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Sets of Positive Reals

Source: 2011 MMO Problem #2

September 11, 2011
ratioinequalitiesfunctiongeometric sequencecombinatorics unsolvedcombinatorics

Problem Statement

Let AA be a finite set of positive reals, let B={x/y∣x,y∈A}B = \{x/y\mid x,y\in A\} and let C={xy∣x,y∈A}C = \{xy\mid x,y\in A\}. Show that ∣A∣⋅∣B∣≤∣C∣2|A|\cdot|B|\le|C|^2. (Proposed by Gerhard Woeginger, Austria)
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