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Inequality of cardinals

Source: Romania National Olympiad 2015, grade x, p.4

August 23, 2019
inequalitiesalgebracombinatoricsDiscrete

Problem Statement

Let be a finite set A A of real numbers, and define the sets S±={x±y∣x,y∈A}. S_{\pm }=\{ x\pm y| x,y\in A \} . Show that ∣A∣⋅∣S−∣≤∣S+∣2. \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 .
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