The United Federation of Planets is all about bringing its members together. Federation worlds are spread all across the Alpha and Beta Quadrants, and vary wildly. Each planet has their own climate, history, attitudes, and cultures. However, all of them can still come together and work towards a common goal that benefits them all. Hundreds of species serve side by side in Starfleet and the Federation government. For the most part the interactions between all of the members of the Federation is related to defense or other vital government functions.
The Olympic Games on Earth brings together most of the countries on Earth to compete together in various sporting events. What if the Federation had something similar? Each member planet has not only the unique physiology of their native species, but also that planet’s physical characteristics and the sports they have there. While Vulcans have more raw strength than other species, they might lack as many experienced athletes or be at a disadvantage in the cold. Meanwhile the team from Andoria would likely find that operating in the cold weather conditions of most winter sports comes naturally to them.
Different sports will require different skill sets and most planets will have at least one sport where they have an advantage. But who do you think would wind up with the most wins? Does Team Earth have your vote? Which Federation world would win the most gold medals?