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National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2012 Kyrgyzstan National Olympiad
2
Cyclic product of $1/a_i^2-1$
Cyclic product of $1/a_i^2-1$
Source: Kyrgyzstan 2012, Problem 2
May 2, 2013
inequalities
inequalities unsolved
Problem Statement
Given positive real numbers
a
1
,
a
2
,
.
.
.
,
a
n
{a_1},{a_2},...,{a_n}
a
1
,
a
2
,
...
,
a
n
with
a
1
+
a
2
+
.
.
.
+
a
n
=
1
{a_1}+{a_2}+...+{a_n}= 1
a
1
+
a
2
+
...
+
a
n
=
1
. Prove that
(
1
a
1
2
−
1
)
(
1
a
2
2
−
1
)
.
.
.
(
1
a
n
2
−
1
)
⩾
(
n
2
−
1
)
n
\left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n}
(
a
1
2
1
−
1
)
(
a
2
2
1
−
1
)
...
(
a
n
2
1
−
1
)
⩾
(
n
2
−
1
)
n
.
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