A Nice Math Olympiad Problem

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[@GarryBoyer]

You can save the first 3 or so minutes by just letting [t = (x/y)] at the very beginning. Substitute into the original equation, then [ty+y = 6y√t]. Factor out y, then [t+1 = 6√t]. Square, then [t^2 + 2t + 1 = 36*t] which is much simpler than the quadratics you had. With the quadratic formula, [t = 17 ± 12√2].