Here’s an old IMO question (or maybe it’s off one of the short or long lists) that I haven’t made any progress on:

Show that for any natural numbers \(a\) and \(n\) the sum

\[

1 + \frac{1}{1+a} + \frac{1}{1 + 2a} + \cdots + \frac{1}{1+n a}

\]

is not an integer.

An obvious and cool corollary is that the harmonic numbers are never integers.