Large language models (LLMs) are increasingly used to screen and rank job applicants, creating incentives for candidates to strategically manipulate algorithmic hiring systems. We study prompt injection in automated résumé screening, defined as subtle self-promotional text that introduces no new qualifications but is designed to influence LLM evaluations. Using controlled experiments, we show that prompt injection reliably improves applicant rankings when résumé quality is homogeneous and few candidates inject. However, its effectiveness rapidly diminishes as more candidates inject, collapsing when manipulation becomes widespread. When candidate quality is heterogeneous, prompt injection is less effective on average, but can occasionally allow lower-quality candidates to outrank higher-quality ones, raising fairness concerns. Overall, LLM-based screening is most vulnerable when manipulation is rare and candidate quality differences are small.