some putnam question
Level
2
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
no
no enough information
i don't know ;)
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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 =4 f (P), whence f (P) = 0.