5.2.4.31 Various Problems on the Plane

Condition: in the triangle \ (a b c \) points \ (m, n, k \) are located respectively on the sides \ (a b, b c, a c \) so that \ (a m: m b = 1: 2 \), \ (c n: n b = 1: 3, a k = k c \). Sifts \ (m n \) and \ (b k \) intersect at point \ (p \). Find the relationship \ (m p: p n \) and \ (b p: p k \).