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Putnam 1949 B1

Source: Putnam 1949

March 20, 2022
Putnaminterval

Problem Statement

Each rational number pq\frac{p}{q} with p,qp,q coprime of the open interval (0,1)(0,1) is covered by the closed interval [pq−14q2,pq+14q2]\left[\frac{p}{q}-\frac{1}{4q^{2}}, \frac{p}{q}+\frac{1}{4q^{2}}\right]. Prove that 22\frac{\sqrt{2}}{2} is not covered by any of the above closed intervals.
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