TITLE: Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$
QUESTION [6 upvotes]: Let $x,y,z>0$ and $x+y+z=1$, then find the least value of 
$${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$$
I tried various ways of rearranging and using AM > GM inequality. But I couldn't get it. I am not good at inequalities. Please help me.
I wrote $x$ as $1-(y+z)$ and I took $x+y$ as $a$ and the others as $b$ and $c$. And I am trying.

REPLY [3 votes]: For the sake of alternatives:
It is easy to check that $$\frac{x}{2-x} \ge \frac{18x-1}{25} \quad\forall x < 2,$$ (equivalent to $(3x-1)^2\ge 0$). Applying the above inequality for $y$ and $z$ also, then taking the sum of the three inequalities, we are done.