IMO 1984 LL GDR29

Origin: GDR

Problem

Let Sn = {1, . . . , n} and let f be a function that maps every subset of Sn into a positive real number and satisfies the following con- dition: For all A \subseteqSn and x, y \inSn, x ̸= y, f(A \cup{x})f(A \cup{y}) \leq f(A \cup{x, y})f(A). Prove that for all A, B \subseteqSn the following inequality holds: f(A) \cdot f(B) \leqf(A \cupB) \cdot f(A \capB).