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IMO Shortlist
1966 IMO Shortlist
13
Prove the inequality on fractions
Prove the inequality on fractions
Source:
September 27, 2010
inequalities
n-variable inequality
IMO Shortlist
IMO Longlist
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers. Prove the inequality
(
n
2
)
∑
i
<
j
1
a
i
a
j
≥
4
(
∑
i
<
j
1
a
i
+
a
j
)
2
\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2
(
2
n
)
i
<
j
∑
a
i
a
j
1
≥
4
(
i
<
j
∑
a
i
+
a
j
1
)
2
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