Obel1x wrote: Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\] There can't be equality, but it holds with this: The ineq is homogenous so $abc=1$ may and then $LL\ge 3 \root 6 \of{\prod_{\rm cyc}\frac{(5a^2+5c^2+8b^2)}{4}} > 3 \root 6 \of{\prod_{\rm cyc}( 0.5(a+b)^2+0.5(b+c)^2)}\ge 3 \root 6 \of{\prod_{\rm cyc}((a+b)(b+c)}\ge RL$ purely $AM-GM$

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