Here is a link to a list of U.S. troops deployed to various nations around the world: PDF. These are permanent deployments openly admitted to by the U.S. military. When U.S. Special Forces drive off a bridge in Mali, as recently happened, we discover that U.S. troops are in Mali in greater numbers than we knew, but those troops aren't listed here or considered in the calculation below. No secret forces are considered here, no allied forces funded or trained or armed by the United States, and of course no drones.
Here is a link to the number of athletes participating in the 2012 Summer Olympics from countries around the world: Link.
Many nations have sent very small delegations. Many nations have a very small U.S. troop presence. In many nations the U.S. troop presence falls just short of or exactly equals the size of the Olympic team.
In 69 nations, there is a larger U.S. military presence than the nation's Olympic team. This count excludes the oceans of the world, in which over 100,000 U.S. troops are stationed, but which of course don't have Olympic teams. The count includes, however, Diego Garcia, which could have an Olympic team if we hadn't removed all the people to make room for the military base. And it includes other nations that have been demoted to U.S. territories. It also includes South Korea, despite the U.S. military not releasing the numbers, because the U.S. military has many times the number of troops (and growing) there than South Korea has athletes on its Olympic team.
The question arises, of course, why a nation that is trailing many poorer nations in education, health, security, sustainability, and infrastructure is paying to create such a global presence of representatives with guns rather than athletes with Chinese-made sports uniforms.
Here are the 69 nations:
Northern Mariana Islands
United Arab Emirates