{"id":93049,"date":"2026-08-11T18:40:51","date_gmt":"2026-08-11T10:40:51","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/93049.html"},"modified":"2026-08-11T18:40:51","modified_gmt":"2026-08-11T10:40:51","slug":"p1014-noip-1999-%e6%99%ae%e5%8f%8a%e7%bb%84-cantor-%e8%a1%a8","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/93049.html","title":{"rendered":"P1014 [NOIP 1999 \u666e\u53ca\u7ec4] Cantor \u8868"},"content":{"rendered":"<h2>P1014 [NOIP 1999 \u666e\u53ca\u7ec4] Cantor \u8868<\/h2>\n<h3>\u9898\u76ee\u63cf\u8ff0<\/h3>\n<p>\u73b0\u4ee3\u6570\u5b66\u7684\u8457\u540d\u8bc1\u660e\u4e4b\u4e00\u662f Georg Cantor \u8bc1\u660e\u4e86\u6709\u7406\u6570\u662f\u53ef\u679a\u4e3e\u7684\u3002\u4ed6\u662f\u7528\u4e0b\u9762\u8fd9\u4e00\u5f20\u8868\u6765\u8bc1\u660e\u8fd9\u4e00\u547d\u9898\u7684&#xff1a;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260811104049-6a7afc31a3c58.png\" alt=\"\" \/><\/p>\n<p>\u6211\u4eec\u4ee5 Z \u5b57\u5f62\u7ed9\u4e0a\u8868\u7684\u6bcf\u4e00\u9879\u7f16\u53f7\u3002\u7b2c\u4e00\u9879\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">1\/11\/1<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1\/1<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7136\u540e\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">1\/21\/2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1\/2<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">2\/12\/1<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2\/1<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">3\/13\/1<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">3\/1<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">2\/22\/2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2\/2<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u2026\u2026\u3002<\/p>\n<h3>\u8f93\u5165\u683c\u5f0f<\/h3>\n<p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">NN<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span>&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">1\u2264N\u22641071 \\\\le N \\\\le 10^7<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8193em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">7<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff09;\u3002<\/p>\n<h3>\u8f93\u51fa\u683c\u5f0f<\/h3>\n<p>\u8f93\u51fa\u8868\u4e2d\u7684\u7b2c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">NN<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span> \u9879\u3002<\/p>\n<h3>\u8f93\u5165\u8f93\u51fa\u6837\u4f8b #1<\/h3>\n<h4>\u8f93\u5165 #1<\/h4>\n<p>7<\/p>\n<h4>\u8f93\u51fa #1<\/h4>\n<p>1\/4<\/p>\n<h3>\u8bf4\u660e\/\u63d0\u793a<\/h3>\n<p>\u5bf9\u4e8e\u5168\u90e8\u6d4b\u8bd5\u6570\u636e&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">1\u2264N\u22641071 \\\\le N \\\\le 10^7<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8193em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">7<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<ul>\n<li>2024-11-18 0:30 \u6570\u636e\u4e2d\u52a0\u5165\u4e86\u6837\u4f8b&#xff0c;\u653e\u5728\u4e0d\u8ba1\u5206\u7684\u5b50\u4efb\u52a1 2 \u4e2d\u3002<\/li>\n<\/ul>\n<p><span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">include<\/span><span class=\"token string\">&lt;bits\/stdc&#043;&#043;.h&gt;<\/span><\/span><\/p>\n<p><span class=\"token keyword\">using<\/span> <span class=\"token keyword\">namespace<\/span> std<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">typedef<\/span> <span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> ll<span class=\"token punctuation\">;<\/span><span class=\"token comment\">\/\/ \u4e25\u683c\u8981\u6c42 (\u5c06long long \u7c7b\u578b\u53d6\u522b\u540d\u4e3all)<\/span><br \/>\nll a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1010<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">1010<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">int<\/span> <span class=\"token function\">main<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\nios <span class=\"token double-colon punctuation\">::<\/span> <span class=\"token function\">sync_with_stdio<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><span class=\"token comment\">\/\/ \u63d0\u9ad8cin\u3001cout\u7684\u8fd0\u884c\u901f\u5ea6<\/span><br \/>\ncin<span class=\"token punctuation\">.<\/span><span class=\"token function\">tie<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\ncout<span class=\"token punctuation\">.<\/span><span class=\"token function\">tie<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token comment\">\/\/ \u8f93\u5165\u91cd\u5b9a\u5411<\/span><br \/>\n<span class=\"token comment\">\/\/freopen(&#034;in.txt&#034;, &#034;r&#034;, stdin);<\/span><br \/>\n<span class=\"token comment\">\/\/ \u8f93\u51fa\u91cd\u5b9a\u5411<\/span><br \/>\n<span class=\"token comment\">\/\/freopen(&#034;out.txt&#034;, &#034;w&#034;, stdout);<\/span><\/p>\n<p>ll n<span class=\"token punctuation\">;<\/span><br \/>\ncin <span class=\"token operator\">&gt;&gt;<\/span> n<span class=\"token punctuation\">;<\/span><\/p>\n<p>a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">for<\/span><span class=\"token punctuation\">(<\/span>ll j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">1000<\/span><span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">if<\/span><span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">%<\/span> <span class=\"token number\">2<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">else<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> j <span class=\"token operator\">\/<\/span> <span class=\"token number\">2<\/span> <span class=\"token operator\">*<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token keyword\">for<\/span><span class=\"token punctuation\">(<\/span>ll j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">1000<\/span><span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\nll x <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> y <span class=\"token operator\">&#061;<\/span> j<span class=\"token punctuation\">,<\/span> cnt <span class=\"token operator\">&#061;<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">if<\/span><span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">%<\/span> <span class=\"token number\">2<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">while<\/span><span class=\"token punctuation\">(<\/span>y <span class=\"token operator\">&gt;&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\nx<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\ny<span class=\"token operator\">&#8212;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token operator\">&#043;&#043;<\/span>cnt<span class=\"token punctuation\">;<\/span><br \/>\na<span class=\"token punctuation\">[<\/span>x<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>y<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> cnt<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token keyword\">else<\/span><span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">while<\/span><span class=\"token punctuation\">(<\/span>y <span class=\"token operator\">&gt;&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\nx<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\ny<span class=\"token operator\">&#8212;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token operator\">&#8212;<\/span>cnt<span class=\"token punctuation\">;<\/span><br \/>\na<span class=\"token punctuation\">[<\/span>x<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>y<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> cnt<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">for<\/span><span class=\"token punctuation\">(<\/span>ll i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">1000<\/span><span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">for<\/span><span class=\"token punctuation\">(<\/span>ll j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">1000<\/span><span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token comment\">\/\/cout &lt;&lt; setw(2) &lt;&lt; a[i][j] &lt;&lt; &#034; &#034;; <\/span><br \/>\n<span class=\"token keyword\">if<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">{<\/span><br \/>\ncout <span class=\"token operator\">&lt;&lt;<\/span> i <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;\/&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> j <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token 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