Problem 3: Nice inequality

by henderson, Feb 11, 2016, 6:07 PM

$\bf\color{red}Problem \ 3 \$ Let $a,b,c>0$ . Prove that:
$(a^{2}+bc)^{3}(b^{2}+ac)^{3}(c^{2}+ab)^{3}\geq 64a^{3}b^{3}c^{3}(a^{3}+b^{3})(b^{3}+c^{3})(c^{3}+a^{3}).$
$\bf\color{red}Solution$ Because by $AM-GM$ $\prod_{cyc}(a^2+bc)^2=\prod_{cyc}((a^2+bc)(b^2+ac))=\prod_{cyc}(a^2b^2+c^2ab+c(a^3+b^3))\geq8\prod_{cyc}\sqrt{abc(c^2+ab)(a^3+b^3)}$
and after squaring of the both sides we are done!

This post has been edited 6 times. Last edited by henderson, Sep 12, 2016, 2:44 PM

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Problem 3: Nice inequality

by henderson, Feb 11, 2016, 6:07 PM

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