International Mathematical Olympiad 2025 Problem 3

Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon \mathbb{N}\to \mathbb{N}$ is said to be bonza if

$$f(a) \; \text{ divides } \; b^a - f(b)^{f(a)}$$

for all positive integers $a$ and $b$.

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