Light beam and mirrors
Source: St Petersburg Olympiad 2012, Grade 11, P6
September 29, 2017
geometrycombinatorics
Problem Statement
On the coordinate plane in the first quarter there are non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than times.