Problem

Source: Iranian third round 2019 finals Combinatorics exam problem 2

Tags: combinatorics



Let $T$ be a triangulation of a $100$-gon.We construct $P(T)$ by copying the same $100$-gon and drawing a diagonal if it was not drawn in $T$ an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in $T$.Let $f(T)$ be the number of intersections of diagonals in $P(T)$.Find the minimum and maximum of $f(T)$.