I am thinking of this interesting problem, seems to be easy but finally its not.
Let $\nabla$ the euclidean connection on $ \mathbb{R}^n$. Let $\Omega= \{ f : \mathbb{R}^n \rightarrow \mathbb{R}^n | \ \ f \ \ is \ \ diffeomorphism, f_*(\nabla_XY)=\nabla_{f_*X}f_*Y \}$. Then the functions in $\Omega$ are of the form $Ax + b$ where $A \in GL_n(\mathbb{R})$ and $b \in \mathbb{R}^n$.
Of course in this situation, $f_*$ is a linear trasformation, so we know that $f_* \in GL_n(\mathbb{R})$. But what can we say about $f$?
Hint: Viewing vector fields on $\mathbb R^n$ as functions $X:\mathbb R^n\to \mathbb R^n$, the connection is given by $(\nabla_XY)(x)=DY(x)(X(x))$ and the push forward is given by $(f_*X)(x)=Df(x)(X(x))$. Using this, you can work out what compatibility with the connection actually means.