Question 48
If $a$, $b$, $c$ are positive real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 3$, prove that
$$
\frac{a+b+c-1}{\sqrt{2}} \ge \frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}
$$
When does equality hold?
$$
\frac{a+b+c-1}{\sqrt{2}} \ge \frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}
$$
When does equality hold?