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Part of 2001 Miklós Schweitzer

Problems(1)

Miklós Schweitzer 2001 Problem 1

Source:

2/12/2017
Let f ⁣:2S→Rf\colon 2^S\rightarrow \mathbb R be a function defined on the subsets of a finite set SS. Prove that if f(A)=F(S\A)f(A)=F(S\backslash A) and max⁡{f(A),f(B)}≥f(A∪B)\max \{ f(A), f(B)\}\geq f(A\cup B) for all subsets A,BA, B of SS, then ff assumes at most ∣S∣|S| distinct values.
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