BMO 2023 | Problem 3 - 10 May 2023 - Balkan Mathematical Olympiad

Problem 3.
For each positive integer $n$ , denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$ ). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$ , then $P(n)$ is also a positive integer with
$\omega(n)\ge\omega(P(n)).$

Greece