We investigate whether competitive selection processes generate gender inequality in the context of a prestigious graduate fellowship program. All applications are scored remotely by expert reviewers and the highest ranked are invited to an in-person interview. The data show a very large gender gap in success rates: women's success rate is 36% lower than men's. About one third of this gap is due to the lower grades of female candidates, which is surprising given women's higher GPA in the population of college graduates. Adjusting for GPA and a rich set of fixed-effects, women's success rate remains 16% lower than for comparable male candidates. We show that this gap is explained by reviewers engaging in gender balancing. Namely, reviewers favor the minority gender in each field of study but, except for STEM, all fields are female-dominated. Our simulations show that the interview plays an important role, but the quantitative scoring has a more profound effect on the award allocation. Merging administrative records on the population of graduates from a large university, we document an important gender gap in participation. We find that high-GPA female graduates are much less likely to apply to the fellowship program. The combination of the gender gaps in participation and success in the program imply that high-GPA female graduates are almost 50% less likely to obtain a fellowship than their male counterparts.