Advice on running a competitive game

Has anyone ran a competitive game in 5e? I'm thinking of doing a short campaign loosely based on the 1904 st. louis marathon. So I'd like for the players to encounter a series of shenanigans, not unlike the real competition, that would make it bascially a linear dungeon. Question is, how could I get the party to be competing against each other while keeping them in the same encounter/room while still being a race? Thanks!