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Baltic Way
2015 Baltic Way
2
Algebra .Inequality
Algebra .Inequality
Source: Baltic Way 2015
November 8, 2015
inequalities
Baltic Way
n-variable inequality
Problem Statement
Let
n
n
n
be a positive integer and let
a
1
,
⋯
,
a
n
a_1,\cdots ,a_n
a
1
,
⋯
,
a
n
be real numbers satisfying
0
≤
a
i
≤
1
0\le a_i\le 1
0
≤
a
i
≤
1
for
i
=
1
,
⋯
,
n
.
i=1,\cdots ,n.
i
=
1
,
⋯
,
n
.
Prove the inequality
(
1
−
a
i
n
)
(
1
−
a
2
n
)
⋯
(
1
−
a
n
n
)
≤
(
1
−
a
1
a
2
⋯
a
n
)
n
.
(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.
(
1
−
a
i
n
)
(
1
−
a
2
n
)
⋯
(
1
−
a
n
n
)
≤
(
1
−
a
1
a
2
⋯
a
n
)
n
.
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