{"task": {"agent_timeout": 600, "task": "omnimath_3597", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nAn empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn  on each of its six faces. A [i]beam[/i] is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\n[list=]\n[*]The two $1 \\times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \\cdot {2020}^2$ possible positions for a beam.)\n[*]No two beams have intersecting interiors.\n[*]The interiors of each of the four $1 \\times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.\n[/list]\nWhat is the smallest positive number of beams that can be placed to satisfy these conditions?\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": []}