An Inequality Made Up Of Greek Alphabets

Calculus Level 5

Let \(0<\alpha,\beta,\gamma<\frac{1}{2}\) be real numbers with \(\alpha+\beta+\gamma=1.\) The minimum value of \(\delta\) which satisfies the inequality

\[\alpha^3+\beta^3+\gamma^3+4\alpha\beta\gamma \leq \delta\]

can be expressed in the form \( \frac{a}{b} \), where \(a\) and \(b\) are positive coprime integers. Find \(a+b\).

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