<s>[INST] Prove the inequality: $\left(\frac{a}{a+2b}\right)^2+\left(\frac{b}{b+2c}\right)^2+\left(\frac{c}{c+2a}\right)^2 \geq \frac{1}{3}$ 
Let $a,b,c$ are all positive real numbers. Prove
$$\left(\frac{a}{a+2b}\right)^2+\left(\frac{b}{b+2c}\right)^2+\left(\frac{c}{c+2a}\right)^2 \geq \frac{1}{3}$$

Can anyone give a hint?
 [/INST] By Cauchy-Schwarz,
$$\sum \limits_{\text{cyc}} \left ( \dfrac{a}{a+2b} \right )^2 \geq \dfrac{1}{3} \left(\sum \limits_{\text{cyc}} \left ( \dfrac{a}{a+2b} \right )\right)^2 = \dfrac{1}{3}\left( \sum \limits_{\text{cyc}} \left ( \dfrac{a^2}{a^2+2ab} \right) \right)^2 \geq \dfrac{1}{3} \left ( \dfrac{(a+b+c)^2}{(a+b+c)^2} \right )^2 = \dfrac{1}{3}.$$
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