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Mediterranean Mathematics Olympiad
2018 Mediterranean Mathematics OIympiad
1
Inequality with sin
Inequality with sin
Source: Mediterranean math olympiad 2018
June 5, 2018
inequalities
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be more than one real numbers, such that
0
≤
a
i
≤
π
2
0\leq a_i\leq \frac{\pi}{2}
0
≤
a
i
≤
2
π
. Prove that
(
1
n
∑
i
=
1
n
1
1
+
sin
a
i
)
(
1
+
∏
i
=
1
n
(
sin
a
i
)
1
n
)
≤
1.
\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.
(
n
1
i
=
1
∑
n
1
+
sin
a
i
1
)
(
1
+
i
=
1
∏
n
(
sin
a
i
)
n
1
)
≤
1.
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