I was about to go to sleep, but I haven’t posted in a while and this one is quick. I don’t think I have anything very insightful to say about how I came across this solution, since it came to me without much fiddling around. So, rather than trying and failing to give a good explanation, I’ll just leave a hint. The full solution will be included with my next post.
2009-A1. Let be a real-valued function on the plane such that for every square in the plane, . Does it follow that for all points in the plane?