{"task": {"agent_timeout": 600, "task": "omnimath_3953", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nTwo rational numbers \\(\\tfrac{m}{n}\\) and \\(\\tfrac{n}{m}\\) are written on a blackboard, where \\(m\\) and \\(n\\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \\(x\\) and \\(y\\) written on the board and write either their arithmetic mean \\(\\tfrac{x+y}{2}\\) or their harmonic mean \\(\\tfrac{2xy}{x+y}\\) on the board as well. Find all pairs \\((m,n)\\) such that Evan can write 1 on the board in finitely many steps.\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": []}