{"id":841,"date":"2019-01-30T13:27:27","date_gmt":"2019-01-30T05:27:27","guid":{"rendered":"https:\/\/0.mnihyc.com\/blog\/?p=841"},"modified":"2020-02-09T03:08:04","modified_gmt":"2020-02-08T19:08:04","slug":"yzoj-p2697-%e7%94%bb%e5%9c%86","status":"publish","type":"post","link":"https:\/\/0.mnihyc.com\/blog\/archives\/841","title":{"rendered":"YZOJ P2697 \u753b\u5706"},"content":{"rendered":"<h1 style=\"text-align: center;\">YZOJ P2697 \u753b\u5706<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\u9650\u5236\uff1a1000MS \u00a0\u00a0\u00a0\u00a0 \u5185\u5b58\u9650\u5236\uff1a131072KB<\/p>\n<p style=\"text-align: center;\">\u96be\u5ea6\uff1a <span style=\"color: #ffcc00;\">\\(5.1\\)<\/span><\/p>\n<ul>\n<li>\n<h3><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5728\u521d\u4e2d\u6570\u5b66\u8bfe\u4e0a\uff0c\\(Alkri\\) \u5b66\u4e60\u4e86\u5706\u7684\u76f8\u5173\u77e5\u8bc6\uff0c\u4ed6\u5bf9\u4e0e\u5706\u6709\u5173\u7684\u95ee\u9898\u66f4\u52a0\u611f\u5174\u8da3\u4e86\u3002<\/p>\n<p>\\(Alkri\\) \u60f3\u5728\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfb\u7684<strong>\u7b2c\u4e00\u8c61\u9650<\/strong>\u4e2d\u4f9d\u6b21\u753b \\(n\\) \u4e2a\u4e0e<strong>\u4e24\u5750\u6807\u8f74\u5747\u76f8\u5207<\/strong>\u7684\u5706\uff0c\u5176\u4e2d\uff0c\u7b2c \\(1\\) \u4e2a\u5706\u7684\u534a\u5f84\u4e3a \\(r\\)\uff0c\u4e4b\u540e\u7684\u6bcf\u4e2a\u5706\u90fd\u6bd4\u4e0a\u4e00\u4e2a\u5706\u5927\uff0c\u4e14<strong>\u4e0e\u4e0a\u4e00\u4e2a\u5706\u76f8\u5207<\/strong>\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u6240\u6709\u6574\u6570 \\(2 \\leq i \\leq n\\)\uff0c\u7b2c \\(i\\) \u4e2a\u5706\u7684\u534a\u5f84\u5927\u4e8e\u7b2c \\(i-1\\) \u4e2a\u5706\u7684\u534a\u5f84\u4e14\u4e0e\u7b2c \\(i-1\\)\u4e2a\u5706\u76f8\u5207\u3002<\/p>\n<p>\u4f8b\u5982\u5f53 \\(n=3\\) \u65f6\uff0c\u4e09\u4e2a\u5706 \\(C_1, C_2, C_3\\) \u5982\u4e0b\u56fe\u6240\u793a\uff08\u7531\u4e8e \\(C_3\\) \u6bd4\u8f83\u5927\uff0c\u672a\u753b\u5b8c\u6574\uff09\uff1a<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-842\" src=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/circle.png\" alt=\"\" width=\"354\" height=\"339\" srcset=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/circle.png 354w, https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/circle-300x287.png 300w\" sizes=\"auto, (max-width: 354px) 100vw, 354px\" \/><\/p>\n<p>\u73b0\u5728\uff0c\\(Alkri\\) \u5f88\u597d\u5947\uff1a\u7b2c \\(n\\) \u4e2a\u5706\u7684\u534a\u5f84 \\(R\\) \u5230\u5e95\u6709\u591a\u5927\uff1f\u4ed6\u77e5\u9053 \\(R\\) \u4e0d\u4e00\u5b9a\u662f\u6574\u6570\uff08\u771f\u806a\u660e\uff01\uff09\uff0c\u5e76\u4e14\u53ef\u80fd\u975e\u5e38\u5927\uff0c\u6240\u4ee5\u53ea\u9700\u8981\u4f60\u4fdd\u7559 \\(R\\) \u7684\u6574\u6570\u90e8\u5206\uff08\u5411\u4e0b\u53d6\u6574\uff09\u7684\u672b\u5c3e \\(p\\) \u4f4d\u6570\u5b57\u5373\u53ef\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u8f93\u5165\u4ec5\u4e00\u884c\uff0c\u5305\u542b\u4e09\u4e2a\u6574\u6570 \\(n\\)\uff0c\\(r\\)\uff0c\\(p\\)\uff0c\u610f\u4e49\u5982\u9898\u76ee\u6240\u8ff0\u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u683c\u5f0f<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u8f93\u51fa\u4ec5\u4e00\u884c\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u7b2c \\(n\\) \u4e2a\u5706\u7684\u534a\u5f84 \\(R\\) \u7684\u6574\u6570\u90e8\u5206\u7684\u672b\u5c3e \\(p\\) \u4f4d\u3002\u6ce8\u610f\u5f53 \\(R\\) \u7684\u6574\u6570\u90e8\u5206\u5b9e\u9645\u4f4d\u6570\u8d85\u8fc7 \\(p\\) \u65f6\u9700\u8981\u8f93\u6ee1 \\(p\\) \u4f4d\uff08\u5373\u9700\u8981\u4fdd\u7559\u524d\u5bfc0\uff09\uff0c\u5982\u679c\u5b9e\u9645\u4f4d\u6570\u4e0d\u6ee1 \\(p\\) \u4f4d\u5219\u4e0d\u7528\u8865\u524d\u5bfc 0 \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u8f93\u5165\u6837\u4f8b<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">10 5 5<\/pre>\n<ul>\n<li>\n<h3><strong>\u8f93\u51fa\u6837\u4f8b<\/strong><\/h3>\n<\/li>\n<\/ul>\n<pre class=\"\">08989<\/pre>\n<ul>\n<li>\n<h3><strong>\u6837\u4f8b\u8bf4\u660e<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7b2c \\(10\\) \u4e2a\u5706\u7684\u534a\u5f84\u6574\u6570\u90e8\u5206\u4e3a \\(38808989\\)\uff0c\u8981\u6c42\u8f93\u51fa\u6574\u6570\u90e8\u5206\u7684\u672b\u5c3e \\(5\\) \u4f4d\u6570\uff0c\u56e0\u6b64\u8f93\u51fa \\(08989\\) \u3002\u6ce8\u610f\u4fdd\u7559\u524d\u5bfc 0 \u3002<\/p>\n<ul>\n<li>\n<h3><strong>\u6570\u636e\u89c4\u6a21\u4e0e\u7ea6\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-843\" src=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/constant1.png\" alt=\"\" width=\"571\" height=\"453\" srcset=\"https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/constant1.png 571w, https:\/\/0.mnihyc.com\/blog\/wp-content\/uploads\/2019\/01\/constant1-300x238.png 300w\" sizes=\"auto, (max-width: 571px) 100vw, 571px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><!--more--><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>\u901a\u8fc7\u7b80\u5355\u7684\u6570\u5b66\u8ba1\u7b97\u53ef\u5f97\u4e0b\u4e00\u4e2a\u5706\u7684\u534a\u5f84\u662f\u524d\u4e00\u4e2a\u5706\u7684\u534a\u5f84\u7684 \\(3+2\\sqrt2\\) \u500d\u3002<del>\uff08\u4e0d\u8981\u95ee\u4e3a\u4ec0\u4e48\uff0c\u9898\u89e3\u8bf4\u7684<\/del><\/p>\n<p>\u90a3\u4e48\u7b54\u6848\u5c31\u662f \\((3+2\\sqrt2)^{n-1}r\\) \uff0c\u9996\u5148 \\(50pts\\) \u5c31\u5230\u624b\u4e86\u3002\u56e0\u4e3a \\(n\\) \u5f88\u5927\u7684\u65f6\u5019\u80af\u5b9a\u4f1a\u7206\u7cbe\u5ea6\uff0c\u6240\u4ee5\u65e0\u6cd5\u5f00\u4ec0\u4e48 <span class=\"lang:default decode:true  crayon-inline\">long double<\/span>\u00a0 \u66b4\u7b97\u3002<\/p>\n<p>\u9ad8\u7cbe\u5ea6\u662f\u80af\u5b9a\u8981\u5199\u7684\uff0c\u90a3\u4e48\u600e\u4e48\u5199\u5c31\u662f\u4e00\u4e2a\u6bd4\u8f83\u5173\u952e\u7684\u95ee\u9898\u3002<\/p>\n<p>\u53ef\u4ee5\u65b0\u5b9a\u4e49\u4e00\u4e2a\u7ed3\u6784\u4f53\u8868\u793a \\(a+b\\sqrt2\\) \uff0c\u5176\u4e2d \\(a, b\\) \u662f\u4e24\u4e2a\u9ad8\u7cbe\u5ea6\u6574\u6570\u3002\u8fd9\u6837\u5c31\u53ef\u4ee5\u7528\u6574\u6570\u8868\u793a\u51fa\u5e26\u6839\u53f7\u7684\u6570\u4e86\uff01\u8fd9\u6837\uff0c\\((a+b\\sqrt2)(c+d\\sqrt2)=(ac+2bd)+(ad+bc)\\sqrt2\\) \uff0c\u5c31\u662f\u4e24\u4e2a\u7ed3\u6784\u4f53\u76f8\u4e58\u7684\u65b9\u6cd5\u4e86\u3002<\/p>\n<p>\u6240\u4ee5\u7b54\u6848\u5c31\u662f \\((3+2\\sqrt2)^{n-1}r=Ar+B\\sqrt2r=\\cdots\\cdots\\) \u3002\u3002\u3002\u3002\u3002\u3002\u597d\u50cf\u6ca1\u4ec0\u4e48\u7528\uff1f\u597d\u50cf\u8fd8\u662f\u9700\u8981 \\(\\sqrt2\\) \u7684\u8fd1\u4f3c\u503c\u624d\u80fd\u7b97\uff1f\uff1f\uff1f<\/p>\n<p><del>\u8fd9\u4e2a\u65f6\u5019\u5c31\u8981\u60f3\u8d77\u5965\u6570\u73edsxb\u8001\u5e08\u7684\u6559\u8bf2\uff08\u597d\u50cf\u5f53\u65f6\u6570\u5b57\u4e5f\u662f\u8fd9\u4e2a\u6765\u7740\u7684\uff09<\/del><\/p>\n<p>\u53d1\u73b0\u8fd9\u4e2a \\(\\sqrt2\\) \u662f\u771f\u7684\u5f88\u8ba8\u538c\uff0c\u5c31\u5fc5\u987b\u60f3\u529e\u6cd5\u628a\u5b83\u5f04\u6389\u3002<\/p>\n<p>\u6ce8\u610f\u5230\u4e8c\u9879\u5f0f\u5b9a\u7406\u3002<\/p>\n<p>\\((a+b)^{n}=\\sum\\limits_{i=0}^{n} C_n^i a^{n-i} b^i\\)<\/p>\n<p>\\((a-b)^{n}=\\sum\\limits_{i=0}^{n} \\left([i\\;\\bmod\\; 2\\; =\\; 0](C_n^i a^{n-i}b^i)+[i\\; \\bmod\\; 2 \\;=\\; 1](-C_n^i a^{n-i}b^i)\\right)\\)<\/p>\n<p>\u5199\u6210\u8fd9\u6837\u5176\u5b9e\u5c31\u975e\u5e38\u6e05\u695a\u4e86\uff0c\u6240\u4ee5\u628a\u5b83\u4eec\u76f8\u52a0\u7684\u8bdd<\/p>\n<p>\\((a+b)^n+(a-b)^n = 2\\sum_{i=0}^{n} ([i\\;\\bmod\\; 2\\; =\\; 0])(C_n^i a^{n-i}b^i)\\)<\/p>\n<p>\u5728\u8fd9\u91cc\uff0c\u82e5\u4f7f \\((3+2\\sqrt2)^{n-1}=A+B\\sqrt2\\) \uff0c\u90a3\u4e48 \\((3+2\\sqrt2)^{n-1}+(3-2\\sqrt2)^{n-1}=2A\\) \u3002<\/p>\n<p>\u6240\u4ee5\u7b54\u6848\u5c31\u662f \\((3+2\\sqrt2)^{n-1}r=2Ar-(3-2\\sqrt2)^{n-1}r=\\cdots\\cdots\\) \u3002\u3002\u3002\u3002\u3002\u3002\u597d\u50cf\u6ca1\u4ec0\u4e48\u7528\uff1f\u4e0d\uff01\uff01\u6709\u975e\u5e38\u5927\u7684\u7528\u5904\uff01\uff01\uff01\uff01\uff01\uff01<\/p>\n<p>\u56e0\u4e3a \\(3-2\\sqrt2 \\approx 0.17 &lt; 1\\) \uff0c\u6240\u4ee5\u53ef\u4ee5\u8bd5\u7740\u8ba1\u7b97\u4e00\u4e0b \\((3-2\\sqrt2)^{14} \\approx 2 \\times 10^{-11} &lt; 10^{-10} &lt; \\frac{1}{r}\\) \uff0c\u4e5f\u5c31\u610f\u5473\u7740 \\(n\\) \u8db3\u591f\u5927\uff08\u5176\u5b9e\u4e5f\u5c31 \\(15\\)\uff09\u65f6\uff0c\\((3-2\\sqrt2)^{n-1}r &lt; 1\\) \uff0c\u6240\u4ee5\u7b54\u6848\u5c31\u662f \\(2Ar-1\\) \u3002<\/p>\n<p>\u8fd9\u6837 \\(n\\) \u5c0f\u7684\u65f6\u5019 <span class=\"lang:default decode:true  crayon-inline \">long double<\/span>\u00a0 \u66b4\u7b97\u4e00\u4e0b\uff0c\\(n\\) \u5927\u7684\u65f6\u5019\u9ad8\u7cbe\u7b97\u4e00\u4e0b\u5c31\u53ef\u4ee5\u901a\u8fc7\u672c\u9898\u4e86\uff01\uff01\uff01<\/p>\n<p>\uff08\u8fd8\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4fdd\u7559\u5c0f\u6570\u4f4d\u6570\u90a3\u91cc\u524d\u5bfc 0 \u7684\u5224\u65ad\u8981\u5341\u5206\u8c28\u614e\uff0c\u6211\u8c03\u4e862d\u624d\u8c03\u51fa\u6765w<\/p>\n<p>&nbsp;<\/p>\n<pre class=\"lang:default decode:true \">#include &lt;cstdio&gt;\r\n#include &lt;cstdlib&gt;\r\n#include &lt;cstring&gt;\r\n#include &lt;climits&gt;\r\n#include &lt;cmath&gt;\r\n\r\nint P;\r\n#define MAXDIG (P+(SETDIG&lt;&lt;1))\/SETDIG\r\n#define SETDIG 3\r\n#define SETDIGN 1000\r\n\r\nstruct _bnum\r\n{\r\n\tint num[1550],len;\r\n\tinline _bnum CutIDig()\r\n\t{\r\n\t\t\/\/this-&gt;PrintNum();\r\n\t\tif(len&gt;MAXDIG)\r\n\t\t{\r\n\t\t\tfor(register int i=MAXDIG;i&lt;len;i++)\r\n\t\t\t\tnum[i]=0;\r\n\t\t\tlen=MAXDIG;\r\n\t\t}\r\n\t\t\/\/this-&gt;PrintNum();\r\n\t\t\/\/printf(\"%d %d\\n\\n\",len*SETDIG,(int)log10(num[0]));\r\n\t\treturn *this;\r\n\t}\r\n\t_bnum(){memset(num,0,sizeof(num));len=0;}\r\n\t_bnum(const char str[])\r\n\t{\r\n\t\tmemset(num,0,sizeof(num));len=0;\r\n\t\tregister int ls=strlen(str)-1;\r\n\t\tfor(;ls-SETDIG&gt;=0;ls-=SETDIG,len++)\r\n\t\t\tfor(register int j=ls-SETDIG+1;j&lt;=ls;j++)\r\n\t\t\t\t(num[len]*=10)+=(str[j]-'0');\r\n\t\tfor(register int j=0;j&lt;=ls;j++)\r\n\t\t\t(num[len]*=10)+=(str[j]-'0');\r\n\t\tlen++;\r\n\t}\r\n\t_bnum(const int&amp;n)\r\n\t{\r\n\t\tmemset(num,0,sizeof(num));len=1;\r\n\t\tnum[0]=n;\r\n\t\twhile(num[len-1]&gt;SETDIGN-1)\r\n\t\t\tnum[len]=num[len-1]\/SETDIGN,num[len-1]%=SETDIGN,len++;\r\n\t}\r\n\t_bnum operator = (const _bnum&amp;ot)\r\n\t{\r\n\t\tlen=ot.len;\r\n\t\tmemcpy(&amp;num,&amp;ot.num,sizeof(num));\r\n\t\treturn *this;\r\n\t}\r\n\t_bnum operator += (const _bnum&amp;ot)\r\n\t{\r\n\t\tregister int flen=len;\r\n\t\tif(ot.len&gt;flen)\r\n\t\t\tflen=ot.len;\r\n\t\tfor(register int i=0;i&lt;flen;i++)\r\n\t\t{\r\n\t\t\tnum[i]+=ot.num[i];\r\n\t\t\tif(num[i]&gt;SETDIGN-1)\r\n\t\t\t\tnum[i+1]++,num[i]-=SETDIGN;\r\n\t\t}\r\n\t\tif(num[flen])\r\n\t\t\tflen++;\r\n\t\tlen=flen;\r\n\t\tthis-&gt;CutIDig();\r\n\t\treturn *this;\r\n\t}\r\n\t_bnum operator -= (const int&amp;n)\r\n\t{\r\n\t\tnum[0]-=n;\r\n\t\tfor(register int i=0;i&lt;len;i++)\r\n\t\t\tif(num[i]&lt;0)\r\n\t\t\t\tnum[i+1]-=(1-num[i]\/SETDIGN),num[i]=(SETDIGN+num[i]%SETDIGN);\r\n\t\t\telse\r\n\t\t\t\tbreak;\r\n\t\twhile(num[len-1]==0 &amp;&amp; len&gt;0)\r\n\t\t\tlen--;\r\n\t\tif(len==0)\r\n\t\t\tlen=1;\r\n\t\treturn *this;\r\n\t}\r\n\t_bnum operator * (const _bnum&amp;ot)const\r\n\t{\r\n\t\t_bnum ans;\r\n\t\tregister int flen=this-&gt;len+ot.len;\r\n\t\tfor(register int i=0;i&lt;this-&gt;len;i++)\r\n\t\t\tfor(register int j=0;j&lt;ot.len;j++)\r\n\t\t\t\tans.num[i+j]+=this-&gt;num[i]*ot.num[j];\r\n\t\tfor(register int i=0;i&lt;flen;i++)\r\n\t\t\tans.num[i+1]+=ans.num[i]\/SETDIGN,ans.num[i]%=SETDIGN;\r\n\t\twhile(ans.num[flen])\r\n\t\t\tans.num[flen+1]=ans.num[flen+1]\/SETDIGN,ans.num[flen++]%=SETDIGN;\r\n\t\twhile(ans.num[flen-1]==0)\r\n\t\t\tflen--;\r\n\t\tans.len=flen;\r\n\t\tans.CutIDig();\r\n\t\treturn ans;\r\n\t}\r\n\tinline void PrintNum()\r\n\t{\r\n\t\tstatic char outbuf[2048];\r\n\t\toutbuf[0]=0;\r\n\t\tstatic char ots[8];\r\n\t\tsprintf(ots,\"%%0%dd\",SETDIG);\r\n\t\tfor(register int i=len-1;i&gt;=0;i--)\r\n\t\t\tsprintf(outbuf+strlen(outbuf),ots,num[i]);\r\n\t\t\r\n\t\tregister int ls=strlen(outbuf),si=ls-P;\r\n\t\tif(si&lt;=0)\r\n\t\t\tfor(si=0;outbuf[si]=='0';si++);\r\n\t\tfor(register int i=si;i&lt;ls;i++)\r\n\t\t\tprintf(\"%c\",outbuf[i]);\r\n\t\tprintf(\"\\n\");\r\n\t}\r\n};\r\n\r\n\r\nstruct _cs2\r\n{\r\n\t_bnum a,b;\r\n\t_cs2(){a=b=0;}\r\n\t_cs2 operator *= (const _cs2&amp;ot)\r\n\t{\r\n\t\t_cs2 ans;\r\n\t\tans.a=this-&gt;a*ot.a,ans.a+=(_bnum)2*this-&gt;b*ot.b;\r\n\t\tans.b=this-&gt;a*ot.b,ans.b+=this-&gt;b*ot.a;\r\n\t\treturn (*this=ans);\r\n\t}\r\n};\r\n\r\ninline _cs2 _pow(_cs2 M,int n)\r\n{\r\n\t_cs2 ans;ans.a=1,ans.b=0;\r\n\twhile(n)\r\n\t{\r\n\t\tif(n&amp;1)\r\n\t\t\tans*=M;\r\n\t\tM*=M;\r\n\t\tn&gt;&gt;=1;\r\n\t}\r\n\treturn ans;\r\n}\r\n\r\n\r\nint main()\r\n{\r\n\tint N,R;scanf(\"%d%d%d\",&amp;N,&amp;R,&amp;P);\r\n\tif(N&lt;=15)\r\n\t{\r\n\t\tregister long long ans=pow(3+2*sqrt(2),N-1)*R,pw10=pow(10,P);\r\n\t\tstatic char ots[8];\r\n\t\tif(ans&gt;=pw10)\r\n\t\t\tsprintf(ots,\"%%0%dlld\",P);\r\n\t\telse\r\n\t\t\tsprintf(ots,\"%%lld\");\r\n\t\tprintf(ots,ans%pw10);\r\n\t\treturn 0;\r\n\t}\r\n\t_cs2 ans;ans.a=3,ans.b=2;\r\n\t(((_pow(ans,N-1).a*(_bnum)2)*(_bnum)R)-=1).PrintNum();\r\n\t\r\n\treturn 0;\r\n}<\/pre>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>YZOJ P2697 \u753b\u5706 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