We have 2020 distinct numbers arranged in a circle. We say that a pair of numbers A, B is dominating if A and B are not next to each other on the circle, and on one of the two arcs between A and B, all numbers are smaller than both A and B. Find the number of dominating pairs, and show that the answer is the same, no matter how the numbers are arranged.
I've experimented with smaller cases, drawing a circle of 9, 11, 13 numbers and connecting the dominant pairs. For a while, I thought the number of pairs was just n/2 rounded up, but as the n increased, the number of pairs increased much more. I fear experimenting manually with larger numbers, because that leaves much more room for error. I long for a logical way to solve this.