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Recent advancements in reinforcement fine-tuning have significantly improved the reasoning ability of large language models (LLMs). In particular, methods such as group relative policy optimization (GRPO) have demonstrated strong capabilities across various fields. However, applying LLMs to ship trajectory prediction remains largely unexplored. In this paper, we propose ShipTraj-R1, a novel LLM-based framework that reformulates ship trajectory prediction as a text-to-text generation problem. (1) We design a dynamic prompt containing trajectory information about conflicting ships to guide the model to achieve adaptive chain-of-thought (CoT) reasoning. (2) We introduce a comprehensive rule-based reward mechanism to incentivize the reasoning format and prediction accuracy of the model. (3) Our ShipTraj-R1 is reinforced through the GRPO mechanism guided by domain-specific prompts and rewards, and utilizes the Qwen3 as the model backbone. Extensive experimental results on two complex and real-world maritime datasets show that the proposed ShipTraj-R1 achieves the least error compared with state-of-the-art deep learning and LLM-based baselines.