Question 117
Let $a, b, c$ be positive real numbers with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Prove the inequality
$$
\sqrt{a b+c}+\sqrt{b c+a}+\sqrt{c a+b} \geq \sqrt{a b c}+\sqrt{a}+\sqrt{b}+\sqrt{c}
$$
$$
\sqrt{a b+c}+\sqrt{b c+a}+\sqrt{c a+b} \geq \sqrt{a b c}+\sqrt{a}+\sqrt{b}+\sqrt{c}
$$