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A new variant of an old problem on rational approximations to square roots

Source: Baltic Way 2018, Problem 17

November 6, 2018
number theory

Problem Statement

Prove that for any positive integers p,qp,q such that 11>pq\sqrt{11}>\frac{p}{q}, the following inequality holds: 11−pq>12pq.\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.
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