2013 IMO Problems/Problem 6
Revision as of 12:50, 21 June 2018 by Illogical 21 (talk | contribs)
Problem
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