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Sets and set intersections: 2011 USAMO #6

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April 28, 2011
AMCUSA(J)MOUSAMOlinear algebramatrixinequalitiesfunction

Problem Statement

Let AA be a set with ∣A∣=225|A|=225, meaning that AA has 225 elements. Suppose further that there are eleven subsets A1,…,A11A_1, \ldots, A_{11} of AA such that ∣Ai∣=45|A_i|=45 for 1≤i≤111\leq i\leq11 and ∣Ai∩Aj∣=9|A_i\cap A_j|=9 for 1≤i<j≤111\leq i<j\leq11. Prove that ∣A1∪A2∪…∪A11∣≥165|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165, and give an example for which equality holds.
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