I thought up the following problem for a prize exam about 16 years ago. (I don't claim it wasn't known before; that's just when it occured to me.)
None of the contestants answered it. A few years later I put it on another prize exam, and Rennie Mirollo gave an elegant solution (which, alas, I have since forgotten).
THEOREM. Given five points on a circle (say, A, B, C, D, and E in circular order), the pentagram with those vertices (that is, the figure with edges AC, CE, EB, BD, and DA) is regular (that is--for instance--has all five angles ACE, CEB, EBD, BDA, and DAC equal) if and only if its five triangular "tips" all have the same area (figure out what I mean by tips by drawing the picture).