MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2003 China Team Selection Test
3
Hard inequality
Hard inequality
Source: China TST 2003 Quizzes
April 1, 2006
inequalities
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
be positive real number
(
n
≥
2
)
(n \geq 2)
(
n
≥
2
)
,not all equal,such that
∑
k
=
1
n
a
k
−
2
n
=
1
\sum_{k=1}^n a_{k}^{-2n}=1
∑
k
=
1
n
a
k
−
2
n
=
1
,prove that:
∑
k
=
1
n
a
k
2
n
−
n
2
.
∑
1
≤
i
<
j
≤
n
(
a
i
a
j
−
a
j
a
i
)
2
>
n
2
\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2
∑
k
=
1
n
a
k
2
n
−
n
2
.
∑
1
≤
i
<
j
≤
n
(
a
j
a
i
−
a
i
a
j
)
2
>
n
2
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