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/volume2/chapter2.tex",3 "problem_type": "calculation",4 "problem": "例4 求 $y=(3 x-1)\\left(\\sqrt{9 x^2-6 x+5}+1\\right)+(2 x-3)\\left(\\sqrt{4 x^2-12 x+13}+ 1\\right)$ 的图象与 $x$ 轴的交点坐标.",5 "solution": "分析:仔细观察所给的式子, 发现\n$$\ny=(3 x-1)\\left(\\sqrt{(3 x-1)^2+4}+1\\right)+(2 x-3)\\left(\\sqrt{(2 x-3)^2+4}+1\\right),\n$$\n从而找到了解题途径.\n解因为 $y=(3 x-1)\\left(\\sqrt{(3 x-1)^2+4}+1\\right)+(2 x-3) \\left(\\sqrt{(2 x-3)^2+4}+1\\right)$.\n令 $f(t)=t\\left(\\sqrt{t^2+4}+1\\right)$, 易知 $f(t)$ 是奇函数, 且 $f(t)$ 是增函数, 所以\n$$\ny=f(3 x-1)+f(2 x-3) \\text {. }\n$$\n当 $y=0$ 时, $f(3 x-1)=-f(2 x-3)=f(3-2 x)$, 所以\n$$\n3 x-1=3-2 x .\n$$\n解方程, 得 $x=\\frac{4}{5}$.\n故图象与 $x$ 轴的交点坐标为 $\\left(\\frac{4}{5}, 0\\right)$.",6 "remark": "",7 "figures": []8}