Let $a, b, c, x, y$ and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}$. If $xy+yz+zx=1000$ and $x+y+z=2016$, find the value of $xyz$.
Source: 2016 Nigerian Senior MO Round 2
Tags: complex numbers, algebra, symmetry
Let $a, b, c, x, y$ and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}$. If $xy+yz+zx=1000$ and $x+y+z=2016$, find the value of $xyz$.