ELMO 2025 Problem 3

Let $n$ be a positive integer and $p$ be a prime. In terms of $n$ and $p$ , find the largest nonnegative integer $k$ for which there exists a polynomial $P(x)$ with integer coefficients satisfying the following conditions:

• The $x^n$ coefficient of $P(x)$ is $1$ .
• $p^k$ divides $P(x)$ for all integers $x$ .

Aprameya Tripathy