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Tourists and t-shirts

Source: All-Russian Olympiad 2006 finals, problem 11.8

May 6, 2006
ceiling functionfunctioncombinatorics proposedcombinatoricsProbabilistic Method

Problem Statement

At a tourist camp, each person has at least 5050 and at most 100100 friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) 13311331 different colors, and any person has 2020 friends whose t-shirts all have pairwisely different colors.