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Achieving Gold-Medal-Level Olympiad Reasoning via Simple and Unified Scaling

Yafu Li, Runzhe Zhan, Haoran Zhang, Shunkai Zhang, Yizhuo Li, Zhilin Wang, Jiacheng Chen, Futing Wang, Xuyang Hu, Yuchen Fan, Bangjie Xu, Yucheng Su, Xinmiao Han, Chenxi Li, Haodi Lei, Yufeng Zhao, Zejin Lin, Qianjia Cheng, Tong Zhu, Xiaoye Qu, Ganqu Cui, Peng Ye, Yun Luo, Zhouchen Lin, Yu Qiao, Bowen Zhou, Ning Ding, Yu Cheng

arXiv:2605.13301v1cs.AIcs.CL

TL;DR

The paper asks whether a compact, post-trained reasoning model can reach olympiad-level mathematical and scientific performance through one unified recipe. It combines behavior-shaping SFT, two-stage RL, and test-time scaling, producing gold-medal-level results while sustaining reasoning trajectories beyond 100K tokens.

  • Problem

    The paper examines whether a compact reasoning backbone can reach olympiad-level performance through a domain-unified pipeline for mathematical and scientific problems.

  • Method

    The recipe combines reverse-perplexity curriculum SFT, two-stage RL with verifiable and proof-level rewards, and test-time self-verification and refinement.

  • Results

    SU-01 achieves gold-medal-level performance across mathematical and physical olympiad competitions, reaching 35 points on both IMO 2025 and USAMO 2026 with test-time scaling.

  • Takeaways & Limitations

    The results support specializing a broadly capable compact backbone toward expert-level proof reasoning while retaining scientific transfer beyond olympiad benchmarks.

  • Takeaways & Limitations

    The base model still benefits from additional search and self-correction on harder, brittle proof attempts that it fails to solve in a single pass.

Abstract

from arXiv · show

Recent progress in reasoning models has substantially advanced long-horizon mathematical and scientific problem solving, with several systems now reaching gold-medal-level performance on International Mathematical Olympiad (IMO) and International Physics Olympiad (IPhO) problems. In this paper, we introduce a simple and unified recipe for converting a post-trained reasoning backbone into a rigorous olympiad-level solver. The recipe first uses a reverse-perplexity curriculum for SFT to instill rigorous proof-search and self-checking behaviors, then scales these behaviors through a two-stage RL pipeline that progresses from RL with verifiable rewards to more delicate proof-level RL, and finally boosts solving performance with test-time scaling. Applying this recipe, we train a 30B-A3B backbone with SFT on around 340K sub-8K-token trajectories followed by 200 RL steps. The resulting model, SU-01, supports stable reasoning on difficult problems with trajectories exceeding 100K tokens, while achieving gold-medal-level performance on mathematical and physical olympiad competitions, including IMO 2025/USAMO 2026 and IPhO 2024/2025. It also demonstrates strong generalization of scientific reasoning to domains beyond mathematics and physics.

1 Introduction

The paper presents a unified pipeline that uses SFT to reshape proof-search behavior, two-stage RL to scale and refine it, and test-time scaling for difficult olympiad problems. Applied to a 30B-A3B backbone, SU-01 achieves strong proof-oriented and official-competition results while sustaining reasoning trajectories beyond 100K tokens.

  • SFT: 338K trajectories shorter than 8K tokens train SFT to instill disciplined proof-search, self-verification, and self-refinement behaviors.The curated mixture spans mathematical, scientific, coding, and instruction-following sources.
  • Reinforcement learning: Two-stage RL first scales SFT-induced behaviors with reliable binary rewards, then targets proof quality using proof-level generative scoring and self-refinement.Coarse RL uses verifiable prompts and efficient outcome checking; refined RL shifts beyond answer correctness toward complete-proof quality.
  • Evaluation: 57.6% on IMO-ProofBench with direct generation and 70.2% with TTS show SU-01’s strong proof-oriented performance, approaching competitive commercial systems.On answer-verifiable benchmarks, SU-01 nearly matches the similar-size Qwen3.6-35B-A3B baseline across the listed tasks.
  • Official competitions: 35 points on both IMO 2025 and USAMO 2026 with TTS meet the IMO 2025 gold line and exceed the USAMO 2026 gold line by 10 points.Direct SU-01 exceeds the IPhO gold lines for 2024 and 2025 and clears the bronze-medal lines on IMO 2025 and USAMO 2026.
  • Unified pipeline: The recipe combines rigorous long-form SFT, coarse and refined RL, and test-time verification and refinement for olympiad-level problem solving.The pipeline applies the same reasoning-centric approach across mathematical and scientific problems.
  • Inference-time scaling: Beyond 100K tokens, SU-01 sustains reasoning trajectories that condition on drafts and error analyses while repeatedly verifying and repairing candidate proofs.These TTS traces illustrate how the model’s reasoning capability emerges at inference time.

2 Instilling Rigorous Reasoning via SFT

The SFT stage reshapes a capable post-trained reasoning backbone toward rigorous proof-search, self-verification, and self-refinement. It uses filtered, diverse reasoning data and a reverse-perplexity curriculum while carefully controlling optimization and training duration to preserve competence.

  • Motivation: SFT specializes a post-trained backbone toward rigorous proof-search rather than rebuilding instruction-following, problem-solving, and scientific capabilities from scratch.The backbone is selected for competitive mathematical and physical scientific reasoning, but its solutions are not always organized around rigorous proof-search patterns.
  • Data Curation: The training mixture spans olympiad mathematics, scientific reasoning, instruction-following, and coding sources to improve generalization.Mathematical prompts include Evan Chen’s materials, the Shuzhimi Forum, AoPS, competition books, and DeepMath problems with difficulty at least 6; scientific prompts come from NaturalReasoning.
  • Data Curation: Generated trajectories are filtered for contamination and noise, with outputs longer than 8,192 tokens removed to keep the supervised signal usable and stable.DeepSeek-V3.2-Speciale generates the long-form reasoning trajectories before filtering and length truncation.
  • Behavioral Training: Self-verification and self-refinement traces teach the model to check whether proofs are justified and improve solutions using identified issues.These traces are generated for mathematical solutions in addition to direct solution trajectories.
  • Optimization Stability: For trajectories capped at 8,192 tokens, four epochs are usually sufficient to recover most capability after the initial behavioral shift when the data mixture and learning rate are controlled.Validation truncation rate is also treated as an operational indicator of SFT sufficiency.
  • Optimization Stability: Reverse-perplexity SFT sorts examples from high PPL to low PPL within each epoch, presenting mismatched proof-search patterns before more familiar examples.The ordering is intended to support behavioral adaptation before consolidation.

3 Boosting Reasoning Capability with RL

The RL stage uses a two-level pipeline: coarse RL strengthens answer-seeking and search with verifiable rewards, while refined RL targets complete, auditable proofs through process-level rewards, self-refinement, and replay. Training uses separate prompt pools and progressively moves from reliable answer verification to proof-quality judgment.

  • Coarse RL: 8,967 verifiable prompts train coarse RL with RLVR and GSPO to improve search, coverage, and direct solving performance.GSPO uses group-level sequence optimization aligned with outcome rewards.
  • Coarse RL: Coarse rewards prioritize canonicalized final-answer matching, Math-Verify8 checks, and generative verification for unresolved cases.This ordering keeps automatic checks high-precision while recovering correct equivalent answers that rule-based parsers miss.
  • Refined RL: Refined RL replaces answer correctness with proof-quality evaluation, using a binary generative proof reward on complete solutions or proofs.DeepSeekMath-V2 supplies the proof reward for mathematical prompts, except physics prompts.
  • Self-refinement: When average proof reward falls below τref = 0.5, failed responses become repair tasks that critique gaps and produce corrected proofs.Self-refinement mirrors the test-time pattern of proposing, inspecting, locating gaps, and correcting a solution.
  • Experience replay: Experience replay preserves rare valid proof trajectories, selects the lowest-entropy stored trajectory, mixes replay at ρ = 0.25, and removes it after reliable on-policy reproduction.Replay uses a GSPO-style update and is targeted rather than exhaustive.

4 Achieving Gold-Medal-Level Reasoning via Test-time Scaling

Test-time scaling addresses olympiad problems that require extensive search, revision, and rigorous proof validation beyond a single generation. SU-01 therefore uses repeated solve–verify–refine computation to repair incomplete or logically flawed arguments.

  • Motivation: Olympiad solutions require more than correct conclusions because hidden gaps or logical fallacies can invalidate an otherwise promising proof.The paper motivates test-time scaling by emphasizing complete and rigorous proofs rather than answer-only correctness.
  • Motivation: Repeated solve–verify–refine stages allocate additional inference computation for exploration, lemma checking, counterexample search, and exposition repair.A single generation has finite context and thinking budgets, which may be exhausted before a proof is fully closed.
  • Method: The TTS procedure generates an initial rigor-focused solution, refines weak steps into a complete proof, and then verifies the refined candidate.This loop follows a verification-and-refinement paradigm and prioritizes proof rigor over merely reaching a final answer.

5 Experimental Results

The experimental evaluation spans answer-verifiable reasoning, proof-oriented and scientific reasoning, and official olympiad problems. SU-01 shows strong performance across these settings, with test-time scaling substantially improving proof and competition results.

  • Evaluation setup: Evaluation covers three families: answer-verifiable tasks, non-verifiable or proof-oriented tasks, and official olympiad competition problems.Answer-verifiable benchmarks include AnswerBench, AMO-Bench, AIME 2025/2026, and FrontierScience-Olympiad; proof-oriented tasks include ProofBench and FrontierScience-Research.
  • Answer-verifiable reasoning: 77.3%: SU-01’s average across five answer-verifiable benchmarks, nearly matching Qwen3.6-35B-A3B at 77.4%.The paper attributes this result to a simpler unified post-training recipe and substantially lower training cost.
  • Proof-oriented reasoning: 70.2%: SU-01’s test-time-scaled IMO-ProofBench score, up from 57.6% in direct generation, including 91.0% on basic and 49.5% on advanced problems.This places the 30B-A3B model close to much larger frontier systems.
  • Scientific reasoning: 11.7%: SU-01 achieves the best similar-size overall score on the harder FrontierScience-Research benchmark spanning physics, chemistry, and biology.It leads the similar-size block on Physics, ties for best Chemistry, and ranks second on Biology.
  • Physics olympiad: 25.3 and 21.7 points: test-time scaling raises SU-01’s IPhO 2024/2025 scores above the gold lines of 20.8 and 19.7 points.Without test-time scaling, SU-01 already scores 23.5 and 20.3 points, respectively.
  • Mathematical olympiad: 35 points: test-time scaling brings SU-01 to the IMO 2025 gold line and 10 points above the USAMO 2026 gold line.Direct generation scores 21 and 15 points, clearing the bronze lines; across both competitions, test-time scaling yields full-credit solutions to ten of twelve problems.

6 Analysis and Discussion

The analysis separates answer recovery from rigorous proof completion, showing that staged training and proof-level supervision primarily improve harder non-verifiable problems. It also finds that reverse-perplexity ordering stabilizes SFT and that test-time refinement sustains complex proof repair.

  • Reasoning Progress: ProofBench reveals a gap between answer recovery and olympiad-style rigor, especially on the Advanced split, where the starting model scores much lower than on AnswerBench.AnswerBench evaluates correct final answers, whereas ProofBench grades complete solutions for rigor, justification, and proof completion.
  • Reasoning Progress: SFT changes AnswerBench from 69.2 to 59.8 while raising ProofBench-Basic from 33.8 to 57.6 and ProofBench-Advanced from 6.2 to 14.8.The shift is attributed to longer proof-search, self-checking, and refinement behavior.
  • Reasoning Progress: RL lifts AnswerBench to 77.2, ProofBench-Basic to 76.7, and ProofBench-Advanced to 25.2, scaling the rigorous reasoning behavior introduced by SFT.The coarse RL stage uses verifiable rewards to recover and improve direct solving ability.
  • Reasoning Progress: SU-01 improves ProofBench-Advanced from 25.2 to 38.1 while keeping AnswerBench at 77.5 and ProofBench-Basic at 77.1.Refined RL combines proof-level generative rewards, self-refinement prompts, and experience replay for harder non-verifiable proofs.
  • Test-Time Scaling: Long refinement traces show the model can use candidate proofs and structured critiques to localize flaws, preserve useful arguments, and synthesize corrected proofs.This indicates inference compute supports complex refinement contexts rather than only long first-pass generation.

7 Related Work

Prior work has developed post-training methods and specialized systems for reliable reasoning and olympiad mathematics. This paper presents a unified post-training recipe that gives a 30B-A3B model rigorous proof behavior and olympiad-level performance.

  • Post-training for Large Reasoning Models: Post-training has become a main mechanism for converting pretrained language models into reliable reasoning systems.Prior approaches include self-improvement from successful attempts, instruction tuning, preference optimization, and reinforcement learning.
  • Toward Olympiad-Level Reasoning: Specialized symbolic and neuro-symbolic systems have addressed olympiad reasoning, with AlphaProof/AlphaGeometry 2 reaching silver-medal-level IMO performance.AlphaGeometry combines a neural language model with symbolic deduction for geometry problems.
  • Toward Olympiad-Level Reasoning: The paper’s contribution is a simple, unified recipe that enables a 30B-A3B model to develop rigorous proof behavior and reach olympiad-level performance.The approach uses self-verification, refinement, and test-time scaling.

8 Conclusion · Appendix

The report presents a simple, unified recipe for strengthening a compact post-trained reasoning model into a mathematical and scientific reasoner. SU-01 combines curriculum SFT, two forms of RL, replay and refinement, and test-time scaling.

  • 8 Conclusion: SU-01 starts from a broadly capable 30B-A3B post-trained reasoning backbone.The backbone is the starting point for the proposed recipe.
  • 8 Conclusion: Reverse-perplexity curriculum SFT is used to strengthen the backbone’s reasoning behavior.The report identifies reverse-perplexity curriculum SFT as a core stage.
  • 8 Conclusion: Efficient coarse RL uses outcome verification to further train the model.Outcome verification is the reward mechanism specified for the coarse RL stage.
  • 8 Conclusion: Refined RL applies proof-level rewards to train more rigorous reasoning.Proof-level rewards distinguish the refined RL stage from coarse outcome verification.
  • 8 Conclusion: Self-refinement and experience replay are included as additional training components.Both mechanisms are listed among the recipe’s combined stages.
  • 8 Conclusion: Test-time scaling uses self-verification and refinement to improve reasoning at inference.The report explicitly associates test-time scaling with self-verification and refinement.

A Implementation and Evaluation Details · B Problem-Solving Prompt

The appendix specifies the implementation and evaluation settings needed to interpret and reproduce SU-01’s training, inference, reward-serving, and grading procedures. It also defines the fixed prompt used across RL training and inference, requiring complete, rigorous solutions with LaTeX formatting and boxed final answers where applicable.

  • A Implementation and Evaluation Details: The appendix collects settings for reproducing the reported training, inference, reward-model serving, and evaluation procedures.It covers modeling, optimization, decoding, serving, and grading settings.
  • A Implementation and Evaluation Details: The reported settings are extracted from launch scripts and evaluation protocols.Infrastructure-only commands are omitted when they do not affect objectives, data semantics, inference, rewards, or evaluation criteria.
  • B Problem-Solving Prompt: Unless otherwise specified, SU-01 uses one fixed problem-solving prompt across all RL training and inference stages.The prompt is explicitly presented as the default for both training and inference.
  • B Problem-Solving Prompt: The prompt asks SU-01 to solve olympiad problems while showing complete reasoning and proof.This establishes the required response format before the listed formatting instructions.
  • B Problem-Solving Prompt: Solutions must use LaTeX to represent variables and formulas during the solution process and in the results.The formatting requirement applies to both variables and formulas.
  • B Problem-Solving Prompt: Final answers must be placed in \boxed{} when a problem asks for specific values, and proofs must be clear and rigorous.The prompt distinguishes formatting for value-finding tasks from standards for proof tasks.

C SFT Training Details

The SFT stage initializes from P1-30B and trains on 8 GPUs using a reverse-perplexity-ordered mixture with curriculum order preserved. It runs for four epochs with specified optimization settings, including Adam, cosine learning-rate decay, warmup, weight decay, and momentum parameters.

  • Implementation and curriculum: SFT initializes from P1-30B, runs on 8 GPUs, and uses slime with rollout shuffling disabled.Disabling rollout shuffling preserves the reverse-perplexity curriculum order throughout training.
  • Implementation and curriculum: The training mixture follows the reverse-perplexity ordering described in Section 2.1.The curriculum order is preserved throughout training rather than randomized during rollout.
  • Optimization settings: Four epochs use batch size 128, Adam, learning rate 1 × 10−5, cosine decay to 1 × 10−6, warmup fraction 0.1, and weight decay 0.1.The stated momentum parameters are β1 = 0.9 and β2 = 0.95.

D RL Training Details · E Inference and Reward-Model Serving Details · F Compared Models

The paper trains with a two-stage, 200-step GSPO RL pipeline and serves iterative verification-and-refinement inference with specified generation, stopping, and reward-model configurations. It compares SU-01 against larger and similarly sized contemporary models using public technical and benchmark sources.

  • D RL Training Details: 200 RL steps comprise 96 coarse-RL steps followed by 104 refined-RL steps over balanced verifiable, proof-reward, self-refinement, and replayed-experience prompts.Training uses slime on 64 GPUs, continuing from the SFT checkpoint; each rollout uses batch size 128, eight samples, and a 160k-token response limit.
  • D RL Training Details: 20% of training queries use self-refinement data, selecting rollout groups with average reward below 0.5 for correction.The refinement prompt contains the original problem and previous failed answer; experience replay follows ExGRPO without reward shaping.
  • E Inference and Reward-Model Serving Details: SGLang serves inference by default, while Nemotron-Cascade-2 and DeepSeek-V3.2-Speciale use vLLM.The setup follows a model-agnostic verification-and-refinement framework.
  • E Inference and Reward-Model Serving Details: The inference loop iteratively generates a proof, receives structured verifier feedback, and revises the proof until acceptance or budget exhaustion.Default generation settings are maximum length 160,000 tokens, temperature 1.0, and top-p 0.95.
  • E Inference and Reward-Model Serving Details: API-constrained generation lengths reach 128,000 tokens for GPT-5.5 API and 65,535 tokens for Gemini 3.1-Pro.These limits replace the default maximum when the respective APIs impose smaller request constraints.
  • E Inference and Reward-Model Serving Details: TTS accepts a candidate after 5 consecutive verification successes, stops after 10 consecutive failures, and permits at most 30 solve–verify–refine cycles.These explicit stopping rules govern repeated exploration and correction within one run.
  • E Inference and Reward-Model Serving Details: The DeepSeekMath-V2 proof-reward server runs on 32 GPUs with 32k maximum context and data-parallel degree 64, using three-step speculative decoding.Each step uses one top-ranked EAGLE draft candidate with four draft tokens, improving verifier efficiency while preserving evaluation correctness.
  • F Compared Models: Table 1 compares answer-verifiable performance across larger models—DeepSeek-V3.2, GPT-5.5, and Gemini 3.1 Pro Thinking—and similarly sized models including GLM-4.7-Flash and Nemotron-Cascade-2-30B-A3B.The comparisons draw on public technical reports, official model cards, and official benchmark reports; the supplied passage also lists Qwen3.6-35B-A3B and Gemma-4-31B in the similar-size group.

G Evaluation Details … Solution

The paper evaluates answer-verifiable tasks with layered automatic checking and proof-oriented tasks with benchmark-specific expert or model judging. Its included solutions establish the IMO 2025 Problem 1 classification and complete an analytic geometry proof for Problem 2, while also presenting a partial bonza-function classification.

  • G Evaluation Details: Answer-verifiable tasks use rule-based checks first, followed by gpt-oss-120b generative verification when unresolved.Checks include canonicalized answer matching and symbolic or expression-level verification with Math-Verify16; FrontierScience-O uses its dedicated prompt.
  • G Evaluation Details: Proof-oriented benchmarks follow official protocols: ProofBench uses scores {0, 1, 6, 7} with Gemini-2.5-Pro, while FrontierScience-R uses GPT-5-high.IPhO uses physics-specific prompts and competition-style criteria; IMO and USAMO use expert-annotated ProofBench-style grading with three independent gold-medal experts.
  • H Model Solutions for IMO 2025 and USAMO 2026: The model-solutions section presents generated IMO 2025 and USAMO 2026 solutions alongside their final human-expert verdicts, excerpting USAMO 2026 Problem 2 because its response is substantially longer.This section is the source of the human-expert scoring reported in Table 4.
  • 5. Conclusion: For every n ≥3, the IMO 2025 Problem 1 construction-and-proof solution concludes that the only possible sunny-line counts are k = 0, 1, 3.Explicit line families attain all three values, while sunny-line incidence bounds and induction exclude k = 2 and all-sunny coverings for n ≥4.
  • 3. No covering by all sunny lines for n ≥4: The IMO 2025 Problem 1 proof bounds sunny-line coverage and shows that n distinct sunny lines cannot cover Pn when n ≥4.For odd n, equality would force every line into one of only three classified extremal lines, contradicting n ≥5.
  • 16. Conclusion: The included IMO 2025 Problem 2 solution uses similarity-invariant coordinate normalization, derives the relevant centers and intersections analytically, and proves the target tangency.It establishes that the distance from the circumcenter of △BEF to the line through H parallel to AP equals the circumcircle’s radius.
  • Solution: The bonza-function solution derives f(1) = 1 and f(a) | a^a, then narrows the admissible triples (f(2), f(3), f(5)) to four cases.The four triples are (1, 1, 1), (2, 1, 1), (4, 1, 1), and (2, 3, 5), grouped into two families for further analysis.

3. Analysis of Family A (f(3) = 1)

Assuming f(3)=1, the analysis shows that f(q)=1 for every odd prime q, every f(n) is a power of 2, and these powers satisfy a linear bound in n.

  • Prime values: f(q)=1 for every odd prime q.Writing f(q)=q^k, any k≥1 would imply q | 3^q−1 and hence q | 2 by Fermat’s little theorem, contradicting that q is odd.
  • Prime-factor restriction: Every f(n) is a power of 2.If an odd prime r divided f(n), then Lemma 2 would give r | n, while applying the defining divisibility to (n,r) yields r | 3^n−1 and thus r | −1, a contradiction.
  • Divisibility constraint: f(n) divides 3^n−1, so its exponent is constrained by the preceding 2-adic valuation.Since f(n)=2^e(n), the defining relation with (n,3) gives 2^e(n) | 3^n−1.
  • 2-adic valuation: For odd n, v2(3^n−1)=1; for even n, v2(3^n−1)=v2(n)+2.The even case is established by writing n=2^αm with m odd and inducting on α.
  • Growth bound: f(n)≤4n for every positive integer n.For even n, the exponent bound gives 2^v2(n)+2=4·2^v2(n)≤4n; for odd n, 2≤4n.

4. Analysis of Family B (f(2) = 2, f(3) = 3, f(5) = 5) … Alice prevents Bazza from winning

The paper proves that Family B forces f(n)=n, establishes the sharp universal bound f(n)≤4n, and solves the subsequent olympiad problems by characterizing infinite divisor-sum orbits and game-winning strategies. In the inekoalaty game, Alice wins for λ>1, Bazza wins for λ<1, and λ=1 is a draw.

  • 4. Analysis of Family B (f(2) = 2, f(3) = 3, f(5) = 5): For every prime p, strong induction eliminates k=0, k=2, and k≥3 from f(p)=pk, leaving f(p)=p.The argument uses congruences, Bertrand’s postulate, and a p-adic valuation contradiction.
  • Family B; 5. Upper bound summary: f(n)=n in Family B, so every bonza function satisfies f(n)≤4n.For arbitrary n, every prime divides n−f(n), forcing n−f(n)=0.
  • 7. Conclusion: The smallest real constant c satisfying f(n)≤cn for every bonza function and n∈N is 4.This conclusion combines the universal upper bound with the function h attaining the ratio 4.
  • Solution; 2. Fixed points; 3. Special case: 12 | N: For the three-largest-proper-divisors sequence, the unique fixed-point divisor pattern is d2=2, d3=3, d4=6, defining F by ν2(N)=1, 3|N, and 5∤N.The solution introduces f(N) through divisor pairing and analyzes the fixed-point equation 1/a+1/b+1/c=1.
  • 4. Lemma on odd numbers; 5. Lemma for even numbers not divisible by 12: Every infinite orbit has all terms even, and any even term not divisible by 12 with an infinite orbit must lie in F.Odd terms strictly decrease, while the even-case analysis excludes alternatives through divisibility and parity contradictions.
  • 7. Sufficiency - every such number works; 8. Final characterization: An initial value a1=12^tK with K∈F generates an infinite sequence, because each application reduces the exponent of 12 by one while preserving membership in F.Induction proves sufficiency, complementing the necessity argument for infinite orbits.

Conclusion … Solution

The paper resolves the game’s outcome by λ and solves the IMO 2025 tiling problem exactly, reducing it to a diagonal configuration and matching a 2n−2 construction with a lower bound. It also begins the USAMO 2026 solution by reducing the floor-function expression to the fractional part of x.

  • Conclusion: Alice wins iff λ > 1/√2, Bazza wins iff λ < 1/√2, and λ = 1/√2 is a draw.The conclusion states the outcome for all positive λ.
  • Solution: For the 2025 × 2025 grid, the uncovered squares form a permutation matrix, while the remaining squares must be covered by non-overlapping axis-aligned rectangles.This reformulates the IMO 2025 problem with n = 2025.
  • 1. Reduction to the diagonal configuration: A column permutation maps the uncovered permutation matrix to the diagonal without changing the number of tiles or the grid-covering properties.This establishes the reduction to the diagonal configuration.
  • 2. A construction using 2n −2 tiles: The diagonal configuration admits a tiling with 2n −2 rectangles: n −1 vertical tiles cover the lower triangle and n −1 horizontal tiles cover the upper triangle.The two tile families are disjoint, and diagonal squares remain uncovered.
  • 3. Lower bound: at least 2n −2 tiles: Every tile leaving the diagonal uncovered lies entirely within either the lower triangle L or the upper triangle U.A tile crossing both regions would contain a diagonal square.
  • 3. Lower bound: at least 2n −2 tiles: The n −1 cells immediately below and above the diagonal are each pairwise separated across tiles, forcing at least 2n −2 tiles in total.The lower-bound witnesses are DL = {(i, i −1)} and DU = {(i, i + 1)}.
  • 4. Minimum number of tiles: The construction and lower bound coincide, so the minimum number of tiles is exactly 2n −2 for n = 2025.The paper explicitly concludes exact optimality after substituting n = 2025.
  • H.2 USAMO 2026 / Solution: For USAMO 2026 Problem 1, writing x = ⌊x⌋ + {x} = m + θ reduces ⌊kx⌋ to km + ⌊kθ⌋ and the expression to dependence on θ ∈ [0, 1).The supplied solution begins by isolating the fractional part of x.

Step 2. Value on the interval · Solution

The expression equals H_n−1 throughout the interval [1−1/n,1), and this is its global maximum, attained exactly when {x} lies in that interval. For USAMO 2026 Problem 2, the solution proves Annie can always win by induction, targeting a final value below 4 times the initial sum.

  • Step 2. Value on the interval: On [1−1/n,1), the expression is constant and equals H_n−1.Here H_n is the n-th harmonic number.
  • Step 2. Value on the interval: For arbitrary θ∈[0,1), writing N=⌊nθ⌋ and β=nθ−N reduces the global bound to proving A(N)≤H_n−1.The decomposition uses t=β/n∈[0,1/n).
  • Step 2. Value on the interval: The gcd decomposition and rearrangement inequality bound the weighted remainder sum, yielding A(N)≤H_n−1 for every N.Equality in the intermediate estimate requires gcd(N,n)=1 and a decreasing residue ordering.
  • Step 2. Value on the interval: The equality chain F(θ)≤A(N)≤H_n−1 is sharp precisely when the fractional part of x lies in [1−1/n,1).For N=n−1, the remainder condition is verified directly, giving F(θ)=A(N)=H_n−1.
  • Solution: The USAMO 2026 problem asks whether Annie can repeatedly merge adjacent powers of two so that the final number is less than 4 times the original sum.The solution states that such a winning sequence always exists.
  • Solution: The induction maintains the claim that a list with sum S and k=⌊log_2 S⌋ can be reduced to F≤4·2^k.If S is not a power of two, this implies F<4S; if S is a power of two, the proof yields F≤2S.
  • Solution: When adjacent equal powers exist, merging them into 2a preserves the sum and permits direct induction; otherwise, the solution merges a carefully chosen adjacent unequal pair and analyzes whether the sum’s logarithmic floor changes.The unequal merge replaces x<y by 2y, changing the sum to S′=S+(y−x).
  • Solution: In the floor-increasing subcase, the proof applies a stronger block-sum lemma to keep intermediate values controlled before further merges.The lemma gives final block values bounded by 2^(k+1) when its sum condition holds.

Proof of the Lemma … 3. Case I: N is prime

The paper develops several olympiad solutions using induction, complex geometry, digitwise carry recurrences, and Fibonacci-preserving reductions. The highlighted results include a power-of-two reduction lemma, tangency and distance equalities in geometry, a closed count of solitary integers, and a Fibonacci characterization in the prime case.

  • Proof of the Lemma: The lemma uses strong induction to show that any power-of-two list with 2^L < T ≤ 3·2^(L−1)−1 can be reduced to a number ≤2^(L+1).The base case analyzes two terms, while the induction merges suitable adjacent terms and handles the restricted case containing 2^L.
  • 1. Complex numbers on the unit circle: The complex-plane geometry solution places the circumcircle as the unit circle, derives equations for ℓB, ℓC, the tangent at N, and the intersections Y and Z, then rotates to N = 1.The rotation gives bc = 1, b = e^(iα), c = e^(−iα), and k = cos α, simplifying the later algebra.
  • 5. A key algebraic identity: The identity (a + 2k)^2 = QT + ∆ establishes the circumcenter of △AYZ, and matching distances confirms its circumradius.The subsequent algebra verifies that the proposed point is equidistant from A, Y, and Z.
  • 7. Cartesian description of triangle R: The triangle R is described by three boundary lines, symmetry places its incenter on the x-axis, and equal distances yield p = 1/2(1 + cos α) and r = (1 + 2k)/(2(1 + k)).The interior inequalities and vertex calculations determine the relevant side of each boundary line.
  • 10. Distance OI: The distance computation gives L = r(2 cos 3α − t), while ∆ implies 2 cos 3α − t = ε|∆|, so OI equals a sum or absolute difference of the radii.Therefore the circumcircle of △AYZ and the incircle of R are tangent.
  • 1. Angle Chase: Angle chasing proves ∠DFE = β, ∠EDF = γ, and ∠DEF = α, so △DEF is similar to △ABC; complex similarity and concyclicity then establish OM = ON.The proof uses a direct similarity after reflection if necessary and shows the origin lies on the three relevant circumcircles.
  • 3. Case I: N is prime: In Case I, the reduction forces k = 3 and equation a^2 + b^2 + 1 = 3ab; all resulting solutions are pairs of consecutive odd-index Fibonacci numbers.The base solutions with a = 1 are (1, 1) and (1, 2), and the induction concludes that all solutions of the displayed equation are Fibonacci pairs.

4. Case II: N is composite and odd (N > 2) · 5. Conclusion

For composite odd N > 2, the divisibility condition forces N = 9 and yields only (a,b) = (1,8) or (8,1). Combined with the even and prime cases, this proves that a and b must be Fibonacci numbers.

  • 4. Case II: N is composite and odd (N > 2): For composite odd N > 2, a² + b² + 1 ≡ 2 (mod 4).
  • 4.1. N cannot have two distinct odd prime factors: If N had two distinct odd prime divisors, then 4 | φ(N) | a² + b² + 1, contradicting the modulo-4 condition; hence N is a prime power.
  • 4.2. The prime p cannot be ≡1 (mod 4): The prime base cannot satisfy p ≡ 1 (mod 4), because then 4 | φ(N) | a² + b² + 1, again contradicting the modulo-4 condition.
  • 4.3. Analysis modulo 3: Modulo 3, the argument excludes p ≡ 1 (mod 3), while reducing the remaining case to p = 3.For p ≡ 1 (mod 3), 3 | φ(N) and ab ≡ 0 (mod 3), but a² + b² + 1 is not divisible by 3; for p ≡ 2 (mod 3), y² + y + 1 has no solution modulo p.
  • 4.3. Analysis modulo 3: For p = 3 and e ≥ 3, 9 | φ(N) | S but S ≡ 3 (mod 9), so e = 2 and N = 9.The congruence ab ≡ −1 (mod 9) and a + b ≡ 0 (mod 3) gives S = a² + b² + 1 ≡ 3 (mod 9).
  • 4.3. Analysis modulo 3: For N = 9, the divisibility condition holds only for (a,b) = (1,8) and (8,1), and both entries are Fibonacci numbers.Among the positive factor pairs of ab = 8, (1,8) gives S = 66 with 66/6 = 11, while (2,4) fails; symmetry gives the second solution.
  • 5. Conclusion: Across the even, prime, and composite cases, the hypothesis φ(ab + 1) | a² + b² + 1 forces a and b to be Fibonacci numbers.The prime case has solutions (F2n−1, F2n+1) and symmetric pairs for n ≥ 1, alongside the degenerate pair (1,1); the composite pair (1,8) also fits the Fibonacci sequence.
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