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Algebra Difficulty 6.1 National olympiad Prove it

Example 1.1.3. Let a,b,ca, b, c be positive real numbers. Prove that
(1+xy)(1+yz)(1+zx)≥2+2(x+y+z)xyz3\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right) \geq 2+\frac{2(x+y+z)}{\sqrt[3]{x y z}}

Solution

1.0. AM−GMA M-G M inequality
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Solution. Certainly, the problem follows the inequality
xy+yz+zx≥x+y+zxyz3\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{x+y+z}{\sqrt[3]{x y z}}
which is true by AM-GM because
3(xy+yz+zx)=(2xy+yz)+(2yz+zx)+(2zx+xy)≥3xxyz3+3yxyz3+3zxyz33\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)=\left(\frac{2 x}{y}+\frac{y}{z}\right)+\left(\frac{2 y}{z}+\frac{z}{x}\right)+\left(\frac{2 z}{x}+\frac{x}{y}\right) \geq \frac{3 x}{\sqrt[3]{x y z}}+\frac{3 y}{\sqrt[3]{x y z}}+\frac{3 z}{\sqrt[3]{x y z}}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.