<s>[INST] Given $x,y,z >0$, $1/x+1/y+1/z = 4$, prove that $ 1/(2x+y+z)+1/(x+2y+z) +1/(x+y+2z) \le 1$ Given $x,y,z >0$, $1/x+1/y+1/z = 4$, prove that 
$$ 1/(2x+y+z)+1/(x+2y+z) +1/(x+y+2z) \le 1 .$$ 
Any hints or direction will be appreciated.
 [/INST] By C-S
$$\sum_{cyc}\frac{1}{2x+y+z}\leq\frac{1}{(2+1+1)^2}\sum_{cyc}\left(\frac{2^2}{2x}+\frac{1^2}{y}+\frac{1^2}{z}\right)=1$$
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