Say that $\Omega \subset \Bbb R^n$ is star-shaped if there is $p \in \Omega$ with the following property : for all $ q \in \Omega$, the segment $[p,q] \subset \Omega$.
We know that
If $\Omega$ is star-shaped, then $\Omega$ is contractible, that is homotopy equivalent to a point. In particular, $\Omega$ is connected and simply-connected.
For proof of this statement see Lemma 2.11 and Theorem 3.4 here
Now we have that:
A topological space X is simply connected if and only if X is path-connected and the fundamental group of X at each point is trivial.
Por proof of this see Theorem 2.3 here
Therefore for a star shaped region, the fundamental group is trivial.
As you asked for a hint I would try prove these claims without clicking the links first.