COMC 2016 C Problem 4

Two lines intersect at a point Q at an angle \theta^{\circ}, where 0<\theta<180. A frog is originally at a point other than Q on the angle bisector of this angle. The frog alternately jumps over these two lines, where a jump over a line results in the frog landing at a point which is the reflection across the line of the frog’s jumping point.The frog stops when it lands on one of the two lines.

(a) Suppose \theta=90^{\circ}. Show that the frog never stops.

(b) Suppose \theta=72^{\circ}. Show that the frog eventually stops.

(c) Determine the number of integer values of \theta, with 0<\theta^{\circ}<180^{\circ}, for which the frog never stops.