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Poland Contests
Poland - Second Round
1971 Poland - Second Round
6
(a_1+a_3+\ldots a_{2n+1})/(n+1) >= (a_2+a_4+\ldots a_{2n})/ n
(a_1+a_3+\ldots a_{2n+1})/(n+1) >= (a_2+a_4+\ldots a_{2n})/ n
Source: Polish MO Second Round 1971 p6
September 8, 2024
algebra
inequalities
Problem Statement
Given an infinite sequence
{
a
n
}
\{a_n\}
{
a
n
}
. Prove that if
a
n
+
a
n
+
2
>
2
a
n
+
1
f
o
r
n
=
1
,
2...
a_n + a_{n+2} > 2a_{n+1} \ \ for \ \ n = 1, 2 ...
a
n
+
a
n
+
2
>
2
a
n
+
1
f
or
n
=
1
,
2...
then
a
1
+
a
3
+
…
a
2
n
+
1
n
+
1
≥
a
2
+
a
4
+
…
a
2
n
n
\frac{a_1+a_3+\ldots a_{2n+1}}{n+1} \geq \frac{a_2+a_4+\ldots a_{2n}}{n}
n
+
1
a
1
+
a
3
+
…
a
2
n
+
1
≥
n
a
2
+
a
4
+
…
a
2
n
for
n
=
1
,
2
,
…
n = 1, 2, \ldots
n
=
1
,
2
,
…
.
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