so GrAo DUC vA nAo rAo rAv NII.IH xY rnr cHeN rrec sINH ctor Lop 12 THpr voNG NAna Hoc zot4 - 2ors rixn NSi,y thit 24 th6ng nf,m 2014 MOn: TOAN ru6i thi th* nh6t Thli gian: 180 phft pi (khong k6 thdi gian giao dA) of cnixn rrnlc tnt gim c6 0I trang, thi sinh khdng phdi chdp di vdo giAy thi) Bii aiam) Cho c6c s6 duong Chimg minh Beri x,y, zthoi mdn tli6u kiQn x + y * z=1 L- * -=-l+ ,ing < e 3xz - , +y 3z' + x + y +z 3y' +z+x -;-]- aieml f thoe m6n di6u ki0n f(x) +2f (xy)= f(x +Zxy),Vx e Chimgminhreng f(x+y)=f(x)+f(y),Vxe ,Vye i Cho hdm BAri siS i , Vy e i arcm1 Cho hai dudng trdn (O1), (O) cilt tpi hai tti€m A, B vi PrP2 h mgt ti6p tuy6n chung cira hai ttuimg trdn d6 (P1 thuQc (Or), Pz thuQc (Oz)) Gqi Qr vd Qz lAn luqt li hinh chi6u vudng g6c cria Pr, Pz l6n dulng thing OrOz Dulng theng AQr cdt (Or) tAi ili6m thf hai M1, Dulng thing AQ2 cit (Oz) tai diem thri hai M2 Chimg minh reng ba cli6m M1, M2, B thing hang Bifi aiaml BCn hinh vudng c6 diQn tich bAng 6, d[t ba da gi6c c6 diQn tich cung b[ng Chimg minh ring vdi ba da gi6c dE d{t hinh vu6ng, ludn tim dugc hai da gi6c mi di€n tfch phin chung cria chring kh6ng nh6 hon l .{ - HET - so GIAo Drrc vADAo rAo rAyNIi\H xi, rnr cHeN Hec srNH cror Lop tz rHpr voNG riNn NAvr Hgc zot4 -zots i\gny thi: 25 thfng nIm 2AL4 Mdn thi: TOAX -Budi thi thfr hai Thbi gian: 180 phrit (khdng kA thdi gian giao dA) DE CHINHTHTIC @A SOm cd 0l trang, thi sinh kh\ng phai chdp di vdo gidy tht) Bni (5 diAm) Giai h0 phuo-ng trinh 5)=64 I"'(3y+s L"V (y'+ 3y + 3) = 12 +5 lx Bei (5 dihm) (l Cho day sO (",) th6a mdn: J*'=5 Lt" +r) ' *n= n2xn+r * (+" * Z)XnXn*, Vn ) 1, n e N Tim limxn Bii (5 diem) Chring minh ring vdi n nguy6n ducrng kh6ng c6 nghiQm nguy6n duong (x; y) Bli vi n > thi phuong trinh 4xn + (x+1)2 =yz adml Cho tam gi6c ABC c6 ba g6c A, B, C deu nhgn vd AB > AC Ggi D, E, F hn luqt ld chdn c6c ducrng cao cta tam gifucABC vE ttr A, B, C xuiSng c4nh di5i diQn.Ducrng thlng EF cfu BC t4i P, dudng thang qua D vd song song vdi ef c6t c6c dudng tfrang AC va Ae tuong img tpi Q R Gqi M ld trung ei6m cpnh BC Chr?ng minh ring b6n Oi6* P, Q, R, M ctng thuQc mQt dudng trdn I H6t a {/ri, SO GIAO DUC VA DAO TAO TAY NINH xi Tm CHQN HQC SINH cror LOP 12 THPT voNG riNn NAvt Hec 2or4 -zors TTUONG oAN cnAu rm vrON roAN - su6r rnr rrrtl NnAr trang I Bili = Chrfrng minh rlng f(x + y) = f(x) + f(y), Vx f (0) Cho v Vdi f(; e R, Vy e IR ' f (Zxy), Yx e v x*0,dflt x-u; y= 2u e lR'' VY € R = f(2x) Vai, f(x +2xY)'Vx thi f(u + v) R,Vy e R - f(u) + f(v), Vu * 0, Vv e R (1) V6i x =0 thi f(x+y) =f(y)=f(0)+f(y) =f(x)+f(v), Vv e]R' (2) (1) vd (2) suy ra: f(x + y) - f(x) + f(y), Vx e R, Vy e IR i; Bili Q diam) Cho hai tlulng trdn (O1), (Oz) cit t4i hai Oi6m A, B vi PlP2li mQt fi6p tuy6n chung cira hai dudng trdn d6 (P1 thuQc (Or), Pz thuQc (Or).Ggi Qr vi Qz fa" foqt n trintr chi6u vu6ng g6c cria Pr, Pz l6n tlulng tneng OrOz.Dudng tning AQr cit (Or) t+i aiOr" tfr1i hai M1, Dulngtneng AQz cit (O2) tli aiam thrri hai M2 Chrfrng minh ring ba di6m Mr, Mz, B thing hing (hl) (h2) trang r 1., ( I trang trang so cIAo DUC YADAo rAo rAvNIi\H xV rHr cHeN Hgc sINH cr6r Lop NAna 12 rHpr voNG riNn Hec zor4 -zots i\giy thi: 25 thfng nlm 20L4 M6; thi: ToAll - Buoi thi thfr hai Thli gian: 180 ph fit (kh6rg kA thdi gian giao ai) DE CHINHTHTTC @A gom cd 0I trang, thi sinh kh1ng phai chdp di vdo gidy tht) Bei L (5 diem) Giei h0 phuong trinh [*'(3v L*V Bili2 + $'+ s 5):64 3y + 3) : l2+5 lx (5 diAm) (1 Cho day sO (*,) th6a mdn: J"'=5 Lr'+1) ' *n = r2Xn*r + ( 4n+2)XnXn+1 Vn > 1, n e N Tim limxn Bei (5 diem) Chrlng minh rang vdi n nguy6n duong vd n > thi phucrng trinh 4xn + kh6ng c6 nghiQm nguy6n ducrng (x; y) Bni (x+l)2 =y2 aidml Cho tam girlc ABC c6 ba g6c A, B, C ddu nhgn vd AB > AC Ggi D, B, F lan luqt ld chdn c6c dudmg cao cta tam gifucABC ve tt A, B, C xuiSng c4nh e16i diQn.Duong thing EF cat BC tai P, dudng tfring qua D vi song song v6i EF c6t c6c dudng thing AC vd AB tuong img t4i Q vi R Gqi M ld trung eiem cpnh BC Chrlng minh ring b6n AiCm P, Q, R, M ctng thu$c mQt dudng trdn , oo HCt SO GIAo DUC VA DAo TAo TAv NINH KY THI CHQN HQC jsrNu cror Lop 12 THpr voNc NAM HQC 2014 -201s HrroNG nAN csAvr riNu rur vroN roAN - BUor rur rntruar CACH GIAI Bni Q diem) Giei h phucrng trinh I.'(ry L*y Vd'i x = 0, h.6 + ss) = 64 (y'+ 3y + 3) = lZ + 5lx ph"""g Dod hQ tuong duong uoir f_ 64 llv+5s-:' *3 I [r'+3 DAt t- A ),thi x co hQ: y2 +3y - 9+51 x lZv+ 55 = t3 0,5 tr'+3y2+3y-3t+51 C0ngth.o;cffiipt,"ffic.iir,ethi.;6' +6y+ 55 - y3 +3y2 13 1,0 +3t+51 €> (y+1)3 +3(y +1)+ 51 - t3 +3t + 51 (*) ii;=ffi;'e, ffi;tio Hdm'6 (Ho[c tii5n a6i (*) e (y+l-rl[fr+1)2 +(y+l)t y+1=t) Vdi y=t-1 Vfly hQ d5 *r, 0r5 "ry." t y+l =T*' ] = o di5 suy 1r0 thi t3 -3t-52=0t-4