{"task": {"agent_timeout": 3600, "task": "20", "verifier_timeout": 3600, "instruction": "Solve the following problem. Reason step by step, create the file /app/answer.txt, and put your final answer there. \n\nTask description: Please solve the problem with clear, rigorous, and logically sound steps. At the end of your response, state your answer in exactly this format: 'The answer is $C=X$', where X is your calculated numerical bound value. Example: 'The answer is $C=1$'.\n\nProblem: Given $a, b, c \\geq 0$ satisfying $a b+b c+c a>0$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$:\n\n$$\n\\sqrt{\\frac{a}{b^2+b c+c^2}}+\\sqrt{\\frac{b}{c^2+c a+a^2}}+\\sqrt{\\frac{c}{a^2+a b+b^2}} \\geq C \\sqrt{\\frac{a b+b c+c a}{(a+b)(b+c)(c+a)}}\n$$\n\nSolution:", "memory": "", "runnable": false, "difficulty": "difficult", "language": "", "cpus": "", "instruction_truncated": false, "category": "math reasoning", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "ineqmath", "tags": ["IneqMath", "multiple-choice", "bound"]}, "runs": []}