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Nemotron-Cascade 2: Post-Training LLMs with Cascade RL and Multi-Domain On-Policy Distillation

Zhuolin Yang, Zihan Liu, Yang Chen, Wenliang Dai, Boxin Wang, Sheng-Chieh Lin, Chankyu Lee, Yangyi Chen, Dongfu Jiang, Jiafan He, Renjie Pi, Grace Lam, Nayeon Lee, Alexander Bukharin, Mohammad Shoeybi, Bryan Catanzaro, Wei Ping

arXiv:2603.19220v2cs.CLcs.AIcs.LG

TL;DR

Scaling reinforcement learning across diverse reasoning and agentic tasks remains challenging because varied rewards and complex feedback can destabilize training. Nemotron-Cascade 2 expands Cascade RL with multi-domain on-policy distillation, achieving gold-medal performance in IMO, IOI, and ICPC World Finals while delivering best-in-class results across broad benchmarks.

  • Problem

    Scaling RL across diverse reasoning and agentic tasks requires handling varied reward signals and complex environmental feedback without destabilizing training.

    Source: §1 · p. 4

  • Method

    Nemotron-Cascade 2 combines expanded domain-wise Cascade RL with on-policy distillation from strong intermediate teachers within each domain.

    Source: §1 · p. 4

  • Results

    Nemotron-Cascade 2 achieves gold-medal performance in IMO, IOI, and ICPC World Finals and best-in-class results across mathematics, code reasoning, alignment, and instruction-following benchmarks.

    Source: §1 · p. 4, §2 · pp. 5–6

  • Takeaways & Limitations

    The released model weights, training data, and methodological details support reproduction, analysis, and extension of the Cascade RL training paradigm.

    Source: §1 · p. 4

    Human experts may find it difficult to verify all intermediate derivation steps in solutions using extensive analytic geometry.

    Source: §2 · p. 6

Abstract

from arXiv · show

We introduce Nemotron-Cascade 2, an open 30B MoE model with 3B activated parameters that delivers best-in-class reasoning and strong agentic capabilities. Despite its compact size, its mathematical and coding reasoning performance approaches that of frontier open models. It is the second open-weight LLM, after DeepSeekV3.2-Speciale-671B-A37B, to achieve Gold Medal-level performance in the 2025 International Mathematical Olympiad (IMO), the International Olympiad in Informatics (IOI), and the ICPC World Finals, demonstrating remarkably high intelligence density with 20x fewer parameters. In contrast to Nemotron-Cascade 1, the key technical advancements are as follows. After SFT on a meticulously curated dataset, we substantially expand Cascade RL to cover a much broader spectrum of reasoning and agentic domains. Furthermore, we introduce multi-domain on-policy distillation from the strongest intermediate teacher models for each domain throughout the Cascade RL process, allowing us to efficiently recover benchmark regressions and sustain strong performance gains along the way. We release the collection of model checkpoint and training data.

1. Introduction

Nemotron-Cascade 2 scales Cascade RL with multi-domain on-policy distillation in an open 30B MoE model with 3B activated parameters. It achieves gold-medal results in the 2025 IMO and IOI while delivering best-in-class performance across broad reasoning, alignment, instruction-following, long-context, and agentic benchmarks.

  • Cascade RL addresses multi-domain post-training by sequentially training specialized task domains while tailoring rewards, hyperparameters, curricula, and verification requirements to each domain.Domain-specific stages resist catastrophic forgetting, and task homogeneity yields compute savings through more uniform response lengths and verification wall-clock times.Source: §1 · p. 4
  • Nemotron-Cascade 2 integrates on-policy distillation and grouped multi-domain RL into Cascade RL to recover regressions, scale across environments, and improve training efficiency.On-policy distillation uses the best-performing intermediate teacher models within each domain, while similar response formats and verification costs determine which tasks can be trained jointly.Source: §1 · p. 4

2. Main Results

Nemotron-Cascade 2 achieves best-in-class performance across broad reasoning, alignment, instruction-following, and other benchmarks, despite its compact 30B MoE size. It also attains gold-medal performance in IMO 2025, IOI 2025, and ICPC World Finals 2025, while revealing remaining weaknesses on knowledge-intensive and agentic tasks.

Competition Results

  • Nemotron-Cascade 2 is the second open-weight LLM, after DeepSeek-V3.2-Speciale-671B (Liu et al., 2025), to achieve gold-medal performance in both the IMO and IOI.The result demonstrates high intelligence density relative to the model’s compact activated-parameter footprint.Source: §2 · p. 4, Table 1 · p. 5

Main Benchmark Results

  • Nemotron-Cascade-2-30B-A3B outperforms Qwen3.5-35B-A3B and Nemotron-3-Super-120B-A12B, achieving best-in-class performance across mathematics, code reasoning, general alignment, and instruction following.The main evaluation covers mathematical and coding reasoning, knowledge and STEM, alignment, instruction following, long-context understanding, in-context learning, multilingual capabilities, and agentic tasks.Source: §2 · pp. 4–5, Table 1 · p. 5

Competition Results

  • Gold-medal performance on IMO 2025, IOI 2025, and ICPC World Finals 2025 demonstrates the model’s strong mathematical and coding reasoning despite its 30B MoE size.IMO solutions were reviewed by a human expert, while IOI and ICPC solutions were verified with official online-judge test cases.Source: Table 2 · p. 6, §2 · p. 6

Training Pipeline Comparison

  • Nemotron-Cascade-2-30B-A3B outperforms Nemotron-3-Nano-30B-A3B on nearly all benchmarks, supporting the effectiveness of the Cascade RL plus MOPD training pipeline.Both models were post-trained from the same Nemotron-3-Nano-30B-A3B-Base pretrained model.Source: §2 · p. 6

Limitations

  • The model underperforms Qwen3.5-35B-A3B on knowledge-intensive and agentic benchmarks, highlighting areas for stronger knowledge-intensive pretraining and agentic RL.For IMO 2025 P2, an LLM grader using the ProofBench reference solution and marking schema was used because the model’s extensive analytic-geometry derivations were difficult for a human expert to verify fully.Source: §2 · p. 6

3. Supervised Fine-Tuning

Supervised fine-tuning is the first post-training stage, establishing reasoning, conversation, instruction-following, agentic, and software-engineering capabilities through broad multi-domain data. The pipeline uses specialized chat formatting, curated mathematical and coding data, and targeted agentic datasets for software engineering and terminal tasks.

3. Supervised Fine-Tuning

  • SFT establishes foundational reasoning, conversational, instruction-following, agentic, and software-engineering capabilities across mathematics, coding, science, tool use, and general domains.The data spans multi-turn dialogue, knowledge-intensive question answering, creative writing, role-playing, safety, and instruction following.Source: §3 · p. 6, §3.1.1 · p. 6

3.1.2. Chat Template

  • The chat template removes /think and /no_think tags, uses an empty <think></think> block for non-thinking mode, and wraps tool calls with explicit tool tags.Available tools are listed in the system prompt within <tools> and </tools>.Source: §3.1.2 · pp. 6–7, Figure 1 · p. 7

3.2.1. Math

  • 1.8M tool-calling and 2.6M non-tool competition-math samples are combined with 816K mathematical proof samples covering proof generation and verification.Competition-math responses come from DeepSeek-V3.2, DeepSeek-V3.2-Speciale, and GPT-OSS-120B; proof samples are generated by DeepSeek-V3.2-Speciale.Source: §3.2.1 · p. 7

3.2.2. Code Reasoning

  • The coding dataset uses approximately 165K deduplicated prompts and teacher-trace filtering, yielding 1.9M Python, 1.0M C++14, and 1.3M Python tool-calling reasoning traces.Scientific coding adds 1.1M samples spanning biology, material science, physics, chemistry, and mathematics.Source: §3.2.2 · pp. 7–8

3.2.3. Science

  • Science SFT contributes 2.7M samples, while long-context training adds 234K samples averaging 128K and 29K tokens from Nemotron-3-Nano and ChatQA-2.Science data covers physics, chemistry, and biology, with responses generated by GPT-OSS-120B.Source: §3.2.3 · p. 8, §3.2.4 · p. 8

3.2.5. General Chat

  • General chat and instruction-following data include reasoning-on and reasoning-off responses, multi-turn conversations, and responses generated by GPT-OSS-120B, DeepSeek-V3-0324, and Qwen3 models.Safety data adds 4K refusal-oriented samples, while approximately 700K multi-turn conversation samples are synthesized using two GPT-OSS-120B instances in a role-playing setup.Source: §3.2.5 · p. 8, §3.2.6 · p. 8, §3.2.7 · p. 9

3.2.9. Software Engineering Agent

  • Combining agentic and agentless SWE data improves SWE-bench Verified from Pass@1 48.9 and Pass@4 62.8 to Pass@1 49.9 and Pass@4 65.2.The final SWE SFT mix contains 125K agentic and 389K agentless samples, using non-thinking mode for agentic data and thinking mode for agentless data.Source: §3.2.9 · p. 9

3.2.10. Terminal Agent

  • The terminal-agent dataset contains 490K samples, combining adapted math, code, and SWE tasks with seed-based and skill-based synthetic tasks.DeepSeek-V3.2 generates execution-feedback solution traces in isolated Docker environments using the Terminus 2 agent framework.Source: §3.2.10 · p. 9

4. Cascade RL and Multi-Domain On-Policy Distillation

Nemotron-Cascade 2 uses a sequential Cascade RL pipeline that integrates multi-domain on-policy distillation to expand capabilities while mitigating inter-domain interference and recovering regressions. The stages progress from instruction following and multi-domain RL through MOPD, RLHF, long-context, coding, and software-engineering RL.

4.1. Training Framework

  • The cascade orders IF-RL first, then multi-domain RL, MOPD, RLHF, long-context RL, Code RL, and SWE RL to minimize catastrophic forgetting and consolidate specialized capabilities.The ordering is adapted to changed SFT quality, RL-task complexity, and evolving model behavior; MOPD serves as a stabilization point that recovers benchmark performance.Source: §4.1 · p. 10, §4.1.1 · p. 10, Figure 2 · p. 10

4.2. Instruction-Following Reinforcement Learning (IF-RL)

  • Verifiable IF-RL is the first stage and achieves 83.13% on IFBench, using thinking-mode training to maximize instruction adherence before later stages recover alignment regressions.Dynamic filtering supplies effective gradients, while overlong penalty discourages incomplete generations that exceed the maximum sequence length.Source: §4.2 · p. 11, §4.2.2 · pp. 11–12

4.3. Multi-domain RL

  • Multi-domain RL jointly trains STEM MCQA, agentic tool calling, and structured output because the blended domains improve MMLU-Pro, τ2-Bench, and IF-Bench without observed cross-domain degradation.The mixture is approximately 55% MCQA, 30% tool calling, and 15% structured output, with similar response lengths and verification times.Source: §4.3 · p. 12

4.4. Multi-domain On-Policy Distillation (MOPD)

  • MOPD selects domain-specific teachers from Cascade RL checkpoints sharing the SFT tokenizer, applies dense reverse-KL token advantages with truncated importance weighting, and combines math, RLHF, and multi-domain expertise.The method addresses capability drift from conflicting RL stages by assembling a capability-diverse teacher pool without external model families.Source: §4.4 · pp. 12–13
  • MOPD recovers regressed capabilities and converges faster than GRPO or RLHF, reaching 92.0 on AIME25 within 30 steps and improving ArenaHard Hard Prompt from 71.5 to 85.5 after 52 steps.MOPD uses dense token-level advantages, while GRPO uses sparse sequence-level rewards; RLHF requires 160 steps to reach 80.7 on Hard Prompt.Source: §4.4 · p. 14, Table 3 · p. 14

4.5. Reinforcement Learning from Human Feedback (RLHF)

  • Later stages specialize the policy: RLHF improves creative writing and non-verifiable reasoning while preserving other domains, long-context RL isolates long-context data, Code RL uses difficult verified problems, and SWE RL improves both agentless and agentic code repair.SWE RL uses executable OpenHands environments and deterministic compilation or test rewards; Table 4 reports cross-scaffold gains from agentless training.Source: §4.5 · p. 14, §4.5.2 · p. 14, §4.6 · p. 15

5. International Mathematical Olympiad (IMO)

Nemotron-Cascade-2-30B-A3B demonstrates strong olympiad-level mathematical reasoning through self-improving test-time scaling, solving the first five IMO 2025 problems and achieving 72.9 on IMO-ProofBench. Increasing test-time compute improves Advanced performance, while evaluation checks suggest the reported protocol does not substantially overestimate performance.

5.1. IMO 2025

  • Nemotron-Cascade-2-30B-A3B successfully solves the first five problems in the IMO 2025 set using iterative generation, verification, and self-refinement.Expert review notes that some proofs are overly long, expose intermediate reasoning, and contain minor typographical issues.Source: §5.1 · p. 17

5.2. IMO-ProofBench

  • 72.9 on IMO-ProofBench places Nemotron-Cascade-2-30B-A3B within 8 points of DeepSeek-Math-V2-671B-A37B despite using 10× fewer active parameters.The model exceeds QED-Nano-4B’s 54.0 by 18 points, though the comparison is not directly comparable because the judge model differs.Source: §5.2 · p. 17, Table 5 · p. 17
  • Increasing test-time compute raises IMO-ProofBench Advanced from 40.7 at round 1 to 53.4 at round 5, narrowing the gap to DeepSeek-Math-V2 under the same grader.Re-evaluated DeepSeek-Math-V2 proofs score within 4 points of the reported human rating, suggesting the protocol does not substantially overestimate performance.Source: §5.2 · p. 17

6. Competitive Coding · Appendix

Nemotron-Cascade 2 combines test-time scaling and large-scale solution generation to achieve gold-medal-level results at IOI 2025 and the ICPC World Finals 2025. On competitive coding benchmarks, it attains strong Pass@1 accuracy and ELO ratings against much larger models, with further gains from Tool-Integrated Reasoning.

6.1. IOI 2025 and ICPC World Finals 2025

  • At IOI 2025, the method uses a multi-round generate-select-submit pipeline with up to 50 rounds and 40 candidate solutions per round.Candidates incorporate submission history with official judge verdicts and shared insights from highly scored or solved subtasks.Source: §6.1 · p. 17

6.2. Competitive Coding Benchmark Results

  • Nemotron-Cascade-2-30B-A3B achieves strong Pass@1 accuracy and ELO ratings against frontier open-source models exceeding 100B total parameters, as reported in Table 6.The evaluation covers LiveCodeBench v6, LiveCodeBench Pro 25Q1/25Q2, and simulated Codeforces participation.Source: Table 6 · p. 18, §6.2 · p. 19

6.1. IOI 2025 and ICPC World Finals 2025

  • At IOI 2025, the pipeline earns a gold-medal score of 439.28 within at most 2,000 generations and reaches 507.66 within 5,000 generations.It fully solves Problems 3 and 4; Problem 2 exceeds 86 points within five rounds, or at most 200 generations.Source: §6.1 · p. 18
  • At the ICPC World Finals 2025, Nemotron-Cascade 2 solves 10 of 12 problems and achieves the #4 Gold medal placement.Eight problems, excluding Problems A and I, are solved within only 100 submissions after generating up to 1,000 solutions per problem.Source: §6.1 · p. 18

6.2. Competitive Coding Benchmark Results

  • With Tool-Integrated Reasoning, Nemotron-Cascade-2-30B-A3B matches stronger open-source models exceeding 300B total parameters, especially on hard problems.The model may call a stateful Python executor for up to 100 calls, while some larger baselines lack TIR support or perform poorly with Python TIR.Source: §6.2 · p. 19
  • Nemotron-Cascade-2-30B-A3B achieves above 0% on the LiveCodeBench Pro hard split within 8 attempts, including problems described as extremely difficult for humans.Results use avg@8 under a 128K-token thinking budget, temperature 1.0, and top_p 0.95.Source: §6.2 · p. 19

A. Benchmarks and Evaluation Setups · A.1. Math · A.1.1. Non-proof Math

The evaluation covers non-proof mathematics benchmarks spanning recent contests and expert-altered Olympiad problems, using task-specific generation budgets, tool settings, and correctness evaluation. Baseline results come from official reports or recommended-setting evaluations when official numbers are unavailable.

A.1.1. Non-proof Math

  • The non-proof math suite includes AIME 2025, AIME 2026, HMMT Feb 2025, and IMO-AnswerBench, covering 30, 30, 30, and 400 problems, respectively.IMO-AnswerBench contains expert-altered problems with verifiable answers selected from past Olympiad competitions.Source: §A.1.1 · p. 20
  • Nemotron-Cascade-2-30B-A3 uses a 131K-token thinking budget, temperature 1.0, and top-p 1.0 on AIME 2025, AIME 2026, and HMMT Feb 2025.These settings apply to the listed non-proof math evaluations.Source: §A.1.1 · p. 20
  • With tools enabled, the model can call a stateful Python executor for up to 100 tool calls while retaining the 131K-token maximum response length.Tool use is activated by appending a system-prompt postfix.Source: §A.1.1 · p. 20
  • IMO-AnswerBench uses a 256K-token budget, GPT-OSS-120B (Agarwal et al., 2025) for LLM-Judge scoring, and the AnswerAutoGrader prompt (Luong et al., 2025) for answer correctness.The larger budget reflects the greater difficulty of its questions and avoids relying on a rule-based verifier for complicated short answers.Source: §A.1.1 · p. 20
  • Reported performance uses avg@64 for AIME and HMMT and avg@16 for IMO-AnswerBench, following Liu et al. (2024, 2026).The averaging conventions differ by benchmark group.Source: §A.1.1 · p. 20
  • Baseline comparisons use official reported numbers or evaluations under recommended settings when official numbers are unavailable.This establishes the procedure for obtaining baseline results across the benchmark comparisons.Source: §A.1.1 · p. 20

A.1.2. Math Proof … A.5. Long Context and Context Learning

The evaluation spans mathematical proof, code, knowledge and STEM, alignment, and long-context/context-learning tasks, using specialized benchmarks and task-specific inference and scoring protocols. Notably, a boxed-answer prompt improves HLE accuracy by 6–7 points, primarily on mathematics.

A.1.2. Math Proof

  • Math-proof evaluation covers IMO 2025 and IMO-ProofBench, with Nemotron-Cascade-2-30B-A3 using generate-verify-refine test-time scaling for up to eight rounds or until the average proof score reaches 0.99999.The full protocol uses 128 proof generations, 64 verifications per proof, top-32 refinement selection, four refined proofs, and a 256K-token maximum generation length; reduced budgets are used for specified IMO-ProofBench subsets.Source: §A.1.2 · p. 20
  • A.1.2. Math Proof adopts a zero-gated aggregation rule for IMO-ProofBench grading: any zero judge score yields a final score of zero; otherwise, scores are averaged.For DeepSeek-Math-V2 on the Advanced split, this changes the reported score from 73.8 to 57.7 and reduces the discrepancy from the human rating from 11.9 points to 4.2 points.Source: §A.1.2 · pp. 20–21

A.2. Code Reasoning

  • Code reasoning evaluation uses LiveCodeBench, LiveCodeBench Pro, IOI, ICPC World Finals, and SciCode to cover competitive programming and realistic scientific research tasks.LiveCodeBench results use pass@1 averaged over eight generations, while tool-enabled evaluation permits up to 100 stateful Python-executor calls.Source: §A.2 · p. 21

A.3. Knowledge and STEM

  • A.3. Knowledge and STEM evaluates MMLU-Redux, MMLU-Pro, GPQA-Diamond, and HLE with thinking-mode accuracy protocols spanning exact match, pass@1, and expert-level academic reasoning.Nemotron-Cascade-2-30B-A3 uses temperature 1.0, top-p 0.95, and a 128K-token thinking budget.Source: §A.3 · p. 21
  • 6–7 points: appending a boxed-answer instruction improves HLE accuracy, primarily on the math subset, by aligning outputs with the mathematical SFT format.GPT-OSS-120B judges answer extraction and correctness verification under this prompt.Source: §A.3 · p. 22

A.4. Alignment and Instruction-Following

  • Alignment and instruction-following evaluation covers ArenaHard 2.0, IFBench, and Scale AI Multi-Challenge, measuring preference alignment, verifiable instruction constraints, and multi-turn context management.Nemotron-Cascade models use non-thinking mode for IFEval and thinking mode for IFBench and ArenaHard, with temperature 0.6, top-p 0.95, and a 32K-token maximum response length.Source: §A.4 · p. 22

A.5. Long Context and Context Learning

  • Long-context evaluation combines AA-LCR, LongBench v2, and NIAH@1M to test cross-document reasoning, long-context task understanding, and retrieval from 1M-token contexts.AA-LCR uses approximately 100K-token document sets and avg@16, LongBench v2 uses avg@4, and NIAH@1M evaluates four retrieval variants in reasoning-off mode with avg@1.Source: §A.5 · pp. 22–23
  • CL-Bench evaluates context learning through 1,899 expert-developed samples spanning 500 complex contexts and 31,607 verification rubrics, requiring models to acquire knowledge from provided context.Models are evaluated in thinking mode using single-generation pass@1 accuracy.Source: §A.5 · p. 23

A.6. Agentic Tasks · A.7. Multilingual

The evaluation covers agentic tasks spanning tool use, software engineering, customer service, and terminal workflows, alongside multilingual reasoning and translation across selected languages and language pairs. Protocols specify task-specific scaffolds, thought-retention policies, sampling counts, and evaluation metrics.

A.6. Agentic Tasks

  • Agentic evaluation spans BFCL v4, SWE-bench Verified, τ2-Bench, and Terminal Bench 2.0, covering tool use, coding, customer service, and terminal-based workflows.BFCL v4 evaluates web search, memory operations, and function invocation; Terminal Bench 2.0 emphasizes end-to-end operations such as model training, system configuration, and debugging.Source: §A.6 · p. 23
  • BFCL v4 reports pass@1 accuracy from one thinking-mode generation, while SWE-bench Verified reports pass@1 averaged over four non-thinking-mode generations.The BFCL protocol covers Agentic, multi-turn, live, and non-live categories; SWE-bench Verified contains 500 human-verified samples.Source: §A.6 · p. 23
  • τ2-Bench evaluates airline, retail, and telecom subsets with avg@16, avg@8, and avg@8 sampling respectively, while Terminal Bench 2.0 reports avg@5 task success using Terminus-2.The τ2-Bench subsets contain 50 airline, 114 retail, and 114 telecom examples; Terminal Bench 2.0 contains 89 human-validated tasks.Source: §A.6 · p. 23
  • SWE-bench Verified uses OpenHands with complete trajectory retention, a 256K-token context, and up to 200 turns, matching the execution-based SWE-RL training environment.The setup preserves prior file views, search results, commands, model outputs, and intermediate patches to support long-horizon debugging and reduce train–test mismatch.Source: §A.6 · p. 23
  • τ2-Bench retains reasoning after the latest user turn; removing all prior reasoning reduces scores by 3–5 points relative to this policy.The latest-turn policy matches the SFT construction and is reported finally because full thought retention gives similar accuracy with substantially longer contexts.Source: §A.6 · pp. 23–24

A.7. Multilingual

  • Multilingual evaluation uses MMLU-ProX across English, German, Spanish, French, Italian, and Japanese, reporting thinking-mode pass@1 accuracy from one generation.MMLU-ProX expands MMLU-Pro to 29 languages, with six selected for evaluation following Blakeman et al. (2025).Source: §A.7 · p. 24
  • WMT24++ evaluates English-to-German, Spanish, French, Italian, and Japanese translation with XCOMET-XXL, using thinking-mode pass@1 accuracy from one generation.WMT24++ extends WMT24 to 55 languages, while the evaluation follows the five translation pairs selected by Blakeman et al. (2025).Source: §A.7 · p. 24

B. Training Hyperparameters · C. Prompt Templates

This section specifies the training hyperparameters used across SFT and Cascade RL stages, then provides prompt templates for IOI 2025 test-time scaling and HLE answer judging.

B. Training Hyperparameters

C.1. Prompt Templates for Test-Time Scaling on IOI 2025

  • The IOI 2025 test-time-scaling prompt requests Python code in a specified format to solve the provided problem statement.Source: §C.1 · p. 26
  • The IOI prompt supplies accepted solutions under different constraints and prior incorrect submissions with official verdicts and scores as references for improvement.Source: §C.1 · p. 26

C.2. HLE Judge Prompt

  • The HLE judge prompt evaluates a response against a precise correct answer by extracting the final answer, explaining only meaningful differences, and returning a correctness judgment.Source: §C.2 · p. 27
  • The HLE judge also outputs a confidence score from 0% to 100%, using 100 when the response provides no confidence score.Source: §C.2 · p. 27

D. ELO Rating Analysis … 1. Constructions showing that 𝑘= 0, 1, 3 are attainable

The paper evaluates Nemotron-Cascade-2-30B-A3B on 40 recent Codeforces contests and presents an IMO geometry problem with a proof that constructs attainable values k=0, 1, and 3.

D. ELO Rating Analysis

  • The model is evaluated on 40 Div.1 and Div.2 Codeforces contests using up to N=8 submissions per problem, with and without python-tool use.Responses use temperature 1.0, top-p 0.95, and a maximum token budget of 128K; results are reported in Tables 11 and 12.Source: D. ELO Rating Analysis · p. 27, Table 11 · p. 28, Table 12 · p. 29
  • The model achieves high rankings on some difficult Div.1 rounds while maintaining stable performance on easy-medium problems, but struggles with constructive, interactive, and hypothesis-driven tasks.Source: D. ELO Rating Analysis · p. 27

E. IMO 2025 Model Solutions

  • The IMO problem asks which nonnegative k permit n distinct lines covering all points (a,b) with a+b≤n+1, with exactly k lines avoiding three specified directions.A sunny line is neither horizontal, vertical, nor parallel to x+y=0.Source: E. IMO 2025 Model Solutions · p. 30, Model Proof · p. 30

Model Proof

  • The proof reformulates the task as covering S_n with n distinct lines, counting as sunny those whose slopes are neither 0, ∞, nor −1.Source: Model Proof · p. 30

1. Constructions showing that k= 0, 1, 3 are attainable

  • Using n vertical lines, the proof establishes that k=0 is attainable for every n≥3; replacing one vertical line with a suitable sunny line establishes k=1.The vertical lines cover points by their first coordinate, while the sunny replacement covers the remaining point with a=n.Source: §1 · p. 30
  • For n=3, three sunny lines directly cover all six points of S_3, proving that k=3 is attainable in the base case.Source: §1 · p. 30
  • For n≥4, n−3 vertical lines cover the low-a points and three sunny lines cover the remaining six-point set W, yielding exactly n lines and k=3.All three additional lines have slopes different from 0, ∞, and −1, and together cover the whole set S_n.Source: §1 · p. 31

2. Impossibility: no other value of 𝑘can occur … 7. Circumcenter 𝑂of △𝐵𝐸𝐹and the tangency condition

The paper proves that coverings of S_n by n lines admit exactly k∈{0,1,3} sunny lines, using boundary counting and induction, and develops a coordinate proof framework for the circle tangency problem.

2.2. A counting argument for n≥4

  • For n≥4, counting the 3n−3 boundary points forces every covering to contain at least one special side.The special sides are x=1, y=1, and x+y=n+1; any other line contains at most two boundary points.Source: §2.2 · p. 31, §2 · p. 31

2.3. Reducing the problem to n−1

  • Removing any present special side and translating when necessary produces a covering of S_{n−1} by n−1 lines with the same k.The three cases remove x=1, y=1, or x+y=n+1; the first two use translations, while the third restricts directly to S_{n−1}.Source: §2.3 · p. 32

2.4. Base case n=3 and induction

  • The base case n=3 excludes k=2, and induction then yields the complete set {0,1,3}.For k=2, the two sunny lines cannot cover the required remaining points without contradicting the classification of sunny collinear pairs.Source: §2.4 · pp. 32–33

2. Impossibility: no other value of k can occur

  • For every n≥3, a covering of S_n by n lines has exactly the possible sunny-line counts k∈{0,1,3}.The induction combines the base case with a reduction from n to n−1, together with constructions from §1.Source: §2.4 · pp. 32–33

Human Expert Comment

  • The geometry problem asks whether the line through H parallel to AP is tangent to the circumcircle of triangle BEF.The configuration uses two intersecting circles Ω and Γ, their centers M and N, intersections A and B, and points C,D,P,E,F,H defined in the statement.Source: Human Expert Comment · p. 34

Model Proof

  • The coordinate model places M=(0,0) and N=(d,0), represents Ω and Γ by their radii, and identifies C and D on the x-axis.The circles have radii 0<r<R, intersect at A and B, and satisfy V<d<U for distinct intersections.Source: Model Proof · p. 34

2. Second intersections E, F and midpoint S of EF; 3. Unit vectors along and perpendicular to AP

  • The proof parameterizes the second intersections E and F along AP, identifies their midpoint S, and writes EF using unit directions parallel and perpendicular to AP.With e along AP and n perpendicular to it, E=S−ℓe and F=S+ℓe.Source: §2 · pp. 34–35, §3 · p. 35

1. Circumcenter P of △ACD; 4. Orthocenter H of △PMN; 6. Expression for |S−B|²; 7. Circumcenter O of △BEF and the tangency condition

  • The final coordinate setup places the circumcenter O on the perpendicular bisector of EF and reduces tangency to an equality involving λ, β, γ, ℓ, and |S−B|².The orthocenter H is computed from triangle PMN, while the circumcenter condition gives |S−B|²+2λβ=ℓ² before comparing the distance from O to the line through H parallel to AP.Source: §4 · p. 35, §6 · p. 36, §7 · p. 37

8. Verification of (3) … Model Proof

This section verifies the preceding results through coordinate calculations and a complete assessment of the tangency proof, and presents a model proof identifying exactly which initial values generate an infinite sequence.

8. Verification of (3)

  • The final algebraic identity makes (3′) hold, so (3) and (2) follow and the line through H parallel to AP is tangent to the circumcircle of △BEF.The reduction substitutes L = U^2 − d^2 and K = d^2 − V^2; both sides simplify identically.Source: §8 · p. 38
  • The assessment finds the submitted coordinate proof mathematically sound, complete, and free of a fatal logical gap, despite its calculation-heavy presentation.It treats the compressed algebraic steps as routine consequences of established formulas and distinguishes the coordinate route from the reference solution’s synthetic route.Source: §8 · pp. 39, 41, 42
  • The proof earns all seven checkpoint points by establishing the coordinate setup, replacing the official constructions with direct computations, and verifying the final tangency criterion.The assessment awards 1/1, 2/2, 2/2, and 2/2 across the four checkpoints, with no deduction or score cap.Source: §8 · pp. 41–42

Model Proof

  • The Model Proof states that the smallest constant satisfying the bonza-function bound is c = 4.This passage supplies the claimed target constant but does not include the proof details in the provided text.Source: Model Proof · p. 43

3. All terms must be multiples of 6

  • Every term of an infinite sequence must be a multiple of 6; otherwise the recurrence produces an impossible infinite strictly decreasing sequence of positive integers.Odd terms remain odd and decrease, while even terms not divisible by 3 yield smaller terms that remain outside the multiples of 6.Source: §3 · p. 48

5. Characterising admissible M_1 = a_1/6

  • Writing a_n = 6M_n, the recurrence fixes M when M is odd and 5 ∤ M, while admissible even values must be divisible by 12 and then map to 13 · (M/12).The remaining cases produce a next term that is not a multiple of 6 and therefore cannot occur in an infinite sequence.Source: §5 · p. 49, §4 · p. 49

Sufficiency

  • If M_1 = 12^k · d with d odd and 5 ∤ d, repeated even steps reach an odd nonmultiple of 5, after which the sequence becomes constant.The induction tracks the transformation 12^(k−i) · 13^i · d until the fixed point is reached, and all terms remain admissible.Source: Sufficiency · p. 50

Necessity

  • Conversely, if d is even or divisible by 5, the dynamics eventually produce either an even M not divisible by 12 or an odd M divisible by 5, contradicting infinite admissibility.The former yields a next term that is not a multiple of 6, while the latter yields an odd term and hence an impossible decreasing sequence.Source: Necessity · pp. 50–51

6. Final answer

  • a_1 is admissible exactly when a_1 = 2^(2k+1) · 3^(k+1) · d for some k ≥ 0, with d odd and 5 ∤ d.Equivalently, a_1 = 6 · 12^k · d, where d is odd and not divisible by 5.Source: §6 · p. 51

Notation and basic facts · Final answer · Human Expert Comment

The proof establishes a threshold game outcome: Bazza wins for λ<√2/2, the game can be drawn at λ=√2/2, and Alice wins for λ>√2/2. The human expert notes that parts of the proof read like an unfinished thinking process, especially around whether an intermediate claim is necessary.

Notation and basic facts

  • Bazza’s maximal strategy restores the even-turn sum of squares to its largest permitted value, Q_2k=2k, whenever Alice does not win immediately.This invariant supports both Bazza’s win below the threshold and his prevention of Alice’s win at the threshold.Source: Notation and basic facts · pp. 52–54
  • For λ<√2/2, Bazza’s maximal strategy is winning because the slack eventually becomes negative while Alice cannot win earlier.The slack decreases enough under this strategy that some finite turn leaves Alice unable to make a legal move.Source: Notation and basic facts · p. 53

Final answer

Notation and basic facts

  • For λ>√2/2, Alice’s slack-based strategy remains legal because she can play 0 whenever she is not yet ready to force a win.The proof verifies that the slack stays nonnegative and eventually becomes large enough for a winning move.Source: Notation and basic facts · p. 55
  • For λ>√2/2, Alice has a winning strategy that guarantees a win in finite time.She usually plays 0, then chooses a sufficiently large move once the slack exceeds the threshold needed to make Bazza’s next move impossible.Source: Notation and basic facts · p. 55

Human Expert Comment

  • The human expert comments that portions of the proof read like a thinking draft, with the argument seemingly revisiting whether an unproved claim is actually necessary.The comment specifically describes the proof as appearing finished before it resumes reconsidering the issue.Source: Human Expert Comment · p. 56

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