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Math Prize 2012 Problem 16

Source:

September 24, 2012
geometryconicsellipseanalytic geometryabsolute value

Problem Statement

Say that a complex number zz is three-presentable if there is a complex number ww of absolute value 33 such that z=w−1wz = w - \frac{1}{w}. Let TT be the set of all three-presentable complex numbers. The set TT forms a closed curve in the complex plane. What is the area inside TT?
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