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\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0, in a grid nxn, sum and product

Source: CRMO 2014 region 2 p6

September 29, 2018
combinatoricsnumber theorygrid

Problem Statement

Suppose nn is odd and each square of an n×nn \times n grid is arbitrarily filled with either by 11 or by −1-1. Let rjr_j and ckc_k denote the product of all numbers in jj-th row and kk-th column respectively, 1≤j,k≤n1 \le j, k \le n. Prove that
∑j=1nrj+∑k=1nck≠0\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0
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