Exceptional talent accounts for a disproportionate share of innovation, yet many individuals with exceptional ability may never realize their potential. Whether expanding access to advanced training generates learning gains remains an open question. We study this using a randomized controlled trial with 620 highly gifted students from 44 countries, nominated by national Olympiad organizations. Participants were randomly assigned either to an 18-week advanced combinatorics course by Art of Problem Solving or to independent study using equivalent materials. Assignment to the course increased final-exam performance by 0.16 standard deviations. Engagement varied widely: roughly half of assigned students participated minimally, and baseline characteristics explain little of this variation (R² ≈ 0.10). Using random assignment as an instrument for engagement, we estimate learning gains of 0.66 standard deviations among fully engaged students. Among those who later competed in the International Mathematical Olympiad, students assigned to the course performed better on combinatorics problems. Overall, access to advanced training yields large gains when engagement is sustained, but access alone does not reliably induce engagement.