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Beautiful inequality on sets

Source: 2021 China Girls Math Olympiad

August 13, 2021
inequalitiesSubsetsCGMO

Problem Statement

Given a finite set SS, P(S)P(S) denotes the set of all the subsets of SS. For any f:P(S)→Rf:P(S)\rightarrow \mathbb{R} ,prove the following inequality:∑A∈P(S)∑B∈P(S)f(A)f(B)2∣A∩B∣≥0.\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.
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