The International Mathematical Olympiad, now 62 years old, has catalyzed the formation of a remarkable international community that meets for 2 weeks each year to celebrate mathematical problem solving. It has influenced and expanded the greater global mathematical culture, and the value of problem-solving in numerous countries. It is time for the mathematics competition community to reach out and pose questions for serious inquiry. What is the extent of this influence? What are its effects? Reciprocally, what effects have different mathematical traditions had on the IMO? Some significant studies have been done to answer such questions, but several remain unanswered, and most remain unposed. We survey the work that has been done, then some anecdotal evidence, including case studies of the participation of individual countries in the IMO. We then suggest some topics needing a deeper inquiry. The intention of this article is to invite the research community to work with the IMO community in a serious investigation of the half-century of experience afforded us by the IMO.