Nemotron-IMO-Bench 2026 A42

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