{"task": {"agent_timeout": 600, "task": "omnimath_175", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nLet $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$ . For positive integer $k$ , define\n$x_{k}=\\begin{cases}0&\\text{ if }x_{k-1}=0\\\\ \\left\\{\\frac{p_{k}}{x_{k-1}}\\right\\}&\\text{ if }x_{k-1}\\ne0\\end{cases}$ \nwhere $\\{x\\}$ denotes the fractional part of $x$ . (The fractional part of $x$ is given by $x-\\lfloor{x}\\rfloor$ where $\\lfloor{x}\\rfloor$ is the greatest integer less than or equal to $x$ .) Find, with proof, all $x_0$ satisfying $0<x_0<1$ for which the sequence $x_0,x_1,x_2,...$ eventually becomes $0$ .\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": []}