I see. So, assuming C = (0,0), A = (1, 0), B = (1, 1), I have:
BM: y_{BM} = 2 x_{BM} - 1
AP: y_{AP} = \frac{- x_{AP} + 1}{2}
BC: y_{BC} = x_{BC}
So P = (\frac{1}{3}, \frac{1}{3}) and BP:PC=2:1, which is the correct answer :).
However, the 'official' resolution of this problem is the...