Toán học và tuổi trẻ Số 205 (7/1994)

Chia sẻ: Physical Funny | Ngày: | Loại File: PDF | Số trang:20

Thêm vào BST

Báo xấu

64
lượt xem 6
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Toán học và tuổi trẻ Số 205 (7/1994) trình bày về phần nguyên của một số và bài toán chia hết; đường vỏ sò và ứng dụng của nó vào thực nghiệm của phương trình bậc 3; tâm của hệ điểm và khoảng cách giữa tâm của hai hệ điểm; bài công thức nghiệm thu gọn.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Toán học và tuổi trẻ Số 205 (7/1994)

. Bo "GrAo DUC vA DAo rAo 2 N -I (2051 , 1'rii! 'i 'i 1994 '|Ar UrlI t(A NGAY rb 11ANG 'I'riANG ff l) A I " 2 A ar fw tot,n$ -F riass' [Hfl]B[Iitf,lt,llfllEg Ai Gilo WEfu"L d' c'14 Aithfip Aa a'*-"* ALF H,R *'O> Q.s d.O SS I$ RH, rlr) -uQ T I F) Q. Hr Hoc sinh_chryryn lo.dry,_Tin hgc, Vat li, Ngoai ngrt tuudng LA Hbng phong TP. Hb Ch; Mitlh trong ngity ph-dt thu-dng cu6l nam hoi rHr,ruYE\r yAq; lep ro CituyErrt TC)AN TiTtDFIG.D}ISP iIA rr[6[ r
roAN Hec vA rubr rRt MATHEMATICS AND YOUTH fu{UC L{"JC Trang t Ddnh cho cdc ban phd th6ng trung hoc co sd For Lower Secondary School Leuel Friends Nguydn Van Vinh - Phdn nguY6n cria -"otiO vb. cdc bdi todn chia hdt I Tdrug hi0n ffiP : t Giii bdi ki trudc NcuYgN cANH t o.it't Solution of problems in preuious issue Phd tdns bi6n ftP : Cdc bdi cria s6 201 3 NGO DAT TU t Bun cd bidt ? HOANG CHI.]NC Do you know ? Nguydn Cang - Dudng "V6 sb" Nicombde vd ilng dung ctta nci vdo vi6c dung nghiQm thrlc cria phrrong trinh bic a 9 nOl odtto nlEru rge : t Db ra ki ndy Problems in this issue Nguy6n C&nir Tohn, Hoing C6c bii tt T1i205 ddn T10i205 vd Chrlng, Ng6 Dat Trtr, L6 Khic Ltl205, L21205 10 BAo, Ngrry6n IluY Doan, o Thi tuydn sinh vio litp 10 chuy6n todn NguySn Viet IIAi,Ilinh Quang DHSP HA NOi 12 HAo, Nguy6n XuAn HuY, Phan c Hoc sinh tim tbi lluy KhAi, Vrl Thanh Khidt. LC Young friends' search in Maths Hai KhOi, NguY6n Van lvlau. Nguydn Thdi Hd - MQt con drrdng di tdi bdi to6n m6i 14 Hodng L6 Minh. NgaY6n Kh6c o Tim hidu sAu thAm tudn hoc phd th6ng Minh, Trdn \ran Nhung. Further study of school Maths Nguy6n Dang Fhdt. Phan Trl.n Quang Vy - TAm cria h6 didm Thanh Quang, Ta Hdng vd khoAng cich giila tAm ctra hai h6 didm 15 QuAng, D4ng llirng Th6ng, Vt1 o Giii tri todn hoc Drtong Ttrqy, Trdn Thdnh Fun with Muthenlatics Trai, L6 86 Kh6nh Trinh, Ng6 Binh phuong - Ai ho gi Bia 4 Xud.n Trung - Ai cao ai thdp ViQt Trung, D4ng Quan Vi6n' o drg kinh cdi crich day vd hoc totln Kaleidoscope : Reform. of Maths teaching and learning Nguy1n Dic Td.n - DOi di6u v6 bdi c6ng thrlc nghiOrn thu gon Trr-1. sd tba soqn. : 81 Trdn Hrrng D4o, Hh NQi DT:260786 $ian. ftp tri stl: VO NtLl THUY uit. 231 Nguy5n ven CiI. TP Hd cni nninn DT: 356111 Trinhbay; DOAN UbNc
Dirnh cho cilc b?n pnd tn6ng Tnung hgc co s6 '? PHAN NG['Y E N GITA IUTOT SO UA GAG BAI TOAN GTTIA HET NGIIYEN VAN VINH a, -l l6p 7 c{.c ban da bi6t : Cho r li mQt sd tt n = mq * r suy ra Mat khric, \Jhrru ti Uat ki. Ki hi6u [r] (dec ld phdn - q v6i r - q 5m, n6n n = (m + l)q + r nguyOn criar) ld s6 nguy6n l6n nhdt kh6ng vrrot t rflt qu6 r. Ching han [2,8] = 2 ; f-3,781 : -4 ; ru
MQnh dd 5 : Ndu sd nguYAn 6 P c6 mQ.t trong phd.n tich ra thita sd nguyAn fi ctta n ! h - l;1.1*5. .lr\v(izn- 2) hhong s6 s6 s = I *;*;*i*...+- cri chfr s6 tAn ctng 0. phd.i lit. sd nguyan. Trong 100 sd cria 100 ! cd 50 sd chin, 50 b) Ching minh rd.ng tdrtg s6 16, vi vAy chi cdn tim tdt cb cric b6i sd cria 111 +... + 1 hhons phd.i td. s6 5 trong tich d6 cho. S = ;*;*Z * Theo m6nh d6 5, s6 mf cao nhdt crla 5 ctj nguyAn (De thi vao chuyen to5n vdng2, 1973) trong phin tich ra thrla sd nguy6n t6 100 ! 'n!n!n! bing: +T * "'*; 2 Gi6i:Tacd:S-I= lfl.l9e,;.1#s = 24 Viy tich ! tin cing Id 24 chit s6 100 s6 0. Tt mQnh d6 5 ta suy ra m6u s6 z ! chia h6t Vi aB 2. Chtlng minh rd.ng nau (n - 1) ! chia cho mQt luy thrla c:iu,a 2 c6 b5c cao hon cria ttr hdt cho n thi n hh1ng phd.i lir. sd nguy€n t5. sd. Di6u niy chtlng t6 S - 1 khongphAi ln s6 Giii : GiA srl nguqc lai z li s6 nguyOn td, nguy6n, do dd S crlng khdng phAi li s6 nguy6n' suy ra n chl c6 hai rr6c sd ld 1 vi n. Theo m6nh b) Cd thd lam tudng t1t nhu cAu a. dd 4, trong d6y stl 1, 2, t1. - 7 c6 .n-7-- Vi aq 5. Tim luy thita a cao nhd.t cil.a 7 md' |Lnl | = 0 s6 chia hdt cho z. T\l dti suy ra 1000 ! c6 thd chia hdt cho 74. tich : 1.2.3 ...(n - 1) kh6ng chia hdt cho n, tr6i Gi6i : Theo mQnh d6 5, lfly thta o cao nhdt v6i giA thi6t (z - 1) ! chia hdt cho n. Viy n crla s6 ? cd trong khai tridn thnnh tich cac kh6ng phAi li sd nguy6n t6. thrla sd nguy6n t6 crla 1000 ! bing : Vi au 3. Ching minh rd.ng n ! khdng chia hai cho 2n. .1ooo. .7000- r 7ooo1 Gi6i : Theo m6nh dd 5 : sd mfl h cao nhdt "=L;)*L *)*lr*)=764' cria sd 2 cti trong phdn tich ra thr)a sd nguy6n V4y 1000 ! chia hdt .1ro 7164' tdcriaz!li 2
Dilc,8A'n Ng6 Gia TU, Hei Htrng ; Bqch lfdng Y, Nguydn Phuong Lan,9T, PTTH NEng Khidu Hh Tinh ; Dd Dang Tao,9F, chuy6n V.T Ilng Hba, Hi TAy. 16 NcuvEN fildi TZl2Ol t Tim cac chtt sd x, y, z, t tnd Bai T1/201 : Tim x e Z dd s6 25x + 46 lit ilch crta hai sd nguyan li€n tidp. xy.yz=ztz(x*y,z*0) Ldi giai z Cd.ch l. (Cria Dd.m. Khanh Hba, Ldi gini : Ta cd .y y. > 10x. 10Y 9A, PTTH chuy6n Luong Ven Chenh, Phri = 100ry < 1000 do dd ry < 10 YAn). GiA sif cd r, n e Z sao cho \t y" cri t{n cing ld z nln ly - l)z i 10. 25x+46 = n(n.*l) Ta c6 cAc tnrdng hqp sau 10. Khi d6xz*7 -- tOa+b *25x *50 = n2+n+4 (a > l) vd. z = x * a. Vi z > r n6n. +25x * 50 = (n + B)(n - 2) + lO (3) z>4.Tac6zx*1:l0a*b+ Ta c6 (n+ 3) - (n - 2) =5 ndn n * 3 vdn - 2 ho4c ctng chia h6t cho 5, ho4c cing kh6ng z2 - za* l. = lka +b.*#= aeZ chia h6t cho 5. Song, trong cA hai trrrdng hgp 6dd-1
nghi6m). MOt s6 bdn cd ldi giei dring nhrl Qwynh,9T Phan BQi ChAu, Ngh6 An ; Trd.n Nguydn Ngoc Hurug 9A Hoing Hria, Thanh Thi Khu€,98 Quang.Trung, Nguytdn 1tJgO" H6a,To Quang Chinh Nang khidu Thrii ThUy, Hung 9A NK Hoing Hcia, Thanh LISa ; Vu Thrii Binh nhrrng ldi giAi cdn ddi. Huy Hod.ng 8T, PTNK HAi Hung ; Hodng Bd.o DANG FIUNG THANG Nam SArPhong Chdu, Vinh Ph:u; Pham Anh Tutin gNKThudn Thdnh, Hi B6c ; Pham Huy Bai T3/201. Bidt cd,c. canh cila, tum giac Tilng 8A Bd Van Din, Nguydn Hbng HdA 8H ABC ld 3 sd nguy1n li€n tidp. Tim rlQ dtti cd,c Tnrng Vuong, He NOi.. csnh crta tum gi6c ndu : 3A+28:18P Nhi6uban khricgiAi dfngnhungvidt ddi ddng. Ldi gi6i : vO rru rnuv Tri gia thidt srly ra BdiT4lz0l : Cho p lir mdt s6 nguy€n duang ^ 16 hhd.c 5, d ld, mQt sd nguy€n duong khdc nC=24+B + C > A, C >B . Do d6 AB > AC, AB > CB. 2, p, p + B. Ching minh rarug til tQp hqp Tr6n canh AB ldy didm D sao cho AD = AC. tl : t2, p, p + B, dj c6 thd tim duoc {t nhdt -,--\\1800-ArC=180"- Tacri CDB = ^ ^ rce-i 3 cQp @, b) khd.c nhau trong dd a, b ld. cd.c _ ph71n tit khdc nhau thuQc td.p t qp M sao cho ^ =180o-(A+B)=C z ab - 1 khOng lit. binh phuon g cfi.a m\t sd nguy€n. Quy udc cQp @, b) uit. cap (b, a) ld nhu nhau. Ldi giai (cira ban Phqm Huy Tirng 8A, Bd Ven Dan, Ddng Da, Ha NOi ; La Anh Vu l0 CT, Qudc Hoc, Hu6) * TnI6c hdt ta chon duoc cqp (p,p + 8) - | = m2 - Thdt vAy ndup(p + 8) (p+4-nt)Qt+4+m):17* Tr) dd LABC oo LCBD + [n*+-m=7 BC DB ln++*m=17 suy ra P : 5 trrii gii thidt, AB: CB+BC::AB.DB + Ti hai c4p (2, p') vd {2, p + 8) la lu6n chon hay BC2 : AB(AB - ACS 1x1 dudc it nhdt m6t c6p. ThAt vdy n6u trrii lai thi Theo bniruAB - BC : t hoac AB - BC :2 2p - | = y2,2(p +8) - I = x2 a) Ndu AB - BC = 1.+ AB = BC + t vit +x2 -y2 = 76
2d : (c, -b)("t *b,) : D6 thdy Qk) : x2. Tir gie thidt cria bdi ra ta c6 P(x2) "21-bi li da thrlc. Dit hi6u hai sd chinh'phrtong chia cho 4 cd drr 0, : t hodc 3. Vi d 16 n6n 2d : 2(mod 4). Mdu P(x2) = +ort*-l +... + apc *a,, thudn. + DPCM. "d D6 thdy, hdm sd (bidn r) P(x2) lll hdm ch6n Nhfn x6t ldi giei-t6t : Nguydn : C6c ban c 0. Chnng minh. rdrug lRtnne'ux>o (qo*q-")2>2@2+q-2) P(x\:t-" zrlux 0 vh T(x) : S(rl) vr < o Ninh Binh) ; NguydnThdi Hd (Mari Quyri HN) Suy ra : R(C) = T(x): S(rr). Do drj Kh6ng mdt tinh tdng qu6t, coi q > 1. ViSt rB(r) = S(r), vd didu niy chrlng t6 P(r) Id da bdt ding thrlc dl cho drr6i dang thfc (DPCM) (qa _q-ayz > 2@ _q-rf NhQn x6t : Ban Vit Thi B{ch Hit, IIC trrrdng Chuy6n ThAi Binh ctngc,r, -l Ban LO Van An cho ldi giAi sai vi chrra hidu (1) O dring kh6i ni6m da thrlc vi kh6ng hidu dung X6t hdm sd de bar. NcuyEN rHAc urNH f(q) = qo - q-o - ,t, (, -'), q > I 1 _, 1. BdiT7l20l. Tim mQt hdm s6 f(x), xd.c d.inh f'(q) : + q-n) -'r/, (, * tri thuc cia x, sao cho udi x bd.t hi udi mgi gid. 4la(t" Yio>r[2,q.-111'6n i)l : f(f(x)) -x ud f(x) nhdn gid. tri nguy€n hhi x nguy€n. f'(q),-ff rr" +q-o)- (n *i), = Ldi gi6i. (theo Nguydn Thdi Hd 10M :'{i | (oc -n\ *Q -Q'I : Marie-Curie Hd N6i, Nguydn Huy Hoitng llB g L'' ''' qo*l )- trrrdng chuy6n Vinh Lac, Vinh Phri vd L€ Anh tt ,l :;lr"-q)\1-r"+t)>o Vit l0 C? Qudc Hoc Hu6) : V6i m6i z e N ki hi6u E, ld ntta kho&ng (n ; n * 11. D0 thdy, v6i Vfiy f(q) li hdm d6ng bidn trong [1, *oo) m6i r € R* ddu tdn tai duy nhdt z € N sao Do dci : cho r € En. Xdt hnm /(r) duoc x6c dinh bdi : f(q) > f(l) :0, dPcm r* 1 ndu x e Ervdi n ch5;n NcuyEN vAN -r*1n6ur€Envdin16 rraAu fk\ = Bei T6/20l : Xdt hd.m s6 P : R - R th6a 0 mdn : Ndu Qk) lit da thic u6i hQ sd thuc c6 bdc >- 2 thi P(Q@) cung td.mQt d.athic. Chrlng Ta sE chrlng minh hdm f(x) th6a tdt ch cac minh rd.ng P lit mQt da thic. di6u ki6n cria d6 bai. Thet vAy, tnrdc hdt, d6 Ldi grai @ia Nguydn ThAi Hd 10M thdy f(x) xdc dinh tr6n ,R vd f(x) nh6n gi5 tri nguyOn khir nguy6n. Tidp theo, ta chrlngminh Marie-Curie He NQi) : X6t da thrlc f(f(x))=t YxeR.
r : 0 hidn nhi6n cci Dpcm. x V6i TiI cAc c6ng thfc (1), ta lai cd : *Vdir>0tacci; ld lb lc bia c*a a*b * Ndu x € Env6i n chin thi x + 1 € En+i -a-a-: ABC Zbccos2 2cacos 2ab cos2 vd ru * 1 16. Do dd : f(f(x)) : f(x * 1) = a-L- , - :-(r*1)*1:-r (b + c)2 (c + a)2 (a + b)t *Ndu x e E,v6in16 thi -r * 1 < 0 vi lABC, = Z (cos7*cos7*cos7) < x - I e En_r, n- 1 chan. Do dd : f(f(x)) = 13\[i s- (dpcm 6 b) (vi ta lu6n luon * ;.; = i{3 = f(-x) + 1) = -f(x - 1) = -(r - 1+ 1) : -r. * Y6i x < 0 thi -r > 0 vd do d
Hidn nhi6n ta cd P(xo, y,,) :0. Nhu vdy, [,ai vi tam glac OAD vr6ng 6 O, n6n ta dtloc : ta c5 7 didm thu6c Fsao cho P(x, y) = 0 (1). HA. HD = -OH2 = -h2 (5) Thd y bdng x2 vio vd trrii cria (1), ta cd : Suy ra : P1x , x2): 0. Day ld m6t phuong trinh da thrlc : {Pal(?) HP . HA = -2h2 (6) nguy6n bAc 6 d6i V6i r th6a min v6i 7 g1d tri khac nhau cria r n6n P(x, x2) : 0, hay moi (oP-ottxoe-on:-2ffi2 didm thudc Faeu th6a man P(x, y) = 0. Lai : coi P(r, y) ln m6t da thrlc nguy6n bidn y, ta cci Sau khi thay OH = h vd. OP . OQ h2 vdo thd viSt P(x, y) : (y - x2) Q@, y) + l?(r) (2) (6), ta dqg" , trong dii Q @, y) Id thrrong vn R(r) ln phdn oP+oQ=4OH:4h; (7) dri. Mqi didm ctra (P: y - *2 = 0 ddu th6a min hQ thrlc (2) vA. (7) chrlng t6 rnng P(x, y) : 0 n6n ,R(r) = 0 hay (2) trb thdnh _Cac OP = xt vd OQ = x)ld hai nghiOm duong phAn P(x, yt : (y - iS Qk, y). Do P(x, y) ld 1 da thrlc bAc 3 ddi vdi r, y n6n Q @, y) ld m6t da bi6t cta phuong trinh bdc hai sau dAy : thrlc bAc nhdt ddi vdi x, y hay cd dang ax * by x2-4hr+h2:o; (8) * c, vd P(x, y) : (y - x2) (ax * by + c). Cric didm Phrrong trinh ndy cho ta 2 nghiOm : a, F, T kh6ng thudc Fndn tai dri y -x2 * 0, xl hQ +t[51 suy ra ot. + by *c: 0 hay chring thu6c dudng ,2: thingoe*by*c:0dpcm. Nhfn x6t. 1. Bdi torin -rdn dring v6i moi cdc h6 thrlc : lop_= tz gn -{ipn hinh cdnic. loe = e +{s 2. C6 5 ban giAi bdi ndy vd giAi dring. chrlng t6 ring P vd Q li hai didm cd dinh. Dci oAttc vtE,tt ld dpcm. Nhfn x6t : Dring tidc ld 2 ban giAi chrta Bei T10/201. Tu diaru OABC uu1ng d O c6 dat (vi ng6 nhAn, kh6ng chfng minh cdn thAn dudng cao OH c6 dinh, cdn cdc dinh A, B, C de kdt ludn P vd Q ld hai didm cd dinh). Chrra chuydn dQng. Gqi A', B' uit. C' ll.n luot lit hinh chidu cia H bAn OA, OB ud OC. Ching mi,nh cri ldi giAi nio hodn chinh, hodc vi chrra chr.lng rang 6 didm A, B, C, A', B', C' cil.ng nd.m tr€n minh cdn thAn 6 didm d6ng cdu, hoac vi chrlng m6t md.t cd.u ud. mQt cdu ndry luin luOn di qua minh chtra thdt chinh x6c (do khOng bidt srl hai didm cd dinh. dung d6 ddi dai sd). Ldi giAi tuong d6i dat hon Ldi gini : Tt c6c tam giric OAH, OBH vit. c6 ld cfia hai ban : Ddn Thu Huong vd. Pham 99H_ vu0n8_ 6 _ H:_ta cri ; Diruh Trudng,l6p 10 chuyOn torin Trdn Phu, : : oA. oA, oB. oB, oc . oc' = oH2 :h2 HAi Phbng. (d4t oH = h) ) (1) Qua b6n didm A, B, C vD. BanNguydn Thai Hit.10M, Mari Quyri Hd A' kh6ng d6ng ph&ng cd m6t m6t cdu (8) duy NQi, c
ffAn ffuc chinh d trudc F m\t doan a d.d quan v4y Uouit. Io cingkh6ng ddi vi ti 16 E =?Io sdt mOt udt pharug cuc ruh6 AB ddt trudc hinh ud,t sao cho dnh cita AB mit mat nhin thdy d cring kh6ng ddi, cdch md.t mOt doqn I (xem hinh). Hdy ching minh lit d0 b)i gidc thu duoc lit - Khi x, !, z n6i tidp nhau thi LD a, En : In(r * x I y * z) fl = r I x + y + z (l) ---> G: u (I +;)tt fl: \ - Khi x, J, z song song nhau, hi6u di6n thd D lit. khodrug nhin rO ngd.n nhdt cia mdt chung cta mach U, : z.Io vd ddng di6n chinh li I.z Irz ngudi quan sat, L lit dO diti quang hoc cia I:I uvx kinh hidn ui (khodng cdch tit ti€u didm dnh -.trldd cia k{nh uQt ddn ilAu didm udt cia k{nh ntat.t, ," , z. E=Irt1+-+-\+Iz- u \ y x/ (, f 1uit. f ,ldn luot ld. ilAu cu cilu kinh udt ud cia hinh md,t. Tit d6 suy ra c6 ntd.y truitng hcp mit, t E:r+-+-*zrz rz M* yx (2) dO b1i gidc tl sE Oang I - Khi O, ll z) n * x, hi6u di6n thd chung cira Itl t y vd z ld Ur, = z Ir: ddng di6n chinh -SIu,Zt I__+I-:I y (' (,\l1+_\:taccj y/ Eo=1,(r.i)(r*x)izro rz xz --->E =r*x*-*-*z vv (3) Tt (1), (2), (3) suy ra , = *i9 , -), r:1r +r; B"L)" r E,, (x2 +rx +l;1r +ry Hudng din gi6i. Y€ 2 tia sdng di tr) B. T:, = r) m6t tia song song voi truc chinh vd m6t tia V6i r= 1f), r : 2e), I(,: SAta cdy - 6e . qua 07, di qua kinh vdt vd kinh r.rrdt.2 tiall z = 12 !) vi.8,,: 105V. kh6i kinh m6t cci drrdng k6o ddi c6t nhau tai NhAn x6t. C
Ban c6 Uidt ? S.udng "o$ s&" MC0e{fnf v& t?ng dUng etia n6 vio vrgc dung nghrQm thqc cfia phuonglrinh bqc B NGIIYEN CANG A - Dudng "v6 sd" cira NICOMEDE - VE theo cd.ch Nicomdde dd ding : Nhi to6n hoc Nicomdde 6 vtng Tidu A (thd Ta dtrng m6t cAy thu6c (xem hinh 2) 6 tr6n H tflf hai tnr6c c6ng nguy6n) da dua ra m6t d6 c62 didmM vi.M' sao cho MM' :2. Didm drrdng cong dd giAi bai todn chia ba m6t gric bdt ki. Ta dd bidt m6t gdc tir ndo ctng phdn tich duoc thdnh mdt gcic vu6ng vD. m6t gdc nhon. Chia ba m6t gric vu6ng ld vi6c d6. Ta chi ban ddn vi6c chia ba m6t gcic nhon bdt ki. irtllt:iit,i.ii:.l.1it.:.Miiii,i .,.i.iii,iiiii:i, ..'.,.. i . ,i , ,,,, ,,,..,,:' .:',' . ,,,,,,,, , ,,,,,.,,,'.,,. .,,..,.", ,:::: .. ,',,.,,,,' ,:,, ":,:::,:::::::: , ,: , ,,., , .,..:.i , ,,,, , ,:::, ::,:,:::::,: ::::::::, *i,j,i i,.,t,i,i.i.i.i., ll:,lltl:lt::l:],li]lll ll Hinh 2 M s6 chay tr6n drrdng thing vu6ng g6c (D) vai AB di qua C vd didm M' sE vach n6n drrdng conchoide do tr6n thtt6c crf mdt cdi rdnh vd trong rSnh dri cri con chay. Didm A di nhi6n drroc cd dinh. Bay gid ta chri y ln dudng conchoide (N) gifp ta chia ra ba phdn bang nhau g6c BAC. Thgc vAy nhin hinh v6 CE' ll AB. Didrn ,E' 6 tr6n conchoide n6n EE' : 2. Ndu F lir trung didm cira EE'th\ FE = FE' = FC = AC = l. Ydy cdc tam giric AFC vd FCE' le. cAn do dci O hinh v6 tr6n, BAC ld. m6t g
Bdi T7l2O5 : Cho da thrlc : P(x) = ao* arcosx* arncos2x* ...* ancosrux nnn tri € 8"+y =, nhAn giri duong V"r R Chfng minh : 0,, ) 0 RA Cdcl0p PTCS KI NAY Bei T8/205 : Cho AABC Chrtng minh ring, trdn drrdng trbn dle cta DNO I RTJONG GIAN(i c6 cdc carrh a, b, c, Bai rr/205 : rim nem sd khac nhau rrong tffi*$ii',ri? !b1(:?l iyi.r itirjrt9Sr,':t; d6v tinh sau dAy : DAM VAN NIII 1**+**+**):r.*:** Bei T9/205 : Cho didm M bdt ki trong Bidt ring : LABC. N6iA]l{, BM, CM c6tc1.c canh d5i diOn a) Trong ba s6 hang trong ngoic, thi cri m6t tudng rlng tai Ap B 1, C,. Chrlng minh ring : s6 hang Ie BSCNN cria hai s6 hang kia. >4 b) Sd chia Ii sd nguy6n td vi li USCLN cria hai sd ncii tr6n. NGUYEN KHANII NGUYtlN PHAN VAN DUC Bai T10/205 : Mat cdu n6i tidp trl di6n Bni T2l205 : Chrlng minh ring nd'i.t a * b, A.AA4, tidp xric v6i mat d6i di6n dinh A, tai b*c,c*avd Bi tgqi-d,ld drtong thing qua trong tAm mat ddi a(l +bc) -b -c all *co) -c -a di6n dinh B, cua trl di6n B ,BprB ovittuOng gic mat d6i di6n dinh,{ c',ra tf dran-A,$$gq, fi=I}\ b -c a -a Chring minh ring cic dudng th6ng d' d6ng quy. a(l+ab)-a-b :P TRAN DUY HINH a-b thi 3p2 - a2 + 1 : 0 C6c dd Vat li TO HAI Bei L1l205 : Cho ry li truc chinh cria m6t thdu kinh, S li m6t didm s6ng thAt nim tr6n Bii T3/205 : GiAi phtrong trinh ry, S' Ii Anh thAt cria S qua thdu kinh, .F, Id ti6u (\t, +r - \ (\t1 -x, + 1) :2x didm vat cta thdu kinh. Bang ph6p vE hinh hoc, hey tim vi tri quang tAm O c,ia t[au Hntr. lE ou6c nAN Bdi T4lz0i : Tam glec ABC vu6ng 6 A c6 ffi VQ AB = 6cm, AC = 8cm. Biri LZ|%O1 : Cho mach di6n nhrr hinh v6, Goi 1ld giao didm c5c drrdng phdn gi6c, M R, : 2Rl - 6R, R., = 2Ro: 8-R. Hai dbn Dt li trung didm cira BC. Tinh s6 d,o g5c BIM. vd D, gi6ng nhau, cd hi6u di6n th6 dinh mrlc ld, 24V, 86 qua girl tri di6n trd cria dAy n6i, vO HOU eiNH khcia K vd cdc ampekd. Khi K dring chi 1,5A, a , Bei T5/205 : Cho drrdng trdn (O; .B) d{ng a, chi 0. drldng trbn (O', R) sao cho tdm O nim tr6n 1. Hey drrdng trdir (O', R').Ddy AB c.iua dudng trbn x5c dinh s6 (O, R) di d6ng vi tidp xric v6i drrdng trdn chi cria o , (O', R') tai C. Xdc dinh vi tri cria day AB dd vi o,- khi ,* ACz + B0 dat giri tri l6n nhdt. m6. NGUYEN DT,C T,{.N 2. Khi K rii m6 hai ddn iii: C6c l6p PTTH D,) vi ,\ sang D.* blnn Bai T6/205 : Cho 10 sd nguy6n dttong thudng. a1t ,.,t oro, chfng minh ring tdn tai cdc sd Hay tfnh c, € {-1, 0, 1} (i : 1, ..., 10), kh6ng ddng thdi a) Hi6u ,ri r::,,:i:r, 10 di6n thd ngu6ii U'; sr bang O sao cho s6 L crai chia hdt cho 1023 b) C6ng sudt ti6u thu dlnh mrlc cta s6c i:1 bcing ddn' TRAN XUAN DANG LAr rHE H'EN 10
T7l2O5. The polynomial P(x) = @., * olcosr + ... + ancostlx takes positive values for all r. Prove that oo > 0 DAO TRUONG GIANG For Lower Secondary Schools Tll2O5. Find five distinct natural numbers T8l2O5. Let a, b, c,be lengths of sides of A from the following equality ABC. Prove that on the Euler circle of ABC there are two and only two points P such that 1*x + xx * I'-*): ** = ;; (1) p4z16z- a2S + PB21cz- o2) a pg21q2- br)=o Provided that a) One number in the bracket is the least DAM VAN NHI common multiple of the two others Tgl2O5. M is a point, inside L ABC. AM, b) The greatest common divisor of those BM and CM meet the sides opposite to A, B two numbers is a prime number and is equal and C respectively at Ar, Bland Cr. Prove that to the divisor of the left side of (1). lAr *\ IBM*\lfcM > PHAN VAN DUC 'l *.^, *E *", 4 T2l2O5.Show that if a * b, b * c, c * a arrd NGUYEN KHANH NGUYEN a(l +bc) -b -c a(l + ca) -c -a -----b-c:----;-;-= T10/205. The inscribed sphere of a tetrahedron A\*fu touches the face _a(l+ab)-a-b _,, (i : a-b opposite to A, at Bi l, 2, 3, 4). then3p2-o2+1=0 Let d, (i = l, 2, 3, 4) be the straight line TO HAI passing through the center of gravity of the 'Tgl2O5. Solve the equation face opposite to Bi of the tetrahedron (\fi +, BpF3B4 and perpendicular to the face -, (ttl -x * 1) = )y LE QUOC HAN opposite to A, of the tetrahedron ArA,yA,rA,r. Prove that d r, d2, dj, dohavea common point T4l2O5. Given LABC withi = 9ff and. AB :6cm, AC = Scm. TRAN DUY TlINl{ .I is the common point of the inbisectors, M is the middle point of BC. Calculate the value of BIM. Th0ng b6o VU HUU BINH T51206. The center O of the cfucle (O, R) Nhu de thong b6o tr6n sd s(eo0 nerrr 1994 lies on the eircle (O', R'). A, B are points on TC ToSn hpc vA tudi tr6 nhfn ddng c6c Anh, (O, R) such that AB in a tangent of (O,, R') at ph&n 6nh cAc dnh thrlc hoc to6n vd dqy to6n a certain point C.Determine AB such that c0ng nhU c6c sinh ho4t khr{c cia c6c hec sinh AO + BCz is maximal. ve thAy gi6o chc l6p chuyGn to6n. chuy6n tin, NGUYEN DUC TAN c6c trrldng chuy6n, tnldng ning khidu cila c6c dia phUong (Tinh, huydn, thi xe v.v...). Cho ddn For. Upper Secondary Schools nay T6a soan chua nhfln dtlQc anh n6i tr6n cta TG|?O5.. Given 10 positive integers a1t ...t nh'rbu trudng, nhibu dia phuong, d{c bi$t l} c6c oro. Prove that there are c, € {-1, 0, 1} tihh, thanh phd pl'u'a Nam. '10 (i = 1,..., 10), not all 0, such that}cra, is ' Bdt mong nhan , t- drlqc Anh c0a c6c ban dd i:1 ddng tr6n Tqp cht To6n hoc va Tudi tr6. divisible by 1023 , TRAN XUAN DANG TodN Ho. c vA rud: tnf 11
l-- NGAY THr THU NHAT 20 - 7- 19e3 (150 phnt) I Cnu I. Cho da thtlc T P(x,y) = 4xy(*2 +y2) -6(*3 +f +x2 +y +*y2) +g (x2 +y2) 1) Hey phAn tich P(* , y) thdnh nhdn trl' 2) Tim tr6n m4t phing toa dO t$p hqp nhirng didm md tqa th6a mdn di6u ki6n P(x, Y) = 0. dQ @' y) c:{ua chung c< = CAu IL Cho hinh thangABCD' bidt'4.B ll CD vdAB = a, CD = b' DrIdnB thinq /T\>, q,r. gi"o aie- cria hai aria-ng chr'o vd song_ song 6 E vitF. Tinh dO dai irqikf theo o vd*b, v6i AB vd"chrlng cin-ce2 minh canh rilng EF b6n < AD {@b vd BC . Z, cnu III. Chrlng minh ring v6i a > 0, he bdt phrrong trinh sau v6 nghiem : D, [*r.y2-1 1rr'.2x2+a li-u{V
Cdu D(. Tr6n m6t phing toa d6, rn6t didm A(xo , !) drioc goi -la dldm nguy6n ndu x6t 4 sd a2, , . . . a, dd dudc hai s6 khric cfing crf hi6u chia hdt cho B. Nhrr vAy trong tich p xo,YoeZ cci it.nhdt hai hiQurkhric nhau ctng chia hdt GiA sir AtAz . . . An ld, m6t n - gi6c l6i c
FC ND MA Nhfln x6t : ,p . Cho ACDA c6 1+ 7117_ MC 1) Ngay thi thf nhdt g6m nhf,ng bii todn NA.MC chi dOi h6i nhfrng ki6n thrlc PTCS dai trd, v6i FC 'Ti; .fr (1) cho cci ddi h6i ki ning B mrlg trung binh ho4c trulg FD EC ^cDA BD AM 1+ EC BM . AC .^- binh kh6. Tuy vAy vdn cci m6t sd thi s!nh'b! didm O. Nhi6u em da chi ra ring ciu III thrla ED' BM' AC = ED = Bo . Ltr4rzt giethidta>0. I (A) 2) MOt s6 em chring minh ciu,!.l) bing Cho AADN "., ND BM Ca = crich chrlng minh c6c ttt gSde n6i tidp, nhypg kh6ng kh6o n6n da m6c phAi nhtng sai ldm Tt (1), (2) vi (3) suy ra ddnglidc v6 l0gic. Cd em de bi6t dnng ph6p vi (FCIFD):(ECIED) = 1+FCIFD = ECIED. tU dd giei cAu V. 3) n6n ldi giAi khri gon. Vay F cd dinh. 3) Cric bii to6n 6 ngAy thi thf hai ddi h6i 2) Ta s6 chrlng minh ring n6u EA = EB kidn thrlc vi trinh dQ tu duy cao hdn, vi mang thi1vIN ll AB. Th{t vdy, giA srl rdngEA = EB s6c th6i 16 n6t cria chuyOn todn. Do dr5, nhi6u em dat kdt qui k6m mdc dir de lnm bdi rdt t6t mi drldng thingMN c6t dudng th&ng AB tai P. trong ngay ihi thrl nhdt. Chi cri duy nhdt m6t Khi dd, ta crlng ding dinh V M6-n6-la-uyt ttlong em d4t Aidm tdi da trong ngdy thi niy. tU tr6n ldn ltigt cho c6c tam g76eABC, ABM, vit 4) Cd kh6 nhi6u em m6c phAi sai ldm l6gic CBM iln &tac PAIPB - EAIEB = 1, v6 ly. khi giAi cdc cdu \Ti, VIII, IX. Rdt it em ldm Tt MN ll AB, suy ra FM =.FN vi hai tarn drjoc tron ven cau X. gl6cMCD vdNCD cd diQn tich bingnhau. Vdy Tcim lai, ki'thi ndy cho thdy nhin chung, m6t vi tri thich hop cria B tr6n tia Ex li didm c6c em cbn ydu v$ hinh hgc vd c6c bdi to6n dbi , ^l: th6a m6n ding thrlc EB = EA. h6i trI duy vd lfp luQn l6gic. NGUYE,N HUY DOAN c(,[Za+,'lZb-c) Hoc sinh tim tbi +,[Zc(a + b) > 4.! * c2=+S < Ding thirc xay ra 2(45 + c21 Mong nh{n drJdc thd trao d6i cdra o4c b4n. 1.4
l. TAm cria hQ didm : Trong hinh hoc V6cto chring ta dd gdp c6c bii torin nhu : Cho ba didm A? A'A3th!_!uOn cri3aOt di{gn 0 duy =ii4 to-i,*e}l, ___) i=l =i oi' = no?':d nhdt sao cho OA, + OA2 + OA3 : 0 (didm 0 +OO'=O+O=O' chinh lA trong tAm cria tam giSc A, AA). M.t Vdy cho n didm A1,A2...A,rlu6n t6n tai m6t cAu h6i rdt trJ nhi6n d6t ra cho chring ta ld cho z didm A1, A2, ..., An c6 hay chang m6t didm didm O duy nhdt sao cho 2 &, = d Tr gsi a+ i-1 didm O ld trong t6m hay vin t6t ld tdm cria 0 sao cho 2 Oa,: O. Ndu cri thi dung didm 0 h6 didm A,,A2...A, i =1 ra sao ? Vn didm 0 dci c dr, : 6 rr dd 2 oa, =i:1) oa, + o1\*, = va M bdt ki thi i=1 +:1 ___> -__ = nOO, + O4*1 = O n Didm O + : d
Thtrc vdy :2 k : kk a1:f) -ir|ue') nffq i:1 Lr-P -Ze-r:\Ir,lP, i=l i-l k + o,o) = *f) p,q ara')rt - 1 MQ: =- \2,tr ) -i r> n,a?) R * t?:fi) =MQ2 : ,kk i \S,o',Y o,o, Dqt : Al1 - d aij i BF, = b,, i ApBq : "pq ) =*n- (\*4 ):l +2>ffe, ffrrrzt i=l +dt:- 1 (> 2 ci,) _ nLn p:1q=l Met kh6c '.21t[*P, L,I-P, = r (> ,3 t n MPI + MPi k - (MPi-MP,\' :n. it-j="ii), -;n" icr:t t" () a; ) (**) / = 22 ffP, Nr'P, : V6i kdt quA tdng quSt (x*) chung ta c mlt - 2 i
DU$NG "v6 so" NIcoMEor vA UNG DUN6 cfin ruo vAo vlEc DUNG rucnlEM THl,Jc mue p$-iu6Ne Yninum mac s (Tidp theo trang 9) di6u ndy ddn d6n p < 0. Ta giA sitp * 0 dd ,, :2\ f t; cosoo loai trir nghiOm tdm thrldng. Francois Vibte (nhn to6n hoc Phrip d tird t
Gidi ttuip bdi ArHQCiZ Mu6n tim xem ai ho gi, ta dua vdo hai di6u ki6n : - Anh t6n Vo ho kh6ng phAi ld Hodng (1) - Anh ho Ly thi cci t6n triing v6i ho cira anh c