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2004 IMO Shortlist Problems/A5

Problem

If $a,b,c$ are three positive real numbers such that $ab+bc+ca=1$, prove $<cmath> \sqrt[3]{\frac{1}{a}+6b}+\sqrt[3]{\frac{1}{b}+6c}+\sqrt[3]{\frac{1}{c}+6a}\le\frac{1}{abc}. </cmath>$

Solution

https://youtu.be/jmXSmmfO7pQ?si=dxJ6At7KHlcn2NT5 [Video Solution by little fermat]

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