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Use an AI reasoning model as a source of candidate ideas and a skeptical proof checker—not as an authority. Give it the exact problem and assumptions, request a short plan and explicit proof, then audit the argument in a separate turn and test boundary cases. If the result must be independently checkable, formalize it in a proof assistant. The mathematician remains responsible for deciding whether the proof is valid.
How do I use an AI reasoning model to solve an Olympiad problem?
Work in a loop: state the problem precisely, set expectations for the answer, explore more than one strategy, and then challenge the proposed proof independently. This is more reliable than asking for a solution once and treating a confident explanation as proof.
- Normalize the statement. Paste the exact wording. Define every variable and its domain, include all conditions and diagram facts, and say what must be found or proved. Distinguish, for example, positive integers from nonnegative integers; a model may otherwise solve a different problem.
- Set an output contract. Ask for a concise plan, named lemmas, explicit assumptions, and a complete proof. Require the model to distinguish conjectures and exploratory ideas from established steps, and to address all cases.
- Choose reasoning effort deliberately. Start at a moderate setting for initial exploration. Use a higher setting when the problem involves many cases or a deeper search, while tracking the extra latency and token use. Higher effort generally means more internal work, not a guarantee of correctness.
- Request distinct approaches. Ask for two or three materially different strategies, such as an invariant, an extremal argument, or an algebraic approach. Have the model explain the key lemma each strategy needs and why it might apply.
- Develop one route into a proof. Ask for each implication to be justified and for equality or boundary cases to be covered. Treat an unexplained step as a question to resolve, not a gap to bridge with trust.
- Audit in a fresh turn. Give the proposed proof to the model separately and ask it to locate the first invalid inference, omitted case, hidden division by zero, or unjustified use of a lemma. A fresh audit reduces the temptation to merely defend the earlier answer.
- Check independently. Test small cases or use a suitable symbolic or formal tool when useful. If the result needs machine verification, formalize the theorem and proof rather than relying on the chat transcript.
- Write the final solution for a human. Remove failed explorations, define notation once, and explain the key idea before technical details. Keep the verification record if computational or formal checking contributed to the result.
This process does not require a model to expose its hidden chain of thought. Ask for the useful deliverables—a plan, lemmas, assumptions, proof, and checks—rather than a private internal trace. OpenAI’s reasoning guidance describes effort as a control for internal work and recommends giving a clear goal, constraints, and output contract.
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A good prompt makes the mathematical task unambiguous and says what a satisfactory answer must contain. Use this as a starting point, then replace the problem text with the complete statement:
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You are helping with an olympiad-style mathematics problem.
Problem:
[Paste the exact statement, including domains and diagram facts.]
Deliverables:
1. Restate the problem with all assumptions explicit.
2. Propose up to three distinct strategies and identify the key lemma for each.
3. Develop the most promising strategy into a complete proof.
4. Check boundary cases, hidden divisions, quantifiers, and whether every implication is reversible.
5. List any claims that still require independent verification.
Do not present a conjecture as a theorem. Keep the final proof separate from exploration.The square-bracketed prompt fields are instructions to replace, not mathematical notation. For a geometry problem, include any diagram information that is actually part of the hypotheses: a picture can suggest relationships, but a visual appearance alone does not establish collinearity, equality, or order. For a problem asking for all solutions, say so explicitly and require both directions: every listed candidate must work, and every possible solution must be accounted for.
Ask for a short plan before the proof if you want to steer the session. If the model proposes multiple methods, choose one based on whether its central lemma is plausible and verifiable—not on which explanation is longest or sounds most elegant. You can ask it to state what would have to be true for a proposed strategy to succeed before asking it to fill in the details.
How can I check whether an AI-generated proof is valid?
Read the proof as a sequence of obligations. Each line should follow from the stated assumptions, a previously proved result, or an explicitly justified theorem. The crucial question is not whether the answer looks familiar; it is whether every inference is valid and the argument covers the entire problem.
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Audit the assumptions and quantifiers
- Check that the proof uses only hypotheses from the original problem. An unstated assumption—such as a variable being nonzero or a geometric point lying inside a segment—can change the result.
- Track quantifiers carefully. A construction that works for one example does not prove that it works for every case, and finding a candidate does not show there are no others.
- For “if and only if” claims, verify both directions. For an inequality, check the conditions under which equality holds if equality matters to the conclusion.
Look for algebraic and logical traps
- Before dividing by an expression, establish that it is nonzero. If it might be zero, split into cases and handle that case separately.
- Check whether squaring, taking roots, applying a logarithm, or multiplying an inequality preserves equivalence under the stated domains. These operations can add extraneous solutions or reverse an inequality.
- Inspect exchanges of limits, sums, extrema, or other operations. Such a step may need a condition or a separate justification, not just an intuitive resemblance.
- When a proof invokes a lemma, confirm that the lemma has already been proved or is valid under the current assumptions. A familiar name is not a substitute for its conditions.
Ask for an adversarial pass
Use a separate message with the proposed proof, not just “check this.” A focused audit prompt makes the task concrete:
Audit the proof below as a skeptical olympiad jury. Check every inference, omitted case,
use of a lemma, and equality condition. Try to construct a counterexample. If the proof
is valid, restate the critical dependencies; if not, identify the first failing line and
repair it only when the repair is justified.
Proof:
[Paste the proposed proof.]The most useful criticism identifies the first failure and explains why it fails. If the model offers a repair, audit that repair too. Agreement between two AI responses is not independent mathematical verification: they can share the same mistaken assumption or persuasive but invalid pattern. For a result that matters, check the argument yourself or use an appropriately rigorous independent method.
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When should I use computation or a proof assistant?
Use computation to find errors or explore a finite range; use formal proof when you need a machine-checkable argument. Neither small-case agreement nor generated code alone proves a general theorem.
Small-case enumeration and symbolic algebra
For a problem with a finite, manageable set of small cases, enumeration can expose a counterexample to a conjecture or reveal a pattern worth proving. Symbolic algebra can help check expansions, simplify expressions, or test whether a proposed identity is even plausible. Keep the code, variable restrictions, and tested range visible: a search over selected values does not establish what happens outside that range.
Use a counterexample as a diagnostic. If computation contradicts the model’s claim, inspect both the claim and the program’s assumptions before deciding which is wrong. Conversely, a program that finds no counterexample in a sample has not ruled out a counterexample elsewhere. The written proof must still explain why the conclusion holds in every required case.
Formalization and machine checking
A proof assistant can check a formal statement and its proof against a formal library and logic. That requires translating the problem into the assistant’s language, choosing definitions, and supplying a proof that meets the system’s requirements. It is a stronger form of verification than a fluent natural-language explanation, but the formal statement must faithfully represent the original problem: a machine can validate a proof of the wrong theorem.
Google DeepMind’s account of AlphaProof describes a pipeline using Lean formalization, candidate-answer generation, automated refutation, and test-time reinforcement learning to find formal proofs. That illustrates how formal systems can support proof search and checking; it does not mean an ordinary chat response has undergone the same process. For personal work, decide whether the cost of learning the formal tool is justified by the need for a reproducible, machine-checkable result.
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Which model is best for IMO-style math?
There is no universal winner established by a single score. Choose according to the task: general reasoning models are convenient for exploration, drafting, and tutoring; geometry-specialized neuro-symbolic systems may be better suited to construction-heavy geometry; and formal-proof systems are appropriate when machine checking is required.
| Need | Useful system type | Trade-off to consider |
|---|---|---|
| Explore an unfamiliar problem or explain an idea | General reasoning model | Convenient for dialogue and drafting, but its proof still needs review. |
| Search for constructions in geometry | Geometry-specialized neuro-symbolic system | Can combine learned construction proposals with a symbolic deduction engine; task focus may not generalize to unrelated problems. |
| Check a theorem in a formal language | Formal-proof system or proof assistant workflow | Produces machine-checkable artifacts, with extra formalization and setup work. |
For a fair comparison, record the exact problem set, time limit, translation burden, and compute budget, as well as whether a system produced a natural-language proof or a formal one. A model may receive a problem already translated into a language or representation it handles well, or may have substantial computation time. Those conditions affect what a score means.
Published achievements show what carefully designed systems can do, not what a general model will do on an unseen problem from a reader’s workbook. Google DeepMind reported that AlphaGeometry solved 25 of 30 problems in its IMO-AG-30 geometry benchmark in 2024. Its neural component proposed useful constructions, while a symbolic engine checked deductions.
For the 2024 International Mathematical Olympiad, Google DeepMind reported a combined AlphaProof and AlphaGeometry score of 28 out of 42 points, equivalent to the silver-medal range. The pipeline involved manual formalization, and some solutions took days of computation—conditions very different from an unassisted student solving a fresh problem under an exam clock.
Google DeepMind reported that Gemini Deep Think scored 35 out of 42 at IMO 2025, a gold-medal score, with natural-language solutions produced within the competition time limit. That result is notable, but it does not establish that a model will solve every new olympiad problem or that every answer it gives outside that setting is correct.
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Other results reinforce the need to keep benchmark scope in view. OpenAI’s 2024 o1 announcement reported performance at the 89th percentile on Codeforces and a placement among the top 500 USA Math Olympiad qualifiers. Those are specific reported outcomes, not a general guarantee of proof reliability. Epoch AI reported that leading models solved less than 2% of its expert-crafted FrontierMath problems in 2024, a reminder that success on familiar or selected benchmarks does not translate automatically into broad mathematical ability.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What can olympiad AI results actually tell me?
They show that some specialized systems and reasoning models can perform impressively under defined conditions. They do not tell you whether a particular answer is correct, how much assistance or compute was used, or whether the system will generalize to a different problem.
It matters how a result was obtained. A geometry system that proposes constructions and checks deductions symbolically is not the same setup as a general chat model responding to plain text. A formal system may rely on human translation into a proof language. A competition score achieved within a time limit is not comparable without qualification to one obtained after days of computation. Read the method and setup alongside the headline score.
Proof quality matters too. A system can find a correct answer but present an incomplete explanation; another can give a clear proof that is easier for a human to inspect. Google DeepMind cited IMO President Prof. Dr. Gregor Dolinar describing Gemini Deep Think’s 2025 solutions as “clear, precise and most of them easy to follow.” That is useful evidence about the reported natural-language output, but clarity by itself is not a substitute for checking a proof’s steps.
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The most productive expectation is modest: AI can help generate candidate strategies, explain concepts, expose gaps, and suggest checks. Treat those as contributions to your work. Mathematical responsibility stays with the person accepting the conclusion.
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How do I turn the model’s answer into a useful final solution?
Separate discovery from presentation. A chat may contain guesses, false starts, alternative lemmas, or corrections; the final write-up should make the valid argument easy to follow without hiding any unresolved gap.
- Write the assumptions and notation once, using the problem’s exact domains and definitions.
- State the central idea in ordinary language before introducing technical details.
- Arrange the lemmas in the order the proof needs them, and show why each follows.
- Include the case split, equality condition, or converse needed to make the conclusion complete.
- Remove strategies that were abandoned and claims that remain unverified. If computation or formal checking was essential, describe what was checked and preserve the relevant record.
If you are using the interaction to learn, do not immediately ask for a finished proof. First ask for a hint or a key lemma, try the step yourself, and then request feedback on your attempt. This preserves the useful work of solving while still giving the model a role as tutor and critic.
What should I do when the model gives different answers?
Do not resolve disagreement by choosing the more confident or detailed response. Compare the assumptions each answer uses, identify the first point where their arguments differ, and test that precise claim with a proof or counterexample search.
If one strategy relies on a lemma the other avoids, ask for the lemma’s proof and conditions. If the outputs disagree about a computation, inspect the code and the domain it covers. If both appear plausible, return to the original statement and verify that each has answered the same question. A disagreement can be productive because it pinpoints the step that deserves independent scrutiny.
Frequently Asked Questions
Can I use an AI model during an Olympiad or school contest?
Contest rules differ, and many assessments restrict outside assistance or internet-connected tools. Check the specific organizer’s rules before using AI; a model’s usefulness for practice does not imply it is permitted during a live competition.
Is a natural-language proof enough for publication or a formal result?
That depends on the standard required by the venue and the claim. A readable argument may be appropriate for exposition, while a formal or independently reviewed proof can be necessary when readers need stronger verification.
How can I use AI for practice without giving away the solution?
Ask for one hint at a time, such as a promising lemma or a question to consider, and attempt the next step before requesting more. You can also ask the model to critique your proof without supplying a replacement unless you request one.
Should I include an AI-generated solution in my homework?
Follow your instructor’s policy and disclose assistance when required. Even when AI use is allowed, make sure you can explain and defend every step yourself rather than submitting an argument you have not checked.
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