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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 19
Prove this inequality
Prove this inequality
Source: 2009 Jozsef Wildt International Math Competition
April 26, 2020
inequalities
Problem Statement
If
x
k
>
0
x_k >0
x
k
>
0
(
k
=
1
,
2
,
⋯
,
n
k=1, 2, \cdots , n
k
=
1
,
2
,
⋯
,
n
), then
∑
k
=
1
n
(
x
k
1
+
x
1
2
+
x
2
2
+
⋯
+
x
k
2
)
2
≤
∑
k
=
1
n
x
k
2
1
+
∑
k
=
1
n
x
k
2
\sum \limits_{k=1}^n \left ( \frac{x_k}{1+x_1^2+x_2^2+\cdots +x_k^2} \right )^2 \leq \frac{\sum \limits_{k=1}^n x_k^2}{1+\sum \limits_{k=1}^n x_k^2}
k
=
1
∑
n
(
1
+
x
1
2
+
x
2
2
+
⋯
+
x
k
2
x
k
)
2
≤
1
+
k
=
1
∑
n
x
k
2
k
=
1
∑
n
x
k
2
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