#8 Measure your Calibre

The special mean of $x$ and $y$ is given as

$\text{S.M.} = \dfrac{\text{A.M. } + \text{ G.M.}}{2} = \dfrac{\frac{x+y}{2} + \sqrt{xy}}{2} = {\left( \dfrac{\sqrt x + \sqrt y}{2} \right)}^2$

Which is a perfect square integer for only perfect squares $x$ and $y$ with same parity.

Odd parity: $x, y \in \{ 1^2,3^2,5^2, \cdots, 43^2 \}$

Even parity: $x, y \in \{ 2^2,4^2,6^2, \cdots, 44^2 \}$

As $x \le y$ , we have $\dbinom{22}{2} + 22$ pairs for $(x, y)$ in each case.

This the total number of pairs is $2 \left[ \dbinom{22}{2} + 22 \right] = 506$ .