USEMO 2019 Problem 2

Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials) such that

• for any polynomials $p, q \in \mathbb{Z}[x]$ , $\theta(p + q) = \theta(p) + \theta(q)$ ;
• for any polynomial $p \in \mathbb{Z}[x]$ , $p$ has an integer root if and only if $\theta(p)$ does.

Carl Schildkraut