At the end of each school year (purple bar at the top left corner), you will be able to participate in University competitions with your best-fitting students (yellow cup icon at the bottom right of your screen). They have a time limit, so check which fits you at that moment! 
They also have a participation fee, but you can win even a much better prize!
When the competition finishes, you will get a notification with the results of the final ranking and your rewards.
**The competition system randomly assigns other players with similar progress to the same list. Some players could be really accurate with their teachers' performances. The team will revise this system, but please take in mind that the game has some new features and they could suffer some corrections in the future. In the meantime, we will forward this info to optimize teachers' skills: