Q:

Inequality $\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+(1-\frac{x+y}{2})^2\geq (1-x)^2+(1-y)^2$

it's a little problem found by myself 

let $x,y\neq 1$ .be real numbers then we have  :
  $$\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+\left(1-\frac{x+y}{2}\right)^2\geq (1-x)^2+(1-y)^2$$

I tried Jensen's inequality and Am-Gm but without success .
I also tried Karamata's inequality but it fails.
I have a ugly proof using derivative  and it's too long to be explain here .
I'm looking for a contest proof .
Thanks a lot  to share your knowledge and your time.

A:

For $x+y\neq2$ by AM-GM we obtain:$$\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+\left(1-\frac{x+y}{2}\right)^2=$$
$$=\left(\frac{(1-x)^2+(1-y)^2}{2-x-y}\right)^2+\left(\frac{2-x-y}{2}\right)^2\geq$$
$$\geq2\sqrt{\left(\frac{(1-x)^2+(1-y)^2}{2-x-y}\right)^2\left(\frac{2-x-y}{2}\right)^2}= (1-x)^2+(1-y)^2$$

