Nguoithay.vn 50 B Bài 1: Cho a S a a Bài 2: Cho a S a a 10 a a2 a a 8 a2 S a 8a a ( 9 ) a 24 a 9 a S a2 6a a ( 8 a 12 ) a Bài 3: Cho a,b >0 a b 1 ab S ab (ab 15 ) 16ab 16ab Bài 4: Cho a,b,c>0 a b c a2 S b2 b2 c2 c2 b2 b2 c2 c2 ab 16ab b2 (a 17 12 S ab ab 15 17 a b 16 a2 Cách 1: Cách 2: S a2 (12 42 )(a b2 c2 : a2 1 ) (1.a ) 2 b b (b ); c c 17 Nguoithay.vn a2 a2 (c ) a 17 4 ) b Nguoithay.vn (a b c a 17 S b ) c 36 (a b c ) a b c 17 135 17 (a b c ) 4(a b c) 4(a b c) 17 Bài x y z 1 y2 x2 z2 y2 (1.x ) y (12 92 )( x2 z2 (y 82 (x y z x 82 TT : y2 S (x y z 82 x2 82 ) y2 x2 z2 y2 (x ) y 82 1 ); z2 (z ) z x x 82 9 81 ) (x y z ) y z x y z 82 ) x y z 80 x y z 82 Bài 6: Cho a,b,c>0 a 2b 3c 20 S a b c a 2b c 12 18 16 12 a 2b 3c 3a a b c a 20 3.2.2 2.2.3 2.4 52 S 13 1 Bài 7: Cho x,y,z> Tìm giá tr x y z 1 P 2x y z x y z x y 2z 4S 4a 4b 4c Ta có 1 x y x y y ; z y z x y y z 4 x y y z 1 16 x y TT : 2x y z 16 x y 16 x Bài y S Nguoithay.vn z 1 ; z x y 2z 2b 2 z 18 b c 16 x 2y z 16 c x 2y z 1 16 x y z Nguoithay.vn x 12 x x 15 x 12 x 12 R , ta có x 15 15 x 20 2.3 ; 20 x 15 x Bài 9: Cho x, x 8x.8x x x 82 3 8x.8x.82 20 2.5 ; 3 y.8 y.82 3 8z.8z.82 3 8x.8 y.8z 3 82.82.82 4z 2.4 x 4x nên : 12.4 z 8x y 8z 4x y x 12 12.4 y ; z z 82 x 12.4 x ; y y 82 64x 3x x x x x y 8z x 192 Bài 10: x3 y3 xy x3 y3 y3 z3 yz x3 y3 xy S x3 xy x y yz y3 3 xyz xy x y y3 z3 ; xy yz 3xy xy xy z3 x3 zx zx 3 yz yz x y2 z xy x y z 3xy xyz z3 x3 ; yz zx 3zx zx 3 Bài 11 x y xy P x P x y xy 2 y x y xy 2 x y x y Khi cho x=0 y= P = -1/4 Khi cho x=1 y = P = 1/4 x y xy x y xy Bài 12 Nguoithay.vn 2 4 P 3xy zx Nguoithay.vn a3 b a3 Cách 1: b a3 Cách 2: b a3 b b3 c b3 c b3 c c3 a ab bc ca c3 a a b c (a b c ) ab bc ca ab bc ac 3 b c ab 2a ; bc 2b2 ; ca 2a c a c3 a ab bc ac ab bc ac ab bc ac 2(a b2 c ) ab bc ac ab bc ac Bài 13 Cho x,y >0 x 3x 4x A y3 y2 3x 4x A y 3x x y2 x y y2 x x3 y3 3xy(x+y) x3 x x3 y3 3xy x3 y3 x TT : 1 y x3 1 y z 1 1 y xz ; x z z xy y x x Bài 17: y z y z 1 z y xyz z y z yz y z xy x y 1 x y z x y z x y z x y z 4a 5b2 3c 48 a b c Nguoithay.vn S S x y y3 3xy=1 3xy x3 y3 y3 3xy xy x y3 xy 1 Bài 15: Cho x,y,z >0 x y z P= y P x y y y3 y2 4 Nguoithay.vn 4a a 4 4 a a 8 16 a a a a 5b 3c b 10 20; c 12 dpcm b b c c Bài 18 a b c a 2b b 2c c 2a 1 a b b Bài 19 ; a 2b b 1 c c ; b 2c c a 36 a b c b c a b c a b c Bài 20: a b 16 c d a b c a2 x b2 y a c 2a 36 a b c 64 a b c d 16 16 16 ; a b c a b c d c2 z a a b c x y z 64 a b c d Bài 21 a Nguoithay.vn a b b c 1 p a a b 1 a b a b Bài 22 b p b p c ; c a b b c c a b b c a b c c b c c ; a c a Nguoithay.vn 1 p a p b p c a b c a b c a b c a b c Bài 23 a b c a b c a b c a b c Cho x,y,z>0 x y x x2 y z Cách1: P Cách 2: x2 y z y z P x2 y z P y2 z2 z x x y x y z x y z y2 z x z2 x y y; 4 z x x y x y z x y z x y x 2 x; a b c x y z 2 a b y2 z2 z x c x y z Bài 24 y 3z 3z x x 2y x 2y 3z 51 y 3z z x x y x 2y 3z x 2y y 3z 3z x 1 x 2y 3z x y 3z 21 Bài 25 24 1 1 x y 3z 24 x y 3z 3 51 a b2 ab a b Bài 26 p a p b p c 3p p c (12 12 12 )( p a Bu- nhi -a ta có : p a Bài 27 p b Nguoithay.vn p b p c) 3(3 p p) 3p Nguoithay.vn A a 1; b a a b 2; b 15b 16 b b 16 15.4 16 17 A a 1 b a b 21 Bài 28 a b4 a2 b2 a 3b ab3 (12 12 ) a b2 a b2 a b2 2ab a b2 a b4 Bài 29 ( x y 1)2 A xy y x ( x y 1)2 xy y x a A a xy y x ( x y 1)2 a; a 8a ( ) a a A a Có a a 9 a 3 10 A 10 Bài 30 Cho ba s th c a , b, c t phân bi t a Ch ng minh b2 (c a ) (b c)2 c2 ( a b) 2 Gi i: a b b (b c) (c a ) c c (c a ) ( a b ) a (a b) (b c) a b c (b c) VT (c a ) ( a b) (Không c n ch d u = x y ho n u c n cho a= 1,b=0 => c=-1 x y d u =) Bài 31 Cho s mãn a + b + c Ch ng ming r ng a Nguoithay.vn b2 c 2009 ab bc ca 670 a 3b ab3 Nguoithay.vn 2009 2 a b c ab bc ca 1 2 a b c ab bc ca ab bc ca 2007 ab bc ca Bài 32: Cho a, b, c 2007 a b c a b c 670 a b c P a b2 ab bc ca a 2b b2c c 2a c2 3(a + b2 + c2) = (a + b + c)(a + b2 + c2) = a + b3 + c3 + a 2b + b2c + c2a + ab2 + bc2 + ca Mà a + ab2 2a 2b ;b3 + bc2 2b2c;c3 + ca 2c2a Suy 3(a + b2 + c2) 3(a 2b + b2c + c2a) > Suy P a b c t = a + b2 + c2, v i t Suy P t 2t t ab bc ca a b2 c P a b c (a b2 c ) 2(a b2 c ) t 2t t 3 2 P a= b= c= Bài 33 P= 16 x 1 4y z 16x 1 1 y x z x x y z 4y z 16x y z 16 x y 16 x z z y z x y x có =khi y=2x; z=4x; z=2y 4y z 16 x z 16 x y Min P = 49/16 x = 1/7; y = 2/7; z = 4/7 Bài 34 P= : : B B 8x 18y x y 8x x; y x 18y 1 ; y 21 16 y z =>P 49/16 23 8x 18y x y x y y 12 23 43 B 43 x; y Bài 35 Nguoithay.vn x z 4y 1 ; Nguoithay.vn Ch x2 + y2 + z2 x x x 2 x2 3x y 3y z 2 2 x +y +z (x 1)(x 2) 3z 3( x + y +z) 6 = Bài 36 1; +b2+c2 = a b c a a a a 0; b2 b 0; c c 0 a b c a b2 c Bài 37 a2 b2 1.a b b2 c2 12 a2 c2 81 16 b2 a2 ; c2 b 4c 97 Bài 38 Ch p p p a p b p c2 b2 p p c p p a b c 97 a2 a2 b2 a 97 ; 4b c 4a 97 9 hay p a p b p c Bài 39 Cho a, 3(a b2 c ) 2abc 52 1 p a p b p c p a p b abc ( a b c)(a b c)(a b c) (6 2a) 2b 2c 2abc a 2 48 b 16 36 (a b c ) 2 c 2 (a b c ) 2abc a b2 c2 (2) 3(a b2 c ) 2abc 18 hay không? Bài 40 Nguoithay.vn p c abc p 24 ab bc ac 48 (1) (1)and(2) dpcm Nguoithay.vn Cho a, b, c P ( bc 3 a a )1 c b 2 2 ab ( b a c , b2( a ( c ( c (2) b ( c a c b (1) c2 b a ) ) ) ) ( ba ) ) Có a 2 a b c c ( b( a ( a ca ) c b b ) c ) b ca c ) b a ( c b (*) (2), (3) ta có : a ( b( c) a) a ( 2 b 8b)a b 2a b c c ) ( ) ( ) cac 90 ( a ) bc b ( a c a b c nên (*) 8 c b ) (*) 9b) a a b b0 c c c ( a a b b c c ( a a 3 3 bb cc c a 6) a 3b3 b3 () a bc bc ( ) a c aa b c ( a a b Ta có a () b c c 3 4) 2b3 ) ( c b b (**) a 1c c bb2 a9 c 4) a a c ( c b c a b a 2( a c a b 3 b )5 3 ( 3c a 8 a b( c ) a b c a b c Bài 41 Cho a, b, c a b3 c3 3abc *P a b3 c3 3abc Ta có a b3 c3 3abc a b3 c3 3abc (a b c)(a b c ab bc ac) (a b c ab bc ac) (1) có abc ( a b c)(a b c)(a b c ) (1 2a)(1 2b)(1 2c) 4(ab bc ca ) 8abc (1)and(2) a b3 c3 3abc mà ab bc ca a 6abc b Nguoithay.vn a b2 a b2 c2 P 2 c ab bc ca (2) c2 ab bc ca 3 a b2 c2 a b2 c2 10 P 1 6 Nguoithay.vn *P a b3 c3 3abc abc ( a b c)(a b c)(a b c) (1 2a)(1 2b)(1 2c) ab bc ca ) 2abc a b3 c3 3abc P 4(ab bc ca ) 8abc (3) (a b c)(a b c ab bc ac) 6abc a b c ab bc ac 6abc ab bc ca 2abc a b c ab bc ca 6abc Bài 42 Cho ba x2 y2 xyz z2 xy yz zx xyz x y z x y z x y z (6 x)(6 y)(6 z) 216 72( x y z) 24( xy yz zx) 8xyz xyz 24 ( xy yz zx) (1) mà x y z x2 y2 x2 y2 z2 2xy yz 2xz z2 Nên xyz x2 xyz x2 xyz x xy yz xz 36 3xy yz 3xz (2) 24 ( xy yz zx)+ 36 3xy yz 3xz y2 z2 xy yz xz y2 z2 xy yz xz 12 ( xy yz zx) mà x y z 3( xy yz zx) y z xy yz xz x y z 12 3 Bài 43 Cho a 1342; b 1342 a 1342 Nguoithay.vn b 1342 12 36 a b2 ab 2013 a b 0; a 1342 b 1342 11 0; a 1342 b 1342 Nguoithay.vn a 1342 b 1342 a 1342 b 1342 a b 2.1342 a b ab 1342a 1342b 1342 a b 2.1342 a b 2013 a b 3.13422 0 2.2013 a b 2.2013.1342 2013 a b (1) (2) 2.13422 ab 1342a 1342b 13422 a b ab 3.1342 a b 2013 a b 2.13422 3.13422 2013 a b 1342 1342 Bài 44 A x 4 x x x Cách 1: Cách : A x A x x x A 2x 8x 10 A 2( x 2) 2 x 2 2 x x x 2 x 4x ( x 2) 2 A 4( x 2) 8( x 2) 4( x 2) 8( x 2) A 8( x 2) 8 Bài 45: Cho a,b,c s th ab bc ca c a b h a mãn a+b+c=1 Ch ng minh r ng: Gi i: Nguoithay.vn 12 2013 a b Nguoithay.vn Bài 46 Cho x,y,z ba s th 1 x3 y3 1 y3 z3 u ki n xyz=1 Ch ng minh r ng: 1 z3 x3 Gi i: x2 y2 2xy x y x2 x3 y3 xy x y z 1 x y2 1 x z ; x y z y3 z3 y Bài 47 Cho a,b s th a b x3 2xy x y y3 xy x y y xy x y z x ; x y z z3 x3 y x y z dpcm ng minh r ng : a b 2a b 2b a a b a b a b Gi i: a b 2 Bài 48 Cho ba s th c a,b,c th 1 1 8a 8b3 8c3 a b u ki n: Gi i: Nguoithay.vn 13 a b ab a b 2a b 2b a Nguoithay.vn 1 8a ; 2a 4a 2a 2a 4a 2a 1 2b ; VT 4a 2a b3 c c3 a 8c3 2c 1 2 2a 2b 2c 1 8b3 2 2a 2b 2c 1 Bài 49 V i a,b,c ba s th ng minh r ng : a3 b a b2 c Gi i: Cách 1: a3 b b3 c c3 a a4 ab b4 bc c4 ca a b2 c 2 a b2 c a b2 c ab bc ca ab bc ca a b2 c Cách a3 b b3 ab 2a ; c c3 bc 2b ; a 2 ca Bài 50 Cho x,y,z ba s th x2 y y2 z z2 x y x; 2c VT a b2 c (ab bc ca ) a b c a mãn xyz = Ch ng minh r ng: Gi i: x2 y Nguoithay.vn y2 z z y; z2 x x 14 z VT x y z 4 3 4 ... (12 12 12 )( p a Bu- nhi -a ta có : p a Bài 27 p b Nguoithay.vn p b p c) 3(3 p p) 3p Nguoithay.vn A a 1; b a a b 2; b 15b 16 b b 16 15.4 16 17 A a 1 b a b 21 Bài 28 a b4 a2 b2 a 3b ab3 (12 12... Nguoithay.vn x 12 x x 15 x 12 x 12 R , ta có x 15 15 x 20 2.3 ; 20 x 15 x Bài 9: Cho x, x 8x.8x x x 82 3 8x.8x.82 20 2.5 ; 3 y.8 y.82 3 8z.8z.82 3 8x.8 y.8z 3 82.82.82 4z 2.4 x 4x nên : 12. 4 z 8x... 3 y.8 y.82 3 8z.8z.82 3 8x.8 y.8z 3 82.82.82 4z 2.4 x 4x nên : 12. 4 z 8x y 8z 4x y x 12 12.4 y ; z z 82 x 12. 4 x ; y y 82 64x 3x x x x x y 8z x 192 Bài 10: x3 y3 xy x3 y3 y3 z3 yz x3 y3 xy S