Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.
Problem
Source: JBMO Shortlist 2017 C1
Tags: combinatorics, regular polygon, Coloring
ayan.nmath
25.07.2018 18:43
It is not hard to obtain by taking an external point very far on the angle bisector of an angle that $\frac{(2n-1)180^{\circ}}{2n+1}<120^{\circ}\implies n<2.5\implies n=1,2.$(or otherwise, four consecutive sides can be seen.) For, $n=2,$ such coloring is not possible. Thus, the answer is $1.~~\blacksquare$
rt03
25.07.2018 18:45
Clearly $n=1$ works. We claim that it is the largest positive integer for which a coloring is possible. Suppose $n>1$. Note that any $n+1$ consecutive sides can be seen. Consider two consecutive line segments $x,y$ with colors $A,B$. Any $n+1$ consecutive sides containing these line segments must only have the colors $A,B$. The union of all such sets is the collection of all the sides of the regular $2n+1$-gon, excluding the side directly opposite the vertice shared by $x,y$. Hence this side must be colored $C$, and is the only side colored $C$. Take the $n+1$ consecutive sides with $C$ as its starting edge and $x$ as its ending edge; all the sides in between must be $A$. Similarly, take the $n+1$ consecutive sides with $C$ as its starting edge and $y$ as its ending edge; all sides in between must be colored $B$. Now take a set of $n+1$ consecutive sides with $C$ not at one of the ends; for $n>1$ it contains both sides colored $A,B$, invalidating the coloring.