Question 89
Let $x_{1}, x_{2}, \ldots, x_{1000}, y_{1}, y_{2}, \ldots, y_{1000}$ be 2000 different real numbers, and form the $1000 \times 1000$ matrix whose $(i, j)$-entry is $x_{i}+y_{j}$. If the product of the numbers in each row is 1, show that the product of the numbers in each column is -1.