IMO 1976 LL CZS6

For each point X of a given polytope, denote by f(X) the sum

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IMO 1976 LL CZS6

Origin: CZS

Problem

For each point X of a given polytope, denote by f(X) the sum of the distances of the point X from all the planes of the faces of the polytope. Prove that if f attains its maximum at an interior point of the polytope, then f is constant.