{"id":669,"date":"2024-02-14T09:52:28","date_gmt":"2024-02-14T01:52:28","guid":{"rendered":"https:\/\/big-tan.top\/?p=669"},"modified":"2024-02-14T10:35:20","modified_gmt":"2024-02-14T02:35:20","slug":"%e4%b8%96%e7%95%8c%e5%90%84%e5%9b%bd%e5%88%9d%e4%b8%ad%e6%95%b0%e5%ad%a6%e7%ab%9e%e8%b5%9b%e9%a2%98%ef%bc%88%e4%b8%80%ef%bc%89","status":"publish","type":"post","link":"https:\/\/itantan.eu.org\/?p=669","title":{"rendered":"\u4e16\u754c\u5404\u56fd\u4e2d\u5b66\u6570\u5b66\u7ade\u8d5b\u9898\u96c6\u9526\uff08\u4e00\uff09"},"content":{"rendered":"\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-deff50c stk-block-background\" data-block-id=\"deff50c\"><style>.stk-deff50c{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-deff50c:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-deff50c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0e8d1f7\" data-v=\"4\" data-block-id=\"0e8d1f7\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-0e8d1f7-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-0e8d1f7-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-52041d2\" data-block-id=\"52041d2\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e00\uff1a\u7f8e\u56fd\u4e2d\u5b66\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u89e3\u65b9\u7a0b$(x+2)^4+(x-2)^4=256$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e00\u822c\u6765\u8bf4\uff0c\u521d\u4e2d\u6570\u5b66\u6700\u591a\u5c31\u662f\u7acb\u65b9\u516c\u5f0f\uff0c\u8fd9\u91cc\u662f\u56db\u6b21\u65b9\uff0c\u5982\u679c\u5f3a\u884c\u5c55\u5f00\uff0c\u4e5f\u662f\u53ef\u4ee5\u5316\u7b80\u4e4b\u540e\u6c42\u89e3\u7684\uff0c\u8fd9\u9053\u9898\u76ee\u5176\u5b9e\u8003\u5bdf\u7684\u5c31\u662f\u5982\u4f55\u901a\u8fc7\u628a$(x\\pm2)^2$\u89c6\u4e3a\u4e00\u4e2a\u6574\u4f53\u8fdb\u884c\u5316\u7b80\uff0c\u96be\u5ea6\u5176\u5b9e\u5e76\u4e0d\u5927\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u539f\u65b9\u7a0b\u53ef\u4ee5\u8f6c\u6362\u4e3a\uff1a}\\\\&amp;\\Big[(x+2)^2-(x-2)^2\\Big]^2+2(x+2)^2\\cdot(x-2)^2=256\\\\&amp;\\Big\\{\\big[(x+2)+(x-2)\\big]\\big[(x+2)-(x-2)\\big]\\Big\\}^2+2\\Big[(x+2)(x-2)\\Big]^2=256\\\\&amp;(8x)^2+2(x^2-4)^2=256\\\\&amp;64x^2+2(x^4-8x^2+16)=256\\\\&amp;64x^2+2x^4-16x^2+32=256\\\\&amp;2x^4+48x^2-224=0\\\\&amp;x^4+24x^2-112=0\\\\&amp;\\Rightarrow(x^2+28)(x^2-4)=0\\\\&amp;\\because x^2+28\\gt0\\\\&amp;\\therefore x^2-4=0\\\\&amp;\\therefore x^2=4\\\\&amp;\\therefore x=\\pm2\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c138510 stk-block-background\" data-block-id=\"c138510\"><style>.stk-c138510{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:dotted !important;overflow:auto !important}.stk-c138510:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c138510-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-13daa90\" data-v=\"4\" data-block-id=\"13daa90\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-13daa90-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-13daa90-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-c3593e4\" data-block-id=\"c3593e4\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\\bbox[yellow,6pt,border:2px solid red]{\u62d3\u5c55\u601d\u7ef4\u9898\u76ee\uff1a\u7f8e\u56fd\u6570\u5b66\u7ade\u8d5b\u9898\u76ee}$\uff0c\u5df2\u77e5$x_1\u3001x_2\u3001x_3\\cdots x_{123}\u3001x_{124}$\u5747\u4e3a\u5b9e\u6570\uff0c$4^{x_1}=5\u30015^{x_2}=6\u30016^{x_3}=7\\ldots126^{x_{123}}=127\u3001127^{x_{124}}=128$\uff0c\u6c42$x_1\\cdot x_2\\cdot x_3\\cdots x_{123}\\cdot x_{124}$\u7684\u503c\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\u597d\u50cf\u633a\u590d\u6742\uff0c\u5176\u5b9e\u5c31\u662f\u66ff\u6362\u518d\u66ff\u6362\uff0c\u7136\u540e\u5229\u7528\u6307\u6570\u7684\u6027\u8d28\u8fdb\u884c\u6c42\u89e3\u3002\u4ece\u8fd9\u91cc\u53ef\u4ee5\u770b\u51fa\uff0c\u7f8e\u56fd\u4e2d\u5b66\u751f\u7684\u6570\u5b66\u6c34\u5e73\u7684\u786e\u4e0d\u9ad8\uff0c\u5373\u4f7f\u662f\u7ade\u8d5b\u9898\uff0c\u96be\u5ea6\u4e5f\u4e0d\u7b97\u9ad8\uff0c\u6bd4\u4e2d\u56fd\u7684\u7ade\u8d5b\u9898\u76ee\u7b80\u5355\u591a\u4e86\uff0c\u5f53\u7136\uff0c\u8fd9\u4e2a\u9898\u76ee\u8fd8\u53ef\u4ee5\u7528\u9ad8\u4e2d\u7684\u5bf9\u6570\u77e5\u8bc6\u6c42\u89e3\u3002<br><br>$\\begin{split}\\text{\u89e3\u6cd5\u4e00\uff1a}\\\\&amp;\\because 127^{x_{124}}=128\\\\&amp;\\therefore (126^{x_{123}})^{x_{124}}\\\\&amp;\\Rightarrow \\Big(\\big(125^{x_{122}}\\big)^{x_{123}}\\Big)^{x_{124}}=128\\\\&amp;\\Rightarrow \\Bigg(\\bigg(\\Big(\\big(4^{x_1}\\big)^{x_2}\\Big)^{x_3}\\bigg)\\cdots\\Bigg)^{x_{124}}=128\\\\&amp;\\therefore 4^{x_1\\cdot x_2\\cdot x_3\\cdots x_{124}}=128\\\\&amp;\\therefore 2^{2{x_1\\cdot x_2\\cdot x_3\\cdots x_{124}}}=2^7\\\\&amp;\\therefore 2{x_1\\cdot x_2\\cdot x_3\\cdots x_{124}}=7\\\\&amp;\\therefore x_1\\cdot x_2\\cdot x_3\\cdots x_{124}=\\dfrac72\\end{split}$<br><br>$\\begin{split}\\text{\u89e3\u6cd5\u4e8c\uff1a}\\\\&amp;\\text{\u5229\u7528\u5bf9\u6570\u77e5\u8bc6\u8fdb\u884c\u6c42\u89e3\u8fc7\u7a0b\u66f4\u4e3a\u7b80\u5355\u660e\u4e86\uff01}\\\\&amp;4^{x_1}=5\\Rightarrow x_1=log_45=\\dfrac{log5}{log4}\\\\&amp;5^{x_2}=6\\Rightarrow x_2=log_56=\\dfrac{log6}{log5}\\\\&amp;\\cdots\\cdots \\\\&amp;126^{x_{123}}=127\\Rightarrow x_{123}=log_{126}{127}=\\dfrac{log127}{log126}\\\\&amp;127^{x_{124}}=128\\Rightarrow x_{124}=log_{127}{128}=\\dfrac{log{128}}{log{127}}\\\\&amp;\\therefore x_1\\cdot x_2\\cdot x_3\\cdots x_{124}=\\dfrac{log5}{log4}\\cdot\\dfrac{log6}{log5}\\cdot\\dfrac{log7}{log6}\\cdots\\dfrac{log127}{log126}\\cdot\\dfrac{log128}{log127}\\\\&amp;=\\dfrac{log128}{log4}=log_4{128}=log_4{4^{\\frac72}}=\\dfrac72\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-27e0b4d stk-block-background\" data-block-id=\"27e0b4d\"><style>.stk-27e0b4d{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-27e0b4d:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-27e0b4d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cab37a4\" data-v=\"4\" data-block-id=\"cab37a4\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-cab37a4-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-cab37a4-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-243fa6d\" data-block-id=\"243fa6d\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e8c\uff1a\u5370\u5ea6\u4e2d\u5b66\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u5df2\u77e5$a\u3001b\u3001c$\u5747\u4e3a\u5b9e\u6570\uff0c\u4e14$a+b+c=0\uff0ca^3+b^3+c^3=0$\uff0c\u6c42$a^{15}+b^{15}+c^{15}$\u7684\u503c\u3002<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u5982\u679c\u8fd9\u9053\u9898\u662f\u586b\u7a7a\u9898\u7684\u8bdd\uff0c\u7528\u62c9\u739b\u52aa\u91d1\u77aa\u773c\u6cd5\u53ef\u4ee5\u4e00\u773c\u5c31\u770b\u51fa\u6765\uff0c\u7b49\u4e8e0\uff0c\u4f46\u662f\u8fd9\u662f\u8ba1\u7b97\u9898\uff0c\u9700\u8981\u6b65\u9aa4\uff0c\u8fd9\u5c31\u6709\u8f83\u5927\u7684\u96be\u5ea6\u4e86\u3002\u4e0d\u8fc7\u5462\uff0c\u53ef\u4ee5\u60f3\u529e\u6cd5\u53bb\u63a5\u8fd1\u62c9\u739b\u52aa\u91d1\u77aa\u773c\u6cd5\u5f97\u51fa\u6765\u7684\u7ed3\u679c\uff0c\u5c31\u662f\u77e5\u9053\u7ed3\u679c\uff0c\u8bbe\u6cd5\u53bb\u5b9e\u73b0\u800c\u5df2\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\because a+b+c=0\\\\&amp;\\therefore a=-(b+c)\\\\&amp;\\text{\u53d6\u7acb\u65b9\u53ef\u5f97\uff1a}a^3=-(b^3+3b^2c+3bc^2+c^3)\\Rightarrow a^3+b^3+c^3=-(3b^2c+3bc^2)\\\\&amp;\\therefore bc(b+c)=0\\\\&amp;\\therefore b=0\u6216c=0\u6216b=-c\\\\&amp;\\text{\u53ef\u4ee5\u5206\u60c5\u51b5\u8fdb\u884c\u8ba8\u8bba\uff1a}\\\\&amp;(1)b=0\\Rightarrow a+c=0\\Rightarrow a=-c\\\\&amp;\\therefore a^{15}+b^{15}+c^{15}=0\\\\&amp;(2)c=0\\Rightarrow a+b=0\\Rightarrow a+b=0\\Rightarrow a=-b\\\\&amp;\\therefore a^{15}+b^{15}+c^{15}=0\\\\&amp;(3)b=-c\\Rightarrow a=0\\\\&amp;\\therefore a^{15}+b^{15}+c^{15}=0\\\\&amp;\\therefore \\text{\u7efc\u4e0a\u6240\u8ff0\uff1a}a^{15}+b^{15}+c^{15}=0\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-4cfc4c6 stk-block-background\" data-block-id=\"4cfc4c6\"><style>.stk-4cfc4c6{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;margin-bottom:25px !important;overflow:auto !important}.stk-4cfc4c6:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-4cfc4c6-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ecaacd8\" data-v=\"4\" data-block-id=\"ecaacd8\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ecaacd8-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ecaacd8-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-8d9db80\" data-block-id=\"8d9db80\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e09\uff1a\u6e05\u534e\u5927\u5b66\u81ea\u4e3b\u62db\u751f\u9898\u76ee\uff0c$\\text{\u56e0\u5f0f\u5206\u89e3\uff1a}(a^3+a^2+a+1)^2-a^3$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u5168\u90e8\u5c55\u5f00\u591a\u9879\u5f0f\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff1b\u4ed4\u7ec6\u89c2\u5bdf\uff0c$\\bbox[yellow,5pt,border:2px solid red]{a^2+a+1}$\u662f\u7acb\u65b9\u5dee\u516c\u5f0f\u4e2d\u7684\u4e00\u9879\uff0c\u80fd\u5426\u628a\u5b83\u89c6\u4e3a\u4e00\u4e2a\u6574\u4f53\u8fdb\u884c\u5c55\u5f00\u5462\uff1f\u8fd9\u6837\u53ef\u4ee5\u964d\u4f4e\u5c55\u5f00\u591a\u9879\u5f0f\u7684\u8ba1\u7b97\u91cf\uff0c\u8fd9\u5c31\u662f\u8fd9\u9053\u56e0\u5f0f\u5206\u89e3\u9898\u7684\u5173\u952e\u548c\u96be\u70b9\u3002\u8fd9\u9053\u56e0\u5f0f\u5206\u89e3\uff0c\u770b\u4f3c\u7b80\u5355\uff0c\u5b9e\u9645\u9700\u8981\u4e00\u5b9a\u6280\u5de7\uff0c\u4e0d\u6127\u662f\u6e05\u534e\u5927\u5b66\u7684\u81ea\u4e3b\u62db\u751f\u9898\u76ee\u554a\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u539f\u5f0f}=(a^3)^2+2a^3(a^2+a+1)+(a^2+a+1)^2-a^3\\\\&amp;=a^6-a^3+2a^3(a^2+a+1)+(a^2+a+1)^2\\\\&amp;=a^3(a^3-1)+2a^3(a^2+a+1)+(a^2+a+1)^2\\\\&amp;=a^3(a-1)(a^2+a+1)+2a^3(a^2+a+1)+(a^2+a+1)^2\\\\&amp;=(a^2+a+1)(a^4-a^3+2a^3+a^2+a+1)\\\\&amp;=(a^2+a+1)(a^4+a^3+a^2+a+1)\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-1277ba6 stk-block-background\" data-block-id=\"1277ba6\"><style>.stk-1277ba6{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-1277ba6:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-1277ba6-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-396bad7\" data-v=\"4\" data-block-id=\"396bad7\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-396bad7-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-396bad7-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-24fc14b\" data-block-id=\"24fc14b\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u56db\uff1a\u534e\u7f57\u5e9a\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u5316\u7b80\uff1a$\\sqrt{2\\sqrt{xy}-x-y}\\qquad (x\\ne y)$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u6839\u53f7\u4e0b\u9762\u5c31\u662f\u4e00\u4e2a\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff0c\u76f4\u63a5\u5316\u7b80\u53bb\u6839\u53f7\u5c31\u53ef\u4ee5\u5316\u7b80\u3002\u5b9e\u9645\u8ba1\u7b97\u4e2d\u53d1\u73b0\u5374\u662f\u65e0\u610f\u4e49\uff0c\u8fd9\u8be5\u5982\u4f55\u5316\u7b80\u5462\uff1f\u8fd8\u662f\u9700\u8981\u6839\u636ex\u3001y\u7684\u5b9a\u4e49\u57df\u5206\u60c5\u51b5\u8fdb\u884c\u8ba8\u8bba\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{(1)\u5f53$x\\gt0\uff0cy\\gt0$\u65f6\uff1a}\\\\&amp;\\text{\u539f\u5f0f}=\\sqrt{2\\sqrt{xy}-(\\sqrt{x})^2-(\\sqrt{y})^2}=\\sqrt{-(\\sqrt{x}-\\sqrt{y})^2}\\\\&amp;\\therefore \\text{\u8d1f\u6570\u5f00\u6839\u53f7\u65e0\u610f\u4e49\uff01}\\\\\\\\&amp;\\text{(2)\u5f53$x\\leqslant0\uff0cy\\leqslant0$\u65f6\uff1a}\\\\&amp;\\text{\u539f\u5f0f}=\\sqrt{2\\sqrt{xy}+(\\sqrt{-x})^2+(\\sqrt{-y})^2}=\\sqrt{(\\sqrt{-x}+\\sqrt{-y})^2}=\\sqrt{-x}+\\sqrt{-y}\\\\&amp;\\therefore \\text{\u7c7b\u4f3c\u9898\u76ee\uff0c\u770b\u4f3c\u7b80\u5355\uff0c\u5b9e\u5219\u9677\u9631\u633a\u591a\uff0c\u8003\u5bdf\u57fa\u7840\u77e5\u8bc6\uff0c\u4e0d\u5c0f\u5fc3\u4e5f\u5bb9\u6613\u51fa\u9519\uff01}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c5ab077 stk-block-background\" data-block-id=\"c5ab077\"><style>.stk-c5ab077{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-c5ab077:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c5ab077-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-aea0ad9\" data-v=\"4\" data-block-id=\"aea0ad9\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-aea0ad9-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-aea0ad9-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-ef354c2\" data-block-id=\"ef354c2\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e94\uff1a\u5317\u4eac\u5e02\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u89e3\u65b9\u7a0b\uff1a$x^3+x^2=150$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u8fd9\u4e2a\u65b9\u7a0b\u6bd4\u8f83\u7b80\u5355\uff0c\u7528\u62c9\u739b\u52aa\u91d1\u77aa\u773c\u6cd5\u53ef\u4ee5\u6c42\u51fa\u6765$x=5$\u662f\u65b9\u7a0b\u7684\u4e00\u4e2a\u6839\uff0c\u77e5\u9053\u4e86\u8fd9\u4e2a\u6839\uff0c\u90a3\u4e48\u79fb\u9879\u540e$x^3+x^2-150$\u4e00\u5b9a\u5305\u542b$x-5$\u8fd9\u4e2a\u56e0\u5b50\uff0c\u7c7b\u4f3c\u9898\u76ee\u53ef\u4ee5\u4f7f\u7528\u957f\u9664\u6cd5\u6216\u8005\u51d1\u7acb\u65b9\u3001\u5e73\u65b9\u7684\u65b9\u5f0f\u6c42\u89e3\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u89c2\u5bdf\u53d1\u73b0\uff0c$x=5$\u662f\u65b9\u7a0b\u7684\u4e00\u4e2a\u6839\uff0c\u90a3\u4e48\u539f\u65b9\u7a0b\u4e00\u5b9a\u5305\u542b$x-5$\u7684\u56e0\u5b50\uff0c\u7528\u957f\u9664\u6cd5\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff1a}\\\\&amp;\\begin{array}{lll}<br>{\\phantom{00000} x^2\\phantom{0000}+6x\\phantom{000}+30} &amp;\\\\[-3pt]<br>{x-5} \\enclose{longdiv}{x^3+x^2\\phantom{000}-150\\phantom{000}}\\kern-.2ex \\\\[-3pt]<br>\\phantom{00000}\\underline{x^3-5x^2\\phantom{00000000000}} &amp; \\\\[-3pt]<br>\\phantom{000000000}6x^2\\phantom{0} &amp; \\\\[-3pt]<br>\\phantom{000000000}\\underline{6x^2-30x\\phantom{0000000}} &amp;\\\\[-3pt]<br>\\phantom{000000000000000}30x -150&amp;\\\\[-3pt]<br>\\phantom{000000000000000}\\underline{30x-150} &amp; \\\\[-3pt]<br>\\phantom{000000000000000000000}0<br>\\end{array}\\\\&amp;\\therefore (x-5)(x^2+6x+30)=0\\\\&amp;\\therefore x-5=0\u6216\u8005x^2+6x+30=0\\\\&amp;\\because x^2+6x+30=0\\text{\u7684\u5224\u522b\u5f0f$\\Delta=6^2-4\\times1\\times30=-84\\lt0$\uff0c\u65e0\u5b9e\u6570\u6839\uff01}\\\\&amp;\\therefore \\bbox[yellow,5pt,border:2px solid red]{\\text{\u7efc\u4e0a\u6240\u8ff0\uff0c\u539f\u65b9\u7a0b\u53ea\u6709\u4e00\u4e2a\u5b9e\u6570\u6839\uff1a}x=5}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-22aed28 stk-block-background\" data-block-id=\"22aed28\"><style>.stk-22aed28{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-22aed28:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-22aed28-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f799b21\" data-v=\"4\" data-block-id=\"f799b21\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-f799b21-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-f799b21-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-007899f\" data-block-id=\"007899f\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u516d\uff1a\u4fc4\u7f57\u65af\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u5df2\u77e5$x_0\\gt x_1\\gt x_2\\gt x_3\\gt \\cdots \\gt x_n\\in R$\uff0c\u8bc1\u660e\uff1a$x_0+\\dfrac{1}{x_0-x_1}+\\dfrac{1}{x_1-x_2}+\\dfrac{1}{x_2-x_3}+\\cdots+\\dfrac{1}{x_{n-1}-x_n}\\geqslant x_n+2n$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\u4f3c\u4e4e\u6bd4\u8f83\u590d\u6742\uff0c\u4f46\u662f\u5b9e\u9645\u5c31\u662f\u8003\u5bdf\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u5b9a\u7406\u3002\u53ea\u8981\u77e5\u9053\u8fd9\u4e2a\u5b9a\u7406\uff0c\u8fd9\u9053\u9898\u4e5f\u975e\u5e38\u7b80\u5355\uff01\u770b\u6765\uff0c\u4fc4\u7f57\u65af\u7684\u521d\u4e2d\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\u4e5f\u6ca1\u6709\u4e2d\u56fd\u7684\u9898\u76ee\u96be\u554a\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\u6839\u636e\u5df2\u77e5\u6761\u4ef6\uff0c\u5bf9\u4e8e\u4efb\u610f\u81ea\u7136\u6570i=0\u30011\u30012\u30013\u3001n-1\u3001n\u90fd\u6709\uff1a\\\\&amp;x_{n-1}-x_n\\gt0\\\\&amp;\\text{\u6839\u636e\u5e73\u5747\u6570\u5b9a\u7406\uff1a$\\dfrac{a+b}{2}\\geqslant\\sqrt{ab}$}\\Rightarrow{a+b}\\geqslant2\\sqrt{ab}\\\\&amp;\\therefore (x_{n-1}-x_n)+\\dfrac{1}{x_{n-1}-x_n}\\geqslant2\\sqrt{(x_{n-1}-x_n)\\cdot\\dfrac{1}{x_{n-1}-x_n}}=2\\\\&amp;\\therefore (x_0-x_1)+\\dfrac{1}{x_0-x_1}+(x_1-x_2)+\\dfrac{1}{x_1-x_2}+(x_2-x_3)+\\dfrac{1}{x_2-x_3}\\\\&amp;+\\cdots+(x_{n-1}-x_n)+\\dfrac{1}{x_{n-1}-x_n}\\geqslant2n\\\\&amp;\\text{\u6574\u7406\u540e\uff1a}(x_0-x_1)+(x_1-x_2)+(x_2-x_3)+\\cdots+(x_{n-1}-x_n)+\\\\&amp;\\dfrac{1}{x_0-x_1}+\\dfrac{1}{x_1-x_2}+\\dfrac{1}{x_2-x_3}+\\cdots+\\dfrac{1}{x_{n-1}-x_n}\\geqslant2n\\\\&amp;\\therefore x_0+\\dfrac{1}{x_0-x_1}+\\dfrac{1}{x_1-x_2}+\\dfrac{1}{x_2-x_3}+\\cdots+\\dfrac{1}{x_{n-1}-x_n}\\geqslant x_n+2n\\\\&amp;\\text{\u8fd9\u9053\u9898\u76ee\u5176\u5b9e\u5c31\u662f\u8003\u4e86\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u5b9a\u7406\uff0c}\\\\&amp;\\text{\u5c45\u7136\u662f\u4fc4\u56fd\u7684\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u770b\u6765\u8fd8\u662f\u4e2d\u56fd\u5b69\u5b50\u6c34\u5e73\u9ad8\uff01}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-f032fb2 stk-block-background\" data-block-id=\"f032fb2\"><style>.stk-f032fb2{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:dotted !important;overflow:auto !important}.stk-f032fb2:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f032fb2-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-03e6b91\" data-v=\"4\" data-block-id=\"03e6b91\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-03e6b91-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-03e6b91-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-b527c67\" data-block-id=\"b527c67\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\\bbox[yellow,5pt,border:2px solid red]{\\text{\u62d3\u5c55\u601d\u7ef4\uff1a\u6377\u514b\u6570\u5b66\u7ade\u8d5b\u9898\uff0c}\u8bc1\u660e\uff1a\u5bf9\u4e8ea\u3001b\u3001c\\in\\Bbb R^+\uff0c\\dfrac{a}{b+2c}+\\dfrac{b}{c+2a}+\\dfrac{c}{a+2b}\\geqslant1}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7c7b\u4f3c\u9898\u76ee\u4e00\u822c\u6765\u8bf4\uff0c\u9700\u8981\u7528\u5230\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u5b9a\u7406\uff0c\u4f46\u662f\u5982\u4f55\u8fdb\u884c\u53d8\u6362\u5462\uff1f\u9700\u8981\u4e00\u5b9a\u6280\u5de7\uff0c\u5bf9\u4e8e\u8fd9\u9053\u9898\u76ee\u53ef\u4ee5\u8003\u8651\u4f7f\u7528\u6362\u5143\u6cd5\u8fdb\u884c\u4ee3\u6570\u53d8\u6362\uff0c\u8fdb\u800c\u4f7f\u7528\u5e73\u5747\u6570\u5b9a\u7406\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u5e73\u5747\u6570\u5b9a\u7406\uff1a}\\dfrac{x_1+x_2+x_3+\\ldots+x_n}{n}\\geqslant\\sqrt[n]{x_1\\cdot x_2\\cdot x_3\\cdots x_n}\\\\&amp;\\text{\u4ee4}\\begin{cases}x=b+2c\\\\y=c+2a\\\\z=a+2b\\end{cases}\\qquad\\text{\u89e3\u65b9\u7a0b\u53ef\u5f97}\\Rightarrow\\begin{cases}a=\\dfrac19(4y+z-2x)\\\\b=\\dfrac19(4z+x-2y)\\\\c=\\dfrac19(4x+y-2z)\\end{cases}\\\\&amp;\\because\\dfrac{a}{b+2c}+\\dfrac{b}{c+2a}+\\dfrac{c}{a+2b}=\\dfrac{4y+z-2x}{9x}+\\dfrac{4z+x-2y}{9y}+\\dfrac{4x+y-2z}{9z}\\\\&amp;=\\dfrac19\\Big(\\dfrac{4y}{x}+\\dfrac{z}{x}-2+\\dfrac{4z}{y}+\\dfrac{x}{y}-2+\\dfrac{4x}{z}+\\dfrac{y}{z}-2\\Big)\\\\&amp;=\\dfrac19\\Bigg[(\\dfrac xy+\\dfrac yx)+(\\dfrac yz+\\dfrac  zy)+(\\dfrac zx+\\dfrac xz)+3\\times(\\dfrac yx+\\dfrac zy+\\dfrac xz)-6\\Bigg]\\\\&amp;\\geqslant\\dfrac19\\Bigg(2\\sqrt{{\\dfrac xy}\\cdot{\\dfrac yx}}+2\\sqrt{{\\dfrac yz}\\cdot{\\dfrac zy}}+2\\sqrt{{\\dfrac zx}\\cdot{\\dfrac xz}}+3\\times3\\cdot\\sqrt[3]{\\dfrac yx\\cdot\\dfrac zy\\cdot\\dfrac xz}-6       \\Bigg)\\\\&amp;=\\dfrac19\\cdot(2+2+2+9-6)=1\\\\&amp;\\therefore \\dfrac{a}{b+2c}+\\dfrac{b}{c+2a}+\\dfrac{c}{a+2b}\\geqslant1\\\\\\\\&amp;\\bbox[yellow,5pt,border:2px solid red]{\u6377\u514b\u6570\u5b66\u7ade\u8d5b \u8fd9\u9053\u9898\u76ee\u8fd8\u662f\u633a\u6709\u96be\u5ea6\u7684\uff0c\u9700\u8981\u6bd4\u8f83\u5de7\u5999\u7684\u53d8\u6362\u6280\u5de7\uff01}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2003\u2003\u9898\u76ee\u4e00\uff1a\u7f8e\u56fd\u4e2d\u5b66\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u89e3\u65b9\u7a0b$(x+2)^4+(x-2)^4=256$\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a\u4e00\u822c\u6765\u8bf4\uff0c\u521d\u4e2d [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,4],"tags":[],"class_list":["post-669","post","type-post","status-publish","format-standard","hentry","category-3","category-4"],"_links":{"self":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/669","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=669"}],"version-history":[{"count":0,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/669\/revisions"}],"wp:attachment":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=669"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=669"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=669"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}