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Partition of positive integers

Source: Czech and Slovak Olympiad 1971, National Round, Problem 3

July 9, 2024
combinatoricspartition

Problem Statement

Consider positive integers 2,3,…,n−1,n2,3,\ldots,n-1,n where n≥96.n\ge96. Consider any partition in two (sub)sets. Show that at least one of these two sets always contains two numbers and their product. Show that the statement does not hold for n=95,n=95, e.g. there is a partition without the mentioned property.
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