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/volume12/exercise7.tex",3 "problem_type": "calculation",4 "problem": "问题8. 对哪些 $n$, 存在 $n$ 条棱的多面体?",5 "solution": "以多面体的顶点为图的顶点, 以多面体的棱为边, 构成一个连通平面图.\n则 $v \\geqslant 4, f \\geqslant 4$. 由欧拉公式 $e=v+f-2 \\geqslant 6$, 即没有棱数少于 6 的多面体.\n若有 $e=7$ 的图, 则 $3 f \\leqslant 2 \\times 7$, 即 $f=4$, 但四个面的多面体只有 6 条棱, 故无 7 条棱的多面体.\n考虑 $k \\geqslant 4$, 以 $k$ 边形为底的棱雉为 $2 k$ 条棱的多面体,而把 $k-1$ 边形为底的棱雉底角处的一个三面角\"锯掉一个小尖儿\", 得 $2 k+1$ 条棱的多面体.\n综上, $n \\geqslant 6, n \\neq 7$ 时, 有 $n$ 条棱的多面体.",6 "remark": "",7 "figures": []8}