How Many?

Knowing that $\frac{1}{\sqrt{20}} = \frac{1}{2\sqrt{5}}$ , and also knowing that $\frac{1}{2} = \frac{1}{4} + \frac{1}{4} = \frac{1}{3} + \frac{1}{6},$ we can deduce:

$\frac{1}{2\sqrt{5}} = \frac{1}{4\sqrt{5}} + \frac{1}{4\sqrt{5}} = \frac{1}{\sqrt{80}} + \frac{1}{\sqrt{80}}$

and

$\frac{1}{2\sqrt{5}} = \frac{1}{3\sqrt{5}} + \frac{1}{6\sqrt{5}} = \frac{1}{\sqrt{45}} + \frac{1}{\sqrt{180}}$

which generates the $\boxed{three}$ ordered integral pairs: $(x,y) = (45,180); (80,80); (180,45)$ .

@Pi Han Goh can you please post a solution for this