VMO 2022 Problem 5

Consider 2 non-constant polynomials $P(x),Q(x)$ , with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$ . Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$ . Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$