<s>[INST] $\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$ Proof Does anyone know hot to prove this inequality?
Having: $a, b, c \gt 0$
$$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$$
I tried with the AM-GM inequality but I couldn't get any improvement. I'm on a still point and I don't know how to continue.
Furthermore, I don't know how to get rid of the known term. I tried with the AM-HM inequality but I'm still not getting any results.
Also, just some hints would be appreciated, thanks
 [/INST] Because
$$\sum \frac{a^2}{c+a} = \sum \frac{c^2}{c+a}.$$
Therefore, we need to prove
$$2\left(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\right) \geqslant \frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}.$$
Suppose $a \geqslant b \geqslant c,$ then
$$a^2 \geqslant b^2 \geqslant c^2 \quad \text{and} \quad \frac{1}{b+c} \geqslant \frac{1}{c+a} \geqslant \frac{1}{a+b}.$$
Using the Rearrangement inequality
$$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b} \geqslant \frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}.$$
$$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b} \geqslant \frac{b^2}{a+b}+\frac{c^2}{b+c}+\frac{a^2}{c+a}.$$
The proof is completed.
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