We consider incentives for organizing competitions in multiple rounds, focusing on situations where there is heterogeneity among the contestants ex ante, which discourages effort in a single contest. Heterogeneity evolves across rounds depending upon the outcomes of previous rounds. We present conditions under which balance in such a competition can be created, by determining the number of rounds and dividing the prize fund carefully across them, so that full rent dissipation entails. In the model, each round is an all-pay auction where contestants differ in their abilities to gain a momentum from winning. We also discuss the case when negative prizes are feasible, demonstrating that this strengthens the full dissipation result; and we consider a case where the size of the winner's momentum is related to the size of the prize attained, showing that the stronger this linkage, the less of the prize is awarded early on.