{"id":727,"date":"2024-02-18T11:34:21","date_gmt":"2024-02-18T03:34:21","guid":{"rendered":"https:\/\/big-tan.top\/?p=727"},"modified":"2024-02-18T12:11:02","modified_gmt":"2024-02-18T04:11:02","slug":"%e4%b8%96%e7%95%8c%e5%90%84%e5%9b%bd%e4%b8%ad%e5%ad%a6%e6%95%b0%e5%ad%a6%e7%ab%9e%e8%b5%9b%e9%a2%98%e9%9b%86%e9%94%a6%ef%bc%88%e4%b8%89%ef%bc%89","status":"publish","type":"post","link":"https:\/\/itantan.eu.org\/?p=727","title":{"rendered":"\u4e16\u754c\u5404\u56fd\u4e2d\u5b66\u6570\u5b66\u7ade\u8d5b\u9898\u96c6\u9526\uff08\u4e09\uff09"},"content":{"rendered":"\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-2424b72 stk-block-background\" data-block-id=\"2424b72\"><style>.stk-2424b72{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-2424b72:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-2424b72-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-80cf954\" data-v=\"4\" data-block-id=\"80cf954\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-80cf954-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-80cf954-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-fd5fa37\" data-block-id=\"fd5fa37\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e00\uff1a\u52a0\u62ff\u5927\u5168\u56fd\u6570\u5b66\u7ade\u8d5b\u9898\uff0c$\u6c42\u51fa\u6240\u6709x\u6ee1\u8db3\uff1ax=\\Big(x-\\dfrac1x\\Big)^{\\dfrac12}+\\Big(1-\\dfrac1x\\Big)^{\\dfrac12}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u4f3c\u4e4e\u4e0d\u96be\uff0c\u4f46\u662f\uff0c\u505a\u8d77\u6765\u624d\u53d1\u73b0\uff0c\u5316\u7b80\u8d77\u6765\uff0c\u4f3c\u4e4e\u6709\u53c8\u4e0d\u77e5\u9053\u5982\u4f55\u4e0b\u624b\u3002\u8fd9\u9053\u9898\u76ee\uff0c\u6700\u5173\u952e\u7684\u662f\u5982\u4f55\u6539\u5199\u539f\u65b9\u7a0b\uff0c\u4ee5\u53ca\u5982\u4f55\u786e\u5b9ax\u7684\u53d6\u503c\u8303\u56f4\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u539f\u65b9\u7a0b\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a}x=\\sqrt{x-\\dfrac1x}+\\sqrt{1-\\dfrac1x}\\\\&amp;\\therefore x\\gt0(x\\ne0)\uff0c1-\\dfrac1x\\geqslant0\\\\&amp;\\because \u5f53x=1\u65f6\uff0c\u539f\u65b9\u7a0b\u4e0d\u6210\u7acb\\\\&amp;\\therefore x\\gt1\\\\&amp;\\text{\u539f\u65b9\u7a0b\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a}\\sqrt{1-\\dfrac1x}=x-\\sqrt{x-\\dfrac1x}\\\\&amp;\\Rightarrow 1-\\dfrac1x=x^2-2x\\sqrt{x-\\dfrac1x}+(x-\\dfrac1x)\\\\&amp;\\Rightarrow (x^2-1)-2\\sqrt{x\\cdot(x^2-1)}+x=0\\\\&amp;\\Rightarrow \\Big(\\sqrt{x^2-1}\\Big)^2-2\\sqrt{x}\\cdot\\sqrt{x^2-1}+\\Big(\\sqrt{x}\\Big)^2\\\\&amp;\\Rightarrow \\Big(\\sqrt{x^2-1}-\\sqrt{x}\\Big)^2=0\\\\&amp;\\therefore \\sqrt{x^2-1}=\\sqrt{x}\\Rightarrow x^2-1=x\\\\&amp;\\therefore x^2-x-1=0\\\\&amp;\\therefore x_{1,2}=\\dfrac{1\\pm\\sqrt{5}}{2}\\\\&amp;\\because x\\gt1\\\\&amp;\\therefore x=\\dfrac{1+\\sqrt{5}}{2}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u52a0\u62ff\u5927\u7684\u6570\u5b66\u7ade\u8d5b\u9898\u603b\u4f53\u6765\u8bf4\uff0c\u96be\u5ea6\u4e00\u822c\uff0c\u8ddf\u7f8e\u56fd\u7684\u6570\u5b66\u7ade\u8d5b\u6c34\u5e73\u76f8\u5f53\uff01}}\\end{split}$<\/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-d5f54aa stk-block-background\" data-block-id=\"d5f54aa\"><style>.stk-d5f54aa{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-d5f54aa:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-d5f54aa-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9b2bc1d\" data-v=\"4\" data-block-id=\"9b2bc1d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-9b2bc1d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-9b2bc1d-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-45bdb40\" data-block-id=\"45bdb40\"><style>.stk-45bdb40{overflow:auto !important}<\/style><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e8c\uff1a\u571f\u8033\u5176\u5168\u56fd\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u5df2\u77e5$a\u3001b\u3001c\\in \\Bbb R^+\uff0c0\\leqslant a\\leqslant b\\leqslant c\uff0c$\u8bc1\u660e\uff1a$(a+3b)(b+4c)(c+2a)\\geqslant60abc$<\/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\u5462\uff0c\u9700\u8981\u62d3\u5c55\u601d\u7ef4\uff0c\u4e09\u4e2a\u6570\u7684\u7b97\u672f\u5e73\u5747\u6570\u5b9a\u7406\uff0c\u5e76\u6ca1\u6709\u8981\u6c42\u4e09\u4e2a\u6570\u662f\u4e0d\u540c\u7684\uff0c\u5176\u4e2d\u7684\u4e24\u4e2a\u6570\u4e5f\u53ef\u4ee5\u76f8\u7b49\u3002\u672c\u9898\u7684\u53e6\u5916\u4e00\u4e2a\u5173\u952e\u73af\u8282\u662f\u4ee3\u6570\u5f0f\u53d8\u6362\u6280\u5de7\uff0c\u4e0d\u662f\u592a\u5bb9\u6613\u60f3\u5230\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u62d3\u5c55\u5e73\u5747\u6570\u5b9a\u7406\u53ef\u5f97\uff1a}\\\\&amp;\\begin{cases}a+b+b\\geqslant 3\\sqrt[3]{ab^2}\\\\b+c+c\\geqslant 3\\sqrt[3]{bc^2}\\\\c+a+a\\geqslant3\\sqrt[3]{ca^2}\\end{cases}\\Rightarrow (a+2b)(b+2c)(c+2a)\\geqslant27abc\\\\&amp;\\because 0\\leqslant a\\leqslant b\\leqslant c\\\\&amp;\\therefore (a+3b)(b+4c)(c+2a)=(a+\\dfrac13b+\\dfrac83b)(b+\\dfrac23c+\\dfrac{10}3c)(c+2a)\\\\&amp;\\geqslant(a+\\dfrac13a+\\dfrac83b)(b+\\dfrac23b+\\dfrac{10}3c)(c+2a)\\\\&amp;=(\\dfrac43a+\\dfrac83b)(\\dfrac53b+\\dfrac{10}3c)(c+2a)\\\\&amp;=\\frac43\\cdot(a+2b)\\cdot\\dfrac53\\cdot(b+2c)(c+2a)\\\\&amp;=\\dfrac{20}9\\cdot\\left(27abc\\right)=60abc\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u6ca1\u6709\u60f3\u5230\u5427\uff1f\u571f\u8033\u5176\u7684\u6570\u5b66\u7ade\u8d5b\u9898\u5c45\u7136\u8fd9\u4e48\u96be\uff0c\u602a\u4e0d\u5f97\u6b27\u6d32\u81ea\u53e4\u4ee5\u6765\u51fa\u4e86\u90a3\u4e48\u591a\u6570\u5b66\u6cf0\u6597\u5462\uff01}}\\end{split}$<\/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-3d60aa5 stk-block-background\" data-block-id=\"3d60aa5\"><style>.stk-3d60aa5{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-3d60aa5:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-3d60aa5-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a38b217\" data-v=\"4\" data-block-id=\"a38b217\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a38b217-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a38b217-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-2f24955\" data-block-id=\"2f24955\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e09\uff1a\u4fc4\u7f57\u65af\u83ab\u65af\u79d1\u5e02\u6570\u5b66\u5965\u6797\u5339\u514b\u7ade\u8d5b\u9898\uff0c$a\u3001b\\in \\Bbb R^+\uff0ca+b=1\uff0c$\u8bc1\u660e\uff1a$\\left(a+\\dfrac1a\\right)^2+\\left(b+\\dfrac1b\\right)^2\\geqslant\\dfrac{25}{2}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u8fd8\u662f\u9700\u8981\u7528\u5230\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u5b9a\u7406\uff0c\u4f46\u5b9e\u9645\u89e3\u9898\u8fc7\u7a0b \uff0c\u8fd8\u662f\u9700\u8981\u8fdb\u884c\u601d\u7ef4\u62d3\u5c55\uff0c\u7136\u540e\u518d\u8fdb\u884c\u4ee3\u6570\u5f0f\u53d8\u6362\uff0c\u6700\u7ec8\u6c42\u89e3\u4e0d\u7b49\u5f0f\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u7531\u7b97\u672f\u5e73\u5747\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u5e73\u5747\u6570\u7684\u4e0d\u7b49\u5f0f\u5b9a\u7406\u53ef\u77e5\uff1a}\\\\&amp;\\dfrac{x+y}{2}\\geqslant \\sqrt{xy}\\Rightarrow \\dfrac{x^2+y^2}{2}\\geqslant xy\\Rightarrow x^2+y^2\\geqslant2xy\\\\&amp;\\therefore 2(x^2+y^2)\\geqslant x^2+y^2+2xy=(x+y)^2\\\\&amp;\\therefore \\dfrac{x^2+y^2}{2}\\geqslant \\Big(\\dfrac{x+y}{2}\\Big)^2\\\\&amp;\\text{\u4ee4$x=a+\\dfrac1a\uff0cy=b+\\dfrac1b$}\\\\\\\\&amp;\\left(a+\\dfrac1a\\right)^2+\\left(b+\\dfrac1b\\right)^2\\geqslant2\\cdot \\left(\\dfrac{a+\\dfrac1a+b+\\dfrac1b}{2}\\right)^2=\\dfrac12\\cdot\\left(1+\\dfrac{a+b}{ab}\\right)=\\dfrac12\\cdot\\left(1+\\dfrac{1}{ab}\\right)\\\\&amp;\\because \\dfrac{a+b}{2}\\geqslant\\sqrt{ab}\\Rightarrow \\left(\\dfrac{a+b}{2}\\right)^2\\geqslant ab\uff0c\u5df2\u77e5\uff1aa+b=1\\\\&amp;\\therefore ab\\leqslant\\left(\\dfrac12\\right)^2=\\dfrac14\\Rightarrow ab\u7684\u6700\u5c0f\u503c\u4e3a\\dfrac14\\\\&amp;\\therefore \\left(a+\\dfrac1a\\right)^2+\\left(b+\\dfrac1b\\right)^2\\geqslant\\dfrac12\\cdot(1+4)^2=\\dfrac{25}{2}\\\\&amp;\\therefore \\text{\u4ec5\u5f53}a=b=\\dfrac12\\text{\u7684\u65f6\u5019\uff0c\u7b49\u53f7\u6210\u7acb\u3002}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u4fc4\u7f57\u65af\u7684\u6570\u5b66\u7ade\u8d5b\u96be\u5ea6\u4e5f\u4e0d\u4f4e\u54e6\uff01\u8981\u77e5\u9053\u5723\u5f7c\u5f97\u5821\u4e4b\u524d\u662f\u6570\u5b66\u5927\u795e\u6b27\u62c9\u751f\u6d3b\u7684\u5730\u65b9\uff01}}\\end{split}$<\/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-6799047 stk-block-background\" data-block-id=\"6799047\"><style>.stk-6799047{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-6799047:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6799047-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-925d5e5\" data-v=\"4\" data-block-id=\"925d5e5\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-925d5e5-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-925d5e5-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-3c63aea\" data-block-id=\"3c63aea\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u56db\uff1a\u4e2d\u56fd\u5965\u6797\u5339\u514b\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u5316\u7b80\uff1a$\\sqrt[4]{7+\\sqrt{48}}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u65e2\u7136\u662f\u6570\u5b66\u7ade\u8d5b\u9898\u76ee\uff0c\u5f53\u7136\u9700\u8981\u4e00\u5b9a\u7684\u6280\u5de7\u3002\u5177\u4f53\u5230\u8fd9\u9053\u9898\u76ee\uff0c\u9700\u8981\u7528\u5230\u6362\u5143\u6cd5\uff0c\u800c\u4e14\u9700\u8981\u7528\u5230\u5de7\u5999\u7684\u4ee3\u6362\u8fdb\u884c\u964d\u6b21\u624d\u80fd\u5316\u7b80\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u8bbe$\\quad a=\\sqrt[4]{7+\\sqrt{48}}\\quad(a\\gt0)$}\\\\&amp;\\therefore \\dfrac1a=\\dfrac{1}{\\sqrt[4]{7+\\sqrt{48}}}\\cdot\\dfrac{\\sqrt[4]{7-\\sqrt{48}}}{\\sqrt[4]{7-\\sqrt{48}}}=\\sqrt[4]{7-\\sqrt{48}}\\\\\\\\&amp;\\therefore a^4=7+\\sqrt{48}\uff0c\\dfrac1{a^4}=7-\\sqrt{48}\\Rightarrow a^4+\\dfrac1{a^4}=14\uff0ca^4-\\dfrac1{a^4}=2\\sqrt{48}=8\\sqrt3\\\\\\\\&amp;\\therefore \\left(a^2+\\dfrac1{a^2}\\right)^2=a^4+\\dfrac1{a^4}+2=16\\Rightarrow a^2+\\dfrac1{a^2}=4(a^2\\geqslant0)\\\\&amp;\\therefore\\left(a+\\dfrac1a\\right)^2=a^2+\\dfrac1{a^2}+2=4+2=6\\Rightarrow a+\\dfrac1a=\\sqrt6(a\\gt0)\\\\\\\\&amp;\\therefore \\left(a^2+\\dfrac1{a^2}\\right)\\left(a^2-\\frac1{a^2}\\right)=a^4-\\dfrac1{a^4}=8\\sqrt3\\Rightarrow a^2-\\dfrac1{a^2}=2\\sqrt3\\\\&amp;\\therefore \\left(a+\\dfrac1a\\right)\\left(a-\\dfrac1a\\right)=a^2-\\dfrac1{a^2}=2\\sqrt3\\Rightarrow a-\\dfrac1a=\\dfrac{2\\sqrt3}{\\sqrt6}=\\sqrt2\\\\\\\\&amp;\\therefore \\begin{cases}a+\\dfrac1a=\\sqrt6\\\\\\\\a-\\dfrac1a=\\sqrt2\\end{cases}\\Rightarrow 2a=\\sqrt6+\\sqrt2\\Rightarrow a=\\dfrac{\\sqrt6+\\sqrt2}{2}\\\\\\\\&amp;\\therefore \\sqrt[4]{7+\\sqrt{48}}=\\dfrac{\\sqrt6+\\sqrt2}{2}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u8fd9\u9053\u9898\u76ee\u975e\u5e38\u9700\u8981\u6280\u5de7\uff0c\u65e2\u6709\u6362\u5143\uff0c\u53c8\u5229\u7528\u4e00\u4e2a\u6570\u4e0e\u5012\u6570\u7684\u5b8c\u5168\u5e73\u65b9\u8fdb\u884c\u964d\u6b21\uff0c\u96be\u5ea6\u8fd8\u662f\u6bd4\u8f83\u5927\u7684\uff01}}\\end{split}$<\/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-f93f186 stk-block-background\" data-block-id=\"f93f186\"><style>.stk-f93f186{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-f93f186:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f93f186-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e09f2fb\" data-v=\"4\" data-block-id=\"e09f2fb\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-e09f2fb-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-e09f2fb-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-37a0bab\" data-block-id=\"37a0bab\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>$\\bbox[yellow,6pt,border:2px solid red]{\u62d3\u5c55\u601d\u7ef4\u9898\u76ee\uff1a\u4e2d\u56fd\u5357\u4eac\u5927\u5b66\u81ea\u4e3b\u62db\u751f\u9898\uff0c\u5316\u7b80\uff1a\\sqrt[3]{\\sqrt5+2}}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u8fd9\u79cd\u9ad8\u6b21\u5f00\u6839\u53f7\u5316\u7b80\u9898\uff0c\u901a\u5e38\u5e94\u8be5\u4f7f\u7528\u6362\u5143\u6cd5\u5316\u7b80\uff0c\u89c2\u5bdf\u6839\u53f7\u4e0b\u9762\u7684\u6570\u5b57\uff0c\u53ef\u4ee5\u60f3\u5230\u5012\u6570\u4e0e\u539f\u6570\u53ef\u80fd\u6709\u4e9b\u5173\u7cfb\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u8bbe\uff1a}a=\\sqrt[3]{\\sqrt5+2}\\gt0\\Rightarrow a^3=\\sqrt5+2\\\\&amp;\\text{\u5219\uff1a}\\dfrac1a=\\dfrac1{\\sqrt[3]{\\sqrt5+2}}=\\dfrac1{\\sqrt[3]{\\sqrt5+2}}\\cdot\\dfrac{\\sqrt[3]{\\sqrt5-2}}{\\sqrt[3]{\\sqrt5-2}}=\\sqrt[3]{\\sqrt5-2}\\Rightarrow \\dfrac1{a^3}=\\sqrt5-2\\\\&amp;a^3-\\dfrac1{a^3}=\\left(a-\\dfrac1a\\right)\\left(a^2+1+\\dfrac1{a^2}\\right)=\\left(a-\\dfrac1a\\right)\\left[\\left(a-\\dfrac1a\\right)^2+3\\right]=4\\\\&amp;\\text{\u4ee4\uff1a}a-\\dfrac1a=t\\Rightarrow t(t^2+3)=4\\Rightarrow t^3+3t-4=0\\Rightarrow t^3-1+3t-3=0\\\\&amp;\\Rightarrow (t-1)(t^2+t+4)=0\\\\&amp;\\therefore t-1=0\u6216t^2+t+4=0\\\\&amp;\\therefore t=1(t^2+t+4=0\u7684\\Delta=-15\\lt0\uff0c\u65e0\u5b9e\u6570\u6839)\\\\&amp;\\therefore a-\\dfrac1a=1\\Rightarrow a^2-a-1=0\\\\&amp;\\therefore a=\\dfrac{1\\pm\\sqrt{5}}{2}\\\\&amp;\\because a\\gt0\\\\&amp;\\therefore \\text{\u539f\u5f0f}=a=\\dfrac{1+\\sqrt5}{2}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u6bd5\u7adf\u662f\u5357\u5927\u81ea\u4e3b\u62db\u751f\u9898\u76ee\uff0c\u96be\u5ea6\u8fd8\u662f\u6bd4\u8f83\u5927\uff0c\u6ca1\u60f3\u5230\u521d\u4e2d\u5b66\u9738\u4e5f\u80fd\u4f5c\u5bf9\uff01}}\\end{split}$<\/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-498de64 stk-block-background\" data-block-id=\"498de64\"><style>.stk-498de64{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-498de64:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-498de64-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0400e51\" data-v=\"4\" data-block-id=\"0400e51\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-0400e51-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-0400e51-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-6f08bb1\" data-block-id=\"6f08bb1\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e94\uff1a\u7f8e\u56fd\u666e\u7279\u5357\u5927\u5b66\u751f\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u7528\u521d\u4e2d\u7684\u65b9\u6cd5\u540c\u6837\u53ef\u4ee5\u6c42\u89e3\u3002\u5316\u7b80\uff1a$\\displaystyle\\sqrt[8]{2207-\\cfrac1{2207-\\cfrac1{2207-\\cfrac1{2207\\cdots}}}}$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u5bf9\u4e8e\u7c7b\u4f3c\u7684\u65e0\u9650\u5faa\u73af\u5206\u6570\u5316\u7b80\uff0c\u901a\u5e38\u4f7f\u7528\u6362\u5143\u6cd5\u8fdb\u884c\u6c42\u89e3\uff0c\u9274\u4e8e\u8fd9\u91cc\u662f\u5f008\u6b21\u65b9\uff0c\u9700\u8981\u8fdb\u884c\u4e24\u6b21\u6362\u5143\uff0c\u7136\u540e\u591a\u6b21\u964d\u6b21\u8fdb\u884c\u6c42\u89e3\u3002<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\\text{\u8bbe\uff1a}t=2207-\\cfrac1{2207-\\cfrac1{2207-\\cfrac1{2207\\cdots}}}\\\\&amp;\\therefore t\\gt1\\Rightarrow t=2207-\\dfrac1t\\\\&amp;\\text{\u8bbe\uff1a}a=\\sqrt[8]t\\geqslant1\\Rightarrow t=a^8\\\\&amp;\\therefore a^8=2207-\\dfrac1{a^8}\\Rightarrow a^8+\\dfrac1{a^8}=2207\\\\&amp;\\because \\left(a^4+\\dfrac1{a^4}\\right)^2=a^8+2+\\dfrac1{a^8}=2207+2=47^2\\\\&amp;\\therefore a^4+\\dfrac1{a^4}=47\\gt0\\\\&amp;\\text{\u540c\u7406\u53ef\u5f97\uff1a}\\left(a^2+\\dfrac1{a^2}\\right)^2=a^4+2+\\dfrac1{a^4}=49=7^2\\Rightarrow a^2+\\dfrac1{a^2}=7\\\\&amp;\\qquad\\qquad\\left(a+\\dfrac1a\\right)^2=a^2+2+\\dfrac1{a^2}=9=3^2\\Rightarrow a+\\dfrac1a=3\\\\&amp;\\therefore a^2-3a+1=0\\Rightarrow a=\\dfrac{3\\pm\\sqrt5}{2}\\\\&amp;\\because a\\gt1\\\\&amp;\\therefore \\text{\u539f\u5f0f=}a=\\dfrac{3+\\sqrt5}{2}\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u7c7b\u4f3c\u5316\u7b80\u65e0\u9650\u5faa\u73af\u7684\u9898\u76ee\uff0c\u901a\u5e38\u4f7f\u7528\u6362\u5143\u6cd5\uff0c\u7136\u540e\u89c2\u5bdf\u7ed3\u679c\uff0c\u518d\u8bbe\u6cd5\u9010\u6b65\u964d\u6b21\u540e\u6c42\u89e3\uff01}}\\end{split}$<\/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-632a75d stk-block-background\" data-block-id=\"632a75d\"><style>.stk-632a75d{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-632a75d:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-632a75d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4f64412\" data-v=\"4\" data-block-id=\"4f64412\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-4f64412-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-4f64412-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-605c47b\" data-block-id=\"605c47b\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u516d\uff1a\u6cd5\u56fd\u7684\u6570\u5b66\u7ade\u8d5b\u9898\uff0c\u65b9\u6cd5\u6bd4\u8f83\u5de7\u5999\u3002\u89e3\u65b9\u7a0b\uff1a$\\sqrt{a-1}+\\sqrt{3a-5}+\\sqrt{4a-7}=4a-5$<\/strong><br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7c7b\u4f3c\u9898\u76ee\uff0c\u901a\u5e38\u8003\u8651\u6362\u5143\u6cd5\uff0c\u4f46\u662f\u5982\u4f55\u6362\u5143\u5462\uff1f\u9700\u8981\u601d\u8003\u4e00\u756a\uff0c\u65b9\u6cd5\u5f97\u5f53\u5219\u975e\u5e38\u7b80\u5355\uff0c\u5426\u5219\uff0c\u4f1a\u9677\u5165\u590d\u6742\u7684\u8ba1\u7b97\u4e4b\u4e2d\uff0c\u6cd5\u5170\u897f\u6570\u5b66\u4e00\u70b9\u4e5f\u4e0d\u6d6a\u6f2b\uff01<br><br>$\\begin{split}\\text{\u89e3\uff1a}\\\\&amp;\u4ee4\uff1ax=\\sqrt{a-1}\uff0cy=\\sqrt{3a-5}\uff0cz=\\sqrt{4a-7}\\\\&amp;\\therefore x+y+z=4a-5\\\\&amp;x^2+y^2+z^2=8a-13=2(4a-5)-3\\quad\\text{\u8fd9\u4e00\u6b65\u662f\u5173\u952e\uff0c\u5de7\u5999\uff01}\\\\&amp;\\therefore x^2+y^2+z^2=2(x+y+z)-3\\Rightarrow x^2+y^2+z^2-2x-2y-2z+3=0\\\\&amp;\\therefore (x-1)^2+(y-1)^2+(z-1)^2=0\\\\&amp;\\therefore x=y=z=1\\Rightarrow \\sqrt{a-1}=1\\\\&amp;\\therefore a=2\\\\&amp;\\bbox[yellow,6pt,border:2px solid red]{\\text{\u6cd5\u56fd\u8fd9\u9053\u6570\u5b66\u7ade\u8d5b\u9898\u867d\u8bf4\u96be\u5ea6\u4e0d\u5927\uff0c\u4f46\u662f\u4e5f\u9700\u8981\u65b9\u6cd5\u5f97\u5f53\u624d\u884c\uff0c\u5426\u5219\uff0c\u4e5f\u5bb9\u6613\u6389\u8fdb\u9677\u9631\uff01}}\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2003\u2003\u9898\u76ee\u4e00\uff1a\u52a0\u62ff\u5927\u5168\u56fd\u6570\u5b66\u7ade\u8d5b\u9898\uff0c$\u6c42\u51fa\u6240\u6709x\u6ee1\u8db3\uff1ax=\\Big(x-\\dfrac1x\\Big)^{\\dfra [&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-727","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\/727","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=727"}],"version-history":[{"count":0,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/727\/revisions"}],"wp:attachment":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=727"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=727"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=727"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}