2014 Cayley Problem 25

Point P is on the y-axis with y-coordinate greater than 0 and less than 100. A circle is drawn through P, Q(4,4) and O(0,0). How many possible positions for P are there so that the radius of this circle is an integer?

\textbf{(A)}\ 2\quad \textbf{(B)}\ 68\quad \textbf{(C)}\ 66\quad \textbf{(D)}\ 65\quad \textbf{(E)}\ 67

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.