(This is Problem 622 from the College Math Journal, proposed by Michael Golomb.)
Identical beads are distributed among the vertices of a regular n-gon in such a way that the center of mass of the distribution is at the center of the n-gon. Show that:

1. if n is a power of 2, the number of beads at any vertex is the same as the number on the diametrically opposite vertex;
2. this need not be true if n is not a power of 2.

Hint:
(I'll be posting the solution soon.) In the meantime, you may want to use the following theorem about cyclotomic (circle-splitting) polynomials. See, for example, Theorem 6.5.6 on page 280 of Abstract Algebra by I.N. Herstein (2nd ed.).

Theorem: The nth cyclotomic polynomial is the minimal polynomial in Q[x] for the primitive nth root of unity.

Solution: You will need Adobe Acrobat Reader (available for free download from www.adobe.com) to read the solution. Click here.