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International Contests
Austrian-Polish
1979 Austrian-Polish Competition
3
3
Part of
1979 Austrian-Polish Competition
Problems
(1)
Find all positive integers n
Source: 1979 Austrian-Polish Mathematical Competition p3
2/15/2020
Find all positive integers
n
n
n
such that the inequality
(
∑
i
=
1
n
a
i
2
)
(
∑
i
=
1
n
a
i
)
−
∑
i
=
1
n
a
i
3
≥
6
∏
i
=
1
n
a
i
\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i
(
i
=
1
∑
n
a
i
2
)
(
i
=
1
∑
n
a
i
)
−
i
=
1
∑
n
a
i
3
≥
6
i
=
1
∏
n
a
i
holds for any
n
n
n
positive numbers
a
1
,
…
,
a
n
a_1, \dots, a_n
a
1
,
…
,
a
n
.
inequalities