<s>[INST] If $x,y,z>0$ are distinct and $x+y+z=1$ what is the minimum of $\left((1+x)(1+y)(1+z)\right)/\left((1-x)(1-y)(1-z)\right)$? If $x,y,z>0$ are not equal and positive and if $x+y+z=1$ the expression 
$$\frac{(1+x)(1+y)(1+z)}{(1-x)(1-y)(1-z)}$$
is greater than what quantity?
 [/INST] By AM-GM we obtain:
$$\frac{\prod\limits_{cyc}(1+x)}{\prod\limits_{cyc}(1-x)}=\frac{\prod\limits_{cyc}(2x+y+z)}{\prod\limits_{cyc}(x+y)}=\frac{\prod\limits_{cyc}(x+y+x+z)}{\prod\limits_{cyc}(x+y)}\geq\frac{8\prod\limits_{cyc}\sqrt{(x+y)(x+z)}}{\prod\limits_{cyc}(x+y)}=8.$$
The equality occurs for $x=y=z=\frac{1}{3}$, which says that we got the answer: $8$.
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