MathDB
Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2011 China Western Mathematical Olympiad
2
Cauchy-like ineq
Cauchy-like ineq
Source: CWMO 2011 Q6
May 22, 2012
trigonometry
inequalities
inequalities unsolved
Problem Statement
Let
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
, prove that
(
a
−
b
)
2
(
c
+
a
)
(
c
+
b
)
+
(
b
−
c
)
2
(
a
+
b
)
(
a
+
c
)
+
(
c
−
a
)
2
(
b
+
c
)
(
b
+
a
)
≥
(
a
−
b
)
2
a
2
+
b
2
+
c
2
\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}
(
c
+
a
)
(
c
+
b
)
(
a
−
b
)
2
+
(
a
+
b
)
(
a
+
c
)
(
b
−
c
)
2
+
(
b
+
c
)
(
b
+
a
)
(
c
−
a
)
2
≥
a
2
+
b
2
+
c
2
(
a
−
b
)
2
Back to Problems
View on AoPS