MathDB
Sequence and light ray

Source: Iran TST 2007, Day 4

May 28, 2007
analytic geometrygraphing linesslopetrigonometrygeometrygeometric transformationreflection

Problem Statement

Let PP be a point in a square whose side are mirror. A ray of light comes from PP and with slope α\alpha. We know that this ray of light never arrives to a vertex. We make an infinite sequence of 0,10,1. After each contact of light ray with a horizontal side, we put 00, and after each contact with a vertical side, we put 11. For each n≥1n\geq 1, let BnB_{n} be set of all blocks of length nn, in this sequence. a) Prove that BnB_{n} does not depend on location of PP. b) Prove that if απ\frac{\alpha}{\pi} is irrational, then ∣Bn∣=n+1|B_{n}|=n+1.
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