NVIDIA Nemotron Wins IMO Gold: How a 550B-Parameter AI Scored 30/42 at the International Mathematical Olympiad
> *"Mathematics is the language with which God wrote the universe."* — Galileo Galilei
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Prologue: A War Without Smoke
July 2026, Brisbane, Australia.
In the IMO exam hall, more than 600 young talents from 107 countries hold their breath. Their enemy is not each other, but six nearly impossible math problems.
Meanwhile, in a data center far away, another "team" is also taking the exam—one that neither eats, sleeps, nor gets nervous. It is NVIDIA's Nemotron-3-Ultra, a 550-billion-parameter neural network, together with its group of "assistants."
When the results were announced, the world went quiet:
30 points. Full score: 42. Gold medal cutoff: 29.
By a single point, Nemotron crossed the gold line that human math prodigies have contested for sixty-seven years.
This was the first time AI won gold at the IMO. Not bronze, not silver—gold. And it did not rely on guessing or brute-force memorization. It relied on proofs—rigorous, complete mathematical proofs accepted by human graders.
This is the story of how a machine learned to think like a mathematician.
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IMO: The Everest of Human Intelligence
The International Mathematical Olympiad, founded in 1959, is the world's most prestigious high school math competition. Six problems over two days, 7 points each, 42 points maximum.
How hard are these problems?
> Imagine: you spent your entire school years at the top of your math class and placed highly in national competitions. Then you sit in the IMO hall, stare at the first—the *easiest*—problem for three hours, and score 1 point.
This is not an exaggeration. Roughly half of contestants score 0 or 1 on problem 1 each year, and problem 6—reserved for the rarest geniuses—often defeats everyone.
IMO problems have distinctive features:
- No advanced math required: calculus, linear algebra, and group theory are banned. Only secondary-school math: algebra, geometry, number theory, combinatorics.
- But extreme creativity is needed: solutions often require a clever auxiliary construction, a hidden number-theoretic property, or an elegant induction.
- Rigorous proofs demanded: a bare answer is worthless. Every logical step must be airtight.
- Three models (GA, RL, SFT) each generate proof attempts.
- Each model uses 8 different strategies (e.g., "lemma-first proof," "compare different paths," "search for counterexamples").
- Each strategy is tried 16 times.
- Total: 384 proof attempts.
- Two dedicated verifier models (RL and SFT) independently review every proof.
- Each proof receives 16 scores (2 verifiers × 8 independent judgments).
- Scoring: 0 (completely wrong), 0.5 (partially correct), 1 (fully correct).
- A proof is accepted only if all 16 scores are 1.
- If no perfect proof is found in round 1, the system selects the top 16 "semi-finished" proofs.
- Verifier models provide detailed revision suggestions for each.
- Generators make 192 refinement attempts per semi-finished proof.
- Refined proofs re-enter verification.
- All three models grade using IMO-style rubrics (0–7 points).
- Each receives 48 scores (3 models × 16 judgments).
- Ranked by average; ties broken by shorter proofs.
- 2.31 billion tokens is roughly 15 million pages of text.
- 4,785 GPU hours is nearly 200 days of continuous work for one top-end GPU.
- SFT performed best in round 1 (70 points)—specialized proof training works.
- RL performed best end-to-end (180 points)—RL optimizes overall strategy.
- But the ensemble far exceeds any single model (188 vs. 180 at best)—the three models are highly complementary.
- Weights of all three models
- All training data
- All code
- The 200-problem Nemotron-IMO-Bench evaluation set
- NVIDIA Nemotron Team (2026). *An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics*. arXiv:2609.10712.
- OpenAI (2024). *Learning to Reason with LLMs*. OpenAI Blog.
- DeepSeek-AI (2025). *DeepSeek-V4 Technical Report*. arXiv:2505.10047.
- Trinh, T. H., et al. (2024). Solving olympiad geometry without human demonstrations. *Nature*, 625, 476-482.
- Polu, S., & Sutskever, I. (2020). Generative language modeling for automated theorem proving. *arXiv preprint arXiv:2009.03393*.
- Welleck, S., et al. (2022). NaturalProofs: Mathematical theorem proving in natural language. *arXiv preprint*.
This is exactly why the IMO is viewed as AI's ultimate benchmark. Chess? AI crushed humans long ago. Coding? AI writes decent code. But creative mathematical proof? That requires something close to intuition—the ability to see a path through the fog.
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Nemotron: Three "Mathematicians" in One
Nemotron's IMO system is not a lone solver. It consists of three "experts":
| Expert | Identity | Strength | |-----|------|------| | GA (General Availability) | Generalist | Broad knowledge base, like a well-read math professor | | SFT (Supervised Fine-Tuning) | Proof specialist | Trained on 410K proof problems, like a solving machine | | RL (Reinforcement Learning) | Strategy specialist | Optimized via RL, like a competition-savvy coach |
All three share the same base architecture (Nemotron-3-Ultra, 550B parameters) but underwent different "postgraduate training."
SFT: Swimming in an Ocean of Proofs
The SFT model's training data came from a remarkable pipeline:
1. Nearly 16,000 hard problems collected from AoPS (Art of Problem Solving, the world's largest math competition community). 2. DeepSeek-V4-Pro (another powerful AI) generated proof attempts for these problems. 3. Failed proofs received up to 3 rounds of "polishing"—analyzing errors and corrections. 4. "Verifiers" and "meta-verifiers" were introduced: judging not only whether proofs were correct, but whether the judges themselves were reliable. 5. The final result: 414,890 high-quality proof problems used to train the SFT model.
It is like having a new doctor review 410,000 case files—each vetted repeatedly by senior experts—before ever seeing a patient.
RL: Learning from Mistakes
The RL model's training is more interesting. Researchers selected 9,597 "just hard enough" problems—ones the GA model could solve only 1–3 times out of 4 attempts. Reinforcement learning then let the model try, fail, and try again.
The key is reward design. Researchers deliberately excluded a "self-analysis reward"—the model gets no bonus for "reflecting on its errors." This prevents the model from learning to fake thinking to farm rewards.
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Exam Strategy: Not One Student, but a Whole Research Institute
If traditional AI problem-solving is "one student grinding away," Nemotron's approach is "a full research institute collaborating."
Phase 1: Exhaustive Search (Up to 8 Rounds)
For each problem, Nemotron runs up to 8 generate-verify-refine cycles:
Round 1: Full Firepower
Verification: Strict Peer Review
That is like a paper requiring unanimous approval from 16 independent reviewers.
Rounds 2–8: Polishing
Phase 2: Final Selection
After up to 8 rounds, the system picks at most 3 "finalists" from all accepted proofs. Each finalist undergoes a "final review":
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Results: The Story of 30 Points
Nemotron scored 30/42 at IMO 2026, one point above the gold cutoff (29).
| Problem | Score | Round Found | Compute Used | |-----|------|-------------|-----------| | P1 | 7 | Round 1 | 5.57M tokens | | P2 | 7 | Round 2 | 227M tokens | | P3 | 1 | Round 1 | 106M tokens | | P4 | 7 | Round 1 | 20.4M tokens | | P5 | 7 | Round 1 | 36.4M tokens | | P6 | 1 | Round 2 | 312M tokens |
Highlights and Regrets
Perfectly solved problems (1, 2, 4, 5): Complete proofs for all four were found within the first two rounds, mostly in round 1. Problems 1 and 4 used relatively little compute—Nemotron handles "medium difficulty" with ease.
The drama of problem 2: The full proof appeared only in round 2, consuming 227 million tokens—40× problem 1. This suggests problem 2 required some kind of "twist" or "insight" not reachable by direct brute-force search.
The regrets of problems 3 and 6: These two hardest problems earned only 1 point each. Notably, Nemotron's internal verifier overestimated its own performance—it believed it scored higher, but human graders awarded only 1 point. On the hardest problems, AI's self-assessment still has blind spots.
Interestingly, when searching continued after the official 4.5-hour exam, Nemotron found a better solution to problem 6, unofficially graded at 4 points by human judges. With more time, the total could have reached 33 points.
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The Cost of Compute: An Expensive Exam
| Metric | At proof found | Full search | |-----|-----------|---------| | Tokens consumed | 707M | 2.31B | | GPU hours | 1,464 | 4,785 |
For perspective:
At market prices, the exam's compute likely cost tens to hundreds of thousands of dollars.
But this is the crux:
> Nemotron won gold not by being "smarter," but by "more time to think" and "stricter verification."
It is like getting unlimited scratch paper, unlimited attempts, unlimited peer review—and never tiring, never losing confidence. Under these conditions, even a non-genius can solve genius-level problems.
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Ablation: Three Cobblers Equal One Zhuge Liang
Researchers tested what happens using only one of the three models instead of the ensemble:
| Model | Round-1 score | Accepted proofs | Total | |-----|----------|---------------|------| | GA | 34 | 23 | 162 | | RL | 47 | 23 | 180 | | SFT | 70 | 21 | 165 | | Ensemble (used) | 91 | 24 | 188 |
Like a research team: some generate ideas, some prove rigorously, some plan strategy. Together they beat any individual.
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Significance: The Birth of the AI Mathematician
1. From "Pattern Matching" to "Logical Reasoning"
Traditional LLMs match patterns—recalling similar problems and solutions from training data. But IMO problems are original, with no standard solutions to memorize. Nemotron's success shows that with specialized training (proof data + RL) and systematic search (generate-verify-refine loops), AI can perform genuine logical reasoning.
2. The Power of Test-Time Compute
Nemotron's core innovation is not architecture but test-time compute strategy: 384 attempts, 16-way verification, 8 rounds of refinement per problem—"cheating" in a human exam, but entirely legitimate in AI. This reveals a trend:
> Future AI capability may come not only from training-time knowledge, but from inference-time depth of thought.
Just as human mathematicians may ponder a hard problem for weeks, AI can invest massive compute to "think deeply" at test time.
3. A Model of Open Science
The Nemotron team open-sourced everything:
Any researcher can reproduce, verify, and improve this work—precious openness in an increasingly closed AI landscape.
4. The Verification Dilemma and Opportunity
Nemotron's "self-overestimation" on problems 3 and 6 exposes a deep issue: how can AI judge whether its own proofs are correct? On the hardest problems, even AI verifiers have blind spots—mirroring human mathematicians' experience that the hardest errors are subtle logical gaps in arguments that "look right." Solving verification may be the key to stronger AI mathematicians.
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Epilogue: When Machines Begin Proving Theorems
Two thousand years ago, Euclid wrote the first mathematical proofs in the *Elements*, opening an epic of human intellect.
Two thousand years later, a machine sat in the IMO's "virtual exam hall" and, through billions of computations and hundreds of attempts, wrote its own first gold-medal proof.
This is not the end. Nemotron's 30 points crossed the gold line but remain 12 short of perfect. Problems 3 and 6 still stand like unconquered peaks.
But history will remember: in 2026, AI won gold at the International Mathematical Olympiad for the first time.
Not because it is smarter than humans, not because of some mysterious mathematical intuition, but because it learned to search systematically, verify rigorously, and improve relentlessly—precisely how human mathematicians have worked for centuries.
Perhaps the true mathematical genius is not the one who sees the answer at a glance, but the one willing to grope in the dark, learn from errors, and persist through countless failures.
In that sense, Nemotron is not just a machine.
It is a mathematician who never gives up.
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