ldm g6c toa dd vd c5.c t,iafD'A, =E L f,- ,) Di6'm wt (xo ?"- ,) l'.- z)'*ao'1 ; tst lD'8, LD'C) lim c6c tia drrong cta truc hoinh, lo , zo) th6a man di6u ki6n , cria trgc tr.rng vi. truc cao. Goi o ld dO ddi canh cira bdi todn khi vi chi khi : trl di6n d6u da cho, b le d0 dai canh cria hinh 4 Idp phuong, thd thi c5.c dinh A, B, C, D ld tAm O c,.ra ttl di6n ddu (cung ld tdm O o a hinh lAp 2a? = k2 & ld. m6t do ddi cho tnidc) ; (*) phrrong) trong h0 toa d0 D6 Cric D'xyz ndy c6 l= I cdc toa d6 nhu sau : A(b, o, o) B(o, b, o), C(o, V4y: (*) o, b), r)(b, b, b) vit o(*,*,|), tro,s oo , - * - f,)',* . (" - Z)'1- *u' =k2 t-_ a [4' (,,, ',1^'?,,- f,)' ".- 72 - *5OMt**r:U, ^ ,,,,,&,, * 4^-az gOMt: kt - 6 (0,aD,c .iN ) -_t,, .ir:, ^ 3, ^ a2' (**) Ta di ddn kdt luAn OMt - T\k'- ); :\ : ^ dz a,l6 a) Ndu k'- >yrq' a > 0-k, a thi qui tich "\i.,i, )€ cira M li mat cdu (?), tam O, b6n kinh o,b,/EV:{_\\_ +a a2 ,l-' . -- -\ D:- r2 a ,., ,' AA(A-A.q (L-a'o) . nr/6 - --tr thi {M} : ,81;,-;:;,1;, ,:..,,,,,1r:, , b) Ndu k t0} DC thdy ring trong h€ toa.dQ nAy, cdc mEt phing chrla cdc md"t cria trl di6n ddu ABCD or/6 cd c) Ndu k . -;- ,o thi {M} = A, t.l. kh6ng tdn phuong trinh ldn lr.rot ld : mp(ABC):x*y-tz-b:0 tai m6t di6m M ndo dd mp(DBC):-x*y*z-b=0 Z.| O7 = U, mp(DCA)ix-y*z-b=0 Cach 2. (Phuong ph6p v6cto). Ngodi crich t- mp(DAB):x*y-z-b:0 giAi tr€n, ta cdn c + I -) + -) 4+ : (1) MA' MB' + MC' + 114p' = iUO i(*u*lo*"o-b)2; Trrong tu, ta dudc : Nhfln x6t : Ban Td D6ng Vu da d::'a ra 2 crich giii. Cric ban sau d6.y cci ldi giAi tdt d?(t,t, DBC) : : * lo * zo - b)2 ; (2) \guydn T hanh Hdi, l2CT Dio Duy Tt, Quang ir-*" Bjnh, Pham Dinh Trudng l}CT Trdn Ph6, d\(rur, DCA) : lie1lhdrg, vitVu Thi Blch Hd, 11C trtrdng !@" - t, o r, - b)2 ; (3) PTTH chuy6n Th6i Binh. Ncuy0ru oANc prrAl' dZ\r,t, DAB) = * -b)2; (4) !{*o to - "o (Xem tidp trang 13)
- Bhi 1'8/206. Chrlng minh 8n 12:r 182 cos + cos *gg- * cos = Bb ,r- 1 n.{1 * 2 t,r n - Tcos6 b C6c klp'I'rung hoc Co sd 'r'o xllAN Iln I Bei T9/206. M Ia rn6t didm nam trong mdt Bhi T1/206. Chtlng minh rang c 3) cd chinh phuong vidt d dang 2n2 + 2 vd,2nt - l, tAm 0. n€2. Ggi M1, M2, ..., M ldn lriol Ii hinh chidu NGUYBN I,O DONG (vu6ng gd;) ctia M tr& c6c drrdng th&ng chrla Bni T21206. Cho a, b, c ld ba sd thuc sao c6c canh ArA2, A2A3,..., vi A'.4, cria da gi6c cho bidu thfc d v6 tr6i crla (1) luOn lu6n cci dn cho. nghia. Chtlng minh ring : GiA sr1 G ld trong tAm ctia hO c6c didm (M,, t ol(b
, I '.For Upper SecondarY Schools ,T Gn06. LeL Fp.be the k-th tetm of the Fiton.""l sequen& l, 1,2,3,r 5, 8,..' Prove that the number For Lower SecondarY Schools *I 4Fn_2FnFn+f u*t 'l1l2OS. Show thast there exists an infinite is a perfect square. number of perfect squaves of the form 'I'A llON(' QTJANG 2ru2 +'2 and 2n2 - 1', n e Z '17 1206. Consider the function f : R' R NCUYIiN I.F. l)LlN(r satisfying the following conditions : TZlzOS.Let a, b, c e R ; a *b - c 1 A, Dlf@-11:rd1 < lu-ul Yu'ue R'u*u' b*c-a*0,c*a'b*0. 2)f(f (f (a)))=o;where a=lggzrs5 Calculate f(a). a21b-c1 , - o) , cz 1a - b'1 - o2 (" Dr\O'l'RIJONG GIAN(i b*c-a c*a-b a*b-c T81206. Prove that (a +b +c)2 (a -b) (b -c) (c'-o) (1) ' (b+c-o)(c +a-b)(a+b.-c) -,., cos 8n * cosl2n * cosl8n : DAM VAN NIII "^t ,, ,5 . x,,[1 n TSi206. Let be given P, q € Z and -rcosB* 2 =,,b f(x): x2 +px+q. .I'0 XUAN HAI Brove thast there exists an interger ft such that Tgl2OS. M is apoint in the plane of a regular f(k) : fl19e4). f(19e5). folygon At 4 fu > 3) with center O A2... DAN(; IIT]NG-I'IIAN(] Let M1, M2,... Mn be respec-tively. the ortnogonai proj6ctions of M on the lines A7A2, AAs, ...,Aa\, and let G be the center of gravity T41206. Given a convex quadrangle ABCD of tfr" systeni of points (M1, M2,..., Mn) inscribed in a circle of radius B. Prove that AC t'BD provided that AB2 + CD2 : 4R2 Prove that +--->€+ DAO TAM t)OAt+OA2+...+04=O 2) G is the middle Point of OM. T5I2OS. Given an acute triangle ABC. M is ]'RINII BNNG (iIANG a moving point on the line BC. The perpendicular bisectors of BM and CM meet T10/206' a, b, c,, d., e, f are the lengths of the lines AB and AC at P and Q respectively. the. six edges of a tetrahedron of volume V' Prove that the perpendicular to PQ drawn Prove that from M passes through a fixed point. a3 +br+c3 +d3 +e3 +f > SO,{2.V DTJONG QUOC VIET. NGI",YEN KH;\NH NGUYTT,N Dd vui Todn hoc ?\/ CHI DUIUG GHU SG 3 Chi dnng chir sd 3 vi cic ph6p to6n dd vi6t Vi 6 = 3! n6n ta c
Tim hibu siu th6m Todn hoc ph6 thdng ----------------l rq6r vAr riNH cnAr cAc NGHIEM cOa I MOT PHIJCING TRINH BAC CAO PHAN nuv xu.AI BAi vidt nny gi6i thi6u v6i c6c b4n m6t vdi DiEu kidn ciin. Gih. stl th6a m6n ddu bii, quy t5c dd khno sdt tinh chdt cric nghiQm cia khi dd theo dinh li Bezu, ta c6 P(x) : mOt phtrong trinh bdc cao, til dci trong nhi6u : (x - xr)(x - xr) (x - rr). Ro rdng P(r) > 0 trudng hop ta c 0+ todn cu thd. m. < -5. Phrrong trinh bAc n cd d,ang tdng qu6t Dibu kiQn di. Dbo lai giA sil m < -5. Khi P(x): auxn*arxn-l +... + dn-rx +an: Q (|) d 0. Do P (0) : nx - trong dci au * O. Ta ki hi6u P'(x), P"(x) tudng 3 < 0, n6n theo dinh Ii 1, t6n tqi nghi6m x, r.1ng Id dao h}.m bAc nhdt vd bdc hai cia P(r), mi -1 < x2 < 0. R6 ringtdn tai o (md a > 0 p(i)(r) Id dao hdm bdc i cita P(x), i : I,2, .... dri I6n) sao cho f (-a) < 0, 't iy t6n tai nghiOm (xin nhic lai ln P(i) (r) = (P(i- t)1r;'; x, mlr -a l xr < -1. Lai t6n tai b > 0 dri l6n Nhu da bi6t, phtrong trinh (1) crj t6i da ld mn f(b) > 0, do dci rip dung dinh li 1 m6t ldn ru nghi6m thuc. Dd khAo srit sd nghi6m, cung nta suy ra tdn tai nghiOm x, md 0 < x, 0. Di6u d ro ddu kh6ng phAi ld nghi6m ctra (1). Dat m = min (a, b, c), M : max (a, b, c) Ta cci d.p.c.m. Chrlng minh rang Drt6i ddy chring tOi drra ra m6t vdi vi du Bm
R6 ring phuong trinh P(r) : O c6 3 nghi6m doan [a, b]. Gia sit P(o) P(b) * 0. Ki hi6u u(r) xt= d,x2: b,xj: c. TheodinhliRollephttong ln sd ldn ddi ddu trong day P(x), P'(x),..., trinh P'(x) : 3xz - 2(a + b + c)x ]:ob * bc * ca = 0 p(n)(r). Chrtng minh ring ndu goi ft Id s6 cci hai nghi6m ir,irmd o .i, < b < ir. nghiem cia (*) tr6n doan [o, b], thi k: u(a) - ". u(b) - 2m, trong dci nz Ii s6 tu nhi6n ndo ddy. (Nrji Til;.t a*b*c-W crich khdc k cci cr)ng tinh chin 16 v6i u(a) - u(b)). = a*bic+W 2) (dinh li Descartes). Xdt phrrong trinh (*), .r. = trong dci e.oan * 0. Ki hi6u u Id sd ldn ddi ddu cta ddy an, clp ... t o.. Goi ft ld sd nghi€m cria (*), thi suy ra d.p.c.m. k : u - 2m, v6i nz ld s6 tu nhi€n ndo ddy. Dd kdt thric cho bdi vidt, xin drra ra mOt 3) Chrlng minh rdng phuong trinh rs - 5r3 vdi bdi tdp dd cric ban luy€n tdp vdi c6c phrrong -l 4x - 1 : 0 cri 5 nghiOm thrrc. ph6p vr)a trinh biy 6 tr6n dd kh6o s6t nghi6m cira m6t phrrong trinh bAc cao. 4) Cho hdm s6y : x4 - 6x2 + 4x + 6. Chrlng minh rang hdm sd dat crtc tri tai 3 did,m A, B, 1) (dinh li Budan - Fourier). X6t phrrong trinh P(x) : aoxn*arxn- 1+...+o.,_ ,x*an- 6(-), tr6n C vi gdc toa dO O la trong tAm cria tam gi5.c ABC, cll{r sAr Ki TRUOC (Tidp theo) l1C PTTH chuy6n Th6i Binh ; NgO Thitnh BdiLllzl2. Anh sdng Met Trdi chidu song Trung 11A, PTTH Li Tu Trong, Cdn Tho. song udi \tQt btic tudtlg thdng ding uit tao uoi OK phuong ,hang ding g6c nhon a. Ding trudc BdiL2lz0z. C6 m.6t ampe kdc6ruhidu thang tudng, ding nt6t guong phArtg, m6t em. be do, c6 dO ch{,nh xdc cao, iO nhung son ridnlg chidu dnh sd.ng uudng goc u6i mat tudng. Khi biAt cho tilng thang do. Dilng ampe kd nd.y dd d6 mat phi.ng cia guang tao udi ntat d,d.t mot do cudng d0 ddng dien trong m1t doan mach. g6c bang bao nhi€u ? Ndu su dung thang d.o 10mA thi ampe kd chi Ldi giai. O trinir ve b€n canh, Sl ld tia toi, It = 2,95mA. Sau khi chuydn sang thang do 3mA, no chi I, : 2,90mA. Hdi cuitng d6 ddng 1B Id tia phin xa, IN In ph6p tuydn vdi mat di€ru lric trudc-khi mac ampe kd ld. bao nhi€u1 gucJng, IZ.ld phrtang thing drlng. Mat gudng tao vdi m6t ddt mOt gcic bang gric x gi:a IN Ldi giai. Goi tz^, n , ia niou aion itro va dien trd trudc khi m6C ampe ke cia doan *aqh ; Rt, R; ld di€n trd cril t & ,f"g voi khi sri'dun-g thang do 10mA,"r"p" gmA. VOi moi thang do ndu giA srl crrdng d6 ddng di6n qua ampe kd c6 gil. tri bang gi6i han do thi khi dd hi6u di6n th6 giua hai ddu ampe kd ld crrc dai vA kh6ng ddi vdi m6i thang Ao kfrac nhau, do d6 U^o, = 10 Rt - 3R,(t). Ta lai crj 1l (Ro*rB,r = Ur,(2\; 12 (Ro+ Rr) = rr. (3) ; Tt (1), (2) vit(3) suy ,u 1,,:! : ,to (Rz-R,) 7ItI2 vd IZ.IR vu6ng gdc v6i tudng n6n nam ngang vd cung vudng gcic v6i 15 vi tia t6i sonpsqng Ifrf 11\ It Iz= fr;qi= 2'e'7mA v6i tudng. YQy IR r mp SIZ.vit gcic SIN : NhAn x6t : Ccj 15 bei giAi dring. Cdc em 45o. Mudn chiSu m6t didm N l6n IZ ta chidu sau d6.y c
I \nhliPytago nrii rang ndu o, b, c ld cdc canh b6n vi canh huy6n ctra mOt tam gi6c vu6ng thi ta c
Phdn Sulbasutras crlng cd nhitng chi d6n chgm dd.t d mQt khod.ng carh liL b so udi gdc vd vi6c dgng cdc hinh trong dd cti cing ddn cd.y. Hd.y tim dQ dd.i cita ddY. dinh li Pitago nhu viQc giAi bdi toSn sau. Blri todn : D4ng mQt hinh uuOng c6 diQn ich bang di|n tich m.Qt hinh cht nhQt cho trudc. Ldi giei : Cho hinh cht nhAtABCD. "tavE hinh vu6ng ABKH trong ABCD. Sau dci x6c dinh cdc trung didm E vd M c: 'a DH vd CK. Drtng hinh vu6ng AEFJ di qua M. Ldy J ldm tdm vo mQt drJdng trbn cci brin kinh JF cilt BM d W. Hinh vu6ng cci canh bing BW sO cri di6n E tich bing di6n tich ABCD vi theo dinh li Pitago ta cci BW:JW-grt:JP-KMz : (JF - BJ)(JF + KM) : AB.BC Ldi giai : Ndu cAy c
f--* Bu'rit ddu tim hibu todn hoc hi6n dai '-_*- -*--_-1 i\l mAx r$,,&N e$Iu u,n.c ruA Ii .Er/B-,s- I-q"pt'&iY t,tFrt x:rya, rrfr. iI I r- -*--*^--.-_ NGLTYEN ufN orlxc -_ -* l Ban hay tudng tuong nlQt cort lac dA cdn chui Ncii c6ch kh5.c, cdu trfc thd tich y6u hon ld qua m6t c6i t0 t
vO hudng cria hai vectd x : (x, , x2) vd I : At,y2)liLA (* ,y) : xt!t *xz!2, cbn di6n tich crja hinh trinh hdnh t4o bdi hai vecto dci 1d B (x , y) = xt!2 - x2!1 Ban cri thd kidm tra r[ng di6n tich ld mQt cdu trric symplectic. Nhrr vdy trong trudng hSp hai chi6u, cdu truc symplectic chang qua ld cdu trfc di6n tich. MOthi6u thri vi ld cdu tnic symplectic chi tdn tai trong c6c kh6ng gian cci s6 chi6u chin (ban hdy thit chrlng minh ring nc5 kh6ng tdn tai trong kh6ng gian ba chi6u). Va tuong tu nhrr khoing c6ch, tt mdt cdu tnic symplectic cring sinh ia drroc m6t cdu tnic thd tich. Dinh nghia cria cdu trric symplectic d tr6n mdi chi ld tai m6t didm. Dd cci mOt cdu tnic Thd ngudi ta cdn nghiOn crlu hinh hoc symplectic dd lAm gi ? Cau trA ldi xudt xrl trl symplectic trong tohn khOng gian (hay n
CHAY NHU THg NAO ? M6t hudn luy6n vi6n mu6n thir tai todn hoc cita cdu tht bdng d6 cira minh d6 ra l6nh cho cdu thti dd nhrr sau : Anh hay chay mOt ludt sao cho drrdng chay cira anh bao dr'roc chu vi vd chia sin'bcing thdnh t6m phdn bang nhau' Gidi ttdp bdi Kh6ng chay lap lai doan ndo cung kh6ng chav ra ngodi sAn bcing -: C6c ban tiin giup xem cdu thi dd phii chay THAY CHU BANG SO nhrr the ndo dd d6p rlng drroc v6u cdu cira hudn TOA NHOC luy6n viOn. HOC TOAN N(iIIYI]N II.IIJI)II 8'*23'-r81 I Nhin vio cdc cht s6 cira con tinh ta thdy ngay T = 9, H = l ddu :F (hdng tram nghin) = 0 I : Ta lai thdy tdng c6c chrf s6 cira s6 TOAN i HOC vn cira s6 HOC TOAN ld bang nhau ndn rEr aud cua ooAN FCIc $!Nh tdng cdc chti s6 cira hi6u cira hai s6 niy chia hdt cho 9. Do vdy : (8 + 0 + 2 + 3 + * + 8 + 1) : 9 vtiT NAM TAI c.uoc^.Trl hay 4 * x: 9. Nghla li ddu :i (hnng tram) : 5' TIN HOC QUOC TE Vdy ta co : Lhhi'TH0 sAU 9OAN1 O C lOCgOAN 8023581 Cuacthi:TIN I{oC QUOC TE ia,ir thrr Sdu d6 di6n ra tai Stockhonr (Thrr i{0 Tt hdng trarl vd hdng nghin cira con tinh ctia Th SII' Ilie rr ) t tr tr g:r\ 3 d in b6n suy ra.Atr : 3 10-7-lti94, f)oitn Viet Nam cr-i "i hqc Tr) hdng don vi suy ra C = 4 sinh tdt cfr r:leu dcat g'irti I Ti hdng van suy ra A : 7 'ltr$ chtrtntli lar: : Tlil