This video gives some insight into the thought process of Olympiad problem solvers. The first speaker, Dr. Zuming Feng, is a good friend of the Metroplex Math Circle and is a regular speaker. Zach Abel, the first student speaker, is a Dallas native and an alumnus of Greenhill School.
These particular problems are much harder than the level normally discussed at math circles, but the strategies of problem solving can be very similar. Exposure to former Olympiad winners and coaches is a great inspiration to math circle participants.