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

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