{"id":401,"date":"2024-01-08T20:31:23","date_gmt":"2024-01-08T12:31:23","guid":{"rendered":"https:\/\/big-tan.top\/?p=401"},"modified":"2024-02-13T12:02:36","modified_gmt":"2024-02-13T04:02:36","slug":"%e4%bb%a3%e6%95%b0%e5%bc%8f%e6%b1%82%e5%80%bc%e9%97%ae%e9%a2%98%e6%8a%80%e5%b7%a7%e9%9b%86%e9%94%a6%ef%bc%88%e4%b8%80%ef%bc%89","status":"publish","type":"post","link":"https:\/\/itantan.eu.org\/?p=401","title":{"rendered":"\u4ee3\u6570\u5f0f\u6c42\u503c\u95ee\u9898\u6280\u5de7\u96c6\u9526\uff08\u4e00\uff09"},"content":{"rendered":"\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-4c1c980 stk-block-background\" data-block-id=\"4c1c980\"><style>.stk-4c1c980{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-4c1c980:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-4c1c980-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-d490419\" data-v=\"4\" data-block-id=\"d490419\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-d490419-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-d490419-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-45e5204\" data-block-id=\"45e5204\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e00\uff1a$\\displaystyle\u82e5a+\\frac1a=34,$\u6c42$\\displaystyle a^{10}+a^5+\\frac1{a^5}+\\frac1{a^{10}}$\u7684\u503c\u3002<\/strong><br><strong>\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u901a\u8fc7\u89c2\u5bdf\u53d1\u73b0\uff0c\u5df2\u77e5\u6761\u4ef6\u4e3a$\\begin{split}a+\\frac1a=3\\end{split}$\uff0c\u800c\u6240\u6c42\u7684\u4ee3\u6570\u5f0f\u4e2d\u6709$\\begin{split}a^5+\\frac1{a^5}\\end{split}$\u4ee5\u53ca$\\begin{split}a^{10}+\\frac1{a^{10}}\\end{split}$\u7684\u9879\uff0c\u8fd9\u4e9b\u9879\u4e0e\u5df2\u77e5\u6761\u4ef6\u7684\u7ed3\u6784\u7c7b\u4f3c\uff0c\u90fd\u662f\u4e00\u4e2a\u6570\u52a0\u5b83\u672c\u8eab\u7684\u5230\u6570\uff0c\u7531\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f$(a\\pm b)^2=a^2\\pm 2ab+b^2$\u53ef\u4ee5\u770b\u51fa\u6765\uff0c\u5f53a\u3001b\u4e92\u4e3a\u5012\u6570\u7684\u65f6\u5019\uff0c2ab\u5c31\u662f\u4e00\u4e2a\u5e38\u65702\uff0c\u7531\u6b64\u60f3\u5230\u53ef\u4ee5\u7528\u5e73\u65b9\u518d\u76f8\u4e58\u7684\u65b9\u5f0f\u8fdb\u884c\u5347\u5e42\uff0c\u7136\u540e\u518d\u6d88\u5143\u7684\u65b9\u6cd5\u3002<br><br>$\\begin{split}\u89e3\uff1a \\\\&amp;\\because a+\\frac1a=3 \\\\&amp; \\therefore(a+\\frac1a)^2=a^2+\\frac1{a^2}+2=9 \\\\&amp;\\therefore  a^2+\\frac1{a^2}=7 \\\\&amp;\\therefore(a+\\frac1a)(a^2+\\frac1{a^2})=a^3+\\frac1{a^3}+(a+\\frac1a)=3\\times7=21 \\\\&amp;\\Rightarrow a^3+\\frac1{a^3}=21-3=18 \\\\&amp;\u7ee7\u7eed\u5347\u5e42 \\\\&amp;\\therefore (a^2+\\frac1{a^2})(a^3+\\frac1{a^3})=a^5+\\frac1{a^5}+a+\\frac1{a}=7\\times18=126 \\\\&amp;\\Rightarrow a^5+\\frac1{a^5}=126-3=123 \\\\&amp;\u53c8\\because a^{10}+\\frac1{a^{10}}=(a^5)^2+(\\frac1{a^5})^2=(a^5+\\frac1{a^5})^2-2=123^2-2=15127 \\\\\\\\&amp;\\therefore \u539f\u5f0f=123+15127=15250\\end{split}$<\/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-5f8541a stk-block-background\" data-block-id=\"5f8541a\"><style>.stk-5f8541a{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-5f8541a:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-5f8541a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-40546a6\" data-v=\"4\" data-block-id=\"40546a6\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-40546a6-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-40546a6-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-c7e2c48\" data-block-id=\"c7e2c48\"><p class=\"stk-block-text__text\">\u2003\u2003<strong>\u9898\u76ee\u4e8c\uff1a\u82e5$a\u3001b$\u662f\u6b63\u6574\u6570\uff0c\u4e14$a+b+ab=34$\uff0c\u6c42$a+b$\u7684\u503c\u3002<\/strong><br><strong>\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7531\u4e8ea\u3001b\u90fd\u662f\u6b63\u6574\u6570\uff0c\u5982\u679c\u80fd\u591f\u5c06$a+b+ab=34$\u7684\u5de6\u8fb9\u5206\u89e3\u6210\u4e3a$(a+?)(b+?)=$\u5e38\u6570\u5f62\u5f0f\uff0c\u5c31\u53ef\u4ee5\u7528$a\u3001b$\u90fd\u662f\u6b63\u6574\u6570\u7684\u6761\u4ef6\u5206\u522b\u6c42\u51fa$a\u3001b$\u7684\u503c\u3002<br><br>$\\begin{split}\u89e3\uff1a \\\\&amp;\\because a+b+ab=34 \\\\&amp;\\therefore a(1+b)+b=34 \\\\&amp;\\therefore a(1+b)+b+1=35 \\\\&amp;\\therefore(a+1)(b+1)=35 \\\\&amp;\\because a\u3001b\u90fd\u662f\u6b63\u6574\u6570 \\\\&amp;\\therefore a+1\\ge2\u4e0eb+1\\ge2\u4e14\u4e5f\u90fd\u662f\u6b63\u6574\u6570\\\\&amp; \\because\u6b63\u6574\u657035\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e24\u7ec4\u6b63\u6574\u6570\u7684\u79ef\uff1a5\\times7\u62167\\times5\uff0c1\\times35\u621635\\times1\uff08\u4e0d\u7b26\u5408a+1\\ge2\u7684\u6761\u4ef6\uff0c\u820d\u53bb\uff09 \\\\&amp;\\therefore\\left\\{\\begin{array}{c}a+1=5 \\\\b+1=7\\end{array}\\right. \\  \\   \\   \u6216\u8005\\  \\left\\{\\begin{array}{c}a+1=7 \\\\b+1=5\\end{array}\\right. \\\\&amp;\\therefore \u65e0\u8bba\u54ea\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u6c42\u5f97\uff1aa+b=10\\end{split}$<\/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-42b3496 stk-block-background\" data-block-id=\"42b3496\"><style>.stk-42b3496{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-42b3496:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-42b3496-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-73073ef\" data-v=\"4\" data-block-id=\"73073ef\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-73073ef-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-73073ef-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-949ec51\" data-block-id=\"949ec51\"><p class=\"stk-block-text__text\"><strong>\u2003\u2003\u9898\u76ee\u4e09\uff1a\u82e5$x\u3001y$\u5747\u4e3a\u6b63\u6574\u6570\uff0c\u4e14$\\sqrt{x}+\\sqrt{y}=\\sqrt{1998}$\uff0c\u6c42$x+y$\u7684\u503c<\/strong>\u3002<br>\u2003\u2003<strong>\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u4e4d\u4e00\u770b\uff0c\u65e0\u4ece\u4e0b\u624b\uff0c\u4f7f\u7528\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\u53bb\u5de6\u4fa7\u6839\u53f7\u6839\u672c\u4e0d\u884c\uff0c\u6240\u4ee5\u8fd8\u5f97\u4ece$x\u3001y$\u90fd\u662f\u6b63\u6574\u6570\u8fd9\u4e2a\u6761\u4ef6\u5165\u624b\uff0c\u901a\u8fc7\u5206\u6790\u6839\u53f7\u4e0b$x\u3001y$\u7684\u5177\u4f53\u53d6\u503c\u3002<br><br>$\\begin{split}\u89e3\uff1a \\\\ &amp;\\because\\sqrt{x}+\\sqrt{y}=\\sqrt{1998}=\\sqrt{9\\times222}=3\\sqrt{222} \\\\&amp;\\therefore\\sqrt{x}\u548c\\sqrt{y}\u5e94\u8be5\u662f\u540c\u7c7b\u4e8c\u6b21\u6839\u5f0f\uff0c\u5426\u5219\u65e0\u6cd5\u5408\u5e76\u540c\u7c7b\u9879\\\\ &amp;\\therefore \\left \\{\\begin{array}{c}\\sqrt{x}=\\sqrt{222}\\\\ \\sqrt{y}=2\\sqrt{222}\\end{array}\\right. \\ \\ \u6216\\ \\ \\left\\{\\begin{array}{c}\\sqrt{x}=2\\sqrt{222} \\\\ \\sqrt{y}=\\sqrt{222}\\end{array}\\right. \\\\&amp;\\therefore \u53ef\u4ee5\u6c42\u5f97\uff1a\\left \\{\\begin{array}{c}x=222\\\\ y=888\\end{array}\\right. \\ \\ \u6216\\ \\ \\left\\{\\begin{array}{c}x=888 \\\\y=222\\end{array}\\right. \\\\&amp;\\therefore \\ x+y=222+888=1110\\end{split}$<\/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-c1e577b stk-block-background\" data-block-id=\"c1e577b\"><style>.stk-c1e577b{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-c1e577b:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c1e577b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0195eab\" data-v=\"4\" data-block-id=\"0195eab\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-0195eab-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-0195eab-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-6450590\" data-block-id=\"6450590\"><p class=\"stk-block-text__text\"><strong>\u2003\u2003\u9898\u76ee\u56db\uff1a\u82e5$x^2-y^2=16$\uff0c$(x+y)^2=8$\uff0c\u6c42$xy$\u7684\u503c\u3002<\/strong><br><strong>\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a<\/strong>$x^2-y^2$\u662f\u5e73\u65b9\u5dee\u516c\u5f0f\uff0c\u800c$(x+y)^2=x^2+y^2+2xy$\u5219\u662f\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff0c\u7531\u6b64\u53ef\u4ee5\u8bbe\u6cd5\u901a\u8fc7\u53d8\u6362\u6d88\u53bb\u4e8c\u6b21\u9879\u5e76\u6c42\u5f97$xy$\u7684\u503c\u3002<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\\because x^2-y^2=(x+y)(x-y)=16\\\\&amp;\\therefore\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u5e73\u65b9\u5f97\uff1a \\\\&amp;(x+y)^2(x-y)^2={16}^2=256\\\\&amp;\u5df2\u77e5(x+y)^2=8\\ \\\\&amp;\\therefore(x-y)^2=32\\end{split}$ <br><br> $\\left \\{\\begin{array}{c}(x-y)^2=32\\\\(x+y)^2=8\\end{array}\\right. \\ \\ \\Rightarrow\\ \\ \\left\\{\\begin{array}{c}(x-y)^2=x^2+y^2-2xy=32\\  \\ (1)\\\\ (x+y)^2=x^2+y^2+2xy=8\\  \\ \\ (2)\\end{array}\\right. $<br><br>$(1)-(2)\u53ef\u5f97\uff1a-4xy=24\u90a3\u4e48\u53ef\u4ee5\u5f97\u51fa\uff1axy=-6$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-f18b456 stk-block-background\" data-block-id=\"f18b456\"><style>.stk-f18b456{background-color:#F6F6F6 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-f18b456:before{background-color:#F6F6F6 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f18b456-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f35114d\" data-v=\"4\" data-block-id=\"f35114d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-f35114d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-f35114d-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-254e5ae\" data-block-id=\"254e5ae\"><p class=\"stk-block-text__text\"><strong>\u2003\u2003\u9898\u76ee\u4e94\uff1a\u5df2\u77e5$x\u3001y$\u4e3a\u6b63\u6574\u6570\uff0c\u4e14$x^2+y^2=2009$,\u6c42$x+y$\u7684\u503c\u3002<\/strong><br><strong>\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u7531\u4e8e\u90fd\u662fx\u3001y\u662f\u6b63\u6574\u6570\uff0c\u4e14$x^2\u548cy^2$\u90fd\u662f\u5b8c\u5168\u5e73\u65b9\u6570\uff0c\u53ef\u4ee5\u4ece\u8fd9\u4e2a\u65b9\u5411\u8fdb\u884c\u7a81\u7834\u3002<br><br>$\\begin{split}\u89e3\uff1a \\\\&amp;\u9996\u5148\u5c062009\u5206\u89e3\u8d28\u56e0\u6570\uff0c\u53ef\u5f97\uff1a \\\\&amp;2009=7\\times7\\times41\\\\&amp;\\because2009=7^2\\times41\\\\&amp;=7^2\\times(16+25)\\\\&amp;=7^2\\times(4^2+5^2)\\\\&amp;=7^2\\times4^2+7^2\\times5^2\\\\&amp;=(7\\times4)^2+(7\\times5)^2={28}^2+{35}^2 \\\\&amp; \\therefore2009={28}^2+{35}^2 \\end{split}$<br><br>\u2003\u2003$\\therefore \\left \\{\\begin{array}{c}x=28\\\\y=35\\end{array}\\right. \\ \\ \u6216\u8005\\ \\left\\{\\begin{array}{c}x=35\\\\y=28\\end{array}\\right. \\therefore x+y=35+28=63$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-d6ccb5a stk-block-background\" data-block-id=\"d6ccb5a\"><style>.stk-d6ccb5a{background-color:#ffe2c7 !important;border-top-left-radius:18px !important;border-top-right-radius:18px !important;border-bottom-right-radius:18px !important;border-bottom-left-radius:18px !important;border-style:solid !important;overflow:auto !important}.stk-d6ccb5a:before{background-color:#ffe2c7 !important}<\/style><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-d6ccb5a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c66ef14\" data-v=\"4\" data-block-id=\"c66ef14\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-c66ef14-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-c66ef14-inner-blocks\">\n<div class=\"wp-block-stackable-text stk-block-text stk-block stk-4e1954b\" data-block-id=\"4e1954b\"><p class=\"stk-block-text__text\"><strong>\u2003\u2003\u9898\u76ee\u516d\uff1a\u5df2\u77e5$a^2=a+\\sqrt7\uff0cb^2=b+\\sqrt7 (a\\ne{b})$\u6c42$a^2+b^2$\u7684\u503c\u3002<\/strong><br><strong>\u2003\u2003\u89e3\u9898\u601d\u8def\uff1a<\/strong>\u5982\u679c\u901a\u8fc7\u8054\u7acb\u4e24\u4e2a\u65b9\u7a0b\u89e3\u51fa\u6765a\u3001b\u7684\u503c\uff0c\u7136\u540e\u8ba1\u7b97$a^2+b^2$\u7684\u503c\uff0c\u53ef\u80fd\u6bd4\u8f83\u590d\u6742\uff0c\u9700\u8981\u901a\u8fc7\u5f62\u5f0f\u53d8\u6362\u8fdb\u884c\u6c42\u89e3\u3002<br><br>$\\begin{split}\u89e3\uff1a\\\\&amp;\\because\\left \\{\\begin{array}{c}a^2=a+\\sqrt7\\ldots\\ \\ \\ (1)\\\\b^2=b+\\sqrt7\\ldots\\ \\ \\ (2)\\end{array}\\right. \\end{split}$ <br><br>$\\begin{split}&amp;\\therefore (1)+(2)\u5f97\uff1aa^2+b^2=a+b+2\\sqrt7\\\\&amp;(1)-(2)\u5f97\uff1aa^2-b^2=a-b\\\\&amp;\\Rightarrow (a+b)(a-b)=a-b\\ \\ \\ (3) \\\\&amp;\\because a\\ne{b}\\\\&amp;\\therefore{a-b}\\ne0\uff0c\u5c06(3)\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5(a-b)\\\\&amp;\\therefore{a+b}=1\\\\&amp;\\therefore a^2+b^2=1+2\\sqrt7\\end{split}$<\/p><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2003\u2003\u9898\u76ee\u4e00\uff1a$\\displaystyle\u82e5a+\\frac1a=34,$\u6c42$\\displaystyle a^{1 [&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-401","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\/401","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=401"}],"version-history":[{"count":0,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=\/wp\/v2\/posts\/401\/revisions"}],"wp:attachment":[{"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=401"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itantan.eu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}