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Aime 2008 Ii 13

  • Thread starter ehrenfest
  • Start date
  • #1
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Homework Statement


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.


Homework Equations





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.
 

Answers and Replies

  • #2
737
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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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