Let $N\ge 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\dots ,A_k\subseteq \{1,\dots ,N\}$ with ${\sum }_{n\in A_i}\frac{1}{n}=1$ for all $i$. Estimate $k(N)$. Is it true that $k(N)=o(\log N)$?