MathDB

Let

A_{1}A_{2} \dots A_{3n}

be a closed broken line consisting of

3n

lines segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index

i=1,2,\dots,3n

, the triangle

A_{i}A_{i+1}A_{i+2}

has counterclockwise orientation and

\angle A_{i}A_{i+1}A_{i+2} = 60º

, using the notation

A_{3n+1} = A_{1}

and

A_{3n+2} = A_{2}

. Prove that the number of self-intersections of the broken line is at most

\frac{3}{2}n^{2} - 2n + 1

Problem Statement