Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.4 National olympiad Prove it

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Example 1 Let $a, b, c>0$ , and $a b+b c+c a=1$ , prove: $\sqrt[3]{\frac{1}{a}+6 b}+$ $\sqrt[3]{\frac{1}{b}+6 c}+\sqrt[3]{\frac{1}{c}+6 a} \leqslant \frac{1}{a b c}$ . (45th IMO Shortlist Problem)

Solution

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