Nemotron-IMO-Bench 2026 C25

Prove that if $A_1,\ldots, A_m$ are subsets of $\{1,\ldots, n\}$, then $$\sum_{i,j=1}^m |A_i|\cdot |A_i\cap A_j|\geq \frac{1}{mn} (\sum_{i=1}^m |A_i|)^3.$$