Repeating earthquakes generating similar waveforms contribute high-precision in detecting inner core motion and relocating earthquakes. We have found more than 100 high-quality earthquake doublets in the South Sandwich Islands (SSI) region, the Aleutian Islands (AI) region, the Kuril Islands (KI) region, the Tonga-Fiji-Solomon Islands (TFSI) region, and an earthquake nest in Bucaramanga of Colombia. Observation from one SSI doublet recorded at INK shows an inner core travel-time change of ~ 0.1 sec over ~ 6 years, confirming the inner core differential motion occurring beneath Central America. Observations from one AI doublet recorded at BOSA, and from one KI doublet recorded at BDFB, show an inner core travel-time change of ~ 0.1 sec over ~ 7 years and ~ 6 years respectively, providing additional evidence for the temporal change of inner core properties beneath Central Asia and Canada respectively. On the other hand, observations from one TFSI doublet recorded at PTGA, and from one Bucaramanga doublet recorded at WRAB and CHTO, show no temporal change of inner core travel times for the three corresponding ray paths, of which the path in the inner core is nearly parallel to the equatorial plane. Such pattern of observations showing both presence and absence of inner core travel-time change may be explained by the geometry and relative directions of ray path, lateral velocity gradient, and particle motion due to inner core super-rotation. For the Bucaramanga earthquake nest, teleseismic high-precision relocation is demonstrated by applying a double-difference (DD) algorithm to phase picks from EHB bulletin of Engdahl et al. (1998) and waveform cross-correlation (WCC) measurements on some similar events. The DD relocation of all Bucaramanga nest events reveals that about 900 mb>=3.8 events (~ 76% of all events in the last 40 years) focus on a small fault-like volume that has dimensions of 20km x 10km x 10km. A map view of the relocated seismicity shows an approximately 135 degree striking elongated structure. An on-fault view indicates an approximately 10 km thick, 40 degree to the SE dipping band of seismicity. Further improvement of the pick-based DD relocation using WCC measurements on a subset of 34 similar events confirm a vertical fault with a fault zone width indistinguishable from zero. Focal mechanisms of the nest earthquakes indicate a slip co-directional with the strike and dip of the fault. This intermediate-depth vertical fault may represent a contact interface between two subducted slabs belonging to the Caribbean plate and the Nazca plate respectively. The nest earthquakes may then result from the colliding and relative sliding of these two slabs. It is also possible that the fault represents a tear structure within a dipping slab.