International Mathematics Competition
for University Students
2023

IMC2023: Day 2, Problem 10

Problem 10. For every positive integer \(\displaystyle n\), let \(\displaystyle f(n),g(n)\) be the minimal positive integers such that

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

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

Fedor Petrov, St. Petersburg State University

  Solution