Is it the...
All of the above
The intrinsic value is the REAL reward.
Students feel intrinsic value from competing and being a part of a challenge. They feel intrinsic value from participating and being a part of a publication. They feel intrinsic reward from being acknowledged by their peers. They feel intrinsic value and reward from winning and succeeding, after setting their sights on a goal. They get motivation from their peers, and healthy competitive willingness from the possibility of being "famous" for all the right reasons.
They get motivation from within! We just give them a little nudge.