4/20 Test/Reflection

So I took the first day of the 2009 Putnam Exam yesterday. It was, as predicted, much harder than the previous exam I took, as that one was many years earlier. I solved one problem, and made some progress on another, but that was about it. I again found it hard to concentrate sometime after the 2-2.5 hour mark. It is probably a good idea to read up on some sort of "concentration exercises". However, I'm really happy I changed my project, and I think forcing myself to do competition math almost daily is definitely a good thing.

Without further ado, here is the problem I solved:


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?

My initial guess was that yes, this is true. (For the following reason: if it weren't I would probably have to find a counterexample, which looks in this case very difficult and sort of annoying. (It's a little more than that, but if you do enough math problems, you get a sort of intuition for this kind of thing).

So I began to look for a "picture", that would confirm this. Basically a figure with a bunch of squares with some common points, I could apply the equation on each square and get some helpful cancellations.

As it turns out, this approach was successful, with the following solution:

Pick a point P in the plane.

Make it the center of some square ABCD.

Label the midpoints of AB, BC, CD, DA, E,F,G,H respectively. (The picture should look like this abysmal MS Paint diagram)

EFGH is a square (this is very easy to verify), as are AEPH, etc. (these are the "quarter squares")

So applying the function to each point in each quarter square and adding them gives:

f(A)+ f(E)+f(P)+f(H) + f(B)+f(E)+f(P)+f(F) + f(C)+f(F)+f(P)+f(G) +F(D)+f(G)+F(P)+f(H)=0, so

f(A)+f(B)+f(C)+f(D)+2(f(E)+f(F)+f(G)+f(H)))+4f(P)=0

but ABCD, and EFGH are squares, so f(A)+f(B)+f(C)+f(D)+2(f(E)+f(F)+f(G)+f(H)))=0

So f(P)=0, and we can make this construction for any point P, so we are done.