Question 18
On every square of an $n \times n$ chessboard we write one of the numbers $1$ or $-1$. Let $a_k$ be the product of all numbers in the $k^{\text{th}}$ row and $b_l$ be the product of all numbers in the $l^{\text{th}}$ column. Assuming $n = 2007$, can we choose the numbers in such a way that the sum
$$
a_1 + a_2 + \dots + a_n + b_1 + b_2 + \dots + b_n
$$
will be zero? What about $n = 2008$?
$$
a_1 + a_2 + \dots + a_n + b_1 + b_2 + \dots + b_n
$$
will be zero? What about $n = 2008$?