2014 Fermat Problem 25

Points P(r,\ \text{s}) and Q(t, u) are on the parabola with equation y = x^2 - \frac{1}{5}mx + \frac{1}{5}n so that PQ = 13 and the slope of PQ is \frac{12}{5}. For how many pairs (m, n) of positive integers with n \le 1000 is r + s + t + u = 27 ?

\textbf{(A)}\ 28\quad \textbf{(B)}\ 26\quad \textbf{(C)}\ 27\quad \textbf{(D)}\ 29\quad \textbf{(E)}\ 25

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 Fermat contests here.