TR-MO 2007 Problem 2

Some unit squares of $2007\times 2007$ square board are colored. Let $(i,j)$ be a unit square belonging to the $ith$ line and $jth$ column and $S_{i,j}$ be the set of all colored unit squares $(x,y)$ satisfying $x\leq i, y\leq j$ . 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.