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Smallest real in a framed set

Source: Brazilian Mathematical Olympiad 2021, Level 3, Problem 4

February 8, 2022
algebrasquareminimum valuecombinatorics

Problem Statement

A set AA of real numbers is framed when it is bounded and, for all a,b∈Aa, b \in A, not necessarily distinct, (a−b)2∈A(a-b)^{2} \in A. What is the smallest real number that belongs to some framed set?
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