MathNet 2013 Question 42

Let $n \ge 4$ be an even number. At the vertices of a regular $n$-gon we write in an arbitrary way $n$ distinct real numbers. Starting from one edge, we name all the edges in a clockwise way by $e_1, e_2, \dots, e_n$. An edge is called "positive", if the difference of the numbers at its endpoint and its start point is positive. A set of two edges $\{e_i, e_j\}$ is called "crossing", if $2 \mid (i+j)$, and among the four, the numbers written at their vertices, the largest and the third largest ones belong to the same edge. Prove that the number of crossings and the number of positive edges have different parity.

Example of edge indexing for n=6.