2012 AMC 12A 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}\qquad\textbf{(B) }\frac{7}{50}\qquad\textbf{(C) }\frac{4}{25}\qquad\textbf{(D) }\frac{1}{4}\qquad\textbf{(E) }\frac{8}{25}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.