This paper describes a resilient navigation and planning algorithm for the high-speed Indy autonomous challenge (IAC). The IAC is a competition with full-scale autonomous race cars that drives up to 290 km/h (180 mph). However, owing to race cars' high-speed and heavy vibration, GPS/INS system is prone to degradation, causing critical localization errors and leading to serious accidents. To this end, we propose a robust navigation system to implement a multi-sensor fusion Kalman filter. We present the degradation identification based on probabilistic approaches to computing optimal measurement values for the Kalman filter correction step. Simultaneously, we present a resilient navigation system so that the race car follows the race track in the event of localization failure. In addition, an optimal path planning algorithm for obstacle avoidance is proposed. Considering the original optimal racing line, obstacles, and vehicle dynamics, we propose a roadgraph-based path planning algorithm to ensure that our race car drives in in-bounded conditions. The designed localization system was experimentally evaluated to determine its ability to handle the degraded data and prevent serious crashing accidents during high-speed driving. Finally, we describe the successful completion of the obstacle avoidance challenge.