Question 185
At each square of a $2007 \times 2007$ chessboard we put one of the numbers $1$ or $-1$. We denote by $A_i$ the product of the numbers of the $i$-th row, $i=1,2,\ldots,2007$ and by $B_j$ the product of the numbers of the $j$-th column, $j=1,2,\ldots,2007$. Prove that:
$$
A_1 + A_2 + \dots + A_{2007} + B_1 + B_2 + \dots + B_{2007} \neq 0.
$$
$$
A_1 + A_2 + \dots + A_{2007} + B_1 + B_2 + \dots + B_{2007} \neq 0.
$$