Putnam 2009-A1

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 f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane?

Good luck!

Good luck!