Question 39
The numbers $1, 2, \ldots, 2006$ are written around a circle. An allowed operation is to exchange two adjacent numbers. After a sequence of such exchanges, each number was located $13$ positions to the right of its initial position. We split the numbers $1, 2, \ldots, 2006$ into $1003$ disjoint pairs, arbitrarily.
Show that in one of the operations the two numbers of at least one of such pairs were exchanged.
Show that in one of the operations the two numbers of at least one of such pairs were exchanged.