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IMO Shortlist
2009 IMO Shortlist
2
IMO Shortlist 2009 - Problem A2
IMO Shortlist 2009 - Problem A2
Source:
July 5, 2010
IMO Shortlist
three variable inequality
Inequality
inequalities
IMO 2009
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers such that
1
a
+
1
b
+
1
c
=
a
+
b
+
c
\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c
a
1
+
b
1
+
c
1
=
a
+
b
+
c
. Prove that:
1
(
2
a
+
b
+
c
)
2
+
1
(
a
+
2
b
+
c
)
2
+
1
(
a
+
b
+
2
c
)
2
≤
3
16
.
\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.
(
2
a
+
b
+
c
)
2
1
+
(
a
+
2
b
+
c
)
2
1
+
(
a
+
b
+
2
c
)
2
1
≤
16
3
.
Proposed by Juhan Aru, Estonia
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