It's a nice problem; I have no idea how to solve it, though, and cannot find any solutions online.
Let be a continuous function on such that the double integral of , taken over any rectangle of area one, is zero. Does this necessarily imply that is identically zero?
As far as I recall, that is the precise wording of the question; there is not elaboration on what type of integral or what type of continuity it exhibits, but because it is on the Putnam I think the interpretation should be Riemann double integral, and absolute continuity.
Intuitions? Thoughts about a solution? IMO it seems to be true, but have no idea how to prove. Possibly an argument from Fourier, but I don't want to mess with it.