# Proving Olympiad Algebraic Inequalities without Human Demonstrations

arXiv: https://arxiv.org/abs/2406.14219 · HTML: https://mlsanity.com/paper/2406.14219/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 20 Jun 2024.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [LeanDojo: Theorem Proving with Retrieval-Augmented Language Models](https://mlsanity.com/paper/2306.15626/) (2023)
- [MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics](https://mlsanity.com/paper/2109.00110/) (2021)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [Formal Mathematics Statement Curriculum Learning](https://mlsanity.com/paper/2202.01344/) (2022)
- [Learning to Prove Theorems via Interacting with Proof Assistants](https://mlsanity.com/paper/1905.09381/) (2019)
- [FIMO: A Challenge Formal Dataset for Automated Theorem Proving](https://mlsanity.com/paper/2309.04295/) (2023)
- [INT: An Inequality Benchmark for Evaluating Generalization in Theorem Proving](https://mlsanity.com/paper/2007.02924/) (2020)
- [Maintaining a Library of Formal Mathematics](https://mlsanity.com/paper/2004.03673/) (2020)

## Built on by (most cited first, 20 of 20)

- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [HunyuanProver: A Scalable Data Synthesis Framework and Guided Tree Search for Automated Theorem Proving](https://mlsanity.com/paper/2412.20735/) (2024)
- [Solving Inequality Proofs with Large Language Models](https://mlsanity.com/paper/2506.07927/) (2025)
- [Proposing and solving olympiad geometry with guided tree search](https://mlsanity.com/paper/2412.10673/) (2024)
- [Proving Olympiad Inequalities by Synergizing LLMs and Symbolic Reasoning](https://mlsanity.com/paper/2502.13834/) (2025)
- [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://mlsanity.com/paper/2412.15184/) (2024)
- [Large Language Models are Interpretable Learners](https://mlsanity.com/paper/2406.17224/) (2024)
- [Diverse Inference and Verification for Advanced Reasoning](https://mlsanity.com/paper/2502.09955/) (2025)
- [Learning to Disprove: Formal Counterexample Generation with Large Language Models](https://mlsanity.com/paper/2603.19514/) (2026)
- [DreamProver: Evolving Transferable Lemma Libraries via a Wake-Sleep Theorem-Proving Agent](https://mlsanity.com/paper/2604.26311/) (2026)
- [Formally Solving Answer-Construction Problems in Lean](https://mlsanity.com/paper/2505.18492/) (2025)
- [Foundation of Intelligence: Review of Math Word Problems from Human Cognition Perspective](https://mlsanity.com/paper/2510.21999/) (2025)
- [Gold-Medal-Level Olympiad Geometry Solving with Efficient Heuristic Auxiliary Constructions](https://mlsanity.com/paper/2512.00097/) (2025)
- [From LLM-Generated Conjectures to Lean Formalizations: Automated Polynomial Inequality Proving via Sum-of-Squares Certificates](https://mlsanity.com/paper/2605.15445/) (2026)
- [From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier](https://mlsanity.com/paper/2607.07779/) (2026)
- [Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs](https://mlsanity.com/paper/2605.04472/) (2026)
- [Automated Proving of Shannon-Type Entropy Inequalities via Fine-Tuned Language Models and Guided Tree Search](https://mlsanity.com/paper/2606.05729/) (2026)
- [Learning to Coordinate Symbolic Tools: LLM Agents for Verified Sum-of-Squares Certificates](https://mlsanity.com/paper/2608.00326/) (2026)

## Abstract (arXiv)

Solving Olympiad-level mathematical problems represents a significant advancement in machine intelligence and automated reasoning. Current machine learning methods, however, struggle to solve Olympiad-level problems beyond Euclidean plane geometry due to a lack of large-scale, high-quality datasets. The challenge is even greater in algebraic systems, which involve infinite reasoning spaces within finite conditions. To address these issues, we propose AIPS, an Algebraic Inequality Proving System capable of autonomously generating complex inequality theorems and effectively solving Olympiad-level inequality problems without requiring human demonstrations. During proof search in a mixed reasoning manner, a value curriculum learning strategy on generated datasets is implemented to improve proving performance, demonstrating strong mathematical intuitions. On a test set of 20 International Mathematical Olympiad-level inequality problems, AIPS successfully solved 10, outperforming state-of-the-art methods. Furthermore, AIPS automatically generated a vast array of non-trivial theorems without human intervention, some of which have been evaluated by professional contestants and deemed to reach the level of the International Mathematical Olympiad. Notably, one theorem was selected as a competition problem in a major city 2024 Mathematical Olympiad.