Toán học và tuổi trẻ Số 212 (2/1995)

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Toán học và tuổi trẻ Số 212 (2/1995) bao gồm những nội dung về bài toán đơn giản về hình vuông; đề thi học sinh giỏi lớp 9 của Hải Phòng; phương pháp véctơ; bất đẳng thức tích phân; đồng quy và không đồng quy. Với các bạn yêu thích Toán học thì đây là tài liệu hữu ích.

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Nội dung Text: Toán học và tuổi trẻ Số 212 (2/1995)

Bo crAo DUC vA DAo rAo * ser roAN Hoc vrpr NAM 2erzt tAp cni na wcAy rs sANc rsAlc * nr ilrgr uAr roAN DoN crliN vf; mtNH wonc * Al |r?Ul OE THE HIIG $ITH GIIII TI$P 9 GUA IIil PHII]IG * PHI (yNG PITAP IrEC T(I ,\ A- * DONG QUYVA I(HONC DONG QUY
ToAN Hoc vA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang o Ddruh cho ctic bqn Trung hoc Co st Td ng bi€n fip : For lower Secondary School Leuel Friends NCUYEN CANH TOAN Pham. Thd.nh Lud.n - Tr) mdt bdi to6n Phd tdng bi€n tdp : don giAn v6 hinh vu6ng 1 NCo OAr rU o Gidi bui ki trdc IIOAN(i CHI. IN(i Solution of Problems in Preuious issue Cric bii cira s6 208 3 o Db raki ndy Problem in this Issue 10 HOr DONG slEt'I tAP : o D6 thi hoc sinh gi6i l6p 9 cria HAi Phdng 11 Nguy6n Cinh Todn, Hodng o Pham Bdo - Phrrong PhriP v6cto 12 Chtng, NgO Dat TrJ, L6 Kh5c o Tim hidu sdu th|m todn hoc phd th6ng Bfro, Nguy6n Hry Doan, To hetp Young Frien ds Gain Better Nguy6n ViCt Hai, Dirh (Jnd,erstand,ing in Secondary School Maths Quang HAo, Nguy6n Xuin Ngb Thd.nh Long - Tr) m6t bdt ding thrlc Huy, Phan Huy KhAi, Vu ticfr pnan 15 Thanh Khi6t, LO Hei KhOi, o Nguy\ru Cd.nh Tod.tt - D6ng quY vi Ngrry6n Ven Mau, Hodng L6 f.to"g ddng quY Bia 3 Minh, Nguy6n Khic Minh, o Gidi tri todn hoc Bia 4 Trdn Van Nhung, Nguydn DIng Phdt, Phan 'lhanh Trd.n Vi\t Hilng - ThaY chii bing s6 Quang, Ta Hdng QuAng, DangHtng Th6ng, Vl1 Driong Thuy, Trdn Thdnh Trai, L6 86 Kh6nh Trinh, Ng6 Vi6t Trung, Dang Quan Vi6n. Alrh b* : Hoc sinh lrtnng ffl$ frnnphi -Hii Phong lronggtci Ilnhoc Tru sd toa soan : 45B Hhng Chudi, He NQi DT: 213786 BiAn ffip uir tri VU KIM THUY sU : 231 Nguy6n Vrn Cu.TP H6 Chi Minh DT:356111 Trinh bity . THANH LONG
Ddnh cho cdc ban Trung hgc co sd TU NNOT gAT TOAN DON GIAN VE HINH VUONG PHAM THANH LUAN TP Hb Chi Minh N6u bi6t c6ch suy nghi thi tr) m6t bdi to6n Cdch 2 - Dgng CC1 // HF (C, tr6n AD) vd don giin. chung ta cd thd det ra nhi6u bii to6n DDt ll EG (Dr tr6n AB), A AD,D : ADC'C kh6 phong phr1. Xin ldy thi du trl bdi torin khri (hinh 2) don giAn sau dAy. dr, :::: ]]i ::::,:l::: r8 Hinh 1 Hinh 2 Hinh 3 Biri to6n 1 - Cho linh uudng ABCD. Ldry Xet bei to5.n vdi rn6t s6 ui trl dac biat ctn mot clidnt M b(it ki trong hinh uudng do. Duitng didm M, thang d1 qua M cat AB ta,i E, cat CD tai G, 1) Khi M = O (tdm hinh vudng), ta cd : duitng thang d, qua M uuOng g6c uoi d, cat Bii to6n 2 - Cho hinh uuOng ABCD. Hai BC tai F, cot AD tai H. Ching ninh riing dudng thang d, uit d, qu.a tdnt O crta hinh EG: FH, uu6ng ud uu6ng goc uoi nhau, cat cd.c canh Bdi to6n cci nhi6u crich giii. Sau d{y ld y hinh uudng tai cdc didnt tuong ing E, G, F, chinh cria 2 cach H (linh 3). Chtng ntinh rang : C6ch 1 -HaES r CD vdET r AD. a) S(AEOH) : S(EBFO) : S(FCGO) : AESG : AFTH (cgc) (hinh 1) 1 S(OGDHt = 4. STABCD) b) EFGH Id, linh uu|ng. MEJE.F: Hinh, 4 Hinh 5 Hi.n.h. 6
2) Ndu M = I, trung didm cria BC, vd dudng B}.i to6n 7 - Dung hinh uuOng ABCD biet thing d, qua A thi F : I vd H : A. Ta ddt dinh A uir hai didm M, N tr€n BC ud CD. drroc bdi to6n 3 : Blri to6n 8 Cho hinh uuong ABCD, tren Biri to6n 3 - Dqng hin.lt uuOng ABCD bidt canh BA uit. CB ddt BP = BQ. Hq BH L CP. d.inh A ud. trung didm M cia BC. Ching minh rang DH t HQ. Goi ! gid.i: Qua M drrng d, r AM. Tr6n d, Goi ! gid.i - K6o dei BH cit AD tai I, suy dung ME : MG : AMIZ. Ha MB .l- AE. ra dttoc IB : CP til dd cci AI : BQ vd IQCD (hinh 4) Id hinh chtt nhdt. Nam dinh I, H, Q, C, D cung ndm tr6n mdt dudng trbn (hinh 8). 3) Ndu M ld m6t didm bdt ki J tr6n canh BC vd dubng thing d. qua A thi F = J vd Blri to6n I - Clrc linh uuong ABCD. Lciy H : A : E. Ta dat dr:oc cdc bii to6n 4 vA 5. M bdt ki tren CD. Duisng trdn dudng kinh AM uir dudng trbn dxdng kinh CD cat nhau d didnt Biri todn 4 - Cho hinh uu6ng ABCD uit. didnt M bdt ki thuQc canh BC (khd.c B ud C). thrl ltai N. DN keo diti cat BC tai P. Ching Goi N l,ir giao cio. hai dudng tlmng AM uir CD. minh rdng : PM t AC. Chilng minh rang : Goi ! gid.i - Gsi Q li giao didm cira AB vd 111 dttdng trbn drrdng kinh AM. Suy ra drroc Q, I N, C thing hing, tr) dri chrlng minh drroc AB2 AM2 ANz QB : PC : MC. (hinh 9). Goi y gini - Qu. A drrng d, r AN, cit CD Blri to6n lO * Clrc linh uu\rug ABCD. I tai G. Ap dung hO thrlc trong tarn gidc vu6ng ld nfit didnt bd.t ki thuoc BC. Ha BM t AI. AGN. thinh 5t BM cdt CD tai G. Dung CC t ll AI (C, ffan AD.t, Blri to:in 5 - Cho hinh uuong ABCD co DDt ll BG (Dl tr€n ABt. Xoc dinh ui tri cia I co.nh bd.ng a. QuaA d,ttng hai tia -Lx, By sao dd MNPQ lir tinh uudng. cho xAy : 15", hai tia At uit, Av cat BC ud AD B}i todn 17 - Clto ABCD lit hinh uuong ldn luot ttti Ii uii E. cqnh o., tio At b1it ki nant trong goc uu,1ng a) Cht|ng ntinh riing tam giac AHE co d6 BAD. Dudng phdn gio,c iia goc B,Lx cat BC dd,i dudng cao xud.t phdt tit A khdng ddi. tai M, dttitng phdn giac cia goc D,+x cat CD tat N. Got E ld gia,o d,idnr cia Ax uit. MN. Tint b) Xdc dinh ui tri cia N uir E dd SANET qu.y ttch cia dii:nt E. ld cutc tieu. Bii to6n 72 - Ctrc hinh uudng ABCD co Goi ! gidi - Coi Ax Ie d2, dUng d, -L- Ax, cat canh barug o,. Tr€n AB, AD ltiy hai didru M, N CD tai P. Suy ra AP : AN vd AAPE : AA}iE sao cho tanr gidc ry! c6 chu ui biing 2a. (hinh 6). Clfing minh rang CMI{ -- 45". Sau dAy ld mOt sd bni to6n trrong tu, dudc Blr.i todn 13 - Cho M, N lit trung didm cio, dat ra ti bai torin 1. cac co,nh AB ud BC cio hinlt uuong ABCD. Bhi to6n 6 - Dttng hiruh uu6ng ABCD bidt ND uir CM cdt nltau d P. Ching m.inh rang 4 rlidm niinr trdn 4 conh cio linh uudng. PA : AB. Goi j, gid.i - Qu" 1 dr-tng Ix ll LJ, tr6n Ix l5y D6 nghi ban doc tirn ldi giAi cria c6c bai doan IK' : LJ. Qua J dung JC r KK'. (hinh 7) to6n tr6n. li6n h6 v6i bdi toan 1. Hinh 7 Hinh. 9
M ro: O e Zvdut: lcl € Z n6n tr) (*) suY ra uneZVn€N(Dpcm). Nh4n x6t : 1. Tba soan nhAn drloc ldi giAi cta 105 ban, trong dci c
,W + z,TCliS : 1r,, + y,)rh+fi, Nhfn x6t Cd 129 bai gi6i grli v6 tba soan, tdt cd d6u giAi dring"Ldi gini t6t g6m c5:Vu Cdng lai ta crj di6u phii chrlng minh. Minh Giang (1181 'PTTH Bim Son - Thanh Cactt 2 (ctia da sO c6c ban) : H6a), Dirylt Trung Hii,n.g (11M Marie - Curie Detr = tgo, y : t# ; z : tgY trong dd cr, Ha NOi), Phanr Dinh Trudng (1 1CT PTTHNK Trdn Phf - Hai Phbng), Thanh Hxong d.v a I\.0, i) . Tr] didu kien bg.-r * tgrl * tg;' : (l1PTTH Luong Van Tuy - Ninh Binh), z/ tga tgp tgy suy ra a * F + y : z. Tr] dci thay Nguyan Ph.n Qud.ng (10CT DHTH - He NQi), vdo vd sit dung bidn ddi lrrong gi6c don giAn Tit Minh Hdi (llCT PTTH Ban M6 Thudt, ta d6n ddn vd trrii ld l(sincz + sinp + sinT) - Dac Lict' sin(a * p) - sin(p + y) - sin(7 + a)l/sina sinf vrtrN sinT : 0 do di6u ki6n a * p * Y : n ')ANG Bai T4i208. Cho nant didnt phdn biet A, Nhfn x6t : B, C, D, E uit. nt6t dtong tlto.ng d citng niint Hoan ngh6nh nhi6u ban l6p 9 nhu c6c ban tygng 1$t nfij photgg. Gg O li4didnt sao ch.o Biti Thi Pliuong Nga (Ldp 9 Lo Khi6t, QuAng OA + OB + OC + OD + OE : O (tic O l,d trqng Ngai), Gidp Daig Khoa (9ATAnY6n, Hi B6c), tdnt cia tt€ d.idnt {A, B, C, D, E} ud A', B', Nguyen Anh Tud.n (9T Phan B6i ChAu), C', D', E', O' ldn luot t.it hinh chie'u tuu1ng Npu-yen Thanh Ild (9A Hda Binh), ... da giAi g6c) ffAn d ctio, cac d,idnt A, B, C, D, E uit, O. A,:ng Uai ndy uri kh6ng dung ph6p d4t hiong Ching ninh ro,ng : giac nhrl y dd cria tl'c gib bdi to6n. Ldi giii cta +l+++++ Ean C)udng d tr6n ld rdt hay, vila so cdp I'ta oa' = ;i) Ga,A' + BB' +cc' +DD' +EE') ngin ggn. 2) Mot vdi ban m5c sai ldm trong su-v- luan Ldi gidi (Drra theo Vuong Vu Thang, l0A khi cho rang : "Vi tg4 + tgB * tgC : tgA tgB chuy6n torin DHTH Ha Ndi vd mdt s6 ban tgC ndu A, B, C ld ba gdc crfa tam giric nen khrlc). pf,al cO x: tgA,Y = tgB,z: tgC." Ta giAi bdi todn tdng qurit cho trtrdng hop IIANG rltlN(iI'HAn*G n didm phin bi6t {A' A., ..., An} cci trong tAm Bni T3/208. Ching minh riing tam giac ld O, nghia ld : Ats(' diu khi ud chi khi : /r - = o; (1) 2o^, - 111 -- -!- -L .;i n-A sin2B sin2c ABC 2sin , sinT sin , Qua O dung dudng thing A vu6ng gcic v6i d (d O') vd goi A;' le hinh chiSu vu6ng gcic tren Ldi giii (dua theo Vu Minlt Giang - 118l d cvaA, (i : 1,2, ..., n). - P?TH Birn Son - Thanh Hcia). Nhnn ca hai ( r,6. voi sinA sinB .sinC, drJoc tt sittBsinCr sinCsinA sinAstnB li: v ai l" ZJ "' 't' .i,,e ;i ,B - s,',, c - Dat : I r=r : 4cos ABC l;"= ), oa', uo s' :) oe, li, + , fZZLcos ; cos; (1 ) -t ABC cosT: Vi OAi : O'Ai + OA; (i : 1, 2, ..., z) non : Ma 4cos ,.or, s== s' -1- s trong dcj s' ccj phuong cria d. cdn A+B +cos,4_8. C s++r++co ohrlong cua A I d. Do do =2(cos , T )rorr= s : 0
nn Ldi giai. Goi O, G ldn lrrot li t6m mdt cdu (1) > o1, = o',*) Fd, :i; ngoai ti6p vd trong tdm trl di6n ABCD, ta cd : i:l i=l+ ---> ---) Lai vi OAi : OO' + O'Ai + ---+ Aii 4R2 = OA2 + OBz + OC2 + OD2 : ++-++--++-- (oG + GA\t + (OG + GB,tz + (OG + GCtz + +1+ = : ,o'Ai + oo' - Ai,{i, rr6n : +(oe1 cn).^: +&+ zfcpa+ ca+ c'c+ cn; + GAz + GBz + GC2 + GD2'. Il-n*n : 4ocz ^9+ 2&,=)Fe,*) to6' -4,,, = *{*? +mf, +rn2, +m)y. Ma i=l i:r i:1 fl- m| + rnzu + m! + *ro, (m, * mu * m" * *o), : no6'-2e\, + (Bu-nhia-kdp-xki) Do dd t:1 : 4Rt^9> t -l m, * m,)z Tt (1) va (2) ta dtroc : nOO' -L,it,:e i:1 64(mo 3 nt u hay ld hayR > : *(mo*mu*mr*mo). oi'::2&,, (3) Ddu ding thicxdy ra khi vd chi khi G : 0. Dd Id d.p.c.m. NhQn x6t : 1 - Cd 103 ban tham ga g,ei bdi tt.,6n ndy. 2 - Ban Thd.ng vd nhi6u ban khric cbn cd nhdn x6t sau dAy : K6t quA cria bdi to:in kh6ng thay ddi khi ta thay hinh chidu vu6ng gric bdi *-** hinh chidu song song fo3c thal, he didm {A1, =* Az. ..., A,) trong mat phing bdi he didm trong khong gian, d6ng thdi drrdng thing d thay bdi mat phing d. 3 - Cd m6t sd ban giAi bii torin tr6n bang ,l..i.........i ,.iiiil,.,li.l....,.i.',,..,.t...l...llllll....,.i.........,.,llllt.".....'.,i.,1...u..tl'....i,.,,....:=iiiitri..iiil phrrong phrip toa dO ; tuy nhi6n, thrrc chdt cung Gsi I,1, J ld c6c trung didm tudng rl"g i,iu ld phuong phrip chiSu v6cto nhtr dE n6u 6 tr6n. AB, CD, ), ta c6 G li trung didm cha IJ. Ndu G Nhi6u ban srf dung phuong phep chr.lng minh :0thiri tr) ctic A, cAn GAB, GCD, ta c6 GI L {uy n?p, nhrrng trinh bdy kh6ng sang iria. AB ; GJ t CD, n6n cdc A vu6ng GIA, GIB, 4 - Y tudng co bin trong chrlng minh h6 GIC, GID bing nhau, suy ra AB = CD. Trrong ihrlc (3) la thiSt l4p he thtlc (2). Cci nhi6u c6ch tU, ta ctng cci BC = DA; CA = BD.ydy, tg chrlng minh (2). Ngoai c6ch n6u tr6n, dd chrlng didn AB CD gdn ddu. DAo lai, ndu trl di6n ABCD minh (2) cri m6t sd ban srl dung d6n tich v6 gdn d6u thi G : 0. Va, d&ng thrlc xdy ra khi vi chi khi tt1 di6n ABCD gdn aOu. hudng : chrlng minh ring -ll+ u-1) O'e,S = 0 (Vd Nhfn x6t. Trrf m6t vdi ban thu hep pham vi xdy ra ding thrlc vdi trl di6n d6u, c6n thi tU dci suy "* i: >;A,: o1-ru, ban Nguydn cdc ban d6u gi6i drlng. Ldi gi6i t6t g6m c6 : Vu i Dtc San (11T - Lrrong Ven Tuy, Ninh Binh), Ngoc Tdn, 10CT DHTH Ha NOi vd La Anh Dinh Trudng (11Ao pTiVK Trdn phu Tud,ru l0A CT DHSP Vinh d6 srl dung ph6p !!qy Hei Phbng), Phan Duy Hing (1lCT Ddo Duy d6i xfng qua dudng thing d dd chrtng minh Tt - D6ng Hdi - QuAng Binh). (2) cflng khri gon. oANc vrEN NGUYEN DANG PI.IAT B)'.i T6/208. Tim s6 c6 S chil sd chia hdt Bei T5/208. Gqi R ld, bd,n kinh met ciiu cho 9 sao cho ngoqi tidp, mo, ffib,ffi",molit dO dd.i cac trong .tltuong sd trong ph6p chia sd d,y cho 9 bdng tdng binh phuong cdi chu s6 cila tuydn xudt phd.t ldn luot tir cac dinh A, B, C, sd tiy. D cia mQt ttl di€n ABCD. Chung minh bdt dang thtlc : _ Ldi giii : theo Nguydn Dang Thd.ng,9NK, ThuAn Thdnh, Hd B6c. o Go.isdphAi timlim (0 < a ( 9, 0 < 6, c < 9). J R, t'(mo*ntr*mr**o) Theo ddu bdi ta cd abc : g(a2 +b2 +c\ (l) khi nd.o thi xd.y ra dang thic ? hay 9(1 La + b) + (a + b * c) = 9(a2.+ b2 + c\ (Z)
Vi tEA i 9 n6n suy ra a*b + c i 9. Tinh ; LA Quang Nd.m,9CT, Drlc P},6 ; Luong Y|ya1-b*c:9,L8,27 Le Ti, Nguy6n Hitu H6i A, Trd.n Thi Ngqc HAi,9T, chuy6n Le Khi6t, QuAng Ngai. 1. Ndu a* bt c = 27 suyra a = b = c - 9. Ta thdy ngay (1) kh6ng dudc th6a men rd NcuvEN 2. N6u a*b -lc = 18 t'ac6: c = L8 - (a+b) Bai T7l208. Cho at, d2, ..., an td cdc s6 (2) + lla * b * 2 : a2 + b2 + c2 (3) duong rc, tdt cd. dbu khOng c6 udc rtguyan 6 Thay c vdo (3) ta c A = (2rn * tllz - 4(4m2 - 13m + 28) : 57 - l2m2 1 1 _+...*q. I **)1, Phrrong trinh (7) cd nghiQm khi vd chi khi (1*E* a>0 , 1 1, 3 5 15 57 - l2m2 > 0 m2 = tn : O, 1, 2 / 1 +;b +... \ +-\ f,l") -2'4- 8 Mat kh6c vi phuong trinh (7) ddi h6i cci nghiQm nguy6n n6n cdn ld A phAi lis6 chinh (vi t + a+ ...+0m =' :o**' I-d < ;Lrroi l-a phuong. Ta thdy chi c
Ldi giai kil.a nhibu bq.n) : Luc, 947 Ddm Doi - Minh }J.hi; Le Quang : i__ Nd.m,9CT Ddc Phd - Qu6ng Ngei; Pham Huy o Dat a: ',/" +r[JrTF ti Tilng,9ABdVan DAn, Hi N6i vdNguydnNgqc DOng,9NK Thu6n Thdnh, Hn B6c. 3r-------:- NGUYEN KHAC MINH Vidt lai phrrong trinh d6 cho dudi dang : BAd T9/208 z Cho tam gidr ABC, uA cdc phd.n gid.c trong AAr,BBy,CCt Gid srt AA, x3 -gqbx - (r3 +6') : o cdt BlCr tqi K, BB, cdt CrA, tai E, CC, cd,t ,y,3 - @ +b)3 - Sablx - (o +b)l = o ArB, tq.i F. Chilng minh rd.ng ndu AK : BE .dt - @ + b)11r2 + (o + b\x + a2 - ab + b2) = O : CF thi ABC ld tam. gidn d,bu. el ^ lr.: a *b - l*, + (o * b)x -t a2 - ab tb2 = O (7) Fhuors trinh (1) cd tr=(o * q2 -4@2 -ab *b') = -3(o -b)2 < O (a * b do s2 + f > O) n6n nri v6 nghi6m. Tt dd suy ra phtrong trinh de cho cci duy nhdt nghiOm: x:a*b: -!s +{-Cz+P + Vs -1;z+-F o Cci : x3 +x * 1 :0 (2) o . 1 1. rr+3 Ex-2.\-r) :0.vi Ldi giai : Theo tinh chdt dttdng ph6n giric ,1.t , Lz 31 ta d5 dAng cci : (a)" * \-Z)-: 108 r 0 n6n, theo tr6n, BAt: ac BCt: cbc ft: cdng thrlc phuong trinh (2) cci duy nhdt nghiOm b +" , : tinh dttdng phAn giric "*, -tf 1 [-gf- 31 x= -Z*1 I 1 ,* + B 2accos - 2 o Vi r : 0 khdng ph6i ld nghidm lu : ----------r- suv ra cria ' a*c phrrong trinh 2013 - ttu2 - 1: 0 (3) n6n (3) B ,1't 1 2BCt. BAr. cos :(;)"+3.5.--2.r0= 0 (4) Dat, =;, 1 BE: , 2ac B BCr+ =;+2b +c 'cos, BA1 tt (4) c 0 n6n pt (b), 2s.c 2bc theo tr6n, cd duy nhdt nghidm : Ndua>bthi a*2b*c- : 2a*b*c r T1o+15 + TT0-15 Tn dd suy ra, pt (3) = W- tl*r. BA cci duy nhdt nghi6m : vdA > B tric cost > cos Vay BE.> AK 1 .5+5 15+W i ,-: EF]TE- N Tuong trl ndu b > a thi A7{ > BE AK : B,E n6n b : a. - Nh{n x6t : 1. Ccj rAt it cec ban hoc sinh Md, Tuong tn EB = CF n6n a : c. bac TIICS grli ldi gi6i.cho bii todn, Suy ra AABC d6u. 2. Phdn l6n cic ban grli ldi giAi da dtng phriong ph6p kh6o srit hdm sd dd chrlng minh NhQn x6t : Gi6i t6t bai ndy g6m c6 cd.c tinh duy nhdt nghi6m cria phuong trirrh b4n : 13 +}rx -2s = 0. (6) IE C6rug Son L2Ar, PTTH Yi6t Ditc, LA 3, MQt s6 ban cho ldi giAi kh6ng hodn chinh Tl^rdnAnh 108, DHTH,NguydnVu Hung 1lD do khQng chrlng minh hoic chtlng minh sai PTTH chuy6n DHSPNN, Phan Linh 9A PTCNN, Pharn Huy Ti^rng,9A THCS BdVen tinh duy nhdt nghi6m cria pt (6). M6t s6 ban Dnn (Hn NQi), Phqm Quang ltlinh l2T PTTH kh5.c "qu6n" kh6ng gi6i phdn hai cira bdi torin. H4 Long (QuAng Ninh), L€ Van Manh llCT 4. C6c ban cci ldi giei t6t (chi tinh trong sd Chuy6n Hoing Ven Thu (Hba Binh), Nguy1n c6c ban lA hoc sinh b6c THCS) : Nguydn L€ Chi Dttng 88 Chuy6n Vi6t lri (Vinh Phd), ffd Thu Thd.o 10CT P?NK (Hei Hung), Ds.o L!,
12A PTTH ChuY6n, Dod,n Minh Ditc 8D vd BY' ; trong d6 BY nim trong g
\t x. y z ax*by+cz 2S vd. t : tt + t2* t, : lflB +f/0; ' "-,,' -:u-: a b -: c o2 q62 4"2 o2 y62 a""2 Crlng cd thd dua vio dd thi vdn tdc - thdi gian dd gi6i. NhQn x6t. Cdc em sau dAy cri ldi giai tdt : Nguydn Vu Quang, 11L, PTTH Ddo Duy Tr), a2 +b2 +c2 - QuAng Binh ; Nguydn Hbng Nh.d.n, 10A4, 2bs PTTH L0 Quy DQn, QuAng Nam - Dd Ning ; Do dci, v6i didm Mo Doitn Dinh Trung, t0L, PTTH Amsterdar4 - : " aZ +b2 +cZ Hanoi ; Td Huy Cudng,11A1, PTTH Chuy6n 2cS " Th6i Binh ; Nguy4n Thd.i, 10T, Phan Bdi d2 +b2 +cz Chdtr, Vinh (Nghc An) ; Nguydn Drtc Tud.n (trong dd S la di6n tich tam gi6c ABC) Vinh, IOCL, PTTH Ban MO Thu6t, DhcLilc; Hit. Huy Hi.rng 10ACT, DHSP Vinh, Ngh6 An ; Vdi didm Mr, ta cd : Nguydn Thanh Hbng, 11A3, PTTH L6 Hdng x Phong, Nam Dinh (Nam }Jir) ; Nguydn Hbng M,:-: La + Minh, B.IOACL Dai hoc Tdng hop Hd Ndi ; (b2 +c2 -a2) V6 Thanh Tilng,10CT, PTTH Qu6c Hoc Hud. 2S + OK !(b2 !c2 -a2) Bdi LZ|ZO8.,Tr€n u6 ctra mQt quat di1n c6 2aS cd.c kl hiau - 110v.60w.50H2. Khi n6i hai d?iu ,. - + +c2 -a2) 1b2 dd,y ctra quat uito 2 cgc crta chidc pin c6 sud,t 2bs diAn d1rug 7,5V ud. di€n trd trong nh6 kh)ng =Mr dd.ng kd thi cudng dQ ditng di€n chay qua d + (b2 +c2 -a2) quat \it 0,0254. 2cS 1. Tint di4n trd Rn uit. hQ sd cOng sudt d{nh + (b2 +c2 -a2) mic cita quat di€n. hay - tuy theo gcic A uhq hay tir : 2. Mudn ditng quat diQn bAn d d diQn xoay Trrong tu, ta duoc cdc didm M2 vd M3. chibu 220V - 50Hz tuguiti ta nrdc n6i ti6p cho no mQt tu di)n. Tint diAn dung thich hop ctta b2 +c2 +a2 * 0, c2 +a2 -b2 *0, a2+b2 -c2 *0 tu diln. vi AABC kh6ng vu6ng. Nhin x6t : 1. Bdi torin ndy tuy don giln Hudng d6n -'o = 3I : UOn giii.'' l) R,, ""'-' ' nhrrng it ban ccj ldi giii honn chinh. Kh6ng cci ban nio chi ra drloc chinh x6c vi tri hinh 'PPRo hoc cria nhtng didm phii tim bing phuong k: cosg = fJI = rJz**p - phdp dtrng hinh hoc nhu dd chi ra, trr) ban LA Huy Khanh. __+ cosp _ tlFE" _ e 2. Trong vi6c tinh c6c khoAng c6ch tt cric u -11 didm M, ddn c6c canh crfa tam gi6c ABC m6t Ro sO ban da tinh khri gqn nhd nhAn x6t : 2)Z:-:110- cosP s(ABC) : I s(M,CA) + - s(MBC) | *zL: {Zr4 :(M4B) = 10{85 (o) NGUYEN DANG PHAT Pd* BAi Lfl208. M|t uQt chuydn dQng chd.m I_ : 1(A) ; dd.n dbu ; x1t 3 doaru duitng liAn ildp bang nhau Uorncorp trudc khi ditng tai thi doan o giua ruo di trong ls. Tim. thiti gian uQt di 3 doan dudng bang lur. - u6l = WT-t{ = 8o{7 * nhau kd tr,2n. lZ, - z.l = 8o{7 (o). Hu6ng d6n. Ap dung v6i cdng thfc v6 Y\2t,:' 10V85 < 8o{7. (,,) chuydn d6ng chAm ddn d6u. Goi v1, v, va li ---Zc Zr:8N7 +e - +:10,48pF vdn t6c ctra vAt 6 ddu doan drrdng thrl uhdt, @L, ddu doan dudng thrl hai vd.ddu doan drrdng Nhfn x6t. Cric em sau ddy cri ldi gini dring thrl ba : -v! = 6as ; - uZ= 4^r; - rZ = 2asvd' vi t6t : Dao Li, l2A, PTTH Chuy6n Thrli v3-v2 Binh ; Phan Hod.ng VieL l2A, Qu6c hoc Quy tZ = = 1. Tr] dci suY ra Nhon, Binh Dinh ; Vfi Ngqc Thd.ng, 12 H6a, PTTH Ha N6i - Amsterdam. t, = (t[2^+ 1) vn t, =({3 - rlTXrlT + t) oK.
Cdc l6p THCS Tll2l2 z Tim s6 .Y, bidt ring Fidi W=(x+y+21a"{vain€N) NGUYEN DUC TAN Hb CHi MINH TP Dii RA Ki I{AY Bni Tgl2l2: C6c didm Ao , Bo , Co , Do tudng BlniT2l212 : Tim tdt ch s6 th{c a, b, c dd rlnglitrgngtAm cta citcmdt BCD, CDA, DAB, ax + by * czl + lbx + cy + azl + | cx + ay + bzl ABC ctla trl diQn ABCD. Goi A, , Bt , Cr, D,l I = lrl +lyl +lrl lddingthrlcdringVx,y,zaR tudng rlng li cdc didm ddi xrlng cria c6c dinh A, B, C, D c0a ttl di6n qua m6t didm O cho DANG VAN TTANC, HAI HTJNG tnJ6c trong kh6ng gian. Chrlng minh ring c5c B,di T31212 : Gi6i vd bi6n luAn phrrong drrdng thing Ao A1, Bo Bt, Co C t, D, D, ddng trinh sau theo tham sd o .' quy. x3 - 3x2 +3(o + lF - @+ 1)2 : g LtJu xuAN riNn, rgaNs rr6a oANc ilUNc THANG, rIA NoI Bai T10/212 : Cho tam gi6c ABC cdn 6 A, T B,di 4D1.2: Hinh vudng ABCD cci canh Tr6n AB ldv 1 didm D vi tr6n BC ta ldy m6t didm ,E Lho hinh chidr: ciua DE lon Bi bing don vi. Tr6n canh AB vdAD chon hai didrr,M 1 "ro vA N sao cho chu vi tam gil'c AMN bing 2. Chring minh ring : dtrdng vudng g6cv6i Chrlng minh ring hai tiaCM vit CN chia dudng ;rC. ch6o BD thdnh ba do4n th&ng md dd ddi ba DE tai E lu6n di qua mQt didm cd dinh. do4n niry l6p n6n tam gi6c vu6ng c2 " ) dting c6 K.K2K3 ring Bidt khi d L. (*) (6 1) Tinh E, r vd gi6 tr! mgi di6n trd ngoii. dey lal ld phdn nguy6n c.&.aavd {o} = a - tal 2) Nhich con chay C A R3 sang ph6i hay ld phdn phAn cria a.) sang tr5i dd c6ng sudt m4ch ngoii gi6m ? TRAN VAN VUONGTA NOT TRAN VAN MINH, HA NOI 10
For upper secondary schools PROBTEMS IN THIS ISSUE .,:,,,: 't6lzlz. Prove that there exist positive integers x, y, z satisfying For lower secondary schools xx+yY - 7P where p is an odd prime. T71212. Find the number xyz such that T7l2l2. Prove that the system of equations 1fr=(x+y*z)a" *2=y3+y2+y+a (n is a natural number). y2=23+22+z-Fa TZl2l2. Find all real numbers a, b, c so that 22:x3+x2+x+a I ax +by *czl + lbx +cy *azl * | cx+ay+bzl has unique solution. : lxl +lyl +lrl T8l2l2. The sequence an is defined by holds for every x, y, z €. R. a1 =1 '131212. Solve the equation 1 * ar.,_,, x3 - 3x2+ 3(a* l)x - (a + 1)2 &., = 1 * " {--;-I}foreveryn > 1. = g, whereaisaparameter. Prove that T4l2l2 The square ABCD has unit sides. 8., = 1 +2(n -2lloqin0)foreveryn) 1 M and N are two points respectively on the (where [a] is the integral part of a, and {a} : sides AB and AD so that the perimeter of a - [a]). triangle AMN is equal to 2. Prove that the Tgl2l2. Ao, B,,, C., D. are respectively the lengths of the three segments obtained by ce}lers^of'gravityof thesides BCD, CDA, DAB, cutting the diagonal BD by the semi lines ABC of a tetrahedron ABCD. Let A,. B,. C,. CM and CN are those of a right triangle with prllegnectively the points d, q;C, q through a given point Oin"ymmetric'of space. Prove tiran ^f area not greater . that the iines AoAr, BoB1, CoC1, DoD, are \2 + l2)) concurrent. TSl2l2. In the plane let- be given two .Tl0l2l2:Uet be given an isosceles triangle parallelograms ABCD, A'B'C'D having a with base BC. Take a point D on AB and- a common vertex D. Prove that the two triangles point E on BC so that the projection of DE on AB'C and A'BC' have common center of 1 gravity. -2 to ; gC. Prove that the line BC is equal p-erp-endicular to DE at E passes through a fixed point. ob rur rrec srNH crdr rvroN ToAN Lop s cuA uAr pubNc Vbng 1 :2811211993 Vbng 2 :281721L998 (Thdi gian ldm bii : 150 phrit, (Thdi gian ldm bdi : 150 phrit, kh6ng kd ch6p dd) kh6ng kd ch6p d6) Bei 1. cho s6 M : iE. H6y tim sd n l6n Bei l. chrlng minh ring phuong trinh nhdt dd ding thric c
PHUoITG pnAp wffimffiffi PHAM BAO Hd N1i Trong to6n hoc khi md rOng mOt khrii ni6m lt +\+:/ *2(e, er* er. er*e.n) >g ndo dd thi ddng thdi vdi su rnE r6ng dci ta cd ,ByS . e, : cos(n - Bl: -"otB, m6t phrrong ph,ip m6i, m6t cdrrg cu mdi dd gihi cac bii torin. (nz.n) = -cosC; (%.e) : -cosA Trong hinh hoc thi uecto ld m6t vi du. Khi md r6ng khrii ni6m "do4n thing" (v6 hrI6ng) Ta co cosA + cosB t"r"C
oat lc?l : lCbl:o lcV,1 :u AC tj4, g: D6 thdy BD, ---> :(-l',a,b) _"norrg*,} , ' @ € R) ta ccj AQ: x& OC OA *AC : OA *xAB : Mat phing = OA+x(OB-gA)_- BC.D ld mat OC=(l-x)OA*x.OB phing ch6n Chor : a I - x : p ta cci di6u cdn chrlng tr6n ba truc toa minh xem hinh d9 n6n c6 0 phrrong trinh ld xvz bl'I-t-,-{- tl--T--_\ -+1+ oab- =.1 vectd phdp ctra mAt ph&ng la + .1 1 1, n= (-,-,;) \o o o/ . Goia ld gdc ldm bdi BD, vd nr[t phing BCl D thi sin.r : -.- cos(BD. . z) : Cdn chu y rang vdi m6i didm C nhdt dinh .I ,{ra?iF 21 tr6n A-B thi cd mdt sd r duy nhdt do dd cAp -a2 +-62 (a, pt ld duy nhat. *2. Trroqg tu-ghd trUgng hSp_!: hjglu di6n , a2 62. OE theo OB , OC vd OD theo OA , OE (xem / +-\ \ -b: a2l hinh 4b) ta cci di6u cdn chfng minh Chir i : M6t h6 quA rdt thudng gap cria (I) a2 62 Id : Trong kh6ng gian cho 2 doan th&ngAB vd nrd -*->2 n6n sina l6n nhdt khi CD ndu 2 di6"m : M trdn AB, N tr6n CD cung a2621 ry^;: + . *, = 2trtc o : b vd.sina : ;khidcihinh b*o-J chia l-B vit CD theo ti ,o AB CD = r thi ++ h6p ld hinh lnp phrrong vA gric a x5.c dinh nhu MI{=(1 -r)AC+xBD (I',). d hinh 3. Dd chrlng minh (I') chi cdn-|dy qrgt di{ryr Qua c6c vi du tr6n ta dd thdy c5"ch vAn dung O tny y ldm g6c ; phdn tich MN: ON - OM "phuong vd d6 ddi" trong phuong phdp vecto. rdi vdn dung nhu trtrdng hgp 1. Tuy nhi6n dd thdy duoc phrlong hrrdng vdn Sau dAy ld m6t sd vi dq 6p dung. dung vd ldi giAi duoc ng5n gon cdn chri y ddn nhring bdi torin co bAn sau dAy (vi khu6n khd -Vijq {: Trong,mat ph&ng chq3 vecto OA, OB , OC cci OA + OB * OC: 0 MOt bai brio ccj han ki ndy chi xin n6u bdi torin thrl drrong thing d cdt OA, OB vd OC keo dai tai nhdt). A', B', C'. Chrlng minh hd thfc : Bii to6n I : "Ph5"n tich mdt vecto theo c5.c OA OB vectd cd sd". 1._!rong kh6ng giarr hai chi6u cho h6 co so aE*ag+oe:a (O , OA, OB) Chrlng minh rang vdi moi didm _aFoe GiAi : Dat :=:=- : nL-*:n C thudc drrdng thing AB ta cd ' uA OC' ++ OC:a,OA+{J.OB : p (nt, n, p € R). Ta cci Oe ++++++ v6ia+P: lvdct,PeR (I) OA' : m OA ; Og' : n Q-B ; OC' _=p OC theo (I) ta c6 OC' : a OA' * p OB' vdi 2.-f,rorlg kb6ng gian ba chi6u cho h€ ct+lJ:L (O , OA, OB , Oq Chfng minh rang voi moi +++ _h.y - p 99 : am. OA + pn OB thay didm D thuOc mat phing ABC ta c6 : OC:-OA-OB --+ + ++-->+ P (-OA - OB) : --t -> OD: a.OA+B.OB +y .OC am.OA +pn OB * p : vdia+P*Y: lvd'a,P,Y €R. (II) : -clnl, : -Bn l3
p,,p ring vdi x : d[, (t: --'I): nL" -- n 1) ChrJng minh , thi doan va MN ngSn nhdt DD 2) r
Tim hidu sdu thdm todn phd th6ng TunnOr eAroAr.rc THUc TfcH pHAN.., NC6 THANH LoNG Hd Nii Tich phdn vd nguy6n ham la mOt phdn m6i b cria chrrong trinh CCGD. Vi thd de girlp th6m c6c ban phdn ndo khi hoc vdn d6 niy, t6i xin ,a'[ ppu, r, :,,,Ai,f{Es.li1} trinh biy nhrlng tim tdi cria t6i khi cdn hoc phd thorig tt m5t bii toan tuydn sinh. Vi thd (3) tr6 thinh Tni6c hdt tdi n6u lai m6t sd kidn thrlc cdn n A. st dung. ri* jrrr8 ,,u# r,,, > rrtE,try, 1. Dinh nghia tich phAn (s5.ch giAi tich 12) il+et:l n +6r:l fi, 2. Ndu f ui., p lA. 2 hdm li6n tuc, xdc dinh hh fri," fG) ) tL .ptE)9-9)2 rsr tr*!l--r ) I p1a* < [ p@:) dx (r; Tt (4) ndu thay o.1: f(€),bi: p(€) ta cd : ililllt\ , ,2 3. Neu f Id hnm li6n tuc, x6c dinh tr6n la., 6l vd nr < f(x) < M thi : Ze'G,), l> fG;)vrE,tl >f'zGt)r=t i:t rct h (-r ) nt(b - "l " d.x < M(b - a) (2) Tr) (6-t ;a d6 ddng cci ngay (5). Dau bang [f(r) xdy ra khi : (Ban doc trr chrlng minh 2 tinh chdt ndy). f(E; e19,1 : fG ) : e(E) =...: f(E) , : elE,,) Ta x6t bdi to6n sau : hay f(x) : cp(x) (c = hing sd). Bhi to6n : Chr.lng minh r5ng ndu f vd p ld Vay bai todn duoc chfng minh. 2 hdm li6n tuc, xric dinh tr6n [o, b] thi Bdy gid tt bai to6n ban ddu, n6u th6m giA : bbh ! f a; o* [,e2 67 a*, ([ no sg; ax)2 rz) thiSt : f, p '. \a, bl [0, 1] thi f(r) < /(r) vd - hb p2@) < p(x), n6n ta cci : I Ptfl rt, * J f@) dx (CAu IVa - Dd 106 - DO thi tuydn sinh) Vi6c chrlng minh bdt dang thrlc nay khdng bbail khcj, c6i xudt phSt cua bd't ding thfc niy ld mdt bdt ding thrlc dai s{ : Bunhiacdpxki. va I p21x) O, = {9(x) d,x. Vi th6 ta cci ngay : Cho 2n s6 a;, bi (i : l, n), ta co : Bii toSn I : Chrlng minh rdng ndu f vit.9 { ("? + al+...+"1)(a1 +b7 + ,+bl) > le 2 him li6n tUc tr6n la, b1 vh (arb, + arb"*...+ atpt)z (4) f : [o' bl - [0, 1] uit. 9 : fa,6l - [0, 1] thi hbh Ddu bang xAy ra khi : a, : b, : a" : b, :...: err'. b, I f@ a, I e@ dx ,- ([f@eg1ax)z {z) Trong doan [o, 6] chia lim z doan nh6 bang nhau b8i e^c didrn chia Vd: A:X,,
Cho nr d6y sd I aitbit,fi (i: t") kfro"g Ldy gi6i han khi 11 + oo ta cci : bhb dm, ta cd : (q + q + ..+,f)(bT +bT +...+al}l ... I fo) a, I e@ dx < (b-a) [ ro) etO tcr (13) { (ft +fi *...*ff) > (oPr.. h+ozbz,..fz+... Vdy ta cri : * anbn ...f,,)* (9) Biri torin 5 : Cho f vi, p ld 2 hdm li6n tuc, x6c dinh, cirng tinh don di6u tr6n [o, 6]. Chtlng ta cung cri bdt d&ng thuc tich phdn tudng minh rang : r-tng md r6ng : .b b b Biri todn 3 : Cho f, g, ..., hldnt hdm sd li6n tuc tr6n [tz, b] vd.: [ 1qa* [ e@ ctx < 16 -") [ f@)e@)d.x u f : {a, b) ---R*, g : fa, bl -,R+ ,, h '. lo,, bl -R+. Tif bei to6n 5 vd (2) ta cci : ndu Chrlng minh rang : b hlth nt, < f(x) +.nt,(b -o.) < [ S1r1 a*. I f"@) o, . I,P"'(!) dx ... { tr'"1*l rtt > u .. Dl NOn ta cci : lh \ Bii to6n 6 : Cho f ud p ld 2 hdm li6n tuc, / nr, erxt tr.1\ dxl (1or xac dinh vd cirng tinh don di6u tr6n la, bl, th6a l" I ) nr6n : nr, < f(x) ud nr, < p(r) Bii to6n p, ..., hldm hdm s61i6n 4 : Cho f, " tuc tr6n [0, 1] vd: Chrlng minh rang : bh {f [0, 1] *[0, tl,9:10,1] -[0, 1], ..., h:lO,' il 1]. Chrlng minh rang: -[0, a) nL7. I e@ d* < [ f@) 9(x) d.x (t4) I r(*) a* . J r@) d* [ h@) dx > bb 00() D nr, I f@) d.x = f ffrl e@) rlx (ts) (,^ )'' Trrong tu ti bei toSn 5 ta ccj bdi to6n nhu lJ ft*t,t{*l ... h(x) dxl r II t sau l,o ) : Biri to6n 7 : Cho f vd, I ld 2 hdm sd 1i6n Nhtng bdi to6n tr6n d6u cri ngudn g6c tr) bdt dang thric Bunhiac6pxki, bAy gid tt mOt tuc, xac dinh, cung tinh don di6u tr6n [0 ,1]. d&ng thric khdc nhrr Ch6busep ta se tim cd.c Chrtng rninh rang : bdi toan tudng tu. Bdt ding thtlc Ch6busep tll phat bidu : ! p1a,* { p@),t, = [ f@)p(x) dx o6) lor=rr
ffing quy vi khOng dbng quy NGIIYEN CANH TOAN HaL NOi B4n cci nhAn thdy ring to6n hoc tidn l6n (goi Id k6p vi dd le ti sd cria hai ti sd) trrong theo con drrdng m6 r6ng tr) nhrrng tnrdng hop trl tr6n B.E, C,F' thi ta cci cdn drroc cr)ng mOt dic bi6t ra thdnh nhttng trrrdng hqp tdng qurit s6 k d6 hay lai ld nhrlng s6 l, m khac k nhung khi ft : 1 thi cting cd I : m : l. Vdy taphAi : ti cdng thric h:ong giric trong tam gi6c vu6ng tinh h theo c6c ti s6 x6c dinh c6c di6m D, E, ddn c1.c c6ng thrlc lrrgng gi6c trong tam giric thrrdng, tt sd hrlu ti ddn sd thrtc, tr) hinh hoc F, DB p, EE Fd r, r6i ho6.n vi vbng Oclit ddn hinh hoc phi Oclit (md hinh hoc Oclit rc: EA: e, Vg: chi la mOt trudng hop dac bi6t). Va ban ccj chri quanh c6c cum chit ABC, DEF, LMN, oqr dd y ring giita cdi "chung" vd cdi "ri6ng" vrla cci cd cdc ti sd kep , :#,# va mtrt trrii ngudc nhau (tam giSc vuOng kh5.c tam giSc thudng) vrla cci mat thdng nhdt vdi rc Nre * : LT: 17r. Tr) A, C, vd. D, taha czlc dudng nhau (tam giric vu6ng ctng ld m6t tam gi5c, chi dac bi6t d ch6 ccj m6t gcic beng 90o). Nhrrng vu6ng g6cAA', CC'vit.DD' xu6ng dudng thing srJ chri y nhu vdy cci thd ddn ban ddn ph6t BE. Ta cd : minh ra c6i "m6i" ddy. Trong bdi ndy, ta hdy trA M TECE EACB thrl m6 r6ng tinh chdt "ddng quy" cria ba dttdng rD: DT=re DT=rc Dts p-l thing ndo dci xudt ph:it tt ba dinh cria m6t leD+DB 1 _a1\ _____/ 1 tam gi6c. Md r6ng nhu thd ndo nhi ? Cd 16 c1.c - q DB q\ p'') - pq ban da qu6 quen vdi c6ch nghi rang "d6ng quy" Bang cdch tinh tuong tu, ta cd vd "khdng d6ng quy" Ii trrii ngrJQc nhau, n6n NTA cim thdy kh
(Ba., nio chrla Dinh 1i I lA trli.Jng hqp dac bi6t c[ra dinh li 2 khi k : 1. bidt dinh li 1 thi cil Ndu k: -l thi k' : -1, n€n ta cd th€m dinh lt. chdp nhAn vi teit hc,n Dinh ti 3. Trong m\t tam gidc, ndu ba dadng thdng lhn nfia. tu chitng minh luctt xudt phrit tit ba dlnh, md cd cdc chan Gtc ld giao didm lay ). vdi dudng thdng chaa canh ddi di€n) thdng hdng thi cdc ddi D6 m6 r6ng clinh phAn cila ching cilng c6 cric chAn thdng hdrtg. li ru)ir. ta hay x6t R5 rlng lA ta clA c6 nht'Ing ph6t minh nho v6i v6n ki6n drlclng c16i phAn AD' th(rc phd thong. C6 dltdc phet ninh niy li nhtt ta de bi6t ciaAD (l't.2). Goi 11A nhin hai kh6i ni6m "r16ng quy" vA "kh6ng ildng quv" v6n clr:gc chAn rlildng phAn gi6c, coi nhil mall thllSn (tr5i nguQc) v6i nhau theo n-rOt c6ch mtli ta c6 circ g6c (dinh girhp th6ng nhdt chr-'rng lai vtii nhau (ci hai ddu trA thenh "k - hu6ng) a. /j nhrt d h.2 cit nhau"). C6ch sr-ry nghi ki€u nhtl vAy nim trong c6r mi v\ AI li phAn gi6c chung cira c6c gtic BAC vi; DAD' . Ta co : ngllc,i ta goi la "tu duv bi€n chitng". Nhtl vQy, c6ng ddu c! dAii I thu6c vd "tu rluy bi6n chitng". N6 giirp chrhng ta nghi vdn dng DR ,AB.ADsin(-o\ OB c6 m6t ctinh li md rQng clinh li 1, nt1 drJdng cho chfrng ta tim P: DV : - AC "iro tiri. D6 tin tdi, chrhng ta di clirng suv cli6n l6gic vi cu6i cirng "irf tim clttclc clinh li 2. VQy. c] clAy. tt1 duv I6gic d6ngvai tro "thtla lAC.ADsir{3 hdnh" d6 giAi quydt nghi vdn khoa hoc n6r tren. Trong khi 1 giii quydt. ta dA dilng m6t thir thuat ta thay thd vi€c so s6nh D-B fB 'eD'sln(-P) ,an ,;np MA NZ r D-T I AC sirw hai ti s6 ,U vd 6, xern chrlnE c6 bing nhau hav kh6ng bing !tL''ttt/ -AD'ACsirrt viQcnghi6ncttutis6Kcirachirng.Ki6utlrduvdirngthtthuat 21\r' l nny c6 th6 goi le "111 duy kr thudt" vi n6 gi6ng v6i tll Lluy cira ( AB\ y Ay : pp. : l-l . Flo:in vi vong quanh. ta co : ngudi ch€! tao ra c6ng to dd chi vi6c nhin vio s6 ghi tr6n c6ng \"' ) ro lil bi€it lLldng di6n ticlr thu. Nhtt vAy. trong bai bio niv. tr,t r duy sSng tao to6n hoc li rn6t srl tdng hQp hAi hoa cua ba loai / BC \ qq' : I tll Lllry : bi€n chitng (girlp ph6t hi€n vdn rt6), l6gic (giirp gini --l l'")- . dd), ki thlrat (lim tt6 cling vi€c giii quy6r vdn dd). qu1,6t vdn 1('ll' IAB.BC.CA\: Morrtipkhic.tasethavthcrrrvai trocr'rzrcActll (hl\thllit ,,: .dorrotVtqr)lp'qr't:l^r.r,,r1nl:1. ro,rn.rrrduyquinli.trrduvkinh16trongsanglaoto6nhoc. L-Bl \ / Ve cA tu duy hinh tur,ng cirng rdt czin. Ndu chi vui c16'u ldnr havk/r':'I n&r dit k: - p'Q'r. nhilng bii to5n khir thi chda drh d6 rdn luyen tLI cluy s:4ng tao torn hoc vi iAm xong mQt bii tohn kh6 16r. n€ir kh6ng c6 ai Ta c6 clinh li : ithriy hay s6ch) ra th6m cho chfing ta nhllng bai to./rn kh6c thi Dinh li 2. Trong m1t tam g/dc, ndu ba dadng thiiltg lhn t, ,itrit ngfiigp,,. Trong bAi niy, ta 1LJ ra ld1, bdi toirn x6t ,r6i luot xudt phdt ti ba dinh mit k - cdt nhau thi cdc d'di phdn q.an hE gi.r, k ua t,vir sau khi lim xong bii ro6n do. ta phar nrinh ra clinh li 3. N6i "ph6t minh" cirng kh6ng cci gi quir d5ng ctia chtirg,se k' - cdt nhau voi k' : !. vi ban cflng kh6 tim cl dAu ra dinh li c1o. Gidi dip bdi LAM THE NAO Gii sd frlnfr vu6ng tron tLtong lA ABCD. sr-t cLta Gqi M, N. P. Q ldn lLlgt la cAc trung didm cta AB, BC, CD, DA. Ta thAy cr-la s6 hinh vu6ng thu hep MNPQ th6a mAn y6u cau cda d6 ra (xem hinh ve) Svrpo :7 Srsco ;cao MP:1m, 1 ngang NQ = 1m. Thay chfr bing sd Hdy.thay cAc-chr.t kfAc nhau bing cAc chLt sd kh6c nhau thich hQp d6 con tinh chia sau dAy ld dfng: tuAr:HOi:T[ BiNH I,HiJONG TRAN VIE.f HUNG. 56C T}{AN(i ISSN,: 0866 - 8035: S6p ch'tr tqi Tiung tAllr Vi tinh vi .Ci6i,:l ,1ssod Chi so 12884 , : In t+i Xuon$ Chd ban:,:in Nhu::iudt,tr6n,:,Gjdo :,d,ud, , ,,,SJ01,r:ttglrir] Ma sd : 8BT14NI5 tn xon$..,+e i.:.l:uu chi$u::::th*fl,g,2li99s i, irEr*r trr:im rlhng