If a^{2}-bc=\alpha, b^{2}-ac=\beta, c^{2}-ab=\gamma, then what is \frac{a\alpha+b\beta+c\gamma}{(a+b+c)(\alpha+\beta+\gamma)} equal to?

Explanation

Substitute \alpha, \beta, \gamma in numerator: a(a^2-bc) + b(b^2-ac) + c(c^2-ab) = a^3+b^3+c^3-3abc. The denominator is (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc. Thus, the ratio is 1.

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