Question 19
Let $n$ be a positive integer. The numbers $1, 2, 3, \ldots, 2n$ are randomly assigned to $2n$ distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.
Show that one can choose $n$ pairwise non-intersecting chords such that the sum of the values assigned to them is $n^{2}$.
Show that one can choose $n$ pairwise non-intersecting chords such that the sum of the values assigned to them is $n^{2}$.