{"task": {"agent_timeout": 600, "task": "omnimath_4380", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nThere is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual),\nso that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured\nmembrane so that no two friends have the same colour, then there is a way to do it with $2021$\ncolours, but not with $2020$ or less.\nTwo friends $A$ and $B$ can decide to merge in which case they become a single bacterion whose\nfriends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are\nnot friends.) It turns out that no matter how we perform one merge or two consecutive merges,\nin the resulting population it would be possible to assign $2020$ colours or less so that no two\nfriends have the same colour. Is it true that in any such population $P$ every bacterium has at\nleast $2021$ friends?\n\n## Instructions\n\nSolve the mathematical problem above and write your final answer to `/workspace/answer.txt`.\n\n**Important**: Write only your final answer to the file, not the full solution process.\n\n### Guidelines\n\n- Provide your final numerical answer or mathematical expression\n- Write the answer as plain text (no special formatting needed)\n- Be precise and clear in your answer\n- The answer should directly respond to what the problem asks for\n\n### Example\n\nIf the problem asks \"What is 2 + 2?\", your answer file should contain:\n\n```\n4\n```\n\nYour answer will be evaluated against the correct solution using an automated grading system.\n", "memory": "", "runnable": false, "difficulty": "hard", "language": "", "cpus": "", "instruction_truncated": false, "category": "math", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "omnimath", "tags": ["omni-math"]}, "runs": []}