Tạp chí Toán học và Tuổi trẻ: Số 227 (Tháng 5/1996)

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Tạp chí "Toán học và Tuổi trẻ - Số 227" ra tháng 5/1996 gồm những bài viết: Tìm hiểu những hệ thức vecstơ trong tam giác, đề thi tuyển học sinh đại học quốc gia 1995, số Ả Rập, trò chơi đoán cầu, điền số vào hình vuông,... Với các bạn chuyên ngành Toán học thì đây là tài liệu tham khảo hữu ích.

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 227 (Tháng 5/1996)

Bo GrAo DrJC vA DAo rAo * ser roAN Hoc vrET NAM tep cui RA NGAY 1s IIANG THANG x DE THr TUYEN SrNH DAI HQC QUOC CIA 1995 THI TUYEN * nd Trrr rrryfiru srNlH Lop to DAr rroc su prram uA nor XSO L2 A RAP * rnd crlcrr ooArrr cAr-r * DIEN SO VAO HINH VUONG .^^2\t Giam doc S& Cia" dwc - Ddo tclo Hd Npi NguyZn Kim Hodn cing hgc sinh gi6i tntdng Hd N)i - Amsterdant
\t TOAN HQC VA TUOI TRE MATHE,MATICS AND YOUTH MUC LI.JC Trang Tim hi|'u sAu him todn hoc phd thing To help young friends gain better understanding in school Maths Tdng hi|n ftp : My Du1, Tho - Tim th6m nhttng h6 thfc v6r:to NGIIYiN ('ANII'IOAN trong tam gi6c Phii td ng bidn fip : N(iO DA't'T'[, Gidi bni ki trudc: I I()ANG (ll ItIN(r Solutions of problents in preuious issue Cdc bii cira sd 223 o Kdt qr,ri ki thi chon hoc sinh gi6i qu6c gia l6p I 8 nOl o6ruo BIEN rAP : o D'A ru ki niil Nguy6n CAnh Todn. Hoang Probl.ents in tltis issue Chring, NgO Dat Tt1, Le Kh6c T1 227,.... 7101227, L11227, L21227 10 86o, Nguydn Hry Doan, DA ili tuydn sinh 1995 Doi hoc Qu6c gia Hd N6i. 12 Nguy6n ViOt Hei, Dinh Quang Hdo, Nguyen XuAn Huy. Phan Di tti tttydn sinlt ldp 10 Do,i hgc Su pltqnt 15 Huy KhAi. Vu Thanh Khidt, LC c Bun c'6 hidt Hei Kh6i. Nguy6n Van Mau, Hodng L6 Minh, Nguy6n Kh6c Do you know ? Ngttydn Co.o Thang - Sd A Rap Bia 3 Minh, Trdn Vdn Nhung, Nguy€n Dang Phdt, Phan o Giii tri lttdn hoc. Thanh Quang, Ta H6ng Fun u'ith Mat hetnatics. QuAng, Dang Hirng Th6ng. Vu o OnE hinh cdi coch da3, ud ltQc toan. Bia 4 Drrong Thuy, Trdn Thnnh Kaleidoscope .' reform of Maths teaching and learning Trai, LO 86 Kh6nh Trinh, Ng6 Viet Trung. Dang Quan Vi6n. DOng Ky Pltong - MSt d6 to6n sai Binh Phuorzg - GiAi d6p bdi Trb choi dodn cdu. Ngd Hdn - Didn s6 vdo hinh vu6ng. Trtt sd ti.ta soan : 45B Hirng Chudi, Hh NQi DT:8213786 BiAn fip uit. tri su ; VU KIM THUY 23r Nguy6n van Cir, TP H6 Chi Minh DT: 8356111 Trinh bdY r TRONG THI0P
a E]} sr=fl Trorg s6ch gi6o khoa vi trong mOt s6 dd ra tr6n b6o THTT da cung cdp cho chring ta m6t s6 hQ thrlc v6cto trong tam giric sau d6y : Cho AABC ggi G, H, O, I ldn ltrgt li trgng t6m, tr{c tAm, tAm H Z-|f,, vbng trbn ngqg! tidp,,nQi lgp cgg AABC .(1)GAtgB+GC50 \=- (2) tgAHL+tsB HBltgP HC + + tr: O. rG =O= + (3) sinA .,tA + sinB . /B {ginC .lC Q* _.., ts (4) s$zA . OA + (5)OH:OA*OB+OC *t2I]_i"OB + sin2C . OC : 0 =E t=- FZ- [-, . : --' 0. - Khi M : 11 li trtlc tAm LABC thi CE, BF la 2 dudng cao : bo CE .=: - E (c)
Mai Xud.n Hidu,9A, D6ng f)a, Quy Nhon, Binh Dinh. ?rDz Thanh Phrt,7A;Trdn Thi My An, Pham Vtt Viil Hoitng,9A, Luong Vdn Chri'nh, TX Tuy Hda, Phti YOn. Phgnt Minh Hitng,9T, Nguy6n Du, Gd Vdp, Tp H6 Chi Minh. Luong Thd Nhd,n,7CT, Chuy6n, Bac Li6u. TO NCiIJYF,N Bei T1/223. Gidi ph.uon.g trinhnghiqnt. nguy€n Bni T21223. Ch.ilng ntinh riing ndu cho *4 _ y4 + z4 + 2xzz2 + SxZ + 4,zZ * 7 = 0. truoc sd nguyAn 6 p udi p > 5 thi c6 nt1t sd tt1 Ldi giai : (Tra.n. Tdt bat,8A,, Chu Ven An, nhibn r khdng ddi uoi2 < r
Nh4n x6t. Crj 55 bai giei, tdt cA d6u cd bd Ninh Binh), Phant Cao Hu4 Linh (94 PT xung di6u ki6n z > 5 vi ddu gini dring. Ldi gi6i Chuy6n Ngu DHSPNN Ha NQi). Ngo Kian tdt gdrn co '.I'rd.n Thanh Phti (74 Luong Van Cuitng (9T Lc. Khidt - QuAng Ng6i), Nguydn Chrinh, Tx Tuy Hda - Phu Y6n), Nguydn Lan. VanQuang (9T PTTH Lam Son - Thanh Hda), Anh (9A Trung Nhi - Ha hlQi), Nguydn Van Nguydn Hoirng Chuong (8 Torin THCS Nang Quang (9 CT PTTH Lam Son - Tx Thanh Khidu Thrii Nguy6n - Bic Th6i). Hcia), Coo Xu.d.n Sinh (9 Todn. Nga Li6n, Nga DANG VII",N Son - Thanh H6a), V{r Anh Tudn (9 Torin, B,niT5l223: Cho dudng tritn (O), ddy BC" THCS Nang Khi6u Thai Nguyen - B6c Thdi), Didnt A chuydn dQrug bAn duong trbn. Gqi M la I Nguydn Anh Tiri (9A PTTII Tu Nghia I - QuAng Ngai), Trdn Tki My An (9A Lrrong Van trung d,idnt cilo. AC, H lit clrd,n dudng uuOttg gdc ha ti M xu,6ng duitng thd,ng AB. Tint quy tictz Ch6nh, Tx Tuy Hba - Phri YOn), Pham Cao Hu6 d.idnt H. Linh (.9AP"l Chuy6n Ngtt, DHSPNN -HaNOi), Loi#ai(tomt5t). Dang Thd Hitng \8 Todn NK - Tx Ninh Binh), Nguydru Van Trung (8 Torin THCS Tran Dang DoOM !AC,Ovd Ninh - Nanr Dinh), NguS'1ru \ri& Hoitng ()A C c6 dinh n6n quj THCS Bic Hn - He Tinh). tich crja M ld drrdng I DANti VII:.N trbn, dudng kinh OC. Goi 1li tdm cita B,iiT41223. Cl,.o tanr giac ABC'nltpn. Ggi I, drrdng trbn ndy, D ld O th.eo thtt tu lit tdnt cac duitn.g trdru nili tidp, giao didrn kh6c C ngoa.i tidp tant giac ABC. Tia CI cat dudng tri>n (O ) tai didm thu lni K soa cho IK = OK. Ho. cdc cira drrdng trbn (I) duitng eao AD, BE ud gqi F ld trttng didnt cfia vdiBC, DKla dudng AB. Chttng nrinh tant. gidc DEF dbu. kinh cria drrilng trbn Ldi giii. Do CK id (/).!l{D c6;lia! r'd D cd dinh n6n K cd dinh' phAn gi6c gcicACB n6n Y\ DMK : DMH : 1v (Ban doc t{ CM) ndr' H, M, K thing hang. K 4(nr chinli gitta lA cung AKB,suyrarK4 = -\ = lv n6n qui tich Do,B va K c6 dinh, BHK KB.Do l ld tdm dudng ctia l{ lh duong trdn cci drtdng kintr lb' BK. trdn n6i lidp-tam-gi6e Nh?n xdt. Cac ban giAi t6t bdi nAy. ABC vd KCB = KAB Phqm. Minh Hilng, 9T Nguy6n Du, Gir VdP, (ndi tjrlpr cung ch6n TPHCM, IIit Minh. Ngoc, 9/15 Trdn Hrtng Dao, kDt, n6n-trf Jaql Bi6n }iba, Ddng Nai, Mai Xttdn Hidu,9AD6ng ".,rrg gtrac r4-C, jg-cp : AIK Da, Quy Nhon, Binh Dinh, Va Chi Thitn.h' Ng6 : JAQ+ ICA: l-AB KiAn Cudng, Nguvdn Trung Kiet,9T, L0 Khidt, + KqB_= IAB + KAB QuingNgai, Thdnh Dung,9T LA H6ng Phong, = ,IA-t(. Suy ra tam Qu6ng Nam - Dh Ning, Nguydn NhQt Quang, gldcKAI c6ndinh K,hayAK: IK = OK,vdta cci tanr gir4c d6u OA7( (vi OA = OK = AK), vA 9A NK H6ng LInh. Hd Tinh, Plnnt Trung Tltdnh, 9T NK Vinh. Nguydn Thd.i Bd.o 81^, iok = 60,,. vay frE = f,raG = Di6ir XuAn, Di6n ChAu, Ngh6 An, Hoit'ng Mtnh Dung, gA THCS Xi mang Bim Son, L€ Duc : sd7F. : fik = 60'). Mat kh6c, do ,t'li Ninh, gF Lam San,IIbng Phuang DOng,9C, Le trung di€'m c0a A-B ndn tt cdc tam gi6c vudng Thd.o {iD THCS Nang khidu Thanh Hcia, ADB, AEB ta cd : FD - FA = FB : FE,sugra Nguydn ThdVinh,8T NK Y Yen, NguydnVan c6c tam g-tg.+ID B--EAA cAn dinh F vd, : D-frQ= : 180" - DF)I--DFA - L&0('- 980:- ZABC) - Trung, Hd. Thanh Tud.n, Triin Minh Toitn, Trd,n Dinh Hilrtg, Doitn Phuang, 8"f, Triin - (180" - 2 BAC) : 2( ABC +- BAC) - 180'r * Ngoc Anh, Nguydn Hbng Dung, Mai Ngqc Kha, : 2(1800 - 600) - 1800 = 60". Hon nita, FD : l-8, 9T Trdn DangNinh, Nam Dinh, Vir Drlc Luong, n6n tam g16c DEF d6u, dpcm. 9 Hcia Chuy6n Xudn Thriy, Natn }Id, Vu Ngoc Nhfn x6t. C:ic b4n sau diy cci ldi giAi t6t : Hdi, 8T Chuy6n Thi xd Th:ii Binh, Trd'n Thi Hod,ng Aruh Thu (9A THCS D6ng Da, Quy Vidt. Anh, 8A,, Ddo Trd.n Minh, 9A,, I{6nEl Nhon - Binh Dinh), Pham. Thu Huong (94 Bdng, Hii Ph6ng, Nguydn Dinh Son,9A, Kim THCS H6ng Bang - Hei Phbng), Phqnr Minh Anh, Kim M6n, H&i Hrlng, Nguydn Van Hidu, Hilng (9 Todn Nguy6n Du, Gb Vdp - Tp Hd Chi 9T Chuy6n D6ng Anh, Nguydn Anh Ti,9T,Tit Minh), Pha,m Hd,i Trung (9 Chuy6n To6n, Li6m, Nguydn Hod.ng D{tng, 7C, He NQi - Nang Khidu Ti6n Son - Hd B6c), Nguydn Minh Amsterdam, Nguydn Minh Hoiti,8A, Chu Van Hod.i (8AChu Van An - Hd Noi), Ng uydn Phudc An, Nguydn Dftc Thang, 9A Cet Linh, Nguydn Hoan (9A THCS Ddng Da - Tp Qui Nhon), Lan Anh, Nguydn LaVd.n,9A Trung Nhi, ?riin. Hodng Thd Hirng (8 To6n Neng Khi6u Th! xa ThanhVinh, SA, Nguydn Hod.ng Minh,9A Bd
t. I Van Dan, Ha N6i, Nguydn Ngqc Thanh, Th.anh Son, Nguydn Xud,n Trttng 9A,, Gia Cdm, Vi6t *33w> I Tri, Vinh Phi, Nguydn Nhu'Chudn, 8NK > t6ryz(x *y.t z) (Dpcm) ThuAn Thanh, Nguydn Hdi Ydn, Pham Trung Ddu ',=,, xAy ra * y * z) = I Dung,9T Nang khidu, Bdc Giang, Pham, Hd.i 4rrO I Trung,9T Ti6n Son, Hd B5c.- I VTI KIM TFIUY ='dy, {x +y + z) = xz2 - 2v2 *x = ! = z. I Bdi 6223. Cho bdn sd a, b, c, d uir sd t4 nhian NhQn x6t : 1. Tda soan nhAn duoc ldi giei I n thut ntd.n cdc dd,ng thtc sau cira 100 ban g:rli t6i. Trong s6 drj c c. Tt (1) suy ra bn > dn. Ndu n chin thi e,, b, c, d > O do dcj b > d, suy ra Bairs/22j. chox,y,z= .chinsninh: en>-An+C)-A.Vh,yA=C [#, ;] Ndun 16 fi (1) suyra b > dncn > an nc > a. sinx-sinv sinv-stnz , sLnz-stnr., Ydya:c. I stnz. slnx stny I Tuong td ndu c < o ciing suy ra o : c. Tr) dci .1r b:d. NhQn x6t : Bdi todn niy thu6c loai d6. Tba Ldi gidi (cria cta s6 cdc b+n). DAt sinr : o, soan nhdn duoc nhi6u ldi giAi dirng trong dri cd cdc ban : Trdrr Nant Dun g 10CT NShQ An, Cao siny : b, sinz = c thi a, b, c €. ,] . Sir dung Qu6c Hidp l0A DHTH Ha N6i, Trdn Nhu [*, ding thfc Quang gr 'fhrJa Thi6n - FIud, Nguydn IIuy Binh a-b b-c c-a (a-b)(b-c)(c-a) 10'\ CT Chuy6n Lang Son, Nguydn Tudn -.--. +- cab -+ Dtang 10 Phan BOi Chdu, NghQ An, L€ Minh Trudng 11A, Qudc hoc Hud, Triin l{itu Luc ta vidt ( 1) dudi dang : 1OCT Dno Diry Tr), Quang Binh, ,\gayin Hodt | (o -b)(6 -c)(c -a) I I 1r Dftc IOA Hdng Quang, Hii Hung, Vit Linh I ab. - l= \' - nT)- t (2t Huybn Trang 9 Nang khidu Thi xd Ninh Binh, Triin Qudc Cuing 9 Todn, Trdn DangNinh, D6 Coir
gi5c, dtng bdt d6.ng thrlc Tr6btrsev, ...) d6u c ThAt vf,y : Hi*t rt_l ZsinL.M n : )2"i"f,.o"(?.xr+ia) : ..(1,, i=o *" sin(2r, .LT14 - sin (* -;) = n * -l nd, = Icos (*, T)sin7 = 0
ld rndt tanr giric ddu. Chfng nlinh tttng tu' t'dt cA c6c mat cria trl di6nABCD li nhungt'aru giiic Kdt quA b,fts''ni = Z\n)r khong rldi d6u, do da LACO li mot td di6n d6u . ,. . Nh4n x6t : 1) Cdc ban sau dAy cd lili giai Nhfln x6t : 1) Tdt ca c6c ban tham gia giai teit. cuns nsan gQn nhu ldi giAi n6u trdn DQng bai to6n ndy d6u cho ket quA dung ; tuy nhi6n An'h rt,1n,-lap*i0'r PTTHNK Trdn Phri, I{ai czic ban chua biet sit dung gric cci huong gitta hai pi,o"n , Flinri Ant, Trrtin.ldp 10T Lanr Son' duong thing trong urat phang tv6i chti y rang ii,.n't Hcia ; Ngrr.r'inViet Dftng, 10T, Phan BQi tI6 lon cria gcic ndy sai kh6c nl6t b6i ctia;r)' Cnar, Nghe Ai'; Phttnt Anh Dt?c, -11 Torln 2) Chung ta cung da gap bii toan tinh tdng FrtiiNtt Hai riu"g. Dtrottg van Yin, 111' Phan - -- B6i ChAu. Nghd An. 9t Zifrc" day lA t5i $ai dtra vao t,e tnqt lug"g * trong s6 221 t thdng trong drrong iron. Sau day la hai loigiAi khzic' )cosz (, U=rt) tuc,ng trr Li,htqi 2"ryln grdc;Etidp A9{B 'ta dtroc : 11/i995r cira t4p chiT.H. vd T.T. (trang l-2t' 'e'i= ABf' v ir A' AB):-A'BB-cgng t fc lr -8,,8-A b N(it IYiIN I),.\N(i I'I I'\'I a : 1.,f e = vit IIAB' = IIBA'' Y4Y AA t ' BAi T10/223 Cho tt? diAn truc tdnt ABCD co truc tdnt l.it H. ndu A,,A' : B,,B'thi suY ta: A,y''B : B,,B'A (-'o,c ctuitng cao AA,,. B8,,, CC,, ud DD,, keo {gcgl vA do dci: BA' = A3'. TiI do ta drloc : ,AA cldi cat ma.t cdlt ngoai tidp tu cliAn d cac didnt HAB' : HBA' , vi do dci '. HA : HB' Chririg ttang ing A', B', C' ud D'. Chrittg ntinlt ritng ininh trrong tU : HA = HB = HC : HD, nghil nirt-A,y''-= B,,B'.- C,,C)' : D,,D' thi ABCD lit ld 11trirng v6i tAm O mat cdu ABCD, vi do dci nt1t tri di1n dbu. ABCD Ii m6t tt-I di6n d6u. Ldi gidi I Liti girii 3. Sir dung kdt quA cua bai to:iu {dda theo T9/145"n0u l6n mdt tinh chdt cira tri dien trdc tAm : Ndu H lA trtic tAm cia ttl di6n tnlc tAIn Ngu.yitt Ngoc ARC:D thi ta cri : IIrntg. 11T. HA' IIB, HC" HD' 1 Lanr Son, 'i r \ HA,,= HB,,: IIC\,= TIq,= 3 'lhanh llciat / , "r5o, Vi vay : A,,A' : B,,B' : C,,C'= D,,l)' +:5 IIai drrong I ,,\ cao AA,, uit. H.r \ HA,,: HB,,= lt(|,, = HD,, BB,,' naur 1A tto ---+--- -- ef{ :1 (TAm r.nat cdu n6i tiep ttl di6n trttc t1t't'n ABCDT t,rong lllat, uro itr ttl di6n d6u dci suy ra dudc ABCD ld mdt rAIII{t di qua 3) Rdt dring ti6c, cci hai b4n da fBQ nhan canh AB ctia rnngA,,H : A,'A', B,,H = B,,B' I do nt6c sai ldm tt't di€n ; rrrat cao ABK c6t mAt cdu Aqq? theo dudng (ABII) LCD=K tron ngoai tiSP tam giric ABK ! I I goi Ii chAn mat Nt;t;Yl,tt I),'\N(i lll L\ l' cao tr6n canh Biili Lll223. Cho n cti1n trd 8,,fi: '''R,, C:D. Mat phAng chfa mflt cao ABK c6t mat cdu ntac song song. Tin.h : ngoai ti6p ttl di6n ABCD theo dudng trdn ngoai 1t Di6n bi tuong r!tong theo R t' Bia't tiep trl g:rac ABA'B ' (xern h.v6), do dci ta cci : R-t 2Rz 3fi; In - t1R',,-, nR, HA. HA' : HB. HB' ; (1) : zRa= 3Rr qnr: - "ry, Rt- = l,,ai v\ ABA,,B,, crirtg la mQt ttt gitic nOi tidp, zi Sa aian tro chn ntoc sortg song cld dtoc nC,n ta co : d.iin trd tttotlg cltong nhd tlrua di€n tra tlui rt' HA. HA,,- IIB. HB,,; (2t ld 3 lon. Tril ve doi ve (1) vA tzt,La dddc : B, HA . A,A' : HB B,,B' (3) Hu6ng dAn giei D4t A, d6 dAng rut ' 1r ffi = Suy ra ra (dp duns trnh chat ctia ti l€ thric) ft : 1' Ap dtrng : A,A, = 8,,8, AHA: HB cong thttclinh di6n tro ttlc,ng dttong suy ra Chting minh tttong ttr, ta di den-ket lt!6n : 2R, A,A' = B,,B' : c,tC' = D,,D'*HA = HB = R,a=;*a1, ll) =HC=HD (tArn D Giasit R,u nh6 thua r,, Id k ldn, tim drloc al{ - O mAt cdu ABCD) Til dd suy ra '. A,,la tAm drrdng trdn ngoai R, R,, = #(2) vi tr) (1) vd 92) rrit ran = 2h - 1' ABCDr tiep'Nhtt vAv. A. vila la trtrc titn, vr)a Ia tAm Yai k =3 suv ra n : 5 :cfin trric songsongc:it' clrrong trbn ngo'ai tiep tam gSdc BCD, n6n BCD di6n trd R,.k,. Il} F{. Rs. 6
NhAn xdt Cac enr cd ldi giai t6t : Br)i Monlt llilng.9A Qu6c hoi: Quy Nhon ; Ditth Pltuong Loott.9A., chuy6n Vi6t Tri, Vinh Phri ; Hoitrtg Tai trang thdi il : r r: T - r, : 11.22 Truttg Hiefu, 9L. PTNK I{Ai Hrrng ; Trdn Ttti 'fir. 704, PTNK NgO Si Li0r1 Hd Bat: : Nguyirt Vort Trong.9L, Nang khieu Y Yon Nanr Ha ; Vo ('lti Tlitnlt,9T, chuvOn LO Kli6t, Qudng Ngai ; Vh Sr Nonr. 9A t.ruitng Nang Khietr, Ilfc Tho IIa Tinh ;'l'ran trigot Anh. {}'l', Trin [)ang Ninh. Nanr Dinh. Nanr I1a , l:Ioing Tilng 9(,T, r::1r !J NK Vinh Yen. Vinh l,hu ; I* I-,
t rfir euA ri rnl cHeN HQC sINH GIGI Qucic cIA I MON TOAI\ L6P 9 . NAM HQC Lees - L9e6 Ki thi chqn hgc sinh gi6i qudc gr4 nam hoc - V6 Trdn Manh. Nsd Anh Tudn (NehQ Ant 199t--i99dMon toan l6p 9 ^cttroc"td chrlc vio - Nzuv6n CAnh(Hn Tnhg, NguY6n Phan Linh, I nsnv 16/3/1996. Thi sinh dU thi drroc chia thinh Trinh --'i ciai Miirh Nsoc TInh). Iiii 6a"s i bane A bAne B. Nam hoc ndy, du thi khuidn khich (tir 11 ddn 12,5 didmt 6 bins A, cd 27 tinh thanh phd. v6i sd thi sinh cci 51 em. d,Jlit257 em. du thi d bAneB cti 26 tinh, v6i so - Trinh LO Tudn. Dd Hdne Son, NguY6n I tlii sinh dtr thi ld,2O4 em. C-6c em drJqc ph6p lim Hrru Nsuv6n. L6 Minh Tudn, frdn Ydn Ly,L! biri trons 180 phut, khOng kd thdi Sian giao d6' Minh fudn tfhantr ph6 H6 Chi Minh). I Hoi ifons c]rdm thi chbn hoc sihh ei6i qudc - D6 HoAngDiQp'. Nguy6n Manh He, D6Vidt fla iip trufig hqc phd th6ng da td chilc chdm ven (He Tav) fhi tthedv 291311996. - Cao Xuin Hba DoAn Thri.i Son (HAi Phdngt. Can'cf vdo kdt quA chdm thi d ttng b6ng, - Vrl Minh HAi (Nam (Thanh \.iI Hd) ' ddns chdm thi dd H6i quvdt dinh nhrr sau : - NsuvSn Htru Tudn(Thrla Hciat Kdt ;ud : Bdne A : c6 110 siai, trons dci cci - Tidi Nhu Ouane Thi6n - HuO 2 eidi inat, zo giat nni, 37 gid,i 6o vd 51 giai - l€ Ki6m Ai. TrdnThai Anh Au, Nguv6n Thd i, khuven khich. Duv (OuAnsNam - Oa NanO - Nzuv,5n Nmc DoaniL -(Vinh?hri) Edns B : co 33 eidi. trons. dci cri I eidi nhdt, Neb floanEl,ons, D6 Minf, Quan -- 2 gid.i rini, s gidi 6a vh.zz {.ai^khuydn khich. TrdriThi tc (Bic Thrii) Danh sach cac ent trinP Pmt : - Nsuv6n H6nsHei. DaneDinh Hanh, Nguvdn Bd.ng A : Gidi nhdt (ti{fi ddn 20 di6'm) ccj Thi'-ThEnh Hans.Trdn H6nE Quane tHn Bfut. 2em: Nzuv6n Vdn Anh, Dao Thi Phtrong Lan, lt NEuv6nKhuvdn lam (nam) tinh Thanh H
BANG A BANG B Bai f . a. Tim tdt ch c5c sd c
B,di TBl227 : Tint cac s6 d(qng o (a > 2) thoo ntrin tl - t:\dt t.J +ot: + I I lri r-l t t.\N( \'l NI I ' { \'(hi l,rt CAC,I.OP TI{CS Bii Tgl227 : ('lto trt diOn OAB(: t:uong o O, B,i:;i Tll227 ; Gioi phtong trinh nghiAnt co chiitt coo OH = lt r:it dd diti coc cottlt ctio tant ngu.vdn; di1n r:ubrtg ' OA = o. OB = b, OC - c. Cluing ntinh rilnl; acotgA 4 bc.otgB + ccotgc > :lh .r, -ri._y *.3r - 2y -S =0 ('lrong do A, B, C l.it r(tt' 51ot:
j i PR$BT-frMS XN TH,{S ISSUE circunrcircle of triangle ADC again at E. Conrpare BC' with BE. Frlr Lower Secondary Schools Frlr Upper secondary schrxrls Tl1227 Find integer solutions of equation '161227 Solve the equation .rl-.rl.y*3r-2y-5:A. xa+4 --5r=0 T21221 Solve the equation r;. _2 4.rl +21'1 - 8.r: + 3r * I = 0 T7t227 "the sequence {ani is defined by : 2-{5 T:r1227 Lr:t be given two equal sets : o,, : --f . 0rr r = o,,(4ai- 10o,,+ 511, Vn > 0. \x, I. z, r) = {1930, 1945, 1975, 1995}. Find the general term on. Determine x, J, zil so that Tgl227 Find the numbers o > 2 satisfying A = (x - .y)z + $ - z)2 * (z - t)z * (t - x)) I- ,. tl rrl rtt =.! _ attains its least value. T41227 LeL I, G be respectively the incenter " t4 +ofl+1 I . and the center of gravity ol' a triangle, the TSl227 Let be given a tetrahedron OABC, lengths of the sides of which are 2,3, 4. Find right at O the altitude of which OH : lt and th'e the length of the segSnent IG. lengths of the edges OA : o, OB : b, OC : c T51227 Let be given two circles iO, R.t and Prove that. ocotgA + bcotgB + ccc:tgC > 3h, /O', R'), R' > R, which cut each other at. two w,here A B, C arc the angles of the triangle ABC . points A, B. The semi-line OA cuts the circle T1Ol227. tJse vector calculus to prove that rO' t again at. C. the setli-line O A cuts the circle the Euler circle of a triangle (the 9-points tOt again at D. The semi-line BD cuts the circle) is tangent to the incircle of the triangle. TTM'TTTEM NT{IJNG A tren canh d6i di6n hoac phdn k6o dni. ddng ".. qtr-v t,ai 1 di€im L vi {tit 1t tirt,,, truil.\ I i Nhd dinh li ham sd sin trong c6c tam gi6c B-L'-A++ tg LB + tS-i LC' + tg2 LA : 0. AOE, BOE ta cci : 2 AEREBR Th6m ntta xin mili cAc ban hiy chrlng minh cac ---: sin0, sin-E, ' sino,= sinl;r -8,) dnng thtlc yau (O. H, G, I ki hi6u nhu trtl6c) : + AE sinO, {9} cosA cos,B +sinA"i.rc' oB + EB sinO,-- - +'-;'oA ""tc o?=d Nhtingsino, = Sin AOC : sin2B tgcic d tAn) sinAsinB '" " l^ vd sino. - sin( BOCI : sin2A r 1o) (rfu = cefr:Stf ,da+"tgc."tsA.da + AE sin2B +rtgA.ilgB.OC. AF : sin2C: + r cos(C-Ai + tudng rO = .in2A , sin2a rll)3GH: cos(.B-Ct .--.OA+. i.OB+ sl rL6slnL. SlnAstn( - Chu , dirng cA khi LABC "" co g6c tu ttlc O + .,y'. niln) ngoal talt) grac. '--- Bt oc +costA -2 sin2B -* sin2C + sinAsinB VA.vA(/ =.ir.,2A OB +-^ .OC A .i, =' sin2A.o7+ sin2B . o-B+ sin2c:.o? = d + t12)2OI= ', + TiSp tuc c5,ch lam tr6n cd:c ban cci thd ttr tirn B cos, 7,OA+ cos, den cdi'dAng thfc sau. . t6t,CIfng nrinh ring 6 LAB( cdc duong ntii .B^ C nr6i dinh. voi_tiep didnr ctia vong tron n6i-ti6p sln srn: Z-')4 trdn canh ddi di6n ddng quy tai 1 diem K vh /+ B+ C- + = - C_: Agg+--_ ___' EOC tgi +tgrKB+tgrKC:0. ('7) cos, cos, cos, cos, fu O, la tamwrrgfron bangtifbgrtuAcriaA,4nC-Qhrlng O aay t6i chi mdi xoay quanh rndt hrrdng dd rninh- sinB.O,B +sinC.O,C--sinA.O ,A=0 (8) tirn ddn c.ic hO thrtc. Cci rdt nhi6u hudng kh6c C.hutng minh rang d LABC cric drtdng ndi m5i nta. Mong c6c ban tirn th6m nhi6u nta cdc h6 dinh voi ti6p didm cta vdng trdn bdng tidp g
I I I Dii THI ruYhN sINH lees MON ToAN I DAI HQC QU6C GIA HA NOI (Thdi gian lim bei 180 Phft) I I PHhN ei,r euOc J=8-3r-2x2, I CduI : Chohdmsdy" : x-Z - !^)' (1) @ Y=2*9x-
Sry @r-t)2 ^ x2-?n+t Ta chrlng minh bdt ding thric sau : .A +sinZB +sin7C 3 srnZ - t2t
ABBCCA kdp S= \-:;"- - Jra - u -Y?1dx - 1. Gsi cic tiSP didm ld A(xyJ) € z -l 63 99 117 I 1P,):f :8*'Jl--?'x2 -[tz*9x-2rz1dx=i- 8- 8 = , B(r2,12) e (P2) | J :2 *9x -?'xz I a. Cach I : Theo y nghia hinh hoc cria dao -2. 2 Phrrong trinh dtidng th&ng di qua hdn, ta thu drloc hai phrrong trinh cira cung mOt x-x" J-!" dddng thing : '' (x,,. Y,,) vd llP(nt, n) cci dang * = --; I : t&t- 3)(, -rl) + I -,\x,t - 2rc4 Vi BC ll PQ vir y:(9 -4.x,$x -.r:) 2+9r--Zxi Pq = (2, -4) n6n drrdng thing chfa BC Tt d6 thu dudc ho : ,4 chinh ld dtrdng th&ng di [-a -4r, - 9 -4-,'' qua /?(-3,5!rd song ) ,"/ \, song vdi PQ. VaY l^i*8:zri+2 !1'', : virtacriA(-l' 9)'B(2' 12)' ,/' //\ \' phtiong trinh dudng T$&5r, - 2 6l __-,.\ \^/ th&ng chrla BC ld : Thay r = -1 vio phuong trinh tiSp tuydn ta eri : rt _2 = y-5 r*3 ! : x+10 = or +b+ o,: 1,b : 10 4 Cd.ch. 2: Dtrdng thingJ : ax * b tidp xric v6i 2r+y+1=0 dtrong cong y - 8 - ax - kz khi vd chi khi ltongtrr phrrongtrinh drrdngthingdi quaAB : o"x*b:8*k-?.i cci nghiOm k6P e+ y-3 *6x + iy 33 : 0 x-2 "---^ 'j-- L,: (a +3;z -8b +64 - Q
DE THI Tuysx srxn r6p l0 cHUyBu roAry - TrN D+r rrQC su prraM (DAr HQC QUciC GrA HA Norl Ndm hgc 1995 - 1996 NGAY THI THU NHAT :17.7-1995 Thdi gian lDrm biri : 180 phrit CAU l. Cho biSt : . C_ag^4. Cho^tanr girlc cAn ABC (AB - AC), a: r! + fir +$1r-Tlf gdc BAC = a. Gqi D1h. E theo thr-f ttr ld trung va :, fi +/ +yl/TT7. o ilidnr ctia cac canh AB vd AC. Tr6n tia d6i cni tia DE ldv mOt didm M tuv v kh6ns trirns vdi Gia thi6t rang tich ry drrons. Hev tinh b theo o. D ; Tr6n lia a'di cria tia ED'ldy mot fiidm N sao CAu 2. Cho phuong trinE : + 4a(a + 2)x - 4(a2 - 1) : 0, cho MAN: *2 . Hui dtrdng thang MB vit 90" + , *t - dri, trong !+"o1.8)f' Id tham s0. NC c6t nhau tai P. Tinh E6c BPC 1) Giei phuong trinh voi a : -112. Cnu 5. Ban An mudntidn vAo tnr6c m5i s6 2) ciai phuonr trinh theo o. trongdiy 1,2,3,... n nr6t ddu "*" hoac m6t ddu CAU 3.'1t Chfng minh rins tich ctia g so '-" d6: dtroc m6t dav tinh ccj kdt qda la'sd trr nguy6n li6n ti6p ludi chia hdt cEo 128. nhi6n nh6 nhdt cci t}d dtroc. Sau k}i di6n ddu . 2) Vdi sci ttr nhi6n rz tuyy cho tnroc, chting rrrinh xong vi thttc hi6n chinh xdc c6c ph6p tinh. ban $ng sd. iz = nLn +1\6 + ii...'r"-+-7i+-7i An drroc ket qud ld 3. Theo em l0 vd y>0-thi'6>0, v>0"thi'6>o tq ccj ta oa chia hdt cho 8. Tlong 8 sd nguyOn tien iiAp.'tuon b : r[dr=T co _4 s6 chAn li6n ti6p. Ta x6t ri6ng tich cria 4 s6 N6u r
tdng cria hai s6 chinh phuongthi c 2 .Do vAy nd d = 2q r6i chia hai v6 cho 4 : lubn lA hon s6 (dpcm). gk + 45 x 7 : p2 + qz (p, q € N).(3) Cau 7. Tra ldi : khons thd. Ve tr6i trone (3) la id [6 n6n trong hai s6 p vA q, 5t ccj nr6t"sd 16, rn6t sd chin, nglia la xAy *:l'tr'";'li.f ,frl*H*P,f,lii1ffn'lujvffi i lit nrr6c tit binh A sang binh B ra o2 + a2 | (mod 4), trong khi 3ct"-fruua""Orroc 8k + 45 x ? ==3 (rnod4), v0 Ii. Uane dongnz g5o I vh ry g{o\l,lrcng dd n-r, n quv rrdc nl > 0 n6u dong ttr A sang fi va nl "Ach e T(Ta C0lu 4. ?n gia thidt suY ra .;) -o: : . 0 ,id, drrs tU .B *t ge, trrong Er d6i v6i nl. Khi dci ), +22: (s0" soo -; (1) xAv ---J -'-;rf, radinelhfc : + ilZ -,[2) = t \Iat kfrac trong tam gtrdc AMD cd (nt--n)r[Z+Zn-1=0(l).qqv- ra(1) I : At * Mt: Dr = (180" - a)12: gO" -; \2) N6u nL * n thi tt : ,t1-: h" i t\t(n '; - n) € v6 Ii vi rl2- IA sd v6 Tt (1) vit' Q) suy ra M r : Az(3) : n Q, ll2 (vft li). -- VAv nh6i1)cciV6inLm6i n =nguy6n drrong ta cri ti. Vi tamddc ABCATT id DE Ii dridng trung Cd;.i 8. : binh n€n MDA = AEN. Di6u dri cirng vdi (3) cho ta : LMDA - AAllN, z1{*rafi):6€;.frk =# suY ra MD -DA : M *M! : Y* -EN 1 LMDB D tu' - zttn-\ti:tl = q;fu, #O=# ,AN: DB - Tr] bdt ding thfc d6 suy ra cdn k6p em' L CEN (crj hai gdc D vd E b6rig nhau KihiouS,, : 1 *,, *IE + "' +17' St 2) xen gitla hai caP canh trtong ring ti duns bdt dine thfc k6p I'ta chtlng minh vdi n le).,:IiI d
ob rur ruypx srNH rees... ('I'idp theo trang 11) .- 4sin2rcos}x - 4sin)xcosrcos2r _ =lill1 vd : o{2 nln DM : g'[' AB .r+o x4 Ydy MA: MB : MC : M; =+ :4linr sin/xcosx rI cos.r - cos2r- -- l: TiSp theo ta r-{r x2 I x2 .l cci SrxAx .Je SITAR: Sc,qa: a2 2sin ,-sin, sin:o T, "tri 3 =4lirrr x2 - e' x4 -6 a2{5 .r .o --:2.4lim-=a .-;, sDCa: snBC: 2 , 4 2. a. Vi DM t : DB : DC : a (ABC) vdDA ,- 17 a3t[2 n6n hinh chi6u cria chfng tr6n m6t phing 12 (ABC) -bang nhau. Do dd MA : MB : MC. Vi A ABC cdn n6n nci Id tam vu6ng cd"n" = s? Str,tt,*sr,qn+SIx'A+sr)8, :o2t2!Fl 1 (truns tuv6n CM : ABt = 2 Ap dlrng cdng , :Y(r thfc - ban kinh Vayr: AB = at[2 3v a(2-'{51 mAt cdu n6i tidp), taco:r: s*:-z B ^^ cb 6i6t ;66 scm n0p HO th6ng sd vd chir sd thOng dung drroc goi S9A nap cci ngudn gdc tt dAu ? Thrlong mai ld h6 thdng sd A Rap. Ngrrdi ta phdn bi6t hai cira An Dd phat tridn manh vd ndn vAn minh loai h6 th6ng s6. Su phdn loai ndy can crl vdo cta An D0 cung phAt tridn rdt sdm. KhoAng the cach tinh gi6 tri cr,ia s6. Loai tht? nhdt cci c6ch ki VII vd VIII d An Do cci nhi6u c6ch viet s6. tinh gia tri theo dang chtr s6 vd vi tri cria chrr trong dci cri h6 th6ng sd Hindu. Do tinh don giAn sd dci trong dang bidu di6n, dtroc goi Id h6 thdng va khoa hoc cta h6 th6ng sd ndy, nci d6 truy6n sd theo vi td. Didn hinh ctra loai ndy ln h6 thdng b6 sang cdc vung l6n cdn. Thd ki IX nri lan sang sd A Rap md ta dang dung. Loai thrl hai cci cd.ch A nap Tai dny nhd to6n hoc Mohamnrecl tinh gi6 tri beng c5.ch c6ng ddn c6c chtr s6 ccj ibn-Musa al- Khowarizmi dA vidt thd"nh s;iich mAt trong dang bidu di6n. Quen thu6c vdi v6 hO thdng sd Hindu bAng ng6n nglr A Rap. chung ta ld hd thdng s6 La Mn ma ta thrrdng Cdc chri s6 duoc viSt theo kidu chrr cai A Rap dung dd danh s6 thf tu. Trong h6 thdng sd La (qua thdi gian hdng thdki, chrr sd dtroc viet nhti Ma, quy t6c cdng d6n dirng ld quy t6c chinh. Chi ngdy nay) . Trong thdi gian ndy, ng6n ngu A nap ccj m6t ngoai 16 ld ndu chrf sd b6n trai cci gi:i tri li m6t ng6n ngu dang thinh hdnh d chdu Au nh6 hon chrt sd b6n phAi thi gi6 tri ctla nhcjm cung vdi ng6n ngu Latin. Vi vQy h6 thdng n:)y hai chtt s6 dci bang gi5 tri cria chtr sd b6n ph6i du nhdp sang TAy Au vdo thd ki XIi. Thd ki XIII trtl di gi6 tri ctra chtt sd b6n trdi. XIV cci chri s6 nci sang Italia. Mai cho ddn thd ki X\ri mdi dtroc drlng gitta ld I cci gi6 tri 1, cht sd cudi V cd gi6 sir dung 6 cac n:.i'6c kh6c cta TAy Au. Ngudi tri Id 5, n6n gizi tri cira cA hai IV la 5 - | : 4. chAu Au muon sd Hindu qua ngudi A Rap nen Vay XIV c6 gia tri ld 14. hs gsi le s6 A RAp. Cdch ggi niy sai voi lich srl HO thdng sd A nap rdt quen thu6c v6i chung nhtrng dd trd thdnh t6n chinh thrlc. Trong c6ch vidt s6 cta ngudi Hindu, sd kh6ng duoc ki hi6u ta. Nri con dtioc goi la h6 co sd 10. Cdch tinh gia ld m6t ddu chdm hoac m6t khuy6n nh6. TrAi tri cira bidu di6n sd theo c6ng thrlc sau : qr-ra thdi gian, sd khdng duoc vidt thdnh khuy6n a.a2...a,,: at. 10t I * ar.lUt-2 + ... + a,,.10,, trdn to bang c6c chtt sd kh6c. Vi du bidu di6n sd N(iUYiiN (lAO ]'tInN(i 252 : 2.102 +b.10r + 2.too. .tt.tu thnt
Gidi &ip bdi TRO CEOI DOAN ChU vio di6u ki6n "s6 qud. cdu xanh uit sd qud. cd.u d6 d4ng trong mdi hitm dbu khdng drtrug DrIa u6i cd.c 6 phia ngoiti m6i hbm" Nam suy luAn vd lim nhrr sau : Hdm thrl ba, 6 ngodi chu da ghi c