How dare they pick a past APMO problem?

BogG wrote: Let $a,b,c \in \mathbb{R^+}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show $\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}$. http://www.mathlinks.ro/Forum/viewtopic.php?highlight=APMO&t=82809

As a recall,someone said in this forum :"the Olympaid problem is recycling."

square it,we have to prove $\sum\sqrt{(ab+c)(ac+b)}\ge\sum\sqrt{ab}+\sum\sqrt{a^2bc}$ by Carlson ineq,$LHS\ge\sum(\sqrt{ab^2c}+\sqrt{ac})=RHS$

Let $f(x)=\sqrt x$. On $(0,\infty)$,$f(x)$ is concave. From $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1 $ we have: $a+b+c\geq 9$ $ab+bc+ca=abc\geq 27$ Let $a\geq b\geq c$ We have: $[a,b,c,\dfrac{abc}{9},\dfrac{abc}{9},\dfrac{abc}{9}]\succ [\dfrac{bc+a}{4},\dfrac{bc+a}{4},\dfrac{ca+b}{4},\dfrac{ca+b}{4},\dfrac{ab+c}{4},\dfrac{ab+c}{4}]$ And from Karamata, we are done.