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239 Open Math Olympiad
2001 239 Open Mathematical Olympiad
2
Inequality with geometric mean and 1+\frac{1}{x(1+x)}
Inequality with geometric mean and 1+\frac{1}{x(1+x)}
Source: 239 2001 J7 S2
May 19, 2020
inequalities
Problem Statement
For any positive numbers
a
1
,
a
2
,
…
,
a
n
a_1 , a_2 , \dots, a_n
a
1
,
a
2
,
…
,
a
n
prove the inequality
(
1
+
1
a
1
(
1
+
a
1
)
)
(
1
+
1
a
2
(
1
+
a
2
)
)
…
(
1
+
1
a
n
(
1
+
a
n
)
)
≥
(
1
+
1
p
(
1
+
p
)
)
n
,
\! \left(\!1\!+\!\frac{1}{a_1(1+a_1)} \!\right)\! \left(\!1\!+\!\frac{1}{a_2(1+a_2)} \! \right) \! \dots \! \left(\!1\!+\!\frac{1}{a_n(1+a_n)} \! \right) \geq \left(\!1\!+\!\frac{1}{p(1+p)} \! \right)^{\! n} \! ,
(
1
+
a
1
(
1
+
a
1
)
1
)
(
1
+
a
2
(
1
+
a
2
)
1
)
…
(
1
+
a
n
(
1
+
a
n
)
1
)
≥
(
1
+
p
(
1
+
p
)
1
)
n
,
where
p
=
a
1
a
2
…
a
n
n
p=\sqrt[n]{a_1 a_2 \dots a_n}
p
=
n
a
1
a
2
…
a
n
.
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