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n integers in a circle, a_1 + a_2 +...+ a_k <= 2k -1

Source: 15th -a QEDMO problem 9 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 30, 2021
combinatorics

Problem Statement

Iskandar arranged n∈Nn \in N integer numbers in a circle, the sum of which is 2n−12n-1. Crescentia now selects one of these numbers and name the given numbers in clockwise direction with a1,a2,....,ana_1,a_2,...., a_n. Show that she can choose the starting number such that for all k∈{1,2,...,n}k \in \{1, 2,..., n\} the inequality a1+a2+...+ak≤2k−1a_1 + a_2 +...+ a_k \le 2k -1 holds.
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