# Putnam-AXIOM: A Functional and Static Benchmark for Measuring Higher Level Mathematical Reasoning in LLMs

arXiv: https://arxiv.org/abs/2508.08292 · HTML: https://mlsanity.com/paper/2508.08292/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 30 of its 50 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 30)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Generative Agents: Interactive Simulacra of Human Behavior](https://mlsanity.com/paper/2304.03442/) (2023)
- [MMMU: A Massive Multi-discipline Multimodal Understanding and Reasoning Benchmark for Expert AGI](https://mlsanity.com/paper/2311.16502/) (2023)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Large Language Models Cannot Self-Correct Reasoning Yet](https://mlsanity.com/paper/2310.01798/) (2023)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [AGIEval: A Human-Centric Benchmark for Evaluating Foundation Models](https://mlsanity.com/paper/2304.06364/) (2023)
- [ChemCrow: Augmenting large-language models with chemistry tools](https://mlsanity.com/paper/2304.05376/) (2023)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [NLP Evaluation in trouble: On the Need to Measure LLM Data Contamination for each Benchmark](https://mlsanity.com/paper/2310.18018/) (2023)
- [xCOMET: Transparent Machine Translation Evaluation through Fine-grained Error Detection](https://mlsanity.com/paper/2310.10482/) (2023)
- [SciBench: Evaluating College-Level Scientific Problem-Solving Abilities of Large Language Models](https://mlsanity.com/paper/2307.10635/) (2023)
- [Data Contamination: From Memorization to Exploitation](https://mlsanity.com/paper/2203.08242/) (2022)
- [ConvFinQA: Exploring the Chain of Numerical Reasoning in Conversational Finance Question Answering](https://mlsanity.com/paper/2210.03849/) (2022)
- [PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition](https://mlsanity.com/paper/2407.11214/) (2024)
- [LLM as a Mastermind: A Survey of Strategic Reasoning with Large Language Models](https://mlsanity.com/paper/2404.01230/) (2024)
- [Functional Benchmarks for Robust Evaluation of Reasoning Performance, and the Reasoning Gap](https://mlsanity.com/paper/2402.19450/) (2024)
- [ARB: Advanced Reasoning Benchmark for Large Language Models](https://mlsanity.com/paper/2307.13692/) (2023)
- [Compression Represents Intelligence Linearly](https://mlsanity.com/paper/2404.09937/) (2024)
- [MACM: Utilizing a Multi-Agent System for Condition Mining in Solving Complex Mathematical Problems](https://mlsanity.com/paper/2404.04735/) (2024)
- [Why Has Predicting Downstream Capabilities of Frontier AI Models with Scale Remained Elusive?](https://mlsanity.com/paper/2406.04391/) (2024)
- [Multilingual Mathematical Autoformalization](https://mlsanity.com/paper/2311.03755/) (2023)
- [Pretraining on the Test Set Is All You Need](https://mlsanity.com/paper/2309.08632/) (2023)
- [Achieving >97% on GSM8K: Deeply Understanding the Problems Makes LLMs Better Solvers for Math Word Problems](https://mlsanity.com/paper/2404.14963/) (2024)
- [Towards Explainable Evaluation Metrics for Natural Language Generation](https://mlsanity.com/paper/2203.11131/) (2022)

## Built on by (most cited first, 11 of 11)

- [An Investigation of Robustness of LLMs in Mathematical Reasoning: Benchmarking with Mathematically-Equivalent Transformation of Advanced Mathematical Problems](https://mlsanity.com/paper/2508.08833/) (2025)
- [ContinualSkillBench: Can LLM Agents Truly Evolve Their Capabilities?](https://mlsanity.com/paper/2608.03874/) (2026)
- [LiveMathematicianBench: A Live Benchmark for Research-Level Mathematical Reasoning with Proof Sketches](https://mlsanity.com/paper/2604.01754/) (2026)
- [ReX-MLE: The Autonomous Agent Benchmark for Medical Imaging Challenges](https://mlsanity.com/paper/2512.17838/) (2025)
- [Putnam-like dataset summary: LLMs as mathematical competition contestants](https://mlsanity.com/paper/2509.24827/) (2025)
- [Even GPT-5.2 Can't Count to Five: The Case for Zero-Error Horizons in Trustworthy LLMs](https://mlsanity.com/paper/2601.15714/) (2026)
- [VisualFLIP: Do Predictions Depend on Task-Critical Visual Evidence in Multimodal Reasoning?](https://mlsanity.com/paper/2606.07872/) (2026)
- [VeRA: Renewing Reasoning Benchmarks with Executable Specifications](https://mlsanity.com/paper/2602.13217/) (2026)
- [Mitigating Reasoning-Induced Misalignment via Safety-Direction Penalty](https://mlsanity.com/paper/2608.23497/) (2026)
- [Objects Without Morphisms: What LLMs for Mathematics Do Not Represent](https://mlsanity.com/paper/2610.03551/) (2026)
- [Agent Behavior as Code: Efficient and Robust LLM Agents with Programmatic Specifications](https://mlsanity.com/paper/2610.04824/) (2026)

## Abstract (arXiv)

Current mathematical reasoning benchmarks for large language models (LLMs) are approaching saturation, with some achieving > 90% accuracy, and are increasingly compromised by training-set contamination. We introduce Putnam-AXIOM, a benchmark of 522 university-level competition problems drawn from the prestigious William Lowell Putnam Mathematical Competition, and Putnam-AXIOM Variation, an unseen companion set of 100 functional variants generated by programmatically perturbing variables and constants. The variation protocol produces an unlimited stream of equally difficult, unseen instances -- yielding a contamination-resilient test bed. On the Original set, OpenAI's o1-preview -- the strongest evaluated model -- scores 41.9%, but its accuracy drops by 19.6% (46.8% relative decrease) on the paired Variations. The remaining eighteen models show the same downward trend, ten of them with non-overlapping 95% confidence intervals. These gaps suggest memorization and highlight the necessity of dynamic benchmarks. We complement "boxed" accuracy with Teacher-Forced Accuracy (TFA), a lightweight metric that directly scores reasoning traces and automates natural language proof evaluations. Putnam-AXIOM therefore provides a rigorous, contamination-resilient evaluation framework for assessing advanced mathematical reasoning of LLMs. Data and evaluation code are publicly available at https://github.com/brando90/putnam-axiom.