<s>[INST] Algebraic proof of a simple inequality Prove $\frac{a+b+c}{2} ≥ \frac{bc}{b+c} + \frac{ca}{c+a} + \frac{ab}{a+b}$
I feel like this problem would have a much simpler solution than just expaning out and simplifying. Any pointers are appreciated
 [/INST] By C-S
$$\sum_{cyc}\frac{ab}{a+b}=\sum_{cyc}\frac{1}{\frac{1}{a}+\frac{1}{b}}\leq\sum_{cyc}\frac{1}{\frac{(1+1)^2}{a+b}}=\sum_{cyc}\frac{a+b}{4}=\frac{a+b+c}{2}.$$
Another way:
It's $$\sum_{cyc}\frac{ab}{a+b}\leq\frac{a+b+c}{2}$$ or
$$\sum_{cyc}\left(\frac{a+b}{4}-\frac{ab}{a+b}\right)\geq0$$ or
$$\sum_{cyc}\frac{(a-b)^2}{a+b}\geq0.$$
Done!
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