OpenAI Formal Math Olympiad Problem Solving

OpenAI has demonstrated that its models can solve formal versions of high-school mathematics olympiad problems. By generating proofs in a formal system, the models can provide verifiable solutions to complex problems sourced from competitions such as the American Mathematics Competitions (AMC12), the American Invitational Mathematics Examination (AIME), and the International Mathematical Olympiad (IMO).

Formal Proof Generation and Tactics

OpenAI's approach utilizes formal systems where proofs are constructed using "tactics." Tactics are high-level search procedures that generate low-level artifacts—comparable to assembly code—which are required by the formal system to verify a proof. This allows the model to move from a high-level mathematical statement to a verifiable formal proof.

Solved Problem Examples

The models successfully solved a variety of problems across different mathematical domains, demonstrating specific reasoning capabilities:

Algebra and Arithmetic

  • AMC12 (2000 Problem 5): The model proved a statement regarding absolute values ($╴ x - 2 ╵ = p$) by inventing the necessary term abs (x - 2) = -(x - 2) to progress the proof.
  • AMC12B (2020 Problem 6): The model solved a problem involving factorials and perfect squares by directly proposing n + 1 as the solution witness.
  • MATH Dataset: The model proved a statement about linear functions $f(x) = Ax + B$ and $g(x) = Bx + A$ using contraposition and by proposing 0 as a witness for the goal.
  • AIME (1984 Problem 1): The model handled a problem involving arithmetic progressions and summations over a range of 98 terms.

Geometry and Inequalities

  • IMO (1964 Problem 2): The model proved a triangle inequality involving sides $a, b, c$ by inventing arguments for the nlinarith tactic, specifically using the non-negativity of squares (sq_nonneg).
  • IMO Longlist (1990 Problem 77): The model solved a complex inequality by employing the Cauchy-Schwarz inequality and introducing necessary "cuts" (inventing terms like $0 ≤ (c + a) * (c + a)$) to satisfy the formal prover.

Technical Implications

The ability to solve these problems highlights the model's capacity for "invention" within a formal proof context. In several instances, the model did not simply apply existing rules but proposed specific witnesses (e.g., $n+1$ or $0$) or invented intermediate lemmas and terms (such as specific squared differences) that were essential to completing the proof. This indicates a level of strategic planning and mathematical reasoning required to navigate the constraints of a formal verification system.

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