Problem:
Twenty distinct points are marked on a circle and labeled through in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original points.
Solution:
Suppose , and are the labels of the three vertices of a triangle with . Note that , so one of or must be 2 , and furthermore, the other two primes must be twin primes. Thus must be one of
In particular, for any pairs of vertices , where , there are exactly two locations for the middle vertex that yield a triangle. There are pairs of vertices for every from 1 to 19 . Hence there are triangles satisfying the given conditions.
The problems on this page are the property of the MAA's American Mathematics Competitions