Toán học và tuổi trẻ Số 222(12/1995)

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Mời các bạn tham khảo Toán học và tuổi trẻ Số 222(12/1995) sau đây để nắm bắt những nội dung về học Toán học như thế nào? Phép chiếu vuông góc với việc xác định khoảng cách giữa hai đường thẳng; một số ứng dụng tích vô hướng của hai véctơ.

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Nội dung Text: Toán học và tuổi trẻ Số 222(12/1995)

Bo GrAo DUc vA DAo rAo * HOr roAN HQC vIET NAM #sD S .r,. 1995 'Fffis TI{U t7't1,i.y?,lilllii,6ir;4I-,+Etrl,i TAP CHI RA NGAY 15 HANG THANG * rrec roAN Nrru 1116 ruAo * pHEp cnlEu vuOnc q6c vfi vlf;c x6c DIHH KHof NG cfctl 7 ct0'6 tt6t DU0NG TtlfiNG i1ttl I I rr!fiT r#f, KHofi G0H8 KHAI gg fDtr?&€ErEEr B6tfl#Egt&& I i i ii I' --l DQi tuydn bdn 9 IIdi Phdrtg 1991 - 1995 inh : B,{uc ouc rnitrn
TOAN HOC VA TUOI TRE MATHEMATICS AND YOUTH MUC LT]C Trang Ddnh cho cdc ban Trung hoc Co s0. For lower secondary school leuel friends Tdng biin fip : . NGUYEN CANII TOAN Nguydn Van Vinh - Hgc to6n nhrl thd ndo. 1 Phd tdng bidn tdp : Gini bdi ki trudc NGO DAT TU Solutions of problems in previous issue IIOANG CIIONG C6c bdi ctra sd 218 3 Db ru ki ndy Problents in this issue 9 HOr o6t'tc atEru r4e : o LA Ngqc Thd.nh. Vinh - Srla sai thdnh chrra dting 10 a Tim hidu sAu th€m Todn hoc phd thdng Nguy6n CAnh Todn, Hoing To help young friends gain better understanding Chring, Ngd Dat Ttl, L6 Kh6c in Secondary school maths BAo, Nguy6n Huy Doan, Thdi Vidt Thd.a - Ph6p chidu vu6ng g6c vdi vi6c x6c Nguy6n Vi6t Hai. Dinh Quang dinh khoAn g c1"ch gi{ia hai drtdng thing 11 HAo, Nguy6n Xuin Huy, Phan Huy KhAi, Vri Thanh Khidt, Lo Lich sfu todn hoc Hai Kh6i, Nguy6n Ven Mau, History of m.athematics Hoing L6 Minh, Nguy6n Khic Huu Lian - Acsimrit 13 Minh, Trdn Van Nhung, o T0 Xudn Hdi - MOt sd rlng dung tich v6 htidng Nguy6n Ddng Phdt, Phan cia hai v6ctd. 15 Thanh Quang, Ta Hdng o Todn hoc vd diti siing Qu6ng, Dang Hirng Th5ng, Vu Mathentatics and life Duong Thuy, Trdn Thdnh DQng Hirng Thd.n g - Mat mi kh
-aTl \:t Hoc to6n nhrr th6 ndo, dri ld v6n dO kh6 r6ng. Trong bdi ndy, xin bdn vd mOt khia canh. Ddi v6i ldi giAi cria rnOt bdi to6n cci ba y6u cdu c
Sau khi giAi xong Bd.i todn 2, chdc c6c ban Dudng thing DE cd hO sd gric ld tgm", s6tuh6i: Tt dd suy ra : drrdng thing DE vudng g6c Ndu didm.I ldy tr6r, canh AB thi ^Bf'D : 90". v6i dtidngthhngBM. Srl dungphttong ph6p toa VAy thi d0 giAi bdua{n 7, ban cd thd nhAn x6t ngay Ndu 1ln mot didm bdt ki tr6n tiaAB (1khac duoc, gcic CDE = nL(' cd thd dttoc xdc dinh mdt A) thi ket quA cdn dring hay kh6ng ? c5ch tuy y chi cdn nf khl.c 90o vd 0" (dd tgm" vi cotgzrzo xdc dinh) ta vdn cci hai drrdng thing Cdu trA ldi ld : ket quA v6n con dung. Vd DE vit BM vudng gcic vdi nhau. chting ta di tdi : Chring ta di tdi : Blri todn 3 :, Cho hinh uudng ABCD cd Blri to6n 4 : Cho hinltuudng ABCD cd diruh. dinh.. I lit nt Ot didnt tltay ddi tran ila AB (I khac I ld. nt|t d,idnt clruydn d6ng ffAn dudng thang AB A). Tia DI cdt tio CB tai E. Dudng thd,ng CI cat (I kltoc N. Dudng thang DI cat duitng tlnrug CB duitng thang AE tai M. Duitng tha.ng BM cat tai E. Dudng thang CI cat dudng thang AE tqi dudng th&ng DE tai F. M. Dxdng thang BM cat duitng thang DE tai F. Tint. qui' tich didnt F. Tint quy tich ctio didnt F. Trong tnrdng hop didm B nam gltta2 didmA Bay gi.ir chring ta x6t bdi todn dd6i mot gdc vd 1, c6c ban chi vi6c chon didm J tr6n canh A-B d6 khdc. sao cho AJ : BE. Sau dri, chtlng rniqblgdng tu nhtt bdi to6n 1 se nhAn dttoc kdt qrh, BFD : 90('. Quj tich ld nrla dudng trbn dridng kinh BD (b6 didnr A). Vi6c chuydn tit : 6di tod.n 1 sang cdc biti todn 2 vd bir.i todn 3 s6 dttoc dinh hinh ngay, ndu nhrl c6c ban sit dur'g phuong phop too. d0 dd giei biri todn 1. That vay, f, chon h6 truc toa Xeit m6t vi tri cria diem l tr6n tia d6i cua tia dO nhrr hinh v6, AB. Ndu thav ddi c6ch xem x6t cdu triic cira vi dd cho gon ta hinh v6, *"* trD ld phdn gi6c cira gclc vudng chon don vi IBE c: .a tam gi6c 1BE, cdn B-F la duilng cao l:DC=DA. thuOc canh huy6n, chting ta di tdi : Dat CDE : n{'. ^ Bhi to6n 5: Clto tant g;nc ARC (C: 90"). 'la co, trong Va dudng phdn giac CD ua cluong cao CH. Ti ne toa oo oa D dung DE ua DF ld.n lttot uu)ttg goc udi canlt CB uit. canh CA. (E 1.l"1n CB vd F trdn CA). chon,'c6c didm A (0, 1);A (1, 1) ; Chung ntinh ra,ng cac dudng tlrung CH, AE uit BF cdt nltau tai ntOt didnt. C(1,0);D(0,0); Crlc bni todn 2, 3, 4. 5 trrong ttr vdi bii todn a E (l,tgnt') ; l l 1. Vi6c giAi bdi to6n 1 c
I ,,2 o ) l}Okz +n >- 104 + 1 .- Vinh Phri, Hd Tdy, Ha NOi, Hii Phbng, HAi bc : * - tooh2 > \ok+ 1)2 - tookz : Httng, Thdi Binh, Naru Hi, Ninh Binh, Thanh :ZOk + I Hda, NghO An, Hd Tinh, QuAng Binh, QuAng Tr!, Thta Thi6n - Hud, QuAng Nam - Dd Ning, .ribc < 100-2Ok+1 < 100 -k < 4 QuAng Ng6i, Phf Y6n, Dak Lak, TAy Ninh, Do dri n2 = l0Oh2 + bc < 1600 + 99 = 1699 Khdnh Hba, Ddng Nai, TP. Hd Chi Minh, Vinh NhAn thdy 422 = 1764 Long, Trd Vinh, Minh HAi. -t'o N(;UYt,N 412:1681. 1681 th6a man di6u ki6n ddu bdi. VAy dci ln BAi T3/218. Gid.i phuang trinh : sd cdn tirn. x:+(\x+1* \2-l Nh{n x6t : Rdt d6ng c6c ban tharn gia giAi bdi niy. Tdt cA d6u giAi dring. Kh6ng cci gi khri Ldi giei Di6u ki6n : x * -1. khan trong vi6c tim tdt cd c\c s6 chinh phrrong Phrrong trinh dang xdt th6a m6n di6u ki6n bdi to6n. Dd ld c6c sd { 121, 144, 169, 196, 441,484, 961, 1681 ). D6ng tidc, .. x .) 2*2 kh6ng cci ban ndo "dio sdu" bdi to6n nhtr viy. '-('-r.1)-+r+1-1:o Cd:c ban sau ddy cri ldi giAi t6t : Dd Minh .xl)'42 *(r*t)-+r+r-l=o Chdu 9T D6ng Anh Ha NOi, Phant Dinh Qudc Hung 7 Trdn DAng Ninh, Nam }J,it, Nguy6n Xudn NguyAn 9T LC Khi6t, QuAng Ngdi, Id :($ +r)2:z Xudn Hoitng 9NK Hi6p Hba, Hd B6c, Nguy6n Hbng Dung 9T Trdn Dang Ninh Narn Dinh, [,2 NguydnDicThanh 9TquAn 3TP Hd ChiMinh, Mai Dic Phuong 7T Bim Son, Thanh H6a,Vfi. L.l*t:{2 ) (1) tx' Tud.n Anh, TIICS Thdi Nguy6n, B5c Th6i, l-1-. +1=-t[2 Q) Phan Thi T'hu Hiing 9 To6n, Bi6n Hoa, Ddng [x+l Nai, Ngrrydn. Qudc Tlmng 9C DOngAnh IIe l.Iii, GiAi ra (2) v6 nghiOm vi cd bi6t thric Am. VAy Nguy1n Quang IIdi 8A Quang Tmng, IIa NOi, phuong trinh cd hai nghiOm la nghiGm ctia (1), Ng6 Qudc Anh,7 Ban M6 Thu6t, DhcL6c, Trdn cld ld : Anh Kian, ST NK Bim Son, Thanh IIda. x1= {2-r*{z'E-t r)ANG r rrlN(i 'rr riN(i 2 Bld T2l218. Cho ltai sd x, y thoo nldn dang tlfic x2: ,{1 - t +,[Ni7 Z*2+1.+Y.:+ Nhf;n x6t. C
Gia Ttt, HAi Hrrng), Tdng Ngoc Trt (9A, BdVen aia tam gid.c ABC. Gqi O 1, Orlhn luot ld tdrn cac Ddn, Ha NQi), Ngzy6n Mqryh Hirng (9T, Ndng dudrug trdn nQi tidp crc nin gior,+Ctt, BCH. Tlm khidu Nghi - LQc, NghO An) Trdn Dtc Soz (8 ui tri cfia C dd Oro2dq.t dQ diLi l6n nhdt. Chuy6n Ba D6n, Quing Ninh, QuAng Binh), Ldi gi6i: NgO Van Giang (9A THCS Nfi Ddi, Kidn Thuy, Cd,ch 1 : Hni Phbng) , D6 Minh Chai @h:uy6n DOng Anh, He Noi) Goi r li tAm dudng trbn nOi tidp cria LABC. DaNc vrEN Bei T4l2 18 . Cho tant gid.c ABC u6i AC > AB. Gqi M lit trung d$&cia BC. TrAn canh BC ldjr didm D sao cho BN:= CS*lr€n tiaAM ldy didnr N sao cho ABN : ACB. Gqi O la giao didm. crta AD uoi BN, til N kA NK ll OM cdt BC tai didm K. So sdruh BD uoi CK. Ldi giai. ?r) c6c gie thidt vd gric bing nhau a HM O' suy ra cdc c4p tam gidc d6ng dang (g.g) sau dAy : DO ddng chrlng minh duoc AABC^AACH^ACBH rr CH 12 CH -r-CB'r-AC ri+r) 1 1 =;:cHZ ("rr*;A): crP @C + cB2) _ I _ , CH ,AB,\- I _ CB2.AC2 -\ca.AC) -r 1 LOAB ^ LMAC ; LABN ^ LACD tr] dcj ta Suy ra ,2 = ,1+ 4 ( 1) c6 cdc ti 16 thfc : BO AB BN Ma Mat khric LO pSt vu6ng tai If (ban doc t{ MC AC -CD'=-- CM) + Opl= OrH2 + OrIP :2r2 (2) -= BO BA/ Tr) (1) vi (2) suy raO oM llNK ndn rOrl6n nhdt khi vd chi Hon nta khi r ldn nhdt. u*: BK. Bff Xdt LOAB ta c6 BM : MC n6n ta cc :'#. suy ra : idE = 18oo - 12r+frr\: 13bo . CD : BK. Tr) dd ta c6 BD = CK. Yqy O thu6c cung chrla gcic 1350 drrng tr€n NhQn x6t. 1) C6 223 bai giai, tdt cA ddu giAi doanAB thu6c nita met phing bd.4.B cr)ng phia dfng. Ldi giei tdt gdm cd : Nguy4n Hbng Dung vdi nrla dudng tron dd cho. D6 thdy khi.O Id didm (9T, Trdn Dang Ninh, Nam Dinh) Nguydn chinh gita cung dd thi r l6n nhdt. Lric dd C ld di6m Thinh. (9T - Phan B6i Chdu, NghO An), Vr7 chinh gita cria nila drrdng trdn drrdng kinh,4B. Phong Hdi (8 To5n Bim Son, Thanh Hcia), Cd.ch 2; (Hrrdng d6n) Phqnr Vd,n Tidn (Trung tdm giSo duc thudng Chrlng minhAC = A.i\y', suy ra O ld tnre tAm xuy6n, Ddm Doi, Minh Hii), Pham Manh Hilng LCOIO,> CO L O1O2. Tt dd suy ra (9 Torin, Nguy6n Du, Gd Vdp, TP Hd Chi Minh), LCOK = L.OrOrK+ CO = OtOz. Phan Trang Xud,n (8, Phan Chu Trinh, Di6n Kh6nh, Kh6nh Hita), Pham Thi Thu Hdng (Trudng B6i Drrdng Gi5o Duc, Bi6n Hda), Ddo Duy Nanr (8 Tori.n L0 Quy DOn, Long Khdnh, D6ng Nai), Trdn Manh Qud.tt. (9 To6n, Nguy6n Tri Phtrong, HuO, La Tlwnh Binh (10A, DAi hoc Sti Pham Vinh), N g0 Van Gidng (9A, THCS Nfi D6i, Kidn Thuy, Hii Phbng) DANG VIEN Beri T5/218 : Cho nfta duong tritn duitng kinh AB tr€n d6 c6 d.idnr C. Hq duitng cao CH 4
Mat khde SO : SA = SB. Tt) dri suy raCOldn Ldi giSi : V6ip = 2 tachon z = 2, vdi p = 3 nhdt khi vd chi khi CS I6n nhdt. Tr) dci suy ra C ld chon z : 3. X6t p * 2,3. Ta hdy chrlng rninh didm d chinh girra nita dtrdng trrn drrdng kinh AB. nhAn x6t sau : Nhin x6t : GiAi tdt bai niy cri cdc ban : Ndu (a, p) =I vd, n2 *o (modp) vdi Vn thi Phant Thu Huong 9A, Hdng BAng, Dang Anlt p-l Tudn, 9T Trdn Phf, Hei Phdng, Ngl Thd.ntt a 2 =-1(modp) (1) Trung,8T Chuy6n cdp II Phf Tho, Ta Xud.n ThQt vdy v6i m6i h a \1, 2..., p - 1 ) d6 chrlng Dttc 9Al cdp II D6t Vi6t Tri, Vinh phd, Br)i minh ring tdn tai duy nhdt & ' € { 1, 2, ..., p - ll Manh Hilng, 9H TtrrngVtrongNguydn COng Minh, (modp). \11 n2*a (modp) v6i mgi n sao cho kh' 9A DAo Phrrong Narn, 8A B6'VitDdn, D6 Minh =a n6u le ;c li'. Tt dd (p - 1) ! = (1.1')(2.2')... (k.k')... Chiu,9T Chuy6n D6ng Anh, Bil.i ViAJ Hd,9C lr-l Ngoc LAnr, Gia LAm, Ha NQi, Trd.n Ngoc Anh, Nguydp Tidn Trung, 9T Trdn Dang Ninh, Nam : a 2 (rnodp) Dinh, Nanr }Id, Nguydn Tidn Hbo,8T Bim Son, md (p - 1) ! = -1 (modp) theodinh liWinson. LA Huy Binh, gT Lam Son, Thanh H6a, Trd,n Do dcj ta cd (1). Nhan x6t duoc chrlngminh. Nant Ditng, 9T, Phan BOi Chdu, NghO An, Doo Trd lai bii to6n giA st P(il lpY n. Xudn Hung 72 td 5 Ddng Vinh, phrrdng TAn Khi dd Vz, n2 * 2 (modp), n2 * 3 (modp) vi Giang, Hd Tinh, Tr:iln Duc_Soz, 8 Chuy6n Ba n2 * 6 (modp). Theo nhAn x6t tr6n suy ra D6n, QuAngBinh, Nguydn Hxu Ngtti, gTL Chuy6n P-t P-l PTTH QuAng "14, Nguydn Dic Thitnh,g, PTCS cdp 2 Colette quAn 3 TP Hd Chi Minh, CZo Anh 2 t :-1 (modp),3 2 :-1 (modp) p-L Dic,9A, chuy6n Mac Dinh Chi, TAy Ninh. VTI KI M I'I IIIY -6- z = - 1 (modp) mdu thudn v6i srr ki6n p-r Bati T6/218 Gidi phuong trinlt 6 z = -1 (modp). 2cos(x - 45") - cos(lc - 45")sin2x - Nhgn x6t : i) Mdu ch6t cria bdi torin ld -3sin2x*4=0 (1) chrlng minh nhAn x6t (1). Nhi6u ban dd phrit Ldi giai (ctra da sd cdc ban) hiQn duoc di6u dci nhung chrlng minh cbn dAi hoAc khdng chrlng minh. (1) -cos(r - 45")(2 - sin2r) * +3(2- sin2r) -2=O ii) Trl crich giSi tr6n d6 dnngchrlng minh bii *-(2- sin2r)[3*cos(r - 45,')f = 2 Q) torin tdng quat v6i da thrlc Ma 2-sinZx>2-l=l P(r) = (x2 -a)1x2 -b)(xz -ab) a * b. 3*cos(r-45") >3-l=2 Cci b4n cho ring khing dinh cria bii toan Vqy (2 - sin2r)[3 *cos(x - 45.,)] >- 2,Yx. vdn dring ndu thay bang da thrlc Suy ra : P(x) = 1xz -211x2 -31 Nhung di6u ndy kh6ng dring ching han vdi p = 5. Ta cd Yru, n2 : 0, 1, 4 (mod5) do d,6 n2 - 2* O, n2 - 3 #0 (mod5) v6i moi n. Sdban tham giagiAibii torin niykhOngnhi6u. Cric b4n cd ldi giAi dfng li : LA Van An llCT Phan BQi Chau, NghQ An, LA Anh Vil tZ Qu6c hoc Hud, Nguy6n LO Lr/c 10CT TP Hd Chi Minh, Trd.n NguyAn Ngqc *r-45":180"+K360o llCT DHTHHN, L€ Tud,n Anh I1CTDHTH €x = 225('+K360(', KeZ. IJN, Phan Dung Hirng QuAng Binh, Nguydn Nhfn x6t. 1) MQt sd ban cho ldi gi6i truc Xudn Thdng, QuAng Tri. ti6'p bing crich phin tich ra thrla s6 : DANG IITJNG TLIANG D4tr-45t'=f,thi Bei T8/218 z Cho sd nguyAn n > 3. Chting (1) *-(1*cos/)(2cos:2t+4cost - 7) :0 ntinh rd.ng, tbn tai hai lrcd.n ui hhdc nhau N(iI.JYEN VAN MAT] (s1, sr, . . . sr) uit. (.t1, tz, . . ., tn) Bei T7i218. Cho da thic cta (1, 2,..., n) sao ch,o P(x) = (xz -2)@2 - B)(r2 - 6) s, *2s, +. . . + nsu= t, +ztz+. .. *ntr. Chilng ntinh riing u6i mgi sd nguyAn fi p d.bu Liti gini (Theo Trinh Hfiu Trung, 1.1T Lam tinr duoc sd nguy1n duorug n d.d P(n) chia hdt Son, Thanh H6a) : Xdt Bdi tod.n sau : "Cho sd cho p. nguyan n >- 3 uit cho n sd tfutrc doi mQt khdc
nhau o,t, &2t ...t an Ch,rtng nlirlh, rang, tbn tai Anh Tud,n, NguyLn Phuong, Nguydn Quang hai ltodn ui @ t, b2, ..., 6J vir Hdi, Plrym Quang Hung (10CT, 12CT PTTH ("t,"2,,..,c,r) c&a (ar,cl2,...,an) thfu Hirng Vuong, Vinh Phu) vd L€ Quang Minlt nrdn dbng tltiti cdc dibu kiQn sau : (12Ks PTNK B6c Thrii) (i\ bi* crVi : llt NGUYEN KHAC MINH ilil Bni Tgl2 18 Cd.c canh AC, AD, BC ud. BD cfia (,,) > a,b,=)ap, tt.diQn ABCD tidp xic udi m.4t cd.u (SJ bd.n il j=l i:l kfnh f, td.nt I n?i.m tuAn cq.nh AB. Cbn cd.i canh Ctr.ing minh: o Vdi n = 3 ta cci CA, CB, DAuir. DB thi tidp xrtc udi m.et cd.w (S') (az, a3, ar) uit, (a.r, a, or) lA hai horin vi th6a bdn hinh. r, tdm J nd.nt trhn canh CD. Chnig nzinh hA tfuhc sau dd.y : min (i) vd (ii). gg+ 19112 _ 4r:z) = CD4 @Bz _ 4pz1 d0 thdy Ldi giei (cria nhi6u ban) (oz, oz, . . ., ar,, a r) uit. (au, a r, ez, . . ., an _ l) GiA sir m6t ld hai ho6n vi th6a rnsn (i) vd (ii). cdu S,,(/, f) ti6p Chgn o' : i,Vi: TI vDr thay didu kiQn (i) xnc vdi AD, AC, bdi di6u kien "nhg" hon : BD, BC ldn ltiot '(bt,b2,...,bu) uit {c,c2,...,c71) la hai 6 M, N, P, Q,TA hodn vi kh6c nhau", trJ Bei to6n tr6n ta cci Bdi c6:BP=BQ; da ra. IP = IQ -/n6n NhQn x6t : 1. Dai da sd cdc ban grli ldi giei L IBP A.fBQ. ^ : -^ :: Suv ra tdi tda soan di giAi bii ra theo phtrong phrip cria ldi giAi tr€n. IBD IBC, I Chrlng minh 2. Danh s5,ch cdc ban cd ldi giai ngSn gon : tudns tu. ta Nguydn. Thi Khdnh Truybn (11CT1 Tnrdng drtgc : IAD = IAC. Do d6 : L ABD : L, ABC + Luong Thd Vinh, Bi6n Hba, Ddng Nai) ; AD = AC ; BD : BC. Ddi vdi rnit cdu Nguydn L€ Lqc, Triin Thidn Anh, Khuu Mintt S'rr, (J, r), chring minh tuong tu, ta drloc : Cdntr (l}CT, 11CT DHTH T.P H6 Chi Minh) ; DA: DB vdCA: CB. Nhu viy: Pharu Phi D6ttg, Pltan Anh. Huy, HbVd,n Hitu, AC : AD = BC: BD (: A). Dd.o DuyAn (10A1, 11A1, 11A2, 11A3 P?TH LC Tt dd d6 ddng thdy ringl ld trung didrn cta Quy DOn Dd Ning) ; Coo Thd Anh, V6 Tlnnh AB vd J ld trung didm cria CD. Tirng, La AnhVil (llCT,12CT Qudc hoc Hu6) ; D1t AB - 2m, CD = 2n, ta c
Nh{n x6t: Nhi6u ban tham gia giai bei Cdch 2 (drra theo Trd.n Qud Ld.m, l2CT, toSn niry, tdt ch d6u gi6i dring, trr) m6t ban kdt tnldng PTTH Ddo Duy Tt, QuAng Binh)' Iu{n sai (ABCD Id trl diQn d6u l). Tuy nhi6n, ldi - Tru6c hdt, chrlng minh ring cric tam gi6c gi6i ctia mQt sd b4n cbn rtldm ri, chrta gon. ArBoCa, BFAova Cr,4.oB" ln d6u vi d$ng ra NGUYEN OANC TTTAT ngoii tam glilc A.BoC(,. Bei T10/218. Ld.y cdc canlt. BC, CA uit. AB ThAt vdv. tac6l. -C++++ cfia ntQt tant gid.c ABC tdttt d,d,y, d,1rug ra phia : zAi), C'A + B'C = f(C'B) + f(B'A) = ngoir.i tant gid.c ABC ba tant giac d.bu A'BC uit C'AB. G,li Ary At i Brt Btud. Co Crliin luot lit --->_> li trun g d.idm cdc cqnh BC, B'C' ; CA, C'A' AB, uit, ArBo = f(Are), trong d6 f = Qa6s" A'B''c&a hai tam giac ABC, A'B'C'" Chtrug ph6p quay v6cto, gtic *600. Do dd ArBoCo li mQt ntinh rd,ng citc d.oqn thang A,,A, BoB tuit. CuC, lam gi6c d6u vd cd hrrdng 6m. dbng quy uit. bitng n ltau. Chrlng minh trrong tt!, c{c tam giric B rC oAu Sau ddy tir ntQt sd cd.ch gid,i hhd.c nhau crta vd C,A.$^ crlng li nhitng tam giric ddu vi cci hrr6ng dri aUng ra ngodi tam g1ilcAoBoCo. biti toa.n. rrd.y. Lbi . giiti - D6i vdi tam gi5cA.rB..,Co, ta tr6 vd bdi to6n cd didn quen thu6c, vatrrici
B tr€n thanh cing (xent. linh uA). Ltic d?iu dqt NhAn x6t. C6c em cci ldi giei tdt: Nguydn thanh AB nant ngang uir, uiAn bi dtng yAn d ui Qudc Nguyan ll Ly PTTH Phan BOi ChAu, Vinh, NghQ An; Nguydn Qudc Khd.nl, 1lF C Li, PTTH Hirng Vrtong, ViQt Tri, Vinh Phri ; ?o Huy Cubng 12A1, PTTH chuy6n, Thrii Binh ; Bili Thd Dfing, tlA, PTTH Van Giang, ChAu Giang, HAi Hung. M.T tri M, klti d6 ld xo L, bi nen m6t doqn a, ld xo B,in L2l2l8 C6 3 diln ffi Rt : SOQ - ; 15A Lrbi ndn ntdt doqn ar. Tqi tltiti didm t = o quay thanhAB xung quanh bI tdi ui tri thanh thang Rr: 10Q - 5A ; dtlng. Rr: 20Q - 20A; 1. Hdry ching ninh, lit uiAn bi sE dao dQng Trong d6 gid. tri sau lit. ditng cq.o nhdt mit dibu hba. cac d.iQn trd c6 thd chiu duoc. i 2. Clrc bidt biin d6 daa dQng crta uiAn bi td 1. Xd,c dinh cum liAn hdt, c6 thd c6 gita ba t 9,8cnt uir. gia t6c trgng trudng ld g,B ntls2. Hay diAn trd nd.y, chiu duoc ngubru Il = 180V. tinh chu hi dao dQng cria ui1ru bi. 3. Cho bif't nt = 7Ng, a r = 15 cnt uit a, = 10 cnt. 2. Gid stt chsn lian hdt Rt ll @2+ R) nr.dr Hoy tinh dQ ctng cilo crii tb xo L, uair. n6i tiip udi cunt b6ng ddn loai J}V= 4OW. Tint cach nmc dd cdc b6ng ddn sdng binh thudng hhi Htrdng d6n giei. 1. Khi thanhAB nam ngang ntdr toitn cunt uito ngubn di€n kh6ng ddi V : 220V, vd vi€n bi dfng ydn 6 M ta c6 lt,a, : k.o." (I). Khi thanhAB thfrng dfng, chon viirl ca-n 6ang sao cho cunt R = R ll (R. + R) hh6ng bi chdy. t lim gdc toa do vd tr6n O-r hridng xu6ng dtr6i. Bd qtta gid tri di€n trd cia ddy n6i. Xdt vi6n bi 6 vi tri ccj toa d6 r, hop luc tac dung Iludng cl6n gidi. l6u vi6n bi cd hinh chi6u l6n truc Or : 1) Trong dri 8 li6n ket girra 3 di6n tr6 c6 4 l- : lt,(.ttt -.r + rr) - li6n ket sau diiy chiu duoc Ur* ( 180V: -h.(u..*r-:c,,)+t1lg 1, ,?, + Il- + R3 (Unr,o = J00Vl chir -1i d6,r, 1.1i , 2t R.ll (R, + R2) (Un,o* = 200V) tt - -(1, + A2)x + (h, * h-,)x,, + ntg. SlIir + (R.ll R7) (u_", :275:v) TaiO,f:0vdr=0suyra 4) R1+ \R1 ll R) (Un',,,-* = 183,37.) ltLR .f: = - 2t Li6n ket.R : Rl ll (R" +l?r) cci U^ : " u, * \2) 150y vi v4y khi m6c n6i tidp B vdi cum.fcing virF : - "- (kr + k)x (g).Apdungdinhtudt ddn dd dtta vao ngu6n Il = 220V, md mu6n cho h, * lt- li€n k6t R khOng bi ch6y thi hi6u di6n thd tr6n Niuton suy ra ,"*'i;' r=0 + cum bdng dbn ph6.i cri giri tri nh6 nhdt ln ZOV. r = Asin(ot + f\ vdi *2 = (kt * h,)lnr.: Ta cci R : 15Q vd U* * Uo = 22OV vd, ngoiti ra, dd dbn sdng binh thrrdng thi UD phAi li b6i viOn bi dao d6ng di6u h6a. cta U., - 30V-UO = nUdvdiz nguy@n dUong. I 2) Luc t= O, x = .T,, vd u = 0, rrit ra Tt dd suy ra chi v6i n = 4-UD = l2OV- \ ,p = -[ uit A. = ;--:=- nlg kt+k. (4). Tt dd Un = 100y. Vd cOng sudt ti6u thu cria mOt khdi bdng ddn ld ra T; U^ SUy a) =1i chu ki dao d6ng U,l= P'rs= 1". * rO = 200W-eriSbcing T=2n mic song song. Nhtr vAy : cdn m6c n6i ti6p 4 = 0,628s 3) Tt (1) va (4) rrit ra \E nhcim bcing ddn, m6i nhdm cri b bting m6c song song. -42 m8 Nh$n x6t. Cd 2 em c6ldi giei dring : frjr I + a,LZ*a.^ 4Nlnr ; - A Nguydn Thanh Tilng tZAy THPT Chi Linh, , a| nL8 HAi Hung ; DOng Minh TuQ, l0 chuy6n li, b-- ,va . - al +a2 A 6Nlm, THPT Dio Duy Tr), Ddng Hdi, QuAng Binh.
B,niTJl1222: Gi6i h0 phuong trinh : fx -sz2x -i, +] =o ly-7iy-3x*d=0 RA CAC LOP THCS Ki NAy lz-sfz-Jy+y3=s NGUYEN \i*i"ffi" Bni TU222 : Chrlng minh rang BiLi'181222: B6n s6 a, b, c, d c6 tich bing vd th6a min ding thrlc : 1 9n3 +9n2 +3n - l6kh6ngchiahdtchoB4Bv6i ntoin e Z a *b *c td,: I *1*! *! . abcd NcuyEN otic'rAN Chrlng minh ring bdn sd a, b, c, d phAn lim (TP EO Ch; MiNh) hai cap giA srl la cap (a, b) vd (c, d) sao cho : BdiT2l222 : Cho a, b, cld} s6 htru ti th6a rnan a.b=c.d=1. {abc: 1 NGUYEN LE DONG (TP Hb Chi Minh) 4a b -1--; c 62 c2 a2 I i- -; c' o- = -a *;1- B.di Tgl222: Drrdng trdn (D n6i ti6p tam [D- o c glac ABC tidp xric vdi canh BC 6 didm D. Gsi J Chtlng rninh ring m6t trong 3 s6 o, {r, c ld vi K ldn tuot U trung didm cria BC vd AD. binh phrrong cria m6t sd htu ti. Chrlng minh rang dudng thing JK di qua 1. rYI,N KTINNI I NCTIYIIN 116 QUANG VINH NC;t (Hdi Phbngt NghQ An) Ba,i Tr 0/222_rO ln rgit didlg niry[:ongli! diQn BdiT3l222: Tirl. urOt h6 thrlc glttaa vd b li ABCD saocho OAt+ OBf OCi ODt= O,6dA di6u ki6n cdn vA dir dd hC bdt phtrong trinh sau At, B t, C 1, D tli hinh chidrrcria O tr6n c6cm6t diy cri nghi6m. t.' (BCD), (CDA), (DAB), GBC). Chrlng minh rang : l3r- - 4xv *y: < o) oAl +oB1 +oc'+oD' < 4r lrr +r1, -.tf >b trong d
PROBLEMS IN THIS ISSUE For upper secondary schools For lower secondary schools. TBl222. Consider the function f(r) : cos2r + acosr *bsirw. Tll222. Prove that 923 + 9n2 + 3n - 16 is Determine a, b so that f(x) ) -1 for every r. not divisible by 343 for every n e Z. T71222. $olve the system of equations : TZl222. Let a, b, c be three nationai numbers ' satisfying lx-\ix-32*/:0 abc: 7 lY-sr2Y-Sx+d=0 ) Ia b c 62 c: +- l_J_-L_--r- a- lr-sfr-3y*yr -.s. T8,lzzz.tl,et o, b, c, d be ?our numbers 'b2' a)- a b c Prove that one"2' ofthese three nurnbers o, b, satisfying c is a squareofa rational nunrber. fabcd = 1 't81222. Find a relation between a and b )1111 which is a necessary and sufficient condition for the systenl of inequations la+b-rc*d:A+ O+;+A. Prole that the set {o, b, c, d'} can be divided )t*'-4xy-tf b ab=cd=\. to have a solution. Tgl222. The incircle of triangleABC touches T41222. M is a point inside the triangle the side BC atD. LetJandKbe respectively the ABC. Let I, J, K be respectively the points of rnidpoints of BC afi, AD. Prove that the line J/( intersection of the semi-lines AM, BM, CM passes through the center of the incircle. with the sides BQ CA, AB. The line passing TtOl222. O is ggointjgside $e tetrahedrqg through M parallel to BC cuts IK, IJ ABCD such that OA1 + OBt + OCl + OD, = O, respectively at E, F. Prove that ME : MF. T51222. A point M moves on a semi-circle where AI ,Bt,Cj,Dt are respectively the with center O and diameter AB.I,c-t N be the point of intersection of the semi-lineBManC the orthogonal projections of O on the faces (BCD), tangent Ax to the semi-circle, P be the second (CDA), (DAB), (A.BC). Prove that point of intersection of the seuri-circle, with the oAl+oB1+oc'+oD'
Tim hidu siu th6ng , th6mri toin A.hgc phd , pxrfip clmfiu vuONG c{ic vdrvrEC xigDF,H 'l , d \a rffif;OAriG CACH GIIfuL IIAI I]UONG TIftNG T}IAI vlfr THAo (Trutrng Phan BQi Chd.u, NghQ Ant Bni vidt nny gidi thi6u eung cric ban rlng Ki hi6u A1,f_1, Dy M1, K, ld hinh chi6u cta dung cria ph6p chidu vu6ng gcic trong vi6c x6.c A, B, D, M, K trdn (P), chfi y ring hinh chi6,u dinh khoAng crich giria hai drrdng thing ch6o cira CN viN chinh ldN. nhau. Qua dci ldm phong phri th6rn crich giAi loai Theo giA thidt CD t (ABC) * CD L CB, todn ndy, ddng thdi cho thdy mdt cdch nhin vd CD tCA.DodriND. llCDvdND, lA,B, taiN. sU van dr,rng sring tao phdn li thuydt de ddoc hoc (xem hinh 2b). Ta cd : d.(BK, CN )'= aQl, b,k,\ ; trong chrtong trinh hinh hoc d(AM, CN1 : d(N, ArMr). Suy ra cdn chtlhg bidt ring ph6p chidu vu6ng gcic bio t6n . Ta c6c-{oan nrinh d/N, BrKr) : d(N,AtM). Suyracdnchrlng ti5d tfu1ng cung 4}ruong vi ndu mitth d6l. B.K.'= dN. A.M.\.' MA : KMB thiM,A, : K . M'8, v6iM,, A,, B' Tt gi4 thidt*l4 Ie trln'g.diAp _rp 3 M, le tudng rlng li frnh cta M, A, B qua ph6p chi6u trung diern B,D, ; cflng nhrr th6 N,ld tning vu6ng gcic. Xdt hai dttdng thing ch6o nhau AB vd CD. didm A.BL, m{t khSc CX : + }CO Ta dung 1 m@t ph&ng (P) vuOng g1cvdi AB tai 1 O (O € AB). (Xem hinh 1) NI(, = -gND, (K € NDI). Tt c6c d&ng thfc trdn d6 ddng suy ra K, lA trong tAm A ArBlDr.Met kh5.c LAtKrBr cdn t+i Kr n6n srry ra d(N, BlK): d(N, ArMr)(dpcm). Biti tod.n 2. C]l,o tir di6n d6u ABCD canh a. Gqi M ld trung didm cqnh AB, N li trung didm canh CD. Hey tim khoAng crich gifla hai dudngthdng BNvdCM. Gid.i :Y\ABCDla,tn di6n d6u, n6n tt gii t . Gie sir aoun mlrtl#"&Ja, .,rrorrg gcic chung cir.aAB, CO ;hinh chidu criaA, B, frTron (P) 15 thi6ttasuyraBN L CD. ?a chidu t(r di1nARCD 4',.8_', N'. Ro ringMN =_Ory.'(h. v6). Nhu vdy l€n m4t phingP vu6ng d(AB, CD) : d (O,-A'B) tTa ki hieudGB, CD'b gric v6i BN tai N. Ro '(hinh kho.rng cich gita 2 drrong thangAB'va CD, 1) rdng CD c (P). d(O, A'B') li khoAng crich girra didm O vi. A'B'. Goilf ln chen dttdng cao k6 ttA t6i mdtBCD. PPP .-. Trong lrlng lqi to6n, cci thd chon mnt phing (P) qua C, ho6c D hay 1 didm dac bi-et nair A'O tr€il Khi dd A -A' B - N, C -*C,D *D, H --->N dudng thing CD Cti thd ta x6t cdc bei to6n s"u , P Biti todlt T. Hinh trl di6nABCD cd canh CD (Ki hi6u A-Attitc A, li Anh c:iua A qua ph6p vu6ng_gtic_v6i m6t phing Ae C, tW la trung didm ii'6:x,"d,- frifrfb1)' rt cia thist suv ra : "i*DB,Niltrytg didnr c,fa AB, K li *i) didrn tr€n CD sao Ta cd NA, = HA ="G-": O (MC, NB) = cho cr( : i"o. I d(N, CM,). Chf f rihe MA = MB + MrAr .'-l-.1 = M':N(Mt Lrtlrt/ e ArN) (x"em hinh 3b). - \asur.uruuuu,r._ Chrlngmiuhr:ang _ Yd. d(N, CM,) - NII, trong tam giSc vu6ng khoAngcrichgirta hai CNM, ta cri ici : r' duong thing BK va [M, 1 ^ CN bang khoAng NH2 NC NMi cach gitta hai dttdng thing AM vit CN -:-J 4616 =_-I._:_+ )at (hinh 2a). o' a' Gid,i.Tachidl tf ari10 di6n ABCD l6n mat NH=-o phing(P) vu6nggcic vdi CN tai N. YQy d.(MC,NB) =E# 11
Bii t&n 3. (Dd 128. B0 d6 thi T\ryo'n sinh DHCD). Gidl: (xem hinh 6). Goi giao ctia AB, vit Trong m4t phing P cho hinh vu6ng ABCD ATBIaO.ViABBIAT c4nh o. Goi O li giao didm cria hai drrdng ch6o. Ie hinh vu6ng n6n Tr6n drrdng thing Or vu6ng gdc vdi m6t phing AB. -l A.B Ta xic ABC tai O ldy I dinfi mat'ph.ing P didm S. Goi a li drrlaAB, ve.t-AlB. gdc do m6t b6n Khi dd hinh chi6u hinh chdp cria B, l6n (P) ln SABCD tao vdi Bl. Dd xric dinh d,iy. Hey xfc hinh chidu cila C dfnh drrdng ffin(P), tt O trong vu6nggric chung m4t CBA, ta k6 ciraSAviSD vi Or t ArB, rdi tt C tinh d0 ('l dei k6 tia Cy ll ArB {r drtdngvu6nggic chring cit nhau t4i C'thi C'Id hinh chidu cria chung dci. C fi6n (P). Gid.i : GoiM, Nli trungdi6'm eriaAB vdCD Ta c6 d(ArB, B9) : d(O,BrC') trlc ln bing (xenr hinh 4), khi dd O la trung didm MN vi drrdngcao cr&a L,OB TC' k6 tt O. Tam giric ndy c
Ka A"rWco d,c r,ila aa o 1,9.c mGsrffiffi (nnGHnMftDE) (khad ns 287 - 212 trutdo @Q0 Archimbde ngudi Hy lap, Id. con trai nhd cAn. Ndu vudng mi6n toi.n bing ving thi trong thi6n viq hoc Pheidias. 6ng sinh tai Syracuse, lrrong cria n
le gidi han chung ctia cdc chu vi c6c da giric d6u thdi danh chidm thdnh M6gare d b6n canh. n canh n6i ti5p, ngoai tidp v6i drrdng trdn khi n Qudn Syracuse idm ctl dttoc ba nim bi vdy h6m. ddn t6i vO cung. Theo c6ch xem x6t 5y 6ng de Vdo mdt ki 16 thdnh quAn ddn Syracuse mAi 16 tinh ddn n : 96,tt dri thu clrroc b6i, cring thdn linh vi ca mtia. Marcellus vdo 10 1 m6t d6m cho quAn ng{m tAm, tr) Mtigare sang, 3lt. n .3i lang 16 leo vdo thinh Syracuse tt phia ddt li6n, vd trong tlng dung 6ng cl6 ldy gdn dfng drrqc lQnh phAi bit sdng Archimdde, khOng duoc "rr = gi6t. Qudn, dAn Syracuse dang ngtr say srta, 22 = I = 3,14 Ie m6t sd cci dO chinh x5c dir dirng , kh6ng kip trd tay, bi tin s6t v0 kd. Archimdde dang mAi ngdi tr6n ddt vE nhtng vbng trbn, cho cdc c6ng trinh thdi dd. Archimdde ciing dua nhac thdy bcing mOt t6n linh La mA chay ddn. ra drrdng xoin 6c Archimbde r : ag trong hO T6n linh sdn lai h6i : "Archimede ddu ? " toa dO cdc (r, p). Ong cbn tinh ra bidu di6n ctra Archimbde bAo : "Kh6ng dttoc dung ddn cac drrdng tidp tuydn cho dudng xo6n dc niy t4i m6t vbng trdn cta ta !". T6n linh s6t ru6t, kh6ngbidt didm cira nci. Archirndde cung dtta ra ti6n d6 ddy la Archimdde, vung gddrrr ch6m chdt cu g1a Archimdde : cho hai sd dttong a, b vdi a < 6 ; S0 75 tudi. ThAnh Syracuse bi chiem vd trd thdnh tdn tai sd nguydn dttong z sao cho na > b. TiAr, thu6c dia La mA ! d6 niy dirng ldm co sd cho ph6p chia hai sd. V6 sau, ti6n d6 duoc md rOng cho mQt tnldng khOng nhdt thidt lA trudng sd : MOt trrrdng T l. MOt pha trong Chi6n tranh Puniques drtoc goi ld Archiurdde nduA, B e T vdA < B thi t6n Gustave Flaubert (1821 - 1880), nhd van hi6n tai so nguy6n drrong n dd nA > B. thtlc bdc thdy cira Phap, trii hi6n rdt sinh d6ng Archirni:de drroc cdc:rhi to6n hoc Hy lap, La trong tidu thuydt lich sri Salantnft6 (1862). m5, Ai c6.p, Trung 4 A" dO duong tfrii va aOi Flaubert cung ld tdc g1h cria tidu thuydt Madante Bouary (1857) rdt ndi tidng. sau xem nhtt ddng phuc vD. ding kinh trong vAo bdc nhdt. C6c nhi nghiOn crlu thdi nay vd lich 2. Goi o = trgng ludng vudng mi6n bdng srl to5.n hoc xem Archimdde ld dt?ng ddu 6 thdi vdng cAn ldn ddu, cd dai. 0rrg suy nghi trr do, t6o baoira hi6n dai. b : treng Itrong vudng miQn bang vdng cdn Di6u s6ng gi5.ld trong to6n hoc Archimdde dA Iiin sau, vAn dung nhttng phuong ph6p tudng tU nhu : ti trgng vdng, u = thd tich vtrong mi6n, d ph6p tinh vi phAn va ph6p tinh tich phdn, nir d' = titrongnddc, (d' : l). hon 2000 nS.m sau Newton ld Leibnit moi ph6t Taccia:ud.,b:a* ud' ftheo nguy6n li tridn l€n. Archimbde) Trong giai doan thit hai ctia chi6n tranh Tt dd suy ra Puniclues trldng La ml Marcellus denr phSo a thuydn vi qudn b6 tdn c6ng ch6p nhoang thinh a-b:ud.':u d Syracuse. Khi ph6o thuydn 6p sat thdnh, q=(a-b)d. Archirudde chi huy qudn Syracuse ddi nhtrng chr\m dd: m5i tAng n6ng hon 2 ta ntdi, ph6 vd C6ch tinh ndy mi6n drJoc cho ta ph6p do thd thtry6n dich, rdi thri nhitng cdn cdu vdi ngodrn tich u. sit l6n nhdc bdng thuydn len mi lang ra xa. Ban doc cd thd d6 ddng lim quan srit hai QuAn b6 leo l6n thinh cfng bi c6c m6y bAn dri ph6p cAn mQt cuc s6t khong nhung nu6c vi cci vd mfy lia qu6t nOn trri boi. Chfng khiSp via, nhfng ntr6c. 56 thdy ngay sd khric nhau giila kinh hdn. Marcellrrs bu6c phAi chu.ydn til tdn hai trong lttong thu dtroc. Cf thil md xern ! c6ng sang bao vdy l;iu ddi thdnh Syracuse, ddng I I.JIT LI ITN TIN BUbN loa soan talr chiTOAN HQC VA TUdl THEv6 cirng thLldng tidc b6o tin: Nhd gido oAO tnUONG GL\NG Sinh ngdy 19.10.1954 Qu6 qudn : Xa ThUOng N6ng, huyen Tam Thanh, Vinh Phri Gi6o vi6n truong THPT Chuy6n Hing VLIong, Vinh Ph(r, cdng t6c vi6n thin thidt cOa TC ToAn hoc vii tudi tr6, do mic bdnh hidm nghdo dA tr; trdn hdi 14 gio 10 ph0t ngay 11.11.1995 tai 86nir vi6n Bach Mai, Hd N0i. Tda soan xin chia budn cirng gia dinh vd nhd trUdng. ToA}I FI1)C VA rudl tNO
Hgr c$ ring dqlng ricn v0 Husr{g cul HAI vECrCI rO xuAN riAr (Truimg PTNK Hdi Hung) 'liong r:hdrlng trinlr hinir lroc lrip 10, khii ni6m vi vectrj Irfi, =i +l+iriLt,:7+fr +f116n chiinr m0t phlin quan rrong. '1'r()ng tri)i viit rury. t6r xin trinh h:ry nrdr sii ttng dung nhti cira -t'iCt t VO LII.IONG. MJS(i=:*!-*4*5:j" .--'" +, -' J' - *l ;-_ * 'lirki: het xin nhiic lai dinh nghia. Cho hai vectrt * 2a .c * 2b.c +Zt,.c't )h..,, - Fr4 n p14:3 *t *d ** +'z? + 7= -y,).6': (-r,._r,j . 'I ich vo htrdng &a cnring, ni fiieu - E- 7i.+u-,7* 2,., ii r4::4. i* z,-, F 6r + dudc xic dinh bdi f rf=:|i*5ir- tri, - t6j2) rroac D;ing thr-tc clin chrlng ,-6* {F+7[ : o -+. (u. - -++il-\ I i.Ti: .t/t. r;i .."1rI)_y,iit7lD) n goc giCra haivectoTi6- b.\:t)ll"+(n Iti(u lhrlc .r'.6': +],,\,, l:r co )--l' .--'.':]1 (b.u1):?t,a)+2'70"1 . I !-;*,4'+ il El : rt tr.r:r rl0 : t:,.l;, lLr(t.r:rc,ic,sqZh) = jl 2. l'f ng tltrttl, norrg gidi phultry rirth ra hi phuottg rinh l;l.lr,1= fii+ri -Sg{E-aE Vi du I : Giii phrlrlng trinh .r v; +T + .f3--; = 2,,t tz fT Seu tlirl' ll:r ntirt stj ing dung cira tich vO lrttitng. -. liili : Dat + v? G /- ILF= (fxTT.{3*:T;. rtri ao ,-.6': tr/.r + 1 - 1- Ihtg dqng dd chtirtg = f.FTl . t/1V, +l;t + ({3--x;: ntittIt dilrt;; thlit :2{f:lT Vi dr.r 1r ('lio DAlt('cri D,, do,iF= dl . Fl *aldcqrng ruy€n ,,\tl : c. l'|(' : a. ('z\: [), trCn x Vr+T r+l .,1/l l:^iy rliinr,4./. ('hilng nrinh I '/3_l " 3-r r:inr cl.(,t'l: : dzANI2 + *xl-3r2+.r+l:0 + *Jtll) + 1al + b2 + ::.(I- l)("t2- 2r- l):0€xr = 1. x2.j: I + {7 Vi du 2 : Girii + t:1A,\.t.BM hS phrlctng tnnlr M,t t-A+rt'll fxr+vl= -v(.r+2.1 (lirii :tii;isit. =I-. 1;rcr'{'[A= rl; lrr +, 'tv: - 2)'z * UO ?\ lBM | . (',1 + t,,tlt[ t . cR l3.r +Sy2 +tty +8)'z = Zt + 4z + 2 *t=l| : (;iii: r. . *e .rTI- l;tMl + lBMl [.r(r +y) +],(y +z) = 0 lttMl ra+tnrult.clr l' *,.r1r'.I1 lr - brl/iA/ll *,rll,IAlll * 1t;* {x(r + l) +y(22 + l) = 0 .JrJ.I/1 . 1ttrt11Ct .t'h1 | +i*+v;:++ir+z,12:1x+ I ;:+122+ I ,1: D,t 2C)l . Ch : u2 * bl - cl nc.n Xr,t a-t (r,r,). F-: (x+y.rr+z). -: (x+ l. 22+ I) c) : tt1 . lSli l? + u? V1iv l! . i(LI 11 + +d+=> . l) : ++ J t O_.r. c = (1. 4h. : c. + 1i + b2 - (.)) . lAMl . IBM I .*1fIn, -.=L=o,,:-; V[ du 2 : () l:] t;inr duitng trr)n n0i tidp AIBC viri 6(' - a, cA : b, A3: c. gh6ng '- o'n,innt '" ht +ol)2 +()(-) : c:'rc canh I * Neu d * othi hvicc6nstuvtn+c .:1 : t2l' Xr'l hai truilng hdp c -: 2h'vi c': - 2b-ta co - .'a ab 1l (;irii : SLl dung rinh chtit r x:0.y:-,2.:-- cita clrc dUr)ng phin giAc ()A, ZZ ,t OB, ()('currg vrti nhiin x['t 1*.y ,'_ ,."-- Vlry he co hai nghiem tn (0, 0, - )r, Ur,),)l tl'"'idu ltadirlangchrlngntinh t\' ! ! 3. t'hrg dung nong chtlntg minh bdt ddng h*c .rfi.:i . Vi dy I : (lho 8 s6 thucxl,_r2,.rj,x.1,rs,16,rr,r*. (ihi.tng minh auac u.ci;t+b.O-lt+e ,, I ' ring it nhit mot trong.r sa) 'l't tlti ', . ,' i ., i ,t .rl.t.t + -r2r1 ;.rlxs +.r)16 ;.(1.r7 + .1116 ; .(-lr5 + rlrT + .r4r8 + .rs.r7 * 1,,.rTt+h.o1+,..ai'): -- g \ , + -rnr* kh6ng im. (iiii:X.'t4vecr,r + zaboi4 dt + 26r111s . (ft: + 2cud' . drl : *)'l i : (x,,x),{ :-(.rgJ, {: c',,."J, :-('r, xu) 1'a co yt . r't = -r1t + x2r6, ..., E, . o . r.'a = x{1 + xi( v3. v 4 = rtrT + "r6x8 vi (fA - dj : itl vi (dA - Oltlt : ,z 4, it nhdt m6t trong cic g6c giila 4 vcctrr kh6ng vuot qu[ 90" +2O1 .Ob:rt.F*oR2-c? n€n it nhlit m6t tr()ng (r tich v6 hu6ng la kh6ng Am. 'l'tl(1ng tU, 1a co Vi d\t 2 : Ch0ng rtinh MRC co c.{c rrung4ruydn trng viri u2otl) + b2OR2 + c2()('! + ,tb1()A: + t)B: - c2t + bctOl)2 + circ canh lB vi B('w6ng g6c thi ta c6 crrs8 > +()('1 - a!1 + t:u1.{){')+ {)At lil: o* g + h+t11.tO.,12+ ; bot* + co('2) = rtttc(t; * b * t: )n# -# *# =, (;irii : D:ir irt' :ia-7= itu.o4,: + *--*,= !.-. ,l.- * .l- * : tltl :,-: .|11in lrjr phr'urg ch,r hei hinh vu6ng clng hu6ng Lr(),r., J ,,r. ntn ( ju - r) ( ..,_J.! ,lB('l ) v;t,l ,!i ,1',/),. ('hLtng minh iing 1 1..- l) o0 2-:, ''r+:)' a + e o.t' :;f.a- +c-lcrx/r =,r1; :i 1 u | : ++ ( ,ll1 + r't'1 : tl1i + Dt)i lj',--- (iirii: l)ir ,tll:i,tat =t , -- -' ', y' -"--:. q +ldz 4 ld2>5(cosi) lA. -t rl-lA 4t,:4,a7,,:1,,a,:i lt t,tA.1A 11,\ivrAil, =7+t-+- I I)in!: ra khi vir chi khi rh(tc \ ('('t:dt +b; +(' rr] : rfi lnv )
mfrrmfr HH6n c6ils ^a z t? \ 2 HHRI ,t MOT UNC DUNG BAT NGO CUA SO HOC DANG srir.rc THANG (Dqi hqc Qudc gia - Hd Nli) Thdgidi ctra nhfrng con sd thAt huy6n Ao kj'la cho ta mOt con s6. Tcim lai vi6c chuydn mQt brlc va hAp d5n. Da cci nhi6u v6 dep bi dn drroc ph6t di6n crj nghia ld chuydn mQt d6y s6 a1,a2, ...,an. hi6n vi cdn bidt bao di6u bi dn chfa drtng trong Dd girr bi mdt thdng tin ta bidn ddi (me hcia) ddy sd 1,2,3,... Sd hoc, vi nrr hoing cria todn hoc m6i sd a thinh sd b theo c6ch sau. m4c du da 3000 tudi md s6c dep lQng l6y cria nci Ldy hai sd nguy6n t6 khd l6np, vdpr(c6 vdn cdn quydn ru bi6t bao nhi6u tAi n5ng. V6 dgp khoAng 100 chrr s6) nhAn chring v6i nhau dd ctia s6 hoc ldrn say ldng nhi6u ngridi, tt nhfrng drroc s6 ffi = prp.t. Chon mQt sd ft nguy6n t6 v6i nhd to6n hoc l6i l4c nhdt ddn ddng dAo c'i.c ban tp(m) 6 dd p la)r5m Ole g(ntt = (p r - 1)(p2 - 1) y6u to5"n. Phai chang su hdp ddn cria sd hoc li d Sd o dtioc md hda thdnh sd 6 : aK (modn). ch6 nhi6u rnOnh d6 cria nci drioc phrlt bidu rdt don CQp sd (m, k) gqi lar chia khoa md. h6a. giAn ai crlng hidu drroc mi dd chfng minh thi phAi Vi6c m5 hda sd a (t:(.c ld tim dti ciia ok khi stl dlrng nhrrng c6ng cg to6n hgc hi6n dai nhdt, chia cho m) drtac thqc hiQn nhanh chring vA d6 nhrrng ki thuat "cao ctldng" nhdt ? Met khAc, ddng nhd srl dgng thudt torin Oclit chay tr6n nhi6u ban tr6 chic dd ciim nhdn srr thich thri khd rn6t chrrong trinh rniu. tA khi tirn ra dUoc m6t ulch gi6i don giAn bdt ngd Khi nhin drro.c s6 b dd tim lai s6 o (gi6i ma) cria urQt bdi to6n sd hoc khd, tridng chrlng phAi ta drta tr6n nhQn x6t sau ddy : ding ddn c5c kidn thric cao si6u. NhQn xdt : Gih. stt ft' li sd th6a rndn kk' = I Trong bii b6o ndy tOi mudn gidi thiCu vdi cdc (mod tp(nz)). Khi dd bk' a (mod ne) ban moi v6 dep khac cira sd hoc thttc hon, trdn = ThAt vAy ta cd bk' - skk' (mod rz) tuc hon md cd 16 it ban duoc bidt tdi. Dd ld vi6c rlng dung cira s6 hoc viro m6t linh vdc rdt quan X6tA - okk' - a: a(akk'-l - 1) trong cta ddi sdng : linh vrrc b6o mAt th6ng tin. Ndup, I o thiA i p,. N6up, i cz thi Hi6n nay nhi6u td chrlc qudn stl, kinh td, tdi okk'-l-1: qe(nr)-12 api-l - 1; pt chinh hay c€rc ca quan chinh phii khi truy6n di (theo dinh li Fecma nh6). VAy ta ludn c
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