If $\space x_1+x_2+\cdots+x_n=1$ and all $x_1,x_2,\cdots,x_n$ are positive and real numbers, prove:$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$$ Additional:we are allowed to use Cauchy(better to use it more than other inequalities),
AM-GM and other simple inequalities. However if you think the problem could not solved with allowed inequalities,use anything you want to solve question.
Things I have tried so far:
I can re-write inequality as:
$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}$$
Using Cauchy inequality:
$$\left(\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\right) \left(\sum ^n_{i=1} x_i\sqrt{1-x_i} \right)\geq \left(\sum ^n_{i=1} x_i\right)^2$$
Now my problem simplifies to proving this:
$$\frac{1}{\left(\sum\limits_{i=1}^n x_i\sqrt{1-x_i}\right)}\geq \frac{1}{\sqrt{n-1}}$$
$$\sqrt{n-1}\geq \left(\sum ^n_{i=1} x_i\sqrt{1-x_i}\right)$$
And I stuck here.