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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 23
Prove this inequality
Prove this inequality
Source: 2009 Jozsef Wildt International Math Competition
April 27, 2020
inequalities
Problem Statement
If
x
k
∈
R
x_k \in \mathbb{R}
x
k
∈
R
(
k
=
1
,
2
,
⋯
,
n
k=1, 2, \cdots , n
k
=
1
,
2
,
⋯
,
n
) and
m
∈
N
m \in \mathbb{N}
m
∈
N
then[*]
∑
c
y
c
(
x
1
2
−
x
1
x
2
+
x
2
2
)
m
≤
3
m
∑
k
=
1
n
x
k
2
m
\sum \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq 3^m \sum \limits_{k=1}^n x_k^{2m}
cyc
∑
(
x
1
2
−
x
1
x
2
+
x
2
2
)
m
≤
3
m
k
=
1
∑
n
x
k
2
m
[*]
∏
c
y
c
(
x
1
2
−
x
1
x
2
+
x
2
2
)
m
≤
(
3
m
n
)
m
(
∑
k
=
1
n
x
k
2
m
)
n
\prod \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq \left (\frac{3^m}{n}\right )^m \left (\sum \limits_{k=1}^n x_k^{2m}\right )^n
cyc
∏
(
x
1
2
−
x
1
x
2
+
x
2
2
)
m
≤
(
n
3
m
)
m
(
k
=
1
∑
n
x
k
2
m
)
n
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