Tài liệu Thtt so 342 thang 12 nam 2005

e0 otno DUC vA DAo rAo ruHn xuAr aAru clRo ouc ilt*i*****"" t, ttttY , ' Tofiruffi &ffi " T$p thd c6n b$ gifio viGn trt{$ng THCS L& Qufil$n Qu&n CSer ffiiffy, Hh N$i ; if$$;-****t TRU#rus Tffic$ [-E *il,f mom, *ar&m cAur cunv, r*e ru*l lAp niirn 199-5 r'r'ri ten goi 'l'ruirng ChuyCn pho tht)ng TIICS hu1,0n'l'iL l-iem, IIt\ Noi.'1'hi'rng 9 nrlm 1997 tru'ing clLroc ddi tOn thi\nh 'li'u'ting 11ICS Le Qu! I)6n, clLrAn CiiLr Ciial'. l)ia di0rr: l)hrj'[i) IiiCu. Nghia'lAn, Ciiu CiiA1.'I-hring 8 nrrnr l{){)J tlttr)ng sirp nhap xii trtLr)ng '11 ICIS Nghia t)o vir chLryc'n ve cluirrrg Ngrrr Cn Virn I ILtr'cn, phLltrne Nghia .'l'hirnh (iiay. o l()0ri girio vi0n diit chLrin. (r-5fl giiio i'i0n dat tlett cltLrin. r'l rrlr'rng dilng driLr cltran CiiLr (iiay vC thr\nh ticll Iroc sinh gi6i crip clLran. thi\nh pho. Li clcrn vi di diiLr tlrng tlttAn vc ting dLtng cirttg nghe' l)o. clLran Criiu tin hoc vi)o cong tirc cpriin li vir da1' hoc. hoc sinh dorit IItry chLtttrtg Virng Oll,nrpiu . Nrlrr 200i1, trLtong cti I l-triin Sirtslrprrrc. . Nhiiu hoc sinh ctl cira trrLr)ng dat giiii Ikrc sinh gioi QuOc gia, c6 nhiitrg cm dirt girii Qucic t0 rion'l'txin.'l'in txrng il6 cri em N,qtt-t',',t 'l'r'ortg ('rlrilr clout IIuv chttong Vang 'lixin Qtrcic te nairn 2003 tai Hi6-utrr-i6ng Nguydn Thi Thei . Nhit Iirn J-i0n tuc lir trrrirnc ticn ti,-1rr rtriit slrc cip rhirnh phti, Nam hoc l0()-l - l0u+ ciLLoc-nhin hlng khcn cua Ilo (iriro rlirc ri l)io tao. Nirnr hoc 20()4 - 100-i drLtrc nhan bilng khcn cirrr Chinh phLr. NhiCLt ltiim liein trLtilng dat clanh hi0u trtLilng tiOn ticn ruat sic clip thl\nh phci r'0 thil rluc thc thrro, LiOn clOi miinh cap qLrin, thirnh pho. ClCrng cloir.r co sir xrrAt silc, Chi hur r'lmg rnanh. Sach mdi Tidng Anh ddnh cho Tidu hoc: LET'S LEARN ENGLISH Hdy hoc tidng Anh cirng v6i c5c bqn Nom vitMai trong sSch Let's Learn English! Ddy ld bQ s6ch tidng Anh dLioc bi€n soan theo chrtong trinh mdi drto. c ban hdnh cria 86 Gi6o duc va Ddo tao. BO s5ch duo. c thidt kdtheo phong c5ch hiOn dai, kdt hop voi tranh Anh minh hoa ng6 nghinhr, tinh td, rdt hdp dAn v€r c6c em hoc sinh tidu hoc. Nhd xudt bAn GiSo duc hEp ticvot Nhd xudt bAn SNP Panpac Singapore xudt bAn bQ s5ch ndy. C5c S& Gi6o duc - Ddo tao, c5c Cdng ty SSch - Thidt bi c6 thd ddt mua tai: Hd NQi: Phdng phSt hdnh SGK, 1S7B-GiAng Vo DT: (04) 8562071 Fax: (04) 8562493 Dd N&ng: Phdng ph5t hdnh SGK, 15-Nguy@n Chf Thanh DT:(0511) 894504 Fax: (0517) 827368 TP Hd Chi Minh: Phong phdt hanh SGK, 231-Nguy€n Van CU, Q5 DT: (08) 8358423 Fax: (08) 8390727 BAtt NHd l'inr ooc so s,ts (1.2006) nn cd ni rnt ('udc' |'HI vul T'ol.lrl 2006 rNc DUNc c0n MQr I BAI TOAN DAI SO NHdDINH l-i uiiru HOANG NGOC DAN D)n, Hd NOi) (GV THCS L0 QuY TRf\G HQC C(} s(} li Vidte: N€'u phrong trinh bac lrui axl + bx + c = 0 (a * 0) c6 lni nghi1nt Ta dd biet dinh bc x 1, -Yt tlri J7 * -rr = -; , -r txt = a' Nhd dinh li nhy, ta dd giii duo. c rdl nhidu bhi to6n dai'sd. Trong bhi viet nhy, chring toi xin bat ddu tir mdt bii to6n sau ddY. Blri to6n mo dAu. Cho Phrong trinlt o.r2 + bx * c = 0 (a *0) c6 lrui nglri€m.Yt, x2' 5,,= .ri'+xl frr e N. )' Clulrng minh rdng oSn*z*b9,,*, +cS,, =0 (*) -i-s,,. duo. c nhd rlng dung bei todn ffen. Hcil' 1i"1' 'r17 + lni nghi€m x77 ' = 8' Tir d6 tinh duoc Su = I 136' Biri to6n 2. Tim da thirc bdc 7 cd h€ sd rilrurt = " (E fi Ldigi(ii. Dat .r, =(/i.rl +.r2 .\E .*, ,u ngltilnt. ,tac5 =G. Jl .*Z=1. 12-o*+l=0. bli Suy ra o? -7o5 + l4cr3 -7(x=+ 15 105cr 15cr7 - lo5.rs + 2lox3 - - 34 = 1o5x- O. 34. ttgut|tr kh1ng chia h|t cho 5. gitii. a) Tru6c hdt ta chring minh 5,, ph6P quy nap : phuong bang Loi eZ Vdirr=2:52=34e Z. Gii st S1,e V' vi S1*1 e Z (k € N* ), ta cdn chring mrnh Ss2 e Z to6n m& ddu ta cd : ThAt vAy, theo bhi toiln mo ddu ta c5 : Sr*z-651*1 +51=0 li Sr*2 = 6S1n1 - 51' Do 51 vh Sp*1 e Z nan tfc til kdt qut trdn c5 5L*2eZ. V4yS,, Do d6 x1 vI x2 ld hai nghiOm cfia phuong trinh Theo I =11 t5' W:) . 52 rrsrtv€tt vd ' -"7 / 6\' Vdin=l:S1=6e2. cia Loi gitii. Theo bhi to6n md ddu ta c6 5,,*2 t25,,*l - 25" = 0 vdi S, = 2, '.-u. , .S = rJ -"2-[!sJ * r7 =(7/11 .[ ,/t ,7-.,r so' Tir d5 suy ra h€ thric (*). Biri to6n l. Cho x1 r'd x1 ld pltrmug tririr x2 - 2x - 2 = 0. 2' Bii toin l.'Cia slt x1 vit x,ld hai nghi€m ctia phrrong tririlt .r2 - 6x + I = 0. Cltltttg ntitit rdrtgS,, = 11" + '\)tt (n E N- ) /a Du6i dAy chring t6i xin trinh bhy m6t so bii toiin gi6i - 1.^s+14.-3. - Mat kh6c 15x7 = (ri'*' +.rf*r )(.r, +.r2)-.r,-rr(-ri' +.ri) , -l S,, = 0 vdi S1 = cr, 52 = a2 - lO5crs + 210cr3 VAy da thrlc cdn tim li Loi girii.Tac6 ^ = '{ln+2*'Y2n+2 J,,+2 Ail+lO - 0S,,*l Til d6 tinh du-d. c 1 1 Sr = xt' *.Y2' = ct'' hay Ddt = -4s S,,*2 eZ(voirre N*;. b)Tir ket qui 6(65,,-S,,-1)-S,, 65..,, S..," ,l+l -S,,= il+l = : = Suy sd du. ra 35S,, -5S,,-t -S,,-l S,,*2 vd -S,,-1 chia cho 5 c6 cing Ta c6 S,,, -S,,*3, ciing s6 du. S,,n6, -S,,19, ... chia cho 5 c5 Me Sr - 6, Sz = 34, St = 198 ddu kh6ng chia het cho 5 ndn (r e Sn NI') khdng chia hot cho 5. Bii torln 4. Tim so' ngLrydtt l(tn rtha't klfing . rlttLi (1r vLtcrr Lt)t gi(ii. D[t r, = 4 + JB ,.yz= 4 x, .-v, = I , ,rt +.r2 = 8. - JtS . t'o v) r., lh hai nghiOm cfra phuong trinh .r'-8x+ I =0. Dat S,, =.{1" + -rr" 1ll e N*) Theo bii to6n mo ddu ta c5: 5,,, -{_t f{, -0 .r1 Ta tinh dLtrrc 51 - 8. S: = 62. Sr = Sr = 3842, Ss = 30248, 5r, = 239 112, S7 V4y .rl = - .v]. Mi o. r] -(o-fi)' 0 nen S,, Vi o< c N. l0 <(1 - 4.,f:)" -ro" -.\' "-) .-1 sLry b) Chfng minh rang S,,=-Li'+.ri (tt € I\-) IA so nguyOn. c) Tim so du trong phdp chia Bii Sr1n1.5 cho 5. 2. X6t phLrcrng trinh -r3+,,-.t + b.r + L = 0 (c, b e Q). a) ChLing minh rAng o = 5. b = 3 ld cep so h['u ti duy nhdt lim cho phtrong trinh da cho c6 ba nghiem trong d6 cd mor nghiOm lz\ 2+."6. b) Ki hiOu -r1 , rr,.rj li ba nghi0m cira phucrng e N'), . ,l. trinh .rr+7r.r- I =0(r,6rp e V,tipli). ChLing minh ring voi moi rr e N thi S,, =-rl" 1-1r" \rh S,,+l =.r,"t' *.rr"*l li ngul,en i') ngLry€n t6 cirng nhau. ciic so Biri 4. Chfng minh rang trong bidu diOn thap phAn cua so (Z+ 4.",5)"r,6i r c, N'c6 it nhar l chlr sO 9 ngay saLr dau phii1,. Bni 5. Chri'ng minh ring phdn thAp phAn ciia so (-s + J26)" v6i ri e N' U6t cldu bang r chtr so gi6ng nhau. Bni 6. a) Chirng minh rang (: Js l" , (u ,',,6 )" Ltn-l *l ? \-./\ I (vdi rz e N ) li so tu nhiOn. ] nen o Tir dd N' bing phuong j _ 4^,8:--L:.a.1 7i -lJ3 ll + l)-r+ I = 0. -5(rrr2 Chfng minh rAng 5,, e il.. c) T)m so du cLra ph6p chia 5.11i,5 cho -1. Bni 3. Giii su-r1,.r, 1i\ hai n_ehiem cua phucrng ./r s;7 < 1 87-+888 1874887. c'Liu day. a) Chfng minh rang phuong trinh luon c6 hai nghiOm phAn bi0t (.r1 r,) .r,). hay tinh Sl. S. , .Sr VAy so ngr-ry0n lcin nhdt khong vuot qu6 , _.'1 (+ i Vts) tAp .sau trinh trOn. Dat.!,, =,rln +.r.n +.r;n (li 1874887 < 1874888 -.r27 l-1874888. Do d6 r bii Bni 1. Cho phucrng trinh ,rl + Jl5) , +.u6)" co it nhat r chlr so 9 Xin mdi c6c ban hay fng dung- bii tozin mE ddu cle giAi cdc _,1 c5 Khi d6 m)r S,, e N' nen (2 ngay sau dau phdy. b) Tim tat ce cdc gii,'r tri cLia ri dd r1,, th so chinh phuong. Bni 7. Tlm chii so don vi trong bidu dien thAp ra I (l t +Jz)".s,, .(7r +J3)" ' 10"' phan cua so 1 , -\/? (S,, <(7+aJl) S,,-,;; h0 thfrc tluy hdi" LI (ts, .,070,1''' , 1rs , uAlo;t'. bii "LJirg dung cita mot cia tdc gia Nguyen Dirc Tludng, (Cdc ban cci thd xern thOm 'flI't"I'so 320, thing 2-2004). s.tt'tt ttri't "rtrYlll\ {-lrdi)N 'l'}{tt(} (lHliYliu +lf 't{),\\ H(}(l \/A T'tj()l'l'R}, r QUYE,\i t) Ra doi trql .liu siich ,,:.: }ir hoc vir Tuoi tre" Todn ll ,,iir\,..|i{){l :.,"'' ,, : ;i'r ;11' . ' . Lr'i ., :. C6c 1iL rurr ap u 1n ii:r\\,0i,1?l%xif,-"'J;J f,ir'e,,'rt,i,iiiioChlnitr'tti Toiritt lt,tc vri Tuoi ltr tQuyll I ). Tuydn chon niLy r',;!'i;+ ,,1:.,i,,.\ ,, .:, "'. ,'. .,,; 'i,i 'i.,,:' ". :i .i' ".''"rr'"1\'. ' ;Gr.tip se ld hiru ich vdi ciic thAy co gi6o ctay torin,i6c em hoc sinh vi moi ngudi y€u to6n' Sdch khci 19 x 2lcm, cldy 300 tang, gi6 bdn le 34.000 clong / cLr6n. Ban doc c6 th€i mua sdch tai Hi vong siich srich cua NXB Girio dtlc: l87B GiAng V6. TP Ha Noi 231 Nguydn Van Cir. Q.5. TP' H'j Chi N{inh l5 NguYOn Chi Thanh. TP. Dn Ndng nhi Cric cOng ty Sfch & Thiet bi trudng hoc c6c tinh, thhnh. xep thanh ba chuong, mOt Tap chi Toin hoc vir Tudi dd thf .t uvei' de. Chuven -pltilt girii nhai Phtottg 187B Giing Vo, lfrione trinh bi/ m6t luc trr cluy vir kr nang grAi todn. Chtiyen de thti hai lh Todtr lrnc t'ir Diti sdilg, se dem lai cho ban.ciich nhln ,nOi vd toan hoc ciui'nhung ring dung.phong ghu., au ann" ri thiet thuc cira n6. Chuyen de thLl ba Lrclr stitTr,,iir /roc nhu nhfng liit cii girip ban nguoc ddng tt oi elon clen vdi nhLtng su ki6n vi con ngudi xAy nen liu dai todrr hoc hom nav. tri: Hi Noi; DT/FAX :01.5144212 Email : [email protected] Cdc doc giai c5 the g'iiTltrr r.hut'e'tt lidrr (khong n€n giri trong phong bi) vd tda soan theo giti: jSOOOo/.ro,i(cla gom ci cudc bLru dien) hoac li€n h0 truc tiep khi muai'6i so lttong til 20 cu6n tro len de duoc trir phi phrit hirnh. Crim on c6c ban. THTT yEepy frffi{}y*H rt liillErr ?uEH$ S$e! rwx TIIV*H $rffiffi vm* ngp 1$ \ &h i H{-xl :tl{}5--I{X}6 (Thdi giun lim hii : 150 Phrtt) Biri I (3 di6m) c6 nghiCm du-v nhit. s* l,1 Thrrc hiQn phdp tinh 2) cho bi6u thirc voi ^= (Jir .,81fi-G I va tirn gid nj '+Jl ll * *".ir, -v>0vd.r+4. RLrt gon bieLr thtrc .6qS ctra .l : - r de '4 3) Giai phuong trinh vd hC phLrong trinh sar't a)9x2 : -9x+2:0; 12.6 ,, l-f--l I l h){ , + rn-6=O' nghiQm d6Lr drn. Bhi 3 (3 rlitim) Cho M ld di6rn bAt ki tr6n n[ta drro'ng tron t6rn 0 dr.rong kinh AB.- 2R (M \!Ong trirng vo-i l' B)' tron fc .a-. titip tuy6n Ax, By, Mz cia-tt[ra dtto-ngy'y'"vd tai ltrot Ar,1l1^lA1 cit Mz thing d6 Duong F. o,,'ons"ttrarP.",aM cit try tai C vd dtrong thang BM cil A"x t4\ D. Chilng minh: Til gi6c AOMI{ n6i ti6p dLto-c trong rnQt . Bli cua n6 di (l; 2). 2) Tim rr d6 nt2 .i l) Tim c6c gi6 tri cira m dC hhm s5 y ='(2ry + l)x + 3 nghlch bi6n vd d6 thi di6m ; -(htt+l)x+ a) Tim c6c nghi0m c[ta phu'ong trinh tlteo rr' Ul f im cdc gla tri cua rrr de phtrcrng trinlr c6 hai drrdng trdn. b) Tarn gi6c NOP lir tam gi6c vu6ng' .., Ca. Jiern ,v vi P lAn ltLcrr la trtrng diem ctra doan tltang A D vd BC Bdi 2 (3 tlitim) qLra ' a) t 9 14 I----r. [.r Y 3) Cho phtrong trinh trQ Phuong trinh tzx-t'=m y=zJ1 \+r-n,2 4 (1 diiim). Cho tan.r gi6c ABC'yy6lq,t?i,4 AD ldrduong phAn gi6c ctra g6c = c,AC = b vd AD= d. co Chilng minh: .Ell - b* l ' Cho biet AB , (Ccic ban xem ldi giai ct s6 sau) e !^ l , * LU GIAI DE THI TUYEN STNH LOP 10 H0 THpr cHuyrn rnuouc DHKHTN.DHQG HA NQr NAtt[ HqC 2005-2006 pi CAu I. D4t x HQ di MON ToAN tni ildng ffAn THTT sii 341, thting tt ndm 2005) vo NG I +y: ;xy =P. ^s cho c6 dang : It* P=3 1 D_1 1" (s =2 H6trdnc6c6cnghiCrn te: ] . =x =t= 1; ^ l.P:r l'.s=- 4 | ., ^ l t6n tai x, y. I _ - + kh6ns lP=1 Vfiy nghiQm cria hQ <16 cho ld (x, y) : ( I , 1) Cdu II. Phuo'ng trinh (PT) dd cho tuong cluong v6'i IIirth I : I t _x_4./ffi e e (x+3-4Jx+3 <+ I _2J3_2y=g + )+ Q-2x-2J3 (Jx+: - 2)2 +(Jt-x -l)2 [Jr*:-z=o lJ3-2r-t=o <+x: x +l)=g =o l, t III. PT da cho c6 d4ng Ciu ', 1 x' + 171,,' + 2xt + 3(x+y)l : 1740. Ch[ ), ring v6'i s6 x nguy6n, x c6 thti : c6 dang sau: r: 17k + .77 tK e- b. r vdi r e {0, 7,2,3,4,5,6,7, 8} vd 1 Tn'd6 suy ra x' e {17k, 17k+l, l7k+4,17k+9, 17k+8, 17k+l6, 17k+2, 17k+l5, 17ft+13\. Nhf,n th6y rang vi5 ph6i ld 1740. khi chia cho l7 c6 s6 dLr IA 6. Trong khi d6 vC tr6i khi chia cho 17 trong rngi trudng hqp d6u kh6ng c6 s6 du ld 6. Vay PT cl6 > frdd,= ddD dOM > oM> o'M. 4 . IdD = Iffu : IC, ndn IB = IE. Mat Kh6C IB : ID, dO d6 L,BED VU6Ng (1) tai B hay BD I BE Tt' gi6c ADBF nQi ti6p vit IB = 1D, n€n (2) IA: IF suy ra AF ll BD (2) Ti'(1) vd suy ra AF L BE. Cflu V. Di6u kiQn dd cho c6 thiS vi6t lai ld xy2. x2l._ + y. I2=). cho kh6ng c6 nghiQm nguy6n. Cdu IV. l) (Xem h.l). Ta c6 OMIIO'D (OM vd O'D cirng vu6ng g6c v6j cD) Tfi' gi6c ACBE n6i ti6p vd IA zz Bi6u thri'c P c6 d4ng : P= \* z . Dbtl= z *o *ya l, ta thu Clugc bdi to6n sau : Vfly h€ c6 c6c nghiCm (x, Y) "V6'i x. y. t lit circ s6 du'o-ng thoa mdn xY' + Yf + tx'= 3. (0. r): (r. 0): Tim girl tri l6'n nh6t cira bi6u thric 1 'o-- *o * 14 +t4 " Theo BET Cauchy cho b5n s6 duong, ta c6 * f*o Ding thftc xity I suyrax+/> l,a-, .r'+.tl+l>4t.r2 31xa * -ltru 11) + 3 > 4(ry'2 + ytz + tx21: I 4 t'Y++[ tt ,{->J- =-. :).{ -)'\ +V /*Yt*r+-3' Viy I =: Pn.,tn dat x: -2 vd.x = 2 ld hai nghiQm cria PT cld cho. crha yt - : (4x+y)(x3 * - 4xy + Y-1= g. 4*4 + y4 1') = ,y': VII. Do x3 + h€ c6 d4ng : CAu xy(3y' 1 n6n PT thri' hai y' - ry') ) y:1 = he c6 nghidm (0' 1). b)-y=O )x= I =hQc6nghiOm(1,0)' (*) c)3y2-4xy*x2=0 a) x = 0 V6'i x + 0 chia cA hai v6 cira PT (*) cho x2 ta I : o' -^?\+ \x/ \x/ :0= t1:1,fi: !. Ddt L= t tacot? *qt+l x3 nhan trusc * V6'i , 3(L)'z /- = 1 tatim duo.cx :y: 1. x *V6'i /=l x 3 *=+,y:L. 1lB" ll?j ruuvw^ tatimduo.c > | : . ,lr*2J1+zJr;rJ, , d4t duoc I '---=. '12 *y > 1 vd 4xy > 0, n6n r__-_= >z+2+ 2,ll+2+o =P> '14+2J3. V6y Pn,in : .lq+2J1 d4t dugc khi x ^15- + JTi =2 c> x : t2' Thil lai ta thAy 1+2xY l- p< ",lz+zJZ +ztll+2J2 vdy P,u* -4 (loai) t2= 2 <) - ,' * ,2 + 2xY : 0) I /r 12 -2 < x < 2. -+ '2 = r/4-Y'=-. + y2) Ding thL1c xity ra khi x : 0 ho4c Y : 0. 2)Ta co p2 = 2+ 2(x + y) + 2,!*4* + y1* +ry I = (# #) rakhix:Y: Mat kh6c (*+y)2 +l ,4y,' )ro n,o +t4 : Ciu VIII. I ) Tir' (x + y)2 < 2(x2 suyra *+y< 0. +l>4xy2 r'o + y4 (r. r)r li hoac y : 0. P' = EFl ==45o135o. n6n ffitr= Do EFP Theo dinh li 0 AD Ciu tX. 1) LaY di6m P' kh6c Phia v6i di6m P AOi vO'i duo'ng thing AB sao cho LBPP' vu6ng cAn (tai B) (h. 3). Ta c6 LBPC = LBP'A (c.g.c) : llinh 3 9go. Pythagore pA2 = Apa + P'P2 = PC2 + 2P82. 2) Tru6c h6t ta ch{i'ng minh.nh4n x6t sau (h' 4)' Nhfrn xit: Gid sw I ld diijm niim trong hinh chir nhQt ABCD. Qua I kd cac dudng thdng MN, PQ tuong ang song song vdi AB, AD' Goi diAn tich hinh chir nhqt IPBN h 51, diQn tich hinh ld 52. Sz khi vd chi khi chrt nhqt IQDM Ta cd 51 : chdo AC. I thu1c duong chin ciic curtg nho c6 s5 do lir a.2a. 3a.....7cr. Do vay d0 dAi c6c d6y d6 chi nhAn 7 gi6 tri kh6c nhaLr. Liy 6 dinh cua (f0 thi sO aay n6i hai dinh trong 6 dinh d6 ld (6 x 5) : 2: 15. Vi 15 dAy ndy,i6 c6c dQ ddi nhan kh6ng qLr6 7 gi6 tri khAc cLra (F1) I r-: --" l'!.- I l I :l nhau tr6n phAi c6 ba dAy cirng d6 ddi. Trong ba clAy do luon co hai dAy khong chung dAu mut (vi neu lrai clay b6t ki trorrg. ba dAv do d6tr I ThAt vAy : CiA siL l thLrQc dtLd'rrg ch6o lC Vi hinh chiL nhAt chia hinh chir dLLd'ng cheo c[ra nhAt theinh lrai phan co di0n tich bang uhau u6rt .tr -- .S:. Ngtro-c lai, gia su 51 : 52. su) ra 1^r rc ,VC' '- -l,\l --IP l,{A :> LMAIcn r\,V1C - iii-,q = fifc. Do ,NL 1, A thdng hdng n6rt l, 1, C'thing hdng. 1.\'1P - -1'ro- I\t.lO' lai bdi to6n (h. 5). chung dAu m[rt thi ba dAy bang rrltaLr d6 tao thnniim6t tam gi6c ddtr. do do so dirrh ctra (F1) chia h6t cho 3, tr6i voi gia thiet). D6 thay'hai clAv barre nhaLt cita mQt duong trdn kh6ng chirrrg dIu nrLrt thi 4 dAu ntitt cita chirng ld 4 dinh c[ra m6t ,hinh thang (cAn) Til 96. suy ra trong 6 dinh bAt ki cira (H) lu6n co 4 dinh lA c6c dinh cira rr6t hirrh thang. 2) PhAn tich 13860 :2.2.3.3.5.7.11 v\ m.n: 13860 n6n m phii ld Ll6'c s6 c[ra 13860 tirc ld tich cira rr6t s6 nhAn t[r trorrg 7 rrhAn tir tr'3n, cdn n ld tfch cila c6c nhAn t[L con lai. NliLr rr c6 chfla nhAIr tir'2 (hoac 3) thi 1 r chrl'a 2- (hoac 3-t r i nstto-c iai 116 i. khorls. lot phai q.talt. nhArt tr.t'con lai. I : I I Vi i i l) i :lt ,"1 I 't 'i 1 ! D6 thAy trlr gi6c NBMQ lA hinh chil nhAt. Qua P vd p kd cdc drLo'ng thang song song v6'i c5c canh cira hinh vtr6ng. Do P thuQc dLLong cheo AM cia lrinh chir nhit ABMR n6u Snrpx Srln.s' P thLrQc dLLd'ng chdo C// cira hinh chit nh4t NBCH n6n Ss1pl = Sprur. VAy : Sr,rn., = S p1.gt. - Sr2n.s =SgrH Theo nhAn x6t trdn. suy ra Q thLr6c duong . ch6o PD cira hinh chit nhAt SPTD, ttrc PQ di qLra c1i6m D. CAu X.. l) C6c dinh cLia (tI) chia duo'ng tro-tr ngoai ti6p n6 thdnh l4 cttng bing nltau, rn6i 6 tlri: ..^ Do do n6Lr ki hiOtr a1 = 2). crt- 32' o3 = 5' ct-1: 7. as - 11 thi rrr ld tfch cua m6t s6 nhAn t[L trong c6c s6 a1, a2, a3, at, a5 cdll n li tich c6c I cung c6 s6 do lA cr : ..lll I'l Cdc dAy n6i hai dinh vA),. chi co c6c trud't-tg ho-p sau C6 I phAn s6 co tLl s6 li I : (rnALr s6 ln 3 860). 2) Co 5 phAn so c6 til s6 ld I trong 5 nhAn tiL at. a2, ..., a5 , (miLr ld ticlr cira 4 nh6n t[l'con lai). 3) C6 10 phAn s6 co tiL s5 lA tich cua hai nhArr tiL trong sd a1, a2,..., a5. (mdu ld tich c[ra 3 nlrAn t[r'con lai). 4) C6 10 phAn s6 c6 tir s6 ld tich c[ra ba nhAn tiL (rn6u ld tich cira hai nhArr t[L cdn lai)' )) L_o 5 pllall so^ c(5 tir s6 ld tich c[ra 4 nhAn t[r' cdrt mAu ld nhin t[r'con lai. 6) C6 1.phAn s6 c6 tir s6 ld tich c[ra ci 5 nhAn tlr (mAu s6 ld 1). Vay sO phArr sO toi giAn !!n tltoa nrdn nt.n: 13860 ld 1 + 5 + 10 + l0+ 5 + I :32. C6c ph6n s6 trer., du'o-c chia thdnh tirng cdp nghich-dAo c[ta trhau vd kh6c I n6n s6 phAn s6 lon horr I lA 1? "-: -- 16. 2_ PHAM VAN HUNG (GV DHKHTT{ - DHQG Hd N1i) qw{vf;E$ sKI HB&l^g{* wqE"e wd mnel EwoT s6 wm Teeffi &&fi sffi &B&,B o D6 THANH SON (GV Kh6i chto'An Toan - T'in' DHKI{TN - DHQG Hd lV|i) Bdi virit ndy de cAp rn6.t phuo-ng plrap giai rrqt s6 a.alg t"o6l.Ye dai s6 nhLr chLrng rninh b6t ding thLrc. tlm giil tri nho nhat c[,u bieu thfLc chfra cdc cin thirc bAc hai, giAi phuong trinh, "' VAn d6 ndy lnong rnoil Ban doc co tlie aa biEt trong rnO1s6 tdi.liQLr hiQn.hdnh (dtro-idane cai Uiirip). rrhtLng con 1an rnan clitta dtrqc hq rlrong' ilui ,iet nfy .n g6rrg lie thonglroa vAn d€. gitrp ban ds.c- cg.the van dung pltuong pirap-xac dinlitoa d6 vectcr, hoqc tqq dQ di6rr.gini quyEt'.4. dqr'ng toin noi tr€n. TruS'c h6t chirng ta nhic_l4i vdi di6rn co-bdn ve vecto trong h6 toa d6 Descartes vu6ng goc O'rv' Cho ba vecto ri : (r,. )',); r'; : D9 dei c6c vecto' i, i, ii' Ta x': 1 ,/.t:'*yi hay > 22 + -tl Dang thirc xdy ra khi vd chi '\l '11 -r1 + .t . Ket qua do go. i z; tt i cho ta thi6t : qLran h-e gitia cdc bi6u thrirc dai s6 ^t; '.w=:l/. ) n' , hay lfp nghiAnt l4ti ring tong ciic vecto. bing khong (hodc co ttt6t vectct bdng t6ng cdc vectrt ct)n tqi), rii slr dlmg bar dangihi,t'c igoD ,i aa tl,'ti ha L'(l-nh ctto tttttt giac' hoqc BDT v€ do dii dtrine eap khuc di di d,in k0t qua bdi tocin. O6 lirn 16 cho phuo'ng ph6p, ch[rng t6i xin dua ra m6t sd thf d9 minh hoa saLr dAy. Thi dg l. Ch{rng minh ritng vdi mqi sd thttc x co rl x) 1 2r' 2. Vdi a, b ld cdc sd thryc titY 1t. J L htrng nttnn rong Thi dr,r -6b+9+ ! 2 +'l /_n . l1ub 1 13a' J ''-' -4q + 4 > 15.,,41 t3 9 J 1 (l) Ld'i gini. Ta bi6n d6i cAc bi6Lr thirc +(b-3)) ,A-J.4.$ ,fn=]d r4. vectct ttr * v )'rv ta c6 di€u cAn ,- Ta thiil tdp ctrc vecto cd tpa d0 th{ch hop trAn hA fftLc toa dq Descartes Oxy sao cho dA ddi cdc vecto d6 ttrong L?ng biing',[7,J8 ,,.. Sau do l--;- tam. gi6c tr : ch[rng rninh. Dang thfrc xAy ra khi -t 0. trong bdi to6n Phuo'ng phiry. Khi gqp cac,biti rotrn dqi s6 mt) nt6i biAu fitlrc' dtro'i cltitt can bcic hui uA.'[8,... rltrrtc biAtr ctidn t]ruii tlqng t6ng cuct hai hinh phtrong Tir BDT rn6i dang x6t i'6'i d0 ddi cua c6c vecto" trong rndt phlng toa do. l--il dLL6'i *I* 2 )'l khi ,,f; lf 1', biiit ring voi dii:Lt kiQn (*) thi a + v f 1 aung lAn luot ld , Ld'i gi6i. Vrl trai cira BD-L c6 th6 vi6t (r-l)2+1. Tr6n m{t phine toa dO O-r1' chsn n : (O + 1), l)l ;; =(11-r) l) ; i, :(2,2). Khi d6 u=i+i; i+i+rr,=0 (.r:, -y:) thoa rndn diAtr ki6n (hoac i'-=ti-r n ) (*). Khi d6 rr-, =' (-r:. -l't)i x-' -2-t+ 2 > 2J 2' /n . 10ab 13a2 39 .2 I 2A\ (b-a\"t +lb--l 3/ \ u(,,!J ;rta, D;A(0,3); : B (2,0) vd lAp c6c vecto' -AN @, b-3) Xdtc6c ai6^ ; Mfr : (u-,,t-+), EM=(,-r,+) Khi MN= d6lN= b2 + (b -3)2 '""' 9 -4a+4 AB + g1 > BC. suy ra F(x) > L Do <16 gi6 tri nho nhdt cua hiun s6 F(x) li I, tl4t tlugc khi va chi khicosx:0ex: n*kn(ke Z\. 2 Thi dU 4. Giai phaong trinh tnt- tt r' I..l -2x+5-'l / -6x+lo | : J5. Ldi giii. PT da cho c6 thd vi6t lqi dudi dang (r) l=,r Oxy. lil;-rf-+-fr,-:Y.r Tr€n mpt phing tga tlQ lQp c6c vecto (x - 1,2), i: (x - 3, I), suy ra i,=i-n: ii: GiA srh' M1 ld AMr>AH=Ef ' 13 BDT (l) cluqc chfng minh. Ding thuc xiy ra khi M ld giao tli6m cria (A voi lB, cdn l/ ld (o-r)2 +(b-d)2 giao cli6m crta AB v6i 0;kz: ac vd ldp c6c vecto VB cdn 8 b c =fii d - >0khi d6b=kp; Trong tam gi6c OAB (O li g6c tqa d0) md IoB :6oo ta c6 BDT : AB > !@o*ob, trongd6 ecosx. cosx) ; fe':1-sirur, -cosx); eA : @osx + sinx, 0). Ta c6 AB = J7 lcosxl ; BC = l; cdn CA = lsinx + cosrl. Theo BDT tam gi6c d d = k2c. X6t c6c di6m A(a,6) thuOc nfa duo'ng thing y = k1x vA B(c, d) thu6c nria tluong thing ! : kzx (v6i x > 0) tr6n m4t phing t1a dQ Oxy. gini. Tr6n m4t phing tqa d0 Oxy, ta chon cliCrn l(cosx, 0); B(0, cosx) ; C(-sinx, 0) : rdng bu$c cfia bdi ra c6 ll+-. ac I hQ \5'5l' . It t-a 18. {3x+2t-6=o y=, - w(9.9).,0. d6b:9. L trong d6 a, b, c, d ld cac td duorg thoa mdn Loi giii. Di6u ki€n l3x+2y-6=0 Tqa d0 cira N ld nghiCm cria .:(,tj;F.G;7) 2 , .l r--i-i OA: tlaz +bz ;OB=.Jc'+d' ; = J@-c)2 +(b-d)2 Ti'd6 ta c6 5 < v6i moi.x 'a J b) Vx'+4 +- r,6imoix2 '3 -r; /./-) r-^ t -- 6 - 22 . (11D. Chon A (0,2); B(-1, -1); M (x,0); con M1 ld ili6m x6c dinh bbi MB=28fri). rcb2 -18b+9 + >- J29 v6i moi s5 thqc a, b. Nhu cdu hqc vd str dung ti6ng Anh d.ang tdng l€n o hdu h\t cdc ntrdc. Tap chi THTT dd c6 chuyAn mttc Ti6ng Anh qua cdc bdi todn vd ldi tir 1.1998. l'lhieu bqn co nguyQn .u,onq gidi "chttv\n muc nav n|n song dong hon' Bat dau tit tA ,ay, chuy\i mttc mtti rh0 hiin ditng nhr tln soi cua no. Toa soqn sd nhan cdc loi giai bdng 7|ing ,anh md cac ban gti v0 va chon dang nhiins loi siai 6t. Cdc ban sd vira nang cao kidn"thtlc iotin, kidn thttc riing'Anh vira tang thAm nhirng hiAu bi€t vO xd h\i. Mong cac ban nhiet tinh huong ang. (FID: Chqn A(0,3); B(2, -2); M(b,3b), N(4, 0)) 10x2 -24x -l8xy+ +16+ -) -oyz+z 1 -122 + 40 > + v6i mgi x,y, z. (tID.' Clrqn A(-2,6); B(4,0); M(0, z) N(2y,3Y); P(3x, x)). 6'E '5=--4u+ I > -;-vo-l r- 2\l 5q' lnQl 4. 6J' ; fl* 3. a) P D b) increases by 8% and their monthly expenses on food; electricity, water and gas; car maintenance increase by 3%;6%o andgo% respectively, calculate -ff -ii1=s gi5 tri nho nh6t cia cd'c bieu thfi'c : i) their new monthlY savings, ii) their percentage increase or decrease in their monthlY savings. = JTlri"xl+lsinx-cosxl; Q: 'u 4..13*k *e**n , 4a2 -2J1a+3 -TJi +1 (HD.' Chqn l(0, 1); B (.,6,0) ; M(a,.6.o )). c)R= 69 (UNIT 1) others $350. i) Calculate their total rnonthly expenses' ii) Express their monthly savings as a percentage of their ittcome. b) Given that their combined monthly income 2. Giai phuong trinh l.,/i'.4,.r3 so Problem a) Peter and Jane earn a combined monthly income of $3000. They have to meet the following expenses each morrth: food $640; electricity, water and gas $110; telephone $40; hire purthase payments $180; housing loan 5360;^ insurance $75; car maintenance $165; 0x2 5b2 -8ab +5a2 + rfi ctlir vn cnuvEN tutEc c0 cActt ttPC MoI 4 ltJ ciasrtuiu . Vocabulary income hire nH6NG TIN THEM rfr vd euA cu6c rHI GIar ToAN - v;fr Li TRON TAP CHi TOAN HOC & TUOI TRE Trong sO l+t (11.2005) do so xudt khi xi'li tren mIv da bo sot =O ai6, c[ra b4n Vd Van Tudn. X"ir"r duo. c thong b6o B4n Vd Vdn Tudn,l6'P 8A5, THCS \guY6n : Du, Kr6ng Buk, DIk LIk dat GiAi Nh6t m6n ioa" tron[ cuQc thi niry. Chirng t6i dd giri Rang chilng nh$n vd gi6i thu6'ng t6i b4n. Tlrdnh that xin t5i ban Tu6n vi tloc gii' THTT thu nlr6p. (gom cA khodn ti6n kh6c) lu'o-ng vA purchase trh.g6P nori ki6m duoc (ti6n) bing ldm viQc payment loan insurance khoin ti6n Ph6i trA vay, muo-n, sd ti6n vaY sO tiAn n6P bAo hi6rn car maintenonce s6 tiOn b6o du6'ng, bAo savtng .t quin xe ,.^ tret Klem Solution (Cho cdc ban gu'i v6;. VU KIM THUY LSI GIAI CEC I}AI TOAru THI HOC SINH GIOI QTIOC GTA THPT ruAm HoC 2rtO4.2O{,5 nAxc n vU oiNu soa (GV DHSP Hd tv1i) t{GAy lnil'NrrAr. Bdi l. Xdt cdc sd thrc x - 3.,f +l Hdy tim gid ctia bi€u thwc tri x, y thoa mdn ctiiu ki€n; =lr[n'4-, lctn nhat vd gia tri nho nhat p:x,y. .Ldi giii. (Xem lo.i giAi bdi I, bAng A, THTT s6 340, th6ng 1 0-2005). Bii 2..Trong mdt phdng, cho tam giac ABC ngoai ti€p dadng trdn tdm L Goi M, N t,d p tin Itrot ld tdm cia durmg tritn bdng ti€p gr5c A, da.ong trdn bdng fiAp gAc B vd dtidng ird"n bdng ti€p g6c C cua tam gidc d6. Goi Ot, Ozvd Oj u:q"S tTng ld tdm cua cdc dugng trdn (INp), (lPM) vir (lMNl. ChLing minh rdng l) Cdg dydng trdn (INP), (tPM) vd (IMN) c6 () Mrili ) ; han kinh bang nhatt ; 2) Cac dwctng, thdng MO1. NOt vit pOj cdt nhau-tsi m.or diam (xyz) ki hi€u drcng rr'dn di qua ba di|rn X, Y, Z). duo'ng thing NP, PM, MAr. Vi th6, ciic dLrdns tron (11/P), (IPM vd (IMAI c6 b6n kinh bin[ nhau (dpcm). 2) (h 2). Gqi O Id tem du.orrg trdn (MNp). d6, theo chri'ng minh tr6n, ta c6 O1, Oz, 03 [.hi {9: l*g v6i O trro.ng fi'ng qLra c6c duong thang NP, PM, MN. d6 suy ra trung di6m M1 ctta OO1 ciu-tg d6ng thoi ld trung di6rn cLia Np, trung dii)n M2 cia OO2 cfrng ddng thd.i ld trung di6m cita ptv{, trung di6m M3 cfia OO3 cir-rgrl6ng thoi ld trung di6m cta MN. Do d6, ta c6 .Tu 06 =2Mtq -_ NM OA =2M@; = pM PBM Lli ttinh I gini. l) (h t). Vi rltro.ngph6n giiic rrong vd^ duong phdn gi6c ngodi xudt ph6i tu. cirng mQt tlinh c[ra rn6t tam giitc vu6ng g6c v6,i nhai n6n suy ra I ld truc t6m cta tam giic MI,,rp. Do 116 c6c ;lvl' Theo y6u cdu cira bdi to6n, ta cAn ch[mg minh I ! lTr-,LaVn.Sl>0. Kh6ng m6t t6ng qu6t, gi6 sil I li s phAn I .l JJ Hi6n nhi6n T Vi tAp c6 nhi6u n Zrl : lrrl + llrl - lll u I1l (1) Vi 71 u L1c L n6n lly u Zrl < lll' Do d6, til (1) ta o L r-, l/ et S: Ir n lr1 a L't, - al J L1 o V1' th6. tir'(4) ta dugc di6u cAn chri'ng minh' NGAY I'IiT. II,.\I duong (x, y, n) thoa mdn (do )) ?,r, rrt ;lrt+llll ILI - JJ.l'I -1"l"l rr > lzl) (2) L4ic6 lTl o L1r-tVtl: lI1 n I1l+ lZll - l(Ir n L)w V1l hO tht?c *!+!! =3. Ldi gi6i. Vi6t lai hC thrlc cia *l dang nt rrl r--{ (1) nl Ta c6 c6c nhfln xdt sau : NiiLr x < n Yd Y < n thi )lhi)n xdt 2; N6u r < n vd Y > n th\ Nhqn xdt - \-l r 'l nl. ttl. l; -' +'-<1. - 'l 1+nt r,l nl e N * to - Nlnttxel J. Ndtr x > n thi {. nl. {nl. ' r Ch{rng minh; Y\, . N* ve.y > x n6n tir x > suy ra x) n* 1 vdY2 n+ 1. Do d6 : !.n+1>z (2) nl. Yl rr+1>2 nl Cong tu'ng v6 cdc Uat Oing thrl'c (2) duo-c b6t darrg tlr(rc cAn ch[lng rninh' Dr-ra ra x: (3) vi vdo c6c nhAn xdt ndu tr6n, tiL (3), ta (l) sLty n. Khi d6, ti (l) ta c6 (3) dA bdi du6'i GiA sir (x, y, n) ld bQ s6 tr1 nhi6n thoa mdn (1)' Vi hC thilc (i) d6i x[Lng v6'i x, y n6n kh6ng m6t t6ng qu6t, giA sLi'.r- < y. 171 du'o-c , ltrt (4) 1i tir nhAt. Ta c6 lrtl l1 llrl--lvl>o (dolrl>ll1). n! 3 ' < lL']. Vi v{v, tir'(2) vd t) o Lt ,-t Vl' cilng d6ng thdi dat di6m gioi o m6n Vdn: ) V1l V- Bni 4. Hdy tim tiit ca cdc bd ba s6 ngt'ryAn 1, -3 = n I1) w V1c n L)'-'t Do d6 l(Ir (3) ta duo. c mon Vdt l{: Htnt suy ra (71 lTl L1 c- tr/. Ti'd6y, v6'i luu y 1+ .,1 ul t-=2 l=3,haY n! n! 141 l1 ray > n* 7. Khi d6, ta c6 Suy y! >n+1>2. nl vith6 Xdt c6c truo'ng hgp sau ( ..t lL=n (4)<:1nl +l I Ti'd6, ln+l=2 t ta duo.'c x : ay:2 0. Khi d6, ddy (x,,) lh ddy hing : xn: 0 Yn 2 1. Vi viy, (xr) ld ddy h6i tu vdn:1. ua 1. y, n) ld b0 s6 tu nhi6n th6a thi phAi c6 : (x, y, n): (1,2, 1) ho4c n): (x, y, l',lgu'o. (2, 1, 1). c lqi, bing c6ch ki6m tra truc ti6p, (l). d6 thdy ca hai b6 so vu'a ndu ddu thoa m6n .. V0y, hai b0 s5 n6u trdn ld t6t ce c6c bQ s6 cAn tirn theo yOu cAtr cira de bdi. Bii bAn 5. Hdy tim tdt ca cdc hdm tii .f *dc dirh fip s6 thLrc R , lciy gia tr! trong R - flx) + fu) - xy Bni 6. Cho ddy sii thtrc (",), n : 1, 2, 3, ... rdc dinh boi : xt : a vd xr-1 : 3*3r-7x2n+5x, vdi n:1,2,3, 3 l vd littt x, =-. h6i tu n-->+a Ldi gif,i. Xdt hdm sO 1r; = 3r3 - 7r2 + 5*. Khi d6 c6 thii vitit nC thirc x6c dinh ddy (x,) dudi d4ng xn+t : /(x; v6'i mgi n : ,2, 3, ... 1 .1 f '(x):9x' - / J\ .tre 0:- i.Yn>1. j,/ '' Ta c6: l.r,,r -ll=l3xl -7xl +5-r, -11 : (r, -t)' .13x, -11 Y, > 1 (2) /l\ Vi x, e l0;;1, Yn , 1 n6n 13x,,-11 < I \ Vn> 3/ til'(2) suy ra 1. Do d6, lr,*r -11.(rn-t)2 Vn>t r, d6, bing quy n4p theo d5 deng chring minh dugc rdng lrn-11.@-1)"-', Vn> I (3) /A\ Viaelor-lnenla-1,<1. \. 3i .rr. l)' I Do tl6 lirrr (a - Vi thO, ti' (3) suy ra ddy (x,) l4x + 5. TiI d6, ta c6 cta hdmflx): bdng bi6n thi6n sau rni do,tn'(l) J Ti Chang minh rdng ddy sii (x) cd gidi hqn hiru hqn khi n -) +@. Hdy tim gi6i hqn d6. J oTrrdnlhqp3 o+d) ,/ ^t Ti' d6, do l(0) : suy ra: 12 4l hu] x.,'=-. neu n)+@ J V27s \ 0, \ 1 (4\ /' linr x, =l n-->+6 4 r[rJ=] "U *.1 /.+J )" i n6u 4 a= J 1) a e [o, lal \ J/ r Chri i,. Cd fia giai biti toan biing cach khao sdt t[nh don di€u vd bi chdn cua ddy s6 (x,). pu0 uiuit vu0ue illlto uit itii'tit t'tlto !, UAIIG FIi.IAU Mdt cira sO hinh vu6ng canh ld a = l00cm bi vd kinlr cAn duo. c ctrin giO t4m thoi. Nguoi ta ding ba hinh nlro bing nhau' s6p x6p chom len nhau dd che kin cu'a s6. Ddnh cho bgn ctgc 1) NCu ba hinh nho ddu li hinh vu6ng thi canh c,la cliirng it nhAt tlreo b4n lA, bao nhiOu ltintr ctrintr xac-dtin xentimet) vd sap xdp nhu the ndo ? 2) N6u ba hinh nho ddu ld hinh trdn thi duong kinl crha chirng it,nhht theo bpn ld bao. nhi0u (tinh chinh x6c d6n xentim6t) vd sdp xOp nhu thd nAo ? Girii elSp: PHAN CHIA DA GIAC ]'HA|{H cAc uixu Tli',DoN(; DANG (Di diing trong THTT sA SSg thdng 9'2005) diju. Ta th61 tam gi6c cira tam gi6c thuo'ng biQt cl[c rl6u ld truo'ng ho.'p hqp dac biQt trud'ng ld (xem h.1), lrinh vu6ng phdn chia n6n (xern lr.2), cira hinh binh hdnh voi hinh dang duo.'c thdnli nhidu hinh nh6 ddng l) X6t c6c hinh da gi6c .r: b = ct-l -t ban dAu. J l Xdt da gi6c dAu n dinl, md n > 4 thi goc u d dinh cira n6 ph6i thoa rnSn 90n < cr < 180" ndn fta > 180o v6'i sd nguydn k> 2, do tlo dinh c[ra da gi6c nho dugc chia ra kh6ng th0 ndm t4i {linh hoi-c tr6n canh c(ra da gi6c ban dAu, nghia ld kh6ng th6 phdn cliia duo-c. 2) Du6'i dAy trinh bdy mQt t6 d4ng da gi6c thdnh k (k>2) 1tOi, tOml co th6 phAn chia duo-c ia ei6c nho bing nhau vd tl6ng dang v6'i da gi5c Uan-Aiu (ke cAlintr dOi xf'ng v6'i n6 qua m6t truc). a) Hinh tam gi6c, k = m2 (h. 1 i'ng voi m: 3)' b) Hinh binh hinh, k: mz (h'2 irng vdi m=3). b b Hinh 3 a3a Llinh 4 d) Hinh chfr' l, e) Hinh chir V, k= k: 4n (h. 4 v6i n= l). v6i n: l)' 4n (h. 5 b 2a Hinh 6a Ilinh l Hirth 2 c) Hinh binh hdnh c6 hai c4nh k€ ld ma vit 6 = qJm ,vdi k- m> | (h. 3 i'ng voi m:3). Hinh 5 g) Hinh thang c6n, k = m' (h.6a vd h. 6b tmg voi m:2 vd m = 3). l3 T chia Ph6p 27 vd du ld 4 127 7)' cho thuong ld 14 vd du ld l1 (t+ > 13)' NOu ta titip theo. i7 thi thuong chac chin sE nho hon 1 7' (N6u phdp chia k'ii6ng c6 du thi thuong do ld mot Lr6'c s6 cira l93. re ri no nho ho-n 17 n6n no s6 c6 it nh6t mQi thira s6 nguydn t6 nh6 ho-n 17)' Khi do thiLa sii ndy cirng ld mQt u6c s6 c[ra 193' chia 193 ino =O nguyCn tO Nhung di6u ndy dd kh6ng xAy ra Va.v phdp chia i9: cho 17 phdi c6 du, tiLc ld 193 kh6ng t f .t.,iu net cho 17. D6rr dAy c6 th6.du'ng laivi 193 kh6ng thtl chia htit cho nhiLng so ngtry6rr t6 l6'n tro'n t-Z trjrc td ta chi cAn chia 193 cho nhirng s6 nguy6n tO nho hon hay bang.J93 ' Ta co tlt6 f.3t1ua" ring 193 ln nguyen t6 vi no kh6ngth6 phAn tich lhdnh c6c thia s6 rrguy6rr to ' Nguo'i ta kh6ng nhfn thAy m6t su tuAn hodn nao"trong day so ngLry6n t6. Trim l6n tr6n c6c ;;",,;;;.1-i6 nio kh6ng? Fermat da tre ld'i khS rhanh ch6ng rang cua sO d6"khOng phAi ld ngLry0n to vi no lii tich to)' ngttyen so (la nhu'ng I 12503 r'6i 898423 Mersenne dd nghl r[ng c6c s6 co d4ng M,, = 2P -1 (trong d6 P td nguY€n tO;, ta tO ,,juyen t6. Nhung nam 1903, nhd to6n hgc Mi p.X- Cote (1861-1921) chilng t6 rang s6 Mu; :261 - 1 kh6ng phAi ld ngLrydn t6' Pierre de Fermat (1601-1665) thi nghi rdng c6c s6 dqngFn= 22'+l li nguy6n t6' Nhung ndm 1 '732, Fulet (l1O'7-1783) chirng to rAng ^5 Fs = 22' +1 kh6ng phni ld nguy€n t6' vi n6 chia hiit cho 641 . Vi6c tim c6c s6 ngtryerr to lon trong mQt tho'i gian ddi chi lh rn6t iro choi cua nhi6u nha to6n hqc. C6c nhd to6rr hoc Euler, Goldbach, Waring' Wilson, Leibniz, Vinogradov v r" da nghi6rr c[ru v6 s6 ngLry6n to. Nhtrn.g ket qua cia sp' ' nghi0n ciru d6 citng chtta nhi€u' . , t6 nguytu Ngdy nay vi6c nghi6n ciru v6 s6. t6 O,rq'; [i.h il.,i.l., boi-t,-t ki€n ld c6c s6 nguy6n th6ng c6 ich trong vi€c rnd h6a vd gi6i rnd c5c diQp th6ng tin. Xa* lgZO, t6 nguyOn t6 tOn nhAt do nhd to6n hoc Ph6p Edouard Lucas (1842-189i) thi6t lpp ld: ,"', 1'7.4.1643, thAy tu Marin Marseune (1588-1648), ngud'i ham thich to6n hgc vd bgn Ngdy Fermat h6i Fermat: So 1008955981 69 co ph6i ln m6t s6 nguy€n t6 .ttu Pierre d; tlt 183 460469231 731 68730i715884 10572? N6 c6 39 chir s6. Sau d6, nhb' m6y tinh diQn t[r, nguoi ta dI tirn duo-c c6c s6 nguY6n t6 sau dAY 232t7 - I c6 687 ch['s6 (ndm 1957) : ?ll2l3 _ I c6 3376 chir sO lnam iO0:; - 1 c6 6533 chir s6 (ndm 1978) 1441e7 - I co 3395 chLr sO lnam t OZO) 286243 - 1 c6 25g62chirs6(nam 1983) 22r60et - I c6 65050 chfr's6 (narn 1985) 285e433 - 1 c6 258'716 chfi's6 (nim 1994) )2t'701 1 c6 3-tg632 chfr,s5 (narn 1996) zt3e826e - 1 c6 420921 chff's6 (ndm 1997)' kh6ng SO cuOi ndy khai tri6n ra vd dSnh m6y 2125'778'1_ 1 aoo',[ irhu c6 m6t su huYAn bi do. :170 d6 khoang c6ch sc c6 chidu ddi 947 m6t' gan c6 th6 xem th6m c6c bii vA cac nin,ooi x6c dinh trcn rHrr ,{ii"-i;'ti :l s6 s6 s (t 2.2003) vd 333 (3.2oos). 15 Dcr" ki nay Bdi T71342. Cho clud'ng tron (O), hai d6y cung CA, CB khong di qua 6m O vd BA + BC. Duong thdng qua tli6m I vuong g6c v6i duong thdng OB cat duo'ng thang CB rai di6m N. Cqi M ld trung diem.crlia lN. Dudng thing BM cit dtro'ng tron (O) ldn nfr'a tai D. Goi OE ld cluong kirrh cua du'or1g tron di qua c6c diiim B, D, O. ChLlng minh rlng ba di6m A, C, E thang hdng. cAc t op rHCS eur vAN cHr (Gl'THCS Ltro'ng Thd l.'inh.Ou1, .\ho'n, Bdi Tll342 (L6p 6). Tim tfit ca c6c s6 tu nhi6n rz sao cho A:2005n + ,200s + 2005n chia hrSt cho 3. PHAM DU,C THANG Hirt, Binh Giang,Hai Daong) (GL/ THCS l'fi Bdi T21342 (L6p 7). Cho tam giAc ABC voi trEa=I.a : 36o. Tr6n tia phdn giric cua g6c ABC l6y di6m 1/ sao cho ECN : t 2o. Hay so s6nh cl6 ddi cria CN vd CA. IRAN vAN HINH (G't/ THCS Nom Giang, Nam Trtc, Nam Dinh) Bii T31342. Tim moi nghiCm nguy6n cia Binh Dinh) cAc r-6p rr{pr Bdi T81342. Cho tgp ho. p M gitn 2005 s6 duong: a7, a2, ..., a2A05.Xet t6t cd c6c t4p hqp con I; kh6c r5ng c'ia M, goi s; ld t6ng c6c s6 thu6c mdt tdp hqp con I; n6i tr6n. Chf'ng minh ring c6 th6 chia tap ho. p tit cd circ s6 ,; duo. thdnh. l4p nhu vdy thdnh 2005 tap hgp con kh6c r6ng kh6ng giao nhau sao cho ti s6 cLia hai s6 bdt ki thu6c cirng m6t t{p - 5x'- 4y'- 5:0. CU IJUY TOAN (SI'C6ng ngh€ I'dt li€u 02, K50, DHBK Hit \Oi) BdiT4l342. Tim gi6 tri p: a2oo5 + cLia bi6u thirc 62oos + ,2oos trong do a, b, c ld c6c s6 thuc kh6c 0 th6a mdn ' ' c6c iidu ki6n : vrlra duo-c (GI' khoa Toan DH I'inh) BdiT9l342. Day s6 (xn) @ : 1,2,...) duoc x6c dinh nhu'sau : x1 : I, vA x,r(xn +l)(r,, +2)(;r, +3)+l n:1,2... nl Dqtyn: )-= vo-i i=1 [l*l*l t" lr3 +b3 +c3 =2e. t NCUVEN rtgN rteN (GL'THPT Gia L'idn B, Ninh Binh) B^i TSl342. Chirng minh b6t dEng thri,c sau v6'ix ld s6 thuc kh6ng Am : 2J1 _++Jx3.2o trong d6 a, b, c ld c6c s6 duong vd o ld sti hiru ti l6n hon 1. Ding tht'c x6y ra khi neo ? NGUYEN TAT THU (GV THPT Le Quy D6n, BiAn Hda, D6ng Nai) Bni T11/342. Goi r vd R lAn lu'ot ld brin kinh .:, vd ngoai ti6p c[ra tam gi6c ABC. Chfing minh ring : du'orrg tron noi ti€p cos,4.cosB.cosC < J 2 2R"" Ding thfc xity rakhi ndo ? NGUYEN TRQNG QUAN (5V44C1, DH Thuy Loi, Hd NQi) Bdi T121342. Trong kh6ng gian cho mQt mit cAu (,$ vd m$t tlud'ng thing A khdng cdt.('$' di6m M trdn I dp'nI ba ti0p diQn b6t ki Oru 'rO, v6i mat cAu (S) vA goi A. ^8, C lir c6c ti6p di6m' Chrirrg minh'ring mAt ptling (ABC) luon tli qua mQt duo'ng thing c6 clinh khi M di d6ng tr6n A' DANC THANH HAI (GL' Hqc ti€n Phdng kh6ng Kh6ng qudn, Hd Tdy) cAc PB vAr Li I r{ Bdi Lll342. I{ai ld xo nhe Lr, Lz gi6ng hQt nhau c6 cirng d6 cirng.k:20N/m vd d0 ddi tU nhi6n /o : 401m. Ta b6 tri mot he nhrr hinh v6: ca" fO xo ludn lu6n thing dfi'ng, r'r^4t llq lpi gifr'a hai lo xo c6 kh6i luo.'ng m: l00g' Hai dAu Ion lqi cria hai lo xo g6n cO dinh viro I vA B v6'i AB=l.Li,yg=lOm/s-. Tr)r vi tri cArr bing.tai thd'i di6m : 0 nguoi ta truYen cho,u^dt y t * A mQt vAn t6c ban dAu vs: 40cm/s theo phucrng thang dung. a) 'Chring to vat dao dQng didu hoa. Vi6t phuong trinh dao PR0BLEI'|S FOR LOWER SECONDARY SCHOOLS Tll342 (for 6th grade) Find all whole ntlmbers n such that the sum A :2005n + n2oo5 + 2005n is divisible by 3. T21342 v,t) b) KhoAng circh AB = / ctn thoa mdn di6u kien gi o m6i trub'ng hgp sau d6 trong qu6 trinh vdt dao dQng thi : + Hai lo xo lu6n lu6n gi6n; + Hai ld xo lu6n ludn ndn; + Lo xo 11 lu6n lu6n giSn con lo xo 12 lu6n (for 7th grade) Let ABC be a triangle witl'r FEe =trdB:zeo ' On the ray bisecting the angle ABC take the point N so that EeN = 72o. Compare the rneasures of C// and CA. T31342. Find all integral solutions of the equatiott 1 r ) (x- + y-+ l)- - 5xz - 4Y2- 5 = 0' T41342. Find tlie value of the expression p:r2oo5+b)oos+c2oo5 where a, b, c are real numbers, distinct from O, satisfu ing the fol lowing conditions [l t.l_ );-i-it" dQng d6. ( ltl THls lssuE lo3 I a+b+c *t3 +c3=2e' T5t342. Pr&e the following inequality for nonnegative real numbers x : cE +L+ J'r'+ Jx ( J;+e I ' When does equalitY occur ? T6t342. Let ABC be a triangle with AB = AC, lu6n n6n. NGUYEN MINH TUAN (GV THPT YAn Thdnh 2, ren rhanh, Nsh€ An) LZt342. MQt ampe k€ mic ntii tiiip voi m6t v6n t 0' Thdnh that xin 16i c6c tdc giit vd b4n dgc' EAe : 80o. Take the Point M inside the triangle so that fue :20o, frdA 30o' Find : the measure of frEe . T71342. Let be given a circle (O) with center O, two chords Ci, CA not passing through O, l, perpendicular BA + BC.The line passing through N' Let M be at CB line the cuts OB, line to the the circle cuts BM line The AN. of midpoint the (O) at B and D. Let E be the point such that OE is a diameter of the circle passing through B' D' O. Prove that the points A, C, E are collinear' FOR UPPER SECONDT\RY SCHOOLS TBt342. Let M be a set consisting of 2005 positive numbers at, a2, ..., a2oo5'Consider all non empty subsets Ti of M and let s; be the sum of the numbers belongin gto T1' Prove that the (Xem ti€P trang 27) t7 Do d5 p li mot s616. Nhung khi dd 2O04pa lh so chia cho 16 du 4 cdn ,r,4 +,r4 +ka ld so chia cho 16 chi c5 thd du 0, 1, 2,3 n€n (2) khong th6a mdn. Vay khong c5 cdc phAn s6 duong ,r, 1,, z th6a Gi&i bal [ti truse m6n bdi to6n. Nhin x6t. l) Vdi cdch giai tr€n, rrong d0 bii c6 thd thay 2004 bang m6t s6 tu nhien bat ki mi khi chia cho l6 cd s6 du ldn hon 3. 'fulSlt (Ltip !lii .ti+ }ir + ;r, It - Xem thêm -