Let $n\ge 1$ and $\{I_j\}_{j=1}^{n}$ is a set of non-degenerate subintervals of $[0,1]$. Then show that : $$ \overline\sum \dfrac{1}{|I_j\cup I_k|}\geq n^2$$ Here $\overline\sum$ denotes summing over $j,k$ with $I_j\cap I_k\neq \emptyset$.
This is a problem from the Miklos Schweitzer competition. But, I don't have an idea about it, can someone give me a hint or something? An idea on how to proceed would be great. Thanks a lot.
See here for a very elegant solution.
Moreover, here you can find a list of the problems from this competition which have been asked on the AoPS forum (but some of them have not been solved there).