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Balkan MO Shortlist
2008 Balkan MO Shortlist
N6
N6
Part of
2008 Balkan MO Shortlist
Problems
(1)
determine n such that terms of a_n are perfect squares
Source: Balkan MO ShortList 2008 N6
4/5/2020
Let
(
x
n
)
(x_n)
(
x
n
)
,
n
=
1
,
2
,
…
n=1,2, \ldots
n
=
1
,
2
,
…
be a sequence defined by
x
1
=
2008
x_1=2008
x
1
=
2008
and \begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*} Let the sequence
a
n
=
x
n
+
1
n
S
n
a_n=x_n + \frac{1}{n} S_n
a
n
=
x
n
+
n
1
S
n
,
n
=
1
,
2
,
3
,
…
n=1,2,3, \ldots
n
=
1
,
2
,
3
,
…
where
S
n
S_n
S
n
=
=
=
x
1
+
x
2
+
…
+
x
n
x_1+x_2 +\ldots +x_n
x
1
+
x
2
+
…
+
x
n
. Determine the values of
n
n
n
for which the terms of the sequence
a
n
a_n
a
n
are perfect squares of an integer.