IMO 1970 LL FRA23

Let E be a finite set, PE the family of its subsets, and f a

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IMO 1970 LL FRA23

Origin: FRA

Problem

Let E be a finite set, PE the family of its subsets, and f a mapping from PE to the set of nonnegative real numbers such that for any two disjoint subsets A, B of E, f(A \cupB) = f(A) + f(B). Prove that there exists a subset F of E such that if with each A \subsetE we associate a subset A′ consisting of elements of A that are not in F, then f(A) = f(A′), and f(A) is zero if and only if A is a subset of F.