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Aime 2008 Ii 13

  1. May 20, 2008 #1
    1. The problem statement, all variables and given/known data
    A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let R be the region outside the hexagon, and let S = [itex] \{ 1/z |x \in R} [/itex]. Then the area of S has the form a [itex]\pi +\sqrt b[/itex], where a and b are positive integers. Find a+b.


    2. Relevant equations



    3. The attempt at a solution
    This should only require high school math although there is probably a solution using the fact that 1/z is a Mobius transformation or something else in complex analysis.

    The hexagon is contained in the closed of radius 1/sqrt(3) center at the origin, which means that S is contained inside of the closed disk of radius \sqrt3 centered at the origin. So basically we need to figure out what to subtract off of 3 \pi. Anyone know how to do that?

    Please just give a hint.
     
  2. jcsd
  3. May 20, 2008 #2
    What you are trying to do is a good idea, but there doesn't seem to be an easy way to find the cutoffs, as you've noticed.

    I caved and peeked at what others did. To solve it without calculus, you need to know what an inversion is. That said, there is a fairly simple solution with calculus that a lot of high school students are probably more familiar with than with inversions.
     
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