Triple Cube Sum

Algebra Level 5

Suppose that the following holds for complex numbers \(a,b,c\): \[\begin{align} a + b + c & = 6 \\ \frac{a-b}{c} + \frac{c-a}{b} + \frac{b-c}{a} & = \left(1 + \frac{a}{b}\right) \left(1 + \frac{b}{c}\right) \left(1 + \frac{c}{a}\right) \\ \frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} + \frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2} & = -2 \end{align}\] What is the value of \(a^3 + b^3 + c^3\)?

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