The Taylor expansion of the metric at the origin in geodesic coordinates
I asked this many years ago as a beginning graduate student. One general method of solving it is using Jacobi fields, and it can be found for example in Schoen-Yau's book. If I recall correctly, it is listed as an execrise problem in Morgan-Tian's book. The first one proving it is apparently Riemann himself.
I do not know if my original method has any hope (it has been four years, so maybe not). Hopefully this helps other people.