Agentic AI
LLM Agent Solves Olympiad Geometry Problems with Reinforcement Learning, Beating Human Gold Medalists
A new AI system achieves gold-medal-level performance on International Mathematical Olympiad geometry problems by iteratively proposing and verifying constructions, using a fraction of the training data required by competing approaches.
Researchers have developed InternGeometry, an LLM-based agent that solves challenging geometry problems at International Mathematical Olympiad (IMO) level—surpassing the average performance of human gold medalists on a benchmark of 50 recent problems.
The Core Innovation#
InternGeometry tackles geometry's weakest point in LLM reasoning: generating useful auxiliary constructions—additional lines, points, or shapes that unlock problem solutions. Rather than relying on brute-force data synthesis and search like existing expert systems, the agent uses an interactive loop: it proposes geometric propositions and constructions, checks them with a symbolic engine, reflects on the feedback, and uses those insights to guide the next round of proposals. A dynamic memory system allows the model to conduct over 200 interactions with the symbolic engine on a single problem.
Training Efficiency Through Complexity Boosting#
The team introduced Complexity-Boosting Reinforcement Learning (CBRL), which gradually increases problem difficulty across training stages. This paces the learning process and appears to dramatically improve data efficiency. InternGeometry trains on just 13,000 examples—about 0.004% of the 325 million examples used by AlphaGeometry 2—yet achieves superior results.
Performance on IMO Benchmarks#
Built on the InternThinker-32B language model, InternGeometry solved 44 of 50 IMO geometry problems from 2000–2024, earning a score of 88% that exceeds the average gold medalist score of 40.9 points (out of 42). The system can also generate novel auxiliary constructions not found in published human solutions to the same problems.
What This Means for AI Reasoning#
The work suggests that LLM agents, when equipped with robust feedback loops and well-designed training curricula, can reach expert-level performance on symbolic reasoning tasks without the massive synthetic datasets previously thought necessary. The approach may point toward more efficient training methods for other formal problem domains.
