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min sum of products of elements of 2 dijoint sets whose union is {1,...,8}

Source: 2020 Greek JBMO TST p4

November 14, 2020
algebraSumProductinequalitiesmin

Problem Statement

Let AA and BB be two non-empty subsets of X={1,2,...,8}X = \{1, 2, . . . , 8 \} with A∪B=XA \cup B = X and A∩B=∅A \cap B = \emptyset. Let PAP_A be the product of all elements of AA and let PBP_B be the product of all elements of BB. Find the minimum possible value of sum PA+PBP_A +P_B.
PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3
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