Successful IMO 2025 Runs + Suggestions and a Challenge Problem

[xx205]

Excellent work! Last week, I successfully used the code in this repo to reproduce the solutions for IMO 2025 p1 and p2 (using the version with extra prompts). However, I have a few suggestions:

1. It would be helpful to allow setting a minimum number of successful agents, so the process can stop once that threshold is reached. When I ran p1, all 10 agents kept running until each one found its own solution. This seems unnecessary, considering that each correct solution is verified 5 times. For my own runs, I set the minimum number of successful agents to 2.
2. The code could be modified to run with an OpenAI-compatible API, which Gemini natively supports.
3. Requests could be made in streaming mode to avoid timeout errors, which I frequently encountered during my runs.

(I have already implemented the above changes in my version.)

Furthermore, I've found that while Gemini 2.5 Pro can successfully solve 5 of the IMO 2025 math problems using the code in this repo, it seems unable to fully solve the final question from the 2024 Chinese National College Entrance Examination (Gaokao) math paper (New Curriculum Standard, Type I). The problem is as follows:

*** Problem Statement ***

Let $m$ be a positive integer. Let the sequence $a_1, a_2, \ldots, a_{4m+2}$ be an arithmetic progression with a non-zero common difference. If after removing two terms $a_i$ and $a_j$ (where $i < j$), the remaining $4m$ terms can be partitioned into $m$ groups, with each group consisting of 4 numbers that form an arithmetic progression, then the sequence $a_1, a_2, \ldots, a_{4m+2}$ is called an \textbf{(i, j)-divisible sequence}.

Two distinct numbers $i$ and $j$ are chosen randomly from the set $\{1, 2, \ldots, 4m+2\}$, with $i < j$. Let $P_m$ be the probability that the sequence $a_1, a_2, \ldots, a_{4m+2}$ is an $(i, j)$-divisible sequence. Prove that $P_m > \frac{1}{8}$.

As a side note, without tool assistance (specifically, without using a web search to find a blog with the solution), I have not yet found any publicly available AI that can solve this problem completely, including o3, o3-pro, Gemini 2.5 Pro, and Grok 4.

[isCopyman]

Which is harder, this Gaokao question or an IMO question, in your opinion?

[forfrt]

i ran the script without changing anything, and got the following response for every run:

Error: Could not extract text from the API response.
Reason: 'parts'
Full API Response:
{
  "candidates": [
    {
      "content": {
        "role": "model"
      },
      "finishReason": "STOP",
      "index": 0
    }
  ],
  "usageMetadata": {
    "promptTokenCount": 751,
    "totalTokenCount": 13113,
    "promptTokensDetails": [
      {
        "modality": "TEXT",
        "tokenCount": 751
      }
    ],
    "thoughtsTokenCount": 12362
  },
  "modelVersion": "gemini-2.5-pro",
  "responseId": "mnydaMjEH72Hz7IPvbGSwQc"
}
>>>>>>> Error in run 0: 'parts'

it seems to me that the gemini-2.5-pro model stopped thinking by hitting an Internal, Unseen Limit. did the gemini-2.5-pro changed its setting?

the original prompt is printed as follow:

>>>>>>>>>>>>>>>>>>>>>>>>>> Run 0 of 10 ...
>>>>>> Initial prompt.
{
    "systemInstruction": {
        "role": "system",
        "parts": [
            {
                "text": "\n### Core Instructions ###\n\n*   **Rigor is Paramount:** Your primary goal is to produce a complete and rigorously justified solution. Every step in your solution must be logically sound and clearly explained. A correct final answer derived from flawed or incomplete reasoning is considered a failure.\n*   **Honesty About Completeness:** If you cannot find a complete solution, you must **not** guess or create a solution that appears correct but contains hidden flaws or justification gaps. Instead, you should present only significant partial results that you can rigorously prove. A partial result is considered significant if it represents a substantial advancement toward a full solution. Examples include:\n    *   Proving a key lemma.\n    *   Fully resolving one or more cases within a logically sound case-based proof.\n    *   Establishing a critical property of the mathematical objects in the problem.\n    *   For an optimization problem, proving an upper or lower bound without proving that this bound is achievable.\n*   **Use TeX for All Mathematics:** All mathematical variables, expressions, and relations must be enclosed in TeX delimiters (e.g., `Let $n$ be an integer.`).\n\n### Output Format ###\n\nYour response MUST be structured into the following sections, in this exact order.\n\n**1. Summary**\n\nProvide a concise overview of your findings. This section must contain two parts:\n\n*   **a. Verdict:** State clearly whether you have found a complete solution or a partial solution.\n    *   **For a complete solution:** State the final answer, e.g., \"I have successfully solved the problem. The final answer is...\"\n    *   **For a partial solution:** State the main rigorous conclusion(s) you were able to prove, e.g., \"I have not found a complete solution, but I have rigorously proven that...\"\n*   **b. Method Sketch:** Present a high-level, conceptual outline of your solution. This sketch should allow an expert to understand the logical flow of your argument without reading the full detail. It should include:\n    *   A narrative of your overall strategy.\n    *   The full and precise mathematical statements of any key lemmas or major intermediate results.\n    *   If applicable, describe any key constructions or case splits that form the backbone of your argument.\n\n**2. Detailed Solution**\n\nPresent the full, step-by-step mathematical proof. Each step must be logically justified and clearly explained. The level of detail should be sufficient for an expert to verify the correctness of your reasoning without needing to fill in any gaps. This section must contain ONLY the complete, rigorous proof, free of any internal commentary, alternative approaches, or failed attempts.\n\n### Self-Correction Instruction ###\n\nBefore finalizing your output, carefully review your \"Method Sketch\" and \"Detailed Solution\" to ensure they are clean, rigorous, and strictly adhere to all instructions provided above. Verify that every statement contributes directly to the final, coherent mathematical argument.\n\n"
            }
        ]
    },
    "contents": [
        {
            "role": "user",
            "parts": [
                {
                    "text": "*** Problem Statement ***\n\nA line in the plane is called *sunny* if it is not parallel to any of the $x$-axis, the $y$-axis, and the line $x+y=0$.\n\nLet $n\\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both the following:\n*   for all positive integers $a$ and $b$ with $a+b\\le n+1$, the point $(a,b)$ is on at least one of the lines; and\n*   exactly $k$ of the lines are sunny."
                }
            ]
        }
    ],
    "generationConfig": {
        "temperature": 0.1,
        "topP": 1.0,
        "thinkingConfig": {
            "thinkingBudget": 32768
        }
    }
}

[xx205]

@forfrt Maybe you can try my modified OpenAI sdk compatible streaming API version.

[xx205]

Which is harder, this Gaokao question or an IMO question, in your opinion?

In my opinion, the problems in the Gaokao are simpler than those in the IMO (International Mathematical Olympiad), but these large models do not seem to agree.

[godcreator02]

Hi, I wonder how long does this program take to solve the first two IMO problems? Hours or Minutes?

[yichenhuang]

Hi, I wonder how long does this program take to solve the first two IMO problems? Hours or Minutes?

One run takes about 30 to 60 min and you might need to run 1 to 5 times (depending on luck).