Applications to elite US colleges have more than doubled over the past 20 years, with little change in the number of available seats. We examine how this increased competition has affected the admissions advantage that legacies and athletes (LA) receive. Using data on Harvard applications over 18 years, we show that non-legacy, non-athlete (NLNA) applications grew considerably and that LA applications remained flat. Yet, the share of LA admits remained stable, implying substantial increases in admissions advantages for legacies and athletes. We develop a simple theoretical model of university admissions to frame our empirical analysis. Viewed through the lens of the model, stability in the share of LA admits implies that elite colleges treat the number of LA admits and overall admit quality as complements. Our empirical analysis reveals that, if the admissions advantages for LA applicants had been constant throughout this period, there would have been a large increase in the number of minority admits.