BALTIC-WAY 2008 Problem 20

Let $M$ be a point on $B C$ and $N$ be a point on $A B$ such that $A M$ and $C N$ are angle bisectors of the triangle $A B C$. Given that

$$\frac{\angle B N M}{\angle M N C}=\frac{\angle B M N}{\angle N M A}$$

prove that the triangle $A B C$ is isosceles.