On one hand, the Olympics date back to ancient Greece, and therefore aren't something that anyone - even a world organization - should be able to copyright. On the other hand, do we really want thousands of competition wannabe's using the word Olympics? Dance Olympics, Midget Olympics, Gay Olympics (imagine the uproar if someone had a Straight Olympics...), etc. I'm not saying the Redneck Olympics are necessarily unworthy of the name - I'm just a little afraid of what else people might start using the word Olympics for.
Maybe change the rules so organizations can use the word Olympics, but only for local or statewide competitions? That shouldn't infringe on anything important.