{"trustable":true,"prependHtml":"\u003cstyle type\u003d\u0027text/css\u0027\u003e\n .input, .output {\n border: 1px solid #888888;\n }\n .output {\n margin-bottom: 1em;\n position: relative;\n top: -1px;\n }\n .output pre, .input pre {\n background-color: #EFEFEF;\n line-height: 1.25em;\n margin: 0;\n padding: 0.25em;\n }\n \u003c/style\u003e\n \u003clink rel\u003d\"stylesheet\" href\u003d\"//codeforces.org/s/96598/css/problem-statement.css\" type\u003d\"text/css\" /\u003e\u003cscript\u003e window.katexOptions \u003d { disable: true }; \u003c/script\u003e\n\u003cscript type\u003d\"text/x-mathjax-config\"\u003e\n MathJax.Hub.Config({\n tex2jax: {\n inlineMath: [[\u0027$$$\u0027,\u0027$$$\u0027], [\u0027$\u0027,\u0027$\u0027]],\n displayMath: [[\u0027$$$$$$\u0027,\u0027$$$$$$\u0027], [\u0027$$\u0027,\u0027$$\u0027]]\n }\n });\n\u003c/script\u003e\n\u003cscript type\u003d\"text/javascript\" async src\u003d\"https://mathjax.codeforces.org/MathJax.js?config\u003dTeX-AMS_HTML-full\"\u003e\u003c/script\u003e","sections":[{"title":"","value":{"format":"HTML","content":"\u003cp\u003eA chip was placed on a field with coordinate system onto point \u003cspan class\u003d\"tex-span\"\u003e(0, 0)\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eEvery second the chip moves randomly. If the chip is currently at a point \u003cspan class\u003d\"tex-span\"\u003e(\u003ci\u003ex\u003c/i\u003e, \u003ci\u003ey\u003c/i\u003e)\u003c/span\u003e, after a second it moves to the point \u003cspan class\u003d\"tex-span\"\u003e(\u003ci\u003ex\u003c/i\u003e - 1, \u003ci\u003ey\u003c/i\u003e)\u003c/span\u003e with probability \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e1\u003c/sub\u003e\u003c/span\u003e, to the point \u003cspan class\u003d\"tex-span\"\u003e(\u003ci\u003ex\u003c/i\u003e, \u003ci\u003ey\u003c/i\u003e - 1)\u003c/span\u003e with probability \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e2\u003c/sub\u003e\u003c/span\u003e, to the point \u003cspan class\u003d\"tex-span\"\u003e(\u003ci\u003ex\u003c/i\u003e + 1, \u003ci\u003ey\u003c/i\u003e)\u003c/span\u003e with probability \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e3\u003c/sub\u003e\u003c/span\u003e and to the point \u003cspan class\u003d\"tex-span\"\u003e(\u003ci\u003ex\u003c/i\u003e, \u003ci\u003ey\u003c/i\u003e + 1)\u003c/span\u003e with probability \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e4\u003c/sub\u003e\u003c/span\u003e. It\u0027s guaranteed that \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e1\u003c/sub\u003e + \u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e2\u003c/sub\u003e + \u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e3\u003c/sub\u003e + \u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e4\u003c/sub\u003e \u003d 1\u003c/span\u003e. The moves are independent.\u003c/p\u003e\u003cp\u003eFind out the expected time after which chip will move away from origin at a distance greater than \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eR\u003c/i\u003e\u003c/span\u003e (i.e. \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/5fd61412ee5fa98f0bfc5e5c6ba8759a?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e will be satisfied).\u003c/p\u003e"}},{"title":"Input","value":{"format":"HTML","content":"\u003cp\u003eFirst line contains five integers \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eR\u003c/i\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e1\u003c/sub\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e2\u003c/sub\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e3\u003c/sub\u003e\u003c/span\u003e and \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e4\u003c/sub\u003e\u003c/span\u003e (\u003cspan class\u003d\"tex-span\"\u003e0 ≤ \u003ci\u003eR\u003c/i\u003e ≤ 50, 1 ≤ \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e1\u003c/sub\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e2\u003c/sub\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e3\u003c/sub\u003e, \u003ci\u003ea\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e4\u003c/sub\u003e ≤ 1000\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eProbabilities \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u003c/span\u003e can be calculated using formula \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/82c390d84ac7ffe11dca4a9e31682e97?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e.\u003c/p\u003e"}},{"title":"Output","value":{"format":"HTML","content":"\u003cp\u003eIt can be shown that answer for this problem is always a rational number of form \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/463fe0c9c2704a9ad99ef6e35bbee4d4?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e, where \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/fac3942fa80460efb091c0657883fa97?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e.\u003c/p\u003e\u003cp\u003ePrint \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eP\u003c/i\u003e·\u003ci\u003eQ\u003c/i\u003e\u003csup class\u003d\"upper-index\"\u003e - 1\u003c/sup\u003e\u003c/span\u003e modulo \u003cspan class\u003d\"tex-span\"\u003e10\u003csup class\u003d\"upper-index\"\u003e9\u003c/sup\u003e + 7\u003c/span\u003e. \u003c/p\u003e"}},{"title":"Examples","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e0 1 1 1 1\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e1\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e1 1 1 1 1\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e666666674\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e1 1 2 1 2\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e538461545\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}},{"title":"Note","value":{"format":"HTML","content":"\u003cp\u003eIn the first example initially the chip is located at a distance \u003cspan class\u003d\"tex-span\"\u003e0\u003c/span\u003e from origin. In one second chip will move to distance \u003cspan class\u003d\"tex-span\"\u003e1\u003c/span\u003e is some direction, so distance to origin will become \u003cspan class\u003d\"tex-span\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eAnswers to the second and the third tests: \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/38f112fdeeee887cca50994f1ad0201b?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e and \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/e660b6ba15031731f4e77e2916035b02?v\u003d1714408326\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e.\u003c/p\u003e"}}]}