AMC 12 2012a Problem 23

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 2012$ and $0\le y\le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior?

$\textbf{(A)}\ \frac{1}{8} \textbf{(B) }\frac{7}{50} \textbf{(C) }\frac{4}{25} \textbf{(D) }\frac{1}{4} \textbf{(E) }\frac{8}{25}$