{"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":"6505e61a-7d93-54a7-b006-3251e8fc7a57","task_key":"olympiad--test--20daab54~2ddf1b~2d4006~2da544~2d7198daee7847","task_revision_id":"1","upstream_id":"20daab54-df1b-4006-a544-7198daee7847","short_description":"Suppose we have an object with large mass `\\( M \\)`. An object with mass `\\(…","config":"olympiad","split":"test","body":"{\"problem\":\"Suppose we have an object with large mass `\\\\( M \\\\)`. An object with mass `\\\\( m\\\\)` moves within the gravitational field created by `\\\\( M \\\\)`. Assume `\\\\( m \\\\ll M \\\\)`. Assume the angular momentum of `\\\\( m \\\\)` relative to `\\\\( M \\\\)` is `\\\\( L \\\\)`, and the total energy is `\\\\( E \\\\)`. Assume `\\\\( E < 0 \\\\)`. Assume the motion is in the xy-plane with `\\\\( M \\\\)` at the origin. When `\\\\( m \\\\)` is closest to `\\\\( M \\\\)`, it is on the x-axis, and at this time, the velocity of `\\\\( m \\\\)` is in the +y direction. Derive the equation of the curve formed when plotting the trajectory of `\\\\( m \\\\)` in velocity space (with `\\\\( v_x = \\\\frac{dx}{dt} \\\\)` and `\\\\( v_y = \\\\frac{dy}{dt} \\\\)` as the two coordinate axes). Give your answer in the form `\\\\( v_x^2 + (v_y - a)^2 = b \\\\)`, where `\\\\( a \\\\) and \\\\( b \\\\) are expressions in terms of` `\\\\( G \\\\)` (the universal gravitational constant), `\\\\( M\\\\)`, `\\\\( m \\\\)`, `\\\\( L \\\\)`, `\\\\( E \\\\)`, `\\\\( v_x\\\\)`, and `\\\\( v_y \\\\)`.\\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":[]}