math: bat dang thuc(cuc kho, cuc hay)
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a, b, c
a
b
+
b
c
+
c
a
≥
a
2
+ 1
b
2
+ 1
+
b
2
+ 1
c
2
+ 1
+
c
2
+ 1
a
2
+ 1
.
a, b, c, d
a
2
− bd
b + 2c + d
+
b
2
− ca
c + 2d + a
+
c
2
− db
d + 2a + b
+
d
2
− ac
a + 2b + c
≥ 0.
a
2
+ b
2
(a
2
+ 1) (b
2
+ 1) ≥
a
2
+ b
2
(ab + 1)
= ab
a
2
+ b
2
+ a
2
+ b
2
≥ ab
a
2
+ b
2
+ 2
⇒
a
b
+
b
a
=
a
2
+ b
2
ab
≥
a
2
+ b
2
+ 2
(a
2
+ 1) (b
2
+ 1)
=
a
2
+ 1
b
2
+ 1
+
b
2
+ 1
a
2
+ 1
a
2
b
2
=
a
2
b
2
+ 1
+
a
2
b
2
(b
2
+ 1)
≥
a
2
b
2
+ 1
+
b
2
b
2
(b
2
+ 1)
=
a
2
+ 1
b
2
+ 1
.
1 +
a
b
2
= 1 + 2
a
b
+
b
a
+
a
2
b
2
≥ 1 + 2
a
2
+ 1
b
2
+ 1
+
b
2
+ 1
a
2
+ 1
+
a
2
+ 1
b
2
+ 1
=
1 +
a
2
+ 1
b
2
+ 1
2
.
a
b
+
b
c
+
c
a
≥
a
2
+ 1
b
2
+ 1
+
b
2
+ 1
c
2
+ 1
+
c
2
+ 1
a
2
+ 1
2(a
2
− bd)
b + 2c + d
+ b + d =
2a
2
+ b
2
+ d
2
+ 2c(b + d)
b + 2c + d
=
(a − b)
2
+ (a − d)
2
+ 2(a + c)(b + d)
b + 2c + d
(1)
2(c
2
− db)
d + 2a + b
+ b + d =
(c − d)
2
+ (c − b)
2
+ 2(a + c)(b + d)
d + 2a + b
(2)
(a − d)
2
b + 2c + d
+
(c − d)
2
d + 2a + b
≥
[(a − b)
2
+ (c − d)
2
]
(b + 2c + d) + (d + 2a + b)
(3)
(a − d)
2
b + 2c + d
+
(c − b)
2
d + 2a + b
≥
[(a − d)
2
+ (c − b)
2
]
2
(b + 2c + d) + (d + 2a + b)
(4)
2(a + c)(b + d)
b + 2c + d
+
2(a + c)(b + d)
d + 2a + b
≥
8(a + c)(b + d)
(b + 2c + d) + (d + 2a + b)
(5)
2(
a
2
− bd
b + 2c + d
+
c
2
− db
d + 2a + b
) + b + d ≥
(a + c − b − d)
2
+ 4(a + c)(b + d)
a + b + c + d
= a + b + c + d.
a
2
− bd
b + 2c + d
+
c
2
− db
d + 2a + b
≥
a + c − b − d
2
b
2
− ca
c + 2d + a
+
d
2
− ac
a + 2b + c
≥
b + d − a − c
2
a = c b = d
a, b, c
a + b + c = 1
ab
1 − c
2
+
bc
1 − a
2
+
ca
1 − b
2
≤
3
8
4ab
a
2
+ b
2
+ 2(ab + bc + ca)
≤
3
2
4ab
a
2
+ b
2
+ 2(ab + bc + ca)
≤
ab
a
2
+ ab + bc + ca
+
ab
b
2
+ ab + bc + ca
=
ab
(a + b)(a + c)
+
ab
(b + c)(a + b)
=
a(b + c)
2
(a + b)(b + c)(c + a)
3(a + b)(b + c)(c + a) ≥ 2
a(b + c)
2
ab(a + b) ≥ 6abc
a = b = c =
1
3
4ab
a
2
+ b
2
+ 2(ab + bc + ca)
≤
4ab
(2ab + 2ac) + (2ab + 2bc)
≤
ab
2a(b + c)
+
ab
2b(a + c)
=
1
2
b
b + c
+
a
a + c
=
1
2
b
b + c
+
c
b + c
=
3
2
a, b, c
1 +
ab
2
+ bc
2
+ ca
2
(ab + bc + ca)(a + b + c)
≥
4.
3
(a
2
+ ab + bc)(b
2
+ bc + ca)(c
2
+ ca + ab)
(a + b + c)
2
(a + b + c)
2
(a + b + c)
2
+
(a + b + c)(ab
2
+ bc
2
+ ca
2
)
ab + bc + ca
≥ 4.
3
(a
2
+ ab + bc)(b
2
+ bc + ca)(c
2
+ ca + ab)
(a + b + c)
2
+
(a + b + c)(ab
2
+ bc
2
+ ca
2
)
ab + bc + ca
=
(a
2
+ ab + bc)(c + a)(c + b)
ab + bc + ca
(a
2
+ ab + bc)(c + a)(c + b)
ab + bc + ca
≥ 3.
3
(a
2
+ ab + bc)(b
2
+ bc + ca)(c
2
+ ca + ab)[(a + b)(b + c)(c + a)]
2
ab + bc + ca
√
3.
3
(a + b)(b + c)(c + a) ≥ 2.
√
ab + bc + ca
a = b = c
a
0
, a
1
, . . . , a
n
a
k+1
−a
k
≥ 1 k = 0, 1, . . . , n−1.
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
n
− a
0
≤
1 +
1
a
0
1 +
1
a
1
···
1 +
1
a
n
n = 1
1 +
1
a
0
1 +
1
a
1
− a
0
≤
1 +
1
a
0
1 +
1
a
1
a
0
(a
1
− a
0
− 1) ≥ 0,
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
≤
1 +
1
a
0
1 +
1
a
1
···
1 +
1
a
k
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k+1
− a
0
≤
≤
1 +
1
a
0
1 +
1
a
1
···
1 +
1
a
k+1
1 +
1
a
0
1 +
1
a
1
···
1 +
1
a
k+1
≥
≥
1 +
1
a
k+1
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
1 +
1
a
k+1
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
≥
≥ 1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k+1
− a
0
1 +
1
a
k+1
1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
≥
≥ 1 +
1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k+1
− a
0
⇔
⇔
1
a
k+1
+
1
a
k+1
a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
≥
≥
1
(a
k+1
− a
0
)a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
⇔
⇔ 1 ≥
1
a
k+1
− a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
1
a
k+1
− a
0
1 +
1
a
1
− a
0
···
1 +
1
a
k
− a
0
≤
≤
1
k
1 +
1
1
···
1 +
1
k −1
= 1.
a, b, c > 0
3
a
2
+ bc
b
2
+ c
2
≥ 9.
3
√
abc
(a + b + c)
a
2
+ bc
3
abc(a
2
+ bc)
2
(b
2
+ c
2
)
≥
9
(a + b + c)
3
a
2
+ bc
3
abc(a
2
+ bc)
2
(b
2
+ c
2
)
=
=
a
2
+ bc
3
(a
2
+ bc)c(a
2
+ bc)b(b
2
+ c
2
)a
≥
3(a
2
+ bc)
sym
a
2
b
3(a
2
+ b
2
+ c
2
+ ab + bc + ca)
sym
a
2
b
≥
9
(a + b + c)
3
a
3
+ b
3
+ c
3
+ 3abc ≥
sym
a
2
b
a = b = c
a, b, c ab + bc + ca > 0
1
2a
2
+ bc
+
1
2b
2
+ ca
+
1
2c
2
+ ab
≥
2
ab + bc + ca
.
ab + bc + ca
2a
2
+ bc
≥ 2, (1)
a(b + c)
2a
2
+ bc
+
bc
bc + 2a
2
≥ 2.(2)
bc
bc + 2a
2
≥
(
bc)
2
bc(bc + 2a
2
)
= 1.(3)
a(b + c)
2a
2
+ bc
≥ 1.(4)
a(b + c)
2a
2
+ bc
≥
a(b + c)
2(a
2
+ bc)
a(b + c)
a
2
+ bc
≥ 2, (5)
2ca + bc
2a
2
+ bc
+
2bc + ca
2b
2
+ ca
≥
4c
a + b + c
.
a, b, c ab + bc + ca > 0
1
2a
2
+ bc
+
1
2b
2
+ ca
+
1
2c
2
+ ab
+
1
ab + bc + ca
≥
12
(a + b + c)
2
.
0 ≤ a ≤ min{a, b}
1
2a
2
+ bc
+
1
2b
2
+ ca
≥
4
(a + b)(a + b + c)
.
1
2a
2
+ bc
,
1
2b
2
+ ca
,
1
2c
2
+ ab
,
1
ab + bc + ca
1
2a
2
+ bc
+
1
2b
2
+ ca
+
1
2c
2
+ ab
≥
2(ab + bc + ca)
a
2
b
2
+ abc(a + b + c)
a, b, c ab + bc + ca > 0
1
2a
2
+ bc
+
1
2b
2
+ ca
+
1
2c
2
+ ab
+
1
ab + bc + ca
≥
12
(a + b + c)
2
.
2
a
2
+ ab + b
2
(a + b + c)
= (2b + a)
2a
2
+ bc
+ (2a + b)
2b
2
+ ca
≥ 2
(2a + b)(2b + a) (2a
2
+ bc) (2b
2
+ ca).
c
2
(2a + b)
2a
2
+ bc
+
c
2
(2b + a)
2b
2
+ ca
≥ 2
c
4
(2a + b)(2b + a)
(2a
2
+ bc) (2b
2
+ ca)
≥
2c
2
(2a + b)(2b + a)
(a
2
+ ab + b
2
) (a + b + c)
=
4c
2
a + b + c
+
6abc
a + b + c
c
a
2
+ ab + b
2
2c
2
a + bc
2
+ 2ab
2
+ b
2
c
2a
2
+ bc
=
c
2
(2a + b)
2a
2
+ bc
+
c
2
(2b + a)
2b
2
+ ca
≥
4c
2
a + b + c
+
6abc
a + b + c
c
a
2
+ ab + b
2
=
4
a
2
+ b
2
+ c
2
a + b + c
+
6abc
a + b + c
c
a
2
+ ab + b
2
≥
4
a
2
+ b
2
+ c
2
a + b + c
+
6abc
a + b + c
(a + b + c)
2
c (a
2
+ ab + b
2
)
=
4
a
2
+ b
2
+ c
2
ab + c
+
6abc
ab + bc + ca
⇒
2a
2
b + 2ab
2
+ 2b
2
c + 2bc
2
+ 2c
2
a + 2ca
2
2a
2
+ bc
=
(b + c) +
2c
2
a + bc
2
+ 2ab
2
+ b
2
c
2a
2
+ bc
≥
(b + c) +
4
a
2
+ b
2
+ c
2
a + b + c
+
6abc
ab + bc + ca
=
8
a
2
+ b
2
+ c
2
+ ab + bc + ca
a + b + c
−
2
a
2
b + ab
2
ab + bc + ca
⇒
1
2a
2
+ bc
+
1
ab + bc + ca
≥
4
a
2
+ b
2
+ c
2
+ ab + bc + ca
(a + b + c) (
(a
2
b + ab
2
))
≥
12
(a + b + c)
2
.
<=>
(a + b)(a + c)
2a
2
+ bc
+
a
2
+ bc
2a
2
+ bc
− 2 ≥
12(ab + bc + ca)
(a + b + c)
2
2a
2
+ 2bc
2a
2
+ bc
− 3 =
bc
2a
2
+ bc
≥ 1
a
2
+ bc
2a
2
+ bc
− 2 ≥ 0
(a + b)(a + c)
2a
2
+ bc
≥
12(ab + bc + ca)
(a + b + c)
2
(a + b)(a + c)
2a
2
+ bc
= (a+b)(b+c)(c+a)(
1
(2a
2
+ bc)(b + c)
≥
3(a + b)(b + c)(c + a)
ab(a + b) + bc(b + c) + ca(c + a)
(a + b)(b + c)(c + a)
ab(a + b) + bc(b + c) + ca(c + a)
≥
4(ab + bc + ca)
(a + b + c)
2
(a + b + c)
2
ab + bc + ca
≥
4[ab(a + b) + bc(b + c) + ca(c + a)
(a + b)(b + c)(c + a)
= 4 −
8abc
(a + b)(b + c)(c + a)
a
2
+ b
2
+ c
2
ab + bc + ca
+
8abc
(a + b)(b + c)(c + a)
≥ 2
a = b = c, a = b, c = 0
a, b, c 16(a + b + c) ≥
1
a
+
1
b
+
1
c
1
a + b +
2(a + c)
3
≤
8
9
.
a + b +
2(c + a) = a + b +
c + a
2
+
c + a
2
≥ 3
3
(a + b)(c + a)
2
.
1
a + b +
2(c + a)
3
≤
2
27(a + b)(c + a)
.
1
3(a + b)(c + a)
≤ 4 ⇐⇒ 6(a + b)(b + c)(c + a) ≥ a + b + c,
9(a + b)(b + c)(c + a) ≥ 8(a + b + c)(ab + bc + ca)
16abc(a + b + c) ≥ ab + bc + ca ⇒
16(ab + bc + ca)
2
3
≥ ab + bc + ca ⇐⇒ ab + bc + ca ≥
3
16
.
a = b = c =
1
4
x, y, z xyz = 1
x
3
+ 1
x
4
+ y + z
+
y
3
+ 1
y
4
+ z + x
+
z
3
+ 1
z
4
+ x + y
≥ 2
√
xy + yz + zx.
2
(x
4
+ y + z)(xy + yz + zx) = 2
[x
4
+ xyz(y + z)](xy + yz + zx)
= 2
(x
3
+ y
2
z + yz
2
)(x
2
y + x
2
z + xyz)
≤ (x
3
+ y
2
z + yz
2
) + (x
2
y + x
2
z + xyz)
= (x + y + z)(x
2
+ yz) =
(x + y + z)(x
3
+ 1)
x
.
x
3
+ 1
x
4
+ y + z
≥
2x
√
xy + yz + zx
x + y + z
.
a, b, c a + b + c =
√
5
(a
2
− b
2
)(b
2
− c
2
)(c
2
− a
2
) ≤
√
5
c ≥ b ≥ a
P = (a
2
− b
2
)(b
2
− c
2
)(c
2
− a
2
) = (c
2
− b
2
)(c
2
− a
2
)(b
2
− a
2
) ≤ b
2
c
2
(c
2
− b
2
).
√
5 = a + b + c ≥ b + c a ≥ 0
(c + b) ·
√
5
2
− 1
· c
2
·
√
5
2
+ 1
b
2
· (c − b)
≤ (c + b)
√
5(b + c)
5
5
≤
√
5;
P ≤
√
5 (a, b, c) =
√
5
2
+ 1;
√
5
2
− 1; 0
✷
a, b, c > 0 a + b + c = 3
1
3 + a
2
+ b
2
+
1
3 + b
2
+ c
2
+
1
3 + c
2
+ a
2
≤
3
5
1
3 + a
2
+ b
2
+
1
3 + b
2
+ c
2
+
1
3 + c
2
+ a
2
≤
3
5
<=>
3
3 + a
2
+ b
2
+
3
3 + b
2
+ c
2
+
3
3 + c
2
+ a
2
≤
9
5
a
2
+ b
2
3 + a
2
+ b
2
≥
6
5
a
2
+ b
2
3 + a
2
+ b
2
(
3 + a
2
+ b
2
) ≥ (
a
2
+ b
2
)
2
(
a
2
+ b
2
)
2
≥
6
5
(
(3 + a
2
+ b
2
))
(a
2
+ b
2
) + 2
(a
2
+ b
2
)(a
2
+ c
2
) ≥
54
5
+
12
5
a
2
8
a
2
+ 10
ab ≥ 54 <=> 5(a + b + c)
2
+ 3
a
2
≥ 54
a + b + c = 3
a, b, c > 0 ab + bc + ca = 1
1
4a
2
− bc + 1
+
1
4b
2
− ca + 1
+
1
4c
2
− ab + 1
≥ 1
1
4a
2
− bc + 1
+
1
4b
2
− ca + 1
+
1
4c
2
− ab + 1
≥
3
2
.
1
a(4a + b + c)
+
1
b(4b + c + a)
+
1
c(4c + a + b)
≥
3
2
.
1
a(4a + b + c)
4a + b + c
a
≥
1
a
2
=
1
a
2
b
2
c
2
.
2
3a
2
b
2
c
2
≥
4a + b + c
a
+
4b + c + a
b
+
4c + a + b
c
.
4a + b + c
a
=
3 +
a + b + c
a
= 9 +
(a + b + c)(ab + bc + ca)
abc
= 9 +
a + b + c
abc
,
9a
2
b
2
c
2
+ abc(a + b + c) ≤
2
3
,
a
2
b
2
c
2
≤
ab + bc + ca
3
3
=
1
27
,
abc(a + b + c) ≤
(ab + bc + ca)
2
3
=
1
3
.
a, b, c ≥ 0 ab + bc + ca = 1
1
4a
2
− bc + 2
+
1
4b
2
− ca + 2
+
1
4c
2
− ab + 2
≥ 1
abc = 0 abc > 0
4a
2
− bc + 2(ab + bc + ca) = (2a + b)(2a + c) ≤
[c(2a + b) + b(2a + c)]
2
4bc
=
(ab + bc + ca)
2
bc
=
1
bc
.
1
4a
2
− bc + 2
≥ bc.
a, b, c
(
1
a
+
1
b
+
1
c
)(
1
1 + a
+
1
1 + b
+
1
1 + c
) ≥
9
1 + abc
.
abc + 1
a
+
abc + 1
b
+
abc + 1
c
1
a + 1
+
1
b + 1
+
1
c + 1
≥ 9
cyc
1 + a
2
c
a
1
a + 1
+
1
b + 1
+
1
c + 1
≥ 9
cyc
1 + a
2
c
a
≥
cyc
c(1 + a)
2
a(1 + c)
≥ 3
3
(1 + a)(1 + b)(1 + c)
1
a + 1
+
1
b + 1
+
1
c + 1
≥
3
3
(1 + a)(1 + b)(1 + c)
a = b = c = 1
a, b, c
a(b + 1) +
b(c + 1) +
c(a + 1) ≤
3
2
(a + 1)(b + 1)(c + 1)
a + b + c + ab + bc + ca ≤ 3abc + 3 <=> 4(ab + bc + ca + a + b + c) ≤
3(a + 1)(b + 1)(c + 1)
(
a(b + 1) +
b(c + 1) +
c(a + 1))
2
≤ 3(ab + bc+ca + a+ b + c) ≤
9(a + 1)(b + 1)(c + 1)
4
a + b + c + ab + bc + ca ≤ 3abc + 3.
<=>
9(a + 1)(b + 1)(c + 1)
4
≥ 2(a + b + c + ab + bc + ca) + 3abc + 3
2
ab(b + 1)(c + 1) ≤
[ab(c + 1) + (b + 1)] = a + b + c + ab + bc + ca + 3abc + 3
=> ab + bc + ca + a + b + c + 2
ab(b + 1)(c + 1) ≤
9
4(a + 1)(b + 1)(c + 1)
=> (
a(b + 1) +
b(c + 1) +
c(a + 1))
2
≤ [
3
2
(a + 1)(b + 1)(c + 1)]
2
=> Q.E.D
a = b = c = 1.
a, b, c
1
a
2
+ b
2
+
1
b
2
+ c
2
+
1
c
2
+ a
2
≥
10
(a + b + c)
2
c = min{a, b, c}
1
a
2
+ c
2
+
1
b
2
+ c
2
≥
2
ab + c
2
⇐⇒ (ab − c
2
)(a − b)
2
≥ 0
((a
2
+ b
2
) + 8(ab + c
2
))
1
a
2
+ b
2
+
2
ab + c
2
≥ 25
5(a + b + c)
2
≥ 2((a
2
+ b
2
) + 8(ab + c
2
)) ⇐⇒
3(a − b)
2
+ c(10b + 10a − 11c) ≥ 0
a = b, c = 0
a, b c ab + ac + bc = 0
a
2
(b + c)
2
a
2
+ 3bc
+
b
2
(a + c)
2
b
2
+ 3ac
+
c
2
(a + b)
2
c
2
+ 3ab
≤ a
2
+ b
2
+ c
2
a
2
(b + c)
2
a
2
+ bc
=
a
2
(b + c)
3
(a
2
+ bc)(b + c)
=
a
2
(b + c)
3
b(a
2
+ c
2
) + c(a
2
+ b
2
)
≤
a
2
(b + c)
4
(
b
2
b(a
2
+ c
2
)
+
c
2
c(a
2
+ b
2
)
) =
a
2
(b + c)
4
(
b
a
2
+ c
2
+
c
a
2
+ b
2
)
LHS ≤
a
2
(b + c)(
b
a
2
+ c
2
+
c
a
2
+ b
2
) =
c(a
2
(b + c) + b
2
(c + a))
a
2
+ b
2
= a
2
+ b
2
+ c
2
+
abc(a + b)
a
2
+ b
2
≤ a
2
+ b
2
+ c
2
+
abc(a + b)
a
2
+ b
2
≤ a
2
+ b
2
+ c
2
+ ab + bc + ca
(a + b)
2
(a + c)
2
a
2
+ bc
≤
8
3
(a + b + c)
2
.
(a + b)(a + c) = (a
2
+ bc) + a(b + c)
(a + b)
2
(a + c)
2
a
2
+ bc
=
(a
2
+ bc)
2
+ 2a(b + c)(a
2
+ bc) + a
2
(b + c)
2
a
2
+ bc
= a
2
+ bc + 2a(b + c) +
a
2
(b + c)
2
a
2
+ bc
,
a
2
(b + c)
2
a
2
+ bc
≤
8
3
(a + b + c)
2
−
a
2
− 5
ab,
a
2
(b + c)
2
a
2
+ bc
≤
5(a
2
+ b
2
+ c
2
) + ab + bc + ca
3
.
5(a
2
+ b
2
+ c
2
) + ab + bc + ca
3
≥ a
2
+ b
2
+ c
2
+ ab + bc + ca
a
2
(b + c)
2
a
2
+ bc
≤ a
2
+ b
2
+ c
2
+ ab + bc + ca.
a
1
≥ a
2
≥ . . . ≥ a
n
≥ 0, b
1
≥ b
2
≥ . . . ≥ b
n
≥ 0
n
i=1
a
i
= 1 =
n
i=1
b
i
n
i=1
(a
i
− b
i
)
2
a
1
≥ b
1
a ≥ x ≥ 0, b, y ≥ 0
(a − x)
2
+ (b − y)
2
− (a + b − x)
2
− y
2
= −2b(a − x + y) ≤ 0.
(a
1
− b
1
)
2
+ (a
2
− b
2
)
2
≤ (a
1
+ a
2
− b
1
)
2
+ b
2
2
,
(a
1
+ a
2
− b
1
)
2
+ (a
3
− b
3
)
2
≤ (a
1
+ a
2
+ a
3
− b
1
)
2
+ b
2
3
, ······
(a
1
+ a
2
+ ··· + a
n−1
− b
1
)
2
+ (a
n
− b
n
)
2
≤ (a
1
+ a
2
+ ··· + a
n
− b
1
)
2
+ b
2
n
.
n
i=1
(a
i
− b
i
)
2
≤ (1 − b
1
)
2
+ b
2
2
+ b
2
3
+ ··· + b
2
n
≤ (1 − b
1
)
2
+ b
1
(b
2
+ b
3
+ ··· + b
n
)
= (1 − b
1
)
2
+ b
1
(1 − b
1
) = 1 − b
1
≤ 1 −
1
n
.
a
1
= 1, a
2
= a
3
= ··· = a
n
= 0
b
1
= b
2
= ··· = b
n
=
1
n
a, b, c ≥ 0
a
2
+ b
2
+ c
2
= 1
1 − ab
7 − 3ac
+
1 − bc
7 − 3ba
+
1 − ca
7 − 3cb
≥
1
3
1
7 − 3ab
+
1
7 − 3bc
+
1
7 − 3ca
≤
1
2
.
1
7 − 3ab
=
1
3(1 − ab) + 4
≤
1
9
1
3(1 − ab)
+ 1
.
1
7 − 3ab
≤
1
27
1
1 − ab
+
1
3
,
1
1 − ab
+
1
1 − bc
+
1
1 − ca
≤
9
2
,
3 − 3ab
7 − 3ac
+
3 − 3bc
7 − 3ba
+
3 − 3ca
7 − 3cb
≥ 1,
7 − 3ab
7 − 3ac
+
7 − 3bc
7 − 3ba
+
7 − 3ca
7 − 3cb
≥ 1 + 4
1
7 − 3ab
+
1
7 − 3bc
+
1
7 − 3ca
.
4
1
7 − 3ab
+
1
7 − 3bc
+
1
7 − 3ca
≤ 2
7 − 3ab
7 − 3ac
+
7 − 3bc
7 − 3ba
+
7 − 3ca
7 − 3cb
≥ 3,
a, b, c ≥ 0
a + b + c > 0
b + c ≥ 2a
x, y, z > 0
xyz = 1
1
a + x
2
(by + cz)
+
1
a + y
2
(bz + cx)
+
1
a + z
2
(bx + cy)
≥
3
a + b + c
u =
1
x
, v =
1
y
w =
1
z
uvw = 1
u
au + cv + bw
=
u
2
au
2
+ cuv + bwu
3
a + b + c
(u + v + w)
2
a
u
2
+ (b + c)
uv
3
a + b + c
1
2
· (b + c − 2a)
(x − y)
2
0
a, b, c
x, y, z > 0
xyz = 1
1
(1 + x
2
)(1 + x
7
)
+
1
(1 + y
2
)(1 + y
7
)
+
1
(1 + z
2
)(1 + z
7
)
≥
3
4
1
(1 + x
2
)(1 + x
7
)
≥
3
4(x
9
+ x
9
2
+ 1)
1
x
9
+ x
9
2
+ 1
+
1
y
9
+ y
9
2
+ 1
+
1
z
9
+ z
9
2
+ 1
≥ 1
xyz = 1
a, b, c
3(a
2
+ b
2
+ c
2
) + ab + bc + ca = 12
a
√
a + b
+
b
√
b + c
+
c
√
c + a
≤
3
√
2
.
A = a
2
+ b
2
+ c
2
, B = ab + bc + ca
2A + B = 2
a
2
+
ab ≤
3
4
3
a
2
+
ab
= 9.
a
√
a + b
=
√
a
a
a + b
≤
√
a + b + c
a
a + b
.
b
a + b
=
b
2
b(a + b)
≥
(a + b + c)
2
b(a + b)
=
A + 2B
A + B
a
a + b
= 3 −
b
a + b
≤ 3 −
A + 2B
A + B
=
2A + B
A + B
(a + b + c) ·
2A + B
A + B
≤
9
2
(a + b + c)
√
2A + B
=
(A + 2B) (2A + B)
≤
(A + 2B) + (2A + B)
2
=
3
2
(A + B)
⇒ (a + b + c) ·
2A + B
A + B
≤
3
2
√
2A + B ≤
9
2
3 ≤ a
2
+ b
2
+ c
2
≤ 4
LHS
2
= (
a
√
a + c
(a + b)(a + c)
) ≤ (a
2
+ b
2
+ c
2
+ ab + bc + ca)(
a
(a + b)(a + c)
)
9(a + b)(b + c)(c + a) ≥ 8(a + b + c)(ab + bc + ca)
a
(a + b)(a + c)
=
2(ab + bc + ca)
(a + b)(b + c)(c + a)
≤
9
4(a + b + c)
9(a
2
+ b
2
+ c
2
+ ab + bc + ca)
4(a + b + c)
≤
9
2
⇔
6 − (a
2
+ b
2
+ c
2
)
24 − 5(a
2
+ b
2
+ c
2
)
≤ 1
⇔ (6 − (a
2
+ b
2
+ c
2
))
2
≤ 24 − 5(a
2
+ b
2
+ c
2
)
⇔ (3 − (a
2
+ b
2
+ c
2
))(4 − (a
2
+ b
2
+ c
2
)) ≤ 0
a = b = c = 1
a, b, c ≥ 0
1
(a
2
+ bc)(b + c)
2
≤
8(a + b + c)
2
3(a + b)
2
(b + c)
2
(c + a)
2
a
2
(b + c)
2
a
2
+ bc
+
b
2
(c + a)
2
b
2
+ ca
+
c
2
(a + b)
2
c
2
+ ab
≤ a
2
+ b
2
+ c
2
+ ab + bc + ca.
a
2
(b + c)
2
a
2
+ bc
+
b
2
(c + a)
2
b
2
+ ca
+
c
2
(a + b)
2
c
2
+ ab
≤ a
2
+ b
2
+ c
2
+ ab + bc + ca
a, b, c ≥ 0
ab + bc + ca = 1
1
8
5
a
2
+ bc
+
1
8
5
b
2
+ ca
+
1
8
5
c
2
+ ab
≥
9
4
a ≥ b ≥ c
1
8
5
a
2
+ bc
−
5
8
+
1
8
5
b
2
+ ca
−
5
8
+
1
8
5
c
2
+ ab
− 1 ≥ 0
8 − 8a
2
− 5bc
8a
2
+ 5bc
+
8 − 8b
2
− 5ca
8b
2
+ 5ca
+
1 −
8
5
c
2
− ab
c
2
+
8
5
ab
≥ 0
8a(b + c − a) + 3bc
8a
2
+ 5bc
+
8b(a + c − b) + 5ac
8b
2
+ 5ca
+
c(a + b −
8
5
c)
c
2
+
8
5
ab
≥ 0
a ≥ b + c
8b
8b
2
+ 5ca
≥
8a
8a
2
+ 5bc
(a − b)(8ab − 5ac − 5bc) ≥ 0
a ≥ b + c
8ab = 5ab + 3ab ≥ 5ac + 6bc ≥ 5ac + 5ac
(a, b, c) = (1, 1, 0)
a, b, c ≥ 0
a
b
2
+ c
2
+
b
a
2
+ c
2
+
c
a
2
+ b
2
≥
a + b + c
ab + bc + ca
+
abc(a + b + c)
(a
3
+ b
3
+ c
3
)(ab + bc + ca)
a
b
2
+ c
2
=
a
2
ab
2
+ c
2
a
≥
(a + b + c)
2
(ab
2
+ c
2
a)
,
a + b + c
(ab
2
+ c
2
a)
≥
1
ab + bc + ca
+
abc
(ab + bc + ca) (a
3
+ b
3
+ c
3
)
,
a + b + c
(ab
2
+ c
2
a)
−
1
ab + bc + ca
=
3abc
(ab + bc + ca)
(ab
2
+ ca
2
)
,
3
a
3
+ b
3
+ c
3
≥
ab
2
+ c
2
a
,
2
a
3
+ b
3
+ c
3
≥
ab
2
+ c
2
a
.
a
b
2
+ c
2
+
b
c
2
+ a
2
+
c
a
2
+ b
2
≥
a + b + c
ab + bc + ca
+
3abc(a + b + c)
2(a
3
+ b
3
+ c
3
)(ab + bc + ca)
.
a, b, c ≥ 0
1
a
2
+ bc
+
1
b
2
+ ca
+
1
c
2
+ ab
≥
3
ab + bc + ca
+
81a
2
b
2
c
2
2(a
2
+ b
2
+ c
2
)
4
a = b = c, a = b, c = 0
(1)
1
a
2
+ bc
+
1
b
2
+ ca
+
1
c
2
+ ab
≥
3
a
2
+ b
2
+ c
2
a
3
b + ab
3
+ b
3
c + bc
3
+ c
3
a + ca
3
(2)
3
a
2
+ b
2
+ c
2
a
3
b + ab
3
+ b
3
c + bc
3
+ c
3
a + ca
3
≥
3
ab + bc + ca
+
81a
2
b
2
c
2
2(a
2
+ b
2
+ c
2
)
4
.
a
2
(a
3
b + ab
3
)
−
1
ab + bc + ca
=
abc(a + b + c)
(ab + bc + ca) (
(a
3
b + ab
3
))
,
2(a + b + c)
a
2
+ b
2
+ c
2
4
≥ 27abc(ab + bc + ca)
a
3
b + ab
3
,
(a) (a + b + c)
a
2
+ b
2
+ c
2
≥ 9abc,
(b) a
2
+ b
2
+ c
2
≥ ab + bc + ca,
(c) 2
a
2
+ b
2
+ c
2
2
≥ 3
a
3
b + ab
3
,
(c)
a
2
− ab + b
2
(a − b)
2
≥ 0,
a, b, c
a
2
(b + c)
b
2
+ bc + c
2
+
b
2
(c + a)
c
2
+ ca + a
2
+
c
2
(a + b)
a
2
+ ab + b
2
2(a
2
+ b
2
+ c
2
)
a + b + c
.
a
2
(b + c)
b
2
+ bc + c
2
=
4a
2
(b + c)(ab + bc + ca)
(b
2
+ bc + c
2
) (ab + bc + ca)
≥
4a
2
(b + c)(ab + bc + ca)
(b
2
+ bc + c
2
+ ab + bc + ca)
2
=
4a
2
(ab + bc + ca)
(b + c)(a + b + c)
2
,
a
2
b + c
≥
(a + b + c)
a
2
+ b
2
+ c
2
2(ab + bc + ca)
,
a
2
b + c
+ a
≥
(a + b + c)
3
2(ab + bc + ca)
,
a
b + c
≥
(a + b + c)
2
2(ab + bc + ca)
,
a
b + c
=
a
2
a(b + c)
≥
(a + b + c)
2
2(ab + bc + ca)
.
a
2
(b + c)
b
2
+ bc + c
2
+
a(b + c)
a + b + c
=
a(b + c)(a
2
+ b
2
+ c
2
+ ab + bc + ca)
(b
2
+ bc + c
2
)(a + b + c)
,
2(a
2
+ b
2
+ c
2
)
a + b + c
+
a(b + c)
a + b + c
=
2(a
2
+ b
2
+ c
2
+ ab + bc + ca)
a + b + c
.
a(b + c)
b
2
+ bc + c
2
+
b(c + a)
c
2
+ ca + a
2
+
c(a + b)
a
2
+ ab + b
2
≥ 2,
a(b + c)
b
2
+ bc + c
2
=
4a(b + c)(ab + bc + ca)
4(b
2
+ bc + c
2
)(ab + bc + ca)
4a(ab + bc + ca)
(b + c)(a + b + c)
2
.
4a(ab + bc + ca)
(b + c)(a + b + c)
2
2,
a
b + c
(a + b + c)
2
2(ab + bc + ca)
,
(a
2
+ ac + c
2
)(b
2
+ bc + c
2
) ≤
ab(a + b) + bc(b + c) + ca(c + a)
a + b
.(1)
√
ac ·
√
bc +
a
2
+ ac + c
2
·
b
2
+ bc + c
2
≤
(ac + a
2
+ ac + c
2
)(bc + b
2
+ bc + c
2
)
= (a + c)(b + c).
(a
2
+ ac + c
2
)(b
2
+ bc + c
2
) ≤ ab + c
2
+ c
a + b −
√
ab
≤ ab + c
2
+ c
a + b −
2ab
a + b
=
ab(a + b) + bc(b + c) + ca(c + a)
a + b
.
1
a
2
+ ac + c
2
+
1
b
2
+ bc + c
2
≥
2
(a
2
+ ac + c
2
)(b
2
+ bc + c
2
)
≥
2(a + b)
ab(a + b) + bc(b + c) + ca(c + a)
.
(2)
(2)
a(b + c)
b
2
+ bc + c
2
=
ab
1
a
2
+ ac + c
2
+
1
b
2
+ bc + c
2
≥
2ab(a + b)
ab(a + b) + bc(b + c) + ca(c + a)
= 2.
a, b, c > 0
a
2
(b + c)
b
2
+ bc + c
2
+
b
2
(c + a)
c
2
+ ca + a
2
+
c
2
(a + b)
a
2
+ ab + b
2
≥ 2
a
3
+ b
3
+ c
3
a + b + c
a
2
(b + c)(a + b + c)
b
2
+ bc + c
2
≥ 2
(a
3
+ b
3
+ c
3
) (a + b + c)
a
2
+
a
2
(ab + bc + ca)
b
2
+ bc + c
2
≥ 2
(a
3
+ b
3
+ c
3
) (a + b + c),
2
(a
3
+ b
3
+ c
3
) (a + b + c) ≤
a
2
+ b
2
+ c
2
+
a
3
+ b
3
+ c
3
(a + b + c)
a
2
+ b
2
+ c
2
,
a
2
b
2
+ bc + c
2
≥
a
3
+ b
3
+ c
3
(a + b + c)
(a
2
+ b
2
+ c
2
) (ab + bc + ca)
,
a
2
b
2
+ bc + c
2
≥
a
2
+ b
2
+ c
2
2
a
2
(b
2
+ bc + c
2
)
=
a
2
+ b
2
+ c
2
2
2
a
2
b
2
+
a
2
bc
,
a
2
+ b
2
+ c
2
3
(ab + bc + ca) ≥
a
3
+ b
3
+ c
3
(a + b + c)
2
a
2
b
2
+
a
2
bc
.
A =
a
4
, B =
1
2
a
3
b + ab
3
, C =
a
2
b
2
, D =
a
2
bc,
a
2
+ b
2
+ c
2
2
= A + 2C,
a
2
+ b
2
+ c
2
(ab + bc + ca) = 2B + D,
a
3
+ b
3
+ c
3
(a + b + c) = A + 2B,
2
a
2
b
2
+
a
2
bc = 2C + D.
(A + 2C) (2B + D) ≥ (A + 2B) (2C + D) ,
2 (A − D)(B − C) ≥ 0,
A ≥ D
B ≥ C
a, b, c ≥ 0
a + b + c = 1
2
a
2
b + b
2
c + c
2
a + ab + bc + ca ≤ 1
2
a
2
b + b
2
c + c
2
a + ab + bc + ca ≤ (a + b + c)
2
2
(a
2
b + b
2
c + c
2
a) (a + b + c) ≤ a
2
+ b
2
+ c
2
+ ab + bc + ca
2
(a
2
b + b
2
c + c
2
a) (a + b + c) ≤
a
2
b + b
2
c + c
2
a
b
+ b(a + b + c)
a
2
+ b
2
+ c
2
+ ab + bc + ca ≥
a
2
b + b
2
c + c
2
a
b
+ b(a + b + c)
c(a − b)(b − c)
b
≥ 0,
b
a
4
+ 2
a
3
c ≥
a
2
b
2
+ 2
a
3
b
(a − b)
2
(a + b)
2
≥ 4(a − b)(b − c)(a − c)(a + b + c)
a ≥ b ≥ c, a − b = x, b − c = y
x
2
(2c + 2y + x)
2
+ y
2
(2c + y)
2
+ (x + y)
2
(2c + x + y)
2
≥ xy(x + y)(3c + 2x + y)
(x + y)
4
≥ xy(x + y)(x + 2y)
(x + y)
3
≥ 3xy(x + y)
a, b, c
a
2
b
2
+ b
2
c
2
+ c
2
a
2
≥ a
2
b
2
c
2
A =
a
2
b
2
c
3
(a
2
+ b
2
)
+
b
2
c
2
a
3
(b
2
+ c
2
)
+
c
2
a
2
b
3
(c
2
+ a
2
)
x =
1
a
, y =
1
b
, z =
1
c
x
2
+ y
2
+ z
2
≥ 1
x
3
y
2
+ z
2
+
y
3
x
2
+ z
2
+
z
3
x
2
+ y
2
≥
√
3
2
LHS ≥
(x
2
+ y
2
+ z
2
)
2
x(y
2
+ z
2
) + y(x
2
+ z
2
) + z(x
2
+ y
2
)
x(y
2
+z
2
)+y(x
2
+z
2
)+z(x
2
+y
2
) ≤
2
3
(x
2
+y
2
+z
2
)(x+y+z) ≤
2
√
3
(x
2
+y
2
+z
2
)
x
2
+ y
2
+ z
2
x
2
+ y
2
+ z
2
≥ 1
(x
2
+ y
2
+ z
2
)
2
2
√
3
(x
2
+ y
2
+ z
2
)
x
2
+ y
2
+ z
2
≥
√
3
2
x
2
+ y
2
+ z
2
= 1
f = x + y + z − xyz.
m = x+y = x+
√
1 − x
2
− z
2
= g(x)(0 ≤ x ≤
√
1 − z
2
),
g
(x) = 1 −
x
√
1 − x
2
− z
2
,
g
(x) > 0 ⇔ 0 ≤ x <
1 − z
2
2
g
(x) < 0 ⇔
1 − z
2
2
< x ≤
1 − z
2
,
m
min
= min{g(0), g(
1 − z
2
)} =
1 − z
2
m
max
= g
1 − z
2
2
=
2 − 2z
2
.
f = f(m) = −
z
2
m
2
+ m +
1 − z
2
z
2
+ z,
m =
1
z
>
2 − 2z
2
