CMOQR 2014 Problem 4

In $\triangle ABC$ , the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$ . A laser beam exits the point $A_1$ , hits side $AC$ at point $B_1$ , and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$ , then side $BC$ at point $A_2$ , then side $AC$ again at point $B_2$ , then side $AB$ again at point $C_2$ , then side $BC$ again at point $A_3$ , and finally, side $AC$ again at point $B_3$ .

(a) Prove that $\angle B_3A_3C = \angle B_1A_1C$ .

(b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$ .