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7- m^2/n^2 >= a/n^2 if m/n < \sqrt7

Source: 1988 Swedish Mathematical Competition p5

March 28, 2021
inequalitiesalgebra

Problem Statement

Show that there exists a constant a>1a > 1 such that, for any positive integers mm and nn, mn<7\frac{m}{n} < \sqrt7 implies that 7−m2n2≥an2.7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .
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