CMO 2023 Problem 4

Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\operatorname{divs}(n)$ to be the set of positive divisors of $n$.

A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which

$$f[\operatorname{divs}(m)]=\operatorname{divs}(n)$$

Prove that for any such $f$, there are finitely many $f$-cool integers. (The notation $f[S]$ for some set $S$ denotes the set $\{f(s): s \in S\}$.) Remark 1. The original problem statement was "For a fixed non-constant polynomial $f(x) \neq$ $x$, prove that there are finitely many composite $f$-cool integers." Note that this allows $f(1)=1$. Try this problem for an added challenge!