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APMO
2024 APMO
3
Mount Inequality erupts again
Mount Inequality erupts again
Source: APMO 2024 P3
July 29, 2024
algebra
APMO 2024
Problem Statement
Let
n
n
n
be a positive integer and let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be positive reals. Show that
∑
i
=
1
n
1
2
i
(
2
1
+
a
i
)
2
i
≥
2
1
+
a
1
a
2
…
a
n
−
1
2
n
.
\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.
i
=
1
∑
n
2
i
1
(
1
+
a
i
2
)
2
i
≥
1
+
a
1
a
2
…
a
n
2
−
2
n
1
.
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