Problem

Source: 2019 Romania JBMO TST 5.4

Tags: combinatorics, numbers, circle



The numbers from $1$ through $100$ are written in some order on a circle. We call a pair of numbers on the circle good if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.