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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 26
Prove this hard inequality
Prove this hard inequality
Source: 2009 Jozsef Wildt International Math Competition
April 27, 2020
inequalities
Problem Statement
If
a
i
>
0
a_i >0
a
i
>
0
(
i
=
1
,
2
,
⋯
,
n
i=1, 2, \cdots , n
i
=
1
,
2
,
⋯
,
n
) and
∑
i
=
1
n
a
i
k
=
1
\sum \limits_{i=1}^n a_i^k=1
i
=
1
∑
n
a
i
k
=
1
, where
1
≤
k
≤
n
+
1
1\leq k\leq n+1
1
≤
k
≤
n
+
1
, then
∑
i
=
1
n
a
i
+
1
∏
i
=
1
n
a
i
≥
n
1
−
1
k
+
n
n
k
\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}
i
=
1
∑
n
a
i
+
i
=
1
∏
n
a
i
1
≥
n
1
−
k
1
+
n
k
n
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