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APMO
2019 APMO
2
2
Part of
2019 APMO
Problems
(1)
Strange Conditional Sequence
Source: APMO 2019 P2
6/11/2019
Let
m
m
m
be a fixed positive integer. The infinite sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
is defined in the following way:
a
1
a_1
a
1
is a positive integer, and for every integer
n
≥
1
n\geq 1
n
≥
1
we have
a
n
+
1
=
{
a
n
2
+
2
m
if
a
n
<
2
m
a
n
/
2
if
a
n
≥
2
m
a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}
a
n
+
1
=
{
a
n
2
+
2
m
a
n
/2
if
a
n
<
2
m
if
a
n
≥
2
m
For each
m
m
m
, determine all possible values of
a
1
a_1
a
1
such that every term in the sequence is an integer.
number theory
APMO