Home > Combinatorics, Geometry, Problem Solving > IMC 2014 Day 1 Problem 5

## IMC 2014 Day 1 Problem 5

Let ${A_1A_2...A_{3n}}$ be a close broken line consisting of ${3n}$ line segments in the Euclidean plane. Suppose that no three of its vertices are collinear and for each index ${i=1,2,...,3n}$, the triangle ${A_iA_{i+1}A_{i+2}}$ has counterclockwise orientation and ${\angle A_iA_{i+1}A_{i+2}=60^\circ}$, using the notation modulo ${3n}$. Prove that the number of self-intersections of the broken line is at most ${\displaystyle \frac{3}{2}n^2 -2n+1}$.

IMC 2014 Day 1 Problem 5