Question 29
If $\alpha$, $\beta$, $\gamma$ are positive real numbers such that $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = 3$, prove that
$$
\frac{\alpha + \beta}{\alpha^2 + \alpha\beta + \beta^2} + \frac{\beta + \gamma}{\beta^2 + \beta\gamma + \gamma^2} + \frac{\gamma + \alpha}{\gamma^2 + \gamma\alpha + \alpha^2} \le 2 .
$$
When is equality valid?
$$
\frac{\alpha + \beta}{\alpha^2 + \alpha\beta + \beta^2} + \frac{\beta + \gamma}{\beta^2 + \beta\gamma + \gamma^2} + \frac{\gamma + \alpha}{\gamma^2 + \gamma\alpha + \alpha^2} \le 2 .
$$
When is equality valid?