VMO 2022 — Academic year 2021–2022, both days Question 5

Let $P,Q$ be nonconstant polynomials with nonnegative integer coefficients. Every coefficient of $P$ is at most 2021, while at least one coefficient of $Q$ exceeds 2021. Suppose $P(2022)=Q(2022)$ and $P,Q$ have a common nonzero rational root $p/q$, where $p,q\in\mathbb Z$ are coprime. Prove that
$$|p|+n|q|\le Q(n)-P(n)\qquad(n=1,2,\ldots,2021).$$