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Frameable polygons

Source: INMO 2020 P5

January 19, 2020
number theoryTilingdescenttrigonometryNiven theoremrationalGalois Theory

Problem Statement

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer n⩾3n \geqslant 3 is called frameable if it is possible to draw a regular polygon with nn sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.
(a) Show that 3,4,63, 4, 6 are frameable. (b) Show that any integer n⩾7n \geqslant 7 is not frameable. (c) Determine whether 55 is frameable.
Proposed by Muralidharan
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