7月15日至16日，第67届国际数学奥林匹克竞赛（IMO 2026）在上海举行。dots 团队受 IMO 组委会邀请，携 dots-note-3.0 的内部版本参加了 IMO 2026 比赛。
From July 15 to 16, the 67th International Mathematical Olympiad (IMO 2026) was held in Shanghai. The dots team was invited by the IMO Organizing Committee and took part in IMO 2026 with an internal version of dots-note-3.0.
本届 IMO 参赛队伍由来自 117 个国家的 666 名选手组成，参赛国家数和选手数均创历史纪录。
This year's IMO brought together 666 contestants from 117 countries, setting records for both the number of participating countries and the number of contestants.
经组委会组织的官方评阅，dots-note-3.0 六道题均获得满分 7 分，取得满分金牌成绩（42/42分）。在本届比赛中，全球仅有 7 位人类选手获得满分。
Following official marking organized by the committee, dots-note-3.0 received the full 7 points on all six problems, earning a perfect-score gold medal (42/42 points). Only seven human contestants achieved a perfect score at this year's competition.
“我们惊喜地发现，模型最终给出了一条正确且漂亮的证明思路。这种解法结构紧凑、逻辑自然，即使放在 IMO 解答中，也属于较为简洁优雅的一类。”
“We were delighted to find that the model ultimately produced a correct and beautiful proof strategy. The solution is compact in structure and natural in its logic; even among IMO solutions, it would rank as one of the more concise and elegant approaches.”
——刘涵祚，两次 CMO 金牌
—Liu Hanzuo, two-time CMO gold medalist
“模型第三题刘邦一侧的归纳法非常优雅，选手常见的做法基本都是转换成图论的连通性来做，但 dots 模型使用了非常巧妙的优雅的归纳法，攻克了这题本质难的部分。我觉得这个想法巧妙简洁，出乎我的意料”
“The inductive approach to Liu Bang's side of Problem 3 is extremely elegant. Contestants' common approach is essentially to transform it into a graph-connectivity problem, but the dots model used a remarkably ingenious and elegant induction to overcome the genuinely difficult part of the problem. I find the idea ingenious and concise—it took me by surprise.”
——汪千桐，CMO 金牌
—Wang Qiantong, CMO gold medalist
IMO 对模型能力的核心挑战，是以全程完整、正确的证明来解决高难度数学问题。
The core challenge the IMO presents to a model is solving highly difficult mathematical problems with complete and correct proofs from beginning to end.
相较于只需要判断最终答案是否正确的 answer 类数学题，IMO 题目的挑战性在于：证明过程往往复杂而漫长，其中的逻辑错误、条件遗漏和论证不严谨很难被自动判断。模型不仅要找到答案，还要保证整个推导过程经得起专业评阅。
Unlike answer-only mathematics problems, where only the final answer needs to be judged, IMO problems demand proofs that are often long and complex. Logical errors, missing conditions, and insufficiently rigorous arguments can be difficult to detect automatically. A model must not only find an answer; it must ensure that the entire derivation withstands expert review.
本届 IMO，dots-note-3.0 对六道题均提交了完整解答，经 IMO 组委会官方评阅认证，六题全部获得满分（42/42 分）。详细解答情况如下：
At this year's IMO, dots-note-3.0 submitted complete solutions to all six problems. Following official evaluation by the IMO Organizing Committee, all six solutions were certified as receiving full marks (42/42 points). Detailed solutions are shown below:
过去，大模型验证证明过程正确性的一类方法是利用形式化验证系统。这种方法通常需要人工先将题目翻译成形式化语言（例如 Lean），再输入给模型进行求解。然而，真实世界中的许多复杂问题难以被完整形式化，这条路径在通用性上仍存在局限。
One class of methods used in the past to verify the correctness of proofs produced by large models relied on formal verification systems. This approach typically requires a person to translate a problem into a formal language, such as Lean, before passing it to the model. Many complex real-world problems, however, are difficult to formalize completely, which limits the generality of this path.
本次参赛的 dots 模型不依赖人工翻译、改写或将题目转换为形式化语言，能够直接读取组委会提供的原始 LaTeX 题目，以 Agentic 方式端到端运行，结合自然语言推理与 Python 代码能力完成解题。
The dots model used in this competition does not depend on human translation, rewriting, or conversion of the problems into a formal language. It directly reads the original LaTeX problems provided by the organizers and operates end to end in an agentic manner, combining natural-language reasoning with Python code capabilities to solve them.
与此同时，这种方式也带来了新的挑战：由于无法依靠形式化语言天然保证证明过程正确，模型必须具备优秀的自我审查能力，这也是我们在 dots 模型训练过程中重点优化的能力。
At the same time, this approach introduces a new challenge: without the inherent guarantees of a formal language, the model must possess excellent self-review capabilities. This is one of the abilities we focused on optimizing during the training of the dots model.
在本次比赛解题过程中，dots-note-3.0 会主动通过多样化的手段来批判反思已生成的证明，识别其中的推理错误、逻辑跳跃和不严谨之处，再针对性完成自我持续优化。通过多轮并行推理、审查、修正和自我方案整合，模型最终形成一份完整、严谨、能够接受专业评阅的证明。
While solving the competition problems, dots-note-3.0 actively used diverse methods to critique and reflect on its generated proofs, identify reasoning errors, logical leaps, and insufficiently rigorous steps, and then continually improve them in a targeted way. Through multiple rounds of parallel reasoning, review, correction, and synthesis, the model ultimately produced complete and rigorous proofs capable of withstanding expert evaluation.
IMO 2026 的成果源于 dots 团队的一个核心技术方向判断：递归自我批判能力是解决难以验证任务的关键。
The IMO 2026 result grew out of a core technical conviction held by the dots team: recursive self-critique is essential for solving tasks that are difficult to verify.
我们的方法主要由Proof、Verify 和 Refine 三个环节组成：系统生成候选证明，再检查其正确性和完整性，并对存在问题的部分继续修改，最终形成提交给评审的证明。其中Verify部分会根据已有证明判断其正误并给出改进建议， 对后续的Refine部分起到了重要的作用。
Our method consists primarily of three stages: Proof, Verify, and Refine. The system generates candidate proofs, checks them for correctness and completeness, and continues revising any problematic parts before producing the proof submitted for evaluation. In the Verify stage, the system assesses the existing proof, determines whether it is correct, and provides suggestions for improvement, playing an important role in the subsequent Refine stage.
模型不仅需要能够生成高质量的结果，更重要的是主动审视自己的推理过程。模型必须反复检查自己的论证、发现潜在缺陷、质疑假设、改进论证链条——这个持续的自我批判循环才是实现迭代优化的必要前提。
A model must do more than generate high-quality results; more importantly, it must actively examine its own reasoning. It must repeatedly inspect its arguments, identify potential flaws, question its assumptions, and improve its chain of reasoning. This continuous cycle of self-critique is a necessary precondition for iterative improvement.
我们相信，递归自我改进的前提是递归自我批判。没有对错误和不足的主动识别，就不可能有真正的优化；反之，有效的自我批判能力才能驱动模型不断逼近正确答案。
We believe that recursive self-improvement begins with recursive self-critique. Genuine optimization is impossible without the active identification of errors and shortcomings. Conversely, effective self-critique is what enables a model to move steadily closer to the correct answer.
这种能力的意义超越了竞赛数学的范畴。现实世界中的复杂任务——科学研究、深度分析、策略规划、开放式创作——都很难被完整形式化，其结果也无法通过简单规则立即验证。在这些模糊且难以自动判断的领域，模型能否通过不断的自我质疑、自我修正而逐步改进，成为决定性因素。
The significance of this capability extends far beyond competition mathematics. Complex real-world tasks—scientific research, deep analysis, strategic planning, and open-ended creation—are difficult to formalize completely, and their outcomes cannot be immediately verified by simple rules. In these ambiguous domains, where automated judgment is difficult, a model's ability to improve through continual self-questioning and self-correction becomes decisive.
如何在复杂、模糊空间中培育可靠的递归自我批判能力，是我们未来持续探索的核心方向。
Cultivating reliable recursive self-critique in complex and ambiguous spaces will remain a central direction of our future research.
本次参加 IMO 评测的模型，是 dots-note-3.0 的内部版本，note 是目前 dots3 系列中最轻量级的模型。完整的 dots3 系列将包括 note、jazz 和 aria 三种不同尺寸，分别面向不同复杂度、响应速度和计算成本的使用场景。
The model evaluated at the IMO was an internal version of dots-note-3.0. Note is currently the lightest model in the dots3 family. The complete family will include note, jazz, and aria in three different sizes, serving use cases with different requirements for complexity, response speed, and compute cost.
我们预期 dots-note-3.0 将会在接下来的一段时间开源给社区，来加速 AI 社区的进步。
We expect to open-source dots-note-3.0 to the community in the near future to help accelerate progress across the AI community.
我们感谢 IMO 组委会的邀请，并为模型提交的解答组织专业、严谨的评阅工作。
We thank the IMO Organizing Committee for its invitation and for arranging the professional and rigorous evaluation of the model's submissions.
我们也向所有参加 IMO 2026 的选手、教练和工作人员表示祝贺与敬意。IMO 首先属于这些热爱数学、勇于挑战困难问题的年轻人。他们展现出的创造力、坚持和对数学的热情，始终是推动数学与科学向前发展的重要力量。最后，感谢所有参与 dots-note-3.0 训练、评测与工程建设的团队成员。
We also extend our congratulations and respect to every contestant, coach, and staff member who took part in IMO 2026. Above all, the IMO belongs to the young people who love mathematics and have the courage to take on difficult problems. Their creativity, perseverance, and passion for mathematics remain an important force advancing mathematics and science. Finally, we thank every team member who contributed to the training, evaluation, and engineering of dots-note-3.0.
这是一次阶段性的成绩，也是 dots 探索通用推理与自我验证能力的新起点。
This result is both a milestone and a new starting point for dots' exploration of general-purpose reasoning and self-critique.