BALTIC-WAY 2012 Problem 11

Let $A B C$ be a triangle with $\angle A=60^{\circ}$. The point $T$ lies inside the triangle in such a way that $\angle A T B=\angle B T C=\angle C T A=120^{\circ}$. Let $M$ be the midpoint of $B C$. Prove that $T A+T B+T C=2 A M$.