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239 Open Math Olympiad
2010 239 Open Mathematical Olympiad
6
Six positive numbers with product of 1
Six positive numbers with product of 1
Source: 239 2010 S6
July 29, 2020
algebra
inequalities
Problem Statement
We have six positive numbers
a
1
,
a
2
,
…
,
a
6
a_1, a_2, \ldots , a_6
a
1
,
a
2
,
…
,
a
6
such that
a
1
a
2
…
a
6
=
1
a_1a_2\ldots a_6 =1
a
1
a
2
…
a
6
=
1
. Prove that:
1
a
1
(
a
2
+
1
)
+
1
a
2
(
a
3
+
1
)
+
…
+
1
a
6
(
a
1
+
1
)
≥
3.
\frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.
a
1
(
a
2
+
1
)
1
+
a
2
(
a
3
+
1
)
1
+
…
+
a
6
(
a
1
+
1
)
1
≥
3.
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