math-ai/BlueMO
BlueMO 🚀 BlueMO: A Comprehensive Collection of Challenging Mathematical Olympiad Problems from the Little Blue Book Series BlueMO is a comprehensive and challenging dataset comprising mathematical olympiad problems paired with detailed solutions, meticulously curated from the esteemed "Little Blue Book" (小蓝书) series (Second Edition)—a vital resource for Chinese students training for national and international olympiad math competitions. Designed to… See the full description on the dataset page: https://huggingface.co/datasets/math-ai/BlueMO.
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1{2 "source_file": "./raw_volume-zh/volume14/exercise12.tex",3 "problem_type": "calculation",4 "problem": "问题1. 在一个圆周上如此选定 $n$ 个点,使得它们两两之间连以弦之后, 任何三条弦之间除端点外不交于同一个点, 问: 这时圆内一共有多少个交点?",5 "solution": "圆上每 4 个点构成一个凸四边形, 它的两条对角线 (弦) 交于一点, 因此每 4 点组成的集合对应一个交点, 由于没有三条弦交于一点,所以不同的 4 个点对应于不同的交点.\n反之, 设点 $P$ 是弦 $A_1 A_3$ 与 $A_2 A_4$ 的交点, 那么 $P$ 是 4 点集 $\\left\\{A_1, A_2, A_3, A_4\\right\\}$ 对应的点.\n从而交点个数就是这 $n$ 个点中取 4 点的取法总数 $\\mathrm{C}_n^4$.",6 "remark": "",7 "figures": []8}