KöMaL 2014 A631

Let $k\ge1$ and let $I_1,\ldots,I_k$ be non-degenerate subintervals of the interval $[0, 1]$. Prove $\sum \frac1{|I_i\cup I_j|} \ge k^2$ where the summation is over all pairs $(i,j)$ of indices such that $I_i$ and $I_j$ are not disjoint.