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National and Regional Contests
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PEN M Problems
30
30
Part of
PEN M Problems
Problems
(1)
M 30
Source:
5/25/2007
Let
k
k
k
be a positive integer. Prove that there exists an infinite monotone increasing sequence of integers
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
such that
a
n
divides
a
n
+
1
2
+
k
and
a
n
+
1
divides
a
n
2
+
k
a_{n}\; \text{divides}\; a_{n+1}^{2}+k \;\; \text{and}\;\; a_{n+1}\; \text{divides}\; a_{n}^{2}+k
a
n
divides
a
n
+
1
2
+
k
and
a
n
+
1
divides
a
n
2
+
k
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
.
Recursive Sequences