MathDB

Consider a convex polygon

A_{1}A_{2}\ldots A_{n}

and a point

M

inside it. The lines

A_{i}M

intersect the perimeter of the polygon second time in the points

B_{i}

. The polygon is called balanced if all sides of the polygon contain exactly one of points

B_{i}

(strictly inside). Find all balanced polygons. [Note: The problem originally asked for which

n

all convex polygons of

n

sides are balanced. A misunderstanding made this version of the problem appear at the contest]

Problem Statement