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help: Find all positive real numbers

Source: 2013 China Mathematical Olympaid P3

January 12, 2013
inequalitiesceiling functiongeometryinequalities proposed

Problem Statement

Find all positive real numbers tt with the following property: there exists an infinite set XX of real numbers such that the inequality max⁡{∣x−(a−d)∣,∣y−a∣,∣z−(a+d)∣}>td \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td holds for all (not necessarily distinct) x,y,z∈Xx,y,z\in X, all real numbers aa and all positive real numbers dd.
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