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2019-IMOC
N5
N5
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Problems
(1)
two operations on number form infinite sequence
Source: IMOC 2019 N5
8/19/2021
Initially, Alice is given a positive integer
a
0
a_0
a
0
. At time
i
i
i
, Alice has two choices,
{
a
i
↦
1
a
i
−
1
a
i
↦
2
a
i
−
1
+
1
\begin{cases}a_i\mapsto\frac1{a_{i-1}}\\a_i\mapsto2a_{i-1}+1\end{cases}
{
a
i
↦
a
i
−
1
1
a
i
↦
2
a
i
−
1
+
1
Note that it is dangerous to perform the first operation, so Alice cannot choose this operation in two consecutive turns. However, if
x
>
8763
x>8763
x
>
8763
, then Alice could only perform the first operation. Determine all
a
0
a_0
a
0
so that
{
i
∈
N
∣
a
i
∈
N
}
\{i\in\mathbb N\mid a_i\in\mathbb N\}
{
i
∈
N
∣
a
i
∈
N
}
is an infinite set.
number theory
Sequences