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Cyclic sums

Source: Moldova TST 2014, Third Day, Problem 4

March 31, 2014
combinatorics proposedcombinatorics

Problem Statement

On a circle n≥1n \geq 1 real numbers are written, their sum is n−1n-1. Prove that one can denote these numbers as x1,x2,...,xnx_1, x_2, ..., x_n consecutively, starting from a number and moving clockwise, such that for any kk (1≤k≤n1\leq k \leq n) x1+x2+...+xk≥k−1 x_1 + x_2+...+x_k \geq k-1.
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