Here are three robotics competition problems that can be easily replicated in a school or community venue and provide a challenge for students
http://robotics.nasa.gov/students/lego03.php
http://robotics.nasa.gov/students/4_can.php
http://robotics.nasa.gov/students/challenge.php
http://robotics.nasa.gov/

Two different targets with a moving base.
https://www.youtube.com/watch?v=a3y0BXH8MCY

Marble release.
https://www.youtube.com/watch?v=wrdN7Y0dT9Y