<s>[INST] Proving $\sum_{cyc}\frac{ab}{a+b+2c} \le \frac{1}{4}(a+b+c)$ for positive real $a$, $b$, $c$ Prove that $$\frac{ab}{a+b+2c} + \frac{bc}{b+c+2a} + \frac{ca}{c+a+2b} \le \frac{1}{4}(a+b+c)$$
for positive real numbers $a$, $b$, and $c$.
 [/INST] By AM-HM
$$\sum_{sym} \frac{bc}{2a+b+c} \leq \sum_{sym} \frac{bc}{4}(\frac{1}{a+b}+\frac{1}{a+c})=\frac{a+b+c}{4}.$$
Seems like you are new to inequalities. To study this topic, I recommend you this book.
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