Problem #1252


Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac\pi 2$ and $0\leq y\leq \frac\pi 2$. What is the area of the subset of $S$ for which $\sin^2 x - \sin x\sin y + \sin^2 y\le \frac 34$?

$\mathrm{(A)}\ \frac {\pi^2}9 \qquad \mathrm{(B)}\ \frac {\pi^2}8 \qquad \mathrm{(C)}\ \frac {\pi^2}6 \qquad \mathrm{(D)}\ \frac {3\pi^2}{16}  \qquad \mathrm{(E)}\ \frac {2\pi^2}9$

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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