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Sending non-intersecting teams to an olympiad

Source: Tuymaada 2018 Senior League/Problem 7

July 20, 2018
combinatorics

Problem Statement

A school has three senior classes of MM students each. Every student knows at least 34M\frac{3}{4}M people in each of the other two classes. Prove that the school can send MM non-intersecting teams to the olympiad so that each team consists of 33 students from different classes who know each other.
Proposed by C. Magyar, R. Martin
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