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Geo FE with orthocenter

Source: Philippine MO 2023/8

March 19, 2023
geometryfunctional equationcombinatorial geometryPMOfunctionnice

Problem Statement

Let S\mathcal{S} be the set of all points in the plane. Find all functions f:S→Rf : \mathcal{S} \rightarrow \mathbb{R} such that for all nondegenerate triangles ABCABC with orthocenter HH, if f(A)≤f(B)≤f(C)f(A) \leq f(B) \leq f(C), then f(A)+f(C)=f(B)+f(H).f(A) + f(C) = f(B) + f(H).
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