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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 22
Prove this inequality
Prove this inequality
Source: 2009 Jozse Wildt International Mathematical Competition
April 26, 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
), then
(
a
1
a
2
)
k
+
(
a
2
a
3
)
k
+
⋯
+
(
a
n
a
1
)
k
≥
a
1
a
2
+
a
2
a
3
+
⋯
+
a
n
a
1
\left (\frac{a_1}{a_2} \right )^k + \left (\frac{a_2}{a_3} \right )^k + \cdots + \left (\frac{a_n}{a_1} \right )^k \geq \frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots + \frac{a_n}{a_1}
(
a
2
a
1
)
k
+
(
a
3
a
2
)
k
+
⋯
+
(
a
1
a
n
)
k
≥
a
2
a
1
+
a
3
a
2
+
⋯
+
a
1
a
n
for all
k
∈
N
k\in \mathbb{N}
k
∈
N
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