Real world trampolines lose energy since they are damped springs with much internal friction. How much energy does the sumo wrestler lose on each bounce in this situation? b) How can a gymnast keep a constant bounce height in a real world trampoline?

The answer to the first question is that, the sumo wrestler loses energy equal to the absorbed energy of the trampoline plus the air friction drag loss. With the second question, the gymnast should maintain energy equal to the Potential energy gymnast has at the peak of bounce height by exerting little force to compensate for the energy spent on the internal friction of the trampolines action.