<s>[INST] Prove that $\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$ Let $a,b,c$ are positive real numbers such that, $a^2+b^2+c^2=3$. Prove that 
 $$\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$$
 [/INST] As indicated in tipshoni’s comment, the inequality to be shown is equivalent to 
$$
\sum_{cyc}\frac{a^2+b^2}{a+b} \geq 3 \tag{1}
$$
This problem is question number 44 in chapter 1 of Vasile Cirtoaje’s book
"Algebraic Inequalities" (available here).

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