Once you have spoken to both chocobos at remiem temple you will learn that you can take place in a chocobo race. Speak to the chocobo on the right hand side to start the race. The idea is to get to the middle before the opponent with collecting as many treasure chests as you can and avoiding the coloured poles. If you hit a pole your prize could be reduced to a mere Potion.
If you go straight to the middle without collecting any chests you will receive a Cloudy Mirror.
1 chest earns you an Elixer.
2 chests earns you a Megalixir.
3 chests earns you 30 Wings of Discovery
4 chests earns you 30 Pendulums.