{"id":1121670,"date":"2025-01-16T02:55:01","date_gmt":"2025-01-16T10:55:01","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-blog-post&p=1121670"},"modified":"2025-01-16T02:55:49","modified_gmt":"2025-01-16T10:55:49","slug":"li-zhang-rstar-math","status":"publish","type":"msr-blog-post","link":"https:\/\/www.microsoft.com\/en-us\/research\/articles\/li-zhang-rstar-math\/","title":{"rendered":"\u5bf9\u8bdd | \u5f20\u4e3d\uff1a\u5927\u9053\u81f3\u7b80\uff0crStar-Math\u8ba9\u5c0f\u8bed\u8a00\u6a21\u578b\u5177\u6709\u66f4\u5f3a\u7684\u63a8\u7406\u80fd\u529b"},"content":{"rendered":"\n

\u7f16\u8005\u6309\uff1a\u8fd1\u5e74\u6765\uff0c\u4eba\u5de5\u667a\u80fd\u5728\u8bed\u8a00\u5904\u7406\u3001\u89c6\u89c9\u751f\u6210\u7b49\u9886\u57df\u7684\u8868\u73b0\u4ee4\u4eba\u60ca\u53f9\uff0c\u4f46\u5728\u590d\u6742\u7684\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e0a\u4ecd\u9762\u4e34\u6311\u6218\u3002\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u63a8\u51fa\u7684\u65b0\u7b97\u6cd5 rStar-Math\uff0c\u901a\u8fc7\u5f15\u5165\u7c7b\u4f3c\u4eba\u7c7b\u7cfb\u7edf\u7684\u6162\u601d\u8003\u548c\u63a8\u7406\u601d\u7ef4\uff0c\u663e\u8457\u63d0\u5347\u4e86\u5c0f\u8bed\u8a00\u6a21\u578b\uff08SLMs\uff09\u7684\u6570\u5b66\u63a8\u7406\u80fd\u529b\u3002rStar-Math \u6253\u7834\u4e86\u201c\u53ea\u6709\u5927\u6a21\u578b\u624d\u6709\u9ad8\u6027\u80fd\u201d\u7684\u56fa\u6709\u89c2\u5ff5\uff0c\u8bc1\u660e\u4e86\u5c0f\u8bed\u8a00\u6a21\u578b\u7ecf\u5de7\u5999\u8bbe\u8ba1\uff0c\u540c\u6837\u80fd\u5b9e\u73b0\u5353\u8d8a\u7684\u63a8\u7406\u6548\u679c\uff0c\u751a\u81f3\u5728\u90e8\u5206\u5965\u6570\u7ea7\u522b\u7684\u6311\u6218\u4e2d\u8d85\u8d8a\u4e86\u73b0\u6709\u5927\u8bed\u8a00\u6a21\u578b\u3002<\/p>\n\n\n\n

\n\n\n\n

\u5f53\u524d\u5927\u8bed\u8a00\u6a21\u578b\uff08LLMs\uff09\u867d\u7136\u5df2\u5177\u5907\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u7684\u80fd\u529b\uff0c\u4f46\u5176\u5e38\u7528\u7684\u63a8\u7406\u89e3\u9898\u601d\u8def\u7c7b\u4f3c\u4e8e\u4eba\u7c7b\u7cfb\u7edf1\uff08System 1\uff09\u7684\u76f4\u89c9\u601d\u7ef4\u6a21\u5f0f\uff0c\u901f\u5ea6\u5feb\u5374\u6613\u51fa\u9519\u3002\u5728\u6b64\u80cc\u666f\u4e0b\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u7684\u7814\u7a76\u5458\u4eec\u63d0\u51fa\u4e86\u5927\u8bed\u8a00\u6a21\u578b\u53ef\u5411\u4eba\u7c7b\u7cfb\u7edf2\uff08System 2\uff09\u504f\u5411\u89c4\u5219\u7684\u601d\u7ef4\u8303\u5f0f\u8f6c\u53d8\uff0c\u901a\u8fc7\u6162\u4e14\u6df1\u5165\u7684\u601d\u8003\u6a21\u5f0f\u6a21\u62df\u4eba\u7c7b\u63a8\u7406\u7684\u8fc7\u7a0b\u3002\u4e3a\u6b64\uff0c\u7814\u7a76\u5458\u4eec\u8bbe\u8ba1\u51fa\u4e86 rStar-Math \u7b97\u6cd5\uff0c\u501f\u52a9\u8499\u7279\u5361\u6d1b\u641c\u7d22\uff08MCTS\uff09\u4f7f\u5176\u5b9e\u73b0\u7c7b\u4f3c\u4eba\u7c7b\u7cfb\u7edf2\u6df1\u5ea6\u601d\u8003\u7684\u81ea\u6211\u8fdb\u5316\u673a\u5236\uff0c\u5c06\u5c0f\u8bed\u8a00\u6a21\u578b\u7684\u6570\u5b66\u63a8\u7406\u80fd\u529b\u63a8\u5411\u4e86\u65b0\u7684\u9ad8\u5ea6\u3002\u5728\u7f8e\u56fd\u6570\u5b66\u5965\u6797\u5339\u514b\u7ade\u8d5b\u4e2d\uff0crStar-Math \u5e73\u5747\u89e3\u51b3\u4e8653.3%\uff088\/15\uff09\u7684\u95ee\u9898\uff0c\u6210\u7ee9\u8dfb\u8eab\u53c2\u8d5b\u9ad8\u4e2d\u751f\u7684\u524d20%\u3002<\/p>\n\n\n\n

rStar-Math \u7684\u81ea\u6211\u8fdb\u5316\u673a\u5236\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f\u8fd9\u79cd\u81ea\u6211\u8fdb\u5316\u80fd\u5426\u5728\u5176\u4ed6\u573a\u666f\u590d\u7528\uff1f\u5c0f\u8bed\u8a00\u6a21\u578b\u6570\u5b66\u63a8\u7406\u80fd\u529b\u7684\u63d0\u5347\uff0c\u5bf9\u73b0\u5b9e\u4e16\u754c\u6709\u600e\u6837\u7684\u610f\u4e49\uff1f\u9762\u5bf9\u4eba\u5de5\u667a\u80fd\u6570\u5b66\u63a8\u7406\u80fd\u529b\u7684\u8bf8\u591a\u96be\u9898\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u662f\u5426\u8fd8\u6709\u66f4\u6df1\u5165\u7684\u7814\u7a76\u8ba1\u5212\uff1f\u9488\u5bf9\u8fd9\u4e9b\u95ee\u9898\uff0c\u6211\u4eec\u548c\u53c2\u4e0e rStar-Math \u7814\u7a76\u7684\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u9ad8\u7ea7\u7814\u7a76\u5458\u5f20\u4e3d\u5c55\u5f00\u4e86\u6df1\u5165\u5bf9\u8bdd\uff0c\u4e00\u540c\u63a2\u5bfb rStar-Math \u80cc\u540e\u7684\u601d\u8003\u4e0e\u6280\u672f\u7ec6\u8282\u3002<\/p>\n\n\n\n

\"\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u9ad8\u7ea7\u7814\u7a76\u5458\u5f20\u4e3d\"
\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u9ad8\u7ea7\u7814\u7a76\u5458\u5f20\u4e3d<\/figcaption><\/figure>\n\n\n\n

Q\uff1arStar-Math \u8fd9\u9879\u7814\u7a76\u8bd5\u56fe\u89e3\u51b3\u4ec0\u4e48\u5173\u952e\u95ee\u9898\uff1f\u4e3a\u4f55\u80fd\u5728\u4e1a\u754c\u5f15\u53d1\u5e7f\u6cdb\u5173\u6ce8\uff1f<\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>rStar-Math \u7814\u7a76\u805a\u7126\u4e8e\u4e00\u4e2a\u6838\u5fc3\u95ee\u9898\uff1a\u5728\u6570\u5b66\u63a8\u7406\u9886\u57df\uff0c\u5927\u8bed\u8a00\u6a21\u578b\u662f\u5426\u80fd\u591f\u901a\u8fc7 System 2 \u6df1\u5ea6\u601d\u8003\u5b9e\u73b0\u81ea\u6211\u8fdb\u5316\uff0c\u4ece\u800c\u5728\u8d44\u6e90\u53d7\u9650\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u5e45\u63d0\u5347\u63a8\u7406\u80fd\u529b\uff0c\u663e\u8457\u964d\u4f4e\u9ad8\u6027\u80fd\u4eba\u5de5\u667a\u80fd\u7684\u5e94\u7528\u95e8\u69db\u3002\u6211\u4eec\u5e0c\u671b\u901a\u8fc7\u8fd9\u4e00\u63a2\u7d22\u7a81\u7834\u73b0\u6709\u6a21\u578b\u7684\u63a8\u7406\u80fd\u529b\u6781\u9650\uff0c\u4e3a\u63d0\u5347\u5927\u8bed\u8a00\u6a21\u578b\u80fd\u529b\u5f00\u8f9f\u65b0\u65b9\u5411\u3002<\/p>\n\n\n\n

\u968f\u7740 OpenAI o1\/o3 \u6a21\u578b\u5728\u6570\u5b66\u548c\u4ee3\u7801\u7b49\u590d\u6742\u63a8\u7406\u4efb\u52a1\u4e2d\u5c55\u73b0\u51fa\u5353\u8d8a\u6027\u80fd\uff0c\u6d4b\u8bd5\u65f6\u63a8\u7406\uff08test-time scaling laws\uff09\u65b9\u6cd5\u5f00\u59cb\u88ab\u89c6\u4e3a\u63d0\u5347\u5927\u6a21\u578b\u6027\u80fd\u7684\u65b0\u8303\u5f0f\u3002\u7136\u800c\uff0c\u8fd9\u4e9b\u6a21\u578b\u7684\u5177\u4f53\u6280\u672f\u8def\u7ebf\u5c1a\u4e0d\u660e\u786e\u3002\u6211\u4eec\u7684 rStar-Math \u662f\u9996\u4e2a\u516c\u5f00\u53d1\u5e03\u7684\u6280\u672f\u7814\u7a76\u62a5\u544a<\/strong>\uff0c\u586b\u8865\u4e86\u8fd9\u4e00\u9886\u57df\u7684\u7a7a\u767d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0crStar-Math \u5e76\u672a\u50cf\u9886\u5148\u7684\u5927\u6a21\u578b\uff08\u4f8b\u5982 GPT-4o \u6216 o1\uff09\u4e00\u6837\u8fdb\u884c\u84b8\u998f\uff0c\u800c\u662f\u5229\u7528\u6781\u4e3a\u6709\u9650\u7684\u8d44\u6e90\uff0c\u5728\u4ec5 7B \u53c2\u6570\u91cf\u7684\u6a21\u578b\u4e0a\u5b9e\u73b0\u4e86\u66f4\u9ad8\u7684\u6570\u5b66\u63a8\u7406\u6c34\u5e73\uff0c\u751a\u81f3\u5728\u67d0\u4e9b\u5965\u6570\u7ea7\u522b\u6d4b\u8bd5\u4e2d\u8d85\u8d8a\u4e86 o1 \u7684\u6570\u5b66\u63a8\u7406\u8868\u73b0\u3002<\/p>\n\n\n\n

rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking<\/strong><\/p>\n\n\n\n

\u8bba\u6587\u94fe\u63a5\uff1ahttps:\/\/arxiv.org\/abs\/2501.04519 (opens in new tab)<\/span><\/a><\/p>\n\n\n\n

Q\uff1a\u4e1a\u754c\u5bf9 rStar-Math \u7684\u53cd\u9988\u5982\u4f55\uff1f\u6709\u6ca1\u6709\u54ea\u4e9b\u53cd\u9988\u7ed9\u4f60\u7559\u4e0b\u4e86\u6df1\u523b\u7684\u5370\u8c61\uff0c\u751a\u81f3\u4fc3\u4f7f\u4f60\u548c\u56e2\u961f\u5bf9\u76f8\u5173\u7814\u7a76\u6709\u4e86\u65b0\u7684\u60f3\u6cd5\uff1f<\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>rStar-Math \u7684\u201c\u7206\u706b\u201d\u8ba9\u6211\u4eec\u65e2\u60ca\u559c\u53c8\u610f\u5916\u3002rStar-Math \u4e0d\u4ec5\u5728\u5b66\u672f\u5708\u5185\u5f15\u53d1\u4e86\u5e7f\u6cdb\u5173\u6ce8\uff0c\u8fd8\u201c\u7834\u5708\u201d\u5438\u5f15\u4e86\u5f88\u591a\u5a92\u4f53\u7684\u76ee\u5149\uff0c\u8d85\u51fa\u4e86\u6211\u4eec\u7684\u9884\u671f\u3002\u8bb8\u591a\u5a92\u4f53\u5bf9 rStar-Math \u7684\u610f\u4e49\u90fd\u7ed9\u4e88\u4e86\u9ad8\u5ea6\u8bc4\u4ef7\uff0c\u5c24\u5176\u5f3a\u8c03\u5b83\u4ec5\u7528 7B \u5c0f\u6a21\u578b\u5b9e\u73b0\u590d\u6742\u63a8\u7406\u7684\u80fd\u529b\uff0c\u5c55\u73b0\u51fa\u4e86\u4f4e\u8d44\u6e90\u3001\u9ad8\u6548\u7387\u7684\u8def\u5f84\u3002\u76f8\u6bd4\u5f53\u524d\u8bb8\u591a\u9700\u8981\u5e9e\u5927\u7b97\u529b\u652f\u6301\u7684\u5de5\u5177\uff0c\u8fd9\u4e00\u6210\u679c\u964d\u4f4e\u4e86\u5bf9\u8ba1\u7b97\u8d44\u6e90\u7684\u4f9d\u8d56\uff0c\u65e2\u8282\u80fd\u73af\u4fdd\uff0c\u53c8\u4e3a\u5c0f\u6a21\u578b\u66f4\u5e7f\u6cdb\u7684\u5b9e\u9645\u5e94\u7528\u6253\u5f00\u4e86\u5927\u95e8\u3002<\/p>\n\n\n\n

\u76f8\u5173\u9886\u57df\u7814\u7a76\u4eba\u5458\u5bf9\u6211\u4eec\u4ec5\u901a\u8fc7\u56db\u8f6e\u81ea\u6211\u8fdb\u5316\u5c31\u80fd\u53d6\u5f97\u5982\u6b64\u4f18\u5f02\u7684\u6548\u679c\u8868\u793a\u60ca\u559c\u3002\u6211\u7559\u610f\u5230\uff0c\u793e\u533a\u79f0\u8d5e rStar-Math \u5728\u8499\u7279\u5361\u6d1b\u7b97\u6cd5\u4e2d\u5229\u7528 Q-value \u81ea\u52a8\u6807\u6ce8\u63a8\u7406\u6b65\u9aa4\u5956\u52b1\u5206\u6570\u7684\u8bbe\u8ba1\u7b80\u6d01\u6709\u6548\u3002\u4f5c\u4e3a\u7814\u7a76\u8005\uff0c\u770b\u5230\u8fd9\u4e9b\u8bc4\u4ef7\u8ba9\u6211\u611f\u5230\u975e\u5e38\u6b23\u6170\uff0c\u56e0\u4e3a\u6211\u59cb\u7ec8\u76f8\u4fe1\uff0c\u6700\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u5f80\u5f80\u662f\u6700\u7b80\u6d01\u7684<\/strong>\uff0c\u8fd9\u4e0e\u201c\u5927\u9053\u81f3\u7b80\u201d\u7684\u7406\u5ff5\u4e0d\u8c0b\u800c\u5408\u3002<\/p>\n\n\n\n

\u5f53\u7136\uff0c\u4e5f\u6709\u8d28\u7591\u58f0\u97f3\uff0c\u4f8b\u5982\u6709\u4eba\u5bf9\u5c0f\u6a21\u578b\u80fd\u8fbe\u5230\u5982\u6b64\u4f18\u5f02\u7684\u6548\u679c\u611f\u5230\u96be\u4ee5\u7f6e\u4fe1\u3002\u8fd9\u4e9b\u53cd\u9988\u8ba9\u6211\u4eec\u610f\u8bc6\u5230\uff0c\u4ee3\u7801\u548c\u6570\u636e\u7684\u5f00\u6e90\u4e0d\u4ec5\u662f\u9a8c\u8bc1\u7814\u7a76\u6210\u679c\u7684\u6700\u4f73\u65b9\u5f0f\uff0c\u4e5f\u6709\u52a9\u4e8e\u89e3\u7b54\u793e\u533a\u7684\u7591\u95ee\uff0c\u63a8\u52a8\u4eba\u5de5\u667a\u80fd\u7814\u7a76\u7684\u4ea4\u6d41\u4e0e\u521b\u65b0\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u8ba1\u5212\u5c3d\u5feb\u5f00\u6e90 rStar-Math \u7684\u4ee3\u7801\u548c\u6570\u636e\uff0c\u4e00\u65b9\u9762\u4e3a\u5176\u4ed6\u7814\u7a76\u8005\u63d0\u4f9b\u4fbf\u5229\uff0c\u540c\u65f6\u4e5f\u5e0c\u671b\u501f\u6b64\u4fc3\u8fdb\u66f4\u5e7f\u6cdb\u7684\u5408\u4f5c\u548c\u8ba8\u8bba\uff0c\u5171\u540c\u63a8\u52a8\u8fd9\u4e00\u9886\u57df\u7684\u8fdb\u6b65\u3002<\/p>\n\n\n\n

Q\uff1arStar-Math \u4e2d\u5f15\u5165\u4e86\u81ea\u6211\u8fdb\u5316\u7684 System 2 \u63a8\u7406\u65b9\u6cd5\uff0c\u8fd9\u79cd\u81ea\u6211\u8fdb\u5316\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff0c\u5728\u5176\u4ed6\u573a\u666f\u4e2d\u662f\u5426\u5177\u6709\u53ef\u590d\u7528\u6027\uff1f <\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>rStar-Math \u7684\u81ea\u6211\u8fdb\u5316\u8fc7\u7a0b\u4e0e\u4eba\u7c7b\u7684\u6df1\u5ea6\u601d\u8003\u6709\u8bb8\u591a\u76f8\u4f3c\u4e4b\u5904\u3002\u4eba\u7c7b\u9762\u5bf9\u590d\u6742\u95ee\u9898\u65f6\uff0c\u6df1\u5165\u601d\u8003\u80fd\u52a0\u6df1\u5bf9\u95ee\u9898\u672c\u8eab\u53ca\u591a\u79cd\u89e3\u9898\u8def\u5f84\u7684\u7406\u89e3\uff0c\u66f4\u7cbe\u51c6\u627e\u5230\u89e3\u51b3\u65b9\u6848\u3002\u8bb8\u591a\u7a81\u7834\u6027\u6210\u679c\u6b63\u662f\u6e90\u81ea\u8fd9\u6837\u7684\u6df1\u5ea6\u601d\u8003\uff0c\u800c\u8fc7\u7a0b\u4e2d\u79ef\u7d2f\u7684\u6210\u529f\u4e0e\u5931\u8d25\u7ecf\u9a8c\u5219\u63a8\u52a8\u4e86\u8ba4\u77e5\u4e0e\u80fd\u529b\u7684\u63d0\u5347\u3002<\/p>\n\n\n\n

\u4ece\u6280\u672f\u5c42\u9762\u770b\uff0crStar-Math \u901a\u8fc7\u8499\u7279\u5361\u6d1b\u641c\u7d22\u7b97\u6cd5\u5b9e\u73b0\u4e86\u7c7b\u4f3c\u4eba\u7c7b\u7684\u6df1\u5ea6\u601d\u8003\u81ea\u6211\u8fdb\u5316\u673a\u5236\u3002<\/strong>\u6211\u4eec\u5206\u522b\u8bad\u7ec3\u4e86\u4e00\u4e2a\u6570\u5b66\u7b56\u7565\u6a21\u578b\uff087B \u53c2\u6570\uff09\u548c\u4e00\u4e2a\u5956\u52b1\u6a21\u578b\uff087B \u53c2\u6570\u7684\u63a8\u7406\u6b65\u9aa4\u504f\u597d\u6a21\u578b\uff09\uff0c\u5bf9\u4e8e74.7\u4e07\u9053\u6570\u5b66\u9898\u8fdb\u884c\u6df1\u5ea6\u63a8\u7406\u3002\u6bcf\u4e00\u8f6e\u63a8\u7406\u7ed3\u675f\u540e\uff0c\u9ad8\u8d28\u91cf\u89e3\u9898\u6570\u636e\u4f1a\u88ab\u6536\u96c6\uff0c\u7528\u4e8e\u66f4\u65b0\u6570\u5b66\u7b56\u7565\u6a21\u578b\uff0c\u63a8\u7406\u6b65\u9aa4\u504f\u597d\u7684\u53cd\u9988\u5219\u4f1a\u8fdb\u4e00\u6b65\u4f18\u5316\u5956\u52b1\u6a21\u578b\u3002\u968f\u7740\u8fed\u4ee3\u63a8\u8fdb\uff0c\u4e24\u4e2a\u6a21\u578b\u4e0d\u65ad\u53d8\u5f3a\uff0c\u518d\u6b21\u5bf9\u76f8\u540c\u6570\u5b66\u9898\u8fdb\u884c\u66f4\u6df1\u5ea6\u7684\u63a8\u7406\u3002\u901a\u8fc7\u5faa\u73af\u8fd9\u4e00\u8fc7\u7a0b\uff0crStar-Math \u4e0d\u4ec5\u663e\u8457\u63d0\u5347\u4e86\u63a8\u7406\u80fd\u529b\uff0c\u8fd8\u5c55\u73b0\u51fa\u7c7b\u4f3c\u4eba\u7c7b\u6df1\u5ea6\u601d\u8003\u540e\u81ea\u6211\u8fdb\u5316\u7684\u7279\u8d28\uff0c\u6301\u7eed\u7a81\u7834\u81ea\u8eab\u5c40\u9650\u3002<\/p>\n\n\n\n

\u8fd9\u79cd\u57fa\u4e8e\u6df1\u5ea6\u601d\u8003\u7684\u81ea\u6211\u8fdb\u5316\u673a\u5236\u5177\u6709\u5e7f\u6cdb\u7684\u9002\u7528\u6027\uff0c\u6211\u4eec\u6b63\u5728\u79ef\u6781\u63a2\u7d22\u5c06\u5176\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\uff0c\u4ee5\u6316\u6398\u5927\u6a21\u578b\u7684\u66f4\u591a\u6f5c\u529b\u3002<\/p>\n\n\n\n

\"rStar-Math
rStar-Math \u6982\u89c8<\/figcaption><\/figure>\n\n\n\n

Q\uff1arStar-Math \u7684\u7814\u7a76\u5bf9\u73b0\u5b9e\u4e16\u754c\u5c06\u4ea7\u751f\u600e\u6837\u7684\u5f71\u54cd\uff1f\u5b83\u80fd\u591f\u4e3a\u54ea\u4e9b\u884c\u4e1a\u6216\u573a\u666f\u5e26\u6765\u4ef7\u503c\uff1f<\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>rStar-Math \u7684\u8bbe\u8ba1\u7406\u5ff5\u548c\u6280\u672f\u601d\u8def\u4e0d\u4ec5\u5728\u6570\u5b66\u63a8\u7406\u9886\u57df\u5927\u653e\u5f02\u5f69\uff0c\u8fd8\u80fd\u5728\u5176\u4ed6\u590d\u6742\u63a8\u7406\u573a\u666f\u4e2d\u53d1\u6325\u91cd\u8981\u4f5c\u7528\u3002\u4f8b\u5982\uff0c\u5728\u4ee3\u7801\u751f\u6210\u4e0e\u8c03\u8bd5\u9886\u57df\uff0c\u5b83\u53ef\u4ee5\u663e\u8457\u63d0\u9ad8\u4ee3\u7801\u751f\u6210\u7684\u51c6\u786e\u6027\uff0c\u5feb\u901f\u8bc6\u522b\u5e76\u4fee\u590d\u6f5c\u5728\u95ee\u9898\uff1b\u5728\u667a\u80fd\u4f53\u89c4\u5212\u65b9\u9762\uff0c\u5b83\u652f\u6301\u66f4\u590d\u6742\u7684\u591a\u6b65\u9aa4\u4efb\u52a1\u6267\u884c\uff0c\u8d4b\u4e88\u667a\u80fd\u4f53\u66f4\u5f3a\u7684\u7075\u6d3b\u6027\u3001\u6548\u7387\u548c\u81ea\u4e3b\u51b3\u7b56\u80fd\u529b\u3002<\/p>\n\n\n\n

Q\uff1arStar-Math \u7684\u7814\u7a76\u662f\u5426\u8bc1\u660e\u4e86\u5c0f\u6a21\u578b\u53e0\u52a0\u4f7f\u7528\u53ef\u4ee5\u5728\u4fdd\u6301\u4f4e\u7b97\u529b\u548c\u4f4e\u80fd\u8017\u7684\u540c\u65f6\uff0c\u8fbe\u5230\u751a\u81f3\u8d85\u8d8a\u5927\u6a21\u578b\u6027\u80fd\uff1f\u8fd9\u5bf9\u4e8e\u4eba\u5de5\u667a\u80fd\u6a21\u578b\u7684\u666e\u53ca\u548c\u52a0\u901f\u843d\u5730\u610f\u5473\u7740\u4ec0\u4e48\uff1f<\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>\u6211\u4eec\u7684\u7814\u7a76\u5e76\u975e\u5426\u8ba4\u6a21\u578b\u89c4\u6a21\u4e0e\u6027\u80fd\u4e4b\u95f4\u7684\u5173\u8054\u3002\u901a\u5e38\uff0c\u5728\u76f8\u540c\u8bad\u7ec3\u65b9\u6cd5\u548c\u9ad8\u8d28\u91cf\u6570\u636e\u7684\u652f\u6301\u4e0b\uff0c\u89c4\u6a21\u66f4\u5927\u7684\u6a21\u578b\u80fd\u5e26\u6765\u66f4\u4f18\u7684\u7528\u6237\u4f53\u9a8c\u3002\u7136\u800c\uff0crStar-Math \u7684\u521b\u65b0\u4e4b\u5904\u5728\u4e8e\uff0c\u8bc1\u660e\u4e86\u5c0f\u6a21\u578b\u5728\u7279\u5b9a\u5782\u76f4\u9886\u57df\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u5353\u8d8a\u7684\u63a8\u7406\u80fd\u529b<\/strong>\u3002\u901a\u8fc7\u5de7\u5999\u7684\u8bbe\u8ba1\u548c\u65b9\u6cd5\uff0c\u5373\u4f7f\u5728\u8d44\u6e90\u53d7\u9650\u7684\u6761\u4ef6\u4e0b\uff0c\u5c0f\u6a21\u578b\u540c\u6837\u53ef\u4ee5\u62e5\u6709\u5f3a\u5927\u7684\u7ade\u4e89\u529b\uff0c\u751a\u81f3\u5728\u67d0\u4e9b\u65b9\u9762\u53ef\u4ee5\u4e0e\u5927\u6a21\u578b\u4e00\u8f83\u9ad8\u4e0b\u3002<\/p>\n\n\n\n

\u5f53\u524d\uff0c\u9ad8\u8d28\u91cf\u7684\u5927\u6a21\u578b\u5f80\u5f80\u4f9d\u8d56\u5e9e\u5927\u7684\u7b97\u529b\u548c\u6602\u8d35\u7684\u8ba1\u7b97\u8d44\u6e90\uff0c\u800c rStar-Math \u5c55\u73b0\u51fa\u4e00\u6761\u4f4e\u7b97\u529b\u3001\u4f4e\u80fd\u8017\u7684\u521b\u65b0\u8def\u5f84\uff0c\u4f7f\u5c0f\u6a21\u578b\u5728\u7279\u5b9a\u9886\u57df\u5b9e\u73b0\u5b9e\u7528\u4ef7\u503c\u3002\u8fd9\u4e00\u7a81\u7834\u663e\u8457\u964d\u4f4e\u4e86\u9ad8\u8d28\u91cf\u4eba\u5de5\u667a\u80fd\u5e94\u7528\u7684\u63a8\u7406\u6210\u672c\u3002<\/p>\n\n\n\n

Q\uff1a\u76ee\u524d\u4eba\u5de5\u667a\u80fd\u5728\u89e3\u51b3\u6570\u5b66\u63a8\u7406\u95ee\u9898\u65b9\u9762\u8fd8\u9762\u4e34\u54ea\u4e9b\u56f0\u96be\u548c\u503c\u5f97\u6df1\u5165\u7814\u7a76\u7684\u65b9\u5411\uff1f\u9488\u5bf9\u8fd9\u4e9b\u95ee\u9898\uff0c\u4f60\u548c\u56e2\u961f\u662f\u5426\u6709\u8fdb\u4e00\u6b65\u7684\u7814\u7a76\u8ba1\u5212\uff1f\u6b64\u5916\uff0c\u4f60\u8fd8\u770b\u5230\u4e86\u4eba\u5de5\u667a\u80fd\u9886\u57df\u4e2d\u54ea\u4e9b\u503c\u5f97\u5173\u6ce8\u7684\u95ee\u9898\u548c\u7814\u7a76\u8d8b\u52bf\uff1f<\/strong><\/p>\n\n\n\n

\u5f20\u4e3d\uff1a<\/strong>\u5c3d\u7ba1\u4eba\u5de5\u667a\u80fd\u5728\u6570\u5b66\u63a8\u7406\u65b9\u9762\u5df2\u53d6\u5f97\u663e\u8457\u8fdb\u5c55\uff0c\u4f46\u8981\u6210\u4e3a\u6570\u5b66\u5bb6\u89e3\u51b3\u590d\u6742\u95ee\u9898\u7684\u5f97\u529b\u52a9\u624b\uff0c\u4ecd\u9762\u4e34\u4e0d\u5c11\u6311\u6218\u3002\u4e0d\u8fc7\uff0c\u6211\u4eec\u5bf9\u6b64\u6301\u975e\u5e38\u79ef\u6781\u7684\u6001\u5ea6\uff0c\u4e5f\u5728\u79ef\u6781\u63a2\u7d22\u65b0\u7684\u8def\u5f84\uff0c\u529b\u6c42\u653b\u514b\u8fd9\u4e9b\u96be\u9898\u3002<\/p>\n\n\n\n

\u81f3\u4e8e\u4eba\u5de5\u667a\u80fd\u9886\u57df\u7684\u672a\u6765\u53d1\u5c55\uff0c\u6211\u4eec\u8ba4\u4e3a2025\u5e74\u5c06\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8f6c\u6298\u70b9\uff0c\u4eba\u5de5\u667a\u80fd\u6280\u672f\u5c06\u52a0\u901f\u6e17\u900f\u5230\u5404\u884c\u5404\u4e1a\uff0c\u771f\u6b63\u91ca\u653e\u5b9e\u7528\u4ef7\u503c\u3002\u6211\u4eec\u4e5f\u975e\u5e38\u5173\u6ce8\u8fd9\u4e00\u8d8b\u52bf\uff0c\u5e0c\u671b\u63a2\u7d22\u5f53\u524d\u7684\u521b\u65b0\u601d\u8def\u662f\u5426\u80fd\u8de8\u8d8a\u9886\u57df\u7684\u754c\u9650\uff0c\u89e3\u51b3\u66f4\u591a\u590d\u6742\u7684\u4efb\u52a1\u3002\u8fd9\u79cd\u8de8\u9886\u57df\u7684\u6f5c\u529b\u662f\u6211\u4eec\u7814\u7a76\u7684\u91cd\u8981\u65b9\u5411\uff0c\u4e5f\u8ba9\u6211\u4eec\u5145\u6ee1\u671f\u5f85\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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