Nemotron-IMO-Bench 2026 N26

Let $p$ be a prime number and let $k$ be a positive integer. Let $$t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.$$a) Let $f\in \mathbb{Z}[X]$ be a monic polynomial of degree $k$ with $p\mid f(0)$. Prove that there exists $n\in\mathbb{N}$ for which $p\mid f(n)$ and $p^{t+1}\nmid f(n).$\b) Prove that the statement above is sharp, i.e. there exists a monic polynomial $g\in \mathbb{Z}[X]$ of degree $k$ such that $p\mid g(0)$ and if $p\mid g(n)$ for some $n\in\mathbb{N}$ then $p^t\mid g(n)$ also holds.