Regular vegetation patterns in semiarid ecosystems are believed to arise from the interplay between long-range competition and facilitation processes acting at smaller distances. We show that, under rather general conditions, long-range competition alone may be enough to shape these patterns. To this end we propose a simple, general model for the dynamics of vegetation, which includes only long-range competition between plants. Competition is introduced through a nonlocal term, where the kernel function quantifies the intensity of the interaction. We recover the full spectrum of spatial structures typical of vegetation models that also account for facilitation in addition to competition.