Earlier this evening I wrote:

The quadratic mean, q, is defined as: (a^2 + b^2) / 2 = q^2 The citardauq mean, c, is defined as: (1/a^2 + 1/b^2)/2 = 1/c^2

Do some algebra, and get c^2 = (a^2 b^2)/(q^2), which shows that c can't be an integer unless q is an integer. No, wait, no it doesn't; q could be a rational number or the square root of a rational number. Back to the drawing board. Sigh.

That was stupid. Since all numbers are positive, I can simply take the square root of both sides and get c = ab/q. And by construction, q can only be an integer or the square root of an integer. So I've proven that the citardauq mean of two positive integers can only be an integer when the quadratic mean is an integer. QED.