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Reflecting a ray wrt the sides of an equilateral triangle

Source: APMO 2018 P4

June 24, 2018
geometrynumber theoryAPMOcombinatoricsHi

Problem Statement

Let ABCABC be an equilateral triangle. From the vertex AA we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle α\alpha, it leaves with a directed angle 180∘−α180^{\circ}-\alpha. After nn bounces, the ray returns to AA without ever landing on any of the other two vertices. Find all possible values of nn.
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