The fraction (2M-N)/(2M+N) must reduce to the square of a fraction, otherwise P will not be rational, let alone integral.
Multiplying the numerator and denominator by (2M+N):
(2M-N)/(2M+N) = [(2M)^2-N^2]/[(2M+N)^2]
The denominator is square, therefore the numerator must be a square to satisfy the problem. Then for each pair (M,N) there must be an X such that X^2 + N^2 = (2M)^2
From the Pythagorean triple (24,32,40): M=20, N=24 yields:
P = 24/4*sqrt[(40-24)/(40+24)] = 6*sqrt[1/4] = 3
This shows there are M,N,P which exist to satisfy the problem.