Problem 7: Wonderful inequality

by henderson, Feb 16, 2016, 7:44 PM

$$\bf\color{red}Problem \ 7 \ $$If $a,b,c$ are positive reals, prove that
\[(a+b+c)^5\ge 81abc(a^2+b^2+c^2).\]$$\bf\color{red}{Solution} $$Multiplying both sides by $(a+b+c),$ we need to prove that
\[(a+b+c)^6\geq 81abc(a+b+c)(a^2+b^2+c^2)\]By $\text{AM-GM},$
$(a+b+c)^6=\left(a^2+b^2+c^2+2(ab+bc+ca)\right)^3\geq 27(a^2+b^2+c^2)(ab+bc+ca)^2\geq 81abc(a+b+c)(a^2+b^2+c^2).$
So, the proof is completed.
This post has been edited 1 time. Last edited by henderson, Sep 12, 2016, 2:38 PM

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