There are some positive integer pairs(x,y) that satisfy $\frac{1}{ \sqrt{x}}+\frac{1}{ \sqrt{y}}=\frac{1}{ \sqrt{20}}$ How many different possible values of the product x and y are there?
I found one value when $x=y$
$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x}}=\frac{1}{\sqrt{20}}$
which simplifies to
$\frac{2\sqrt{x}}{x}=\frac{1}{\sqrt{20}}$
which gives $x=y=80$
therefore one possible value of the product is $80^2$
but I don't know how to prove this is the only solution or if there are more
suggestions, help, and solutions would all be appreciated.
taken from the 2019 IWYMIC/SAIMC(Question 7) https://chiuchang.org/imc/wp-content/uploads/sites/2/2019/08/SAIMC-2019_Keystage-3_Individual_Final.x17381.pdf