IMC 2023 · Problem 10

For every positive integer $n$, let $f(n), g(n)$ be the minimal positive integers such that

$$1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!} = \frac{f(n)}{g(n)}.$$

Determine whether there exists a positive integer $n$ for which $g(n) > n^{0.999n}$.