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China Contests
China Team Selection Test
2010 China Team Selection Test
2
China 2010 quiz4 problem 2
China 2010 quiz4 problem 2
Source:
September 12, 2010
inequalities
inequalities unsolved
Problem Statement
Find all positive real numbers
λ
\lambda
λ
such that for all integers
n
≥
2
n\geq 2
n
≥
2
and all positive real numbers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
with
a
1
+
a
2
+
⋯
+
a
n
=
n
a_1+a_2+\cdots+a_n=n
a
1
+
a
2
+
⋯
+
a
n
=
n
, the following inequality holds:
∑
i
=
1
n
1
a
i
−
λ
∏
i
=
1
n
1
a
i
≤
n
−
λ
\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda
∑
i
=
1
n
a
i
1
−
λ
∏
i
=
1
n
a
i
1
≤
n
−
λ
.
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