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.
39.7k
1{2 "source_file": "./raw_volume-zh/volume2/exercise7.tex",3 "problem_type": "calculation",4 "problem": "问题10 求所有满足下面方程的函数 $f: \\mathbf{R} \\rightarrow \\mathbf{R}$ : 对任意实数 $x 、 y$, 有\n$$\nx f(y)+y f(x)=(x+y) f(x) f(y) .\n$$",5 "solution": "取 $x=y=1$ 代入题设方程, 得 $2 f(1)=2(f(1))^2$, 所以 $f(1)=0$ 或 $f(1)=1$. (1) 若 $f(1)=0$. 取 $y=1$ 代入原方程, 得 $x f(1)+f(x)= (x+1) f(x) f(1)$, 所以 $f(x)=0$. (2) 若 $f(1)=1$, 取 $y=1$ 代入原方程, 得 $x+f(x)=(x+1) f(x), x(f(x)-1)=0$. 所以 $f(x)= \\begin{cases}1, \\text { 当 } x \\neq 0 \\text { 时; } \\\\ \\text { 任意实数, 当 } x=0 \\text { 时.\n}\\end{cases}$.",6 "remark": "",7 "figures": []8}