Problem

Source: INMO 2020 P5

Tags: number theory, Tiling, descent, trigonometry, Niven theorem, rational, Galois Theory



Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. Proposed by Muralidharan