TITLE: Inequality on length of intervals
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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).