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/exercise19.tex",3 "problem_type": "calculation",4 "problem": "问题1. 已知三个数 $5,12,18$. 每一次操作是从这三个数中任意选出两个 $a, b$, 并用 $\\frac{\\sqrt{2}}{2}(a+b), \\frac{\\sqrt{2}}{2}(a-b)$ 代替它们.\n问: 是否能进行有限次操作, 使得得到的三个数为 $3,13,20$ ?",5 "solution": "因为\n$$\n\\left[\\frac{\\sqrt{2}}{2}(a+b)\\right]^2+\\left[\\frac{\\sqrt{2}}{2}(a-b)\\right]^2=a^2+b^2,\n$$\n所以, 每一次操作后, 这三个数的平方和保持不变 (即三个数的平方和是不变量). 而 $5^2+12^2+18^2=493,3^2+13^2+20^2=578$, 于是, 无论如何操作, 数 $5,12,18$ 不会变为数 $3,13,20$.",6 "remark": "",7 "figures": []8}