Brains vs. Bytes: Evaluating LLM Proficiency in Olympiad Mathematics

We're truly grateful for the reviewer's insightful and encouraging evaluation, which affirms the significance and clarity of our work. We especially appreciate the praise for our original focus on logical rigor over just final answers, which addresses a critical gap in current benchmarks. Their recognition of our rigorous methodology, particularly the systematic fallacy categorization and the value of human expert evaluation, confirms the strength of our approach. We're also excited by the reviewer's insight into how our research could inform the development of improved training strategies for LLMs.

Addressing "Reasons to Reject"

LLMs' Lack of Mathematical Reasoning is Well-Defined, But Evaluation on Existing Benchmarks is Missing:

The reviewer correctly notes that LLMs' lack of mathematical reasoning is a known area of research, and our paper provides further detailed insights from experts. We acknowledge the suggestion to evaluate on existing benchmarks. However, as discussed in our paper, many current benchmarks, such as GSM8K and MATH, largely consist of routine problems that have been widely circulated. Not only are the problems and their solutions repetitive, but even individual solution steps are often highly formulaic. This makes these benchmarks less suitable for measuring LLMs' capabilities in generating solutions that require creativity and novel insights, which is a core focus of our work with Olympiad-level problems. Our aim was to specifically probe the limits of LLMs beyond mere recall or mechanical application of common methods.

Response to Questions to Authors

1. Have you considered using the same rubric of fallacy recognition and analysis for existing Olympiad benchmarks?

The main issue with applying our rubric to other available Olympiad benchmarks (like OmniMath and OlympiadBench) is their primary focus on problems with a concrete final answer. A significant portion of our work, and a key strength, lies in evaluating proof-based problems which are not covered by the existing benchmarks. We chose IMO shortlist problems specifically for their originality, multi-step nature, and the creativity required for their solutions, qualities that are less consistently present in other lower-tier contests or benchmarks.

2. How do you plan to share the results? I did not see any of the code or prompt? Did you employ any prompting strategy?

Yes, we plan to release the annotated dataset publicly. We are currently working on adding more comprehensive annotations and aim to make it available by the end of summer.

Regarding our methodology, we employed a simple one-step prompting strategy for generating proofs. The specific prompts used for our automated analysis section are provided in the appendix of the paper. While we did not systematically investigate different prompting strategies, our manual inspections suggest that the choice of prompt does not significantly alter the fundamental findings. If a model lacks the intrinsic capability to generate rigorous proofs for Olympiad-level problems, minor variations in prompting strategies generally do not substantially improve the quality of the response.

We thank the reviewer for their positive feedback and for highlighting the value of our work.
The reviewer raises an important point about our methodology: the possibility that logical reasoning might occur within the Chain-of-Thought (CoT) but fail to appear in the final proof.

CoT outputs were not consistently available across all models we evaluated. Even when available, systematically analyzing raw CoTs is highly labor-intensive and challenging. CoTs are often significantly longer than the final response, lack clear structure, and frequently contain failed attempts or irrelevant detours, making rigorous, large-scale human evaluation impractical.

That said, we disagree with the hypothesis that models may internally produce correct reasoning within their CoTs but fail to express it in the final answer. While we did not conduct a comprehensive human evaluation of all CoTs, we manually examined a subset in cases where CoTs were available, motivated by the same concern raised by the reviewer. In these cases, we found no evidence of complete, sound proofs within the CoT that were then omitted in the final output. Although some CoTs included additional calculations or exploratory reasoning, these did not amount to rigorous or correct arguments that were distinct from the flawed logic present in the final response.

Furthermore, as the reviewer rightly notes, if models were indeed capable of correct internal reasoning, they should be able to distinguish between valid and invalid proofs. However, as shown in Section 5, models failed to reliably identify even blatantly incorrect proofs containing naive errors. If a model were capable of constructing a sound proof internally, we would expect it to succeed at such verification tasks. Its failure to do so strongly challenges the notion that correct reasoning is happening internally but simply not being communicated.

In summary, we believe the errors observed in the final proofs reflect a deeper issue: a fundamental limitation in the models’ ability to perform rigorous reasoning, rather than a mere failure in surfacing internally correct thought processes.

We thank the reviewer for their careful reading and constructive feedback. We appreciate the acknowledgment of our robust empirical evaluation, the utility of our fallacy categorization, and our work's contribution to identifying a critical gap in current LLM evaluation methodologies. We address each of the reviewer's points below:

Addressing "Reasons to Reject"

1. Dependency on specialized human assessment: We agree human assessment is costly and limits scalability. This dependency is a key finding of our work. Our goal was to rigorously assess LLM proof correctness in Olympiad math. Section 5 shows current automated methods, including LLM-as-a-judge, cannot reliably perform this. Human expertise remains indispensable for verifying complex mathematical proofs. Our findings highlight the critical need for more sophisticated automated evaluation methods, e.g., theorem provers, to address this need. Our findings also underscore the need for better approaches to training LLM for mathematical reasoning.

2. Comprehensiveness, overlaps, and applicability of fallacy categories: Fallacy categories were empirically derived from LLM solutions to IMO problems, representing pervasive reasoning failures in this domain. Each category highlights a distinct logical flaw, though some overlap. We acknowledge they may not be exhaustive or universally relevant beyond Olympiad math. For example, heuristics might be allowed for final answer tasks. Examples:

   • "Proposal Without Verification": Claiming "every even natural number is a sum of two prime numbers" without proof. Its core flaw is lack of justification.
   • "Inventing Wrong Facts": Citing a non-existent "Theorem 2 from 'Graphs and Random Structures' by Paul Erdos." To avoid ambiguity, multiple fallacy labels were allowed for consistent classification.

3. Broad definition of "partially correct" solutions: We agree our 'partially correct' definition needed clarity. A solution was partially correct if it realized at least a high-level step with details (minor errors possible), but failed subsequent crucial steps. Olympiad solutions can be decomposed into key steps, each with a high-level idea and derivations. Partially correct solutions realize one or more high-level steps but fail to complete all.

4. Data leakage and its impact on heuristic approaches: We acknowledge probable data leakage from public IMO problems. Our study focuses on proof rigor, not just final answers. Models' low performance on correct proofs indicates data leakage isn't significant for genuine problem-solving. If models recalled solutions, proof generation would be higher. Most proofs show fundamental mistakes, relying on pattern recognition or heuristics, not sound reasoning: One might find the form of a sum like $1+2+3+...+n$ by just testing examples and guessing the final pattern, but unless the guessed pattern is proven correct, it does not meet the bar as an Olympiad-level proof. Memorization affects final answers, not our proof of correctness metrics.

5. Section 5 (Automatic Evaluation) brevity and alignment with other findings: Section 5's brevity reflects its objective: to demonstrate LLMs' unsuitability for verifying rigorous Olympiad proofs. For Olympiad proof verification, a model must: 1) Distinguish correct from blatantly incorrect proofs, especially with obvious mistakes. 2) Detect correct solutions as correct significantly more often than labeling incorrect ones as correct. Our LLMs failed both. A simpler schema isolated this point, avoiding arguments that low performance was due to schema complexity.

6. Lack of actionable next steps for research: Our findings point to two critical research directions. First, novel benchmarks are needed to rigorously evaluate logical soundness and step-by-step reasoning correctness, not just final answers. High final answer performance doesn't imply solution correctness. Second, current training (e.g., RL) often rewards only final answer correctness, leading to shortcuts and hallucinations. Future research must explore improved training paradigms that explicitly reward sound reasoning steps, moving beyond insufficient LLM-as-a-judge or final-answer-only verification.

Thank you for the detailed response. I appreciate you taking the time to address the concerns raised. While your clarifications are helpful, I find that several of my original points of concern remain insufficiently addressed.

My concern about human evaluation still remains. The limitations of LLM-as-a-judge or similar automated methods are generally known (lots of prior work on this). Stating that the dependency on expensive human evaluation is a "key finding" of your work seems illogical. Also, framing a methodological limitation as a primary finding feels like a post-hoc justification. So the core issue remains that the scalability and replicability of your own analysis are severly constrained.

With flaw categories overlapping it is hard to pinpoint and control why the model made a certain mistake.

I find the argument about data leakage unconvincing. The concern is not necessarily that models are recalling entire proofs (or generations), but it is about exposure to the problem set (or similar problems with, for instance, on policy learning methods). The connection between leakage, heuristics, and the observed fallacies is not clear or very convincing.

You suggest that your work points toward the need for "novel benchmarks" and "improved training paradigms." While correct, these are generic recommendations applicable to nearly any paper in the field.

Regarding the thought tokens (when available), This seems like a critical omission. The distinction between a model lacking the ability to reason and a model struggling to articulate a rigorous proof is fundamental. Analyzing the chain-of-thought, however unstructured, is essential for probing this distinction.

Regarding the intuition for correct answers without logic, your reply is plausible, but feels like an oversimplification for many abstract problems in the IMO shortlist that do not have simple, guessable numerical answers.

Thank you again for your response.

We thank the reviewer for their positive assessment, recognizing the significance of our findings regarding LLM capabilities in high-level mathematics and the clarity of our writing. We appreciate the confidence in our work. Regarding your question about making the annotated dataset public: Yes, we do plan to release the annotated dataset publicly. We are currently working on adding more comprehensive annotations to the data and aim to make it available by the end of summer.