RMO 2025 Problem 6

Let $p(x)$ be a nonconstant polynomial with integer coefficients, and let $n\geq2$ be an integer such that no term of the sequence $p(0), p(p(0)), p(p(p(0))),...$ is divisible by $n$ . Show that there exist integers $a$ , $b$ such that $0\leq a<b\leq n-1$ and $n$ divides $p(b)-p(a)$ Proposed by Bhavya Tiwari