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Balkan MO Shortlist
2014 Balkan MO Shortlist
A6
A6
Part of
2014 Balkan MO Shortlist
Problems
(1)
BMO 2014 SL A6
Source: Balkan MO 2014 Shortlist
10/1/2016
A
6
\boxed{A6}
A
6
The sequence
a
0
,
a
1
,
.
.
.
a_0,a_1,...
a
0
,
a
1
,
...
is defined by the initial conditions
a
0
=
1
,
a
1
=
6
a_0=1,a_1=6
a
0
=
1
,
a
1
=
6
and the recursion
a
n
+
1
=
4
a
n
−
a
n
−
1
+
2
a_{n+1}=4a_n-a_{n-1}+2
a
n
+
1
=
4
a
n
−
a
n
−
1
+
2
for
n
>
1.
n>1.
n
>
1.
Prove that
a
2
k
−
1
a_{2^k-1}
a
2
k
−
1
has at least three prime factors for every positive integer
k
>
3.
k>3.
k
>
3.
Sequence
algebra