Can someone help me with this algebraic inequality? I found this inequality on twitter and I can't seem to prove the statement.
Prove that for $a,b,c > 0$ that 
$$
\frac{a+b+c}{2} \geq \frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c}
$$
After an hour (and a crick in my neck) I've only been able to turn it into 
$$
a^3(b+c)+b^3(a+c)+c^3(a+b)-2abc(a+b+c) \geq 0
$$
and I'm not even sure if that's much better. 

We have that by HM-AM inequality
$$\frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c}=\frac{1}{\frac1a+\frac1b} + \frac{1}{\frac1a+\frac1c}+ \frac{1}{\frac1b+\frac1c} \le$$ $$\le\frac{a+b}4+\frac{a+c}4+\frac{b+c}4=\frac{a+b+c}2$$
