Struggle with finding equality condition in an AM-GM inquality problem Given $0<x,y,z<1$. Prove that
$$\dfrac{1-x}{1+y+z}+\dfrac{1-y}{1+z+x}+\dfrac{1-z}{1+x+y}\geq 3(1-x)(1-y)(1-z).$$
I've thought that the equality holds when $x=y=z$ and I'm thinking ways to use AM-GM inequality. However, it turns out that
$$3\dfrac{1-x}{1+2x}=3(1-x)^3 \Longleftrightarrow (1-2x+x^2)(1+2x)=1\Longleftrightarrow 2x^3-3x^2=0\Longleftrightarrow x=\dfrac{3}{2},$$
which is totally wrong due to $0<x,y,z<1$! Thank you a lot!

It should be $$3x^2-2x^3\geq0,$$ which is true for $0<x<1.$
Let $x=\frac{a}{1+a},$ $y=\frac{b}{1+b}$ and $z=\frac{c}{1+c},$ where $a$, $b$ and $c$ are positive numbers.
Thus, we need to prove that
$$\sum_{cyc}\frac{(a+1)^2(b+1)^2}{3ab+2a+2b+1}\geq3.$$
Now, by C-S
$$\sum_{cyc}\frac{(a+1)^2(b+1)^2}{3ab+2a+2b+1}\geq\frac{\left(\sum\limits_{cyc}(ab+2a+1)\right)^2}{\sum\limits_{cyc}(3ab+4a+1)}.$$
Thus, it's enough to prove that
$$\left(\sum\limits_{cyc}(ab+2a+1)\right)^2\geq3\sum\limits_{cyc}(3ab+4a+1)$$ or
$$\sum_{cyc}(a^2b^2+2a^2bc+4a^2+5ab+4a^2b+4a^2c+4abc)\geq0,$$ which is obvious.
We can use also another C-S.
Indeed, we need to prove that:
$$\sum_{cyc}\frac{1}{(1-x)(1-y)(1+x+y)}\geq3.$$
Now, $$\sum_{cyc}\frac{1}{(1-x)(1-y)(1+x+y)}\geq\frac{9}{\sum\limits_{cyc}(1+x+y)(1-x)(1-y)}.$$
Id est, it's enough to prove that
$$3\geq\sum_{cyc}(1+x+y)(1-x-y+xy)$$ or
$$\sum_{cyc}(2x^2+xy)\geq\sum_{cyc}(x^2y+x^2z)$$ or
$$\sum_{cyc}(x^2(1-y)+x^2(1-z)+xy)\geq0,$$ which is obvious.
