<s>[INST] Algebraic Inequality involving AM-GM-HM If $$a,b,c \;\epsilon \;R^+$$ 
Then show, $$\frac{bc}{b+c} + \frac{ab}{a+b} + \frac{ac}{a+c} \leq \frac{1}{2} \Bigl(a+b+c\Bigl)$$
I have solved a couple of problems using AM-GM but I have always struggled with selecting terms for the inequality.
I tried writing $a+b+c$ as $x$ and re-writing the inequality as $$\frac{bc}{x-a} + \frac{ab}{x-c} + \frac{ac}{x-b} \leq \frac{1}{2} \Bigl(x\Bigl)$$ 
This didn't help so instead I wrote it as $$2\Biggl(\frac {1}{a} + \frac{1}{b} + \frac{1}{c}\Biggl ) \leq \frac{1}{2} \Bigl(a+b+c\Bigl)$$
Not sure what to to do next.
Any help would be appreciated.
 [/INST] Proof
Just by $H_n \leq A_n$, we have 
\begin{align*}
\frac{bc}{b+c} + \frac{ab}{a+b} + \frac{ac}{a+c}&=\frac{1}{2}\left(\frac{2}{\frac{1}{b}+\frac{1}{c}}+\frac{2}{\frac{1}{a}+\frac{1}{b}}+\frac{2}{\frac{1}{a}+\frac{1}{c}}\right)\\&\leq \frac{1}{2}\left(\frac{b+c}{2}+\frac{a+b}{2}+\frac{a+c}{2}\right)\\&=\frac{a+b+c}{2}.
\end{align*}
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