{"status": "success", "data": {"description_md": "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\n\n$$\\sin^2{(\\pi x)} + \\sin^2{(\\pi y)} > 1$$
What is the maximum value of $P(a)?$\n\n$\\textbf{(A)}\\ \\frac{7}{12} \\qquad\\textbf{(B)}\\ 2 - \\sqrt{2} \\qquad\\textbf{(C)}\\ \\frac{1+\\sqrt{2}}{4} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{5}-1}{2} \\qquad\\textbf{(E)}\\ \\frac{5}{8}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2020 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/20_amc12B_p24"}}