[r]
(1)BAØI TẬP BẤT ĐẲNG THỨC a2 b2 1)Cho a,b>0 vaø a+b=1.Cmr: (Hungary 1996) a 1 b 1 1 HD:C1:Ta coù:VT=a-1+b-1+ 1 a+1 b ab2 (a+b) C2 : AÙp duïng bñt SWaùc ta coù:VT ab2 a2 2a b2 2b 1 C3 : VT 4ab2 4 a 1 b+1 a 1 b 1 Chú ý:Nội dung bđt SWác sau:Cho R; bi i=1,n lúc đó ta có a12 a22 a2 (a a2 an )2 n b1 b2 bn b1 b2 bn a2 a 1 ab2 a VT (a b) VT a 1 9 3 2)Cho x,y R Cmr: 3(x+y+1) 3xy (Columbia 2001) C4 : AÙp duïng bñt Coâsi ta coù: HD:Ta cm: 3(x+y+1)2 3( xy ) 9(x y)2 24(x y) 16 (3x 3y 2)2 a b a a2 b ab2 b a ab b a b2 3)Cho a,b 0.Cmr: ( ) ( ) 1 4)Cho a,b >0.Cmr: 2(a+b)( ) a b a b b a (Czech 1993) 2xy x y ( x y )2 5)Cho x,y R vaø xy >0.Cmr: x+y 2 Lop12.net (APMC 1993) (2) 2(a3 b3 c3 ) 9(a b c)2 6)Cho a,b,c Cmr: 33 abc a b c2 xy yz zx xyz 7)Cho x,y,z >0 Cmr: xyz 3 x y z 8)Cho x,y,z >0.Cmr: 1 x+ (x+y)(x+z) y+ (y+z)(y+x) z+ (z+x)(z+y) 9)Cho x,y,z >0 vaø x+y+z=1.Cmr : 10)Cho x,y,z >1 vaø x y z 1-x 1-y 1-z 1 2.Cmr : x y z x+y+z x y z (Iran 1998) 11)Cho a,b,c >0.Cmr: (a2 b b2 c c2 a)(ab2 bc2 ca2 ) abc (a3 abc)(b3 abc)(c3 abc) 12) a4 b4 c4 a2 b2 b2 c2 c2 a2 a3 b b3 c c3a ab3 bc3 ca3 13) a4 a2 b2 b4 b4 b2 c2 c4 c4 c2 a2 a4 a 2a2 bc b 2b2 ca c 2c2 ab 14)Cmr a,b,c R ,ta coù: a2 (1 b)2 b2 (1 c)2 c2 (1 a)2 2 15)Cho a,b,c >0.Cmr: a2 ab b2 b2 bc c2 a2 ac c2 16) Cho a,b,c,d >0.Cmr: (a+c)2 (b d)2 ad bc (a+c)2 (b d)2 a2 b2 c2 d (a+c)2 (b d)2 17)Cho a,b,c 1.Cmr: a-1 b c c(ab 1) Lop12.net (3) 18)Cho a,b,c>0.Cmr: (a+b)(b+c)(c+a) 19)Cho x,y,z >0 vaø xy+yz+zx=xyz Cmr : 20)Cho a,b,c >0 Cmr : a a 8bc b b 8ac ab(a+b) c 21)Cho a,b,c >0 vaø ab+bc+ca=1.Cmr: 22)Choa, b, c 0.Cm : ab bc ca 1 1+x 1+y c 8ab 6b a bc(b c) 1+z 1 6c b ca(c a) 1 6a c abc 4abc (a b)(b c)(c a) b c (Macedonia 1995) ca ab 1 24) Cho a,b,c >0 vaø abc=1.Cm: (a-1+ )(b )(c ) (IMO 2000) b c a 1 1 1 25)Cho a,b,c>0.Cm: (a+ 1)(b 1) (b 1)(c 1) (c 1)(a 1) b c c a a b 3 x y z 26)Cho x,y,z>0 vaø xyz=1.Cm: (IMO 1998) (1 y)(1 z) (1 z)(1 x) (1 x)(1 y) ab bc ca 27)Cho a,b,c>0 vaø abc=1.Cm: (IMO 1996) 5 a b ab b c bc c +a5 ca 1 28)Cho a,b,c>0 vaø abc=1.Cmr : a (b c) b (c a) c (a b) a b c d 29)Choa, b, c,d 0.Cm : (IMO 1993) b+2c+3d c+2d+3a d+2a+3b a+2b+3c a3 b3 c3 d3 30)Cho a,b,c,d>0 vaø ab+bc+cd+da=1.Cm: bcd cda da b a bc 31)Cho x1 , x 0, y1 , y ,z1 ,z R, x1y1 z1 , x y z 22 Cm : 23)Cho a,b,c>0 Cm: a b+c 1 x1y1 z12 x y z 22 (x1 x )(y1 y ) (z1 z )2 Lop12.net (4) a2 b2 c2 bc ca ab 32)Cho a,b,c>0.Cm: 1 a 2bc b 2ca c 2ab a 2bc b 2ca c 2ab a3 b3 c3 33)Cho a,b,c >0.Cmr: a b c (Canada 2002) bc ca ab (b+c-a)2 (c+a-b)2 (a+b-c)2 34)Cho a,b,c >0.cm: (Japan 1997) (b c)2 a2 (c a)2 b2 (a b)2 c2 (b+c+2a)2 (c+a+2b)2 (a+b+2c)2 35)Cho a,b,c >0.cm: (USA 2003) (b c)2 2a2 (c a)2 2b2 (a b)2 2c2 1 bc ca ab 36)Cho a,b,c>0.Cm : a b c a bc b ac c ab a2 bc b2 ca c2 ab 37)Cho a,b,c>0.Cmr : abc bc ac ab 1 2(a3 b3 c3 ) 2 38)Cho a,b,c>0 vaø a b c 1.Cmr : a b c abc 1 39)Cho a,b,c>0 vaø a2 b2 c2 3.Cmr : (Belarus 1999) 1+ab bc ca 1 40)Cho a,b,c>0 vaø a2 b2 c2 1.Cmr : 1-ab bc ca 1 41)Cho a,b,c>0 vaø a4 b4 c4 3.Cmr : 1 (Moldova 2005) 4-ab bc ca a b c 42)Cho a,b,c>0 vaø a2 b2 c2 1.Cmr : (a a b b c c)2 b 1 c 1 a 1 Lop12.net (5) 176 abcd 27 27 65)Cmr : (a+b-c)2 (b+c-a)2 (c a b)2 (a2 b2 c2 )(b2 c2 a2 )(c2 a2 b2 ) a,b,c R 64)Choa,b,c,d 0vaø a+b+c+d=1.Cmr: abc+bcd+cda+dab 66)Cho x,y,z>0 vaø x+y+z=1.Cmr:x y y z z x 67)Cho x,y,z>0 vaø xy+yz+zx=1 Cmr: (Canada 1999) 27 68)Cho a,b,c 1.Cmr: a2 b2 c2 a2 b b2 c c2 a (Italy 1993) x(1-y )(1 z ) y(1 z )(1 x ) z(1 x )(1 y ) 69)Cho x 0;1.Cmr : (1-x n )m (1 (1 x)m )n 70)Cho a,b,c vaø a+b+c abc.Cmr: a2 b2 c2 abc 71)Cho a,b,c vaø a+b+c abc.Cmr: a2 b2 c2 3abc (BMO 2001) 72)Cho x,y,z>0 vaø x+y+z= xyz.Cmr : xy+yz+zx 9(x+y+z) 73)Cho x y z 2.Cmr : x y z xyz 74)Cho x y z 2.Cmr : x3 y3 z3 xyz 2 75)Cho x y z 9.Cmr : 2(a+b+c)-abc 10 (Vietnam2002) 76)Cho a,b,c>0.Cmr: (a+b)4 (b c)4 (c a)4 (a4 b4 c4 ) (VN 1996) 1 1 77)Cho a,b,c,d>0 vaø: + + + =1.Cmr: abcd (Latvia 2002) 1+a4 1+b4 1+c4 1+d 78)Cho x,y,z>1.Cmr: x x 2yz .y y 2zy .z z 2xy (xyz)xyyzzx (USAMO 1999) 79)Cho a,b,c>0.Cmr: (a2 +2)(b2 +2)(c2 +2) 9(ab+bc+ca) (APMO 2004) 80)Cho a,b,c>0.Cmr:(a5 a2 3)(b5 b2 3)(c5 c2 3) (a b c)3 (USA 2004) 81)Cho a,b,c vaø a2 +b2 c2 abc 4.Cmr : ab+bc+ca-abc (USA 2001) 82)Cho c b a 0.Cmr:(a+3b)(b+4c)(c+2a) 60abc (Turkey 1999) (Macedonia 1999) abc 84)Cho a,b,c>0 vaø a+b+c=1.Cmr: a2 b2 c2 3abc (Poland 1999) 83)Cho a,b,c>0 vaø a2 b2 c2 1.Cmr : a+b+c+ Lop12.net (6) 85)Cho x,y,z>0.Cmr: x y z 2(xy yz) 86) (APMC 1995) Cho x,y>0.Cmr: (n-1)(m-1)(x m+n y m n ) (n m 1)(x n y m x m y n ) nm(x nm 1y y nm 1x) 87)Cho a,b,c>0.Cmr:a2 b2 c2 2abc (1 a)(1 b)(1 c) 88)Cho a,b,c,d vaø:2(ab+ac+ad+bc+bd+cd)+abc+bcd+cda+dab=16 Cmr:a+b+c+d (ab ac ad bc bd cd) 89)Cho x,y,z 0.Cmr: xyz (x y z)(y z x)(z x y) Lop12.net (7) 1+ 1 (a b2 c2 )( ) a b c a2 b2 c2 (Albania 2002) a b c 3 a b c a b bc ca 44)Cho a,b,c>0.Cmr: (Belarus 1997) b c a ca a b bc a b c ab bc 45) Cho a,b,c>0.Cmr: (Belarus 1998) b c a bc ab a b c 46)Cho a,b,c - vaø a+b+c=1.Cmr: (Poland 1996) a b c 10 1 1 1 47)Cho a,b,c>0 vaø abc=1.Cmr: (Bulgaria 1997) 1+a+b b c c a a b c x9 y9 y9 z9 z9 x9 48)Cho x,y,z>0 vaø xyz=1.Cmr: (Rumania 1997) x x y y z z y y x x 3z z 43)Cho a,b,c>0.Cmr: 49)Cho x y z>0.Cmr: x2 y y2z z2 x x2 y2 z2 z x y (Vietnam 1991) 1 1 50)Cho a,b,c,d>0 vaø abcd=1.Cmr:a3 b3 c3 d max a b c d ; a b c d 2 1+ab bc ca 18 51)Cho a,b,c>0 vaø abc=1.Cmr: (Hong Kong 2000) 3 c a b a b3 c3 3+ xyz(x y z x y z ) 52)Cho x,y,z>0 Cmr: (Hong Kong 1997) (x y z )(xy yz zx) ab bc ca a b c 53)Cho a,b,c>0.Cmr: (Moldova 1999) c(c+a) a(a b) b(b c) b c c a a b a+c b d c a d b 54)Cho a,b,c,d>0.Cmr: 4 a+b b c c d d a a-b b c c d d a 55)Cho a,b,c,d>0.Cmr: 0 c+b c d d a a b a+y az ax az ax ay 56)Cho a,x,y,z>0.Cmr: x y zxyz x y z a+z ax ay ax ay az 1 57)Cho -1<x,y,z<1.Cmr: 2 (1-x)(1-y)(1-z) (1+x)(1+y)(1+z) 58)Cho x,y,z>0 vaø x+y+z=3.Cmr: x+ y+ z xy+yz+zx a b c a+b+c 59)Cho a,b,c>0.Cmr: (1+ )(1+ )(1+ ) 2(1+ ) (APMO 1998) b c a abc x y z x+y+z 60)Cho x,y,z>0.Cmr: + + y z x xyz 61)Cho x,y,z>0 vaø x+y+z=1.Cmr: (1+x)(1+y)(1+z) (1-x )2 (1-y )2 (1-z )2 62)Chox,y,z 0vaø x+y+z=1.Cmr: 7(xy+yz+zx) 2+9xyz 63)Cho x,y,z>0 vaø x+y+z=1.Cmr: x3 y3 z3 6xyz Lop12.net (8) Lop12.net (9)