You have a rectangular grid and arbitrary real numbers in its cells. You are allowed repeatedly to multiply the elements in any row or any column by -1. Prove that you can make all row sums and all column sums non-negative simultaneously.
If there is any row or column in the grid with negative sum, multiply it by -1. Since on every step the total sum of the numbers in the grid increases, there will be able to do this procedure only finitely many steps. At the end all row sums and column sums will be non-negative.