<s>[INST] Prove : $\dfrac{a}{ac+1}+ \dfrac{b}{ab+1}+ \dfrac{c}{bc+1} \le \frac 12 (a^2+b^2+c^2)$ $a;b;c\in \mathbb{R}^{+}$ such that $abc=1$
Prove : $\frac{a}{ac+1}+ \frac{b}{ab+1}+ \frac{c}{bc+1} \leq \frac{1}{2}(a^2+b^2+c^2)$
 [/INST] $\frac{a}{ac+1} \leq \frac{a}{2\sqrt{ac}} = \frac{1}{2} a \sqrt{b}$. So it suffices to show that 
$$a^2+b^2+c^2 \ge (abc)^{1/6} (a\sqrt{b} + b\sqrt{c} + c\sqrt{a}) = \sum_{cyc} a^{7/6}b^{2/3}c^{1/6}$$
This follows from adding up the AM-GM inequalities: 
$$7a^2+4b^2+c^2 \ge 12 a^{7/6}b^{2/3}c^{1/6}$$ 
and its cyclic variants.
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