Question 194
Consider two non-constant polynomials $P(x), Q(x)$ with non-negative integer coefficients. The coefficients of $P(x)$ are not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022) = Q(2022)$ and that $P(x), Q(x)$ have a common rational root $\frac{p}{q} \neq 0$ for some $p, q \in \mathbb{Z}$ $(\gcd(p, q) = 1)$. Prove that
$$
|p| + n|q| \leq Q(n) - P(n), \forall n = 1, 2, \dots, 2021.
$$
$$
|p| + n|q| \leq Q(n) - P(n), \forall n = 1, 2, \dots, 2021.
$$