MathDB
Trig Probability

Source: 2020 AMC 12B #25

February 6, 2020
trigonometryAMCAMC 12AMC 12 Bprobability2020 AMC 12B2020 AMC

Problem Statement

For each real number aa with 0≤a≤10 \leq a \leq 1, let numbers xx and yy be chosen independently at random from the intervals [0,a][0, a] and [0,1][0, 1], respectively, and let P(a)P(a) be the probability that sin⁡2(πx)+sin⁡2(πy)>1.\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1. What is the maximum value of P(a)?P(a)?
<spanclass=′latex−bold′>(A)</span> 712<spanclass=′latex−bold′>(B)</span> 2−2<spanclass=′latex−bold′>(C)</span> 1+24<spanclass=′latex−bold′>(D)</span> 5−12<spanclass=′latex−bold′>(E)</span> 58<span class='latex-bold'>(A)</span>\ \frac{7}{12} \qquad<span class='latex-bold'>(B)</span>\ 2 - \sqrt{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{1+\sqrt{2}}{4} \qquad<span class='latex-bold'>(D)</span>\ \frac{\sqrt{5}-1}{2} \qquad<span class='latex-bold'>(E)</span>\ \frac{5}{8}
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