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Macedonia National Olympiad
2016 Macedonia National Olympiad
Problem 5
Macedonia National Olympiad 2016 Problem 5
Macedonia National Olympiad 2016 Problem 5
Source: Macedonia National Olympiad 2016
April 9, 2016
inequalities
Macedonia
Problem Statement
Let
n
≥
3
n\ge3
n
≥
3
and
a
1
,
a
2
,
.
.
.
,
a
n
∈
R
+
a_1,a_2,...,a_n \in \mathbb{R^{+}}
a
1
,
a
2
,
...
,
a
n
∈
R
+
, such that
1
1
+
a
1
4
+
1
1
+
a
2
4
+
.
.
.
+
1
1
+
a
n
4
=
1
\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1
1
+
a
1
4
1
+
1
+
a
2
4
1
+
...
+
1
+
a
n
4
1
=
1
. Prove that:
a
1
a
2
.
.
.
a
n
≥
(
n
−
1
)
n
4
a_1a_2...a_n \ge (n-1)^{\frac n4}
a
1
a
2
...
a
n
≥
(
n
−
1
)
4
n
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