3
$\begingroup$

Let $a,\ b,\ c$ be positive real numbers. Prove that: $$\left(a+\dfrac{bc}{a}\right)\left(b+\dfrac{ca}{b}\right)\left(c+\dfrac{ab}{c}\right)\geq 4\sqrt[3]{\left(a^{3}+b^{3}\right)\left(b^{3}+c^{3}\right)\left(c^{3}+a^{3}\right)}\tag1$$

1 Answers 1

1

By AM-GM $$(a^2+bc)(b^2+ac)=c(a^3+b^3)+ab(c^2+ab)\geq2\sqrt{abc(a^3+b^3)(c^2+ab)}.$$ Thus, $$\prod_{cyc}\left((a^2+bc)(b^2+ac)\right)\geq8\sqrt{a^3b^3c^3\prod_{cyc}(a^3+b^3)\prod_{cyc}(c^2+ab)}$$ or $$\prod_{cyc}(a^2+bc)^3\geq64a^3b^3c^3\prod_{cyc}(a^3+b^3)$$ or $$\prod_{cyc}\left(a+\frac{bc}{a}\right)\geq4\sqrt[3]{\prod_{cyc}(a^3+b^3)}.$$ Done!