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Balkan MO Shortlist
2014 Balkan MO Shortlist
A5
BMO 2014 SL A5
BMO 2014 SL A5
Source: Balkan MO 2014 Shortlist
October 1, 2016
Sequence
Integer sequence
algebra
Problem Statement
A
5
\boxed{A5}
A
5
Let
n
∈
N
,
n
>
2
n\in{N},n>2
n
∈
N
,
n
>
2
,and suppose
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1,a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
is a permutation of the numbers
1
,
2
,
.
.
.
,
2
n
1,2,...,2n
1
,
2
,
...
,
2
n
such that
a
1
<
a
3
<
.
.
.
<
a
2
n
−
1
a_1<a_3<...<a_{2n-1}
a
1
<
a
3
<
...
<
a
2
n
−
1
and
a
2
>
a
4
>
.
.
.
>
a
2
n
.
a_2>a_4>...>a_{2n}.
a
2
>
a
4
>
...
>
a
2
n
.
Prove that
(
a
1
−
a
2
)
2
+
(
a
3
−
a
4
)
2
+
.
.
.
+
(
a
2
n
−
1
−
a
2
n
)
2
>
n
3
(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3
(
a
1
−
a
2
)
2
+
(
a
3
−
a
4
)
2
+
...
+
(
a
2
n
−
1
−
a
2
n
)
2
>
n
3
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