Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy $S(P) \geq 2A(P)$? If P is a rectangle, then $S(P) = 2A(P)$.

Supposing this is true, is there a lower bound with some parameter that would detect, for instance, that for P a triangle, $S(P) = 3A(P)$?

A friend mentioned this question to me several months ago, and it has been bothering me (and several other people) since. I've heard claims of a Fourier-analytic proof, so I believe it isn't open. On the other hand, it sounds like a problem from an IMO or Putnam competition, and it feels like it ought to have a clean solution.