Question 55
(1) $r_1, r_2, \ldots, r_{100}, c_1, c_2, \ldots, c_{100}$ are distinct reals. The number $r_i + c_j$ 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) $r_1, r_2, \ldots, r_{2n}, c_1, c_2, \ldots, c_{2n}$ are distinct reals. The number $r_i + c_j$ 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.
(2) $r_1, r_2, \ldots, r_{2n}, c_1, c_2, \ldots, c_{2n}$ are distinct reals. The number $r_i + c_j$ 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.