MathDB
Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2011 Kyrgyzstan National Olympiad
3
Square difference(KgNM2011)
Square difference(KgNM2011)
Source:
April 29, 2011
inequalities
inequalities unsolved
Problem Statement
Given positive 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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