Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it Soviet Union

Problem:
Show that $x^4 + y^4 + z^2 \geq xyz\sqrt{8}$ for all positive reals $x$ , $y$ , $z$ .

Solution

Solution:
By AM/GM, $x^4 + y^4 \geq 2x^2y^2$ . Then by AM/GM again, $2x^2y^2 + z^2 \geq (\sqrt{8})xyz$ .

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