2024 USAMO Problems/Problem 6

Let $n>2$ be an integer and let $\ell \in\{1,2, \ldots, n\}$ . A collection $A_1, \ldots, A_k$ of (not necessarily distinct) subsets of $\{1,2, \ldots, n\}$ is called $\ell$ -large if $\left|A_i\right| \geq \ell$ for all $1 \leq i \leq k$ . Find, in terms of $n$ and $\ell$ , the largest real number $c$ such that the inequality $\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2$ holds for all positive integers $k$ , all nonnegative real numbers $x_1, \ldots, x_k$ , and all $\ell$ -large collections $A_1, \ldots, A_k$ of subsets of $\{1,2, \ldots, n\}$ . Note: For a finite set $S,|S|$ denotes the number of elements in $S$ .

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2024 USAMO Problems/Problem 6