Prove f(P) = 0 for Any Point P in the Plane | Putnam A1 Question Help

• john562
In summary, the conversation discusses a real-valued function on the plane that satisfies certain equations for every square in the plane. The question asks if this means the function must equal 0 for all points in the plane. The answer explains that it does follow, using a specific example with smaller squares within a larger square to show how adding certain equations and subtracting others leads to the conclusion that the function must equal 0 for all points. There is also a comment asking for clarification on the equations for smaller squares within the larger square.
john562
Question:
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?

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.

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)?

Those are smaller squares within the larger square ABCD. To be specific, they are the four quarters of ABCD

I don't understand how adding them equals zero.

what is this? like the easiest putnam problem ever?

These equations are obtained by considering the square ABCD with center P and its four smaller squares (AEFP, BFGP, CGHP, DHPE) formed by connecting P to the midpoints of the sides of ABCD. By the given property of f, the sum of the values at the four corners of each of these squares must equal 0. Therefore, we can write the equations as shown above, where P is the shared corner of all four smaller squares. By adding and subtracting these equations as shown, we can eliminate the values at the corners and isolate f(P), which must equal 0.

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