Let $r$ be the radius of the circle, then for the smaller square we can apply Pythagoras' Theorem:

$r^2=(\sqrt{x})^2+(\sqrt{x}/2)^2=x+x/4=5x/4$

and for the larger square:

$r^2=(\sqrt{y}/2)^2+(\sqrt{y}/2)^2=y/2$

So $x:y = 2:5$.

*This problem is taken from the UKMT Mathematical Challenges.*