MathNet 2013 Question 14

Suppose that $x$ and $y$ are chosen randomly and uniformly from $(0,1)$. What is the probability that $\left\lfloor\sqrt{\frac{x}{y}}\right\rfloor$ is even? Hint: $\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$.