SCHWEITZER 2014 Problem 2

Let $k\geq 1$ and let $I_{1},\dots, I_{k}$ be non-degenerate subintervals of the interval $[0, 1]$ . Prove that $$\sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2}$$ where the summation is over all pairs $(i, j)$ of indices such that $I_i\cap I_j\neq \emptyset$ .