2019 Fermat Problem 25

In \triangle QRS, point T is on QS with \angle QRT = \angle SRT. Suppose that QT = m and TS = n for some integers m and n with n > m and for which n + m is a multiple of n - m. Suppose also that the perimeter of \triangle QRS is p and that the number of possible integer values for p is m^2 + 2m - 1. The value of n - m is

\textbf{(A)}\ 4\quad \textbf{(B)}\ 1\quad \textbf{(C)}\ 3\quad \textbf{(D)}\ 2\quad \textbf{(E)}\ 5

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.