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National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2015 Iran MO (3rd round)
6
iran inquality
iran inquality
Source: Iranian third round 2015 algebra problem 6
September 8, 2015
inequalities
algebra
Problem Statement
a
1
,
a
2
,
…
,
a
n
>
0
a_1,a_2,\dots ,a_n>0
a
1
,
a
2
,
…
,
a
n
>
0
are positive real numbers such that
∑
i
=
1
n
1
a
i
=
n
\sum_{i=1}^{n} \frac{1}{a_i}=n
∑
i
=
1
n
a
i
1
=
n
prove that:
∑
i
<
j
(
a
i
−
a
j
a
i
+
a
j
)
2
≤
n
2
2
(
1
−
n
∑
i
=
1
n
a
i
)
\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)
∑
i
<
j
(
a
i
+
a
j
a
i
−
a
j
)
2
≤
2
n
2
(
1
−
∑
i
=
1
n
a
i
n
)
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