 Solution

The inverse function theorem says that a continuously differentiable map on an open set, with invertible derivative at a point, restricts to a $C^1$ diffeomorphism between neighborhoods of that point and its image. The inverse derivative is the inverse matrix of the original derivative at the corresponding preimage.

Put $s=1+x+y\ne 0$. Differentiating each coordinate givesThe determinant never vanishes on $\Omega$, so the theorem applies at every point. The image point $(1/3,1/3)$ has preimage $(1,1)$: the equations $x/s=y/s=1/3$ imply $x=y=1$. Therefore