CGMO 2013 Problem 8

Let $n$ ( $\ge 4$ ) be an even integer. We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$ -gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$ . A side is called positive if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called alternating if it satisfies both conditions:

(i) $2 \mid (i+j)$ ; and

(ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$ , then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$ .

Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.

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