IMO 2019 Shortlist C6
Let n ą 1 be an integer. Suppose we are given 2n points in a plane such that no three of them are collinear. The points ...
Category: Combinatorics
Problem
Let n ą 1 be an integer. Suppose we are given 2n points in a plane such that no three of them are collinear. The points are to be labelled A1, A2, ..., A2n in some order. We then consider the 2n angles =A1A2A3, =A2A3A4, ..., =A2n´2A2n´1A2n, =A2n´1A2nA1, =A2nA1A2. We measure each angle in the way that gives the smallest positive value (i.e. between 0˝ and 180˝ ). Prove that there exists an ordering of the given points such that the resulting 2n angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group. (USA)