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Romania TST 2016 Day 1 P2

Source: Romania TST 2016 Day 1 P2

November 1, 2017
combinatorics

Problem Statement

Let nn be a positive integer, and let S1,S2,…,SnS_1,S_2,…,S_n be a collection of finite non-empty sets such that ∑1≤i<j≤n∣Si∩Sj∣∣Si∣∣Sj∣<1.\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1. Prove that there exist pairwise distinct elements x1,x2,…,xnx_1,x_2,…,x_n such that xix_i is a member of SiS_i for each index ii.
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