{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"frontierscience","formal_name":"FrontierScience","introduction":"専門的な科学課題を解く能力を評価するベンチマークです。公開データはolympiadとresearchに分かれ、競技問題と研究課題を区別して扱います。\n\nFrontierScience evaluates the ability to solve expert-level scientific tasks. Its public data separates olympiad and research problems so that competition and research tasks can be examined independently.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/openai/frontierscience","indexing_mode":"noindex"},"task_id":"10483cba-c760-5a08-9a5f-bb49367e36b6","task_key":"olympiad--test--4f1cf869~2d3ea3~2d400c~2d8fa3~2de4d3f6fe1cdf","task_revision_id":"1","upstream_id":"4f1cf869-3ea3-400c-8fa3-e4d3f6fe1cdf","short_description":"Two uniform balls of radius `\\(r\\)` and density `\\(ρ\\)` are in contact with each…","config":"olympiad","split":"test","body":"{\"problem\":\"Two uniform balls of radius `\\\\(r\\\\)` and density `\\\\(ρ\\\\)` are in contact with each other, and they revolve around a uniform ball of radius `\\\\(R_0\\\\)` and density `\\\\(ρ_M\\\\)` with the same angular velocity. The centers of the three balls remain collinear, and the orbit of the revolution is circular. Suppose `\\\\(ρ_M R_0^3≫ρr^3\\\\)` and the distance between the point of contact of the two small spheres and the center of the large sphere `\\\\(R \\\\gg r\\\\)`. We can assume that the large ball is not moving at all due to its huge mass. The only force of interaction between the bodies that we consider here is gravity. Find the critical value for `\\\\( R \\\\)` denoted as `\\\\( R_C \\\\)` such that the two small balls keep in touch. Express your answer in terms of the constants defined above.\\n\\nThink step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.\",\"subject\":\"physics\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/openai/frontierscience","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}