X Vl +Vl +b^ 7l +
o Vx^ llsVx^ 1 4x 8)= 0 3Vx^ l =4 x+
<=> i X >7x^ + 64x + 73 = 0 -2 <=> X = -32 + 3V57
So vdi DKXD, ta diTdc tap nghicm phiTdng trinh: S = ' -1; 1 b) Dieukien:^! < x< 9 , - l < y < 9
Tru" vc theo ve cua hai phifdng trinh cua he da cho, ta diTdc: V^T+T + V9 - y = % / > ^ + V9 - X
• Neu X > y, ta c6: Vx + 1 > V y T T ; ^9 - y > V9 - x
Nen Vx + 1 + 79 - y > Vy + 1 + V9 - x Mau thuan • Néu X < y, ta c6: Vx + 1 < yjy + 1 ; ^9 - y < \l9 - x
Nen Vx + 1 + 79 - y < Vy + 1 + V9 - X => Mau thuan Vay X = ỵ He phiTcfng trinh da cho trd thanh:
-32 + 3V57I
V ^ + V 9 - x = 2V5 O X + 1 + 9 - x + 2V(x + l ) ( 9 - x ) = 20 o x ^ - 8 x + 1 6 = 0 o ( x - 4 ) ^ = 0 o x - 4 = 0 o x = 4
Vay he phiTcfng trinh c6 nghi0m duy nhat (x ; y) = (4 ; 4)
Bai 3: Vdi a, b, c diTdng, ta c6:
1 + â = ab + be + ca + â = (a + b)(a + c) (thay 1 = ab + be + ca) 1 . 1 Ap dung bát ding thuTc Co-si vdi hai so diTdng:
a + b va a + c , ta c6: 1 1 a + b 1 >2 1 1 a + c Va + b b + c V(a + b)(b + c) . 1 1 1 + V i T ? V(a + b)(b + c) 2 U + b a + c j 1 + a < 2 2 a + b a + cy luan tiTdng tiT ta diTdc:
n o
Cty TNHH M i V úvVH Khang Vi§t
< 1 1 + b^ Suy ra: 1 < - 2 • + a + b b + c b va VTTc^ 2 < - c C + c + b a + c Vl + â Vl + b^ V T T ? + \ + b a + c a + b b + c c + b a + c, Dau ding thuTc xay ra o a = b = c
|Bai 4:
(a) (1) CO nghiem khi Á > 0 <=> 4(m - if - 8m(m - 4) > 0
o m^ - 4 m - 4< 0 o ( n i - 2) ' < 8 < : ^ |m-2| <2V2 > *' I, o 2 - 2V2 <m<2 + 2%^
9) He thiJc Vi-et: x, + X2 = - i ^ ^ (2) v^ x,.X2 = "^^""""^^ (3)
2 8
Tir (2) cho: m = 2(x, + X2) + 2 va thay vao (3) ta diTdc:
X1.X2 = •^[2(X| + X2) + 2][2(x, + X j ) + 2 - 4 ] = - (xi + X2 + 1)(X| + X 2 - 1) 8 2 <=> 2X|.X2 = (X| + X2)^ - 1 O X? + X2 = 1 (*) V, Ttf (*) ta nhan thay: Bai 5: 0 < X? < 1 '• = > - ! < X i < 1 va-1 <X2 < 1. 0 < X2 < 1 a) AMB = 90" ^ AM 1 MB
OM = OB, NM = NB => ON la du^dng trung triTc cua doan thing MB => ON 1 MB. Vay AM // ON Tỉ giac OAMN la hinh lhang. => ON 1 MB. Vay AM // ON Tỉ giac OAMN la hinh lhang. ANKH c6: H M 1 K N , KB 1 N H ^ O inrc lam ANKH
^ 0 N 1 H K ^ M B / / H K b) Ta c6: b) Ta c6:
BKH = MBA (so le trong, MB // KH)
MBA = ONB (Cling phu MBN)
ONB = AEB (dong vi, ON // AE) AEB = AFH (linh chat doi xij-ng true) ^ BKH = AFH
Luy$n giai <Si trMc kl thi vao Iflp 10 ba mign B&c, Trung, Nam m6n ToAn _ Nguyjn Difc TSn
c) DatOH = X.
ANKH can tai N ' h :
=> NO la dirdng trung triTc cua doan thing KH ,.j
^ OK = OH ' BH.BN = BỌBK <^ BH.BN = R(x + R)
o BH.OC = R(x + R) (VI BN = NM = OC)
<:> BH'.OC' = R'.(X + R)' o (X' - R')RX = R'(X + R)= (OC' = OM.OH) c=> x' - 2Rx - R' = 0
Á= R^ + R^ = 2R'=> ^/Á= R72
X, - (l - V2)R<()(loai) ; X 2 = (l +V2 )R Vay OH= (l + N/2)R
Bai 6: Difng IE 1 AB tai E, du-cJng IE ciit (1) tai M => ME la trung triTc cua AB.
XctAACM va ABOM c6:
OB = CA (gt), OBM = CAM (hai goc noi tiep cilng chan OM ), MB = MA => AACM = ABOM (c.g.c) => OM = MC
Vay M e (d) la trung trufc cua OC => (d) CO dinh.
Dat xOy = a => BMA = a
Do AMAB can tai M => MBA = 180" - a Ma MOA = MBA MOA =
Suy ra M thuoc lia Oz: zOA =
2 180" - g 2 180" - a (khong đi) Oz CO dinh.
Vay M la diem co dinh. Difdng tron ngoai tiep AOAB luon di qua hai diem
CO dinh O, M => Tarn 1 ciia du^dng tron ngoai tiep tam giac OAB thuoc dúdng trung triTc cua OM.
D E S6 81
Bai 1: (4 diem) Cho phifrtng trinh (an x): -t
2x^ + 2(m + 2)x + m^ + 4m - 4 = 0 (1)
a) Djnh m dc (1) CO nghiem. "' b) ChiJng minh khi (1) co hai nghiem X|, X2 thi: [x, + X j + 3x,X2| < 16.
Bai 2 : (4 diem) Giai phiTdng trinh va he phtfdng trinh: 1 1 1 Cty TNHH MTV DVVH Khang Vỉt a) 1 1- y + z 1 X + z 1 1 — I - z X + y 2 ]_ 3 _1_ 4 'ik-^'- : - a- ..| . (ti • C . f ... J ""f b) 2 72x + 4 + 4 V 2 - x = yj9x^ +16 •U • . :
Tim gia trj nho nhát cua bieu thiJc: A = — + — a b
Bai 3: (3 diem) Cho a, b la hai so diTdng thoa man: 4a + 3b = 11.
- i: |i -V 1 ỉ: rii ;:; r •
Bai 4: (2 diem) Tim taft ca cac gia tri x, y nguyen thoa man: x' - ý = 2010.
Bai 5: (4 diem) Cho tam giac ABC vuong tai A co dinh A, B la co djnh va C thay doi tren nuTa di/dng thang At vuong goc vdi AB tai Ạ Goi I la tam dúdng tron
noi tiep tam giac ABC va P, Q, R Ian \\i6l la cac tiep diem cua du-clng U-on nay
voti cac canh AC, BC, AB. DiTdng thang PQ va AI cit nhau tai D. a) Chufng minh rang bon diem B, D, Q, R nam tren mot difdng tron.