USAMO 2024 Problem 6

Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$ . A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$ -large if $|A_i| \ge \ell$ for all $1 \le i \le k$ . Find, in terms of $n$ and $\ell$ , the largest real number $c$ such that the inequality $$\sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2$$ holds for all positive integer $k$ , all nonnegative real numbers $x_1,x_2,\dots,x_k$ , and all $\ell$ -large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$ .

Proposed by Titu Andreescu and Gabriel Dospinescu