Question 131
(i) If $x$, $y$ are positive real numbers, prove that:
$$
\frac{4}{x+y} \le \frac{1}{x} + \frac{1}{y}.
$$
(ii) If $\alpha$, $\beta$, $\gamma$, $\delta$ are positive real numbers, prove that
$$
\frac{2}{(\alpha + \beta)(\gamma + \delta) + (\beta + \gamma)(\alpha + \delta)} \le \frac{1}{(\alpha + \gamma)(\beta + \delta) + 4\alpha\gamma} + \frac{1}{(\alpha + \gamma)(\beta + \delta) + 4\beta\delta}.
$$
Determine when equality holds.
$$
\frac{4}{x+y} \le \frac{1}{x} + \frac{1}{y}.
$$
(ii) If $\alpha$, $\beta$, $\gamma$, $\delta$ are positive real numbers, prove that
$$
\frac{2}{(\alpha + \beta)(\gamma + \delta) + (\beta + \gamma)(\alpha + \delta)} \le \frac{1}{(\alpha + \gamma)(\beta + \delta) + 4\alpha\gamma} + \frac{1}{(\alpha + \gamma)(\beta + \delta) + 4\beta\delta}.
$$
Determine when equality holds.