Question 108
Some unit squares of $2007 \times 2007$ square board are colored. Let $(i, j)$ be a unit square belonging to the $i$-th line and $j$-th column and $S_{i,j}$ be the set of all colored unit squares $(x, y)$ satisfying $x \leq i$ and $y \leq j$. At the first step in each colored unit square $(i, j)$ we write the number of colored unit squares in $S_{i,j}$. In each step, in each colored unit square $(i, j)$ we write the sum of all numbers written in $S_{i,j}$ in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd. (Özgür Kişisel).