RMO 2013 C5

Let $ABC$ be a triangle which it not right-angled. Define a sequence of triangles $A_iB_iC_i$ , with $i \ge 0$ , as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$ , $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$ , $C_iA_i$ , $A_iB_i$ , respectively. Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$ . Prove that $\angle A = 60^{\circ}$ .