Gemini 2.5 Pro Capable of Winning Gold at IMO 2025

TL;DR

A model-agnostic inference-time pipeline drives Gemini 2.5 Pro to solve 5 of 6 IMO 2025 problems with rigorous proofs, by repeatedly invoking a solver → self-improver → verifier → bug-report-guided refiner loop on the same base model. No fine-tuning, no tools, no web search — pure prompt engineering and iterated verification. The headline mechanism is that the verifier produces a categorized bug report (critical errors vs. justification gaps) and the solver revises against that report, with multiple iterations and a stricter acceptance gate (five clean verifier passes) at the end. Result is a clean demonstration that the verifier-guided refinement loop converts latent capability of a single 2025-era base model into IMO-medal-level proofs without changing weights.

Key claims

A self-verification pipeline with Gemini 2.5 Pro solves 5 of 6 IMO 2025 problems with full rigorous proofs; only Problem 6 fails, and the failure is a concretely identified flawed combinatorial premise [§4, Appendix A.6].
The pipeline is model-agnostic and uses no tools, no web search, no code, low temperature, and the model’s max 32,768-token thinking budget; the only knob is how many independent runs are launched in parallel/serial [§3, §2].
A single forward pass exhausts the 32,768-token thinking budget on a typical IMO problem; Step 2 (“self-improvement”) effectively injects a second budget of equivalent size and noticeably improves outputs [§2.2].
The verifier classifies issues into critical errors (demonstrably false or logical fallacies) and justification gaps (major vs. minor), and produces a per-issue bug report that the solver then revises against [§2.3].
The verifier is reliable but imperfect; the pipeline is robust to false positives via an optional human-style “bug report review” step (Step 4) and to false negatives via iteration plus a final 5-pass acceptance gate [§2.3].
Hint-free solutions for Problems 1 and 2 are achievable but require more sampling than hint-guided ones, supporting the authors’ hypothesis that hints reduce search space rather than provide new capability [§4].
Evaluation uses IMO 2025 specifically to avoid data contamination — the problems were released days before evaluation [§1].

Method

The pipeline runs six steps for each problem and is iterated independently many times in parallel/serial until at least one run yields a solution that survives strict verification. (1) Solver: prompt Gemini 2.5 Pro with a rigor-emphasizing prompt to produce an initial proof. (2) Self-improvement: prompt the model to review and continue its own work, which effectively doubles the thinking budget. (3) Verifier: a separate prompt walks the solution step-by-step, emitting a structured bug report with issues labeled critical error or justification gap (major / minor), each with an explanation. (4) Optional bug-report review removes obvious false positives from the report. (5) The solver is re-prompted with the bug report and revises the solution, with explicit instructions to push back on items it disagrees with (peer-review analog). Steps 3–5 iterate until either issues remain stable (reject) or the solution is clean. (6) Final accept requires the verifier to clear the solution on 5 independent runs.

All runs use temperature 0 (low randomness to avoid drift), the maximum thinking budget per call, and Gemini 2.5 Pro as both solver and verifier. Hints (e.g. “use induction” for Problem 1, “use analytic geometry” for Problem 2) were tested as a way to reduce computational sampling but are not load-bearing for correctness on the problems where unhinted solutions were also found.

Results

5/6 IMO 2025 problems solved with verbatim rigorous proofs in Appendix A; the sixth (Problem 6) is failed with a concretely identifiable flawed combinatorial premise (treating a tile as living entirely in one of two disjoint column regions when it can span both) [§4, Appendix A.6].
Hint-free solutions for Problems 1 and 2 are reported in arXiv v3 alongside the original hint-guided ones; finding the hint-free versions required more sampling but no different capability [§4].
No quantitative benchmark of verifier accuracy is provided; the authors report qualitatively that the verifier “almost always needs some revision” when it flags a critical error, and that some flagged justification gaps are not gaps for a mathematician [§2.3].
Caveat surfaced in the paper itself: the authors point to the matharena.ai “Not Even Bronze” evaluation (ref [3]) showing that raw model capability without the pipeline scores far below medal level — the pipeline, not the base model alone, is what gets the gold.

Why it’s interesting

This is a clean, surgical instance of the verifier-guided refinement axis of Inference-Time Scaling — distinct from the six axes already on that page (scaffold-side RLM, MiroThinker interaction-depth, DMS KV-cache compression, K2.5 PARL parallel-agent, generator-side iterative refinement, TTT-E2E weight updates), because here the extra compute lands as iterated solver↔verifier passes against a structured bug report with the same base model on both sides. It is the natural complement to Training-Free Group Relative Policy Optimization (Training-Free GRPO): both treat the policy update as a context-space operation on a frozen base model, but Training-Free GRPO distills group-relative advantages into an experience library, while this work distills verifier feedback into a single solution. It also resonates with Incorrect Baseline Evaluations Call Into Question Recent LLM-RL Claims — the matharena.ai result the authors cite shows that raw-pass evaluations sharply understate frontier-model capability; the gain here lives in the scaffold, not the weights.