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Balkan MO Shortlist
2008 Balkan MO Shortlist
N5
N5
Part of
2008 Balkan MO Shortlist
Problems
(1)
Infinitely many n exist such that b divides a_n
Source: Balkan MO ShortList 2008 N5
4/5/2020
Let
(
a
n
)
(a_n)
(
a
n
)
be a sequence with
a
1
=
0
a_1=0
a
1
=
0
and
a
n
+
1
=
2
+
a
n
a_{n+1}=2+a_n
a
n
+
1
=
2
+
a
n
for odd
n
n
n
and
a
n
+
1
=
2
a
n
a_{n+1}=2a_n
a
n
+
1
=
2
a
n
for even
n
n
n
. Prove that for each prime
p
>
3
p>3
p
>
3
, the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of
n
n
n