{"task": {"agent_timeout": 600, "task": "omnimath_525", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\n(a) Can 1000 queens be placed on a $2017 \\times 2017$ chessboard such that every square is attacked by some queen? A square is attacked by a queen if it lies in the same row, column, or diagonal as the queen. (b) A $2017 \\times 2017$ grid of squares originally contains a 0 in each square. At any step, Kelvin the Frog chooses two adjacent squares (two squares are adjacent if they share a side) and increments the numbers in both of them by 1. Can Kelvin make every square contain a different power of 2? (c) A tournament consists of single games between every pair of players, where each game has a winner and loser with no ties. A set of people is dominated if there exists a player who beats all of them. Does there exist a tournament in which every set of 2017 people is dominated? (d) Every cell of a $19 \\times 19$ grid is colored either red, yellow, green, or blue. Does there necessarily exist a rectangle whose sides are parallel to the grid, all of whose vertices are the same color? (e) Does there exist a $c \\in \\mathbb{R}^{+}$such that $\\max (|A \\cdot A|,|A+A|) \\geq c|A| \\log ^{2}|A|$ for all finite sets $A \\subset \\mathbb{Z}$? (f) Can the set $\\{1,2, \\ldots, 1093\\}$ be partitioned into 7 subsets such that each subset is sum-free (i.e. no subset contains $a, b, c$ with $a+b=c)$?\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": []}