<s>[INST] the inequality $\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$ How to show that $$\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$$ for $a,b,c>0$?
I tried to prove $$\frac{a^4}{a^3+b^3}\ge \frac {5a}4+\frac{-3b}4$$
but could not continue. 
Give me ideas, please.
 [/INST] Here is a solution from "Secrets in Inequalities" by Pham Kim Hung.

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