MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1992 China Team Selection Test
3
China TST 1992 inequality
China TST 1992 inequality
Source: China TST 1992, problem 6
June 27, 2005
inequalities
inequalities unsolved
Problem Statement
For any
n
,
T
≥
2
,
n
,
T
∈
N
n,T \geq 2, n, T \in \mathbb{N}
n
,
T
≥
2
,
n
,
T
∈
N
, find all
a
∈
N
a \in \mathbb{N}
a
∈
N
such that
∀
a
i
>
0
,
i
=
1
,
2
,
…
,
n
\forall a_i > 0, i = 1, 2, \ldots, n
∀
a
i
>
0
,
i
=
1
,
2
,
…
,
n
, we have
∑
k
=
1
n
a
⋅
k
+
a
2
4
S
k
<
T
2
⋅
∑
k
=
1
n
1
a
k
,
\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},
k
=
1
∑
n
S
k
a
⋅
k
+
4
a
2
<
T
2
⋅
k
=
1
∑
n
a
k
1
,
where
S
k
=
∑
i
=
1
k
a
i
.
S_k = \sum^k_{i=1} a_i.
S
k
=
∑
i
=
1
k
a
i
.
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