{"trustable":true,"prependHtml":"\u003cstyle type\u003d\"text/css\"\u003e\n div.illustration {\n float: right;\n padding-left: 20px;\n }\n div.illustration .illustration {\n width: 100%;\n border-radius: 4px;\n }\n pre {\n display: block;\n margin: 0 0 10px;\n font-size: 13px;\n line-height: 1.42857143;\n color: #333;\n word-break: break-all;\n word-wrap: break-word;\n }\n\u003c/style\u003e\n\u003cscript\u003e\n window.katexOptions \u003d {\n delimiters: [\n {left: \u0027\\\\[\u0027, right: \u0027\\\\]\u0027, display: true}\n ]\n };\n\u003c/script\u003e\n","sections":[{"title":"","value":{"format":"HTML","content":"\n \u003cp\u003eAlice received a cake for her birthday! Her cake can be\n described by a convex polygon with \u003cspan class\u003d\"tex2jax_process\"\u003e$n$\u003c/span\u003e vertices. No three vertices are\n collinear.\u003c/p\u003e\n \u003cp\u003eAlice will now choose exactly \u003cspan class\u003d\"tex2jax_process\"\u003e$k$\u003c/span\u003e random vertices (\u003cspan class\u003d\"tex2jax_process\"\u003e$k{\\ge }3$\u003c/span\u003e) from her cake and cut a\n piece, the shape of which is the convex polygon defined by\n those vertices. Compute the expected area of this piece of\n cake.\u003c/p\u003e\n \u003ch2\u003eInput\u003c/h2\u003e\n \u003cp\u003eEach test case will begin with a line with two\n space-separated integers \u003cspan class\u003d\"tex2jax_process\"\u003e$n$\u003c/span\u003e and \u003cspan class\u003d\"tex2jax_process\"\u003e$k$\u003c/span\u003e (\u003cspan class\u003d\"tex2jax_process\"\u003e$3\\! \\le \\! k\\! \\le \\! n\\! \\le \\! 2\\,\n 500$\u003c/span\u003e), where \u003cspan class\u003d\"tex2jax_process\"\u003e$n$\u003c/span\u003e is\n the number of vertices of the cake, and \u003cspan class\u003d\"tex2jax_process\"\u003e$k$\u003c/span\u003e is the number of vertices of the\n piece that Alice cuts.\u003c/p\u003e\n \u003cp\u003eEach of the next \u003cspan class\u003d\"tex2jax_process\"\u003e$n$\u003c/span\u003e\n lines will contain two space-separated real numbers\n \u003cspan class\u003d\"tex2jax_process\"\u003e$x$\u003c/span\u003e and \u003cspan class\u003d\"tex2jax_process\"\u003e$y$\u003c/span\u003e (\u003cspan class\u003d\"tex2jax_process\"\u003e$-10.0{\\le }x,y{\\le }10.0$\u003c/span\u003e), where\n \u003cspan class\u003d\"tex2jax_process\"\u003e$(x,y)$\u003c/span\u003e is a vertex of the\n cake. The vertices will be listed in clockwise order. No three\n vertices will be collinear. All real numbers have at most\n \u003cspan class\u003d\"tex2jax_process\"\u003e$6$\u003c/span\u003e digits after the\n decimal point.\u003c/p\u003e\n \u003ch2\u003eOutput\u003c/h2\u003e\n \u003cp\u003eOutput a single real number, which is the expected area of\n the piece of cake that Alice cuts out. Your answer will be\n accepted if it is within an absolute error of \u003cspan class\u003d\"tex2jax_process\"\u003e$10^{-6}$\u003c/span\u003e.\u003c/p\u003e\n \u003ch2\u003eSample 1\u003c/h2\u003e\u003cbody\u003e\u003ctable class\u003d\"vjudge_sample\"\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\u003e4 3\n0 0\n1 1\n2 1\n1 0\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e0.50000000\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/body\u003e\n \u003ch2\u003eSample 2\u003c/h2\u003e\u003cbody\u003e\u003ctable class\u003d\"vjudge_sample\"\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\u003e5 5\n0 4\n4 2\n4 1\n3 -1\n-2 4\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e12.50000000\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/body\u003e\n \u003ch2\u003eSample 3\u003c/h2\u003e\u003cbody\u003e\u003ctable class\u003d\"vjudge_sample\"\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\u003e5 3\n-1.20 2.80\n3.30 2.40\n3.10 -0.80\n2.00 -4.60\n-4.40 -0.50\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e12.43300000\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/body\u003e\n "}}]}