 Solution

In normal coordinates, each radial line $u(t)=tv$ is an affinely parametrized geodesic. Its coordinate geodesic equation gives ${\Gamma }_{ij}^k(tv)v^i{v}^j=0$. At $t=0$ this holds for every tangent vector $v$. Since ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$, polarization of this quadratic form gives every coefficient zero. Thus all Christoffel symbols vanish at the centre $p$; they need not vanish elsewhere.