P rompt C o T : Synthesizing Olympiad-level Problems for Mathematical Reasoning in Large Language Models (2025)

Abstract: The ability of large language models to solve complex mathematical problems has progressed significantly, particularly for tasks requiring advanced reasoning. However, the scarcity of sufficiently challenging problems, particularly at the Olympiad level, hinders further advancements. In this work, we introduce PromptCoT, a novel approach for automatically generating high-quality Olympiad-level math problems. The proposed method synthesizes complex problems based on mathematical concepts and the rationale behind problem construction, emulating the thought processes of experienced problem designers. We provide a theoretical analysis demonstrating that an optimal rationale should maximize both the likelihood of rationale generation given the associated concepts and the likelihood of problem generation conditioned on both the rationale and the concepts. Our method is evaluated on standard benchmarks including GSM8K, MATH-500, and AIME2024, where it consistently outperforms existing problem generation methods. Furthermore, we demonstrate that PromptCoT exhibits superior scalability, consistently maintaining high performance as the dataset size increases, outperforming the baselines.

## introduction recent advancements in large language models (llms) have greatly enhanced their capability for solving complex problems through planning and reasoning. particularly in mathematics, strong reasoning models such as openai o1 @xcite and deepseek r1 @xcite have significantly pushed the boundaries of ai from mastering grade school problems @xcite to excelling at olympiad-level challenges @xcite . the remarkable achievements have inspired the community to explore dedicating more computational resources to the inference stage. consequently, the scaling paradigm of llms is shifting from training time to test time @xcite . while powerful reasoning llms, such as deepseek r1, have been open-sourced, the details of math problem acquisition remain obscured in the published literature (e.g., the technical report of deepseek r1), and such data is still kept private. consequently, studies aiming to reproduce o1-like or r1-like performance have to rely on open-source math datasets @xcite , which are often constrained by scale and difficulty. this leaves an open research question (rq): how can we obtain high-quality, sufficiently challenging math problems at scale? we emphasize that rq is crucial for test-time scaling research, as it serves as a prerequisite for effectively initiating the supervised fine-tuning (sft) or reinforcement learning (rl) process. we investigate the automated generation of olympiad-level math problems as a principled approach to addressing rq. prior to our work, several projects have curated math datasets, typically involving problem synthesis procedures. existing synthesis methods can be categorized into three frameworks: (1) direct generation through prompting powerful llms @xcite @xcite , (2) mining from pretraining corpora @xcite , and (3) problem evolution, either through simple-to-simple rephrasing @xcite or difficulty-based progression @xcite . while these efforts progres- ## discussions in addition to the extensive evaluation across multiple benchmarks, we seek to further understand the underlying mechanisms of promptcot. specifically, we explore the following research questions: ## conclusion in this paper, we introduce promptcot, a novel approach for synthesizing olympiad-level math problems. our method integrates rationale generation to guide problem synthesis, inspired by the success of llms in reasoning tasks such as chainof-thought. by leveraging this rationale and underlying mathematical concepts, we generate complex, high-quality problems that are well-suited to improve the reasoning capabilities of llms. through extensive experimentation across multiple mathematical reasoning benchmarks, including gsm8k, math-500, and aime2024, we show that promptcot outperforms existing problem generation methods.