Nemotron-IMO-Bench 2026 N19

A function is called special if it is of the form $x\mapsto a^{f(x)}$ with $a$ a positive integer and $f\in \mathbb{Z}[X]$ satisfying $f(n)>0$ for positive integers $n$. Let $P\in \mathbf{Z}[X]$ be nonconstant and let $f$ be a nonzero function which is a sum of products of special functions. Prove that there is a positive integer $n$ such that $P(n)$ does not divide $f(n)$.