1, sO crAo DUC vA DAo TAo rAv Nrr\H Ky rHr cHeN Hec srNH Gror Lo'p 12 THpr voNG NAtu Hoc zot3 -zot4 rixu I\giy thi: 26 thftng nf,m 2AI3 MOn thi: TOAN BtrOi ttri thir hai TIrd'i gian: 180 phut (khong ke tho,i gian giao dA) or csixn rsrlc @a thi gom co 0l trang, thi sinh khong phai clt6p di vdo giay thi) Bai (4 diem) Giai phrrong trinh: (Z*- 3) +2x+2 = x2 -2x+ Bi i (4 diLrn) Giai Bni hQ phucmg trinh: I*' - 6x + y + 1 lv'-v-x-3 g arcm1 * t I vol rnoi n nguyOn clucrng =1[r, z\ un) Chimg minh ring v6i moi sO nguyOn ducrng n Z thi ,n , ,6 Tim limu,, Cho ddy s6 (r,.,) xac dinh boi: ur=2 vi un*r Bei & aiam) Tim t6t cA cdc s6 tq nhi0n m cho v6i n ld mOt s5 tU nhiOn nAo rlri, ta mn =1(modn) Biri cci thi m=1(modn) g aie4 Cho tam gi6c dOu ABC canh a M AC ho AM * AN = l MB vi N ld hai cli6m di dQng lAn lucrt tren hai canh AB vd NC ring MBCN la tf gi6c ngoai ti6p dugc m6t dudng trdn b) Tim gi6tri lcyn nh6t cria diQn tich tam gi6c AMN theo a a) chring minh IIeI He ve ten thi s inir: So bao danh: s0crAo DUC vAEAo rAo rAy NrNH rY rnr cHeN Hec jsrNH cr6r lop 12 Trrpr voNG riNn NAM HQC 2013 -2014 uudxc oAN cnAna rnr nnoN roaN (Bu6i thi rhrl hai) CACH GIAI Bei I Gidi phwong @ diem) Di6ukien: EAt 3)W= trinh (z* - - 2x + xrl2 t- W.t>o Phucrng x2 thi x'=tz -2x-2 trinh dd cho tucrng ituong v6i t2 - Giai phucrng trinh tr6n tim dugc t Vdi t - 2x- I thi dugc W:2x-1o) Giai phuong trinh trOn tim dugc nghiQm lopi BAi @ diem) [*' =6x+y+11 Girti h0 pltwnng trinlt 1,, He tuong ducnrg -y-x-3 v6i I(": L(v'+ 8)-(y+2)=3-x 27)-6(x-3)-y+Z I ti;:;;i*r*;;.;r; +2)(y2 -3 -Zy+3) Lfv ;i;;*:"*;;;;";; -ri 3>0 n6n x>3=+y> -2=)x vdri I =o non lim(u, - 6) NCU m = thi bai to6n 0r5 - o Suy limun - ,6 0r5 n >t.Z Gi6 sti p ld u6c nguy0n t6 ntrO nhdt cria n, dO Z":1 (modp) (*) Tt (*) ta c6 p ld s0 16, suy n li 0,5 n ld mQt sa qt nhihn ndo itri, ta cd thoi mdn =2,tac6 2" =1(modn) yoi -oi VO,y t6n - u ) 1(", - 6) #(r, rim#(,,-.,6) = sO le, d6 (n, p-1) tai cdc so nguy0n d,b cho an + b(p - l) -1 0,5 = 0,5 0r5 Theo dinh lf Fermat nh6 thi: 2t 2nu.2@-r)b : I (modp).V6 lf N6u m > 2,ta"6 *(m-l)' =[1+(m-1;1('-rt2 = (mod(m-t)r) 0r5 0r5 .- - MAt khSc T6m lpi m/l (mod(m -t)') Bdi to6n kh6ng tho6 man ffr:1 ld c6c gi|tri cAn tim 0r5 0r5 trang Bni ft diem) Cho tsm gidc it1u ABC cgnh o M vd N ld hai iti6m di ctQng ldn lwgt trhn cgnh AB vd AC hui N*4N "6 MB NC = a) Ch*ng minh rdng MBCN ld tir gtdc ngogi ti6p itwgc mQt itwdng trbn b) Ttm gid tr! td,n nhdt crta diQn tich tam gidc AMN theo a ,A XA, B A\ C a) Dat AM = x, AN = y Ggi E, F l6fl luqt ld trung rti6m cria AB, AC ri, suy r" {Y:Y **g MB NC= LANI < NC vay M, N ran ruqt thuQc c6c do4n AE vd AF Ttr gi6thi6t 0,5 ta.o, u YB * = u, IC = I =+ =L*j= NC MB NC = MB :i1YP:XE3P_I9:_ I > f MN2 = AM2 + A}-.{2 _ AI\4.ANI = (a - MB)2 + (a - NC)' - (a - MB)(a - NC) 0,5 = a2 +MB2 + NC2 - 2a(MB + NC) + a(MB + NC) - MB.NC = a2 +MB' +NC2 -2a(MB + NC) + 3MB.NC - MB.NC - a' +MB2 +NC2 -Za(MB + NC) + 2MB.NC - (MB+NC -a)' 0,5 + MN = MB +NC - a )MN +BC = MB +NC + MBCN ld trl gidc ngoqi.ti6p dugc Dudng trdn nQi ti€p tti gi6c MBCN ctng ld o Phucrng x2 thi x'=tz -2x-2... *(6 -,.,6 -.6),nr, Tim tdt cd ctic sd 4r nhihn m cho mn = I (modn) thi m: I (modn) N6u m > vdri I =o non lim(u, - 6) NCU m = thi bai to6n 0r5 - o Suy limun - ,6 0r5 n >t.Z Gi6 sti p ld u6c nguy0n... un*, = clwong Chfing minh rdng vd,i mgi s6 nguyAn t 1[r" 2[ " * un ) vdi mgi n nguy ) dwong n> thi un, Jj n* limun * i u2= I( s) s ;1",.i ):;'^6 0r5 Gia sri b6t ddng thfc dring vdi fl-k,tac6