Extra Credit: Go Figure

January 31, 2010
 

Mathematically Speaking? In the William Lowell Putnam Mathematical Competition, answers are in the form of proofs that illustrate the problem-solving process. Anyone with a solid background in high-school math can do well—in theory.

Can you solve this problem from this year's Putnam test?

Problem:
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 is follow that f(P)=0 for all points P in the plane?

Solution:
Yes, it does follow. Let P be any point in the plane. Let ABCD be any square with center P. Let E; F; G;H be the midpoints of the segments AB;BC;CD;DA, respectively. The function f must satisfy the equations

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

If we add the last four equations, then subtract the first equation and twice the second equation, we obtain 0 =4f(P), whence f(P) = 0.

Remark: Problem 1 of the 1996 Romanian IMO team selection exam asks the same question with squares replaced by regular polygons of any (fixed) number of vertices.