{"task": {"agent_timeout": 600, "task": "omnimath_4173", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nTwo different integers $u$ and $v$ are written on a board. We perform a sequence of steps. At each step we do one of the following two operations:\n\n(i) If $a$ and $b$ are different integers on the board, then we can write $a + b$ on the board, if it is not\nalready there.\n(ii) If $a$, $b$ and $c$ are three different integers on the board, and if an integer $x$ satisfies $ax^2 +bx+c = 0$,\nthen we can write $x$ on the board, if it is not already there.\n\nDetermine all pairs of starting numbers $(u, v)$ from which any integer can eventually be written on the board after a finite sequence of 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": []}