<s>[INST] Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$ I faced problem proving this inequality for positive $a$, $b$, $c$:
$a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$
I tried to simplify it and I got that:
$bc^3 + a^3 c + a b^3 \geq a b^2 c + a^2 bc + a b c^2$
Then I tried to prove it using MMI, but it gave me nothing.
Please just give me a hint, maybe I am missing something important.
 [/INST] An AM-GM proof
By AM-GM inequality $$ab^3+abc^2\geq2\sqrt{a^2b^4c^2}=2ab^2c\tag{1}$$ Similarly we get $$bc^3+bca^2\overset{\text{AM-GM}}\geq2abc^2\tag{2}$$ and
$$ca^3+cab^2\overset{\text{AM-GM}}\geq2a^2bc\tag{3}$$
Hence adding $(1),(2),(3)$ we get the desired inequality.
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