Reasoning-intensive retrieval requires deep semantic inference beyond surface-level keyword matching, posing a challenge for current LLM-based rerankers limited by context constraints and order sensitivity. We propose \textbf{\BracketRank}, a framework that treats document reranking as a reasoning-driven competitive tournament. Our approach introduces three key innovations: (1) adaptive grouping based on model context limits, (2) reasoning-enhanced prompts that mandate step-by-step relevance explanations, and (3) a bracket-style elimination structure with winner and loser tracks. This design ensures robust document advancement while enabling parallel processing across competition stages. Evaluation on the BRIGHT reasoning benchmark shows that \BracketRank achieves \textbf{26.56 nDCG@10}, significantly outperforming state-of-the-art baselines including RankGPT-4 (17.0) and Rank-R1-14B (20.5). On TREC datasets, BracketRank achieves 77.90 nDCG@5 on DL 19 and 75.85 nDCG@5 on DL 20, exceeding all baselines, establishing that explicit reasoning within competitive elimination is a powerful paradigm for complex, multi-step retrieval tasks. https://github.com/DataScienceUIBK/BracketRank

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