Question 20
We have a square board of $2002 \times 2002$ with squares painted black and white, like a chess board. The rows have been numbered from $1$ to $2002$ and the columns from $1$ to $2002$ (the box in row $1$ and column $1$ is black). In each box we write the product of the row number and the column number to which the box belongs. Let $A$ be the sum of the numbers written in the black boxes and $B$ be the sum of the numbers in the white boxes.
Prove that $A - B$ is a perfect square.
Prove that $A - B$ is a perfect square.