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a/b - 1/2ab > \sqrt2 for pos.integers with a/b > \sqrt2

Source: 2022 Czech-Polish-Slovak Match Junior, individual p4 CPSJ

September 3, 2022
inequalitiesnumber theory

Problem Statement

Let aa and bb be positive integers with the property that ab>2\frac{a}{b} > \sqrt2. Prove that ab−12ab>2\frac{a}{b} - \frac{1}{2ab} > \sqrt2
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