NASA does have a number of contests. A number of the big ones are called the Centennial Challenges (http://www.nasa.gov/offices/oct/stp/centennial_challenges/in...) These currently include:

  - Sample Return Robot ($1.5 Million)
  - Night Rover ($1.5 Million)
  - UAS Airspace Operations Challenge (AOC) ($Half a Million)

They've also done other competitions, such as robo-ops for college students (http://www.nasa.gov/centers/langley/news/releases/2011/11-08...) (https://www.wpi.edu/news/20112/rascalrob.html)