Tạp chí Toán học và Tuổi trẻ: Số 233 (Tháng 11/1996)

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Về một tính chất thú vị của hình vuông, xây dựng công thức tính độ dài trung tuyến, ứng dụng tích phân tính giới hạn, định nghĩa 3 đường công Cônic trong mặt phẳng Afin,... là những nội dung chính trong tạp chí "Toán học và Tuổi trẻ - Số 233" ra tháng 11/1996. Mời các bạn cùng tham khảo.

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 233 (Tháng 11/1996)

i{.-i ' i'.i:,}P"./' i. i BO GrAO DUC VA DAO TAO * HOr TOAN HOC vrET NAM RA HANG TUAXC 4 ac ? q ;* "-.-- +.1 - 4 wffi F{#T ffiNr{ €HffiT Ttr{tr rf$ (ffiffi ffiilffiffi w{roNffi --!--. a m xAy DIJNG e0ruCI TF{uc rfrurt nQ nru TRUnTG TTJyEN M UwE DUI\tg n Cf cytfiy nywr E ror Wy 7 m BlilH ffGHIffi B Bu#$rs (ffi${r( Trqoffie ffimT prlfirss BFr$* 0-3 At!'?t m BE rHt QU0G SIA SH$r.I H$C Srr'Ilt Gror rm $ obd/tl x,g@ lsss - tes6 cDodn ngdj sinfi Xac dinh tam NhCrng doy s6 duon6 tron ki lo Ldp 12T bddng chuyAn LA Khidt, Qud.ng Ngdi ndm hoc 1995 - 1996
ToAN HQC VA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang o Ddnh cho cric ban Trung hoc co sb. For Lower Secondary School Leuel Friends tht vi Tdng biAn tdP : LA Qudc Hd.n - Vd mQt tinh chdt ctia hinh vudng. 1 NcuvsN cANH roeN Ph6 tdngbi6n fiP : e Gidi hdi ki trudc NGO DAT TU Solution of Problems in Prouious Issue Cdcbdi eias6229 2 HoANG cIlfNG c Db ra ki ndy Problems in This Issue Tu233,..., T10/233,LU233,L21233 8 nOl oOruc etEH rAP : e D6 Nhu Ngq, - Xdy dung c6ng thrlc tinh dO dai trung tuydn tam gi6c I Nguy6n CAnh Todn, Hodng Chring, Ng6 Dat Ttl, LO Khdc c Dinh cho cric ban chudn bi thi vdo dqi hac BAo, Nguy6n HuY Doan, For College and UniversitY Nguy5n Vict Hai, Dinh Quang Entrance Exam PrePaPers HAo, Nguy6n XuAn HuY, Phan o Nguydn. Thanh Giang - Uttg dgng tich phin Huy KhAi, Vfl Thanh Khidt, L0 tinh gi6i h4n 10 Hai Khoi, Nguy6n VEn M{u, HoingLO Minh, NguY6n KhSc o Nguydn Tltrtc Hd'o - Dlnh nghia 3 drrdng c6nic Minh, Trdn V6n Nhung, trong m[t phing a{in 12 Ding Phdt, Phan Nguy6n o Nguydn Htu Thd.o - Dd thi qudc gia chon Thanh Quang, Ta Hdng hoc sinh gi6i tornn ldp 9 ntrm hoc 1995 - 1996 14 QuAng, Dang Htng Th5ng, Vr1 Duong Thuy, Trdn Thdnh o Gidi tri todn hac Trai, LO 86 Kh6nh Trinh, Ng6 Fun with Mathem'atics Vi6t Trung, D+ng Quan Vi6n. .Binh phuong - Gi6ri d6p bdi : Do6n ngdLy sinh Bia 4 Va Kim HuQ - X6c dinh tdm dtrdng trbn Thanh Tud.ru - Nhirng deY s6 ki 14. Trq. sd tda soqn : - 45B Hlrng chu6i, Ha Noi DT: 8213786 BiAn tQp uit. lri sy: vu KIM THIrY "- --' 2Bl Nguy6n vrn Cil:?;'h Chi Minh DT: 885G111 irirn oav , QU6c sbNc
Ddnh clro cdcbgn fH6 ,\l u( ful ryrT[rrytt ffimTTtt$u$ c*m ,m]]*Huu0r$G lE oudc HAN (Nsh? An) Trong srich gi6o khoa hinh hoc l6p 8 dE n6u Gi6i: Goi P t l6n c6c tinh chdt co bAn cria hinh vu6ng. Trong Ii didm d6i " tll bni brio niy, chring t6i xin n6u th6m m6t tin[ xingciaMqua 0 chdt khric ctia hinh vu6ng vd c6c rlng dung o thi P thu6c phong phri ctla nri. canh BC. Tt N Bhi torin I z Cho hinh uu6ng ABCD uit. cd.c k6 NI{ t MP vit d:6y M, N, P, q tuong ilng tr€n cd.c dudng l6y tr6n dudng t!*"19;,1.c; A r,t fi th8ng NI/ m6t CD, DA, ChTNg didm Q sao cho m.inh rd.ng : MP NQ = MPthlQ, = NQ khiudchi K tY thu6c eanh AO khi MP.r NQ. (xem hinh 3). ,A P unns Gi6,i: Dd Gqi ^E li didm chrlng minh ta ,A vdl ld chdn dudng vu6ng d6i xrlng cria Q qua O K6 MH II AD, g.rg hatit O xudng EN.Ldy B vd C tr6n dtrdng NK ll AB rdi thing.EN sao cho : .IB = IC = IO.Ldy A vd. D chrlng minh hai 0 ddi xrlng v6i C vi B qua O thl ABCD li hinh tam giric vu6ng MHP vd NKQ tlPc ' Illnh 1 vu6ng phAi dung. K6t qu6 sau d6y li su tdng qu6t htia cria bii bing nhau (xem hinh 1). to6n 1. BAy gid, ta hiy 6p dung kdt quA crja bdi to6n Blri to6n l' : Cho hinh chit nhet ABCD c6 1 dd giai crnc bii to6n sau : AB = a, BC = b ud. cd,c didm M, N, p, e nd.m Blri to6n 2 z Cho hinh uu6ng ABCD eanh tr€n cd,c duimg thd.ng AB, BC, CD, DA. Cfulng bdng a ud. mQt didm M chuydn d.6ng ffAn canh BC. Phdn gid.c crta g6c DAM cat CD tqi N. ruinh MP a NQ khi utr. cni nhiffi:I Chtng minh AN < Z . MN. D&ng thtte xa.y Chring minh bii to6n 1' tuong tu nhrr crich ra hhi ndo ? chrlng minh bni to6n 1, xin dinh cho ban doc. I Giii: Dudng Blri todn 4z Chofi giacABCD. Dung hinh th&ng kd tr) M chit nhQt MNPQ ngoai tidp fi giat ABCD d6, vu6ng gdc vdi bidt tt s6 crta hai canh hb nhau bd.ng h (k td, s6 AN citANtaoH duong cho trudc) vi cdt drrdng Gi6i: GiA thing AD tat L A, B-,9,D l,l OGq hinh 2) S /Y :.,I theo thrl tg DAN = NAM nim tr6n c6c n6n D cgnh MN, 1 NH=HI=;AN NP, PQ, QM (xem hinh 4) Iltnh'2 . Theo bdi to6n "^ vd,MN1NP = 1, ttAN t Ml,tacd AN = MI = > Z. MN. K,KAAH T D&ng thtlc x6y ra khi vi chi^,IH khi ff = tf
tsdi TZl22g. Tim nghiQm. nguYAn cia phuong tinh x2 +f +i +f :27144a Ldi gi6i : .c:iua Nguydn Hdi Hd,9b, Chuy6n Van - Torin Ung Hda, Hd TAY. x2 +F +xa +f = 277440 BidiTVzLg Cho x > 0, ! > 0, z > 0 +x27x2 + 1)(x + t) = 24 .92 .s . lg . _29 (t) Chilng ntinh Tt (1) ta suy ra ngay nghiQm r phAi I6n hon L vd x2Id udc chinh phttong ctra 271440' Cdc (xyz + D (;++).:.; +L > x *y * z * 6 rioc chinh phrrong ctiZlt{qo c 2x. Nhin x6t : Hdu h6t cac ldi giai grli ddn d6u dring. Song ldp lufn ddi dbng. C6c b4n au dAy VOyA > ?'x +2y + 22 *1 +L = x I Y * **xyz cO tai giai t6t : Hlr B,6rc : Nguydn Danh. Nann, Nguydi Hilng Cuitng, Trd.n Thi Hd Phuong, , *;)1.* *r)l, + , +;) 7, r,1y*z*6. 9T, NK BEc Giang. Lho Cai : Nguydn Hbng z+ (x (, (z y;, Quang. Vinh Ph6 : NguYdn Dtc Minh, 8A, Ddu bing xAy ra khi vi ctri kfri *y = r, =!, Chuy6n Tam Dtro ; Hd.Vd.n Son, 9T, ChuY6n Phri Thq. H}r Tey : D6 Anh Tud.n,9T, Thudng z 1 ,ru' niry tuong Tin; Nguydn. Mq.nh Hd, 9K, La Lqi, Hd' DOng' '-, " =1. HDr NOi ; Diling NgQc Son, 9CT, Tit Liam' drfongv6ir=Y:z=1. Qu6ng Ninh : Etrt Ann Dtc 8A, TD Uong Bi' Nhan x6t : Bii to6n niy drrgc hing tr6m Hii Phdng : D6 Thity Chi, 8Ar, Hdng Bdng' ban grli ldi giai d6n. Tuy6t d4i da s6 giAi dtng, Ttranh H6a : Hd. Xudn Gid.p, 6Tr; Hoit'n'g Thi ngiriggn, batt nhtr tr-on. Chi cci mQt s6 it ban giei h-i dii."oTrong sd nhi6u ldi giei tdt c6 -: Dinh Hd, Hd Thi Phuong Thd.o, 8T, IIK gim Son. fuam Duong 9A NghQ An., Le Anh Tho,9-A I0r6nh H6a : BitiThanhMai 9T, L0 QuyD6n, Thanh H.6a, Trd.n NguY€n Thq I Hh Tinh, Nha Trang. TP Hd Chi Minh : Nguydn Cd'nt Nguydn Viet Hd 9 Hh B6c, D-d lhity Chi 8A Thgch,8r, HdnBEing, QuQn 5,. H-ai f nang, Trd.n Luu Vd.n 8C Ngqc t>am, Bt4 rd NcUYEN Thanh Hilie 9H HA NOi, Yd Anh Tud.n,9T, BAi T3/229 " Gidi Phuong trinh : Quing ginh, Nguydn. Tud.n. Trung 8T HII EA", Nguydn Thd.i Soz 9! Thanh H5a, @ -sr+2)(x2 * 15x *56) +8 = o Ngiydn"HiyVu, ST Ninh Binh, In lnhVinh LA,i giai. Ta cci : sil, Ha, N[i, rvguydz Dtc Hdi 98 Vinh Ph(r,. - s, + 2)(x2 * * 56) + I = 15x Nguydn Thi Thi.FIa 8A Quing Ninh, La Thd @2 fnd:ng 8H Hn NOi, Ld Trung Ki€n 9T }Jt6, =x4+12f+fi*-fiBx+120= Dinh-Trqng Quang 7C Hn NQi, Trdn Tq Dpt = 1x4 t of - ts#1 + @f + 36P - eo) - 8A, Ha NOi, Ha Thu Hibn Y6n Bdi'.' B4n D6 Nggc Dtlc (6H Trttng vrronglld NQi) - (8r2 14Bx - 120) = x21x2 + 6x - 15) + de phrnt bidu va chrlng minh bii to6ntd-ng.qu6t * 6x(x2 * 6x - 15) - 8(x2 * 6x - 15) = sau : Cho tu >- 3 at, an > 0. Chrtng minh ring = (x2 * 6x - 15\(x2 + 61c -8) = (r + 3 - 2r[6) I' * a2 (r +3 + 2\[6)@ + 3 +{17) : o. vQY -Phuong ,1 irinh c6 4 nghi|m. lit. : x1= -3+2'[6; (at...an+l) (-+...*4) %..*,+ xz = -3-2r[ 6,' rs = -3 + {I7 ; x4 : -B -'ln ' a3 &r, al 1I ... +d1..an_2* ----:-- 7 ar* ... Nh$n x6t. C
(Khr{nh Hda, 8 To6n L6 Quy' D6n, Nha Trang), Nguy6n Hrru Quy6n (Vinh Ph(, 9T Chuy6n Nguydn D6 Thdi NguyAn (Vinh I.ong, 9T, Phri Tho), HdVdn Son (Vinh Phrl, 9T Chuy6n Chuy6n Nguy6n Binh Khi6m, Tk Vinh Long[ Phri Tho), Nggc Bich Phuong (Tidn Giang, 9 Ng4ydn Hbng Quang (Tx Lho Cai), Nguydn Torin NK huy6n Cai LAy), Le Chi ThAnh (Hu4 Khanh Linh (Ha NQi, 9c THCS Ngoc LAm, Gia 9I Nguy6n Tri Phrrong), Nguy6n Hoach Tnic L6m), Hdn Minh Trung (Thanh H6a, 6E Sinh (Binh Dinh, 8A Qudc Hgc Qui Nhon), THCS Nang Khidu, Tp Thanh H6a),Vtt Mqnh Nguy6n Minh QuAn (QuingNg6i,9T Chuy6n Cudng (V[nh Phti, 8A Chuy6n CII Tam DAo), Nghia HAnh), Nguy6n Hoing Chrtong (B6c Dinh Trqng Hilng (Virng Tdu, 9T LO Quy D6n, Th6i, 9 To6n THCS Ntrng Khi6u Tp Thrii Tp Vung Tdu),'Nguydn Cd.nh Tod.n (Iuydn Nguy6n), Trdn Ngoc Cudng(Tp Hd Chi Minh, Quang, 9 To6n Nang Khi6u Le Quy Ddn), ?a BT, Nguy6n An Khuong, Ho
Her NOi BiLi Mqnh Hitng, 9H Trung z q6 Vrrong, Nguy 6n Minh COng, 9A Cflp II YOn Hba, Chri ), ringr,, . i,r"do dci a +b < = S' | Tt Li6m. Suy ra ring o = b. Vay Hmr. tdn tai. Thanh H6a : IIiLn Ngqc Son, 8E Ndng b) Trrong tU ndu r, ( r, t4.cring cri limrrrtdn khidu Thi x5,, Cao Xudn Sinlt, 9T Nga Li6n, Nga - Son. tai. Gqi gidi han ldA. Ta c6 A=t+a -* - Quing Binh : Trd,n Chi frdo PTNK D6ng A={2. Hdi. Quing Tri z Nguydn Hitu Nghi, gTL Nhfn x6t : C6c b4n sau cd ldi gi6i tdt : Chuy6n L0 Quf Ddn. Neuven Ti6n Drlng 11 la Nguy6nPhric NA-qg, Quing Ng6i : Nguydn Minh Qud.n, 9T, Xfiairn 11 HAi Hung, Trdn Nam Dung 11CT Chuy6n Nghia Hinh, QuAng Ngei. Neh6 An, Lo Hdnelia Vinh, D4ng Htru Thg, Kh6nh Hda : Bil.i Thanh Mai, 9T, Va ThY Binh Dinh, Nguy6n Anh Hoa 11A Nant Ha, Dung Hba, Trd.n Tud.n Anh,8T, Le Quy D0n, Phan Anh fiuyliOa N6ng, Ph4m V-en Du 11 Nha Trang. Thanh Hda,-Ddo Ngoc LuAn Hh NQi, Trdn Hfru Luc 11 6uinE BInh. Cd khd nhi6u ldi giai TP Hd Chi Minh z Ch_uy.g Nhd,n Phti, STr sai, kh&ng dirin aaf {rn} li tdng vi bi chen hoec Nguy6n An Khuong,ro" *i*.*rMrHty chrlng minh gi6i han ld i. BidiTsdzg Cho dau s: { *n} tnao md.n L < oANc HUNG rnANc. *r< Zud.xn*r= I +sn =Zoo > 1. Chtng minh B,di T71229 : Cho hd'm sd f(x) li€n tq.c ffan' [0, 7], c6 dao hd,m trong (0, 1) uit. f(O) = f(l) = rd.ng day {xn} n\i fip uit. tim gi6i hqn c&a n6. 0. Chilng ntinh rd.ng tbn tai mQt s6 c e. (0, L) Ldi giAi: C6ch 1 (cria b4n Nguy6n Minh sao cho f(c) = 1.996f'(x). Phrrong 10A Hnng Vtrong Pht Thq). HOi hdt ludn cfia.biti tod'n c9 thary d6i hhdryg ndu f(0) = f(1) = m,'udi m lit sd th1c khat 0 cho 3-(xn-l)2 trudc ? Ta cd : rn+l" = z Ldi giai : (cria La Quang Adry, LlCT - Tt bing quy nap dd d6 thdy 1 < xn < ZYn DHKHfN - DHQG TP Hd chi Minh) ; cao > 1. Khidd Thd Anh,11CT Qudc hgc Ilud ; Truon-g Vinh 1 Ld.n, 10CT PTNK Quing Binh ; N guydn N gsc l*,*r - ,lZl=;l(B - (%-1)1- G - (rl7-t;;21 Phtic,12I PTTH sd 1 Dtlc Phd - Qunng Ng?-r ; ,l Trd.n'Ti,dn Dfi.ng,11T PTTH Amsterdam, Hh NQi Trinh Httu Trung, 11T - LrT Son - = i.lr, - {Zl ,[i - (2 - x,1l I (1) flranfr H6a ; Nguydn Tidn Dung, Phan Anlt. M0 2 - *n < r/7 do dd_tt (1) suy ra lrn*, Huy, LlA1, 12Ar PT"itI 1.6 Quf Don - Dn N6ng) : 1 - rlTl . *r*, -,-z1,{, = fr' t *n -,[21 X6thim s6g(x) = ets%.f(x)x6cdlnht€n [0, Suy ra Tt cdc gshthi6t d6i v6i hdm/(r) suy ra him 11. g(x)lilntgc trOn [0, 1], cd d4o hdm trong (0, 1) lr"-rlzl . (#)'-' lxr-,El "u S(O) = g(1) = 0. Bdi th6, theo dinh Ii 1 -_, Lagrdng, tae6 :3c € (0, 1) sao chog'(c) = O (l). ' Vi lim = 0 n6n suy ra lirnr,, = {2 (U^)"-' -xrlr Md,: g'(x) = eGG L/(,) - re%/(')i vr e (0, 1), vd Cdch 2 (ctra ban NguySn Nggc Hrtng 12? Thanh Hda). Bingquynap d6 thdy I < xn< 2. -x ets% + 0 !r € (0, 1) n6n tt (1) ta c6 f(e) - T:2 X6t hdm s6f(x) =- 1 *r - f'(x) = 1 -r < 0 1 i.6 ir*- f(c) = 0hay f(c) = 1996f (c). (Dpcrn)' Vr € (1, 2) vhy fk) nghieh bidn trong (1, 2). Do dti Klii thay di6u kiQn f($ = flt) = 0 bdi di6u a) N6ur, > r, thiT'(r1 < flxr)'*xr< ro tidp tuc kiQn 1'(0)=-fl1) = m (rn + 0 cho tru6c) thi k6't nhrt vAy ta cd x, > x3 > x5 2 ...vd r, ( ,+ ( ... lufn cta bdi torln s€ kh6ng cbn dring vdif(x) lit Thnnh thtt hai d6y {rr*} vd {rr1* 1} hOi tu vn him bdt ki th6a mdn di6u kiQnf(x) li6n tuc tr6n giAsito=limrr* [0, 1] vA c
2" Ngodi cdc ban dE n6u t6n trong phdn Ldi BAi Tgi229. Goi AA1, BB tlit hai dudn"g cao gSdi, cac ban sau ddy crlng cd ldi giAi tdt : Lfirn cila tam giar nhgn ABC, M-uit M lh.n tuqt ld, Ddng : Phan Thanh Hdi (l2T PTTH Th6ng trung didm, crta cdt doan th&.ng AB ud. ArBr. Long) ; Qu6ng Binh: Trb,n Dilc Thudn g}f Duimg thd.ng CM c&t lai duimg trbn (ArBrC)Z PTTH Ddo Duy Tt) ; Thanh Hria : L6 Vdn Cuimg (11T' PTTH Lam Son) ; I.{am tl}r : T ud, duirng thd.ng CM, cdt tgi duitig-trdn (ABC) d T, Ch&ng mini rd,ng Trddi xilng u6i {€"y6? 7a1Q Hoa (11A PTTH L6 Hdng T qua duimg th&ng AB. Phong) ; Hn NQi z La Tudn Aruh (12ts PTCT: DHKH?N, DHQG HN) Ldi giAi 1. NGUYEN K}IiC MINH (Dtta theo ldi gif,i cta Trdn Bhi T8i229. Tinr cd,t cd. cd,c sd thuc a > Z Tdn D4t, BA, sao cho Chu Van An, n t, ViAr t | G_tz\dt ' -_,t to#*at2+L (1) He NQi). vd Br nim 8 tr6n drrUng Giii (cria da sd c6e ban), trbn dudng kinh AB, tdm Dat r : -Y ^ thi dx - 2(l - P)dt v- M, n6n ta l+tz (L+t1z drrgc : CArBr = AB.M ; (1) vn do dti trqt-t21at 1l dx AA : Jot4+"tz+t=rl AtBrC - ABC; (2) r*+*= (Ki hieu .^ chi ring hai tam gi6c nly ddng 1 :warclg- ,rE4 dang nghich ('ddng a?ng nhtrng rlrq" hriOng)). 2-, Lai vi 7 nim lrg\dudng trot u ,(ArB rC), n6n cri : Suy ra (l) i\(1) vd (3)/'--1-suv ra : v6iu =Y, o - - MABr= CTBlvidoddA, M,T,Brcing :---\\. /: trbn :. tt dri_.----- thu6c m6t drrdnE : X6t hdm s6 f(u) - aretgu - u> MAT = MB.T, hayli: BAT: ^ MBrT; (4) i" , O. - Dttdng thingMtsrld tiSp tuydn tai B, cria Ta cri f(u) = -[; f'@t = Q
tu.ing rlng cta hai tarn giSe ddng dang (ngh!ch) hay lA: , ABC vb,ArBrC, u6n ta drrgc (6), cring trlc li : ocz + GB + GC + GD)z; R2 - " *
BAi L2l22g. vbrvrErrirvn clr{r Cho m.qch d.i6n uE (tidp theo tang 1) dudi, udi Bgy gid,.c.r{c bqn hay dirng cric kdt guA tr6n x,*tr=48f/",di€n "_ glar cac bal tap sau oe : kd G c\ri 0,BA; u6n Rt= 3on .- Bbi tfrrr I : Chc Biri tflp Cho hinh '".uOne AtsAfl ennh wudno ABCD e-anh kdchi z4v. bing o. GQi M, N, P, q, li trune 0 la trunfdidm didm cta'c6r cta'c6c X.) Tinh dietu bd *nfi A-8, Ed:'cd.^iL canh AB; BC, CD, Cdc fiit"?11ai,EaN. :D, DA. Ua" C6c drr6ne thins AI/, AI/. q?,.Cq, qlq tdsrilcdfiGfl nhau tao thinh ffi"s)6cEFG.L{. g crta diqn kdud. gu Chrlnrminh "$ rinc etia udtt hd. I)EFGH in hinh vu6ng 2)SEFGH _ W IO t ),- 2) Tinh di€n Blr! tori,n to6n 2 : Dr.tng Dr hinh vu6ne ABCD bi6t trd x trong 2 vi tri fl ulnnJt vi vl dinhA va tritrirng vi Ert didm N cua Erung crem.{v canh BC. 6tra c4nh,6(_. trudng hqp chuydn x. sang ll AB thi : a) Diin ffi kiun mqch dat 6ng sudt c49 dgi p ri t6.m IO vihai didfiM p-!4.tam lf n&mfran didfrM. Nnim t"r6n naic+nti haicanh r,* ddi di6n crla""a.h+i hinh vu6ng. vu6ns. b) Di€n trd x dqt c6ng sudt cqc dq.i pon*. Bir! t96n-4 : Dgng,hinlr vudng ngoai ti6p mQt trl gi6c eho trttdc (D6 thi vdo chuv6n toair Eludarg d6n gi6i. DHTH vA DHSP, 1972) t') E = 1(r+r) +R2(I -I)+[JI-.>BA=4BI* Bhi torin 5 : Cho hlnh chtr nhdt ABCD vir +40(I -0,2E *24-->I = M mOt didqM chuydn d6ng tr€n canh BC. Phdn g76c DAM c6t canh -tsC tai N, Xric dinh w tri M UAB=R'z(I-tr)+tJv = 32V.-+s =ry *g - ia *3' "'# uu, gi6 tr! nh6 nhdt. tst=1oQ; uv U, (tidp theo tong 9) Rr=-;--= lv I -Is* R. U,. = 600 Q. *r.;( 7--++ fr +fr12 - lfr -fr111 1 UAB - ABz + ACz +; @p _ CBz) o\ D Z)Itas= =32O. a) Khi I r chuydn sang lll-B thi mach ngodi Do dri Ap=g;+-ff,n^, ld: crf di6n tr6I = cri c6ng sudt p th6a +c2 az *3= b2 2-7 #vi rwdn EI = rI2 + p. Tt dd L = E2 4rp > 0 --> DI - b) CdEh 2 ; Goi f]d trung didm canh BC. P** = h *t A = 0. So s6nh v6i bidu thric tdng Ta c
b) lim a, , n+*a nx-1 NcuvBtl utuu otjc *'*!: Bai T fi83 : cho thrlc SE RA Ki XAV + f{x) = xa 4x3 - n 2r?+l S = ! -: c2r2 - tzx + l' H6y tinh tdng v6i n li sd nghi6m vd r, Id Ehi T1/233 : Cho daY s6 nguY6n - ,!!r(xrt - 1)z o th6a rndn nghiQm cfia da thr3cf(r)' ""} ff = * a, : { * on) Yn > 1 ooan rri6 pHrE.. an *z - r = 2 {on n t (Nam Hd) Chtlng t6 ring tdn tai sd nguy6n M kh6ng BAti T8/233 : Cho ba sd thrrc a, b, c th6a m6n ' z sao cho ohu thu6c M *4an*r.an : di6u kiQn az +b2 * c2 = 2. Chrlng minh ring : ld sd chinh phr-tong Yn > 0 1)lo'rb*c-abcl abc' + BDz = P D2 + ABZ + BCz + CA2, Ct riog minh ring tdn tai trong khdng gran Ddu "=" xlY ra khi nno ? . m6t didm M th6a m6n N vAN TR6c AGr= BGz-- CGr= D6o (Qudng Ngai). Trong dd G G2, G3, 4ldn G lrrqt ld trgng tdm Bili T5/233 : Cho tam girie vudng ABC ( cta oic trl d iC\tt''1, MB c D, *1?*-T?fl"lMAB c' )^ = 9V ) drrdng cao AH, trung tpydn BM vir pt ao eia'" CD d"dngguy-tai rnQt didm' Chrlng (Btnh Dinh)' - {5 1 minh ring sinB = z Tt dci suy racdchdtrngm6t tarn gi6c ccj tinh cAc oii vAr li chdt n€u tr.n' BAi L1l2S3 : Tt 2 didm A vi B c6ch nhau vr oueic D,NG 100m. xe 1 vi xe 2 cilng xudt phiit v6i ctr-ng vfln (Bdc Thdi). ;6;;': :,;= L0 m/s' x"e 1 di tLec hrrdnghop v6i eS iAcOOl. niAt ring 2 xe so g?p nhau d C 'H,ey cAc l6P TrIcE --- dinh x6c : Bei T6/233 : Cho m la s6 thgc duong' - ifuang chuydn dQng cria xe 2? Vdi m6i n nguY6n drrong daY sd thgc - Th,di didm 2 xe gAP nhau ? - Tga do didrn c ?" I e,,,i\ f = 6 d,roc x6c dinh nhu sau : pHau soNc ouYd,r. ,1 (Hd N/,i) an,o= ! on,i*t= an,i ('*;;o,,,r) ooi BBi L2l233 : IvtOt diQn tich didm l=0, lr...n-l o - *2. 10-s c drlns c6ch tdrn kim loai phing irOt aat *6t Lt oang"o -= 3cm' H4v 16: dinh lgc en,n2fi+fvdimoineZ+ iJo"iiacsi,r" diQTr tich s. y? t6+ kim loai dd 2) Gie sil m > tr . Chring minh : khi dEt chd'ng trong chdn kh6ng i ltl - NCUYEN DUY TRUY uj on.n. vdi mol -z+&' rL e' (:fhdt Btnh). * 4 i$
rg ) tl a ^ 1)Provethatifnr.,{r"t+L @N$Ill{il we" uQlL itst * ";is?#"a7nir*> fi't 1 prove rhat: FOR LCIWER SECONDARY SChIOCILS u) on,r, < * lfor every n e Z+ n? - T1/233. Let the sequence of integers { :, satisfy _ ff ""}an * an _ +2 I = 2(an + I * an),Vn >- l. Pro_v-e that therq exists an integer M, not ",'*:,,*=#' poll.nomial T71233. Consider the depending on z, such that f(x) = xa +4F -2x2 - 12x I 1 . !.2x!+t is a nerfect TZl233 Provd =o,ru"J#BJ "h , a. that if (a+c)(u+b+c) 4aia +'b'+ c) . T-8/2P3. Thg-non positive integers a, b, c, d Where n is the nurnber of roots and satisfy- the conditions : x; (i = 7 , 2 , ... , n) ate the roots of f(x). )", + 2b2 + Sc2 + 4d2 = 96 (1) T'8/233. The real nurnbers a2 +b2 + c2 = 2. Prove that : o., b, c satisfy lzoz+b2-2d2=6 jz (2) l)la+b*c-abcl + Day lh qich lArn ggn vi. d5 hidu. Tuy n-hi6n, cdn O dAy, theo t6i vi6c viSt CA, B$lh4lrrra trr ccf i:rich trinh bdylhric hop li hon. ub6a*"i hoc sinh thildng vi6t AC,AB ho4c Trddc ti6u, dhring ta'iem c6c tdc giA vidt : CA,AB (kidu ho6n vi vbng quanh). Mat khr4c, "Ggi.I l& trung didrrecanh BC. muc dich li timmocbn b! "khudt". Theo tdi nghi, +-+ ..-t cd thd trinh ba'y theo 1 fiong2 crich sau dAy : Tacri: CA=IA-IC -..) -+ a) Cd.ch I : BA=IA-IB-t f lA trung didm canh BC. fheo $S t$1c Goi Qg dci :* trung didm ctia doan th&ng :2 AI = AB + AC CAz +BAz = 4+^++ lry: * ABz (rA - rc)2+ (rA - rc|t: 4AIz -r8.AC AB2 +ACz +2AB = + ACz + (xem tidp trang 7)
- codnfr cfro cdc 6aru cfrudh 0i tfri r.'do dai froa $ru$ $uru$ t E{cm PHAH vfmrx un#r af,qn* NcuvEN TI{ANH GIANG (Hdi ltwng) o Nhie lai : Dlnh nghia tich phAn (GT12) h Cho/(r) xde dinh tr6n [o, b] Lint = [ 161ax Brr6c 3 : Kdt lu4n Sn 1. Chia dogr [o, &] thenh z p]rdn b&ng nhau - tt+ @' a bdi (n + 1) didrntti",, 1i =ffi1nhu sJu trudnshqp ' ffo= a
+ LimS, = J e.osttx * =Y . n+** g. li = ,. VD4: TInh **) (, * ?-l , F.*\* tr6n [0, 1]. ,yT-l(r Chia [0,.1] thdnh'z phdn blng nhau bdi c6e D4t didm ehia *i=* Q=7e. Tr6n m6i doan ,,='l(r.,*) (,n?.1 vd ?":\: i._ s,=rn Pn=i[* (r+) *...nr" (r+)] f*i x)ldy E, = (i = Ln) ; t;=*;*;-r=1. :_t, * Ta ed tdng tich ph6n X€t harn s6 y = f(x) = ln (l +r) li6n tsc ; ttnl1 tr6n [0: 1] IGi dd S- li tdne tich phdn ctia him f(il trfun [0, 1] coi pH6p chiE [0, 1] thanh z.ph'dn'blng nhau bdi cric didm chia x, = I vd ehgn L :1+ 1 _,"" Ei = i-1 i '1e lxi - p x) (i = Ln) v6i Li = i. "?'uS-e)' Do dd limsn= [tnql*1ax = rtr(.Itu) - 71+*m o l1 dx 1, 11 ,':T!.:[#=*i eari= cosf, (, = [ E,t1) il (;r =l -I OO # ,+ lim n+*o P, =rn2 - a* * : 1+*o I lim do = I # , JTJ" - = 2tnz e?tnz - L -1 ' + dx = -%osirtt d.t x : : .xL 1 0 =+ / [ *=t* ; Ta x6t tdn [a, a7 marig rl du *a him l6v" t{ch phin x6e dinh lt t0 ; 1] o _ - ,). U Do dti Jt lt . l- I sln- xt . xt Sln- zsrn Ltt zxt - r '" -2sint dt = 2- n Tinh tim\l "=+--+ - n-+&z lL *..#** , **7fu* '=;tu# ,a [o':'1*=Z-X:Z , fut ln -'l nsln- I ++-fr1 '^t^yT:"=t nill 1 *cossp2 o , \lDB : (D6 thi DHQG IIaNOi kh6i D - n6m t99b) Tim . I sin{ zri','u n n * Det ' s:fr'l n + lT;Det (r +cos L +"n T. *.o,4o) "z lt ln +co*f, r +.oir%t ' .nnf Tn-n ?n.2n ,,,:*(r +""i +.*3+...+"o r**): nlnlnnrun 7/ n-l *"'4 2n xL 0'-n, ,*"r[l _oLrn$ -h,,T-t=-t._-.=--T--r =n_\:': cos: +coff + co#) 1ao.&- ." " Xdt f(x) = costtx, tr€n [0, tl TXtt ..tL1t thinh z phdn blng nhau bdi crtc Chia [0, 1] 4Stn- orem cnra xi=i(L=on) chgn €i= (i=o;n-L) +...+-nn =*,2,'(T) n 7+ ^llxt cot- n thu6c [r, 1, xiTvdi i = I7 *,a a, = -L. - * n-l ,r-r1 i I , g6s Yi 0,n': =+'t'r(o+'+) -)r"G) o, = a4" "* .; = ; ( .. + cosL+... + 1 Kftf(x) =,ffiLi6n tsc tr6n [0, z] r?. "orL-;o)=sn (xem tidp trang 13) 11
r,$ il]r1ilI ilrilrli $fiIlfiIft rfilfir; IT,illllt ilr ffi l'fi [Jl[i iffil I I NcuYEN THUC HAo @a Nai) Trong hiuh hgc phd th6ng, ngrrdi ta dlnh Ta suy ra nghia drrAng elip vd hypebol, vdi 2 ti6u didm F, Lx, a7 = -_r1ta,bll @) .F.1 ve fing Ai6u ki6n li tdng ho4c hi6u cta [r, b] = Ela,b) l kho&ng cacfr tt didm M ddn F vi .F''khdng ctdi' Thay vdo (2), ta duqc Ndu ia tdng, qui tich ciaM ld drrdng elip ; ndn 62+q2=t (5) l}r hiQu thi quy tich ciraM lA dirdng hypebol' Cbn Ch* !. - Cdihd minh d6 ding ring paratrol thi lb qu! tich didrn M ei'ch ddu mQt OAt OB lh,hai b6n "n,frrg kinh li6n hop, trlc AA', BB' Aldm cO dinh.F vA dridngth&ug cd dlnh A' Cflng dubrug elip' td. tzai dubng h,inh tihn ho.p crta cbn cci dinh nghia chung cho 3 drrdng, vdi mQt ti6u didm .F' va dribng chudn tudng rXng A' Di6u kiQn det ra trh ti sdp cria khoAng c6ch trJ M ddn II" Dinh nghla dudng h5rPebol F vi A ph6i Id hing s6. Quf tich c&aM li elip,. Ch.o ba d.idm. A, A, B kh\ng th&ng himg' Ta hypebol hay parabol ld tny theo,u < l, lt > t, s€ gqi tit hypebol qui tich didnt' M sao cho hiQu hay li pr : l. sf-crta nai iirun phuong d,iQru tich cila hai tam' Nhung trong hinh hoc afin, chring ta kh6ng giar MAB, ruOd &tio thf tg) biing binh thd lim nhu vf;y, vi khod'ttg cd,clt. lit' nt|t khdi pt uong d.i\n iich cila tant gid.c AOB' Trlc li niQm khdng c6 ! nghia gi trong hinh hqc afin' (hinh 2) Thd cho n6n sau dAy, ta sE dirng diQn tich afin (ctra binh hdnh hay tarn gi6c), li mQt bdt bidn oBI42 - OALrt - OAB2 (6) Qcd phrrggg trinh-gecto- bing cdch dat afin, dd dinh nghia 3 c6nic. OivI = x, OA: a, OB = b I. Dinh nghia duirng eliP Phrrong trinh (6) s6 vidt drroc li Cho 3 d.idnt. a, A, B khOng th&.ng hdng. Ta sd gqi l&--"lip quy'tich didnt M sqo cho fx,bf2 -fx,a)z - fa,blz (7) Phrrong trinh theo tga dQ, vdi co sE {o, b}, oAr,I2 + oBlli2 = oAB2 (l) s6 cci drJgc, c6n cri vlLo (4) : trlc lir : Tdng cfia binh Ez_rf=t (8) Ta thdy rlng O li nm ctia drrdng hypebol, phuong diQn t{ch hai didm A thudc hypebol.cbn didm B thi khdng' tam gid.c MOA, MOB Hai dttdng thing chtra OA vd OB ld hai dildng bd,ng binh phuong diQn tich tam gid.c hinh han hqp. OAB (xem hinh 1) Chil !!6t thaYjigu ki0n (6) bing R6 rdng le A, B OAI,I2 - OBIO - OABz (6') thuoc dudngelip. Cbn thi quf tich cria M s6 cd phuong trinh O ln tdm (ddi xttng) Fllnh 1 €2 -'12 = -l (8',) cta elip. Ta h6y tim Dd la dudng phrrong trinh crla drrdng eliP. hypebol liAn hqp (Trlc tarn girnc OAB ln m0 t tant gidc lieghqp) ' criadrrdng (8). Nd Ta h6y d4t x = OiuI, a OA, b = OB Ldy - chrla didm B mi diQn tich binh hdnh (gdp doi) thay diQn tich tam kh6ng chrla didm giric, phrrong trinh (1) vidt dugc li A. Trong hinh vE +fx,blz = {a,bfz lx,a72 (2) dudnglf l}dudng Dd ld phrrong trinh vecto c{ta drrdng elip' cd phrrong trinh Mudn c
trII. Dinh nghia dudng parabol v6i kf hi6u nhu tr6n, ta cri phuong - "Ch.o 3 di&n kh6ng r-,-C"Ol{- trlnn sau : thary hhng O, A E. Quy tich didrn M d lx, a12 + [x,b]2=$fx-a, x -b)2, lt = const. citng mlt phia uoi A hay ld dli uoi ditong thing OB, saa clio binfi. Z(lx, alz+[x, b1z)-tre fx, b]*fx, al+labl)z=g + phuong dien tich Am. (2 -p)([x,a)z +fx,b]\ +2pfx,al[x,b] gtlc OIIA bdrw tfch * 2p[a, bJ[x, a b] p[a, bJ2 O (13) - xj crta dicn tictl OUA - - = ugi diCn tich OAB, nnuon6,1riYltfii,;'#l*o AUW lg rrfut dudng (14) -So! parabol. trong dci A(rr) lA d4ng toin phrlons Ta r"f di6u ki6n A @x) = (2 -p) ([x, a]2 +lx, u1z1 +4i.1x, al oMAz =dW . olB (9) nP,li9rrrfflgn;i2o_, 6 bl (1 b) {Det plia drrdng_lhing oB (hinh B). "u" -va Sgtns (16) OM = x, OA: a, OE'= b vd a ld hing sd Phuong trinh vidt drroc li a- -pfa,b)z (17) lx, a]2 = lx,b)la,bl (10) Ctng vdi cdc ki hi6u nhrr trong qic phdn Chuy6'n sangtga d6 theo (B) va. (a) ta s6 dtioc tr6n, ta se cd phuong trinh sau crla-c6nic, theo ,t :6 - ^2_ (11) toa dO €, r7 cia.M rlring OB p tidp tuy6n tai O. phuong (z-p)(€z+rtz) -zp€rt+zp(€+D -p = 0 (18) ^ , ?.ual.g phtrong cria parabol. UA!4vi6n Trong phuong trinh ( 1 8), ta cci dinh thrlc cria IV. Dinh nghia chung B c6nic dang todn phirongli _ Tr6n dAy, 12-u -u chfng ta d6 dinh l-p z -p =(2-p)2-p=4(t-p) A)= I ' o' nghia ricng fi6i dudng dudne c6nic vi. Ta suy ra ldp phrlong lQp phuong trinh 1) n6ul < 1 thi @ ) o, ta cd gidng elip cta chfns du6i 2) ndu p > 1 thi @ 1 O, ta c
Db THI eudc GIA cHoN E{sc sINH ctot roAx r,op g NAwr rrec Lees - Lee6 NcUYEN u0u ruAo (Hd Nai) Bing A (180 phtit, kh\ng hd thiti gian b. Trong m4t ph&ng tqa dQ xoy (o li gdc tqa giao db) dQ), ngudi ta v6 mQt drrdng trbn cci tAm ld didm Bei l. a. Tim tdt ch c5c sd cti hai cht sd o6 C t3;4), bdn kinh bing 2 don vi' Hey tinh 916 tri nh6 nhdt crla tdng oic sao cho , -'" ,,ld la-ol sd nguyon t5. khoAng crich tt didm M tr6n drrdng trdn tdm C b. V6i 100 s6 t1t nhi6n bdt ki, h6i c L)' x'r xi x; rl-r, +r[*, +... 1fr, -,--___G- Chrlng minh rhng : -+-+.,.+- xi- xi^ xi ttn BAi 3. Cho didmA c6 dlnh vihai didmB, C - di dOng sao cho AB = a, AC = b (a, b ld hai sd drrong-cho tru6c). Ngudi ta vE tam gi6c ddu Bei 3. Cho tam gi6c nhgn ABC vdAD, BE, BCDlao cho A vd D thuOc hai ntla mat ph&ng CF ld cdc phdn giric trong cria nri. Goi S, vi S ddi nhau mi bd la dudng thhngBA.. ldn lrrgt ld diQn tich cria c'5c tam g75c DEF vit Hdyx6c dlnh d0ldn eria g6e BAC khi/D ctj d0 dai l6n nhdt. ABC. a. Chfng minh ring 4So < S. .-qAi 4. a. Cho drrdng fiin (C) nim trong gdc b. Vdi m6i didm M nim trong t am g76c ABC roy (dudng trdn (Q3.h6ng cci didm chung v6i (iVI kh6ng thu6c c5c cqnh cria tam $6zABC), goi cric canh cira gdc xoy) .Hay tim tr6n drrdng trbn a', b', c'ldn lrrqt le d0 dei cl&a cd;c khoAng cdch (C) mQt didm M sao cho tdng eic khoAng crich tt M ddn ede c4nh BC, AC vdAB ; tim tfp hop tt MJI{A hai dudng th&ng chrla c6c c4nh cria nhirng didm M th6a m6n hQ thtlc a' < b' < c'. g6c xoy ln nh6 nhdt. (C) nim trong gdc --&Ai 4. a. Cho dttdng trbn roy (drrdng trbn Glkh6ng.cri didm chung v6i b. Trong m4t phing tqa d0 xoy (o li gdc tga d0), ng[di ta vO mQt dudng trbn cri tAm ln didm c6c canh cir a g6c xoy) . H6y tim tr6n drrdng trdn C (3 ;4), bdn kinh bing 2 don v!. (Cl mQt didm M sao cho tdng c6c khoAng crich Hay tinh gli td nh6 nhdt ctra tdng cdc tr) M-lifu hai drrdng thing chrla c.4c c4nh cta khoAng cach tt didm M tr6n dtrdng trdn tdm C g6c xoy li nh6 nhdt. ntii trdn d6n hai trgc tga dQ or vd oy. 14
DAP AI\ BingA L996 *x2 = (xy *yz *za) *x2 : Bhi 1 Cnu a) Ndu sd o6 th6a mdn di6u ki6n = x(y + x) + z(y + x) = (y +r)(x +z) bii torin thi 6a cring th6a m6n di6u ki6n bii Tt d b (1996 + z1 _ 0+x)(x*)(tt+z)z _ : a.b a.b (1), v6ip nguyOn td vd 1996 *xz $*)(x*) la -bl a -b =p = (g +z)2 0
Bai 3 Cd.u a K{ hi6u BC = oc'[-r- bc ac ob 1 a,AC=b,AB= uu Lr @+b)(c+a) - -- (o+b)(b+c) (a+c)(b+c)) = 3(S * Ser,r - Saor' - Scoa) - 3So c, AD, tsE vd CF lA c6cdrrdngphdn gr6p criX cdc g6c hayS - So 2 3So + S > 4So(dpcm) A, B vir C, Tac6: Tnr6c hdt ta chrlng minh bd d6 : AFACb Cho mQt gtic nhgnroy vd M lir m6t didm nim BF- BC- a - trong g6c xO y. Goi r' vdy' ldn lrrot li kho6ng cach AF BF AF+BF AB tit M d6n cdc canh Ox, Oy ; ggi Oz ln phAn gi6c ciag6cxoy. Chrlngminh ririgr'Jdi 2abc - Ndu N thu6c mi6n trong c,&.a g6c zOy,6rp : c(az + b2) > 2abc dgng crich chtlng minh phdn thudn d tr6n, thi lc& + cb2 t NX, > AIY,. Di6u niy cfng tr6i y6ti-giA thi6t. non : b2c +bc2 + a2c + ac2 + a2b + ab2 > 6abc Vay.N thu6c mi6n trong cfia gcic xOz. Ap dung bd d6 nny dd chrlng minh cAu b : +s-s,=1rTayff|1"*,; s= Ggi O li giao didm ctra ba dttdng phdn gi6c AD, BE vdCF. 2abc Theo giA thiftM nim trongtamg16cABC. : oQ- ""(a +b)@ +c)(c +o)' Vio' < b'n6nMphAi nim trong Lam g; c BCF Vi (a + b)(b + c)(c + o) Vib' < c'n6nMphAinim trong tam $6c CAD = Ydy M th6a m6n di6u kiQn a' < b' < c'+M 2abc * ob2 + ab + acz + azc +bc2 +bzc nim trong LCOD kh6ng kd cric cqnh crla nd. = 2abc + ab(a +b) + ac(a + c) + bc(b + c) Bni 4 n6n 2abc = (a +b)(b + c)(c + a) - lab(a+b)+ Cdu a +a.c(a +c) +bc(b +c)l - Phdn tich : 2abc Tt didmM'bdt ki (a+b)(b+c)(c+a) tr6n drrdng trdn +b) + ac(a + c) +bc@ + c) (C),haM'H t Ox, _ ab(a M'K t Oy. Dltng -7 -a (a +6)(b +c)(e +a) ti6p tuy6n v6i drrdng trbn (C) tai =| M sao cho tidp (b+c)(c+a) (a+b)(b+c) (a+b)(c+a tuydn niry cfit Or 6 16
A, cdt Oy d B, AA = OB, vd cdc didm O vd C thins qua s(3 - 21[2 ; nam v6 hai phia cria ii6p tuydn ndy. 4) .l#iff4':tt$y?*g ?a thdy ri.ng: -# M'K . OB +M'H . AA = 2S*noo hay ,, fea dQ cri" #; il 1*1',u.f "u, hd phriong trlnn : (M'H +M'I{)OA:25*nou fy:x+l x:3-{2 suy raM'I( + M'H =T ly:-x+7-212'y:4-\12 Gqi S le tdng khoAng c{rch tr) M ddn Ox vd. !i sdc cho tnldc vd dudne trdn (Ct cd dinh Oy, ta c6 : n6n tidp tuydn AB xdc dinh vd Ol,: OB kh6nE S: 7 -^[2. ddi. Do dd IvI'K + M'H nh6 nhdt khi Srrru nh6 Bing B nhdt, mudn v&v di6n tich L,M'AB phAi nh6 nhdt piti ! (a, b) : xem ldi giei d bAng A. (vi tam s16c OAB c6 dinh). Mu6n vdv M' M Bd.i 2 (hic4ay-dttdng cag hq tt M'xudngAE bang= 0). C!.u a.' Theo gia thi6't, cric sd 1 - o, > 0 Vi itz la tidF didm'n€n CM l. A? (1) Ap dung kdt qu6 criabdt dingthric C6-i,ta6: IvIat khric L OAB cdn (OA : OB), n6n phAn sdc O phAi vu6ng s6c v'6i AB. -pr.dcTtOz(1)criava-(Z) (ar+(l-o,)\2 t Cdch dune su'y ra CM lfdz. : " o,(1 -r,.,*l Z l=n - Duns'pEan si6c Oz c:iua Edc xOy \tt - T& Cli6 drid"ns thine sofrs soni v6i Oz cdL Triong h!: ar(\ - a) < 4 (Ci a hai didm. Di6m M."mi t'i6p tilvdn tai dd chia mat ph&ne lim hai'phdn. sho cho O va C nam d hai-nrla mat phine d6i rihau cci bb ln tidp ae(7 -rr) tuydn tai M, ta diC'fi phA'i tim. = * Chilne minh QuaTI duns tiSn tuvdn AB. + [or(1 - o1)J . la2! - o)] ... x [o.(1 - os)l \n CNI r- AB v'd CM ll'Oz"n6i Oz" t AB. Oz vita ld drtdng cao, vta li phAn eidc cta s6cAOB n6n A AOB-cdn (OA: OB). ' x ranl- os)r < l,i]' ,-, : liv M' bdt-ki tr6n (C),tac6 (M'K+M'H)OA 25u.eoa > 2Snoo: (I/IE + MF| . OA Tt bdt ding thkJ {*) suy ra : Trong 9 tich sd Suy ra M'K + M'H > ME + MF.YQy ME + dA cho it nhdt phAi cri m6t tich khOng l6n hon 1 Mtr'nh6 nhdt. vi ndu kh6ng tich nio nhrt v6y thi v6 trdi 4i cci Bi&*thn : Di c69 *9y Rhgp, vuong nav tu tht giaotidri M de cira bdt ding thfc (*) sE l6n hon uo ndi d tr6n lu6n lu6n x5c dinh. Bdi trrii vdi kdt quA tr6n. ft)' ''u., Cd.u b: xem ldi giAi d bAng A. to6n lu6n'lu6n cd nehiOm hinh. Khi Bdi 3 (xem hinh v€...) C"nam tr6n Oe thi Dtrng tam giric m6t trons hai ddu ABE sao cho gido didm 6.&a Oz vdi dtldnstrdn (C) dinh E vi dinh C p1rai tiin. ln didm D-hai tim. cing thuOe nrla m6t Cqq 6 ,: Ap.dgng. \6t.qyh_clt1a;didm M c
xAc DlNl-l rAM DUONG TRON LAm thdnAo dd x6c dinh tAm cira mQt drrdng trbn cho tnl6c md chi dirng compa mQt ldn vd dirng thtldc kh6ng qua 6 ldn ? V6 KIM HUE Gidi dfip bdi DOAN NGAY SINh NHITNG cAY sd ni r+ Gqi ngny sinh ctra ngridi mn Th6ng do6n alf 1.23456789 x 8 + 9 : 987654321 t6, than; sinh Id cd. ndrr- sinh li Tgrnn (vi c6c ban cria lf,atrg d6u sinh vdo thd ki 20). 12945678 x 8 + 8 : 98765432 Theo c6c ph6p tinh mn Th6ng d6 d{t ra ta cci : 1234567x8*7:9876543 *33 = { t(a6 xz +tt) 5 +22)10+cd} t oo+rmzrz 123456 x 8*6 =987654 : 10000 x a6 + 77000+100 xda+19331i1n: : aFUOOO +cd6A +tnn + 78933 = 12345 x 8*5:98765 = a5i[fri + 78933. 1234x 8+4 = 9876 Vi kdt quA bao gid cf,ng ld' adc4tntx + 78933 123x8+3=987 n6n b4n ThSng ehi cdn liiy kdt qu6 cria cdc b4n trrl di ?8933. Trong hi6u s6 cbn lai thi 2 chrt s6 12x8*2:98 hdng tr6m nghin vd hing chuc nghin chi ngiy sinn] hai cht-s6, hang nghin vi hdng tram chi 1x8*1:9 th6ng sinh, hai chit sd hing chuc vd hdng don vr chi ndm sinh (da bdt di 1900) cira ngudi 123456789 x 9+ 10: 1111111111 Th6ng dorin. 12345678 x 9 + I = 111111111 Tt kdt quA dri ta tinh drrgc ngiy sinh cira Hdng : 110115-78933=031182 1234567 x 9+8 = 11111111 tric ld Ngdy 3 th6ng 11 nam 1982 123456x9+7:1111111 NgiysinhcriaNgqc: 229813- 78933 = 150880 12345 x 9*6:111111 trie ld Ngny 15 th6ng 8 nem 1980. (fheo Doitn Hd.i Giang,7A, NK Qulnh Lrtu, L234x9+5:11111 Ngh€ An) 123 x 9+4: 1111 NhQn xdt : B4n Phqm Luong Anh Minhl6P 52, Nguy6n Dinh Chidu, QI, TP. Hd Chi Minh 12x9+3:111 ctng dadua ra c6ch dorin ngiy sinh cira Th6ng 1x 9+2=ll nhrr tr6n. Bd Nguydn Thi Tuydt (139 Dudng 30 - 4, P.5, TX Cd Mau, Minh Hii) vd rdt 0x9*1=1 nhi6u ban dA cti giAi d6P t6t T}{ANH -IUAN BiN}I PI{TJONG (Suu thm) Tda sagn"xin trd tdi clzuig cho c6.c bgn a6 *ta hdi ub cii,e utd,n _dE sou : , *Thbihqnrabd,o:Tod,whoeud,tuditrdraryQtth.arie*otxi.uit'ocudithd.ng. * Thbi ho.n.nhQ,n bdi gid,i : Hai thdng tinh tit igi'y cd:6i iit,a thd,ng *zti bd,oph6,,.?.""F - Tim. rn'ua bd.o taanlqe tudi trd d d.d.u : Bg'rt c6 tk€ d1f mua d.d.i hgn' tq'i B*u d'iin dibu hoq'e kian rnua tq.i cac C0n6 ty sath ad. thidtb{ trubng hgc trong 1d. n*6c. !,ba !1\Ort l!,0n5 10 '";e;;n";'si*plai bin auq*. Cdc iq,n 6 rid NOi co thd ntua tq,i bl T'rd.n Huns, Dq'o, '25 Hdn Thuyan, 57 Gid.ng V6... Cd.nt on cd.c ban. ISSN:0866-8035 Sip chrl tai TTVT NhA xudt bAn gi:4o duc In tai nha mey in Di6n Hdng. 57 GiAng Vo Gi6 2.000" Chi sd: 12884 Hai nghin ddng M6 s5: 8BT35MG Inxongvir nQp hlu chieu tn6ng 1111996