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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?

Answer:

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.

Comments:

I don't understand why

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)?