Many Variables

Algebra Level 5

\[\begin{align*} x + y + z &= 1, \\ xa^2 + yb^2 + zc^2 &= w^2, \\ xa^3 + yb^3 + zc^3 &= w^3, \\ xa^4 + yb^4 + zc^4 &= w^4 \end{align*} \]

The real numbers \(w\), \(a\), \(b\), \(c\) are distinct, such that there exist real numbers \(x\), \(y\), and \(z\) satisfying the system of equations above.

If \( (a,b,c) = (13, 17, 19) \), there is a unique value of \(|w|\) such that it is less than 6. Find the value to the nearest tenth.

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