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Turkey MO (2nd round)
2010 Turkey MO (2nd round)
3
Turkey NMO 2010 P3
Turkey NMO 2010 P3
Source:
December 15, 2010
inequalities proposed
inequalities
Problem Statement
Prove that for all
n
∈
Z
+
n \in \mathbb{Z^+}
n
∈
Z
+
and for all positive real numbers satisfying
a
1
a
2
.
.
.
a
n
=
1
a_1a_2...a_n=1
a
1
a
2
...
a
n
=
1
∑
i
=
1
n
a
i
a
i
4
+
3
≤
1
2
∑
i
=
1
n
1
a
i
\displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i}
i
=
1
∑
n
a
i
4
+
3
a
i
≤
2
1
i
=
1
∑
n
a
i
1
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