Proving Olympiad Algebraic Inequalities without Human Demonstrations

TL;DR

This paper introduces AIPS, an autonomous system that generates and proves complex algebraic inequalities at Olympiad level, outperforming existing methods and producing novel theorems recognized in competitions.

Contribution

AIPS is the first system capable of autonomously generating and proving high-level algebraic inequalities without human demonstrations, demonstrating advanced reasoning and theorem generation.

Findings

Successfully solved 10 out of 20 IMO-level problems

Generated numerous non-trivial theorems evaluated by experts

One theorem was used as a competition problem in 2024

Abstract

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.