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- Feb 14, 2012

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Problem:

While there do not exist pairwise distinct real numbers $x, y, z$ satisfying $x^2+y^2+z^2=xy+yz+xz$, there do exist complex numbers with that property. Let $x, y, z$ be complex numbers such that $x^2+y^2+z^2=xy+yz+xz$ and $|x+y+z|=21$.

Given that $|x-y|=2\sqrt{3}$, $|x|=3\sqrt{3}$, compute $|y^2|+|z^2|$.

Attempt:

I don't believe we have to assign all those variables $a, b, c, d, e, f$ so that we have $x=a+bi$, $y=c+di$ and $z=e+fi$ in order to begin attacking the problem, thus I don't know how to solve it.

Any help would be very much appreciated. Thanks.