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Poland - Second Round
2009 Poland - Second Round
1
Inequality with products of a_1,a_2, ... a_n
Inequality with products of a_1,a_2, ... a_n
Source: Polish Second Round 2009
August 4, 2011
inequalities
induction
function
inequalities proposed
Problem Statement
Let
a
1
≥
a
2
≥
…
≥
a
n
>
0
a_1\ge a_2\ge \ldots \ge a_n>0
a
1
≥
a
2
≥
…
≥
a
n
>
0
be
n
n
n
reals. Prove the inequality
a
1
a
2
…
a
n
−
1
+
(
2
a
2
−
a
1
)
(
2
a
3
−
a
2
)
…
(
2
a
n
−
a
n
−
1
)
≥
2
a
2
a
3
…
a
n
a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n
a
1
a
2
…
a
n
−
1
+
(
2
a
2
−
a
1
)
(
2
a
3
−
a
2
)
…
(
2
a
n
−
a
n
−
1
)
≥
2
a
2
a
3
…
a
n
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