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120 rotations around it's center of a regular 360gon

Source: Czech-Polish-Slovak Match Junior 2019, team p5 CPSJ

January 20, 2020
geometrygeometric transformationrotationcombinatorial geometrycombinatorics

Problem Statement

Let A1A2...A360A_1A_2 ...A_{360} be a regular 360360-gon with centre SS. For each of the triangles A1A50A68A_1A_{50}A_{68} and A1A50A69A_1A_{50}A_{69} determine, whether its images under some 120120 rotations with centre SS can have (as triangles) all the 360360 points A1,A2,...,A360A_1, A_2, ..., A_{360} as vertices.
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