2024 USAMO Problems/Problem 6
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Let 
be an integer and let 
. A collection 
of (not necessarily distinct) subsets of 
is called 
-large if 
for all 
. Find, in terms of 
and , the largest real number 
such that the inequality
![\[\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2\]](http://latex.artofproblemsolving.com/3/e/d/3ed5047ee81526de349e5b0029f19c0187c25429.png)
holds for all positive integers 
, all nonnegative real numbers 
, and all -large collections of subsets of .
Note: For a finite set 
denotes the number of elements in 
.
