RU-ALLRUSSIAN 1991 Problem 16

(1) $\mathbf{r}_1, \mathbf{r}_2, \ldots , \mathbf{r}_{100}, \mathbf{c}_1, \mathbf{c}_2, \ldots , \mathbf{c}_{100}$ are distinct reals. The number $\mathbf{r}_i + \mathbf{c}_j$ is written in position i, j of a $100 \times 100$ array. The product of the numbers in each column is 1. Show that the product of the numbers in each row is -1. (2) $\mathbf{r}_1, \mathbf{r}_2, \ldots , \mathbf{r}_{2n}, \mathbf{c}_1, \mathbf{c}_2, \ldots , \mathbf{c}_{2n}$ are distinct reals. The number $\mathbf{r}_i + \mathbf{c}_j$ is written in position i, j of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.