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\sum_{i=1}^{k} x_i \le k - 1, inequality with labeling beads from necklace

Source: Danube 2018 p1

July 22, 2019
inequalitiescombinatorics

Problem Statement

Suppose we have a necklace of nn beads. Each bead is labeled with an integer and the sum of all these labels is n−1n - 1. Prove that we can cut the necklace to form a string, whose consecutive labels x1,x2,...,xnx_1,x_2,...,x_n satisfy ∑i=1kxi≤k−1\sum_{i=1}^{k} x_i \le k - 1 for any k=1,...,nk = 1,...,n
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