Question 84
There are $2n$ distinct points on a circle. The numbers $1$ to $2n$ are distributed randomly over these points. Each point is connected to exactly one other point so that none of the resulting connecting segments cross. If a segment connects the numbers $a$ and $b$, we assign the segment the value $|a-b|$. Show that we can choose the segments so that the sum of these values equals $n^{2}$.