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(1)Bài tập bất đẳng thức -Thày QTuấn
1 Cho :x,y vµ x3+y3=x+y c/m : x2+y2 1
2 Cho :a,b,c lµ cạnh tam giác
c/m: a4+b4+c4 < 2(a2b2+b2c2+c2a2)
3 Cho : a , b , c∈[0 ;1]∧a+b+c=2 c /m:abc ≥(1− a)(1− b)(1 −c)
4 Cho : a , b , c ≥ ;c /m: a3+b3+c3≥3 abc Cho :a,b,c>0 c/m:
a) (1+a/b)(1+b/c)(1+c/a)
b) (1+1/a)(1+1/b)(1+1/c) 64 Cho :a,b,c > c/m:
a+b+ c
3 +
3
a+b+c¿
2
c +1/c¿2≥ 3¿
b+1 /b¿2+¿
a+1 /a¿2+¿ ¿
7 Cho : 0 ≤ x ≤1 ;0 ≤ y ≤ c /m:(1 − x)(2 − y )(4 x+ y)≤2
8 Cho : x+ y+ z=1 c /m:−1
2≤ xy+yz+zx ≤ 1 Cho : x : y ≥ 0∧ x3
+y3=2 c /m: x2+y2≤ 2
10 Cho :x;y [0 ;1]∧ x2+y2=x√1 − y2+y√1 − x2 c /m:(3 x+4 y )≤ 5 11 Cho : a>b>0 C/m : a+
(a −b)b≥ 3
12 Cho a;b;c;d>0 & 1+aa + b 1+b+
c
1+c+
d
1+d≤ C /m :abcd ≤ 81
13 Cho x;y;z 0∧ x (x− 1)+ y( y −1)+z(z− 1)≤4
3 C /m : x+ y+z ≤ 4 14 Cho:a;b;c>0 C/m:
a) a+b+c¿
3
a(b2
+bc+c2)+b (c2+ca +a2)+c (a2+ab+b2)≤1 3¿
b) a3
b2+bc +c2+
b3 c2
+ca+a2+
c3 a2
+ab+b2≥
a+b+c
3 15 Cho a;b;c>0 t/m: ab+bc+ca=1
C/m: a3
b+c+ b3 c +a+
c3 a+b≥
1
(2)17 Cho : a;b;c>0 vµ a+b+c=1 C/m
a) a2+4 b2+9 c2≥36 49
b) ab2c3≤
432
18 Cho a;b;c>0 t/m: a+b+c=1
C/m: (2+1
a)(2+
1
b)(2+
1
c)≥ 125
19 Cho:a;b;c>0 C/m:
a) (2
a+b+c )(
2
b+c+a)(
2
c+a+b)≥ 64
b) (1
a+b+c )(
1
b+c+a)(
1
c+b+a)≥ 27 víi abc=1
20 Cho:a;b;c>0 vµ abc=1 C/m:
a) a+b
2 + c3 ≥ 11
b) ab1 + bc +
1 ca+
3
a+b+c≥ 4
21 Cho a;b;c>0 C/m:
C/m: b+ca +b+c
a + b c +a+
c+a b + c a+b+ a+b c ≥ 15 22 ĐH NNI 2000: Cho:a;b;c>0 , abc=1 tìm P min:
P= bc
a2b+a2c+
ca
b2a+b2c+
ab
c2b+c2a
23 §HQGHN 2000: Cho víi mäi a;b;c t/m a+b+c=0 C/m 8a+8b+8c≥2a+2b+2c
24 SPVinh 98: Cho x;y;z: (x-1)2+(y-2)2+(z-1)2=1 tìm x;y;z cho P max P=(x+2y+3z-8 ) Tìm gtrị max
25 SPVinh 01: Cho a;b;c độ dài cạnh tam giác a+b+c=3 C/m: 3 a2+3 b2+3 c2+4 abc ≥13
26 Cho a;b;c t/m:
¿
a2+b2+c2=2 ab+bc+ca=1
¿{ ¿
c/m: −4
3≤ a ;b ;c ≤
27 HVNH HCM 01D:Cho a;b;c>0 C/m: (a+b+c)(1/a+1/b+1/c)
(3)a) a2+b2+c2≥ ab+bc+ca
b)(ab+bc+ca) ❑2≥ abc(a+b+ c) (SP TP HCM 2000)
29 ĐHĐà Nẵng96:Cho a;b;c cạnh tam giác c/m: 1)a2+b2+c2<2(ab+bc+ca)
2) a2+b2+c2=1; c /m :1
2≤ab+bc +ca≤ 1