VMO 2001 Problem 6

$a_1$, $a_2$, ... , $a_{2n}$ is a permutation of 1, 2, ... , 2n such that $|a_i$ - $a_{i+1}$| ≠ $|a_j$ - $a_{j+1}$| for i ≠ j. Show that $a_1$ = $a_{2n}$ + n iff 1 ≤ $a_{2i}$ ≤ n for i = 1, 2, ... n.