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a point starts at the origin of the coordinate plane, rotates also

Source: Canada Repêchage 2023/8 CMOQR

March 25, 2024
rotationcombinatorics

Problem Statement

A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the xx-direction or is rotated θ\theta degrees counterclockwise about the origin.
(a) If θ=90o\theta = 90^o, determine all locations where the point could end up.
(b) If θ=45o\theta = 45^o, prove that for every location L L in the coordinate plane and every positive number ε\varepsilon, there is a sequence of moves after which the point has distance less than ε\varepsilon from LL.
(c) Determine all rational numbers θ\theta such that for every location LL in the coordinate plane and every positive number ε\varepsilon, there is a sequence of moves after which the point has distance less than ε\varepsilon from LL.
(d) Prove that when θ\theta is irrational, for every location LL in the coordinate plane and every positive number ε\varepsilon, there is a sequence of moves after which the point has distance less than ε\varepsilon from L.L.
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