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National and Regional Contests
China Contests
China Team Selection Test
2006 China Team Selection Test
2
N variables inequality
N variables inequality
Source: 2006 china tst
May 19, 2006
inequalities
function
Cauchy Inequality
inequalities proposed
Problem Statement
x
1
,
x
2
,
⋯
,
x
n
x_{1}, x_{2}, \cdots, x_{n}
x
1
,
x
2
,
⋯
,
x
n
are positive numbers such that
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^{n}x_{i}= 1
∑
i
=
1
n
x
i
=
1
. Prove that
(
∑
i
=
1
n
x
i
)
(
∑
i
=
1
n
1
1
+
x
i
)
≤
n
2
n
+
1
\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}
(
i
=
1
∑
n
x
i
)
(
i
=
1
∑
n
1
+
x
i
1
)
≤
n
+
1
n
2
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