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Polynomial mapping set of divisors to set of divisors

Source: CMO 2023 P4

March 11, 2023
algebrapolynomialCMOCMO 2023

Problem Statement

Let f(x)f(x) be a non-constant polynomial with integer coefficients such that f(1)≠1f(1) \neq 1. For a positive integer nn, define divs(n)\text{divs}(n) to be the set of positive divisors of nn.
A positive integer mm is ff-cool if there exists a positive integer nn for which f[divs(m)]=divs(n).f[\text{divs}(m)]=\text{divs}(n). Prove that for any such ff, there are finitely many ff-cool integers.
(The notation f[S]f[S] for some set SS denotes the set {f(s):s∈S}\{f(s):s \in S\}.)
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