Sports video understanding requires perceiving high-speed dynamics, complex rules, and long temporal contexts. Yet, current Multimodal Large Language Models (MLLMs) remain narrowly focused on single sports, specific tasks, or training-free paradigms. We introduce DeepSport, the first end-to-end trained MLLM for multi-task, multi-sport video understanding. DeepSport shifts from passive frame processing to active, iterative reasoning, dynamically extracting frames to "think with videos." To train our model, we curate a unified 78k-sample dataset via a rigorous three-step text-and-vision distillation pipeline. We then employ a progressive two-stage training strategy: a Sports Curriculum Supervised Fine-Tuning phase to build foundational perception, followed by Agentic Reinforcement Learning with a novel tool-use reward. Extensive experiments on a comprehensive 6.7k benchmark demonstrate that DeepSport achieves state-of-the-art performance, outperforming powerful proprietary and open-source models, while utilizing significantly fewer frames. Furthermore, it exhibits strong zero-shot transferability to unseen sports and broad motion recognition tasks, establishing a highly efficient and generalized foundation for complex video reasoning.