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John and a string of paper

Source: Baltic Way 2021, Problem 10

November 15, 2021
combinatoricscombinatorics proposed

Problem Statement

John has a string of paper where nn real numbers ai∈[0,1]a_i \in [0, 1], for all i∈{1,…,n}i \in \{1, \ldots, n\}, are written in a row. Show that for any given k<nk < n, he can cut the string of paper into non-empty kk pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than 11.
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