<s>[INST] Prove that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$ for positive $a,b,c$ Prove the following inequality: for
$a,b,c>0$
$$\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$$
What I tried is using substitution:
$p=a+b+c$
$q=ab+bc+ca$
$r=abc$
But I cannot reduce $a^2(b+c)(c+a)+b^2(a+b)(c+a)+c(a+b)(b+c) $ interms of $p,q,r$
 [/INST] Cauchy-Swartz:
$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+ \cdots +\frac{a_n^2}{b_n} \geq \frac{(a_1+a_2+ \cdots +a_n)^2}{b_1+b_2+ \cdot +b_n}$
So,
$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b} \geq \frac{(a+b+c)^2}{2(a+b+c)}=\frac{a+b+c}{2}$
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