INMO 2021 Problem 6

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients, and let $\operatorname{deg} P$ denote the degree of a nonzero polynomial $P$. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:

• $f$ maps the zero polynomial to itself,
• for any non-zero polynomial $P \in \mathbb{R}[x]$, $\operatorname{deg} f(P) \leq 1+\operatorname{deg} P$, and
• for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.