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What if FE met rectangles?

Source: 2023 IRN-SGP-TWN Friendly Math Competition P2

July 16, 2023
functional equationalgebrageometry

Problem Statement

Let f:R2ā†’R+f: \mathbb{R}^{2} \to \mathbb{R}^{+}such that for every rectangle ABCDA B C D one has
f(A)+f(C)=f(B)+f(D). f(A)+f(C)=f(B)+f(D).
Let KLMNK L M N be a quadrangle in the plane such that f(K)+f(M)=f(L)+f(N)f(K)+f(M)=f(L)+f(N), for each such function. Prove that KLMNK L M N is a rectangle.
Proposed by Navid.
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