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Number of intersection points on bith sides of a line

Source: Serbia MO 2018 P3

April 2, 2018
combinatoricsinequalitiescombinatorial geometrydiscrete geometrySerbia

Problem Statement

Let nn be a positive integer. There are given nn lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these nn lines on both of its sides is at least ⌊(n−1)(n−2)10⌋.\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor. Notice that the points on the line are not counted. b) Find all nn for which there exists a configurations where the equality is achieved.
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