Suppose n is odd and each square of an n×n grid is arbitrarily filled with either by 1 or by −1. Let rj and ck denote the product of all numbers in j-th row and k-th column respectively, 1≤j,k≤n. Prove that n∑j=1rj+n∑k=1ck≠0
Source: CRMO 2014 region 2 p6
Tags: combinatorics, number theory, grid
Suppose n is odd and each square of an n×n grid is arbitrarily filled with either by 1 or by −1. Let rj and ck denote the product of all numbers in j-th row and k-th column respectively, 1≤j,k≤n. Prove that n∑j=1rj+n∑k=1ck≠0