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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 13
Prove this inequality
Prove this inequality
Source: 2009 Jozsef Wildt International Math Competition
April 22, 2020
inequalities
Problem Statement
If
a
k
>
0
a_k >0
a
k
>
0
[
k
=
k=
k
=
1, 2,
⋯
\cdots
⋯
,
n
n
n
], then prove the following inequality
(
∑
k
=
1
n
a
k
5
)
4
≥
1
n
(
2
n
−
1
)
5
(
∑
1
≤
i
<
j
≤
n
a
i
2
a
j
2
)
5
\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5
(
k
=
1
∑
n
a
k
5
)
4
≥
n
1
(
n
−
1
2
)
5
(
1
≤
i
<
j
≤
n
∑
a
i
2
a
j
2
)
5
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