Achieving Olympia-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning
Authors: Haiteng Zhao
Junhao Shen
Yiming Zhang
Songyang Gao
Kuikun Liu
Tianyou Ma
Fan Zheng
Dahua Lin
Wenwei Zhang
Kai Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Model
Reinforcement Learning
Geometry Agent Prover
Primary area: foundation or frontier models, including LLMs
TL;DR: We propose Complexity Curriculum Reinforcement Learning to train LLMs to solve IMO-level geometry with minimal data, surpassing gold medalists and showing emergent creativity.

Abstract: Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, due to weak heuristics for auxiliary constructions, AI for geometry problem solving remains dominated by expert models such as AlphaGeometry 2, which rely heavily on large-scale data synthesis and search for both training and evaluation.
In this work, we make the first attempt to build a medalist-level LLM agent for geometry and present InternGeometry. InternGeometry overcomes the heuristic limitations in geometry by iteratively proposing propositions and auxiliary constructions, verifying them with a symbolic engine, and reflecting on the engine's feedback to guide subsequent proposals. A dynamic memory mechanism enables InternGeometry to conduct more than two hundred interactions with the symbolic engine per problem. To further accelerate learning, we introduce Complexity-Boosting Reinforcement Learning (CBRL), which gradually increases the complexity of synthesized problems across training stages.
Built on InternThinker-32B, InternGeometry solves 44 of 50 IMO geometry problems (2000-2024), exceeding the average gold medalist score (40.9), using only 13K training examples, just 0.004% of the data used by AlphaGeometry 2, demonstrating the potential of LLM agents on expert-level geometry tasks. InternGeometry can also propose novel auxiliary constructions for IMO problems that do not appear in human solutions.

Published as a conference paper at ICLR 2026
ACHIEVING OLYMPIA-LEVEL GEOMETRY LARGE LAN- GUAGE MODEL AGENT VIA COMPLEXITY BOOSTING REINFORCEMENT LEARNING
Haiteng Zhao Junhao Shen
112, Yiming Zhang .1, Songyang Gao', Kuikun Liu'
Tianyou Ma 1.3, Fan Zheng', Dahua Lin Wenwei Zhang Kai Chen !Shangaai AI Laboratory Shanghai Jiao Tong University Peking University *ICMAT, Spanish National Research Council "MMLa,, The Chinese University of Hong Kong (zhachaiteng zhangwenwei, chenkai jlj.. [ipjlab.rrg.cm org.en
ABSTRACT
Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, due to weak heuristics for auxiliary constructions, AI for geometry problem solving remains dominated by expert models such as AlphaGeometry 2, which rely heavily on large-scale data synthesis and search for both training and evaluation. In this work, we make the first attempt to build a medalist-level L.MM agent for geometry and present InternGeometry. InternGeometry overcomes the heuristic limitations in geometry by iteratively proposing propositions and auxiliary constructions, verify- ing them with symbolic engine, and reflecting on the engine's feedback guide subsequent proposals. A dynamic memory mechanism enables InternGeometry to conduct more than two hundred interactions with the symbolic engine per problem. To further accelerate learning, we introduce Complexity -onssixityBBoosting Reinforcement Learning (CBRL), which gradually increases the complexity synthesized prob- lems across training stages. Built on InternThinnkr-32BB, InternGeometry solves 44 of 50 IMO geometry problems (,000-2024), exceeding the average gold medalist score (40.9), using only 13K training examples, just 0.004% of the data used by AlphaGeometry 2, demonstrating the potential of LLM agents on expert-level geometry tasks. InternGeometry can also propose novel auxiliary constructions for IMO problems that do not appear in human solutions.
INTRODUCTION
Large language model (LLM) agents have demonstrated general problem-solving ability across domains such mathematics and programming. By interacting with tools such code interpreters and LEAN (Moura & Ullrich 2021) multiple times, LLM agents can reason and reflect on the tool execution feedback and progressively solve complex problems. Such a universal paradigm can obtain medalist-level performance on International Mathematical Olympiad (IMO) level problems and is
believed have better generalization ability (Huang & Yang 2025
Luong & Lockhart 2025).
However, when faced with geometry problems, the potential of LLM agents is still underexplored. The IMO-level geometry problems usually require extra-long proving steps, whose solution not only combines various geometry theorems but also contains creative auxiliary constructions that have weak heuristics and require multiple trials, as shown in Figure [ Consequently, current sfate-of-the-an
approaches (Trinh & Luong 2024; Chervonyi et al.) 2025 Chen et al. (025a) mainly adopt expert models learned from large-scale synthesized data to guide the large-scale search with a symbolic engine for finding the geometry proof. Given the success of LLM agents in other mathematical domains, this raises a natural question: Can we adopt LLM agents in solving geometry problems for achieving higher efficiency and generalization?
To answer this question, this paper makes the first attempt to investigate LLM agents for solving IMO-level geometry problems and proposes InternGeometry, an LLM agent that obtains medalist-

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Figure 1: An example of IMO-level geometry problem. (a) The configuration in IMO 2018 Problem 6 appears simple, but its solution relies on sophisticated constructions, as illustrated in (b).
level performance for solving geometry problems. We first identify the limitations in the current open-sourced geometry symbolic engines and build Interneeometr--DA... whose expression capacity covering most of the IMO geometry problems. Taking InterGGeomeery-DDDA. as tool, InternGeometry solves the geometry problem through long-term LLM-tool interactions, where the LLMs continuously propose propositions auxiliary constructions after thinking with natural language, verify those ideas the symbolic engine with formal language, and reflect on the feedback from the symbolic engine at each interaction. In the long-term reasoning process, InternGeometry adopts dynamic memory to maintain the exploration history and the observed geometry properties in compact form, which not only reduces the context without losing key information but also guides diverse explorations in future interactions. By extra-long-horrt horizon L.L--toll interactions with memory, InternGeometry can conquer the weak heuristics of geometry proof and progressively find the feasible solution based on the accumulated geometry properties observed during explorations. Such design also aligns with human experts who obtain insights into auxiliary construction by exploratory probing (Trinh & Luong 2024 Chervonyi et al. 2025).
To train Interneemmetry, we first apply cold start training using 7K examples created by formalizing existing geometry problems and constructing trajectory data. After the cold start, we introduce camplexity-boostiine reinforcement learning (CBRL) framework, a multi-stage stage curriculum RL pipeline (Wang et al., 2025b; Chen et al.) [025b] Zhang et al., |025b) Parashar et al. (2025) ) to further improve training efficiency. Specifically, we build data synthesis pipeline that can generate geometry tasks with a specified complexity (e.g., required proof steps). At each stage, we first synthesize prob- lems at the current complexity level, then perform RL training on the current InternGeometry model, and update the target complexity based on the results to best fit the current model. Over iterations, the synthesized problems become highly challenging, providing the foundation for acquiring expert-level capabilities high-difficulty tasks.
We conduct extensive experiments to verify the effectiveness of InternGeometry. InternGeometry solves 44 out of 50 geometry problems from 2000 to 2024, surpassing the average score of IMO gold medalists (40.9 points) and the score of AlphaGeometry2 (42) and SeedGeometry (43), and it also solves the geometry problem in IMO 2025. Notably, the model attains this performance with only approximately 13K training examples, 0.004% of AlphaGeometry2 and 0.006% of SeedGeometry. Our ablation studies further demonstrate that aag-horizon proof interaction critical the agent's ability: removing proposition proving steps and only allowing the agent to add auxiliary constructions significantly degrades performance, substantiating the importance of long-horizon trial-and-error for the weak-to-strong heuristic transition. Complexity escalation plays pivotal role in RL. convergence: directly training on high-difficllyy data leads to low task completion rates and poor convergence, whereas using data below a certain difficulty threshold substantially impairs generalization to IMO- level tasks. addition, our case studies show that the model can devise novel auxiliary constructions compared to human solutions, exhibiting creativity in geometric reasoning.
2 METHOD
2.1
GEOMETRY PROOF LANGUAGE AND ENVIRONMENT
In previous work such as AlphaGeometry, geometric structures are defined point by point through domain-specific language (DSL). Once the construction is complete, deductive database arithmetic reasoning (DDAR) system is employed to exhaustively search for theorems, deriving all conclusions reachable from the known facts. In this work, build IntemGGeemetry-DDDAA an interactive
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Query
InternGeometry
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Action
Feedback
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Dynamic Memory
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Generate complexity- conditioned training batch
Training Batch
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Synthesize complexity- conditional data based rewards
Reward
Synthesis
Compute rewards
Agent Policy
Figure 2: An overview of InternGeometry and complexity-Boosttinn Reinforcement Learning (CBRL). (a) Interneemmeery performs natural- watural-language reasoning (Think), outputs a structured action in domain-specific language (Action), and receives execution results (Feedback) in each turn. dynamic memory module 20 compresses the multi-turn interaction history to preserve essential actions and outcomes. (b) CBRL optimizes the agent policy by generating synthetic training data with controllable difficulty, assigning binary rewards to effective steps and successful outcomes, and optimizing policy through iterative reinforcement learning.
geometric proof engine based on the open-source DDAR system Newclid (Sicca et al. 2024). To support complex geometric structures, introduce several advanced definition strategies, such as globally optimizing point placements to satisfy constraints. During interaction, an agent not only employs the DSL for both problem specification and auxiliary point construction, but also proposes sub-proof goals that will be subsequently verified by the engine. As an interactive proof engine, InternGeemmery-DDDAA maintains state steps, including the geometric configuration, constructed auxiliary points, and all proven preliminaries and propositions. Further details are provided in the Appendix B
2.2 GEOMETRIC PROOF AGENT
The agent is allowed to perform natural-language reasoning each step and then mark its final action output using an aetion-separating token. Let the agent be denoted by G and the interactive proof engine by C. Given a geometric problem X, step é the agent's output is
[P,, A,] (X, (0HH-)))
(1)
where Pt denotes the slow thain-of-thought reasoning, A, the final formalized code, and H,- the interaction history, which includes each round's s thoughts, actions, and the feedback observations obtained from the environment prior to step t. The module 20 is a memory manager that returns compressed long history to improve the agent's long-horizon capacity.
The code A, is executed by the proof engine E, which is in state f. 1. The execution result O is appended together with the corresponding thoughts and actions to the interaction history as feedback to the agent, guiding its next step of reasoning and action.
O,, G, = C.-1((A))
H, = 11-11111 + Ar, A, O.]
At each reasoning step, the agent may summarize progress, analyze the problem, plan future proof strategies. In its action code, the agent can choose specific operations-sul such as constructing geometric objects, adding auxiliary constructions, verifying whether proposition holds.

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When all targets in the problem have been proven, the geometric reasoning tool determines that the problem is fully solved, aggregates the entire proof process, and produces a complete proof of the problem, thereby concluding the reasoning session.
We posit that long- Long-horizon capability is key to addressing the weak-heuristic challenge of auxiliary construction in geometric proofs. To this end, we introduce dynamic memory management strategy 20 for the agent, as well as a prior-guided rejection sampling method for G.
To shorten the long interaction history H, which can span hundreds of turns, 20 summarizes earlier exchanges, including their thoughts and detailed environment feedback, while retaining only core action outputs and key environment feedback improve the agent's context efficiency, illustrated in Figure 2 Through 2,, the agent obtains a concise overview of its action history and their outcomes c.g., whether an auxiliary construction was successfully added and whether a proposed proposition holds. The last turn feedback remains unchanged, informing the agent of the currently known propositions.
Another major challenge for long- Long-horizon agents is action collapse, where the model falls into poor patterns, such as producing highly repeated outputs or outputs similar to previous rounds (Sinha et al. 2025). To address this challenge, we use an intuitive rejection sampling method during agent inference to avoid such patterns. Denote naive LLM inference as [Pr, Ar] G (X,,((())))) Then, for the sampled value [Pr, As].
If PassCheck([ PassCCheck(Pr.AA)) A.): ( [P.
Else: reject the value of [Pr, A.] and return to the sampling step
Here, PassCheck a rule-based multi-condition checking policy that enforces no repeated actions relative to the history, no excessively long thinking without stop, turn without valid action or with formatting issues, and no use the same action type for too many consecutive rounds.
2.3 COMPLEXITY BOOSTING REINFORCEMENT LEARNING
Before reinforcement learning begins, there is cold-start phase in which we perform supervised fine-tuning on a small amount of synthetic data to help the agent quickly adapt to the task paradigm.
i=1' where X denotes the input, h (H) denotes the compressed history (aligned with the agent's input), and y [P, A] denotes the output, including the thinking content P and the action content A. Let the agent model G have parameters The supervised fine-tuning objective is:
Denote the training dataset as D (x', h', v²) )
Lat Σ
Σ
log Go (vt xt, mn)))))
Based on the supervised fine-tuning result Ozt, the agent exhibits the basic behavior patterns expected in geometric reasoning tasks, such as slow thinking and proactively invoking tools.
The subsequent CBRL phase is an iterative interaction interaction-training loop. In each iteration, the agent first attempts a proof on the training task and then performs online learning using reward signals from its trajectories. Following GRPO (Shao et al. (024), given task X, the training gradient is:
VJr((X,) 0) Gy,x.-CC(1)))
Go (3 X,hx).
Go (ve X.h.)
min
clip( Gows (ye X,he) 1- 111,,,+))) A(X, (ve (2)))) (5)
Goa (ye X, h.)
gVDxL (Go (
where
A (X, v.) = 7i mean (
r 77.
(6)
std (11..))... , represents the advantage at step of the i-th trajectory within batch of K samples. It measures the quality improvement at step t of the i-th generated trajectory relative to the average policy. Here, ri denotes the reward at step of the i-th trajectory. hyperparameter that constrains the policy
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ratio. Gaas is the policy model from the previous iteration. Giet is the initial model. Dxz is the KL divergence, used as a regularizer to constrain optimization the agent model.
Here, the reward is binary value, computed the conjunction of the outcome reward and the step effectiveness reward:
The outcome reward r is the proof is complete; otherwise, it is The step effectiveness reward steps, r* if the steps, r* is if
defined by whether the step's action succeeds. For proposition-proposiii proposed proposition is successfully proven by the engine. For the construction successfully added and used in the final proof of the overall question; otherwise, is O.
Note that the reward in our work is deliberately simple and can be computed by rules automatically, It rewards effective steps in trajectories that succeed while penalizing all steps in failed trajectories and ineffective steps in successful trajectories.
Next, we focus on the curriculum algorithm for CBRL in geometric tasks. One major advantage of our data synthesis pipeline is the fully controllable difficulty of the problems. As illustrated by AlphaGeometry (Trinh et al. 2024), the difficulty of IMO geometry problems for humans is positively correlated with the number of DDAR proof steps. Therefore, we choose the DDAR proof step count the measure of task complexity, denoted as Denote the data synthesis pipeline X. We implement CBRL as follows:
0r arg maxB Jr 0)
(8)
K" = arg max By.
where the goal of optimization is maximize the average absolute advantage during learning, and A(X, y) is the advantage of outcome reward. We then present properties of CBRL. As shown in Wang al. (2025b) and Chen al. (025b)), maximum absolute advantage has the following properties:
Theorem 1.
Given model parameter 0, the K* obtained from Equation 9 approximately optimally accelerates the learning progress.
Theorem 2. For binary rewards, the maximum average absolute advantage is 0.5, which indicates that the task is of moderate difficulty for the
too difficult nor too trivial.
In practice, in each CBRL round, we sample data conditioned on complexity K. perform RL training to the agent, and finally update λ according to learning rate See Appendix for details.
2.4 DATA SYNTHESIS PIPELINE
The proposed data pipeline targets synthesize geometry problems with adaptable levels of difficulty for InternGeometry. Specifically, it comprises two stages: cold start and expert-level problem synthesis for CBRL.
First, due to the scarcity of data in DSL form, we fine-tuned InternThinker 32B (Lyu et al. 2025) as InternGeometry- eerrGoommmttyyrooalieerrrr Formalizer through expert iteration (Anthony et al. 2017) and then exploit large- scale natural language problem data from diverse sources. This process produced a total of 7K formal problem and solution trajectory pairs, which provide cold start for InternGeometry.
However, these paired data are constrained by the imbalanced difficulty distribution with relatively few problems at the expert level. To endow LLM with expert-level problem-solving abilities, we further synthesize problems dynamically during reinforcement learning. We first add auxiliary constructions into randomly constructed problems with statistical prior of given complexity, then leveraging the InternGeometry -DAR to filter valid constructions and goals to form problems. Finally, total of 6K problems are constructed by difficulty based on proof steps during CBRL. See Appendix D for details.
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Table I: Comparison of overall performance on IMO 50 between InternGeometry and SOTA geometry expert models.
Model
Model Type
Training Data
Sampling Setting
IMO 50 Pass@
AlphaGeometty
Expert Model
300M
Ensemble of search trees
42/50
SeedGeometry
Expert Model
230M
N/A Pass
43/50
InlemGeometry
LLM Agent
13K
44/50
Table 2: problem-by- comparison on IMO 50 between InternGeemeery and SOTA geometry expert models.
Year
ID
Split
AG2
SG
IG
Year
ID
Split
AG2
SG
IG
Year
ID
Split
AG2
SG
IG
2000
P1
/
2007
2015
P3
2000
V
2007
2015
P4
2001
P1
X
X
X
2008
P1
2016
P1
2008
2017
2002
P2
2008
P6
2018
P1
V
2002
b
2009
2018
P6
X
2002
2009
2019
2003
P3
X
X
2009
P4
b
X
2019
16
2010
2003
P4
V
V
2010
P4
2020
P6
K
X
2004
2011
2021
P3
2004
V
2012
2021
P4
2004
P5
b
2012
P5
2022
P4
2005
/
2013
2023
2005
P5
V
/
2013
P4
2023
P6
X
2006
2014
2024
2006
P6
X
X
2014
2025
P2
N/A
EXPERIMENT
3.1
EXPERIMENT SETUP
Implementation. We use TiiernThinner3321 (Lyu et al., 2025) the backbone model for our method. For the agent model, we set the maximum number steps to 200 by default, with inference hyperparameters of temperature 0.9 and top-p 0.9. During the test, the pass@K is set to 256.
Dataset and Baselines. We use IMO 50 (Chervonyi et al. 2025) as the test set, which includes all geometry problems from IMO 2000 to IMO 2024. We additionally evaluate InternGeometry the geometry problem from IMO 2025, reported separately in Table [ We use AlphaGeometry (Chervonyi et al. 2025) and SeedGeometry (Chervonyi al. (2255) as baselines. both of which are state-of-the-aar geometry proving methods based on expert models. The performance these baselines is taken directly from the results reported in their respective papers.
3.2 OVERALL RESULTS
We compare the performance of InternGeometry with baselines in Table InternGeometry solved 44 problems in IMO 50, surpassing AlphaGeometry 2 and SeedGeometry. The "split" in the table refers subproblems of questions that contain multiple subquestions. Notably, Interneeometry used only 13K data points-just just 0.004% of AlphaGeometry 2 and 0.006% of SeedGeometry. Furthermore, its test-time scaling budget was also far lower than AlphaGeometry 2, which use ensembles of beam search, and the reported optimal single beam tree configuration is beam size 128, branching number (samples) 32, and beam depth See Appendix [ for more discussion. These comparisons clearly demonstrate the potential of LLM-based agent approaches expert-level tasks.
We list the individual results on IMO in Table We additionally include the geometry problem of IMO 2025 in the table. Interneeometry solved 45 out of 51 problems, covering all problems solved by AlphaGeometry 2, and additionally solving 2018 P6 and 2023 P6. Compared to SeedGeometry, additionally solved 2001 P5 and 2009 P4b, but missed 2006 P1. Notably, the remaining unsolved problems largely involve computations that go beyond the scope of pure geometric proof, and thus fall outside the current expressive range of geometric DDAR systems. See Appendix F for cases.

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Supar
Stea00
Poxss BK (36)
Paoss @K (56
16
44
134
510
9224
2009
4090
0980
30704
Agent Max Step
Total Steps (Samping Time K. K* Agent Steps)
Figure Left: The effect of long-horizon interaction on the proof. As the interaction steps increase, the proving success rate improves significantly, which holds for different sampling times. As sampling times increase, Pass @ K also rises, indicating test-time scalability of InternGiemmety. Right: Extending the agent's trajectory length is more effective than repeated sampling for scaling. The total inference budget is defined as the sampling number K multiplied by the agent's steps. When the maximum length is capped (the blue lines), performance improves with inference budget at a slower rate for shorter trajectories. On the other hand, when the sampling size is fixed (the green lines), increasing the budget by lengthening the trajectory yields efficient scaling.
Table 3: Ablation study on long horizon agents in IntenGeometry.
Propositions
Slow Thinking
Context Compression
Reject Sampling
IMO 50 Pass@256
44/50
35/50
x
X
23/50
/
20/50
38/50
3.3 ANALYSIS FOR LONG HORIZON AGENT
To analyze the effect of long-horizon interaction to the proof, we compare the pass@ K on IMO 50 under different steps increase, the proving success rate improves significantly, which holds for different sampling times. Shorter interactions significantly limit the success rate of agent proofs. As the interaction trajectory grows, the agent can continually explore to develop heuristics about the problem, enabling it better leverage its reasoning ability for generalization. Additionally, the figure illustrates the trend of test-time scaling. As shown, as sampling times increase, Pass @ K also rises, indicating the test-time scalability of InternGeometry.
step setting, and the result is in F It is evident that interaction
We emphasize that extending agent's trajectory length scales performance more effectively than repeated sampling. shown (right), with total inference budget defined as times the number of steps. performance grows slowly when the step cap is 64 but improves much faster when it 200. For fixed K, increasing trajectory length consistently yields higher efficiency, supporting our hypothesis that long horizon interactions enhance heuristics more effectively in geometric proofs.
An ablation on IMO 50 (Table 3) further confirms this. Interneeometryy solves fewer problems when any long- horizon component is removed. Slow thinking and context compression increase solved problems from 20 and 23 to 44, underscoring their key roles in enabling IMO-level reasoning.
3.4 ANALYSIS ON COMPLEXITY BOOSTING REINFORCEMENT LEARNING
In this section, we analyze the effectiveness of the CBRL method. We first present the distribution of the synthetic data obtained during the CBRL process length of problem's proof steps as indicator of its difficulty. Accordingly, on the distribution of proof lengths in the synthetic data generated during model training, as shown in the figure. The figure shows that the difficulty distribution of the synthetic data exhibits fairly uniform improving trend, indicating that our agent training provides a sell-structured curriculum from simple to difficult tasks, which helps the agent master complex combinations of proof skills.
( As noted earlier, we use the
compile statistics
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Proof Langh (Compless of Training Data
IMO 50 Pass (%)
Updale Round
Schedule Updale Round
Figure 4: Left: The distribution of proof lengths in the synthetic data generated during model training, indicating task complexity. The figure shows that the difficulty distribution the synthetic data exhibits fairly uniform improving trend, providing well sell-structured curriculum from simple difficult tasks. Right: Agent's generalization performance on IMO 50 during training. The agent's s overall performance on IMO 50 shows steady upward trend. Notably, there is significant performance jump in the sixth training round.
Table 4: Ablation study on CBRL in InternGeometry.
Training Setting
IMO 50 Pass @ 256
With CBRL
44/50
SFT Cold Start
22/50
Easy Data Only
29/50
Challenging Data Oaly
24/50
Same Data without Schedule
38/50
We further demonstrate the agent's generalization performance on IMO 50 during training in Figure (right). As the training data is continually updated, the agent's overall performance IMO 50 shows a steady upward trend. Notably, there is significant performance jump the sixth training round, indicating breakthrough progress as task complexity is progressively increased in the reinforcement learning process.
We then conduct ablation studies the key components of CBRL. The result is in Table Notably, after SFT cold start, the model's baseline performance on IMO 50 is 22/50. We first analyze the impact of data difficulty on reinforcement learning, revealing the importance of allowing the learning algorithm to autonomously control the difficulty of synthesized data. To strictly control variables, we rum experiments with the same data scale and training steps in InternGeometry. Specifically, we modify the data distribution in the RL phase to: using only low-difficulty data, and using only high-difficulty data. The corresponding results are shown in the rows Easy Data Only and Challenging Data Only in the table. The results indicate that using only low-difficulty data limits the agent's generalization ability to IMO-difficllty problems; on other hand, using only high-difficulty data also leads to suboptimal training outcomes. The latter occurs because learning becomes slow and fails to converge within the same training budget: the agent remains an early trial-and aal--nd-error stage with extremely sparse learning signals. This highlights the efficiency of CBRL.
Finally, we ablate the dynamic difficulty curriculum by uniformly sampling from all synthesized data throughout the Interneometry training phase. The result (Same Data without Schedule) shows that removing the curriculum again degrades performance. Without progressive difficulty, the agent struggles with sparse absent rewards sampled hard problems, preventing it from learning effective strategies under the same training budget. This confirms that CBRL significantly improves data efficiency and training effectiveness in reinforcement learning.
3.5
CASE STUDY
During manual checking, find that InternGeometry shows remarkable creativity certain problems. As shown in Figure [ while most human solvers relied on inversion, trigonometry or complex numbers, the agent solves this problem via elegant geometric construction using classical
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angle chasing and basic theorems. Specifically, InternGeometry first places point T on segment AC such that LBDA- = ZTDC, and defines point K as the intersection of two circles. These two points form an isogonal conjugate pair in quadrilateral ABCD, revealing that the agent can discover this implicit structure through exploration. To further exploit the isogonal property, it then constructs the symmetric points of T with respect each side of the quadrilateral, showing both an understanding of rotational symmetry and an ability to generalize the use of auxiliary points for handling isogonal conjugates from triangles quadrilaterals. Overall, this case study highlights how Interneeometryy generate creative constructions that differ fundamentally from human solutions.
4 RELATED WORK
Reinforcement learning agents in the field of mathematics Currently, large language model agents have achieved remarkable performance in tasks such as Code, Search, GUI, and also Mathematics through reinforcement learning (RL). In mathematics, RL agents based on informal proofs solve problems using general-purpose tools like Python compilers. Examples include OR (Open Reasoning) approaches Singh et al. (2025); ; Li et al. (2025)); Mai et al. (2025): Zuo et al. (2025): Prabhudesai al. (2025 ( Shen et al. (2025); Shang et al. (2055) and PR (Proof Reasoning) approaches Li et al.
(2025a): Simonds & Yoshiyama (2025) Goldie et al. (202); y |ao et al. (202)) Alternatively, agents built on Interactive Theorem Provers (TTPs) specialized for mathematics can handle more complex problems. Works like Xin et al. (2024); Ren et al. (2025); (hann) et al. (2025a); |Wang et (2025a) achieve strong results on benchmarks such as mini (Zheng et al.] (021)) and ProofNet (Azerbayey let al. 2023). However, few agents address geometry problems. Our work targets this gap. developing an interactive geometric prover and showing the potential of data synthesis and difficulty scaling.
Rurriculum learning for agents While several research (Wang et al. 2025b Zhang et al.| 2025b)
Parashar et al. limited, and most approaches rely on highly structured task types. For example, Voyager (Wang et al., 2023) uses manually designed curricula in MineDojo (Fan et al. (022) to teach agents complex skills. WebRL (Qi et al. 2024) iteratively generates increasingly complex task instructions and employs a reward model for automatic success evaluation. In scaling RL for agents and task synthesis, Envgen
2025) study curriculum reinforcement learning, curriculum um agents learning remains
(Zala et al. 2024) trains large language models to generate formal parameters for Crafter (Hafner) 2021) and Heist (Cobbe et al., 2020). enabling dynamic environment and task generation. Unlike these approaches, our method fully automates large-scale synthesis at specified difficulty levels, allowing unrestricted and unbiased curriculum adjustment, especially at higher complexity.
Automatic geometry theorem proving Current Al-based approaches to automated geometric proofs remain largely expert-model driven. State-of-thee the-art systems like AlphaGeometry (Chervonyi et al. 2025 Trinh& & Luong 2024) and SeedGeometry (Chen et al. 2025a) typically decompose a problem into two tasks: (I) auxiliary construction prediction, where the model proposes additional geometric elements (e.g., lines, points); and (2) formal reasoning, where a search algorithm assembles a complete proof using geometric theorems (e.g.. angle bisector, triangle similarity) and logical inference rules. Following this paradigm, most methods train specialized models on large datasets to predict constructions and then combine them with formal search engine for proof generation. Recent work has also explored using large language models for geometric reasoning, but these efforts mainly target elementary- problems and computational task formats.
5 CONCLUSION
LLM-based agents can solve tough math problems, even IMO-level with formal provers, but geometry remains dominated by expert systems like AlphaGeometry2 that depend on massive synthetic data and search. We introduce InternGeometry, a medalist- LLM agent for geometry. It overcomes weak heuristics by iteratively proposing propositions and constructions, verifying them with a symbolic engine, and refining based on feedback. A dynamic memory enables over 200 interaction steps. To speed learning, we use CBRL, which gradually increases synthesized InternGeometry solves 44/50 IMO geometry problems 2000-2024) surpassing the average gold medalist score (40.9), using only 13K training examples-abes about 0.004% of AlphaGeometry22 ' data. It can also generate novel auxiliary constructions unseen in human solutions.
auxiliary-construce
problem difficulty during training. Built on InternThinker-

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Published as a conference paper at ICLR 2026
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Shubham Parashar, Shurui Gui, Xiner Li, Hongyi Ling, Sushil Vemuri, Blake Olson, Eric Li, Yu Zhang, James Caverlee, Dilcep Kalathil, et al. Curriculum reinforcement learning from easy to hard tasks improves Ilm reasoning. arXiv preprint SX1::25066006632, 2025.
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Abhay Zala, Jaemin Cho, Han Lin, Jachong Yoon, and Mohit Bansal. Envgen: Generating and adapting environments via Ilms for training embodied agents. arXiv preprint arXiv:2403.12014, 2024.
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A
THE USE OF LARGE LANGUAGE MODELS (LLMS)
Beyond these technical roles, LLMs did not play a significant part in research ideation, experimental design, or manuscript writing, the authors conceived the study, designed/evaluated experiments, analyzed results, and wrote the paper. Any automated assistance, if present, was limited to non- substantive copy-edttin//LaTee linting. The authors take full responsibility for all contents, including verifying any machine- machine-generated intermediate artifacts, and acknowledge that LLMs are not eligible for authorship. This disclosure follows the ICLR policy on LLM usage and research integrity.
B IMPROVEMENTS IN INTERNGEOMETRY-DDAA.
IntenGeomerry-DDAA builds upon spur-sourced symbolic engines (i.e., Newclid (Sicca et al. 2024) and AlphaGiemetty (Trinh & Luong 2024)) and mainly consist of two components: a deductive database and algebraic reasoning. The former expands the current set of premises toward the proof goal based on geometry rules, while the latter performs angle, length, and ratio chasing using Gaussian elimination. We introduce three main improvements: dynamic diagram adjustment, the incorporation of syntax and rules for handling double points, and the addition of new predicates and rules.
First, open-sourced symbolic engines can only create points one by one based on existing construction definitions, with each point constrained by at most two construction definitions. However, in IMO geometry problem, is often necessary to make global adjustments to previously constructed points so they satisfy more specific requirements (e.g., e,& a line defined two existing points may also need to be tangent to a existing circle). Consider IMO 2003 P4a as an example: "Let ABCD be a cyclic quadrilateral. Let P. Q and R be the feet of the perpendiculars from D to the lines BC, CA and AB, respectively. Show that PQ QR if the bisectors of angles LABC and LADC meet on segment AC". The condition "the bisectors of angles LABC and LADC meet on segment AC" cannot be enforced automatically during point-by-point construction; it is specialized condition that holds only for certain point placements. Such hard configurations require globally adjusting previously constructed points so that multiple geometric constraints are satisfied simultaneously. To address this issue, we use gradient descent to adjust certain specific points so that they simultancously satisfy multiple requirements.
Second, we address the issue of double points (i.e., distinctly named points with identical coordinates) by proving that they represent the same geometric point, which is important technique. Difficult geometry proofs often rely on reformulating intersections or handling degeneracies that naturally create such overlaps-a common trick used by human solvers To this end, introduce new syntax: prefixing a construction statement with ! allows the system to create a point even it shares coordinates with an existing one. In addition, we update the inference module of the symbolic engine to support this extended behavior. We also introduce new predicates and rules for handling double points. Specifically, we define the predicate idc x y to indicate that points X and Y are geometrically considered the same, and we provide rules for determining this relationship.
Additionally, we add several common geometry theorems, such as the Power of Point and Menelaus' theorem into anterGeemeetr-DDAA.
InternGeometr--DA.. serves both as an automated geometry problem solver and as a powerful symbolic tool for InternGeometry.
C INTERACTION BETWEEN INTERNGEOMETRY AND INTERNGEOMETRY-DDAE
InternGeometry has three interaction with DDAR,,, which includs obtain initial state in symbolic engine, adding auxiliary construction and proposing proof steps. Specifically, when giving a formal geometry problem, InternGeometry first output '<build>' tag to construct this problem and retrieve the initial geometric relationships from the symbolic engine. Then, in the following turn, InternGeometry performs thinking and then automatically decides whether to add an auxiliary construction using '<add>' tag propose proof step using '<propose>', Intermeeometry -DAR executes the instructions from Interneeometry and returns feedback, such as successfully proving the proposed proposition or reporting failure when adding new point. After each step, InternGeometry
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engine alone. Published as a conference paper at ICLR 2026
summarizes and compresses the current proof state support long- horizon interaction. When the final goal is proven, Interneeometry reviews the entire proof and briefly summarizes the reasoning process, extracting key steps and auxiliary constructions.
D DETAILS OF COMPLEXITY BOOSTING REINFORCEMENT LEARNING (CBRL)
We provide a detailed introduction to CBRL in this section, as the main paper is space-constrained. As outlined in Subsection 2.3 CBRL adapts task complexity maximize the expected absolute advantage during reinforcement learning. The core intuition is to present tasks that are neither too difficult nor too easy for the policy model-ia tasks that best match the model's current capability. Following AlphaGeometry (Trinh & Luong 2024), which observed that human--prreivee perceived difficulty of IMO geometry problems (measured by average IMO scores) correlates positively with the number of proof steps taken by the DDAR solver, we quantify task complexity using DDAR proof length.
We first introduce the geometry question generation algorithm, then describe the data synthesis pipeline. Finally, we present the CBRL algorithm and explain its motivation.
The question generation procedure, Generate Data (Algorithm [ aims to sample nontrivial geometry questions that cannot be solved by exhaustive search using the
The algorithm iteratively samples a raw geometric structure, Xraw. by randomly instantiating DDAR predicates and points. It then augments this structure with auxiliary constructions to obtain X Both stages are conditioned on a user-specified complexity parameter k, for which we design distinct priors and construction patterns to improve the hit rate of valid problems. We perform exhaustive search on both X yaw and X Conclusions that (i) involve only points in X (ii) are provable in Xodd, and (iii) are not provable Xyaw. deemed nontrivial problems rooted in the raw structure. Among these candidates, we select the most complex one-primarily by proof length, while also considering priors such as predicate distributions. The algorithm repeats continuously generate questions targeting the specified complexity prior k. Because the actual complexity of generated items is only loosely controlled by via stochastic priors, we apply additional post-sampling to better align the dataset with the desired complexity.
Algorithm 1 Generate Data
Require: Complexity x
1: for in Sange(M) do
2: Xraw + RANDDDARCONSTRUCTIONNN
3: Xadd + ADDAUXCCONSTRUCTIONS(....
4: P Praw EXIIAUSTSEARCII(X....
5:
Padd + EXHAUSTSEARCH
6:
C+ .ce Pada Praw conclusion of uses only points in Xraws
7:
C + then
data", ,pl' <- SHLFCTMOSTCCMPLEX (N C) 9: yield data", pl*
8:
& primarily by proof length
10: end
11:
end for
Next, Algorithm A presents the complete Data Synthesis Pipeline. We maintain global cache of all generated questions. Given target complexity к and required sample count, we first check whether the cache already contains enough items within tolerance range around k. If not, we invoke Generate Data to enrich the cache until we can retrieve a sufficient number of questions the desired complexity.
Finally, we introduce Complexity Boosting Reinforcement Learning (CBRL) in Algorithm The idea is straightforward: in each iteration, we sample batch of data from the Data Synthesis Pipeline, run RL on that batch, and record rewards. After processing the batch, we compute the average reward and update by comparing this average to 0.5: increase if the average reward exceeds 0.5, otherwise decrease it. Below, we justify why using 0.5 as the target lead to optimized expected absolute advantage, indicated by Theorem 2
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Algorithm 2 Data Synthesis Pipeline X
Require: Complexity Data Number K
1: global
cache: list of (data, proof length)
2: ok, dataset <- STLLECTAROUNDRANGCCCC.. K)
3: while not ok do
4: for (data, pl) in GENERATEDATA(A) do C.append(datt,, pl))
5:
6: end for
7: ok, dataset + SELECCTAROUDDRNNGECCCC..
8: end while
9:
return dataset
b exactly K' items around target complexity
For a binary reward with P(r 1) p and P(r = 0) 1 p, we have mean(r) p and std(r) = vp(1 p). The expected absolute advantage is
r: P p((1 p)
1-p (11 Vp(1-))
P vp(1 p)
A E
= P.
2/p(1 p). (10)
This quantity is concave in p [0, 1] and is maximized p 0.5. Therefore, maximize the expected absolute advantage by steering the average reward toward 0.5 via k-adjustment.
Algorithm 3 Complexity Boosting Reinforcement Learning (CBRL)
Require: Initial complexity K, Initial policy parameter 0
1: for in (MageeMa do
2: batchdata (1s, DataNumEEchhtter)
3: batchrewards < [.]
4: for minibatch SPLITMiNNBarcHHtuuchhaaa do
5:
rewards, 0 + EVALUATEANDUPDATEE(. minibatch) batchhewardssexeeddrrwwddd
6:
7: end for
8: if AVERRAGG((atchheeaards) 0.5 then
9:
+
else
11:
+
12: end
13:
end for
14:
return
We further explain why maximizing the expected absolute advantage benefits policy learning (Wang al. 2025b Chen et al. (2025b), stated in Theorem The key idea is that the expected absolute advantage serves as the primary learning signal that scales the gradient. The simplified gradient of the GRPO objective is
VoJ(O) I(w) [ (Eg--G((-a) [A, Vo log Go (y x)]
(11)
Thus, the gradient norm is
(+J Exax(N)
|| || [A+ V log Go (ys
x)]
(12)
Assuming that the gradient term Vo log Go (y: x)) is bounded and approximately random in direction, the gradient norm approximately maximized by maximizing E Ex- AA(X, v)]. In other words, increasing the expected absolute advantage yields larger gradients during optimization and thus can accelerate learning.
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E DETAILS OF TRAINING TOKEN
We report the total number of training tokens and token-based comparison to prior work. Our model totally trains approximately 1.91 10 tokens. For reference, AlphaGeometry 2 reports training on up to x 10 tokens. Framed in terms of training tokens, Interneeometry is therefore substantially more data-efficient.
F FAILED CASES
We illustrate failure cases of InternGiemmetry IMO 50 in this section. Notably, the unsolved problems are largely beyond pure geometry, and they primarily rely on numerical or non-geometric analysis.
2001 P1
Let ABC be an acute-angled triangle with O as its circumcenter. Let on line BC be the foot of the altitude from A. Assume that LBCA 2 LABC +-32.
30 Prove that ZCAB+ZCOP 90"
2002 P6
Let be positive integer. Let C₁ C2. Cn be unit circles in the plane, with centers O1, Oz....,
...,,... On respectively. If no line meets more than two of the circles, prove that
n 1)T
Σ IgiciEn
0,0,
4
2003 P3
Each pair of opposite sides of convex hexagon has the property that the distance between their
v3
midpoints is times sum of their lengths. Prove that the hexagon is equiangular.
2
2006
Let ABC be triangle with incenter T. A point P in the interior of the triangle satisfies
LPBA+ LPCA = LPBC + LPCB.
Show that AP 2 Al and that equality holds if and only if P =
2006 P6
Assign to each side of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.
2020 P6
Consider an integer n > 1, and set S of n points in the plane such that the distance between any two different points in at least 1. Prove there line separating S such that the distance from any point of S to & is least (( (A line separates set of points S if some segment joining two points in S crosses É.)
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G DISCUSSION OF INFERENCE COST
Because neither AlphaCieometry 2 nor SeedGeometry has released open-source code or models, we cannot perform direct, controlled comparison. Therefore, we rely on the configurations reported in their respective papers to provide a rough estimate and comparison. Since the SeedGeometry paper does not detail its inference budget, we primarily base our equivalent estimates on AlphaGeometry 2.
According to the AlphaGeometry 2 paper, the method uses a Shared Knowledge Ensemble of Search Trees (SKEST) that integrates classical bean search with several tree-search variants (c.g.. predicting multiple auxiliary points at each node) and runs multiple trees with different search configurations and models in parallel. Consequently, the total inference budget scales with the number of configurations x the number of instantiated trees x the number of models. The reported optimal single-tree configuration uses a beam size of 128, a branching number (samples) of 32, and beam depth of 4. However, within SKEST there are configurations that incur higher budgets, such as beam size 64 with depth 10 or beam size 512 with depth 4. The AlphaGeometry model size 3.3B parameters.
We compare inference efficiency from four perspectives: the equivalent number solutions ex- plored during inference, the overall inference steps, the environment execution cost, and the overall computation cost.
1. Equivalent number of solutions explored. For single beam search, the equivalent number of explored solutions can be approximated as beam size x branching number. Under the optimal single-tree configuration, this is 128 32 4,096, and it can be larger for other configurations (e.g., beam size 512). The total number further scales with the number of configurations x the number of instantiated trees the number of models. By contrast, Interneeometty's 's best-of-K inference explores only K solutions (256 in our experiments), addressing its solution-efficienn
Inference steps. The optimal beam-tree configuration results in 128 32 4 16,384 steps, and the same number of symbolic-enging executions per tree. Other configurations, such as beam size 64 and depth 10 (64 32 10 = 20,480) or beam size 512 and depth (512 x 32 x 65,536), can require more steps. InternGeometry uses simple pass @ 256 parallel-inference setting with up to 200 turns of agentic interaction per pass, totaling 51,200 steps. Overall, the per-tree inference budget of AlphaGeometry 2 is on the same order InternGeometry. However, the total inference cost for AlphaGeometry 2 scales with the number of configurations x the number of instantiated trees x the number of models, leading to larger overall inference steps.
Environment execution cost. Each step in both AlphaGeometry 2 and InternGeometry is executed in an engine. Because Interneemmety's 's total number of steps is smaller than the full SKEST budget of AlphaGeometry 2, it yields fewer total executions. Furthermore, each AlphaGeometry 2 step requires the DDAR system attempt solving the entire problem, whereas each InternGeometry step either adds an auxiliary construction or attempts to prove a subgoal operraaionns that are less expensive than attempting to solve the whole problem.
4. Overall computation cost. InternGeometry's reasoning style and larger model size indeed increase computation. Each InterGeeometry step involves natural-language reasoning, resulting in more tokens and higher compute: for IMO-50, the average number of output tokens per trajectory is 89.6K. The InternGeometry model (32B) is also larger than AlphaGeometry (3.3B). Due the unknown total cost of AlphaGeometry 2, direct comparison is difficult. However, we emphasize that the increased computation from deeper reasoning and larger models should not be viewed as a drawback. Instead, it represents feasible new scaling cimension-alongsiiin training data size and the number of searched solutions-ons that aligns more naturally with LLM-based approaches than with expert-model diagram.
17


Reviews:
[{"review_id": "SJGg1LvoKG", "reviewer": "Reviewer_vjZE", "summary": "This work introduces InternGeometry, a LLM agent designed to solve geometry proof problems at the IMO level. By integrating Complexity-Boosted Reinforcement Learning (CBRL) and dynamic memory mechanisms, the model achieves gold-medalist performance on IMO geometry problems, significantly outperforming strong baselines AlphaGeometry2 and SeedGeometry. Notably, InternGeometry only uses 13K training samples, two orders of magnitude less than AlphaGeometry and SeedGeometry. The experiments are thorough, demonstrating the effectiveness in data efficiency and long-range reasoning.", "questions": "1. Line 40-42: Add references to justify LLM agents can obtain medalist-level performance on IMO-level problems.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Propose the first LLM agent for IMO-level geometry proving, avoiding the use of specialist models.\n2. Propose a dynamic memory mechanism and rejection sampling strategy, enabling up to 200-step interactive reasoning and guiding diverse explorations in interactions.\n3. Solid experiments well justify that InternGeometry outperforms current SOTA models, with exceptional data efficiency. Comprehensive ablation studies validate the necessity of key components like long-range interactions, CBRL, and dynamic memory. The case study justifies the model's creative construction capabilities.", "weaknesses": "1. The title is not clear. Since the manuscript focuses on developing plane geometry prover, the title should contain such information.\n2. Equation 5 extends beyond the page margin.\n3. Experiments on more datasets (e.g., JGEX-AG-231 proposed in AlphaGeometry) and other LLMs (other than InternThinker) can further demonstrate the generalization ability of InternGeometry.\n4. Considering that the interactive reasoning requires many steps, analyzing and comparing the computational resources of InternGeometry, AlphaGeometry, and SeedGeometry during reasoning is necessary.", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "avskq5SetY", "reviewer": "Reviewer_YNAr", "summary": "The paper introduces InternGeometry, an LLM agent system with a symbolic geometry solver (InternGeometry-DDAR) that solves math competition geometry problems. It improves the previous SOTA on 50 IMO geometry problems from 43/50 to 44/50, and achieves this with significantly less geometry problem training data than previous methods.", "questions": "1. What is InternThinker-32B? There is no citation, and I couldn’t find any information about it on the Internet.\n\n2. How are Eqs. (5) and (6) different from GRPO? (other than what appears to be typos)\n\n3. How is CBRL different from RL with a curriculum?\n\n4. Should the top-left cell of Table 1 say “AlphaGeometry2” instead of “AlphaGeometry”?\n\n5. The claim that InternGeometry’s test-time scaling budget is “far lower than that of AlphaGeometry2” (lines 311-312) is not clear to me. InternGeometry does pass@256 and uses a larger model (32B instead of 3.3B), so it’s not clear that it uses \"far less\" computational resources than AlphaGeometry2.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. InternGeometry is the new SOTA on IMO 50.\n2. The fact that the training set of geometry problems is much smaller than in previous approaches is impressive.\n3. The analysis about scaling max steps vs. scaling # samples is insightful.", "weaknesses": "1. The paper would benefit from more discussion regarding inference-time costs of the compared methods. For example, listing the model sizes in Table 1 would be helpful, as well as explaining the different search parameters in AlphaGeometry2’s custom beam search. If available, information about the total # of output tokens or wallclock time etc. would also be appreciated.\n\n2. Similarly, I think it would be nice if the paper also discussed training costs and compared them with previous methods. Currently, it mentions the size of the training set, but other information would also be helpful, such as the total # of tokens in the training set and a version of Figure 4 where the x-axis is the number of training tokens.\n\n3. Writing quality is poor:\n  - Contextualization within previous work is often missing, e.g., it is not clear what the novelty of CBRL is compared to previous work (curriculum learning for RL, GRPO).\n  - There are many grammatical mistakes and unidiomatic uses of English, including in the title and abstract. I recommend using something like ChatGPT to improve the writing.\n  - There are missing citations (e.g., the last two paragraphs of the introduction section have no citations)\n  - Other miscellaneous issues, e.g., labeling imprecise claims as “theorems” (lines 247-251), potential typos in Eq. (5), and missing explanations (e.g., what’s InternThinker-32B? what’s “split” in Table 2?)", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ghBkOz407F", "reviewer": "Reviewer_2RDT", "summary": "While LLM agents and agentic workflows have shown strong performance (gold medal level) in International Mathematical Olympiad (IMO), solving geometry problems is still by far the most challenging task, and existing IMO math agents typically have poor performance on such problems. The main reason is that a large subset of geometry problems typically requires multiple trial and error to find the right auxiliaries to add to the current problem diagram to make it solvable. Solutions like AlphaGeometry exist, these solutions typically require an enormous amount of formal data to either train a custom transformer from scratch or perform continual training on the same data but on already pre-trained models. Then, the trained model on formal data would go in a loop with a deductive symbolic engine, where the model proposes constructing a new auxiliary to the problem at hand, and then the symbolic engine would run over and over again to extensively perform deductive search for reaching the proof goal, and upon exhaustion of the engine, the loop continues.\n\nThe vast amount of search done on formal deductive engines, and the enormous amount of training data make the existing solutions very inefficient. On the other hand, it is widely known that LLM agents and workflows can reach perfect performance on non-geometry problems in either formal or non-formal settings even when fine-tuned on a very limited amount of training data compared to AlphaGeometry. Thus, this work aims to address this timely question of how LLM agents can be designed to tackle geometry problems as well. They propose introducing a novel tool call for dynamic interaction with their enhanced deductive engine, supporting multiple actions such as building the formal geometric configuration, adding auxiliary constructions, and proposing a proposition to be verified. The dynamic interaction with the engine enables leveraging the already strong high-level natural language planning and reasoning of frontier LLMs for optimized interaction with the engine.\n\nTo enable this tool call, they performed cold-start training to teach their base model (InternThinker-32B) how to work with this new tool, and then performed curriculum RL to improve the reasoning with their proposed tool call, and the final solution is InternGeometry which is claimed to perform on par with, or even sometimes outperform the two main baselines: AlphaGeometry and SeedGeometry on the IMO 50 evaluation suite (all geometry problems from 2000-2024 IMO). In contrast to these baselines, InternGeometry is trained on roughly 0.005% of the training data size of other baselines (13k samples, 7k for cold-start, and 6k for RL).", "questions": "1. Could you provide a simplified high-level pseudocode for the working implementation of CBRL so the selection of the curriculum complexity $\\kappa$ becomes clear, as well as explaining the technical challenges of implementing the CBRL algorithm? This would greatly benefit the soundness and clarity of the RL training done in this paper.\n\n2. Please see Weakness 2, and elaborate more on how exactly the cold-start data is generated?\n\n3. A cost comparison between deploying InternGeometry versus AlphaGeometry or SeedGeometry would be greatly valuable for this work. I understand this might or might not be feasible based on the information available from the prior work, but it would be interesting to compare the number of interaction steps between the deductive engine and the LLM with the prior work. Especially since the LLM usage for proposing propositions might greatly improve the interaction efficiency as well.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. The introduction of a new tool call for leveraging strong planning and reasoning of LLMs for dynamic interaction with the formal deductive engine is novel and an exciting idea. The paper did a good job explaining this tool call and motivating it, which makes their high-level approach clear and sound.\n\n2. As mentioned in the summary, much trial and error is typically expected for figuring out the helpful auxiliary constructions for solving the given problem, and this fact would require the proposed agent to deal with the challenge of long-horizon reasoning that demands careful context management of past exploration after each tool call. The paper then proposes to maintain a dynamic memory containing key information from past tool call iterations (e.g., what actions were made in previous turns, what were the outcomes of those actions, plus the current action and current feedback from the deductive engine which contains all successful propositions). The introduction of this dynamic memory is novel and an interesting solution for solving the weak heuristic nature of auxiliary construction in IMO-level geometry problem solving. The ablation also confirms the long-horizon interaction is indeed necessary for obtaining the InternGeometry performance.\n\n3. Only requiring roughly 0.005% of the key baselines' training data (AlphaGeometry and SeedGeometry) to reach comparable performance, and even outperform them on the IMO 50 dataset is an impactful contribution to the LLM math agents community, and this work can be seen as an initial step to harness general-purpose LLMs for solving complex geometry problems.\n\n4. The paper mentions the model, data, and the deductive engine used will be open-sourced, which is of great benefit to the community, and this would make this work reproducible.", "weaknesses": "1. While the RL reward and RL loss are clearly defined, the handling and explanation of the curriculum algorithm lacks clarity, which makes it hard to evaluate the soundness of the proposed curriculum approach. The paper attempts to touch on the theory behind the curriculum algorithm on the surface, and both Theorem 1 and 2 statements are hard to follow and vague, and could be better explained. More importantly, the paper does not explain the CBRL algorithm with sufficient detail and particularly it is not clear how the complexity $\\kappa$ is updated in each CBRL round. The paper only briefly mentions the following (line 253): \"In practice, in each CBRL round, we sample data conditioned on complexity $\\kappa$, perform RL training to the agent, and finally update κ according to learning rate $\\alpha$.\"\n\n2. Data curation for cold-start requires clarity, as the complexity of obtaining and curating such data is actually high. Regarding this, the paper only briefly addresses this around line 262, mentioning that \"First, due to the scarcity of data in formal systems, we fine-tuned InternThinker-32B as InternGeometry-Formalizer through expert iteration (Anthony et al., 2017) and then exploit large-scale natural language problem data from diverse sources. This process produced a total of 7K formal problem and solution trajectory pairs, which provide a cold start for InternGeometry.\" I believe due to the complexity of data curation for this phase, one would want to know the details of how exactly the cold-start trajectory is generated.", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Fca4R6SSqc", "reviewer": "Reviewer_5LSi", "summary": "1. The paper introduces InternGeometry, an LLM-based agent that achieves medalist-level performance on International Mathematical Olympiad (IMO) geometry problems. \n2. Unlike previous systems such as AlphaGeometry2, which rely heavily on large-scale data synthesis and heuristic search, InternGeometry integrates long-horizon reasoning, symbolic feedback, and a Complexity-Boosted Reinforcement Learning (CBRL) framework. \n3. This work performs efficient training using only 13K examples (0.004% of AlphaGeometry2’s data).", "questions": "1. How crucial are the additions of \"dynamic diagram adjustment\" and \"double points\" handling for solving the IMO-level problems? Will be good if authors can add this to improve the quality of the paper. \n2. As mentioned in Equation (9) add a brief explanation of how maximizing \"absolute advantage\" creates a curriculum of moderate difficulty and how the search for ‘k’ is done in each round.\n3. The paper builds upon InternThinker-32B as the foundation for InternGeometry model. However, the current version lacks sufficient details about InternThinker-32B’s architecture, training setup, and reasoning capabilities. Hope authors can add this to the draft and improve the paper quality. \n4. Also, please clarify what aspects of InternThinker-32B enable effective long-horizon reasoning and symbolic interaction in this work.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work introduces a long-horizon agentic reasoning paradigm for geometry, moving beyond static and propose a model toward interactive proof reasoning.\n2. The complexity curriculum addresses sparse-reward issues and ensures stable reinforcement learning progression.\n3. This work achieves state-of-the-art performance on IMO geometry problems with significantly less training data.", "weaknesses": "1. Does Pass@256 with 200 steps implies ~51K LLM–engine interactions per problem? Does this raise scalability and efficiency concerns. \n2. The paper does not report inference time per problem or single-shot success rate (Pass@1); it will make the paper better if we have this presented so we can assess the model’s true reasoning efficiency.\n3. Would be nice if authors can add a baseline between the pretrained InternThinker-32B and the CBRL-trained InternGeometry, this will make it clear on how much improvement gain from CBRL does the proposed method have. \n4. Although case studies are insightful, a deeper analysis of failure cases (the unsolved problems in Table 2) or reasoning trajectories would provide better understanding. Authors can add this qualitative analysis in the draft, which will make the readability better. \n5. Will be great if the paper can mention inference/training compute requirements to reproduce the results. This will help in reproducibility.\n6. To justify the InterGeometry's generalization capabilities, authors may want to include an analysis of the cross-domain performance on mathematical and scientific reasoning tasks, which can improve the usefulness of the paper.", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)