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Intersection of all these sets is empty

Source: IMO Longlist 1989, Problem 60

September 18, 2008
combinatorics unsolvedcombinatorics

Problem Statement

A family of sets A1,A2,…,An A_1, A_2, \ldots ,A_n has the following properties: (i) Each Ai A_i contains 30 elements. (ii) Ai∩Aj A_i \cap A_j contains exactly one element for all i,j,1≤i<j≤n. i, j, 1 \leq i < j \leq n. Determine the largest possible n n if the intersection of all these sets is empty.
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