2020 AMC 12B Problem 25

For each real number a with 0 \leq a \leq 1, let numbers x and y be chosen independently at random from the intervals [0, a] and [0, 1], respectively, and let P(a) be the probability that

\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1 What is the maximum value of P(a)?

\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.