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/volume3/exercise2.tex",3 "problem_type": "calculation",4 "problem": "问题21. 求下列各式的值:\n(1) $\\cos ^2 24^{\\circ}+\\sin ^2 6^{\\circ}+\\cos ^2 18^{\\circ}$;\n(2) $4 \\cos ^2 36^{\\circ}-\\sin 84^{\\circ}\\left(\\sqrt{3}-\\tan 6^{\\circ}\\right)$.",5 "solution": "(1) $\\cos ^2 24^{\\circ}+\\sin ^2 6^{\\circ}+\\cos ^2 18^{\\circ}=\\frac{3}{2}+\\frac{1}{2}\\left(\\cos 48^{\\circ}-\\cos 12^{\\circ}+\\right. \\left.\\cos 36^{\\circ}\\right)=\\frac{3}{2}+\\frac{1}{2}\\left(-2 \\sin 30^{\\circ} \\sin 18^{\\circ}+\\cos 36^{\\circ}\\right)=\\frac{3}{2}+\\frac{1}{2}\\left(\\sin 54^{\\circ}-\\sin 18^{\\circ}\\right)= \\frac{3}{2}+\\cos 36^{\\circ} \\sin 18^{\\circ}=\\frac{3}{2}+\\frac{\\cos 36^{\\circ} \\sin 18^{\\circ} \\cos 18^{\\circ}}{\\cos 18^{\\circ}}=\\frac{3}{2}+\\frac{\\frac{1}{4} \\sin 72^{\\circ}}{\\cos 18^{\\circ}}=\\frac{7}{4}$.\n (2)参照例 3 的解法, 可得 $4 \\cos ^2 36^{\\circ}-\\sin 84^{\\circ}\\left(\\sqrt{3}-\\tan 6^{\\circ}\\right)=1$.",6 "remark": "",7 "figures": []8}