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reflections of ray of light in rectangle mxn, billiard clone (Chile NMO 1994 P7)

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November 21, 2021
geometryreflectionrectangle

Problem Statement

Let ABCDABCD be a rectangle of length mm and width nn, with m,nm, n positive integers. Consider a ray of light that starts from AA, reflects with an angle of 45o45^o on an opposite side and continues reflecting away at the same angle.
∙\bullet For any pair (m,n)(m,n), show that the ray meets a vertex at some point.
∙\bullet Suppose mm and nn are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.
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