2020 AMC 12B Problems/Problem 25

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 $P(a)?$

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

Solution

Let's start first by manipulating the given inequality.

$\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1$ $\sin^{2}{(\pi x)}>1-\sin^{2}{(\pi y)}=\cos^{2}{(\pi y)}$

Let's consider the boundary cases: $\sin^{2}{(\pi x)}=\cos^{2}{(\pi y)}$ and $\sin^{2}{(\pi x)}=-\cos^{2}{(\pi y)}$

$\sin{(\pi x)}=\cos{(\pi y)}=sin{(\frac{\pi}{2}\pm \pi y)}$

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2020 AMC 12B Problems/Problem 25

Problem 25

Solution