Define

\(\textcolor{blue}{f(x) = \frac{1}{x + \frac{1}{x}}}\)

and

\(\textcolor{green}{g(x) = \frac{1}{x - \frac{1}{x}}}\)

Find the square of the largest real solution to the equation

\(\textcolor{purple}{ (g(x))^2 - (f(x))^2 = \tfrac54}\)

AoPS.Morrisville Nov 23, 2019