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International Contests
Baltic Way
2001 Baltic Way
12
12
Part of
2001 Baltic Way
Problems
(1)
Sum of third powers equals 3, sum of fifth powers equal 5
Source: Baltic Way 2001
11/17/2010
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\ldots , a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers such that
∑
i
=
1
n
a
i
3
=
3
\sum_{i=1}^na_i^3=3
∑
i
=
1
n
a
i
3
=
3
and
∑
i
=
1
n
a
i
5
=
5
\sum_{i=1}^na_i^5=5
∑
i
=
1
n
a
i
5
=
5
. Prove that
∑
i
=
1
n
a
i
>
3
2
\sum_{i=1}^na_i>\frac{3}{2}
∑
i
=
1
n
a
i
>
2
3
.
inequalities proposed
inequalities