{"raw_statement":[{"iden":"statement","content":"The citizens of BubbleLand are celebrating their 10th anniversary so they decided to organize a big music festival. Bob got a task to invite _N_ famous singers who would sing on the fest. He was too busy placing stages for their performances that he totally forgot to write the invitation e-mails on time, and unfortunately he only found _K_ available singers. Now there are more stages than singers, leaving some of the stages empty. Bob would not like if citizens of BubbleLand noticed empty stages and found out that he was irresponsible.\n\nBecause of that he decided to choose exactly _K_ stages that form a convex set, make large posters as edges of that convex set and hold festival inside. While those large posters will make it impossible for citizens to see empty stages outside Bob still needs to make sure they don't see any of the empty stages inside that area.\n\nSince lots of people are coming, he would like that the festival area is as large as possible. Help him calculate the maximum area that he could obtain respecting the conditions. If there is no such area, the festival cannot be organized and the answer is 0.00."},{"iden":"input","content":"The first line of input contains two integers _N_ (3 ≤ _N_ ≤ 200) and _K_ (3 ≤ _K_ ≤ _min_(_N_, 50)), separated with one empty space, representing number of stages and number of singers, respectively.\n\nEach of the next _N_ lines contains two integers _X__i_ and _Y__i_ (0 ≤ _X__i_, _Y__i_ ≤ 106) representing the coordinates of the stages. There are no three or more collinear stages."},{"iden":"output","content":"Output contains only one line with one number, **rounded to exactly two decimal places**: the maximal festival area. Rounding is performed so that 0.5 and more rounds up and everything else rounds down."},{"iden":"example","content":"Input\n\n5 4\n0 0\n3 0\n2 1\n4 4\n1 5\n\nOutput\n\n10.00"},{"iden":"note","content":"Example explanation: From all possible convex polygon with 4 vertices and no other vertex inside, the largest is one with points (0, 0), (2, 1), (4, 4) and (1, 5)."}],"translated_statement":[{"iden":"statement","content":"泡泡国的公民正在庆祝他们的10周年纪念日，因此他们决定举办一场大型音乐节。鲍勃的任务是邀请 #cf_span[N] 位著名歌手来参加演出。但他太忙于为表演布置舞台，以至于完全忘记了及时发送邀请邮件，结果只找到了 #cf_span[K] 位可用的歌手。现在舞台数量多于歌手数量，导致一些舞台空置。鲍勃不希望泡泡国的公民注意到空置的舞台，从而发现他不够负责任。\n\n因此，他决定选择恰好 #cf_span[K] 个舞台，使它们构成一个凸集，用大型海报作为该凸集的边，并在内部举办音乐节。虽然这些大型海报会阻止公民看到外部的空置舞台，但他仍需确保他们看不到该区域内部的任何空置舞台。\n\n由于将有大量观众前来，他希望音乐节的区域尽可能大。请帮助他计算在满足条件的情况下所能获得的最大面积。如果没有这样的区域，音乐节无法举办，答案为 0.00。\n\n输入的第一行包含两个整数 #cf_span[N (3 ≤ N ≤ 200)] 和 #cf_span[K (3 ≤ K ≤ min(N, 50))]，用一个空格分隔，分别表示舞台数量和歌手数量。\n\n接下来的 #cf_span[N] 行，每行包含两个整数 #cf_span[Xi] 和 #cf_span[Yi] #cf_span[(0 ≤ Xi, Yi ≤ 106)]，表示舞台的坐标。不存在三个或更多共线的舞台。\n\n输出仅包含一行，一个数字，*四舍五入到恰好两位小数*：最大的音乐节面积。四舍五入规则为：#cf_span[0.5] 及以上向上取整，其余向下取整。\n\n示例说明：在所有具有 #cf_span[4] 个顶点且内部不含其他顶点的凸多边形中，面积最大的是由点 #cf_span[(0, 0)]、#cf_span[(2, 1)]、#cf_span[(4, 4)] 和 #cf_span[(1, 5)] 构成的多边形。"},{"iden":"input","content":"输入的第一行包含两个整数 #cf_span[N (3 ≤ N ≤ 200)] 和 #cf_span[K (3 ≤ K ≤ min(N, 50))]，用一个空格分隔，分别表示舞台数量和歌手数量。接下来的 #cf_span[N] 行，每行包含两个整数 #cf_span[Xi] 和 #cf_span[Yi] #cf_span[(0 ≤ Xi, Yi ≤ 106)]，表示舞台的坐标。不存在三个或更多共线的舞台。"},{"iden":"output","content":"输出仅包含一行，一个数字，*四舍五入到恰好两位小数*：最大的音乐节面积。四舍五入规则为：#cf_span[0.5] 及以上向上取整，其余向下取整。"},{"iden":"note","content":"示例说明：在所有具有 #cf_span[4] 个顶点且内部不含其他顶点的凸多边形中，面积最大的是由点 #cf_span[(0, 0)]、#cf_span[(2, 1)]、#cf_span[(4, 4)] 和 #cf_span[(1, 5)] 构成的多边形。"}],"sample_group":[],"show_order":[],"formal_statement":"**Definitions**  \nLet $ N, K \\in \\mathbb{Z} $ with $ 3 \\leq N \\leq 200 $, $ 3 \\leq K \\leq \\min(N, 50) $.  \nLet $ P = \\{p_1, p_2, \\dots, p_N\\} \\subset \\mathbb{R}^2 $ be a set of $ N $ distinct points in the plane, no three collinear, where $ p_i = (x_i, y_i) $.\n\n**Constraints**  \n1. All points in $ P $ have integer coordinates: $ 0 \\leq x_i, y_i \\leq 10^6 $.  \n2. No three points in $ P $ are collinear.  \n\n**Objective**  \nFind the maximum area of a convex $ K $-gon whose vertices are a subset of $ P $, and which contains no other points of $ P $ in its interior.  \nIf no such $ K $-gon exists, output $ 0.00 $.\n\n$$\n\\max_{\\substack{S \\subseteq P \\\\ |S| = K \\\\ \\text{conv}(S) \\text{ is convex} \\\\ \\text{int}(\\text{conv}(S)) \\cap P = \\emptyset}} \\text{Area}(\\text{conv}(S))\n$$","simple_statement":null,"has_page_source":false}