2013 IMO Problems/Problem 6
Let be an integer, and consider a circle with equally spaced points marked on it. Consider all labellings of these points with the numbers such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels with , the chord joining the points labelled and does not intersect the chord joining the points labelled and .
Let be the number of beautiful labelings, and let N be the number of ordered pairs of positive integers such that and . Prove that