Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.8 AIME, harder Prove it

13. Let $a, b, c>0$ . Prove:
$\frac{a}{\sqrt{a^{2}+8 b c}}+\frac{b}{\sqrt{b^{2}+8 c a}}+\frac{c}{\sqrt{c^{2}+8 a b}} \geqslant 1 .$

Solution

13. First use the Rodenburg inequality, then use the arithmetic mean-geometric mean inequality.

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