imo-3-gpt

Let $\mathbb N$ denote the set of positive integers. A function $f:\mathbb N\to\mathbb N$ is said to be bonza if $f(a)$ divides $b^a-f(b)^{f(a)}$ for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\le cn$ for all bonza functions $f$ and all positive integers $n$.