Find all pairs of integers $m \leqslant n$ such that the following quantity is an integer:

$$
\frac{1}{m}+\frac{1}{m+1}+\ldots+\frac{1}{n}
$$
(Solved by Florent Noisette)

We have $\frac{1}{m}+\cdots+\frac{1}{2 m-1}<m \times \frac{1}{m} \leqslant 1$ so necessarily, $n \geqslant 2 m$. (except if $m=1$, in which case $\mathrm{n}=1$)

In this case, let $r$ be such that $m \leqslant 2^{r} \leqslant n<2^{r+1}$. The 2-adic valuation of the integers from $m$ to $n$ is then maximal only at $2^{r}$, which shows that the sum cannot be an integer. The only solution is therefore $\mathrm{m}=\mathrm{n}=1$.