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A1
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2022 PUMaC Individual Finals A1
Source:
September 9, 2023
number theory
Problem Statement
Let
f
:
Z
>
0
→
Z
>
0
f : Z_{>0} \to Z_{>0}
f
:
Z
>
0
→
Z
>
0
be a function which satisfies
k
∣
f
k
(
x
)
−
x
k|f^k(x)-x
k
∣
f
k
(
x
)
−
x
for all
k
,
x
∈
Z
>
0
k, x \in Z_{>0}
k
,
x
∈
Z
>
0
and
f
(
x
)
−
x
≤
2023
f(x)-x \le 2023
f
(
x
)
−
x
≤
2023
. If
f
(
1
)
=
2000
f(1) = 2000
f
(
1
)
=
2000
, what can
f
f
f
be? Remark: Here,
f
k
(
x
)
f^k (x)
f
k
(
x
)
denotes the
k
k
k
-fold application of
f
f
f
to
x
x
x
.
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