{"id":31198,"date":"2024-04-28T13:21:53","date_gmt":"2024-04-28T05:21:53","guid":{"rendered":"https:\/\/www.wukongsch.com\/blog\/?p=31198"},"modified":"2026-01-30T10:28:04","modified_gmt":"2026-01-30T02:28:04","slug":"stanford-math-tournament-past-papers","status":"publish","type":"post","link":"https:\/\/www.wukongsch.com\/blog\/zh\/stanford-math-tournament-past-papers-post-31198\/","title":{"rendered":"SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u5386\u5e74\u771f\u9898\u4e0e\u7b54\u6848\u89e3\u6790(2026\u6700\u65b0)"},"content":{"rendered":"<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>\n<p>\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\uff08Stanford Math Tournament, SMT\uff09\u4f5c\u4e3a\u4e00\u9879\u6781\u5177\u6311\u6218\u6027\u7684\u9ad8\u4e2d\u751f\u6570\u5b66\u7ade\u8d5b\uff0c\u81ea\u4e3e\u529e\u4ee5\u6765\u4e00\u76f4\u5438\u5f15\u7740\u4e16\u754c\u5404\u5730\u5bf9\u6570\u5b66\u5145\u6ee1\u70ed\u60c5\u7684\u5b66\u5b50\u4eec\u3002\u8fd9\u9879\u7ade\u8d5b\u4e0d\u4ec5\u8003\u9a8c\u53c2\u8d5b\u8005\u7684\u6570\u5b66\u89e3\u9898\u80fd\u529b\uff0c\u8fd8\u80fd\u57f9\u517b\u4ed6\u4eec\u7684\u903b\u8f91\u601d\u7ef4\u3001\u56e2\u961f\u5408\u4f5c\u7cbe\u795e\u548c\u521b\u65b0\u610f\u8bc6\u3002\u901a\u8fc7SMT\uff0c\u5b66\u751f\u4eec\u6709\u673a\u4f1a\u63a5\u89e6\u5230\u66f4\u9ad8\u5c42\u6b21\u7684\u6570\u5b66\u95ee\u9898\uff0c\u4e3a\u5c06\u6765\u7684\u5b66\u672f\u53d1\u5c55\u6253\u4e0b\u575a\u5b9e\u7684\u57fa\u7840\u3002<\/p>\n\n\n\n<p>\u672c\u6587\u65e8\u5728\u5206\u6790\u548c\u5206\u4eab\u5386\u5e74SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u771f\u9898\uff0c\u5e2e\u52a9\u5e7f\u5927\u5b66\u5b50\u66f4\u597d\u5730\u4e86\u89e3\u8fd9\u9879\u7ade\u8d5b\u7684\u7279\u70b9\u548c\u8981\u6c42\uff0c\u4ece\u800c\u5728\u5907\u8003\u8fc7\u7a0b\u4e2d\u66f4\u52a0\u6709\u7684\u653e\u77e2\u3002\u6211\u4eec\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u83b7\u53d6\u771f\u9898\u7684\u9014\u5f84\u548c\u771f\u9898\u7684\u91cd\u8981\u6027\uff0c\u6df1\u5165\u5256\u6790\u771f\u9898\u9898\u578b\u548c\u6240\u8986\u76d6\u9886\u57df\uff0c\u5206\u4eab\u89e3\u9898\u6280\u5de7\u548c\u5e38\u89c1\u9519\u8bef\u5206\u6790\uff0c\u4ee5\u671f\u4e3a\u5e7f\u5927\u53c2\u8d5b\u8005\u63d0\u4f9b\u6709\u76ca\u7684\u53c2\u8003\u548c\u6307\u5bfc\u3002\u901a\u8fc7\u672c\u6587\u7684\u5b66\u4e60\uff0c\u76f8\u4fe1\u5927\u5bb6\u80fd\u591f\u5728SMT\u7684\u821e\u53f0\u4e0a\u5c55\u73b0\u51fa\u81ea\u5df1\u7684\u6700\u4f73\u6c34\u5e73\uff0c\u53d6\u5f97\u4f18\u5f02\u7684\u6210\u7ee9\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"574\" src=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2025\/04\/image-283-1024x574.png\" alt=\"SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u5386\u5e74\u771f\u9898\" class=\"wp-image-31212\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e4%b8%80%e3%80%81smt%e6%96%af%e5%9d%a6%e7%a6%8f%e6%95%b0%e5%ad%a6%e9%94%a6%e6%a0%87%e8%b5%9b%e7%9c%9f%e9%a2%98%e8%8e%b7%e5%8f%96%e9%80%94%e5%be%84%e5%92%8c%e9%87%8d%e8%a6%81%e6%80%a7\"><\/span>\u4e00\u3001SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u771f\u9898\u83b7\u53d6\u9014\u5f84\u548c\u91cd\u8981\u6027<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u8981\u83b7\u53d6<\/strong><strong>SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u771f\u9898\uff0c\u53c2\u8d5b\u8005\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u51e0\u79cd\u9014\u5f84\uff1a<\/strong><strong><\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5b98\u65b9\u7f51\u7ad9\uff1a\u8bb8\u591a\u5b98\u65b9\u7ade\u8d5b\u7f51\u7ad9\u4f1a\u5728\u8d5b\u540e\u516c\u5e03\u90e8\u5206\u6216\u5168\u90e8\u7684\u771f\u9898\u53ca\u89e3\u7b54\uff0c\u4f9b\u53c2\u8d5b\u8005\u4e0b\u8f7d\u548c\u7814\u7a76\u3002<\/li>\n<\/ul>\n\n\n<div class=\"retention-card-new\" data-lang=\"zh\" data-subject=\"MATH\" data-btnName=\"\u7acb\u5373\u514d\u8d39\u8bd5\u542c\" data-subTitle=\"\u9002\u5408\u5168\u74031-12\u5e74\u7ea7\u5b66\u751f\">\r\n    <div class=\"retention-card-l\">\r\n        <div class=\"trustpilot-image\"><\/div>\r\n        <h3><p>\u609f\u7a7a\u56fd\u9645\u6570\u5b66<br \/>\n<span>\u8ba9\u6570\u5b66\u4e4b\u5149\u7167\u4eae\u6bcf\u4e00\u4e2a\u5b69\u5b50\uff01<\/span><\/p>\n<\/h3>\r\n        <p>\u9002\u5408\u5168\u74031-12\u5e74\u7ea7\u5b66\u751f<\/p>\r\n        <a class=\"retention-card-button is-point\" href=\"https:\/\/www.wukongsch.com\/zh\/independent-appointment\/?subject=math&amp;l=c87d1e8c-ec8e-4a2a-83b7-84e0af53d81f&amp;booking_triggerevent=BLOG_DETAIL_MODEL_CTA_BUTTON\" data-buttonname=\"\u7acb\u5373\u9884\u7ea6\u6309\u94ae\u70b9\u51fb\" data-event=\"C_Blog_BLOG_DETAIL_MIDDLE_CTA_BUTTON\" data-expose-buttonname=\"\u7acb\u5373\u9884\u7ea6\u6309\u94ae\u66dd\u5149\" data-expose-event=\"D_Blog_BLOG_DETAIL_MIDDLE_CTA_BUTTON\" target=\"_blank\" title=\"\u7acb\u5373\u514d\u8d39\u8bd5\u542c\">\r\n            \u7acb\u5373\u514d\u8d39\u8bd5\u542c\r\n        <\/a>\r\n    <\/div>\r\n    <div class=\"retention-card-r\"><\/div>\r\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>\u5b66\u6821\u8d44\u6e90\uff1a\u90e8\u5206\u53c2\u8d5b\u5b66\u6821\u7684\u6570\u5b66\u4ff1\u4e50\u90e8\u6216\u6559\u7ec3\u53ef\u80fd\u4fdd\u5b58\u6709\u5386\u5e74\u7684\u771f\u9898\u8d44\u6599\uff0c\u53ef\u4ee5\u5411\u5b66\u6821\u76f8\u5173\u90e8\u95e8\u54a8\u8be2\u3002<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5728\u7ebf\u8bba\u575b\u548c\u793e\u7fa4\uff1a\u52a0\u5165\u6570\u5b66\u7ade\u8d5b\u76f8\u5173\u7684\u5728\u7ebf\u8bba\u575b\u3001QQ\u7fa4\u6216\u5fae\u4fe1\u7fa4\uff0c\u7ecf\u5e38\u4f1a\u6709\u524d\u8f88\u5206\u4eab\u5386\u5e74\u771f\u9898\u548c\u590d\u4e60\u7ecf\u9a8c\u3002<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8d2d\u4e70\u8f85\u5bfc\u4e66\u7c4d\uff1a\u5e02\u9762\u4e0a\u6709\u4e9b\u8f85\u5bfc\u4e66\u7c4d\u4f1a\u6536\u5f55\u5386\u5e74\u7684\u7ade\u8d5b\u771f\u9898\uff0c\u5e76\u63d0\u4f9b\u8be6\u7ec6\u89e3\u7b54\u548c\u89e3\u6790\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e66\u5e97\u6216\u7f51\u7edc\u8d2d\u4e66\u5e73\u53f0\u8d2d\u4e70\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u73b0\u5728\u609f\u7a7a\u4e3a\u5927\u5bb6\u63d0\u4f9b\u90e8\u52062025\u5e74SMT<strong>\u65af\u5766\u798f\u6570\u5b66\u7ade\u8d5b<\/strong>\u771f\u9898\u53ca\u7b54\u6848\u7684PDF\u7248\u672c\u4f9b\u5927\u5bb6\u67e5\u9605\uff1a<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-problems.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of algebra-problems.\"><\/object><a id=\"wp-block-file--media-8ce1ad63-152d-4e29-a5ce-6328e29df4be\" href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-problems.pdf\">algebra-problems<\/a><a href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-problems.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-8ce1ad63-152d-4e29-a5ce-6328e29df4be\">Download<\/a><\/div>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-solutions.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of algebra-solutions.\"><\/object><a id=\"wp-block-file--media-bda64d97-c9d4-4e15-a3f2-7914e0451543\" href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-solutions.pdf\">algebra-solutions<\/a><a href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/algebra-solutions.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-bda64d97-c9d4-4e15-a3f2-7914e0451543\">Download<\/a><\/div>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-problems.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of calculus-problems.\"><\/object><a id=\"wp-block-file--media-ec3255ec-6e95-4a74-b910-8d7dd911d30a\" href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-problems.pdf\">calculus-problems<\/a><a href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-problems.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-ec3255ec-6e95-4a74-b910-8d7dd911d30a\">Download<\/a><\/div>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-solutions.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of calculus-solutions.\"><\/object><a id=\"wp-block-file--media-fe334703-3469-4747-a70f-7a82b0a406e8\" href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-solutions.pdf\">calculus-solutions<\/a><a href=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2024\/04\/calculus-solutions.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-fe334703-3469-4747-a70f-7a82b0a406e8\">Download<\/a><\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u771f\u9898\u7684\u91cd\u8981\u6027\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762<\/strong><\/h3>\n\n\n\n<p>\u719f\u6089\u8003\u8bd5\u98ce\u683c\uff1a\u901a\u8fc7\u771f\u9898\u53ef\u4ee5\u8ba9\u53c2\u8d5b\u8005\u719f\u6089SMT\u7684\u8003\u8bd5\u9898\u578b\u548c\u51fa\u9898\u98ce\u683c\uff0c\u4e3a\u5b9e\u6218\u505a\u597d\u51c6\u5907\u3002<\/p>\n\n\n\n<p>\u8bc4\u4f30\u81ea\u8eab\u6c34\u5e73\uff1a\u901a\u8fc7\u505a\u771f\u9898\uff0c\u53c2\u8d5b\u8005\u53ef\u4ee5\u68c0\u6d4b\u81ea\u5df1\u7684\u6570\u5b66\u6c34\u5e73\u548c\u89e3\u9898\u80fd\u529b\uff0c\u660e\u786e\u81ea\u5df1\u7684\u5f3a\u9879\u548c\u5f31\u9879\u3002<\/p>\n\n\n\n<p>\u8bad\u7ec3\u89e3\u9898\u6280\u5de7\uff1a\u771f\u9898\u5f80\u5f80\u6d89\u53ca\u591a\u79cd\u89e3\u9898\u6280\u5de7\u548c\u65b9\u6cd5\uff0c\u901a\u8fc7\u4e0d\u65ad\u7ec3\u4e60\uff0c\u53c2\u8d5b\u8005\u53ef\u4ee5\u63d0\u9ad8\u89e3\u9898\u901f\u5ea6\u548c\u51c6\u786e\u7387\u3002<\/p>\n\n\n\n<p>\u53d1\u73b0\u5e38\u72af\u9519\u8bef\uff1a\u901a\u8fc7\u5206\u6790\u771f\u9898\u4e2d\u7684\u9519\u8bef\uff0c\u53c2\u8d5b\u8005\u53ef\u4ee5\u603b\u7ed3\u7ecf\u9a8c\u6559\u8bad\uff0c\u907f\u514d\u5728\u6b63\u5f0f\u8003\u8bd5\u4e2d\u91cd\u590d\u72af\u9519\u3002<\/p>\n\n\n\n<p>\u603b\u4e4b\uff0c\u83b7\u53d6\u5e76\u5145\u5206\u5229\u7528\u5386\u5e74\u771f\u9898\u662f\u5907\u8003SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u5173\u952e\u6b65\u9aa4\u4e4b\u4e00\uff0c\u6709\u52a9\u4e8e\u53c2\u8d5b\u8005\u5168\u9762\u63d0\u5347\u81ea\u5df1\u7684\u7ade\u8d5b\u5b9e\u529b\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e4%ba%8c%e3%80%81%e6%96%af%e5%9d%a6%e7%a6%8f%e6%95%b0%e5%ad%a6%e7%ab%9e%e8%b5%9b-2026-%e6%a6%82%e8%a7%88\"><\/span>\u4e8c\u3001<strong>\u65af\u5766\u798f\u6570\u5b66\u7ade\u8d5b 2026 \u6982\u89c8<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p><strong>\u65e5\u671f<\/strong>\uff1a2026 \u5e74 4 \u6708 17\u201318 \u65e5<br><strong>\u5730\u70b9<\/strong>\uff1a\u65af\u5766\u798f\u5927\u5b66<br><strong>\u53c2\u4e0e\u5bf9\u8c61<\/strong>\uff1a\u5c45\u4f4f\u5728\u7f8e\u56fd\u5883\u5185\u7684\u9ad8\u4e2d\u751f<br><strong>\u56e2\u961f\u4eba\u6570<\/strong>\uff1a\u6bcf\u961f 5\u20136 \u4eba<br><strong>\u8d39\u7528<\/strong>\uff1a\u6bcf\u4f4d\u5b66\u751f 30 \u7f8e\u5143*\uff08\u8d39\u7528\u5305\u542b\u4e00\u4ef6 T \u6064\u3001\u5468\u4e94\u665a\u9910\u53ca\u5468\u516d\u5348\u9910\uff1b\u5982\u6709\u9700\u8981\uff0c\u53ef\u7533\u8bf7\u7ecf\u6d4e\u8d44\u52a9\uff01\uff09<br><strong>\u6210\u7ee9\u67e5\u8be2\uff1a<\/strong>\u6210\u7ee9\u67e5\u770b\u76f8\u5173\u4fe1\u606f\uff0c\u53ef\u4ee5\u67e5\u770b\u609f\u7a7a\u201c<a href=\"https:\/\/www.wukongsch.com\/blog\/zh\/stanford-math-tournament-results-post-31218\/\">SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u6210\u7ee9\u67e5\u8be2\u6307\u5357<\/a>\u201d\u8fd9\u7bc7\u6587\u7ae0\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e4%b8%89%e3%80%81%e7%9c%9f%e9%a2%98%e5%88%86%e6%9e%90\"><\/span>\u4e09\u3001\u771f\u9898\u5206\u6790<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u771f\u9898\u5206\u6790\u662f\u5907\u8003SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u91cd\u8981\u73af\u8282\uff0c\u901a\u8fc7\u5bf9\u5386\u5e74\u771f\u9898\u7684\u6df1\u5165\u7814\u7a76\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u4e86\u89e3\u8003\u8bd5\u7684\u9898\u578b\u3001\u6240\u8986\u76d6\u7684\u9886\u57df\u4ee5\u53ca\u89e3\u9898\u6280\u5de7\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u771f\u9898\u9898\u578b<\/h3>\n\n\n\n<p>SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u9898\u578b\u591a\u6837\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\u9009\u62e9\u9898\u3001\u586b\u7a7a\u9898\u3001\u89e3\u7b54\u9898\u7b49\u3002\u5176\u4e2d\uff0c\u89e3\u7b54\u9898\u5360\u636e\u8f83\u5927\u6bd4\u91cd\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u624e\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u548c\u7075\u6d3b\u7684\u601d\u7ef4\u80fd\u529b\u3002\u9898\u578b\u7684\u591a\u6837\u6027\u4f7f\u5f97\u8003\u8bd5\u5185\u5bb9\u66f4\u52a0\u4e30\u5bcc\uff0c\u4e5f\u589e\u52a0\u4e86\u8003\u8bd5\u7684\u96be\u5ea6\u548c\u6311\u6218\u6027\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.\u6240\u8986\u76d6\u9886\u57df<\/h3>\n\n\n\n<p>SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u6db5\u76d6\u4e86\u6570\u5b66\u7684\u591a\u4e2a\u9886\u57df\uff0c\u5305\u62ec<strong>\u4ee3\u6570\u3001\u5fae\u79ef\u5206\u3001\u79bb\u6563\u3001\u51e0\u4f55\u5b66\u7b49<\/strong>\u3002\u5176\u4e2d\uff0c\u4ee3\u6570\u548c\u51e0\u4f55\u662f\u8003\u8bd5\u7684\u91cd\u70b9\u9886\u57df\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u638c\u63e1\u76f8\u5e94\u7684\u57fa\u7840\u77e5\u8bc6\u548c\u89e3\u9898\u65b9\u6cd5\u3002\u6b64\u5916\uff0c\u6570\u8bba\u548c\u7ec4\u5408\u7b49\u9886\u57df\u4e5f\u65f6\u5e38\u51fa\u73b0\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u4e00\u5b9a\u7684\u6d89\u730e\u548c\u7406\u89e3\u80fd\u529b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u4e0b\u9762\u662f2023\u5e74SMT\u76f8\u5173\u771f\u9898\u4ee5\u53ca\u89e3\u7b54<\/h3>\n\n\n\n<p><strong>\u4ee3\u6570<\/strong><br>Compute the number of values of x in the interval [\u221211\u03c0, \u22122\u03c0] that satisfy 5 cos(x)+4 5 sin(x)+3 = 0<br>\u89e3\uff1a\u5f53\u5206\u6570\u7684\u5206\u5b50\u7b49\u4e8e0\u800c\u5206\u6bcd\u4e0d\u7b49\u4e8e0\u65f6\u5206\u6570\u7b49\u4e8e0\u30025cos (x) + 4 = 0\u7684\u89e3\u5f62\u5f0f\u4e3ax =\u00b1arccos(- 4\/5) + 2\u03c0k\uff0c5 sin(x) + 3 = 0\u7684\u89e3\u4e3ax =\u00b1arcsin(- 3\/5) + 2\u03c0k\uff0c\u6211\u4eec\u770b\u5230[2k\u03c0\uff0c(2k + 1)\u03c0]\u5f62\u5f0f\u7684\u6bcf\u4e00\u4e2a\u533a\u95f4\u90fd\u6709\u4e00\u4e2a\u89e3\uff1b[(2k + 1)\u03c0\uff0c(2k + 2)\u03c0]\u5f62\u5f0f\u7684\u65b9\u7a0b\u548c\u533a\u95f4\u65e0\u89e3\u3002\u56e0\u6b64\uff0c\u67094\u4e2a\u533a\u95f4[\u221211\u03c0\uff0c\u22122\u03c0]\u5185\u7684\u89e3\u3002<\/p>\n\n\n\n<p><strong>\u5fae\u79ef\u5206<\/strong><br>For how many real numbers x do we have that log5 (1 + x) = x?<br>\u89e3\uff1a\u6211\u4eec\u9996\u5148\u6ce8\u610f\u5230\u4e0d\u5b58\u5728x\u2264- 1\u7684\u89e3\uff0c\u56e0\u4e3a\u5de6\u8fb9\u5728\u8fd9\u4e2a\u533a\u57df\u5185\u6ca1\u6709\u5b9a\u4e49\u3002\u66f4\u8fdb\u4e00\u6b65\uff0cx = 0\u662f\u4e00\u4e2a\u89e3\u3002\u5bf9\u4e8e\u4efb\u610fx &gt; 0\uff0c\u5c06\u65b9\u7a0b\u6539\u5199\u4e3a1 + x = 5x\u3002\u56e0\u4e3a1 + x\u2264e x\u22645X(\u53ef\u4ee5\u7528\u6c42\u5bfc\u6216\u5e42\u7ea7\u6570\u6765\u8868\u793a)\u4e0d\u53ef\u80fd\u6709X &gt; 0\u7684\u89e3\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u8981\u68c0\u9a8c\u7684\u552f\u4e00\u533a\u95f4\u662f- 1 &lt; x &lt; 0\u3002\u8bbef(x) = 5x\u2212x\u22121\u3002\u5219f(0) = 0,f(- 1) = 15 &gt; 0\u3002\u7136\u800c\uff0cf &#8216; (x) = ln 5\u00b75x\u22121\uff0c\u5f53x &gt; &#8211; log5 ln 5\u65f6\uff0c\u6ee1\u8db3f &#8216; (x) &gt; 0\u3002\u8fd9\u610f\u5473\u7740\uff0cf(\u2212log5 ln 5) &lt; 0\u3002\u7136\u800c\uff0c\u4e2d\u503c\u5b9a\u7406\u5f97\uff0c\u5728- log5 ln 5\u548c- 1\u4e0a\uff0c\u5b83\u4eec\u4e4b\u95f4\u4e00\u5b9a\u5b58\u5728\u67d0\u4e2ax\u4f7f\u5f97f(x) = 0\u3002\u4e8b\u5b9e\u4e0a\uff0c\u8fd9\u4e2ax\u4e5f\u5fc5\u987b\u662f\u552f\u4e00\u7684:\u5bf9\u4e8e\u6240\u6709\u22121\u2264x &lt;\u2212log5, f &#8216; (x) &lt; 0ln 5\u3002<br>\u56e0\u6b64\uff0c\u7b54\u6848\u662f2\u3002<\/p>\n\n\n\n<p><strong>\u79bb\u6563<\/strong><br>How many trailing zeros does the value 300 \u00b7 305 \u00b7 310 \u00b7 \u00b7 \u00b7 1090 \u00b7 1095 \u00b7 1100 end with?<br>\u5c06\u8868\u8fbe\u5f0f\u91cd\u5199\u4e3a\uff1a300 \u00b7 305 \u00b7 310 \u00b7 \u00b7 \u00b7 1090 \u00b7 1095 \u00b7 1100 = 5161 \u00b7 220!\/ 59! .<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"540\" height=\"149\" src=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2025\/04\/image-281.png\" alt=\"\u79bb\u6563\u7684\u89e3\u6790\u8fc7\u7a0b\" class=\"wp-image-31205\" style=\"width:875px;height:auto\"\/><\/figure>\n\n\n\n<p><br>220\u7684\u56e0\u65702 !\u56e0\u6b64\u6709\uff1a161+53-13=201<br>\u4e58\u79ef\u4e2d\u7684\u56e0\u5b505\u548c: 215-54=161<br>\u56e0\u4e3a161 &lt; 201\uff0c\u6240\u4ee5\u7b54\u6848\u662f161\u3002<\/p>\n\n\n\n<p><strong>\u51e0\u4f55<\/strong><br>Triangle \u25b3ABC has side lengths AB = 39, BC = 16, and CA = 25. What is the volume of the solid formed by rotating \u25b3ABC about line BC?<br>\u89e3\uff1a\u8bbe\u4eceA\u5230\u76f4\u7ebfBC\u7684\u5782\u7ebf\u811a\u4e3ad\u3002\u6211\u4eec\u8981\u6c42\u7684\u4f53\u79ef\u53ef\u4ee5\u7528\u65cb\u8f6c\u25b3ACD\u5f62\u6210\u7684\u5706\u9525\u7684\u4f53\u79ef\u51cf\u53bb\u65cb\u8f6c\u25b3ABD\u5f62\u6210\u7684\u5706\u9525\u7684\u4f53\u79ef\u6765\u8ba1\u7b97\u3002\u8bbeCD = x, AD = y\uff0c\u6839\u636e\u52fe\u80a1\u5b9a\u7406\uff0c\u6211\u4eec\u6709<br>(x + 16)2 + y2 = 392<br>\u548c<br>X2 + Y2 = 252\u3002<br>\u7528\u7b2c\u4e00\u4e2a\u65b9\u7a0b\u51cf\u53bb\u7b2c\u4e8c\u4e2a\u65b9\u7a0b\uff0c\u89e3\u51fax<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"506\" height=\"51\" src=\"https:\/\/wp-more.wukongedu.net\/blog\/wp-content\/uploads\/2025\/04\/image-282.png\" alt=\"\u51e0\u4f55\u89e3\u6790\u8fc7\u7a0b\" class=\"wp-image-31206\" style=\"width:858px;height:auto\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">3.\u9898\u578b\u7279\u70b9<\/h3>\n\n\n\n<p>\u6839\u636e\u4ee5\u4e0a\u7684\u771f\u9898\u5206\u4eab\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u771f\u9898\u6709\u4ee5\u4e0b\u51e0\u4e2a\u7279\u70b9\u3002\u9996\u5148\uff0c\u8bd5\u9898\u5f80\u5f80\u6ce8\u91cd\u57fa\u7840\u77e5\u8bc6\u7684\u8003\u67e5\uff0c\u540c\u65f6\u4e5f\u6ce8\u91cd\u5bf9\u601d\u7ef4\u80fd\u529b\u548c\u89e3\u9898\u6280\u5de7\u7684\u6d4b\u8bd5\u3002\u5176\u6b21\uff0c\u8bd5\u9898\u5f80\u5f80\u5177\u6709\u4e00\u5b9a\u7684\u96be\u5ea6\u68af\u5ea6\uff0c\u4ece\u57fa\u7840\u9898\u76ee\u5230\u96be\u9898\u9010\u6b65\u8fc7\u6e21\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u826f\u597d\u7684\u5fc3\u7406\u7d20\u8d28\u548c\u5e94\u8bd5\u80fd\u529b\u3002\u6700\u540e\uff0c\u8bd5\u9898\u5f80\u5f80\u5177\u6709\u4e00\u5b9a\u7684\u521b\u65b0\u6027\u548c\u8da3\u5473\u6027\uff0c\u80fd\u591f\u6fc0\u53d1\u53c2\u8d5b\u8005\u7684\u5b66\u4e60\u5174\u8da3\u548c\u63a2\u7d22\u6b32\u671b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4.\u4fa7\u91cd\u8303\u56f4<\/h3>\n\n\n\n<p>SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u4fa7\u91cd\u8303\u56f4\u4e3b\u8981\u5305\u62ec\u6570\u5b66\u57fa\u7840\u77e5\u8bc6\u548c\u89e3\u9898\u65b9\u6cd5\u3001\u6570\u5b66\u601d\u7ef4\u548c\u89e3\u9898\u80fd\u529b\u3001\u6570\u5b66\u7d20\u517b\u548c\u7efc\u5408\u80fd\u529b\u7b49\u3002\u5176\u4e2d\uff0c\u6570\u5b66\u57fa\u7840\u77e5\u8bc6\u548c\u89e3\u9898\u65b9\u6cd5\u662f\u8003\u8bd5\u7684\u6838\u5fc3\u5185\u5bb9\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u719f\u7ec3\u638c\u63e1\u76f8\u5e94\u7684\u77e5\u8bc6\u70b9\u548c\u89e3\u9898\u6280\u5de7\u3002\u6570\u5b66\u601d\u7ef4\u548c\u89e3\u9898\u80fd\u529b\u5219\u662f\u8003\u8bd5\u7684\u91cd\u8981\u76ee\u6807\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u826f\u597d\u7684\u903b\u8f91\u601d\u7ef4\u3001\u7a7a\u95f4\u60f3\u8c61\u80fd\u529b\u548c\u521b\u65b0\u80fd\u529b\u3002\u6570\u5b66\u7d20\u517b\u548c\u7efc\u5408\u80fd\u529b\u5219\u662f\u8003\u8bd5\u7684\u5ef6\u4f38\u8981\u6c42\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u5e7f\u6cdb\u7684\u77e5\u8bc6\u9762\u3001\u826f\u597d\u7684\u5b66\u4e60\u4e60\u60ef\u548c\u4f18\u79c0\u7684\u7efc\u5408\u7d20\u8d28\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5.\u89e3\u9898\u6280\u5de7<\/h3>\n\n\n\n<p>\u9488\u5bf9SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u771f\u9898\u7279\u70b9\uff0c\u4ee5\u4e0b\u662f\u4e00\u4e9b\u5e38\u7528\u7684\u89e3\u9898\u6280\u5de7\uff1a<\/p>\n\n\n\n<p>\uff081\uff09\u8ba4\u771f\u5ba1\u9898\uff1a\u4ed4\u7ec6\u9605\u8bfb\u9898\u76ee\uff0c\u7406\u89e3\u9898\u610f\u548c\u8981\u6c42\uff0c\u660e\u786e\u5df2\u77e5\u6761\u4ef6\u548c\u6c42\u89e3\u76ee\u6807\u3002<\/p>\n\n\n\n<p>\uff082\uff09\u6316\u6398\u4fe1\u606f\uff1a\u4ece\u9898\u76ee\u4e2d\u6316\u6398\u6709\u7528\u7684\u4fe1\u606f\uff0c\u5982\u6570\u5b57\u3001\u56fe\u5f62\u3001\u5173\u7cfb\u7b49\uff0c\u4e3a\u89e3\u9898\u63d0\u4f9b\u7ebf\u7d22\u3002<\/p>\n\n\n\n<p>\uff083\uff09\u5efa\u7acb\u6a21\u578b\uff1a\u6839\u636e\u9898\u610f\u5efa\u7acb\u6570\u5b66\u6a21\u578b\uff0c\u5c06\u5b9e\u9645\u95ee\u9898\u8f6c\u5316\u4e3a\u6570\u5b66\u95ee\u9898\u8fdb\u884c\u6c42\u89e3\u3002<\/p>\n\n\n\n<p>\uff084\uff09\u7075\u6d3b\u8fd0\u7528\uff1a\u7075\u6d3b\u8fd0\u7528\u6240\u5b66\u7684\u6570\u5b66\u77e5\u8bc6\u548c\u89e3\u9898\u65b9\u6cd5\uff0c\u5bfb\u627e\u89e3\u9898\u601d\u8def\u548c\u9014\u5f84\u3002<\/p>\n\n\n\n<p>\uff085\uff09\u68c0\u67e5\u7b54\u6848\uff1a\u5f97\u51fa\u7b54\u6848\u540e\u8981\u8fdb\u884c\u68c0\u67e5\u548c\u9a8c\u8bc1\uff0c\u786e\u4fdd\u7b54\u6848\u7684\u6b63\u786e\u6027\u548c\u5408\u7406\u6027\u3002<\/p>\n\n\n\n<p>\u603b\u4e4b\uff0c\u901a\u8fc7\u5bf9\u5386\u5e74\u771f\u9898\u7684\u6df1\u5165\u5206\u6790\u548c\u7814\u7a76\uff0c\u53c2\u8d5b\u8005\u53ef\u4ee5\u66f4\u597d\u5730\u4e86\u89e3SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u9898\u578b\u3001\u6240\u8986\u76d6\u7684\u9886\u57df\u4ee5\u53ca\u89e3\u9898\u6280\u5de7\u3002\u5728\u5907\u8003\u8fc7\u7a0b\u4e2d\uff0c\u8981\u6ce8\u91cd\u57fa\u7840\u77e5\u8bc6\u7684\u5de9\u56fa\u548c\u62d3\u5c55\uff0c\u63d0\u9ad8\u601d\u7ef4\u80fd\u529b\u548c\u89e3\u9898\u80fd\u529b\uff0c\u57f9\u517b\u826f\u597d\u7684\u5b66\u4e60\u4e60\u60ef\u548c\u7efc\u5408\u7d20\u8d28\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e5%9b%9b%e3%80%81%e5%b8%b8%e8%a7%81%e9%94%99%e8%af%af%e5%88%86%e6%9e%90\"><\/span>\u56db\u3001\u5e38\u89c1\u9519\u8bef\u5206\u6790<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5728\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u65f6\uff0c\u5c24\u5176\u662f\u50cfSMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u8fd9\u6837\u7684\u9ad8\u7ea7\u6570\u5b66\u7ade\u8d5b\uff0c\u53c2\u8d5b\u8005\u5e38\u5e38\u4f1a\u72af\u4e00\u4e9b\u5e38\u89c1\u7684\u9519\u8bef\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u5178\u578b\u7684\u9519\u8bef\u53ca\u5176\u539f\u56e0\u5206\u6790\uff1a<\/p>\n\n\n\n<p>1.\u7b26\u53f7\u9519\u8bef\uff1a\u5728\u590d\u6742\u7684\u4ee3\u6570\u8fd0\u7b97\u4e2d\uff0c\u5bb9\u6613\u6df7\u6dc6\u53d8\u91cf\u3001\u5e38\u6570\u548c\u8fd0\u7b97\u7b26\uff0c\u5bfc\u81f4\u7ed3\u679c\u51fa\u9519\u3002\u4f8b\u5982\uff0c\u5fd8\u8bb0\u62ec\u53f7\u3001\u6df7\u6dc6\u4e58\u9664\u7b26\u53f7\u6216\u9519\u8bef\u5730\u5e94\u7528\u6307\u6570\u89c4\u5219\u3002<\/p>\n\n\n\n<p>\u4f8b\u5982\u8fd9\u4e2a\u9898For how many real numbers x do we have that log<sub>5 <\/sub>(1 + x) = x?<\/p>\n\n\n\n<p>\u4e2d\u95f4\u6709\u4e00\u6b65\u662f\u5f53x &gt; &#8211; log5 ln 5\uff0c\u8fd9\u4e2a\u8d1f\u53f7\u5728\u8ba1\u7b97\u65f6\u5f88\u5bb9\u6613\u88ab\u5ffd\u7565\u6389\uff0c\u8981\u65f6\u523b\u6ce8\u610f\u8d1f\u53f7\u7684\u5b58\u5728\u3002<\/p>\n\n\n\n<p>2.\u5ffd\u7565\u5b9a\u4e49\u57df\u548c\u503c\u57df\uff1a\u5728\u89e3\u51b3\u51fd\u6570\u76f8\u5173\u95ee\u9898\u65f6\uff0c\u5ffd\u7565\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u548c\u503c\u57df\u9650\u5236\u662f\u4e00\u4e2a\u5e38\u89c1\u9519\u8bef\u3002\u8fd9\u53ef\u80fd\u5bfc\u81f4\u89e3\u96c6\u4e0d\u51c6\u786e\u6216\u5b8c\u5168\u9519\u8bef\u3002<\/p>\n\n\n\n<p>\u6bd4\u5982\u8fd9\u4e2a\u4ee3\u6570\u9898Compute the number of values of x in the interval [\u221211\u03c0, \u22122\u03c0] that satisfy 5 cos(x)+4 5 sin(x)+3 = 0\uff0c\u6211\u4eec\u8981\u7edd\u5bf9\u4e0d\u80fd\u5ffd\u7565\u9898\u76ee\u7ed9\u51fa\u7684\u9650\u5236\u662f\u5728[\u221211\u03c0, \u22122\u03c0]\uff0c\u800c\u4e0d\u662fR\u3002<\/p>\n\n\n\n<p>3.\u8ba1\u7b97\u5931\u8bef\uff1a\u5373\u4f7f\u662f\u57fa\u672c\u7684\u7b97\u672f\u8fd0\u7b97\u4e5f\u53ef\u80fd\u51fa\u9519\uff0c\u5c24\u5176\u662f\u5728\u957f\u4e32\u7684\u8ba1\u7b97\u4e2d\u3002\u8fd9\u53ef\u80fd\u662f\u7531\u4e8e\u6ce8\u610f\u529b\u4e0d\u96c6\u4e2d\u3001\u75b2\u52b3\u6216\u7b80\u5355\u7684\u7c97\u5fc3\u5927\u610f\u3002<\/p>\n\n\n\n<p>4.\u5ffd\u7565\u7279\u6b8a\u60c5\u51b5\uff1a\u5728\u89e3\u51b3\u95ee\u9898\u65f6\uff0c\u53ef\u80fd\u4f1a\u5ffd\u7565\u67d0\u4e9b\u7279\u6b8a\u60c5\u51b5\uff0c\u5982\u96f6\u503c\u3001\u65e0\u7a77\u5927\u6216\u8fb9\u754c\u60c5\u51b5\u3002\u8fd9\u4e9b\u7279\u6b8a\u60c5\u51b5\u6709\u65f6\u5019\u4f1a\u5bf9\u6700\u7ec8\u7ed3\u679c\u4ea7\u751f\u91cd\u5927\u5f71\u54cd\u3002<\/p>\n\n\n\n<p>\u6bd4\u5982\u8fd9\u4e2aFor how many real numbers x do we have that log<sub>5 <\/sub>(1 + x) = x?\u6211\u4eec\u8981\u6ce8\u610f\u5176\u4e2d\u7684\u4e00\u4e2a\u89e3\u4e3ax=0\uff0c\u6240\u4ee5\u6700\u7ec8\u7b54\u6848\u6709\u4e24\u4e2a\u89e3\u800c\u4e0d\u662f\u4e00\u4e2a\u89e3\u3002<\/p>\n\n\n\n<p>5.\u8fc7\u5ea6\u590d\u6742\u5316\uff1a\u6709\u65f6\uff0c\u5b66\u751f\u4e3a\u4e86\u663e\u793a\u81ea\u5df1\u7684\u80fd\u529b\u800c\u8bd5\u56fe\u4f7f\u7528\u8fc7\u4e8e\u590d\u6742\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u95ee\u9898\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u4e0d\u5fc5\u8981\u7684\u9519\u8bef\u548c\u65f6\u95f4\u6d6a\u8d39\u3002\u7b80\u5355\u76f4\u63a5\u7684\u65b9\u6cd5\u5f80\u5f80\u66f4\u4e3a\u6709\u6548\u3002<\/p>\n\n\n\n<p>\u5173\u952e\u8bcd\uff1aSMT\u65af\u5766\u798f\u5927\u5b66\u6570\u5b66\u9526\u6807\u8d5b&nbsp;\u5e38\u89c1\u9519\u8bef<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e4%ba%94%e3%80%81%e5%b8%b8%e8%a7%81%e9%97%ae%e9%a2%98\"><\/span>\u4e94\u3001\u5e38\u89c1\u95ee\u9898<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.\u771f\u9898\u7ec3\u4e60\u4e2d\u5e94\u6ce8\u610f\u54ea\u4e9b\u9519\u8bef<\/h3>\n\n\n\n<p>\u5728\u4f7f\u7528\u771f\u9898\u8fdb\u884c\u7ec3\u4e60\u65f6\uff0c\u53c2\u8d5b\u8005\u9700\u8981\u7279\u522b\u6ce8\u610f\u4ee5\u4e0b\u51e0\u7c7b\u9519\u8bef\uff1a<\/p>\n\n\n\n<p>\u8ba1\u7b97\u9519\u8bef\uff1a\u5305\u62ec\u56db\u5219\u8fd0\u7b97\u3001\u4ee3\u6570\u8fd0\u7b97\u3001\u4e09\u89d2\u51fd\u6570\u8fd0\u7b97\u7b49\uff0c\u8fd9\u4e9b\u9519\u8bef\u901a\u5e38\u662f\u7531\u4e8e\u7c97\u5fc3\u6216\u4e0d\u719f\u7ec3\u9020\u6210\u7684\u3002<\/p>\n\n\n\n<p>\u6982\u5ff5\u9519\u8bef\uff1a\u5bf9\u6570\u5b66\u6982\u5ff5\u6216\u5b9a\u7406\u7684\u7406\u89e3\u4e0d\u51c6\u786e\uff0c\u5bfc\u81f4\u89e3\u9898\u601d\u8def\u504f\u822a\u6216\u7b54\u6848\u5b8c\u5168\u9519\u8bef\u3002<\/p>\n\n\n\n<p>\u903b\u8f91\u9519\u8bef\uff1a\u5728\u89e3\u9898\u8fc7\u7a0b\u4e2d\uff0c\u63a8\u7406\u4e0d\u4e25\u5bc6\u6216\u903b\u8f91\u6df7\u4e71\uff0c\u5bfc\u81f4\u7ed3\u8bba\u65e0\u6cd5\u6210\u7acb\u3002<\/p>\n\n\n\n<p>\u7b26\u53f7\u9519\u8bef\uff1a\u5305\u62ec\u53d8\u91cf\u3001\u5e38\u6570\u3001\u8fd0\u7b97\u7b26\u7b49\u7b26\u53f7\u7684\u6df7\u6dc6\u6216\u8bef\u7528\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u4f7f\u7528\u771f\u9898\u590d\u4e60\u6709\u4ec0\u4e48\u7279\u522b\u7684\u6280\u5de7\u5417\uff1f<\/h3>\n\n\n\n<p>\u4f7f\u7528\u771f\u9898\u8fdb\u884c\u590d\u4e60\u65f6\uff0c\u4ee5\u4e0b\u51e0\u4e2a\u6280\u5de7\u53ef\u80fd\u4f1a\u5bf9\u53c2\u8d5b\u8005\u6709\u6240\u5e2e\u52a9\uff1a<\/p>\n\n\n\n<p>\u5206\u7c7b\u6574\u7406\uff1a\u5c06\u771f\u9898\u6309\u7167\u9898\u578b\u3001\u96be\u5ea6\u6216\u77e5\u8bc6\u70b9\u8fdb\u884c\u5206\u7c7b\u6574\u7406\uff0c\u6709\u52a9\u4e8e\u9488\u5bf9\u6027\u5730\u8fdb\u884c\u590d\u4e60\u548c\u5f3a\u5316\u8bad\u7ec3\u3002<\/p>\n\n\n\n<p>\u6a21\u62df\u8003\u8bd5\uff1a\u5728\u771f\u5b9e\u7684\u8003\u8bd5\u73af\u5883\u4e0b\u4f7f\u7528\u771f\u9898\u8fdb\u884c\u6a21\u62df\u8003\u8bd5\uff0c\u6709\u52a9\u4e8e\u719f\u6089\u8003\u8bd5\u6d41\u7a0b\u548c\u65f6\u95f4\u538b\u529b\uff0c\u63d0\u9ad8\u5e94\u8bd5\u80fd\u529b\u3002<\/p>\n\n\n\n<p>\u9519\u9898\u53cd\u601d\uff1a\u5bf9\u505a\u9519\u7684\u771f\u9898\u8fdb\u884c\u53cd\u601d\u548c\u603b\u7ed3\uff0c\u627e\u51fa\u9519\u8bef\u7684\u539f\u56e0\u548c\u6839\u6e90\uff0c\u907f\u514d\u5728\u7c7b\u4f3c\u95ee\u9898\u4e0a\u518d\u6b21\u72af\u9519\u3002<\/p>\n\n\n\n<p>\u53cd\u590d\u7ec3\u4e60\uff1a\u5bf9\u4e8e\u6613\u9519\u6216\u96be\u4ee5\u638c\u63e1\u7684\u771f\u9898\uff0c\u53ef\u4ee5\u8fdb\u884c\u53cd\u590d\u7ec3\u4e60\u548c\u5f3a\u5316\u8bad\u7ec3\uff0c\u76f4\u5230\u719f\u7ec3\u638c\u63e1\u4e3a\u6b62\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%e5%85%ad%e3%80%81%e6%80%bb%e7%bb%93\"><\/span>\u516d\u3001\u603b\u7ed3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u672c\u6587\u56f4\u7ed5SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u771f\u9898\u8fdb\u884c\u4e86\u5168\u9762\u7684\u5206\u6790\u548c\u603b\u7ed3\u3002\u9996\u5148\uff0c\u6211\u4eec\u4ecb\u7ecd\u4e86SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u80cc\u666f\u548c\u7279\u70b9\uff0c\u5f3a\u8c03\u4e86\u771f\u9898\u5728\u5907\u8003\u8fc7\u7a0b\u4e2d\u7684\u91cd\u8981\u6027\u3002\u63a5\u7740\uff0c\u6211\u4eec\u901a\u8fc7\u4ee3\u6570\u9898\u3001\u5fae\u79ef\u5206\u9898\u3001\u79bb\u6563\u9898\u548c\u51e0\u4f55\u5b66\u9898\u56db\u4e2a\u5177\u4f53\u771f\u9898\uff0c\u5c55\u793a\u4e86SMT\u7ade\u8d5b\u7684\u9898\u578b\u548c\u96be\u5ea6\uff0c\u5e76\u5bf9\u6bcf\u4e2a\u9898\u76ee\u7684\u89e3\u9898\u601d\u8def\u548c\u65b9\u6cd5\u8fdb\u884c\u4e86\u8be6\u7ec6\u7684\u9610\u8ff0\u3002\u5728\u771f\u9898\u5206\u6790\u90e8\u5206\uff0c\u6211\u4eec\u6307\u51fa\u4e86\u5728\u89e3\u9898\u8fc7\u7a0b\u4e2d\u9700\u8981\u6ce8\u610f\u7684\u5e38\u89c1\u9519\u8bef\u548c\u5e38\u89c1\u95ee\u9898\uff0c\u5305\u62ec\u7b26\u53f7\u9519\u8bef\u3001\u5ffd\u7565\u5b9a\u4e49\u57df\u548c\u503c\u57df\u3001\u8ba1\u7b97\u5931\u8bef\u3001\u5ffd\u7565\u7279\u6b8a\u60c5\u51b5\u3001\u8fc7\u5ea6\u590d\u6742\u5316\u3001\u5ffd\u89c6\u56fe\u5f62\u7684\u5e2e\u52a9\u3001\u4e0d\u68c0\u67e5\u7b54\u6848\u4ee5\u53ca\u65f6\u95f4\u7ba1\u7406\u4e0d\u5f53\u7b49\u3002\u9488\u5bf9\u8fd9\u4e9b\u95ee\u9898\uff0c\u6211\u4eec\u63d0\u51fa\u4e86\u76f8\u5e94\u7684\u89e3\u51b3\u7b56\u7565\u548c\u6280\u5de7\uff0c\u5982\u4ed4\u7ec6\u5ba1\u9898\u3001\u4fdd\u6301\u4e13\u6ce8\u3001\u4ed4\u7ec6\u68c0\u67e5\u8ba1\u7b97\u7ed3\u679c\u3001\u7406\u89e3\u6570\u5b66\u6982\u5ff5\u548c\u5b9a\u7406\u3001\u5408\u7406\u5206\u914d\u65f6\u95f4\u7b49\u3002<\/p>\n\n\n\n<p>\u901a\u8fc7\u672c\u6587\u7684\u5206\u6790\u548c\u603b\u7ed3\uff0c\u6211\u4eec\u5e0c\u671b\u80fd\u591f\u5e2e\u52a9\u5e7f\u5927\u5b66\u5b50\u66f4\u597d\u5730\u4e86\u89e3SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u7279\u70b9\u548c\u8981\u6c42\uff0c\u638c\u63e1\u6709\u6548\u7684\u5907\u8003\u65b9\u6cd5\u548c\u6280\u5de7\uff0c\u63d0\u9ad8\u89e3\u9898\u80fd\u529b\u548c\u5e94\u8bd5\u6c34\u5e73\u3002\u540c\u65f6\uff0c\u6211\u4eec\u4e5f\u5e0c\u671b\u80fd\u591f\u4e3a\u6570\u5b66\u6559\u80b2\u5de5\u4f5c\u8005\u63d0\u4f9b\u4e00\u4e9b\u6709\u76ca\u7684\u53c2\u8003\u548c\u542f\u793a\uff0c\u4fc3\u8fdb\u6570\u5b66\u6559\u80b2\u7684\u6539\u8fdb\u548c\u53d1\u5c55\u3002<\/p>\n\n\n<div class=\"retention-card-new\" data-lang=\"zh\" data-subject=\"MATH\" data-btnName=\"\u7acb\u5373\u514d\u8d39\u8bd5\u542c\" data-subTitle=\"\u9002\u5408\u5168\u74031-12\u5e74\u7ea7\u5b66\u751f\">\r\n    <div class=\"retention-card-l\">\r\n        <div class=\"trustpilot-image\"><\/div>\r\n        <h3><p>\u609f\u7a7a\u56fd\u9645\u6570\u5b66<br \/>\n<span>\u8ba9\u6570\u5b66\u4e4b\u5149\u7167\u4eae\u6bcf\u4e00\u4e2a\u5b69\u5b50\uff01<\/span><\/p>\n<\/h3>\r\n        <p>\u9002\u5408\u5168\u74031-12\u5e74\u7ea7\u5b66\u751f<\/p>\r\n        <a class=\"retention-card-button is-point\" href=\"https:\/\/www.wukongsch.com\/zh\/independent-appointment\/?subject=math&amp;l=c87d1e8c-ec8e-4a2a-83b7-84e0af53d81f&amp;booking_triggerevent=BLOG_DETAIL_MODEL_CTA_BUTTON\" data-buttonname=\"\u7acb\u5373\u9884\u7ea6\u6309\u94ae\u70b9\u51fb\" data-event=\"C_Blog_BLOG_DETAIL_MIDDLE_CTA_BUTTON\" data-expose-buttonname=\"\u7acb\u5373\u9884\u7ea6\u6309\u94ae\u66dd\u5149\" 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\u6064\u3001\u5468\u4e94\u665a\u9910\u53ca\u5468\u516d\u5348\u9910\uff1b\u5982\u6709\u9700\u8981\uff0c\u53ef\u7533\u8bf7\u7ecf\u6d4e\u8d44\u52a9\uff01\uff09\u6210\u7ee9\u67e5\u8be2\uff1a\u6210\u7ee9\u67e5\u770b\u76f8\u5173\u4fe1\u606f\uff0c\u53ef\u4ee5\u67e5\u770b\u609f\u7a7a\u201cSMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u6210\u7ee9\u67e5\u8be2\u6307\u5357\u201d\u8fd9\u7bc7\u6587\u7ae0\u3002 \u4e09\u3001\u771f\u9898\u5206\u6790 \u771f\u9898\u5206\u6790\u662f\u5907\u8003SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u91cd\u8981\u73af\u8282\uff0c\u901a\u8fc7\u5bf9\u5386\u5e74\u771f\u9898\u7684\u6df1\u5165\u7814\u7a76\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u4e86\u89e3\u8003\u8bd5\u7684\u9898\u578b\u3001\u6240\u8986\u76d6\u7684\u9886\u57df\u4ee5\u53ca\u89e3\u9898\u6280\u5de7\u3002 1.\u771f\u9898\u9898\u578b SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u7684\u9898\u578b\u591a\u6837\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\u9009\u62e9\u9898\u3001\u586b\u7a7a\u9898\u3001\u89e3\u7b54\u9898\u7b49\u3002\u5176\u4e2d\uff0c\u89e3\u7b54\u9898\u5360\u636e\u8f83\u5927\u6bd4\u91cd\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u624e\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u548c\u7075\u6d3b\u7684\u601d\u7ef4\u80fd\u529b\u3002\u9898\u578b\u7684\u591a\u6837\u6027\u4f7f\u5f97\u8003\u8bd5\u5185\u5bb9\u66f4\u52a0\u4e30\u5bcc\uff0c\u4e5f\u589e\u52a0\u4e86\u8003\u8bd5\u7684\u96be\u5ea6\u548c\u6311\u6218\u6027\u3002 2.\u6240\u8986\u76d6\u9886\u57df SMT\u65af\u5766\u798f\u6570\u5b66\u9526\u6807\u8d5b\u6db5\u76d6\u4e86\u6570\u5b66\u7684\u591a\u4e2a\u9886\u57df\uff0c\u5305\u62ec\u4ee3\u6570\u3001\u5fae\u79ef\u5206\u3001\u79bb\u6563\u3001\u51e0\u4f55\u5b66\u7b49\u3002\u5176\u4e2d\uff0c\u4ee3\u6570\u548c\u51e0\u4f55\u662f\u8003\u8bd5\u7684\u91cd\u70b9\u9886\u57df\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u638c\u63e1\u76f8\u5e94\u7684\u57fa\u7840\u77e5\u8bc6\u548c\u89e3\u9898\u65b9\u6cd5\u3002\u6b64\u5916\uff0c\u6570\u8bba\u548c\u7ec4\u5408\u7b49\u9886\u57df\u4e5f\u65f6\u5e38\u51fa\u73b0\uff0c\u9700\u8981\u53c2\u8d5b\u8005\u5177\u5907\u4e00\u5b9a\u7684\u6d89\u730e\u548c\u7406\u89e3\u80fd\u529b\u3002 \u4e0b\u9762\u662f2023\u5e74SMT\u76f8\u5173\u771f\u9898\u4ee5\u53ca\u89e3\u7b54 \u4ee3\u6570Compute the number of values of x in the interval [\u221211\u03c0, \u22122\u03c0] that satisfy 5 cos(x)+4 5 sin(x)+3 = 0\u89e3\uff1a\u5f53\u5206\u6570\u7684\u5206\u5b50\u7b49\u4e8e0\u800c\u5206\u6bcd\u4e0d\u7b49\u4e8e0\u65f6\u5206\u6570\u7b49\u4e8e0\u30025cos (x) + 4 = 0\u7684\u89e3\u5f62\u5f0f\u4e3ax =\u00b1arccos(- 4\/5) + 2\u03c0k\uff0c5 sin(x) + 3 = 0\u7684\u89e3\u4e3ax =\u00b1arcsin(- 3\/5) + 2\u03c0k\uff0c\u6211\u4eec\u770b\u5230[2k\u03c0\uff0c(2k + 1)\u03c0]\u5f62\u5f0f\u7684\u6bcf\u4e00\u4e2a\u533a\u95f4\u90fd\u6709\u4e00\u4e2a\u89e3\uff1b[(2k + 1)\u03c0\uff0c(2k + 2)\u03c0]\u5f62\u5f0f\u7684\u65b9\u7a0b\u548c\u533a\u95f4\u65e0\u89e3\u3002\u56e0\u6b64\uff0c\u67094\u4e2a\u533a\u95f4[\u221211\u03c0\uff0c\u22122\u03c0]\u5185\u7684\u89e3\u3002 \u5fae\u79ef\u5206For how many real numbers x do we have that log5 (1 + x) = x?\u89e3\uff1a\u6211\u4eec\u9996\u5148\u6ce8\u610f\u5230\u4e0d\u5b58\u5728x\u2264- 1\u7684\u89e3\uff0c\u56e0\u4e3a\u5de6\u8fb9\u5728\u8fd9\u4e2a\u533a\u57df\u5185\u6ca1\u6709\u5b9a\u4e49\u3002\u66f4\u8fdb\u4e00\u6b65\uff0cx = 0\u662f\u4e00\u4e2a\u89e3\u3002\u5bf9\u4e8e\u4efb\u610fx &gt; 0\uff0c\u5c06\u65b9\u7a0b\u6539\u5199\u4e3a1 + x = 5x\u3002\u56e0\u4e3a1 + x\u2264e x\u22645X(\u53ef\u4ee5\u7528\u6c42\u5bfc\u6216\u5e42\u7ea7\u6570\u6765\u8868\u793a)\u4e0d\u53ef\u80fd\u6709X &gt; 0\u7684\u89e3\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u8981\u68c0\u9a8c\u7684\u552f\u4e00\u533a\u95f4\u662f- 1 &lt; x &lt; 0\u3002\u8bbef(x) = 5x\u2212x\u22121\u3002\u5219f(0) = 0,f(- 1)&#46;&#46;&#46;<\/p>\n","protected":false},"author":211806801,"featured_media":31212,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","footnotes":""},"categories":[135358],"tags":[137467],"class_list":["post-31198","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-education-info","tag-math-question-answer"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast 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