Question 148
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 $\{f(s): s \in S\}$.)
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 $\{f(s): s \in S\}$.)