Problem

Source: ELMO Shortlist 2013: Problem C4, by Evan Chen

Tags: inequalities, algorithm, combinatorics unsolved, combinatorics



Let n be a positive integer. The numbers {1,2,...,n2} are placed in an n×n grid, each exactly once. The grid is said to be Muirhead-able if the sum of the entries in each column is the same, but for every 1i,kn1, the sum of the first k entries in column i is at least the sum of the first k entries in column i+1. For which n can one construct a Muirhead-able array such that the entries in each column are decreasing? Proposed by Evan Chen