Brains vs. Bytes: Evaluating LLM Proficiency in Olympiad Mathematics
Authors: Hamed Mahdavi
Alireza Hashemi
Majid Daliri
Pegah Mohammadipour
Alireza Farhadi
Samira Malek
Yekta Yazdanifard
Amir Khasahmadi
Vasant G Honavar
Venue: COLM 2025 (2025)
Keywords: Mathematical Reasoning
Human Evaluation
Reasoning Evaluation
Math Problem-Solving
TL;DR: We evaluate large language models on Olympiad-level mathematics, revealing their inability to produce rigorous and logically sound proofs despite occasional correct final answers.

Abstract: Recent advancements in large language models (LLMs) have shown impressive progress in mathematical reasoning tasks. However, current evaluation benchmarks predominantly focus on the accuracy of final answers, often overlooking the logical rigor crucial for mathematical problem-solving. The claim that state-of-the-art LLMs can solve Math Olympiad-level problems requires closer examination. To explore this, we conducted both qualitative and quantitative human evaluations of proofs generated by LLMs, and developed a schema for automatically assessing their reasoning capabilities. Our study reveals that current LLMs fall significantly short of solving challenging Olympiad-level problems and frequently fail to distinguish correct mathematical reasoning from clearly flawed solutions. We also found that occasional correct final answers provided by LLMs often result from pattern recognition or heuristic shortcuts rather than genuine mathematical reasoning. These findings underscore the substantial gap between LLM performance and human expertise in advanced mathematical reasoning and highlight the importance of developing benchmarks that prioritize the rigor and coherence of mathematical arguments rather than merely the correctness of final answers.

Published as a conference paper at COLM 2025
Brains vs. Bytes: Evaluating LLM Proficiency in Olympiad Mathematics
Hamed Mahdavi, Pegah Mohammadipour, Samira Malek, Vasant Honavar Pennsylvania State University
University Park, PA, USA (hmm5834, pegahmp,,xm6547 sxm6547, vhonavar)epsu.eda
Alireza Hashemi
City University New York
New York, NY, USA
alireza. hashemi coo
Majid Daliri
New York University
New York, NY, USA
daliri.man mijiiimmjjidny.. edu
Alireza Farhadi
Amirkabir University of Technology
Tehran, Iran farhadi@aut.c ac.
Yekta Yazdanifard
Bocconi University
Milan, Italy yekta. yaktany fard@uniboccoom it
Amir Khasahmadi
Autodesk
Toronto, Canada
amir shasahmadi@eutoodesk. com
Abstract
Recent advancements in large language models (LLMs) have shown impres- sive progress in mathematical reasoning tasks. However, current evaluation benchmarks predominantly focus on the accuracy of final answers, often overlooking the logical rigor crucial for mathematical problem-solving The claim that state-of-the-art LLMs solve Math Olympiad-level problems requires closer examination. To explore this, we conducted both qualitative and quantitative human evaluations of proofs generated by LLMs, and developed a schema for automatically assessing their reasoning capabili- ties. Our study reveals that current LLMs fall significantly short of solving challenging Olympiad-level problems and frequently fail to distinguish cor- rect mathematical reasoning from clearly flawed solutions. We also found that occasional correct final answers provided by LLMs often result from pattern recognition or heuristic shortcuts rather than genuine mathematical reasoning. These findings underscore the substantial gap between LLM performance and human expertise advanced mathematical reasoning and highlight the importance of developing benchmarks that prioritize the rigor and coherence of mathematical arguments rather than merely the correctness of final answers.
1

---

Published as a conference paper at COLM 2025
1 Introduction
The release of OpenAl's ol model (OpenAl et al., 2024) marks a significant breakthrough in artificial intelligence research, particularly in the domains of reasoning and problem solving. Building on this achievement, several state-of-the-art models have been infroduced (DeepSeek-AI et al., 2024; Qwen et al., 2025; Google, 2025), which incorporate post-training on chain-of-thought data. These models demonstrate enhanced capabilities in reasoning and solving mathematical problems. Although the exact methodologies and best practices for developing reasoning models remain active area of research, the use of post-training tech- niques has proven pivotal in improving performance on tasks requiring planning, iterative thnnking, and trial-and-erron strategies. By generating reasoning tokens before producing a final answer, these models offer more deliberate and reliable solutions, particularly for complex reasoning tasks such as mathematical problem-solving.
A range of different benchmarks, such as GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), have been developed to evaluate the capability of large language models (LLMs) to rddress mathematical challenges. As these models advance in performance- Qwen2.5-72B achieving 91.5% accuracy on GSM8K and 62.1% 1% on MATH, while OpenAl's O1 attains 94.8% on the MATH dataset-more challenging benchmarks have been intro- duced. These include OlympiadBench (He et al., 2024), OlympicArena (Huang et al., 2024b), CHAMP (Mao et al., 2024), AlphaGeometry (Trinh et al., 2024), MathOdyssey (Fang et al., 2024), and Omni-MATH (Gao et al., 2024), which contain contest-level mathematical prob- lems designed to push the boundaries of LLM capabilities.
Except for CHAMP, which includes "concepts, general math facts, and hints" (Mao et al., 2024) as additional annotations, and AlphaGeometry, which is distinct as it contains prob- lems and solutions in their formal translation form, all other benchmarks rely on final answer correctness, either symbolic or numerical, as the evaluation metric to assess the reasoning capabilities of various language models.
Models may exploit heuristics or flawed reasoning yet still arrive at the correct final answer, rather than relying on logically sound meth- ods. This limitation is less evident in widely-used reason- ing benchmarks like GSMSK and MATH, which mainly in- clude simpler problems where finding the correct solution is often equivalent solving the problem correctly. In con- trast, benchmarks featuring contest-level mooblemssuchh as CHAMP, Olympiadennch, Omni-MATH, MathOdyssey, and OlympicArena-requuit greater emphasis on the validity and soundness of the solution's reasoning, not just the accuracy of the final answer. This distinction becomes particularly crucial when addressing open-ended questions.
This approach presents clear issue:
Find all functions R-
R such that, f(x2-y²) -y²)
x xffx) - yfly).
For all x,
Let's test simple forms. Clearly, f(x) = kx satisfies the given functional equation Thus, a suitable
O
solution is:
flx) = kx
I'm not convinced. Can you provide rigorous proof?
LLM-as-a-Judge is recent method to automatically assess the quality of LLM-generated responses (Li et al., 2025; 2024; Cohen et al., 2023; Adlakha et al., 2024; Liu et al., 2023; Man- akul et al., 2023). Directly evaluating LLM-generated solu- tions, as in Dubois et al. (2023), can be inherently complex for challenging math problems, as verifying the correctness of a solution may itself require significant effort. Applying con- sistent rubrics for automatic grading of generated solutions also nontrivial because the relative importance of different facts may vary across solutions and problems. In the context of math Olympiad problems, some mistakes are fundamental and reflect deeper misunderstandings, while others are more superficial and could be fixed with minor adjustments. There is no clear, objective method to define which mistakes are critical and which are less significant. Sawada et al. (2023) proposes a rubric-based approach where reference solution and evaluation criteria are used to assess the quality of a generated solution. However, this approach faces an obvious limitation: the existence of multiple valid solutions that may not align with the ref-
Figure 1: LLMs often fail to generate logically sound so- lutions for challenging prob- lems.
2

---

Published as a conference paper at COLM 2025
erence solutions. Furthermore, the challenges inherent in directly assessing LLM-generated solutions also persist with this approach.
Given these limitations, our goal in this work is twofold: first, to evaluate the quality of LLM-generated solutions to mathematics Olympiad problems, including identifying their common failure modes and error types; and second, to investigate whether LLMs can verify the correctness of their own solutions. This study is significant for two main reasons. First, evaluating the correctness of the proofs generated by LLMs can reveal whether training strategies that rely solely on the correctness of the final answer are sufficient for guiding LLMs toward producing valid and logically sound proofs. Second, assessing the ability of LLMs to verify their own generated solutions can provide insights into the feasibility of bootstrapping verification processes, potentially enabling LLMs to improve the quality of their proofs.
To achieve our goal, we assembled a group of evaluators assess the quality of LLM- generated solutions for problems from the International Mathematics Olympiad (IMO) shortlist. After carefully analyzing solutions generated by the frontier models such as ol, ol- mini, 03-mini, Gemini 2.0 (Flash Thinking mode), and DeepSeek R1', we observed that only negligible percentage of these solutions were correct or provided meaningful, non-trivial insights into the problems. We found that LLM-ggeerated solutions frequently contained common types of fallacies, which made their reasoning blatantly incorrect. To address this, we defined and categorized these fallacies, then systematically evaluated solutions generated by frontier models, analyzing their correctness and labeling each incorrect solution by fallacy type. We assessed the ability of frontier models to verify solutions. Our findings revealed that, in nearly all cases, the models failed distinguish between correct solutions and incorrect ones containing obvious fallacious arguments. Importantly, identifying such naive fallacies is a simpler task than full solution verification, yet the models still struggled to perform reliably, In summary, our contributions are as follows:
We conducted an extensive evaluation of solutions generated by frontier models on 455 IMO shortlist problems, emphasizing proof correctness rather than merely checking final answers.
To gain deeper insight into recurring mistakes made by LLMs, we systematically identified and categorized common logical fallacies present in their solutions, estab- lishing comprehensive framework for classifying typical errors.
Leveraging these findings, we created a labeled dataset by annotating each solution according to correctness and the type of fallacy exhibited. This dataset supports tasks related to solution verification and offers valuable insights into the current capabilities and limitations of the frontier LLMss
Our analysis reveals that even advanced models frequently struggle to distinguish between valid solutions and those containing evident logical fallacies.
2 Related Work
Various datasets evaluate mathematical reasoning in large language models (LLMs) (Ahn et al., 2024). Some focus purely on arithmetic problems (Yuan et al., 2023), while math word problem (MWP) datasets, like GSMSK (Cobbe et al., 2021) and MathQA (Amini et al., 2019), present natural language scenarios requiring logical reasoning (Wei et al., 2023).
Benchmarks:
Recently, the limitations of LLMs on seemingly simple problems have motivated bench- marks designed to probe robustness and compositionaliit. GSMIK (Zhang et al., 2024) carefully examines model performance on adversarial arithmetic, revealing surprising fragility even on variations closely related to GSM8K. Building on this, Compositional GSM (Hosseini et al., 2024) introduces multi-step arithmetic tasks requiring compositional
When we started this project, models like Claude 3.7, o3-mini, Gemini 2.5, and Grok 3 hadn't 't been released yet, we could only include 03-mini in our evaluations. After the submission process, we conducted evaluations of Gemini 2.5 Pro and have added them to the appendix.
3

---

Published as a conference paper at COLM 2025
reasoning, further highlighting challenges in generalization. Functional MATH (Srivastava et al., 2024) expands the landscape by evaluating systematic generalization and reasoning robustness via functionally diverse tasks, pushing LLMs beyond pattern matching and toward flexible abstraction.
Automated theorem proving (ATP) datasets evaluate models' capabilities in logical theorem proving (Zheng et al., 2022; Yu et al., 2024; Jiang et al., 2024). Recent benchmarks focus on advanced Olympiad-level mathematics, such CONIC10K for conic sections (Wu et al., 2023), GHOSTS and miniGHOSTS for graduate-level mathematics (Frieder et al., 2023), and CHAMP (Mao et al., 2024), OlympiadBench (He et al., 2024), MathOdyssey (Fang et al., 2024), and Omni-MATH (Gao et al., 2024), specifically focusing on competition-level problems. HARP (Yue et al., 2024) provides human-annotated US competition problems, and MatinaMath offers a large-scale collection of math problems and solutions (LI et al., 2024).
LLM-as-a-judge: Utilizing large language models as evaluative judges has gained popular- ity, reducing reliance on human annotations (Stephan et al., 2024; Li et al., 2024; Nasrabadi, 2024; Ning et al., 2024). This paradigm offers adaptable evaluations based on task-specific contexts (Tan et al., 2024; Dhurandhar et al., 2024), and its effectiveness is typically measured against human judgments (Kim et al., 2024; Ye et al., 2024; Liu et al., 2025). Benchmarks like UltraFeedback (Cui et al., 2024), Alpaca Eval Eval (Dubois et al., 2024), Chatbot Arena (Chiang et al., 2024), and MT-Bench (Zheng et al., 2023) evaluate different LLM judging domains. Specifically for mathematical reasoning, REASONEVAL (Xia et al., 2025) assesses answer correctness and reasoning validity, while MATHCHECK (Zhou et al., 2024) uses LLMs for robust evaluation across diverse mathematical tasks. The SMART-840 dataset (Cherian et al., 2024) benchmarks zero-shot mathematical reasoning based on human performance statistics.
Mathematical Reasoning in LLMs: Large language models (LLMs) have shown success in various reasoning tasks, especially when employing prompting techniques like Chain- of-Thought (CoT), which encourages them to generate correct intermediate steps toward solution (Chen et al., 2024; Wei et al., 2023; Kojima et al., 2023). These methods can sig- nificantly boost performance on challenging problems (Havrilla et al., 2024). Furthermore, inference-time techniques like CoT with Self-Consistency (CoT-SC) have been developed to enhance reasoning by generating multiple reasoning paths and selecting the most consistent one (Wang et al., 2023; Wang & Zhou, 2024). Benchmarks like MATH (Hendrycks et al., 2021), GSM-Symbolic and GSM-NoOp (Mirzadeh et al., 2024) have been introduced to provide more controllable evaluations and reveal limitations such sensitivity to numerical variations and irrelevant information, suggesting a potential lack of deep understand- ing of mathematical concepts. These benchmarks show that current LLMs rely more on probabilistic pattern-matching than genuine formal logical reasoning. To further refine LLMs' reasoning, approaches like reward modeling to evaluate solution correctness and self-refinement techniques (Huang et al., 2024a) and decomposing problems into smaller, algorithmic steps (Zelikman al., 2023) are being explored.
General Workflow
In this section, we outline the data collection process. In the first phase, we selected a set of challenging problems to evaluate the quality of solutions generated by the LLMs. We gathered a group of seven evaluators, each either a former national-level Olympiad medalist or holding or doing a relevant PhD in fields like mathematics or computer science, and asked them to analyze the correctness of the LLM--gnnrated solutions.
3.1 Problem Selection Rationale
Through this project, we used IMO shortlist problems. The process of selecting problems for the IMO Shortlist is rigorous and carefully coordinated. Each participating country submits a set of candidate problems, which typically span the main four mathematical fields: algebra, geometry, combinatorics, and number theory. These submissions are reviewed
4

---

Published as a conference paper at COLM 2025
by problem selection committee to ensure they meet key criteria, including originality, mathematical depth, and suitability for the competition. The committee carefully evaluates the problems for their difficulty level, ensuring a balance between accessibility for less experienced participants and sufficient challenge for the most advanced contestants. From this review process, a shortlist is created, containing diverse collection of high-quality problems. This shortlist forms the basis for the final selection of problems used in the IMÓ.
The shortlist problems have some distinct features that make them suitable for testing the mathematical reasoning capabilities of the frontier models:
Shortlist problems highly original, even within the context of contest-level prob- lems. The selection committee ensures that the solution ideas for these problems are novel and unique as possible. As result, while attempting to generate solu- tions, an LLM cannot simply combine standard building blocks from well-known problems arrive at the correct answer.
It is almost always verified whether the problem can be reduced to a well- established result in advanced mathematics, such as undergraduate graduate- level topics or research-level findings. Consequently, an LLM cannot leverage its extensive knowledge to apply an advanced mathematical result for easy solution.
All problems are designed to be solvable using high-school level mathematics, with the challenge lying in the intricacy of the ideas rather than requiring background in advanced mathematics.
The solutions typically involve multiple steps, each requiring nuanced arguments. Solving these problems demands careful planning, systematic thinking, and rigor- ous verification of each step, in contrast to simpler mathematical problems that can be tackled through straightforward algebraic manipulations trial and error.
On the other hand, since IMO and IMO Shortlist problems are highly reputable, there is significant likelihood of data leakage, as frontier models may have been trained on publicly available high-quality mathematical datasets. In this paper, we demonstrate that even such data leakage has occurred, it does not substantially affect the ability of the LLM to solve IMO Shortlist-level problems.
3.2 Classifying the Failure Modes of LLM Solutions
We asked evaluators to present several IMO shortlist and shortlist-level problems to frontier models, including OpenAl ol, ol-mini, 03-mini, DeepSeek R1, and Gemini 2, and qual- itatively analyze the details of the LLM-generated arguments. We found out that when these frontier models generate incorrect solutions, the errors consistently follow common patterns. Specifically, incorrect solutions typically involve blatantly inaccurate mathematical arguments or statements.
After thorough analysis, we identified the following fallacies commonly occurring incor- rect -...-eeeerteedd solutions. To illustrate each error type, demonstrative as well as real examples are provided in the appendix.
Proof by Example. Drawing a general conclusion based on limited number of specific instances without rigorous justification for all cases. This error occurs when statement appears valid in few examples, misleadingly suggesting universal validity.
Proposal Without Verification. Introducing method or strategy without properly justifying its correctness. The model proposes an idea but provides no rigorous argument or proof supporting its validity.
Inventing Wrong Facts. Citing or inventing non-existent theorems, definitions, or facts to justify a claim. Instead of relying on established mathematical facts, the argument relies on fabricated statements.
Begging the Question (Circular Reasoning). Assuming the conclusion's 's truth within the argument itself, thereby creating inherently flawed logical reasoning.
5

---

Published as a conference paper at COLM 2025
Solution by Trial-and-Erro: Offering solutions derived solely from guesswork or testing few random examples without providing reasoning for why selected solutions work or why alternatives are not considered.
Committing substantial arithmetic or algebraic mistakes that criti- cally undermine the overall correctness of the solution. We specifically considered calcula- tion errors severe enough to compromise the validity of the conclusion.
Calculation Mistakes.
3.3
Data Annotation Processs
After defining the fallacy categories, we provided a list of IMO shortlist problems and corresponding model-generated solutions to the evaluators, instructing them to classify these solutions based on their correctness. We employed the following checklist to annotate our data:
1. First, the evaluator read the solution and determined whether it was correct, par- tially correct, or incorrect.
solution was considered correct it fully addressed all aspects of the problem and contained no significant errors in statements or conclusions.
A solution was deemed partially correct if it included some essential steps of a correct solution but omitted other crucial steps or contained significant inaccuracies.
solution was classified as incorrect if it lacked any non-trivial useful informa- tion relevant to solving the problem.
2. If the solution was not correct, the evaluator identified the categories fallacies present. In some cases, multiple fallacies could be identified.
3. If the problem required final answer, the evaluator recorded both the correct final answer and the mnddel-generated final answer.
To ensure consistency, the evaluators' team lead conducted a thorough review of the evalua- tors' outputs in parallel, verifying that the definitions of fallacies applied correctly and consistently across all evaluators. Borderline cases were identified and discussed separately.
4 Human Evaluation Results
We evaluated the models using IMO shortlist problem sets from the years 2009 to 2023, comprising a total of 455 problems. These included 108 algebra, 117 combinatorics, 116 geometry, and 114 number theory problems. The number of problems per annual shortlist varied slightly, typically ranging from 26 to 35. Each set was carefully curated to maintain balanced distribution of difficulty across the four primary mathematical topics. The performance of each model on the IMO shortlist problems is sunmarized in Table 1. As evident from the results, none of these models achieve performance levels comparable to those obtained through calculating final answers' accuracy, as reported in Gao et al. (2024) and Fang et (2024).
Model
Correct (%)
Partially Correct (%)
Incorrect (%)
DeepSeek
3.8
6.7
89.4
Gemini 2.0
0,0
1.1
98.9
ol ol-mini
1.9
3.9
94.2
0.0
0.0
100.0
03-mini
3.3
4.4
92.2
Table 1: Performance of different models on IMO shortlist problems (%)
The observed gap between the outcomes of our evaluation and other methods that focus exclusively the correctness of final answers arises because models may produce incorrect

---

Published as a conference paper at COLM 2025
intermediate steps yet still obtain the correct final result. To investigate this issue, specifically examined problems with concrete final answers within our evaluation set. Table 2 illustrates both the proportion of correct final answers and the conditional probability of having a correct solution, given that the final answer is correct. Interestingly, we found that the frontier models still predominantly generated wrong solutions despite arriving at the correct final answer.
Model
Final Answer Accuracy (%)
Correct Correct Final Answer (%)
DeepSeek
63.2%
0%
Gemini 2.0
43.8%
0%
ol ol-mini
30.8%
12.5%
35.0%
0%
03-mini
48.3%
14.3%
Table 2: Comparison of evaluated LLMs highlighting the gap between final answer correct- ness and overall solution quality. Final Answer Accuracy denotes the percentage of correct final answers, whereas Correct Correct Final Answer represents the percentage of fully correct solutions among instances where the final answer correct.
These results suggest that the models rely on heuristics, shortcuts, and educated guesses rather than constructing logically sound solutions. Such strategies are particularly applicable for Olympiad-level problems, where the final answers often exhibit predictable patterns or can be inferred without deriving a complete solution. As a result, we argue that evaluating model performance based solely on final answer accuracy is fundamentally flawed, as it ignores the critical problem of solution validity and the systematic exploitation of answer predictability.
50
Fallary Ganggries
Invanting
Proposal Without Venticationn
Prosf Example
Solution By And Emro
Regging The Question
Cakcuiation Mistakiss
Reletive Frequency
20
10
DeapStek
Commini 2.0
fonin
ob-mini
Figure 2: Relative frequencies of each fallacy among the LLM-generated solutions for each model.
also helpful to examine how common fallacies are distributed among different models. Figure 2 shows the relative frequency of each fallacy found in the L.M-generated solutions from each model. Relative frequencies are used because multiple fallacies may occur within single solution. Among these fallacies, Inventing Wrong Facts is the most frequent in four models and the second-most frequent another. This observation might be explained by the training methods used for these models, which generally involve reinforcement learning algorithms with rewards based on the correctness of the final answers (e.g,, see the DeepSeek report; (DeepSeek-Ar Al et al., 2024)).
7

---

Published as a conference paper at COLM 2025
Proposal Without Verification is the other common fallacy. Although the internal reasoning tokens are not visible for OpenAl models, an examination of the thinking traces from DeepSeek and Gemini suggests that this fallacy often arises because the model struggles to determine which calculations and statements should be included in the final response. As result, instead of presenting concrete reasoning useful intermediate steps, the model may produce vague mathematical claims, often beginning with phrases like "It is easy to show that..." " "One can show that..." " without providing proper supporting argument.
We also observed that models tend to exhibit different types of fallacious reasoning depend- ing on the problem type. Figure illustrates the relative frequencies of various fallacies in questions with and without final answers. Notably, Proof by Example and Solution by Trial and Error occur more frequently in questions where final answer is provided. This suggests that models often arrive at final answers either through heuristic trial and error or by gener- alizing from a small number of test cases, leading to a higher prevalence of these two types of fallacies in their generated solutions. In contrast, and Proposal Without Verification occur more frequently in generated solutions to problems lacking explicit final answers. Since these problems are purely proof-based, it is natural to observe these particular fallacies more often. To produce valid proofs, models must logically connect the problem's 's initial assumptions and constraints to the required conclu- sion. However, we find that models frequently circumvent this rigorous reasoning process either by introducing incorrect statements or by omitting essential steps in their arguments.
observe that Inventing Wrong Facts
Similar patterns of difference can be observed in relative frequen- cies of fallacies among different problem topics. Figure 4 demon- strates the relative frequencies of fallacies among geometry, algebra, combinatorics, and number the- ory problems. As more geometry questions can be solved only us- ing logical statements rather than algebraic manipulations, Inventing Wrong Facts, Proposal Without Veri- fication, and Begging the Question are more common in geometry problems. A significant propor- tion of algebra problems fall into categories such as functional equa- tions, polynomial equations, or op- timization tasks. We observed that all frontier models tend to avoid generating rigorous analytic solutions, instead relying on trial and error to determine the final answer. This behavior results in higher frequency of the Solution by Trial and Error fallacy in LLM-geeneraed solutions for algebra problems. Similarly, number theory problems involving Diophantine equations functional equations exhibit the same issue. Additionally, we found that the Proof by Example fallacy occurs more frequently algebra, combinatorics, and number theory problems compared to geometry. This trend arises because many problems in these three areas can be framed as proving statements of the form Q(x), where belongs to a specific domain defined by the problem. In such cases, LLMs frequently attempt to verify the proposition Q by evaluating selected examples from its domain rather than constructing a general proof, thus resulting in the Proof by Example fallacy.
Inventing Wrong Facts
Proposal Without Verticatioon
Proof by Example
Solution by Triel and
Relative Frequency (%)
Boophn ng Guestion
Caliuu wdinn Masaiies
0
With Final Answer
Without Finel Answer
Figure 3: Relative frequencies of each fallacy in LLM- generated solutions, comparing questions with and without final answer.
integer-valued
5 Automatic Evaluation Results with Gemini 2.5 Pro
Verification of candidate solution to problem is generally considered an easier task than solving the problem itself. Consequently, a common strategy for training reasoning-oriented LLMs is the generator-verifier schema. Within this framework, generator produces
8

---

Published as a conference paper at COLM 2025
Fallacy Calegory
80
Inventing Wrong Facts
Proposel Without Verification
Proof Example
50
Solution By Trial And Emro
Relative Frequency (%
Begging The Question
Calculation Mistakes
40
30
10
Geometry
Algebra
Combinatorics
Number Theory
Figure 4: Relative frequencies of each fallacy in LLM-ggenrrted solutions among different topics
Model
Real Solutions Correct (%)
Wrong Solutions Correct (%)
DeepSeek
48
43
Gemini 2.0
52
50
o1 ol-mini
31
39
36
45
03-mini
26
31
Table 3: Percentage of correct and incorrect solutions identified as correct by different LLMs during verification. The results illustrate the LLMs' difficulty in accurately distinguishing genuinely correct solutions from clearly incorrect ones containing explicit fallacies.
candidate solutions, and a reward model evaluates these candidates. The reward model can either be hard-coded (e.g., checking only the correctness of final answers), learned reward model trained specifically for evaluation, or another powerful LLM serving as a judge (Plaat et al., 2024). A pertinent question in this context is whether saate-of-the-art models can reliably distinguish fallacious generated solutions from correct, authentic solutions. We approach this question through two complementary analyses:
1. Do LLMs identify correct, authentic solutions as valid more frequently than falla- cious generated solutions?
2. When presented with pairs consisting of correct solution and incorrect, falla- cious one for each problem, are LLMs capable of accurately choosing the correct solution?
To investigate the first question, we gathered all problems from our evaluation dataset that had incorrect solutions generated by LLMs. Corresponding correct solutions for these problems were obtained from the Art of Problem Solving (AoPS) website?. We then prompted the LLMs to analyze each solution and explicitly request final judgment of either correct or wrong. While this verification task can generally be nuanced due to solutions that partially satisfy correctness criteria, our selected examples distinctly represent either clearly incorrect solutions with obvious fallacies or entirely correct solutions. The description of all prompts used in this section be found in the appendix.
Pnttps://artoffrroblaassllii cow.unicmmmmncccccmmmmmmmm. ttte:muuuii////2222 imo shortlist

---

Published as a conference paper at COLM 2025
As shown in Table 3, DeepSeek and Gemini 2.0 produce responses of correct and woong with nearly equal frequency for both genuinely correct solutions and incorrect LLM-generated solutions. Interestingly, the likelihood of identifying genuinely correct solution correct is even lower for the ol, ol-mini, and o3-mini models. These results demonstrate that the models not suitable for use judges, they cannot reliably distinguish genuine solutions from obviously incorrect ones.
To investigate the second question, we applied similar methodology. For each problem, we presented the models with pairs consisting of correct solution and an incorrect, LLM- generated solution. We then prompted the models identify the correct solution after careful analysis, explicitly informing them that exactly one solution was correct and the other was incorrect.
Model
Accuracy (%)
DeepSeek
57
Gemini 2.0
49
o1
50
ol-mini
46
03-mini
52
Table 4: Accuracy of various LLMs in identifying the correct solution when presented with pairs consisting of one correct solution and one incorrect solution generated by an LLM.
As presented in Table 4, models ol, ol-mini, and Gemini 2.0 perform at or below random in distinguishing correct from incorrect solutions. Only DeepSeek and 03-mini perform modestly better than chance, outperforming random selection by 7% and 2%, respectively. These results indicate that the evaluated models currently have limited effectiveness as verifiers for challenging tasks such as IMO Shortlist-level problems.
6 General Insights into Gemini 2.5 Pro Solutions
Our evaluation of frontier LLMs on Olympiad-level mathematics revealed significant short- comings in their ability to produce logically rigorous proofs and engage in genuine mathe- matical reasoning. Models such as OpenAl's ol, ol-mini, 03-mini, DeepSeek R1, indicate reliance on heuristic shortcuts rather than authentic reasoning processes. Additionally, these LLMs demonstrated limited capability in effectively verifying solutions, performing at or near random levels when distinguishing correct proofs from clearly incorrect ones.
These findings emphasize two critical areas for improvement. First, there is a clear need for developing more sophisticated benchmarking methods that evaluate logical rigor and reasoning quality rather than merely assessing final answers. Second, relying solely on final answer correctness or utilizing more powerful LLMs as judges is insufficient; improved training schemas specifically designed to address the logical rigor of the proofs are essential for advancing future models toward human-level mathematical proficiency.
References
Vaibhav Adlakha, Parishad BehnamGhader, Xing Han Lu, Nicholas Meade, and Siva Reddy, Evaluating correctness and faithfulness of Instruction-following models for question answering. Transactions of the Association for Computational Linguistics, 12, 2024. doi: 10.1162/taci a 00667. URL https: pettps://ace http:////caaaanaaaa/aa/ wwtp:222....ll1111.... tacl-1.38/.
Janice Ahn, Rishu Verma, Renze Lou, Di Liu, Rui Zhang, and Wenpeng Yin. Large language models for mathematical reasoning: Progresses and challenges, 2024. URL https://
arxiv.org www.org///b///////ee 00157.
Aida Amini, Saadia Gabriel, Peter Lin, Rik Koncel-Kedziorsk Yejin Choi, and Hannaneh Hajishirzi. Mathqa: Towards interpretable math word problem solving with operation- based formalisms, 2019. URL https:://axxi...rrggeeeooe/ 1955. 13319.
10

---

Published as a conference paper at COLM 2025
Haolin Chen, Yihao Feng, Zuxin Liu, Weiran Yao, Akshara Prabhakar, Shelby Heinecke, Ricky Ho, Phil Mui, Silvio Savarese, Caiming Xiong, and Huan Wang. Language models are hidden reasoners: Unlocking latent reasoning capabilities via self-rewarding, 2024. URL
244..0222.
Anoop Cherian, Kuan-Chuan Peng, Suhas Lohit, Joanna Matthiesen, Kevin Smith, and Joshua B. Tenenbaum. Evaluating large vision-and-language models on children's 's mathe- matical olympiads, 2024. URL https: http:///xxvvv.rrree
15736.
Wei-Lin Chiang, Lianmin Zheng, Ying Sheng, Anastasios Nikolas Angelopoulos, Tianle Li, Dacheng Li, Hao Zhang, Banghua Zhu, Michael Jordan, Joseph E. Gonzalez, and Ion Stoica. Chatbot arena: An open platform for evaluating Ilms by human preference, 2024.
URL https::/aaxxv..rgg/////// org/abs/2403.041 04132.
Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems. arXiv preprint 4rXi::2110.14168, 2021.
Roi Cohen, May Hamri, Mor Geva, and Amir Globerson. LM vs LM: via cross examination. In The 2023 Conference on Empirical Methods in Natural Language
Detecting factual errors Processing, 2023. URL https: http::://opeeree /opp/reveeeeeeeeeeeeeeeeeeeeeeee nett/oorummdd--PnTTTeee
Ganqu Cui, Lifan Yuan, Ning Ding, Guanming Yao, Bingxiang He, Wei Zhu, Yuan Ni, Guotong Xie, Ruobing Xie, Yankai Lin, Zhiyuan Liu, and Maosong Sun. Ultrafeedback: Boosting language models with scaled ai feedback, 2024. URL https://arxivv or org/obag / 01377.
DeepSeek-AI, Aixin Liu, Bei Feng, Bing Xue, Bingxuan Wang, Bochao Wu, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, Damai Dai, Daya Guo, Dejian Yang, Deli Chen, Dongje Ji, Erhang Li, Fangyun Lin, Fucong Dai, Fuli Luo, Guangbo Hao, Guanting Chen, Guowei Li, H. Zhang, Han Bao, Hanwei Xu, Haocheng Wang, Haowei Zhang, Honghui Ding, Huajian Xin, Huazuo Gao, Hui Li, Hui Qu, J. L. Cai, Jian Liang, Jianzhong Guo, Jiaqi Ni, Jiashi Li, Jiawei Wang, Jin Chen, Jingchang Chen, Jingyang Yuan, Junjie Qiu, Junlong Li, Junxiao Song, Kai Dong, Kai Hu, Kaige Gao, Kang Guan, Kexin Huang, Kuai Yu, Lean Wang, Lecong Zhang, Lei Xu, Leyi Xia, Liang Zhao, Litong Wang, Liyue Zhang, Meng Li, Miaojun Wang, Mingchuan Zhang, Minghua Zhang, Minghui Tang, Mingming Li, Ning Tian, Panpan Huang, Peiyi Wang, Peng Zhang, Qiancheng Wang, Qihao Zhu, Qinyu Chen, Qiushi Du, R. Chen, R. L. Jin, Ruiqi Ge, Ruisong Zhang, Ruizhe Pan, Runji Wang, Runxin Xu, Ruoyu Zhang, Ruyi Chen, S. Li, Shanghao Lu, Shangyan Zhou, Shanhuang Chen, Shaoqing Wu, Shengfeng Ye, Shengfeng Ye, Shirong Ma, Shiyu Wang, Shuang Zhou, Shuiping Yu, Shunfeng Zhou, Shuting Pan, T. Wang, Tao Yun, Tian Pei, Tianyu Sun, W. L. Xiao, Wangding Zeng, Wanjia Zhao, Wei An, Wen Liu, Wenfeng Liang, Wenjun Gao, Wenqin Yu, Wentao Zhang, X. Q. Li, Xiangyue Jin, Xianzu Wang, Xiao Bi, Xiaodong Liu, Xiaohan Wang, Xiaojin Shen, Xiaokang Chen, Xiaokang Zhang, Xiaosha Chen, Xiaotao Nie, Xiaowen Sun, Xiaoxiang Wang, Xin Cheng, Xin Liu, Xin Xie, Xingchao Liu, Xingkai Yu, Xinnan Song, Xinxia Shan, Xinyi Zhou, Xinyu Yang, Xinyuan Li, Xuecheng Su, Xuheng Lin, Y. K. Li, Y. Q. Wang, Y. X. Wei, Y. X. Zhu, Yang Zhang, Yanhong Xu, Yanhong Xu, Yanping Huang, Yao Li, Yao Zhao, Yaofeng Sun, Yaohui Li, Yaohui Wang, Yi Yu, Yi Zhengg, Yichao Zhang, Yifan Shi, Yiliang Xiong, Ying He, Ying Tang, Yishi Piao, Yisong Wang, Yixuan Tan, Yiyang Ma, Yiyuan Liu, Yongqiang Guo, Yu Wu, Yuan Ou, Yuchen Zhu, Yuduan Wang, Yue Gong, Yuheng Zou, Yujia He, Yukun Zha, Yunfan Xiong, Yunxian Ma, Yuting Yan, Yuxiang Luo, Yuxiang You, Yuxuan Liu, Yuyang Zhou, Z. F. Wu, Z. Z. Ren, Zehui Ren, Zhangli Sha, Zhe Fu, Zhean Xu, Zhen Huang, Zhen Zhang, Zhenda Xie, Zhengyan Zhang, Zhewen Hao, Zhibin Gou, Zhicheng Ma, Zhigang Yan, Zhihong Shao, Zhipeng Xu, Zhiyu Wu, Zhongyu Zhang, Zhuoshu Li, Zihui Gu, Zijia Zhu, Zijun Liu, Zilin Li, Ziwei Xie, Ziyang Song, Ziyi Gao, and Zizheng Pan. Deepseek-v3 technical report, 2024. URL https: //arxiv. http://awvvvvoooabbbr////ss /212.. 1.12).9933 19437.
Amit Dhurandhar, Rahul Nair, Moninder Singh, Elizabeth Daly, and Karthikeyan Natesan Ramamurthy, Ranking large language models without ground truth, 2024. URL https: // //aaxiiv.org www.iv..orgabb/////// 14860.
11

---

Published as a conference paper at COLM 2025
Yann Dubois, Xuechen Li, Rohan Taori, Tianyi Zhang, Ishaan Guirajani, Jimmy Ba, Carlos Guestrin, Percy Liang, and Tatsunori Hashimoto. Alpacafarm: simulation framework for methods that learn from human feedback. In Thirth-seventh Conference on Neural Infor- mation Processing Systems, 2023. URL https: htttp:://opnneeveww wtw.ppperriiiieeeeeoorrn met/Forum?id=4htur
Yann Dubois, Balázs Galambosi, Percy Liang, and Tatsunori B. Hashimoto. Length- controlled alpacaeval: A simple way to debias automatic evaluators, 2024. URL
https://arxiv. wrg./b//////......o http:://arrxx..rrrr//b///A.... 04475..
Meng Fang, Xiangpeng Wan, Fei Lu, Fei Xing, and Kai Zou. Mathodyssey; Benchmarking mathematical problem-solving skills in large language models using odyssey math data, 2024. URL https: //axivv. http://////awww.o///b/////////// wrgg/bs////.....o/ 18321.
Simon Frieder, Luca Pinchetti, Alexis Chevalier, Ryan-Rhys Griffiths, Tommaso Salvatori, Thomas Lukasiewicz, Philipp Christian Petersen, and Julius Berner. Mathematical capa- bilities of chatgpt, 2023. URL https://arxiis wtwpaaawwv...oo/bbeee//eee 2t///201/1 13867.
Bofei Feian Song, Zhe Yang, Zefan Cai, Yibo Miao, Qingxiu Dong, Lei Li, Chenghao Ma, Liang Chen, Runxin Xu, Zhengyang Tang, Benyou Wang, Daoguang Zan, Shanghaoran Quan, Ge Zhang, Lei Sha, Yichang Zhang, Xuancheng Ren, Tianyu Liu, and Baobao Chang. Omni-math: A universal olympiad level mathematic benchmark for large language models, 2024. URL
wrg.orgg////./ooooooooo.ooo //////////11. 07985.
Google. Gemini model v2 documentation, 2025. URL /ttps://ai..oogle... wwwgge.dee//geminn... dww//gemiii-ap.. docs/mooelssggeminn ni- Accessed: 2025-01-19.
Alex Havrilla, Sharath Raparthy, Christoforus Nalmpantis, Jane Dwivedi-Yu, Maksym Zhuravinskyi, Eric Hambro, and Roberta Raileanu. Glore: When, where, and how to improve Ilm reasoning via global and local refinements, 2024. URL httpt:://///xtit.oo abs/2402. 10963.
Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems, 2024.
Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset, 2021. URL https: //axivv orgg/bb//////...o// https::///axxi.oorr/////10.. 03874.
Arian Hosseini, Alessandro Sordoni, Daniel Kenji Toyama, Aaron Courville, and Rishabh Agarwal. Not all Ilm reasoners are created equal. In Proceedings of the 4th Work- shop on Mathematical Reasoning and Al (MATH-Al) at NeurlPS 2024, 2024. URL https: //ooe/neviee.eet//orun/mniieee www..et/ooorumiicRRcnnnoo..L. Introduces the Compositional GSM bench- mark.
Jie Huang, Xinyun Chen, Swaroop Mishra, Huaixiu Steven Zheng, Adams Wei Yu, Xinying Song, and Denny Zhou. Large language models cannot self-correct reasoning yet, 2024a. URL / http://axxxv...r/a/aaaa
Zhen Huang, Zengzhi Wang, Shijie Xia, Xuefeng Li, Haoyang Zou, Ruijie Xu, Run-Ze Fan, Lyumanshan Ye, Ethan Chern, Yixin Ye, Yikai Zhang, Yuging Yang, Ting Wu, Binjie Wang, Shichao Sun, Yang Xiao, Yiyuan Li, Fan Zhou, Steffi Chern, Yiwei Qin, Yan Ma, Jiadi Su, Yixiu Liu, Yuxiang Zheng, Shaoting Zhang, Dahua Lin, Yu Qiao, and Pengfei Liu. Olympicarena: Benchmarking multi-discipline cogmitive reasoning for superintelligent ai. ar arxi preprint Xiv::2440 12753, 2024b. URL
abs// 12753.
Dongwei Jiang, Marcio Fonseca, and Shay B. Cohen. Leanreasoner: Boosting complex logical reasoning with lean, 2024. URL htttps://axxiv.orggbb///// 13312
Yubin Kim, Chanwoo Park, Hyewon Jeong, Cristina Grau-Vilchez, Yik Siu Chan, Xuhai Xu, Daniel McDuff, Hyeonhoon Lee, Cynthia Breazeal, and Hae Won Park. A demonstration of adaptive collaboration of large language models for medical decision-making, 2024. URL https:://rxxv..rgg/ 2411 00248.
12

---

Published as a conference paper at COLM 2025
Takeshi Kojima, Shixiang Shane Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. Large language models are zero-shot reasoners, 2023. htt.ttps://axxivvoorr 11916.
Dawei Li, Bohan Jiang, Liangjie Huang, Alimohammad Beigi, Chengshuai Zhao, Zhen Tan, Amrita Bhattacharjee, Yuxuan Jiang, Canyu Chen, Tianhao Wu, Kai Shu, Lu Cheng, and Huan Liu. From generation to judgment: Opportunities and challenges of llm-as-a-judge, 16594.
2025. URL
Haitao Li, Qian Dong, Junjie Chen, Huixue Su, Yujia Zhou, Qingyao Ai, Ziyi Ye, and Yiqun Liu. Lims-as-judges: A comprehensive survey on llm-based evaluation methods, 2024.
URL https::/arxiv.org/abb/////
05579.
Jia LI, Edward Beeching, Lewis Tunstall, Ben Lipkin, Roman Soletskyi, Shengyi Costa Huang, Kashif Rasul, Longhui Yu, Albert Jiang, Ziju Shen, Zihan Qin, Bin Dong, Li Zhou, Yann Fleureau, Guillaume Lample, and Stanislas Polu. Nu- //tttps://githhe com/ minamath. htttts::/huggiingfaee http://///////mmaaaaaattt----ee. pdf 2024.
Yang Liu, Dan Iter, Yichong Xu, Shuohang Wang, Ruochen Xu, and Chenguang Zhu. G-eval: NLG evaluation using gpt-4 with better human alignment. In Houda Bouamor, Juan Pino, and Kalika Bali (eds.), Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, Singapore, December 2023. Association for Computational Linguistics. doi: 10.18653 /v htt./222..emlppnnn 153..
Yinhong Liu, Han Zhou, Zhijiang Guo, Ehsan Shareghi, Ivan Vulié, Anna Korhonen, and Nigel Collier. Aligning with human judgement: The role of pairwise preference in large language model evaluators, 2025. URL https://arxiv.. org
wrwp/////gga/b//0333 abs/ 16950.
Potsawee Manakul, Adian Liusie, and Mark Gales. SelfcheckGPT: Zero-resource black-box hallucination detection for generative large language models. In The 2023 Conference on Empirical Methods in Natural Language Processing, 2023. URL https://operreview.. net/ forum??d-RRwFNNDJ3E.
Yujun Mao, Yoon Kim, and Yilun Zhou. CHAMP: A competition-level dataset for fine- grained analyses of LLMs' mathematical reasoning capabilities. In Findings of the Associa- fion for Computational Linguistics: ACL 2024. Association for Computational Linguistics, 2024. doi: 10.18653/v1 http://1/22242/nniii 4024.finndigg--ad.75555 URL h o02/2024. findings acl. 785/.
Iman Mirzadeh, Keivan Alizadeh, Hooman Shahrokhi, Oncel Tuzel, Samy Bengio, and Mehrdad Farajtabar. Gsm-symbolic: Understanding the limitations of mathematical reasoning in large language models, 2024. URL attps://arxxxr wwtprxiww.oog/b/////4/ 241/ 21..22111.2222 05222.
Dom Nasrabadi. Juree not judges: safeguarding Ilm interactions with small, specialised encoder ensembles, 2024. URL httphtts//arxiiiorgg//aeee
08442.
Kun-Peng Ning, Shuo Yang, Yu-Yang Liu, Jia-Yu Yao, Zhen-Hui Liu, Yu Wang, Ming Pang, and Li Yuan. Pico: Peer review in Ilms based on the consistency optimization, 2024. URL 01830.
OpenAl, Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, Alex Iftimie, Alex Karpenko, Alex Tachard Passos, Alexander Neitz, Alexander Prokofiev, Alexander Wei, Allison Tam, Ally Bennett, Ananya Kumar, Andre Saraiva, Andrea Vallone, Andrew Duberstein, Andrew Kondrich, Andrey Mishchenko, Andy Applebaum, Angela Jiang, Ashvin Nair, Barret Zoph, Behrooz Ghorbani, Ben Rossen, Benjamin Sokolowsky, Boaz Barak, Bob McGrew, Borys Minaiev, Botao Hao, Bowen Baker, Brandon Houghton, Bran- don McKinzie, Brydon Eastman, Camillo Lugaresi, Cary Bassin, Cary Hudson, Chak Ming Li, Charles de Bourcy, Chelsea Voss, Chen Shen, Chong Zhang, Chris Koch, Chris Orsinger, Christopher Hesse, Claudia Fischer, Clive Chan, Dan Roberts, Daniel Kappler, Daniel Levy, Daniel Selsam, David Dohan, David Farhi, David Mely, David Robinson, Dimitris
13

---

Published as a conference paper at COLM 2025
Tsipras, Doug Li, Dragos Oprica, Eben Freeman, Eddie Zhang, Edmund Wong, Eliz- abeth Proehl, Enoch Cheung, Eric Mitchell, Eric Wallace, Erik Ritter, Evan Mays, Fan Wang, Felipe Petroski Such, Filippo Raso, Florencia Leoni, Foivos Tsimpourlas, Fran- cis Song, Fred von Lohmann, Freddie Sulit, Geoff Salmon, Giambattista Parascandolo, Gildas Chabot, Grace Zhao, Greg Brockman, Guillaume Leclere, Hadi Salman, Haiming Bao, Hao Sheng, Hart Andrin, Hessam Bagherinezhad, Hongyu Ren, Hunter Lightman, Hyung Won Chung, Ian Kivlichan, lan O'Connell, Ian Osband, Ignasi Clavera Gilaberte, Ilge Akkaya, Ilya Kostrikov, Ilya Sutskever, Irina K Jakub Pachocki, James Lennon, Jason Wei,Jean Harb, Jerry Twore, Jiacheng Feng, Jiahui Yu, Jiayi Weng, Jie Tang, Jieqi Yu, Joaquin Quiñonero Candela, Joe Palermo, Joel Pariih, Johannes Heidecke, John Hallman, John Rizzo, Jonathan Gordon, Jonathan Uesato, Jonathan Ward, Joost Huizinga, Julie Wang, Kai Chen, Kai Xiao, Karan Singhal, Karina Nguyen, Karl Cobbe, Katy Shi, Kayla Wood, Kendra Rimbach, Keren Gu-Lemberg, Kevin Liu, Kevin Lu, Kevin Stone, Kevin Yu, Lama Ahmad, Lauren Yang, Leo Liu, Leon Maksin, Leyton Ho, Liam Fedus, Lilian Weng, Linden Li, Lindsay McCallum, Lindsey Held, Lorenz Kuhn, Lukas Kondraciuk, Lukasz Kaiser, Luke Metz, Madelaine Boyd, Maja Trebacz, Manas Joglekar, Mark Chen, Marko Sintor,Mason Meyer, Matt Jones, Matt Kaufer, Max Schwarzer, Meghan Shah, Mehmet Yat- baz, Melody Y. Guan, Mengyuan Xu, Mengyuan Yan, Mia Glaese, Mianna Chen, Michael Lampe, Michael Malek, Michele Wang, Michelle Fradin, Mike McClay, Mikhail Pavlov, Miles Wang, Mingxuan Wang, Mira Murati, Mo Bavarian, Mostafa Rohaninejad, Nat McAleese, Chowdhury, Neil Chowdhury, Nick Ryder, Nikolas Tezak, Noam Brown, Ofir Nachum, Oleg Boiko, Oleg Murk, Olivia Watkins, Patrick Chao, Paul Ashbourne, Pavel Izmailov, Peter Zhokhov, Rachel Dias, Rahul Arora, Randall Lin, Rapha Gontijo Lopes, Raz Gaon, Reah Miyara, Reimar Leike, Renny Hwang, Rhythm Garg, Robin Brown, Roshan James, Rui Shu, Ryan Cheu, Ryan Greene, Saachi Jain, Sam Altman, Sam Toizer, Sam Toyer, Samuel Miserendino, Sandhini Agarwal, Santiago Hernandez, Sasha Baker, Scott McKinney, Scottie Yan, Shengjia Zhao, Shengli Hu, Shibani Santurkar, Shraman Ray Chaudhuri, Shuyuan Zhang, Siyuan Fu, Spencer Papay, Steph Lin, Suchir Balaji, Suvansh Sanjeev, Szymon Sidor, Tal Broda, Aidan Clark, Tao Wang, Taylor Gordon, Ted Sanders, Tejal Patwardhan, Thibault Sottiaux, Thomas Degry, Thomas Dimson, Tianhao Zheng, Timur Garipov, Tom Stasi, Trapit Bansal, Trevor Creech, Troy Peterson, Tyna Eloundou, Valerie Qi, Vineet Kosaraju, Vinnie Monaco, Vitchyr Pong, Vlad Fomenko, Weiyi Zheng, Wenda Zhou, Wes McCabe, Wojciech Zaremba, Yann Dubois, Yinghai Lu, Yining Chen, Young Cha, Yu Bai, Yuchen He, Yuchen Zhang, Yunyun Wang, Zheng Shao, and Zhuohan Li. Openai ol system card, 2024. URL https: http///wxxvvvr/r/ee///// /bs///b////// 16720.
T
Aske Plaat, Annie Wong, Suzan Verberne, Joost Broekens, Niki van Stein, and Thomas Back. Reasoning with large language models, a survey, 2024. URL https: //arxiv. org/abs/ 2407. 11511.
Owen, :, An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, Huan Lin, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jingren Zhou, Junyang Lin, Kai Dang, Keming Lu, Keqin Bao, Kexin Yang, Le Yu, Mei Li, Mingfeng Xue, Pei Zhang, Qin Zhu, Rui Men, Runji Lin, Tianhao Li, Tianyi Tang, Tingyu Xia, Xingzhang Ren, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yu Wan, Yuqiong Liu, Zeyu Cui, Zhenru Zhang, and Zihan Qiu. Qwen2.5 technical report, 2025. URL //arxiv////vvi..////// orgg/bs///12/ 15115.
Tomohiro Sawada, Daniel Paleka, Alexander Havrilla, Pranav Tadepalli, Paula Vidas, Alexander Kranias, John Nay, Kshitij Gupta, and Aran Komatsuzaki. ARB: Advanced reasoning benchmark for large language models. In The 3rd Workshop on Mathematical Rea- soning and Al at NeurlPS'23 23, 2023. URL https: hatp::://ppneeeiiieee www.:ne/foooo net/forumiid--rr-ee
4Rr.
Saurabh Srivastava, Annarose MB, Anto PV, Shashank Menon, Ajay Sukumar, Adwaith Samod T, Alan Philipose, Stevin Prince, and Sooraj Thomas. Functional benchmarks for robust evaluation of reasoning performance, and the reasoning gap. arXiv preprint XxX::22402.19450, 2024.
Andreas Stephan, Dawei Zhu, Matthias Aßenmacher, Xiaoyu Shen, and Benjamin Roth. From calculation to adjudication: Examining Ilm judges on mathematical reasoning tasks, 2024. URL https: //arxiv wwwaaaiiii.oo/aaaaeeeeee org//bs/2409. 04168.
14

---

Published as a conference paper at COLM 2025
Zhen Tan, Dawei Li, Song Wang, Alimohammad Beigi, Bohan Jiang, Amrita Bhattacharjee, Mansooreh Karami, Jundong Li, Lu Cheng, and Huan Liu. Large language models for data annotation and synthesis: A survey, 2024. URL
1344..
Trieu Trinh, Yuhuai Wu, Quoc Le, He He, and Thang Luong. Solving olympiad geometry without human demonstrations, Nature, 2024. doi: 10.1038/ 04008///11188-02-07777----
Xuezhi Wang and Denny Zhou. Chain-of-thought reasoning without prompting, 2024. URL
https://are http///xxiivvvv/ivvv////////avl/ www.awwwwww....////../aa.... wrg. 2ttpbb//222...0000
Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain of thought reasoning in language models, 2023. URL
2
Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le, and Denny Zhou. Chain-of-thought prompting elicits reasoning in large lan- guage models, 2023. URL https://arviv.org/bb///22 1190// 11903.
Haoyi Wu, Wenyang Hui, Yezeng Chen, Weiqi Wu, Kewei Tu, and Yi Zhou. Conic10k: A challenging math problem understanding and reasoning dataset, 2023. URL https: // //arxiv..rr//bs/223333 http//bs//21111/ 2311.0113..
Shijie Xia, Xuefeng Li, Yixin Liu, Tongshuang Wu, and Pengfei Liu. Evaluating mathematical reasoning beyond accuracy, 2025. URL https: /aax:i/////vv///aa / 05692.
Jiayi Ye, Yanbo Wang, Yue Huang, Dongping Chen, Qihui Zhang, Nuno Moniz, Tian Gao, Werner Geyer, Chao Huang, Pin-Yu Chen, Nitesh V Chawla, and Xiangliang Zhang. Justice or prejudice? quantifying biases in llm-as-a-judge, 2024. URL https://arxxiv.orr 0ts/22410.02736
Longhui Yu, Weisen Jiang, Han Shi, Jincheng Yu, Zhengying Liu, Yu Zhang, James Kwok, Zhenguo Li, Adrian Weller, and Weiyang Liu. Metamath: Bootstrap your own mathematical questions for large language models, 2024. URL https://arxixi http://xxxv..rggb///// 2309. 12284.
Zheng Yuan, Hongyi Yuan, Chuanqi Tan, Wei Wang, and Songfang Huang. How well do large language models perform in arithmetic tasks?, 2023. URL
org/abs/
Albert S. Yue, Lovish Madaan, Ted Moskovitz, DJ Strouse, and Aaditya K. Singh. Harp: A challenging human-annotated math reasoning benchmark, 2024. URL https://arxiv. org/abs/2412, 28819981......
Eric Zelikman, Qian Huang, Gabriel Poesia, Noah D. Goodman, and Nick Haber. Parsel: Algorithmic reasoning with language models by composing decompositions, 2023, URL https://arxivv wrggabs//22/ 2212. 10561.
Hugh Zhang, Jeff Da, Dean Lee, Vaughn Robinson, Catherine Wu, Will Song, Tiffany Zhao, Pranav Raja, Charlotte Zhuang, Dylan Slack, Qin Lyu, Sean Hendryx, Russell Kaplan, Michele Lunati, and Summer Yue. A careful examination of large language model performance on grade school arithmetic. In Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS 2024), Datasets and Benchmarks Track, Vancouver, BC, 2024. URL https: http:////ppeeeeeieeee wttp://wwwwiiieeeeeooommmmm/mmme net//oorumidd-RJJRMwZZZ
Kunhao Zheng, Jesse Michael Han, and Stanislas Polu. Minif2f: a cross-system benchmark for formal olympiad-level mathematics, 2022. URL hww.h:tts/ss/itii
001 10.
Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric P. Xing, Hao Zhang, Joseph E. Gonzalez, and lon Stoica. Judging am--aee with mt-bench and chatbot arena, 2023. URL 05685.
15

---

Published as a conference paper at COLM 2025
Zihao Zhou, Shudong Liu, Maizhen Ning, Wei Liu, Jindong Wang, Derek F. Wong, Xiaowei Huang, Qiufeng Wang, and Kaizhu Huang. Is your model really a good math reasoner? evaluating mathematical reasoning with checklist, 2024. URL https: //axxiv org/ org/bbs/
A Human Evaluation Results with Gemini 2.5 Pro
We repeated the same process for 31 problems from the IMO shortlist 2014 problems with Gemini 2.5 Pro (preview-03-25). As shown in Table 5, Gemini 2.5 Pro significantly outperforms all other models. Qualitatively, Gemini 2.5 Pro solutions are noticeably more coherent and contain fewer basic errors. Additionally, Gemini 2.5 Pro generated a higher percentage of partially correct solutions. Even when solutions were incomplete, they often contained useful derivations and relevant proofs. In contrast, solutions from other models typically lacked useful information meaningful insights.
LLM Solution Model
Correct
Partially Correct
Incorrect
DeepSeek Gemini 2.0 Gemini 2.5 Pro 0l ol-mini o3-mini
3.8%
6.7%
89.4%
0.0%
1.1%
98.9%
25.8%
25.8%
48.4%
1.9%
3.9%
94.2%
0.0%
0.0%
100.0%
3.3%
4.4%
92.2%
Table 5: Performance of different models on IMO shortlist problems (%)
Table 6 indicates that Gemini 2.5 Pro is more likely provide a correct overall solution when its final answer is correct.
Model
Final Answer Accuracy (%)
Correct Correct Final Answer (%)
DeepSeek
63.2%
0%
Gemini 2.0
43.8%
0%
Gemini 2.5
55.6%
16.7%
ol
30.8%
12.5%
ol-mini
35.0%
0%
03-mini
48.3%
14.3%
Table 6: Comparison of evaluated LLMs highlighting the gap between final answer correct- ness and overall solution quality. Final Answer Accuracy denotes the percentage of correct final answers, whereas Correct Correct Final Answer represents the percentage of fully correct solutions among instances where the final answer is correct.
Figure 5 shows the relative frequencies of various fallacies for each model, including Gemini 2.5 Pro. Gemini 2.5 notably makes fewer basic errors such as "Proposal Without Verification" and "Proof by Example," which typically have predictable patterns. Even when employing "Solution by Trial and Error," " Gemini 2.5 generally attempts to check the other possible solutions and rule out alternatives. Interestingly, Gemini 2.5 introduces unique pattern when does "Inventing Wrong Facts". This occurs when the model initially attempts legitimate ideas and derivations but resorts to citing non-existent theorem or result equivalent to the problem statement after failing. This behavior was common in nearly all challenging problems. We suspect that the root cause of this error is agent-based training. Since the model didn't have internet access during the evaluation, it hallucinated non-existent papers, blog posts, and theorems.
16

---

Published as a conference paper at COLM 2025
Fallry Ganegries
50
Inventing Wrong Facts
Proposal Without Veriication
Prosf By Example
Solution
Enror
Regging The Quastion
Cakculation Misakias
Reletive Frequency
20
10
DesgiBekk
Gamiel 2.0
Gamini 2.6
of-iin
obamio
Figure 5: Relative frequencies of each fallacy for Gemini 2.5
Figures 6 and 7 illustrate the relative frequencies of fallacies categorized by problems with or without final answer, and by topic, respectively. The previously mentioned patterns continue to hold for Gemini 2.5.
60
Invoringg Wrong Facts
Proposal Without Venlication
Example
Relative Frequency (%)
60
Bopging the Quest on
Calculation Matakes
40
30
10
Final
Figure 6: Relative frequencies of each fallacy in Gemini 2.5 solutions, comparing questions with and without final answer.
B Automatic Evaluation Results
Table 7 shows each model's ability to classify genuine correct solutions versus LLM- generated incorrect solutions. Ideally, models should classify nearly 100% of correct so- lutions as correct and close to 0% of incorrect solutions as correct. Gemini 2.5 achieves 39% accuracy correct solutions and 4% incorrect solutions, resulting in the greatest distinction among all models. Other models perform near random chance.
Table 8 summarizes model accuracy in distinguishing correct solutions from incorrect solutions when presented together. Gemini 2.5 significantly outperforms other models with 83% accuracy, while others show near-random performance.
Notably, when Gemini 2.5 evaluated solutions generated by itself, its accuracy dropped to 60% from 83%.
17

---

Published as a conference paper at COLM 2025
Fallacy Calegory
70
Inventing Wrong Facts
Proposel Without Verification
Proof Example
60
Solution By Trial And Emro
Relative Frequency (%)
Begging The Question
Calculation Mistakes
Geometry
Algebra
Combinatorics
Number Theory
Figure Relative frequencies of each fallacy in Gemini 2.5 solutions among different topics
Model
Real Solutions Correct (%)
Wrong Solutions Correct (%)
DeepSeek
48
43
Gemini 2.0
50
Gemini 2.5
39
4
ol
31
39
ol-mini
36
45
o3-mini
26
31
Table 7: Percentage of correct and incorrect solutions identified as correct by different LLMs during verification. The results illustrate the LLMs' difficulty in accurately distinguishing genuinely correct solutions from clearly incorrect ones containing explicit fallacies.
C Conclusion
We consistently observed better solutions from Gemini 2.5 compared to other reasoning models. Gemini 2.5 shows clearer understanding of what a mathematical proof is, while other models mainly focus on finding equations or final answers without properly proving them. Because of this, other models often produce incoherent or unclear statements. This is why fallacies like "Proposal Without Verification" and "Proof by Example" happen less often with Gemini 2.5. Even when Gemini 2.5 uses "Solution by Trial and Error," " it generally tries to rule out other possible solutions, although sometimes it doesn't 't fully check all possibilities.
Model
Accuracy (%)
DeepSeek
57
Gemini 2.0
49
Gemini 2.5
83
o1
50
ol-mini
46
03-mini
52
Table 8: Accuracy of various LLMs in identifying the correct solution when presented with pairs consisting of correct solution and one incorrect solution generated by an LLM.
18

---

Published as a conference paper at COLM 2025
Geometry problems remain challenging for Gemini 2.5. The model either attempts to use complex coordinates, resulting in lengthy and unclear derivations, or uses geometric reasoning that still contains basic errors, similar to earlier models. Particularly, Gemini 2.5's geometry solutions often include trivially wrong claims or tautological facts. Despite these weaknesses, Gemini 2.5 significantly improves upon other reasoning models currently available.
D Concrete Examples for Common LLM Fallacies
D.1 Illustrative Examples for Common LLM Fallacies
In this section, we present illustrative examples and comprehensive explanations for each fallacy, facilitating clearer interpretation and deeper conceptual understanding.
Example: Proof by Example
Problem: Prove that 3 is divisible by 8 for all integers a 2 1.
Fallacious Solution: To "prove" this statement, we can test the initial cases:
For
32 - 8, which divisible by 8.
For 2,
3 1 81 = 80, which is divisible by 8.
For π = 3,
36 1 729 1 728, which divisible by 8.
Hence, we "proved" that 32r divisible by for all integers n 2 1.
This argument is not valid proof by induction, as it lacks the necessary inductive step. Without this inductive step, verifying few initial cases does not guarantee the statement is true for all n. While the statement itself happens to be true in this example, the method of proof is fallacious.
Example: Proposal Without Verification:
Two players, Alice and Bob, take turns choosing a number from the (1,2,..., ,I, such that the sum of the chosen numbers never exceeds 2n. The player unable make a valid move loses. Prove that for all n 1, there exists winning strategy for one of the players.
Problem:
Fallacious Solution: Alice always has a winning strategy. She should begin by picking the largest available number, and on each subsequent turn, she should pick numbers that ensure Bob cannot reach the limit of 2n. By doing this, she can control the game and guarantee her victory. This approach is clearly optimal because by keeping Bob under pressure, Alice can always force him into a losing position."
While this strategy seems intuitive, the proposer does not provide any proof that it guaran- tees win for Alice in all cases. For example, there is no justification that Alice's strategy prevents Bob from achieving the same goal or that Bob has no counter-strategy. Furthermore, rigorous argument is provided to show that this method works for all values of n. A proper solution would involve analyzing the game mathematically, determining conditions under which Alice Bob has winning strategy, and proving these results using tools like
19

---

Published as a conference paper at COLM 2025
invariants or backward induction. Without this reasoning, the proposal remains incomplete and exemplifies the fallacy.
Example: Inventing Wrong Facts
Problem: Prove that the harmonic series
00
1
#= 11
diverges.
Fallacious Solution: By the Harmonic Square Root Theorem, the partial sum of the harmonic series up terms given by:
H 2 2 -
nn
where is the Euler-Mascheroni constant. As 11 + oo, the dominant term 2/n grows without bound, while the constant and error term O become negligible. Thus, limy-pou Hμ co, proving divergence of the harmonic series.
The Harmonic Square Root Theorem non-existent. While H, In(n) + +y valid asymp- totic approximation, claiming Hn /n is blatantly false.
Example: Begging the Question
Problem: Prove that the number / is irrational.
Fallacious Solution: Assume /2 is irrational. By definition, an irrational number cannot be expressed as fraction y where and b are integers with b + 0. Since v2 cannot be written as f. it follows that / is irrational.
The argument begins by assuming the conclusion "Assume /2 is irrational". then uses the definition of irrationality to "prove" the same claim, creating logical loop. The conclusion ("v2 irrational") is restated verbatim as the premise, offering no actual proof.
Example: Solution by Trial and Error
Problem: Find all functions f R + IR such that
f(x² y2) - xf(x))y(()), Vx,eE E
-
Fallacious Solution:
Let us try some simple forms of fx): Suppose f(x) = 0. Substituting into the functional equation:
f(x² - - = and xf(x)- (y( 0. This satisfies the equation, so f(xx = is solution.
Now suppose fxx x. Substituting f(x) = into the equation:
f(x2 ) = 2- and -f x² 2
20

---

Published as a conference paper at COLM 2025
This also satisfies the equation, so fxx is another solution. Finally, consider f(x) = kx for some constant k. Substituting:
f(x² y2) - and xf(x) yf(y) -(xx² y²).
This works for any k. So the solutions are f(x) and f(x) kx for any constant k.
This solution seems rely on educated guesses to reach the correct answer but doesn't explain why other functions fail to satisfy the given functional equation.
Example: Calculation Mistakes
Problem: Let x, z > satisfy + y + 6. Find the maximum value xyz.
Fallacious Solution: Using the AM-GM inequality, we know:
+ y + 2 x/yz.
Substituting x + + = 6, we get:
6 2 xxyz.
Simplifying:
2 2 X/yzz.
Now, squaring both sides (instead of cubing):
4 Σ xyz.
Thus, we conclude that xyz <
The critical error lies in Step 4, where the inequality was squared instead of being cubed. Since the inequality involves cube root /xyz, the proper operation should have been cubing both sides: (2)3 8. This mistake leads to the incorrect conclusion xyz < whereas the correct bound is xyz S 27. The solution becomes invalid because squaring distorts the inequality, making the result overly restrictive.
D.2 Real Examples for Common LLM Fallacies
Example: Proof by Example
On flat plane Camelot, King Arthur builds a labyrinth & consisting of n walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where red side and a blue side meet, corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.
Problem:
Let k(2) be the largest number such that, no matter how Merlin paints the labyrinth ②, Morgana can always place at least knights such that no two of them can
21

---

Published as a conference paper at COLM 2025
meet. For each nI what are all possible values for k(2), where is a labyrinth with walls?
Fallacious Solution:
For each integer и 2 1, the labyrinth constructed by King Arthur consists of non-parallel, infinitely extending walls with no three walls intersecting at single point. Merlin paints each wall with one side red and the other blue. The intersections of these walls create doors that connect specific regions within the labyrinth. Understanding k()):
The value k{L) represents the largest number k such that, regardless of how Merlin paints the labyrinth, Morgana can always place at least knights in the labyrinth where two knights meet through the doors. Analyzing Small Cases:
1. For π 1:
There is only one wall, dividing the plane into two regions.
Since there are no intersections, there are no doors.
Morgana can place one knight in each region, so k(L) = 2.
2. For ı
2:
Two intersecting walls create four regions.
There is one door connecting two diagonally opposite regions.
Morgana can place three knights such that no two are connected by the door, k{_) = 3.
3. For 3:
Three walls intersecting pairwise create seven regions.
There are three doors, each connecting two regions.
Morgana can ensure that four knights are placed without any two being connected by door, so k( (()) = 4.
Generalizing the Pattern:
Observing the pattern from the small cases, we see that:
For each n, k(2) n +
This means that matter how Merlin paints the labyrinth, Morgana can always place и + 1 knights such that no two knights can meet through the doors. This result stems from the combinatorial structure of the labyrinth and the way doors are created at intersections.
Conclusion:
For any number of walls n, the largest guaranteed number of knights k(L) that Morgana can place without any two being able to meet always и +
Answer:
For every number n of walls, the maximal is n + 1. Thus, each k k()) equals +
The proof claims that k(인) n + 1 by testing it for small ns and then it generalizes this claim for all natural as without justification.
Example: Proposal Without Verification
social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with user B, user B is also friends with user A. Events of the following kind may happen repeatedly, one at time: Three users A, B, and such that is friends with both B and but B and C are not friends, change their
Problem:
22

---

Published as a conference paper at COLM 2025
friendship statuses such that B and C are friends, but A is longer friends with B, and no longer friends with C. All other friendship statuses are unchanged. Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists sequence of such events after which each user is friends with most one other user.
Fallacious Solution:
To solve this problem, we need to demonstrate that it's possible to transform the given social network into state where each user is friends with at most one other user using the specified operations.
Initial Setup
Number of Users: 2019
Friendship Structure:
1010 users have 1009 friends each.
1009 users have 1010 friends each.
Total Friendships Initially:
1010 1009 + 1009 x 1010 = 2 1010 x 1009
This counts each friendship twice (once for each user in the pair), so the actual number of unique friendships is:
Total Edges = 1010 1009
Objective
Transform the network so that each user has at most one friend. In graph theory terms, we aim to reach state where the graph is matching, meaning it consists of disjoint edges with no overlaps.
Operation Details
The operation allows us to:
Select Three Users: A, B, such that:
is friends with both B and C.
B and C are not friends.
Modify Friendships:
Add: Friendship between B and C.
Remove: Friendships between A and B, and between A and C.
Effect of Each Operation
Total Friendships Decrease: Each operation removes two friendships and adds one, resulting in net decrease of one friendship.
Degree Changes:
A's degree decreases by 2.
and C's degrees remain unchanged since one friendship is removed and another is added.
Strategy to Achieve the Objective
Reducing Degrees:
Repeatedly apply the operation to reduce the degrees of users with more than one friend.
Each operation ensures that a user's degree decreases, moving towards the desired state.
23

---

Published as a conference paper at COLM 2025
Termination Condition:
The process continues until
has more than one friend. If a user still has more than one friend, the operation can still applied because there will always be at least two friends who aren't connected (given the initial high degree and the specific structure of operations).
Final State:
The network will consist of disjoint pairs of friends (a matching) and possibly some users with no friends. This ensures that no user has more than one friend.
Conclusion
Through systematic application of the specified operation, we decrementally reduce the degrees of all users with more than one friend. Given the initial conditions and the nature of the operation, it's guaranteed that we can reach state where each user has at most one friend.
Final Answer: After performing suitable operations, the network can be transformed so no user has more than one friend. Thus, such a sequence exists.
As see, the model has proposed sequence but doesn't do any mathematical verification of its correctness.
Example: Inventing Wrong Facts
Problem: You are given set of blocks, each weighing least 1; their total weight is 2nt Prove that for every real number with 0 S r 2n 2 you can choose subset of the blocks whose total weight is least but at most +
Fallacious Solution:
To prove that for any real number y with 0 < r < 2n - 2, there exists subset of blocks whose total weight is least but at most r 2, we can proceed as follows:
Key Observations:
1. Block Weights and Total Weight:
Each block has a weight of at least 1.
The total weight of all n blocks is 2n.
2. Subset Sums:
The possible subset sums range from 0 (empty set) to 2n (all blocks). Since each block has weight of at least 1, any two distinct subset sums differ by at least 1.
3. Covering the Interval [0, 2n 2):
We need to ensure that for any r in [0,2n 2], there exists a subset sum s such that r < s 5
Proof Strategy: Sequential Coverage
List all possible subset sums in increasing order: so =2, = 2n.
Each subsequent sum satisfies 5;41 s; 1 because each block has weight of at least
Interval Overlap
Consider intervals of the form [s + 2] for each subset sum S;.
24

---

Published as a conference paper at COLM 2025
Since $j11
sj + 1, the next interval [s; 1//5/ 2] starts no more than unit after the start of the previous interval.
Therefore, the union of these intervals [$t + 2] covers the entire range [0, 2m] without gaps larger than 2.
Ensuring Coverage for Any r
For any in [0,2n 2), there exists some subset sum 8 such that 8; r $; + 2.
This guarantees that lies within the interval s; + 2), meaning there exists subset with a sum in [r,
Conclusion By sequentially covering the range 2n] with overlapping intervals of length 2 derived from the subset sums, we ensure that for any r within the specified range, there exists suitable subset whose total weight satisfies r I sum(S) S r 2. Answer By ordering all subset sums and noting that each differs by at least 1, the intervals [s,s + 2] cover [0, 2n 2). Thus, for any between and 2n 2, some subset has total weight at least and at most r + 2.
The model invents wrong fact/ obtervatein and builds its argument around it. The main fact that is used for the proof obviously not true. Consider 21,15,,2,355.
Example: Begging the Question
Problem: A polynomial 2,), in three variables with real coefficients satisfies the identities
p,x, = py,,y,xy-i 2) = P(x, [xx - y,z) = P(yz y, Prove that there exists polynomial F(t) in one variable such that
P(x,y,z) F(x² + y + +2² =2xxxz).
Fallacious Solution:
To prove that there exists a polynomial F(t) in one variable such that
P(x,y.z) = F(x2 + y² + xyz),
we analyze the given conditions and the structure of the polynomial P(x, Step 1: Define the Invariant Let's define the invariant:
t - x + y + xyz.
Our goal to show that remains unchanged under the transformations provided in the problem statement.
Step 2: Verify Invariance Under Given Transformations We need verify that invariant under the following transformations:
1. First Transformation
P(x,y,z) P(x, P(x,y,xy 2).
Compute fx, y, z):
== x2 y2 + (xy 2)2 xy(xy z).
Expanding this:
f= +2 xy2 2xyz + 2 x2y² + xyz = x2 + y2 + + xyz.
Thus, remains unchanged.
25

---

Published as a conference paper at COLM 2025
2. Second Transformation
P(x,y,z) P(x,2x y,z).
Compute fx, z)
1 1= x² + (zx y)2 22 x(2x y)z.
Expanding this:
= +
2xyz + y² - + xyz x2 y² +22 2 xyz.
Again, i remains unchanged. 3. Third Transformation
P(x,y,2) P(yz - x,),
Compute f(yz x,,,
f = (yz x)² +y² +z - (yz x)yz.
Expanding this:
1= x22 2xyz + x² + + +2 -22 + xyz = x² y² + xyz.
Once more, f remains unchanged.
Step 3: Conclude the Existence of F(I) Since f = x y² + xyz is in- variant under all the given transformations, and P(x, y,z) remains unchanged under these transformations, it follows that P depends only on t. Therefore, there exists a polynomial F(t) such that:
P(x,y,2) F(f) F(x² + y² + 22 xyz).
Conclusion Any polynomial P(x, v, 2) satisfying the given identities must be a poly-
nomial in the single variable 1 == x2 + y² 2² xyz.
Answer Because x2 + y² + 2 - xyz is unchanged by the given equalities, P depends only on it. Thus, there an F with:
pxx,y,z) = F(x2 + y² + +2 xxyz).
The generated solution begs the question. The fact P(x,y,2) = F(x² + y2 + + - xyz). is the problem statement itself, and it needs to be proven.
Example: Solution by Trial and Error
Problem: Let Z be the set of integers. Determine all functions f: Z - Z such that, for all integers and b,
f(2a) 2b(b) f(f(a+b)).
Proposed by Liam Baker, South Africa
Fallacious Solution:
To determine all functions f Z - Z that satisfy the functional equation f(2a) + 2f(b) = f(f(a (b) for all integers a and b,
let's explore possible forms of f.
Step 1: Assume Linear Function
First, consider that might be linear function of the form:
f(n) kn C,
26

---

Published as a conference paper at COLM 2025
where k and constants in Z. Substituting into the functional equation:
f(2a) + 2f(bb k(2a) + 2(kb + c) = 2ka + c 2kb 2c 22((a a b) 3c.
On the other hand:
f(f(a+b)) f(k(a + b) c) = k(k(a + b) + + -kcc bb kc
Setting the two expressions equal gives:
2k(a + + 3c k2 (a + b) + kc +
For this to hold for all integers and b, the coefficients must match:
k2 and
3c = kc+ c.
Solving the first equation:
2 - 2k =
= k(k - 2) =0 = k == 0 or k = 2.
Case 1: k = 0
30 = 2c c=
fo, fn) = 0 is solution.
Case 2: k = 2
3c = 3c (always true).
Here, can any integer. Thus, fn) = 2n + is a solution for any integer c. Step 2: Verify Non-Linear Solutions Assuming is not linear leads to more complex conditions that are difficult to satisfy uniformly across all integers. Hence, no non- linear solutions appear to exist beyond the linear forms already identified. Conclusion All solutions to the functional equation are either the constant zero function affine linear functions of the form f(n) 2n c, where is integer constant.
Final Answer: All integer functions equal twice n plus fixed constant. In other words, every f with f() = 2n + for some integer c.
The generated proof only tests the problem's 's equation for special class and concludes the result. This is an example of a solution by trial and error fallacy.
Example: Calculation Mistakes
Problem: Determine all integers n 2 2 having the following property: for any
integers 41, .2, An whose sum is not divisible by n, there exists an index 1 < i < such that none of the numbers
4; + Ajt1 + A;+1 +... + ajtn-1
divisible by n. Here, we let a; = a;nx when > n.
Fallacious Solution:
We will show that among the integers 11 2 2, the property "For any integers a1, #2, ..., an whose
not divisible by n, there is some index 1 < S so that none of the numbers
ai, Ai + aitl,
A; + Ai+1 + .... ai-n-1
is divisible by n² holds if and only if π1 In other words, if n 2 3 then one may construct a counterexample sequence, while for 2 the property always holds.
27

---

Published as a conference paper at COLM 2025
In what follows, describe method to translate the condition into involving a cyclic sequence of residues and then show that counterexamples exist for every 2 3, but not for 2. Reformulating the Problem For given sequence 01, ....,, an, define the cumulative sums
S a1 42
ak for k M, So =0=0. 0.
All congruences below are modulo и. Note that for any index (with indices taken cyclically modulo n, i.e, Ajtn aj), the partial sums
Ai aj + 1:111. .... a; aj
aj
can be written as
S - S; 1, Sit1 S 1, Si S 1-
Thus, none of these numbers is divisible by if and only if
S; * S; 1 for j = 1,2,...,
In other words, in the list So, S Sn (indices modulo + 1), we seek an index f (with f 1) such that the residue S, is isolated that is, no other Sy is congruent to S, modulo n.
Since Z/nZ has n elements, but we are choosing n + values S,, the pigeon- hole principle guarantees that at least residue is repeated. However, for the property to hold, we only need that at least one St be unique among the Sy- Thus, the problem reduces to the following:
For which integers 2 is true that for every choice of integers ay,..., On with Sn # 0 there exists f E {0, 1, such that
S # Sk for all k + t?
If such a always exists, then setting = f ensures that all partial sums starting are nonzero modulo 11.
Counterexamples for 2 3
We now describe how to construct counterexamples when 2 3.
It is possible to construct sequence So, S Sn in Z/ with So 0 and
S F 0, such that every residue appears at least twice.
In such a configuration, no S; is unique, so the desired property fails.
One simple way to construct such a sequence is to alternate between two values (e.g., 0 and x), for example:
So = 0,
S = 0, S2 = x,
S3 = 0, S4 =
S =x
where x # 0 mod n. Then define
ag = Sy - S for 1,...,
This ensures that a1 +... an S # and because every residue among So...., Sn appears at least twice, the property fails.
Thus, for every n 3, one can construct sequence where no index satisfies the given condition.
The Case Now consider n 2. In Z/2Z, there are only two residues: 0 and 1.
The sequence So, S2 has 3 elements in 2-element set.
By the pigeonhole principle, at least one residue appears twice.
Therefore, the third must appear only once. So there exists unique S,-
28

---

Published as a conference paper at COLM 2025
Hence, the property must hold for 2.
concrete check:
If a1 is odd and (2 is even, then 41 = 1, + 02 1.
If aj is even and d2 is odd, then taking i = 2, 82 = 1, and d2 +
So in both cases, the two partial sums are nonzero modulo 2, confirming the property. Conclusion
For all 2 one may construct sequence A1,..., an such that for every i, at least one of the cyclic partial sums is divisible by n.
For = 2, such a counterexample is impossible, and the property always holds.
Therefore, the only integer 2 2 for which the property holds is:
2
Final Answer: The property holds if and only if n = 2.
If i is odd, then S, will be 0, which means the sum of + d2 +... is divisible by n, and this contradicts the problem statement. The model has made calculation mistake in this case and failed to realize that S, f for odd as
29

---

Published as a conference paper at COLM 2025
D.3 Evaluation Prompts
Prompt: Solution Evaluation Prompt
You are an expert mathematician tasked with rigorously evaluating the correctness of a proposed solution to a mathematical problem, specifically in the style of the International Mathematical Olympiad (IMO).
Below, you will receive:
1. The Problem Statement: A description of the problem to be solved.
2. The Candidate Solution: A step-by-step explanation of proposed solution to the problem.
Your evaluation should include:
Logical Analysis: Carefully check each step for logical consistency, correct- ness of mathematical reasoning, and appropriateness of methods used.
Identification of Errors:
Clearly pinpoint and describe any mathematical
errors, flawed logic, unjustified assumptions, or missing steps.
Clarification: Briefly explain why each identified issue is problematic, refer- ring explicitly to mathematical principles or concepts involved.
Provide your evaluation strictly adhering to the following structured response format:
Analysis:
(Provide thorough, step-by-step analysis of the candidate solution. Clearly indicate and where logical fallacies, incorrect assumptions, or mathematical inaccuracies occur. If the solution is fully correct, justify your reasoning thoroughly.)
Final Answer:
Clearly indicate the result of your analysis using exactly one of the following formats:
Correct
Wrong
Prompt: Solution Comparison Prompt
I will present problem along with two candidate solutions. Your task is evaluate the solutions and determine which one is correct. Note that one solution is always correct and one solution always wrong.
The primary criterion for your judgment is the correctness of the arguments provided in each solution.
Please respond using the following template:
Analysis:
(Provide your analysis of the solutions here, discussing the strengths and weaknesses of their arguments.)
Final Answer:
{
Solution 1
or Solution 2
30

---

Published as a conference paper at COLM 2025
D.4
Additional Notes
To mitigate recency bias, the solutions provided by the LLM and the real solutions were randomly permuted during the comparative analysis experiment. Consistent outcomes were observed across various prompts, indicating robustness in the experimental setup. Furthermore, our investigation revealed that automated rating rubrics, such as the LLM- as-a-judge approach, yielded similar comparative results. Notably, frontier LLMs did not assign significantly higher ratings to correct solutions compared incorrect ones. Due to constraints in space, the present discussion is intentionally restricted to the binary evaluation scenario.
31


Reviews:
[{"review_id": "2KQDeFITsb", "reviewer": "Reviewer_1MfA", "summary": "This paper presents the result of evaluating LLM-generated answers to Math Olympiad short-list problems.\nTypical fallacies in the answers are classified to six categories and experts in math or CS annotated the LLM-generated answers with those categories.\nAs a result, it was revealed that state-of-the-art LLMs generate correct answers to less than 4% of the problems.\nThe types of fallacies are fairly similarly distributed across the evaluated LMMs.\nIn addition, by testing whether the LLMs can discriminate between fully correct answers and those with fallacies, it was revealed that those LLMs' judgements are almost random, which suggests that it is inadequate to employ LLM-as-judge for the Olympiad-level math solving.", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "- The paper exposes a significant fact about the ability of current LLMs on high-level math problem solving\n- The writing is clear", "reasons_to_reject": "- I found no significant reason to reject this paper", "questions_to_authors": "Do you have any plan to make the annotated dataset public?", "extra_scores": {}}, {"review_id": "8MOkOFnECX", "reviewer": "Reviewer_qtx8", "summary": "This paper investigates the proficiency of Large Language Models (LLMs) in solving Olympiad-level mathematical problems, emphasizing the logical rigour of proofs rather than just the correctness of final answers. The authors conducted qualitative and quantitative human evaluations of proofs and solutions generated by frontier LLMs, including OpenAI's o1, o1-mini, o3-mini, DeepSeek R1, and Gemini 2.0. The authors mention a schema for assessing the reasoning capabilities of LLMs and categorized common errors. \n\nThe study evaluates models on 455 IMO shortlist problems from 2009 to 2023, covering algebra, combinatorics, geometry, and number theory. The findings reveal that current LLMs fall short of solving challenging olympiad problems with a negligible percentage of solutions being fully correct or providing meaningful insights. The paper highlights that occasional correct final answer from LLMs often stem from pattern recognition or heuristic shortcuts rather than genuine math reasoning. Further, the LLMs struggle to distinguish between correct and flawed solutions, performing at or near random levels in verification. The paper mentions that there is a substantial gap between LLM performance and human expertise in advanced mathematical reasoning.", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "The paper presents a robust empirical evaluation of frontier models on 455 IMO shortlist problems, prioritizing proof correctness over final answer accuracy. Human evaluators rigorously assessed solutions, lending strong credibility. \n\nThe identification and categorization of common logical fallacies in LLM generated mathematical generated solutions provides a good framework for error analysis.\n\nThe research addresses a critical gap by demonstrating that existing benchmarks, which rely solely on final answer correctness, overlook solution validity and systematic exploitation of answer predictability by LLMs. Prioritizing logical rigour, the paper advocates for more sophisticated evaluation methods essential for advancing LLMs.", "reasons_to_reject": "The study relies on highly qualified, and therefore expensive, human evaluators. This raises some concerns. This dependency on specialized human assessment poses a significant challenge for future automation and scalability, as replicating such evaluations for newer model iterations or broader application would be difficult and costly.\n\nThe paper introduces categories for common errors observed in the solutions generated by LLMs. Although these categories offer valuable insights, the discussion lacks an exploration of their comprehensiveness, potential overlaps, or their applicability to reasoning tasks in fields beyond olympiad mathematics\n\nThe definition of a \"partially correct\" solution is described as one that \"included some essential steps of a correct solution but omitted other crucial steps or contained significant inaccuracies\", and is quite broad. This lack of specificity could make it challenging for readers to precisely grasp what qualifies as \"partially correct\" and to understand the consistency with which this classification was applied during the annotation process.\n\nThe authors acknowledge a high probability of data leakage, given that the problems from the IMO and IMO Shortlist are well-known and likely included in the training datasets of leading models. While it is claimed that this leakage does not significantly impact the LLMs' problem-solving capabilities, the observed low success rates could be interpreted as evidence that models are relying on recall rather than genuine reasoning. A more in-depth discussion on how this data leakage might still affect heuristic approaches, potentially leading to correct answers without sound reasoning, would be advantageous.\n\nThe experiments detailed in Section 5, which focus on the verification abilities of LLMs, seem relatively brief and could be more thoroughly developed. Considering that prior research has shown that similar advanced models can achieve better-than-chance verification performance on other benchmarks, this section appears less developed in comparison to the comprehensive analysis provided elsewhere in the paper and does not fully align with the depth of the other findings.\n\nThe paper effectively analyzes current LLM limitations in high-level mathematical reasoning but fails to adequately bridge these findings to tangible next steps for research. It lacks substantive discussion on how the insights could inform concrete questions or guidance.\n\nFurther concerns are mentioned as Questions to the Authors.", "questions_to_authors": "Could you elaborate on the inter-annotator agreement for the classification of fallacies and solution correctness? How was consistency ensured among the seven evaluators, especially for borderline cases?\n\nThe paper mentions that \"frontier models may have been trained on publicly available high-quality mathematical datasets\". Given the observation that models often rely on heuristics and pattern recognition, did the evaluators consider cases where models might have memorized solutions to specific IMO problems or similar problem structures? How was this potential bias mitigated in the evaluation process?\n\nFor the \"Proposal Without Verification\" fallacy , the paper notes that models might struggle to determine what calculations and statements to include in the final response, sometimes producing vague claims like \"It is easy to show that...\". Are there any insights into why LLMs omit these essential steps or introduce incorrect statements instead of providing rigorous arguments?\n\nGiven that \"Inventing Wrong Facts\" was the most frequent fallacy in four models and second-most frequent in another, and \"Proposal Without Verification\" was also common, what are the implications of these specific failure modes for the development of more robust and trustworthy LLMs in mathematical reasoning? Are there any hypothesized architectural or training limitations that might contribute to these particular fallacies?\n\nSimilarly, the paper suggests that relying solely on final answer correctness or using powerful LLMs as judges is insufficient for evaluating logical rigour. What specific characteristics or capabilities would a more \"sophisticated benchmarking method\" or \"improved training schema\" need to possess to effectively evaluate the logical rigour and coherence of mathematical arguments?\n\nThe paper presents a breakdown of fallacy frequencies by problem type (geometry, algebra, combinatorics, number theory). Could the authors provide a deeper analysis of the unique challenges each problem type poses for LLMs and how these challenges correlate with the prevalence of specific fallacies? For example, why do geometry problems lead to more \"Inventing Wrong Facts\" and \"Begging the Question\" fallacies?\n\nCould you provide an example of a \"shortcut\" used by a model that has supposedly not seen the question before, and where the problem cannot be solved by relating it to other similar questions? Specifically, what is considered to be a shortcut in such olympiad problem solvings?\n\n\nGiven that the current training paradigms often lack direct supervision over the sequence of actions leading to the final answer (with rewards primarily based on final answer correctness), it's plausible that the observed fallacies might be a consequence of this training setup rather than a complete lack of understanding. To what extent do you believe these findings indicate that models genuinely lack the ability to solve the questions, as opposed to merely struggling with generating human-interpretable, rigorous proofs?\n\n\nThe paper states that \"Inventing Wrong Facts is the most frequent in four models and the second-most frequent in another... This observation might be explained by the training methods used for these models, which generally involve reinforcement learning algorithms with rewards based on the correctness of the final answers\". Could you elaborate on the specific mechanisms or experimental evidence that support this connection between final answer-based reward training and the prevalence of \"Inventing Wrong Facts\"?\n\nCould you clarify whether all the internal \"thought tokens\" generated by the LLMs were examined by the human evaluators, or only the summarized output after the models' internal thought blocks?\n\nHow were the \"wrong facts\" invented by the models checked and confirmed by the human evaluators to ensure that the models were indeed fabricating information rather than making subtle errors or misinterpreting established concepts?\n\nThe related work section could benefit from discussing some recent work on math reasoning robustness, such as \"GSM1K\" (Zhang et al., 2024), \"Functional MATH\" (Srivastava et al., 2024), and \"Compositional GSM\" (Hosseini et al., 2024). Could the authors comment on how these studies relate to or differ from the evaluation framework presented in this paper?\n\nModern LLMs  undergo various training phases such as pre-training (PT), supervised fine-tuning (SFT), and reinforcement learning (RL). How do the authors believe these different training stages influence the models' mathematical reasoning capabilities and the types of errors observed?\n\nThe paper highlights that the IMO shortlist problems are \"highly original\" and designed so that an LLM \"cannot simply combine standard building blocks from well-known problems to arrive at the correct answer\". However, a desirable feature for LLMs is their ability to combine standard methods and basic principles from easier problems to solve harder, more complex ones. Why did the authors choose not to include such examples in the shortlist to evaluate this specific capability?\n\nThe fallacy \"Proposal Without Verification\" describes instances where a method or strategy is introduced without proper justification. Could the authors provide a more detailed breakdown of how often such proposals, if pursued, ultimately lead to correct versus incorrect final solutions in the dataset?\n\nGiven that models like Gemini 2.0 and o1 exhibit a high performance in arriving at the correct final answer despite their difficulties in producing a fully correct chain of thought, what is the intuition about how these models are able to achieve accurate results without a logically sound reasoning process?\n\nConsidering these findings, what are the actionable next steps for the LLM research? How do these insights translate into concrete research questions or developmental priorities for improving mathematical reasoning capabilities?", "extra_scores": {}}, {"review_id": "psLOVcDehN", "reviewer": "Reviewer_hUEG", "summary": "The paper questions the performance of language models on difficult mathematical reasoning tests. They find that the explanations and reasoning chains produced by the language models often contain mistakes and fallacies, calling into question their true ability for mathematical reasoning", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "I find this type of work to be valuable to the community because it points out a possible overestimation of language model capabilities. I believe that the categorization is helpful for understanding how language models make mistakes.", "reasons_to_reject": "One issue with the methodology is that just because the surface form of the chain of thought reasoning may contain incomplete proofs that does not mean that the model isn’t capable of generating complete logical proofs. Some of the reasoning may be taking place internally to the model and may not be surfaced during generation. Indeed the high performance of these models may indicate that such internal reasoning is happening.\n\nThat being said, the finding that models cannot distinguish correct from incorrect proofs does reinforce the hypothesis that the models are doing incorrect reasoning. The only alternative hypothesis I can think of is that the models have some preferences for their own generations, causing them to favor the incorrect proofs.", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mfDUBoieT0", "reviewer": "Reviewer_ZMU4", "summary": "The paper critically examines the ability of large language models (LLMs) to solve challenging mathematical problems from the International Mathematics Olympiad (IMO). The authors conduct qualitative and quantitative evaluations of LLM-generated proofs, revealing that current LLMs struggle with Olympiad-level problems and often fail to distinguish between correct and flawed mathematical reasoning. The study emphasizes that LLMs often rely on pattern recognition and heuristic shortcuts rather than genuine mathematical reasoning, highlighting a significant gap between LLM performance and human expertise in advanced mathematical problem-solving.\n\nThe study design involves both qualitative and quantitative evaluations, which strengthens the validity of the findings. The methodology seems rigorous, with a focus on identifying and categorizing fallacies in LLM-generated solutions. The use of human evaluators with expertise in mathematics further enhances the reliability of the evaluation.\n\nThe introduction effectively sets the context by discussing recent advancements in LLMs and their performance on mathematical reasoning tasks. The authors clearly outline their methodology, including the problem selection rationale and the process of evaluating LLM-generated solutions.\n\nThe originality of this work lies in its focus on evaluating the logical rigor of LLM-generated proofs rather than solely relying on the correctness of final answers. The study addresses a significant gap in current evaluation benchmarks, which often overlook the importance of sound mathematical reasoning. The identification and categorization of common fallacies in LLM solutions also contribute to the originality of the research.\n\nThe significance of this study is highlighted by its potential impact on the development of more effective benchmarks for evaluating mathematical reasoning in LLMs. The findings underscore the limitations of current LLMs in solving challenging Olympiad-level problems and emphasize the need for training strategies that prioritize the validity and coherence of mathematical arguments. The study also provides insights into the feasibility of bootstrapping verification processes to improve the quality of LLM-generated proofs.", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "1. The paper evaluates LLMs on their ability to produce logically sound mathematical proofs, rather than just focusing on the correctness of the final answer. This is significant because it addresses a critical gap in current benchmarks, which often fail to assess the quality of reasoning. The study systematically identifies and categorizes common logical fallacies in LLM-generated solutions. This provides valuable insights into the limitations of current LLMs and offers a framework for improving their mathematical reasoning abilities.\n\n2. The research work involves human expert evaluation that exposes further insights into LLM behavior\n\n3. The research investigates whether LLMs can verify the correctness of their own solutions, which could lead to the development of bootstrapping verification processes. This has the potential to enable LLMs to improve the quality of their proofs and reduce reliance on human evaluation.", "reasons_to_reject": "LLMs lack of mathematical reasoning is well defined. It is interesting to see the improvement in details provided by experts. However, perhaps the contribution might be strengthened by the evaluation on existing benchmarks as well.", "questions_to_authors": "1. Have you considered using the same rubric of fallacy recognition and analysis for existing olympiad benchmarks? If so, provide more details.\n\n2. How do you plan to share the results? I did not see any of the code or prompt? Did you employ any prompting strategy?", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper critically examines reasoning LLMs ability to solve Olympiad-level mathematical problems and show that they often fail to produce rigorous and logically sound proofs. While LLMs can occasionally arrive at correct final answers, these are frequently the result of pattern recognition or heuristic shortcuts rather than genuine mathematical reasoning. Their qualitative and quantitative evaluations underscore a substantial gap between LLM performance and human expertise in advanced mathematical reasoning, emphasizing the need for benchmarks that prioritize the rigor and coherence of mathematical arguments over just answer correctness. 

Overall, I favor acceptance and request the authors to address the concerns raised by the reviewer (particularly qtx8)

Potential areas for improvement: 
(1) The automatic eval section lacks mention of some important details, should contain verification accuracy (false positive and true positive rates) as a function of inference time compute. 
(2) Evaluating LLMs CoT reasoning on existing Olympiad benchmarks with concrete final answers for comparison. Are the limitations of LLMs reasoning correctness mostly on proof-based problems or also on problems with a concrete final answer.
Decision: Accept