Question 122
Let $Q(x) = a_{2023}x^{2023} + a_{2022}x^{2022} + \dots + a_1x + a_0 \in \mathbb{Z}[x]$. For any odd prime number $p$ define the polynomial $Q_p(x) = a_{2023}^{p-2}x^{2023} + a_{2022}^{p-2}x^{2022} + \dots + a_1^{p-2}x + a_0^{p-2}$. It is known that there are infinitely many primes $p$ for which $\frac{Q_p(x)-Q(x)}{p}$ is an integer for all $x \in \mathbb{Z}$. Determine the largest possible value of $Q(2023)$.