MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2014 Balkan MO Shortlist
A3
BMO 2014 SL A3
BMO 2014 SL A3
Source: Balkan MO 2014 Shortlist
October 1, 2016
Sequence
algebra
Problem Statement
A
3
\boxed{A3}
A
3
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
a
2
=
1
,
a
2
n
+
1
=
2
a
2
n
−
a
n
a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n
a
1
=
a
2
=
1
,
a
2
n
+
1
=
2
a
2
n
−
a
n
and
a
2
n
+
2
=
2
a
2
n
+
1
a_{2n+2}=2a_{2n+1}
a
2
n
+
2
=
2
a
2
n
+
1
for
n
∈
N
.
n\in{N}.
n
∈
N
.
Prove that if
n
>
3
n>3
n
>
3
and
n
−
3
n-3
n
−
3
is divisible by
8
8
8
then
a
n
a_n
a
n
is divisible by
5
5
5
Back to Problems
View on AoPS