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/volume8/exercise5.tex",3 "problem_type": "calculation",4 "problem": "问题1. 空间四点 $A, B, C, D$ 满足 $|\\overrightarrow{A B}|=3,|\\overrightarrow{B C}|=7,|\\overrightarrow{C D}|=11,|\\overrightarrow{D A}|=$ 9 , 则 $\\overrightarrow{A C} \\cdot \\overrightarrow{B D}$ 的取值 ( ).\n(A) 只有一个\n(B) 有两个\n(C) 有四个\n(D) 有无穷多个",5 "solution": "A.\n因为 $\\overrightarrow{A B}^2+\\overrightarrow{C D}^2=3^2+11^2=130=7^2+9^2=\\overrightarrow{B C}^2+\\overrightarrow{D A}^2$, 由 $\\overrightarrow{A B}+\\overrightarrow{B C}+ \\overrightarrow{C D}+\\overrightarrow{D A}=\\overrightarrow{0}$, 得 $\\overrightarrow{A B}+\\overrightarrow{C D}=-(\\overrightarrow{B C}+\\overrightarrow{D A})$, 两边平方得 $\\overrightarrow{A B} \\cdot \\overrightarrow{C D}=\\overrightarrow{B C} \\cdot \\overrightarrow{D A}$, 故 $\\overrightarrow{A B} \\cdot \\overrightarrow{C D}=-\\overrightarrow{A D} \\cdot \\overrightarrow{B C}$, 于是\n$$\n\\begin{aligned}\n\\overrightarrow{A C} \\cdot \\overrightarrow{B D} & =(\\overrightarrow{A B}+\\overrightarrow{B C})(\\overrightarrow{B C}+\\overrightarrow{C D}) \\\\\n& =(\\overrightarrow{A B}+\\overrightarrow{B C}+\\overrightarrow{C D}) \\overrightarrow{B C}+\\overrightarrow{A B} \\cdot \\overrightarrow{C D} \\\\\n& =\\overrightarrow{A D} \\cdot \\overrightarrow{B C}+\\overrightarrow{A B} \\cdot \\overrightarrow{C D}=0 .\n\\end{aligned}\n$$\n$\\overrightarrow{A C} \\cdot \\overrightarrow{B D}$ 只有一个值 0. 故选 A.",6 "remark": "",7 "figures": []8}