$x,y,z,t \in \mathbb R_+^*$: \[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]
Problem
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Tags: inequalities
dragonx111
12.10.2015 23:01
Does anybody have an idea of how to solve this problem?
arqady
13.10.2015 07:08
It's obviously true by BW. We need to prove that $(ab+ac+bc+ad+bd+cd)^3\geq27(a^2b^2c^2+a^2b^2d^2+a^2c^2d^2+b^2c^2d^2)$, where $a$, $b$, $c$ and $d$ are positives. Let $a=\min\{a,b,c,d\}$, $b=a+u$, $c=a+v$ and $d=a+w$ ...
dragonx111
15.10.2015 19:22
Does anybody have a solution, different from arqady's?
luofangxiang
29.03.2016 12:45
dragonx111 wrote: $x,y,z,t \in \mathbb R_+^*$: \[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]