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International Contests
Silk Road
2011 Silk Road
3
Ineq
Ineq
Source:
August 27, 2014
inequalities
trigonometry
Problem Statement
For all a,b,c\in \bb{R}^+ such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
and
(
1
(
a
+
b
)
2
+
1
(
b
+
c
)
2
+
1
(
c
+
a
)
2
)
(
a
−
b
c
)
(
b
−
a
c
)
(
c
−
a
b
)
≤
M
⋅
a
b
c
( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc
(
(
a
+
b
)
2
1
+
(
b
+
c
)
2
1
+
(
c
+
a
)
2
1
)
(
a
−
b
c
)
(
b
−
a
c
)
(
c
−
ab
)
≤
M
⋅
ab
c
. Find min
M
M
M
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