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for every x >= 1/2 exists n such that |x - n^2| <= \sqrt{x\frac{1}{4}}.

Source: 1965 Swedish Mathematical Competition p3

March 21, 2021
algebrainequalities

Problem Statement

Show that for every real x≥12x \ge \frac12 there is an integer nn such that ∣x−n2∣≤x−14|x - n^2| \le \sqrt{x-\frac{1}{4}}.
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