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Probabilistic sum is strictly increasing

Source: Bulgarian Winter Tournament 2024 12.3

January 28, 2024
combinatoricsprobability

Problem Statement

Let nn be a positive integer and let A\mathcal{A} be a family of non-empty subsets of {1,2,…,n}\{1, 2, \ldots, n \} such that if A∈AA \in \mathcal{A} and AA is subset of a set B⊆{1,2,…,n}B\subseteq \{1, 2, \ldots, n\}, then BB is also in A\mathcal{A}. Show that the function f(x):=∑A∈Ax∣A∣(1−x)n−∣A∣f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|} is strictly increasing for x∈(0,1)x \in (0,1).
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