{"task": {"agent_timeout": 600, "task": "omnimath_2", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nA tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other.  Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\\overrightarrow{uv}$ and $\\overrightarrow{vw}$, those two edges are in different colors.  Note that it is permissible for $\\overrightarrow{uv}$ and $\\overrightarrow{uw}$ to be the same color.  The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring.  For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices.\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": []}