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문제

정수좌표를 갖는 점을 격자점이라고 한다. 격자 다각형은 모든 꼭짓점이 격자점으로 이루어진 다각형이다.

만약, 다각형의 두 꼭짓점을 잇는 모든 선분이 다각형 내부(또는 경계)에 있다면, 이 다각형을 볼록 다각형이라고 한다. 즉, 다각형의 내부각이 모두 180도 보다 작은 것이다.

격자점으로 이루어진 집합 S가 주어졌을 때, S의 모든 격자점을 포함하는 가장 작은 볼록 (격자) 다각형을 컨벡스 헐이라고 한다. 컨벡스 헐의 꼭짓점은 모두 S에 포함된 격자점이어야 한다. 만약, 모든 점이 같은 일직선 상에 있다면, 컨벡스 헐은 선분이 될 것이다. (오른쪽 그림)

아래 그림에서 집합에 포함된 점은 굵은 점으로, 컨벡스 헐의 꼭짓점은 X로, 변은 선분으로 나타낸 그림이다.

격자 다각형의 꼭짓점의 일반적인 순서는 다음과 같다.

  1. 첫 번째 꼭짓점은 y좌표가 가장 큰 점이다. 만약, 그러한 점이 2개라면, x가 작은 점이 첫 번째 점이다.
  2. 그 다음 점부터는 시계방향 순서이다.

격자점의 집합이 주어졌을 때, 컨벡스 헐을 일반적인 순서로 출력하는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 P(1 ≤ P ≤ 1000)가 주어진다. 각 테스트 케이스의 첫째 줄에는 집합에 포함된 격자점의 수 N(3 ≤ N ≤ 50)이 주어진다. 나머지 줄은 집합에 포함되어 있는 격자점의 좌표가 한 줄에 5개씩 주어진다. (마지막 줄은 이보다 적을 수 있다) 모든 점은 x와 y좌표가 순서대로 주어지며, 공백으로 구분되어 있다. 좌표는 절댓값이 20보다 작거나 같은 정수이다.

출력

각 테스트 케이스에 대해서, 첫째 줄에는 컨벡스 헐에 포함된 점의 수를 출력한다. 그 다음 줄부터 컨벡스헐에 포함된 격자점을 한 줄에 하나씩 일반적인 순서대로 출한다. x와 y를 공백으로 구분하여 출력한다.

예제 입력 1

4
25
2 1 7 1 1 2 9 2 1 3
10 3 1 4 10 4 1 5 10 5
2 6 10 6 2 7 9 7 3 8
8 8 4 9 7 9 6 2 3 3
5 4 7 5 8 6 4 6 3 7
30
3 9 6 9 3 8 9 8 3 7
12 7 2 6 12 6 2 5 12 5
2 4 12 4 1 3 11 3 1 2
11 2 1 1 11 1 1 0 10 0
4 -1 10 -1 7 -2 10 -2 5 0
7 3 4 5 6 8 3 1 2 6
3
3 1 2 2 1 3
6
1 3 19 1 4 2 2 1 11 2
10 1

예제 출력 1

10
4 9
7 9
10 6
10 3
9 2
7 1
2 1
1 2
1 5
2 7
8
3 9
6 9
12 7
12 4
10 -2
7 -2
1 0
1 3
2
1 3
3 1
4
1 3
11 2
19 1
2 1
[{"problem_id":"2699","problem_lang":"0","title":"\uaca9\uc790\uc810 \ucee8\ubca1\uc2a4\ud5d0","description":"<p>\uc815\uc218\uc88c\ud45c\ub97c \uac16\ub294 \uc810\uc744 \uaca9\uc790\uc810\uc774\ub77c\uace0 \ud55c\ub2e4. \uaca9\uc790 \ub2e4\uac01\ud615\uc740 \ubaa8\ub4e0 \uaf2d\uc9d3\uc810\uc774 \uaca9\uc790\uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub2e4\uac01\ud615\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d, \ub2e4\uac01\ud615\uc758 \ub450 \uaf2d\uc9d3\uc810\uc744 \uc787\ub294 \ubaa8\ub4e0 \uc120\ubd84\uc774 \ub2e4\uac01\ud615 \ub0b4\ubd80(\ub610\ub294 \uacbd\uacc4)\uc5d0 \uc788\ub2e4\uba74, \uc774 \ub2e4\uac01\ud615\uc744 \ubcfc\ub85d \ub2e4\uac01\ud615\uc774\ub77c\uace0 \ud55c\ub2e4. \uc989, \ub2e4\uac01\ud615\uc758 \ub0b4\ubd80\uac01\uc774 \ubaa8\ub450 180\ub3c4 \ubcf4\ub2e4 \uc791\uc740 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\uaca9\uc790\uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569 S\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, S\uc758 \ubaa8\ub4e0 \uaca9\uc790\uc810\uc744 \ud3ec\ud568\ud558\ub294 \uac00\uc7a5 \uc791\uc740 \ubcfc\ub85d (\uaca9\uc790) \ub2e4\uac01\ud615\uc744 \ucee8\ubca1\uc2a4 \ud5d0\uc774\ub77c\uace0 \ud55c\ub2e4. \ucee8\ubca1\uc2a4 \ud5d0\uc758 \uaf2d\uc9d3\uc810\uc740 \ubaa8\ub450 S\uc5d0 \ud3ec\ud568\ub41c \uaca9\uc790\uc810\uc774\uc5b4\uc57c \ud55c\ub2e4. \ub9cc\uc57d, \ubaa8\ub4e0 \uc810\uc774 \uac19\uc740 \uc77c\uc9c1\uc120 \uc0c1\uc5d0 \uc788\ub2e4\uba74, \ucee8\ubca1\uc2a4 \ud5d0\uc740 \uc120\ubd84\uc774 \ub420 \uac83\uc774\ub2e4. (\uc624\ub978\ucabd \uadf8\ub9bc)<\/p>\r\n\r\n<p>\uc544\ub798 \uadf8\ub9bc\uc5d0\uc11c \uc9d1\ud569\uc5d0 \ud3ec\ud568\ub41c \uc810\uc740 \uad75\uc740 \uc810\uc73c\ub85c, \ucee8\ubca1\uc2a4 \ud5d0\uc758 \uaf2d\uc9d3\uc810\uc740 X\ub85c, \ubcc0\uc740 \uc120\ubd84\uc73c\ub85c \ub098\ud0c0\ub0b8 \uadf8\ub9bc\uc774\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/upload\/images\/latice.png\" style=\"height:249px; width:584px\" \/><\/p>\r\n\r\n<p>\uaca9\uc790 \ub2e4\uac01\ud615\uc758 \uaf2d\uc9d3\uc810\uc758 \uc77c\ubc18\uc801\uc778 \uc21c\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\uccab \ubc88\uc9f8 \uaf2d\uc9d3\uc810\uc740 y\uc88c\ud45c\uac00 \uac00\uc7a5 \ud070 \uc810\uc774\ub2e4. \ub9cc\uc57d, \uadf8\ub7ec\ud55c \uc810\uc774 2\uac1c\ub77c\uba74, x\uac00 \uc791\uc740 \uc810\uc774 \uccab \ubc88\uc9f8 \uc810\uc774\ub2e4.<\/li>\r\n\t<li>\uadf8 \ub2e4\uc74c \uc810\ubd80\ud130\ub294 \uc2dc\uacc4\ubc29\ud5a5 \uc21c\uc11c\uc774\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\uaca9\uc790\uc810\uc758 \uc9d1\ud569\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ucee8\ubca1\uc2a4 \ud5d0\uc744 \uc77c\ubc18\uc801\uc778 \uc21c\uc11c\ub85c \ucd9c\ub825\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 P(1 &le; P &le; 1000)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \uc9d1\ud569\uc5d0 \ud3ec\ud568\ub41c \uaca9\uc790\uc810\uc758 \uc218 N(3 &le; N &le; 50)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ub098\uba38\uc9c0 \uc904\uc740 \uc9d1\ud569\uc5d0 \ud3ec\ud568\ub418\uc5b4 \uc788\ub294 \uaca9\uc790\uc810\uc758 \uc88c\ud45c\uac00 \ud55c \uc904\uc5d0 5\uac1c\uc529 \uc8fc\uc5b4\uc9c4\ub2e4. (\ub9c8\uc9c0\ub9c9 \uc904\uc740 \uc774\ubcf4\ub2e4 \uc801\uc744 \uc218 \uc788\ub2e4) \ubaa8\ub4e0 \uc810\uc740 x\uc640 y\uc88c\ud45c\uac00 \uc21c\uc11c\ub300\ub85c \uc8fc\uc5b4\uc9c0\uba70, \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc788\ub2e4. \uc88c\ud45c\ub294 \uc808\ub313\uac12\uc774 20\ubcf4\ub2e4 \uc791\uac70\ub098 \uac19\uc740 \uc815\uc218\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, \uccab\uc9f8 \uc904\uc5d0\ub294 \ucee8\ubca1\uc2a4 \ud5d0\uc5d0 \ud3ec\ud568\ub41c \uc810\uc758 \uc218\ub97c \ucd9c\ub825\ud55c\ub2e4. \uadf8 \ub2e4\uc74c \uc904\ubd80\ud130 \ucee8\ubca1\uc2a4\ud5d0\uc5d0 \ud3ec\ud568\ub41c \uaca9\uc790\uc810\uc744 \ud55c \uc904\uc5d0 \ud558\ub098\uc529 \uc77c\ubc18\uc801\uc778 \uc21c\uc11c\ub300\ub85c \ucd9c\ud55c\ub2e4. x\uc640 y\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"2699","problem_lang":"1","title":"Convex Hull of Lattice Points","description":"<p>A lattice point is a point with integer coordinates. A lattice polygon is a polygon with all vertices lattice points.&nbsp;<\/p>\r\n\r\n<p>A polygon is convex if any line segment between two points of the polygon is inside (or on the boundary of) the polygon. Equivalently, the interior angle at each polygon vertex is less than 180 degrees.&nbsp;<\/p>\r\n\r\n<p>For a set S, of lattice points, the convex hull is the smallest convex (lattice) polygon which contains all points of the set. (The vertices of the convex hull must be members of the set of lattice points). If all the points are on a single straight line, the convex hull will be a line segment (a degeneratepolygon &ndash; see rightmost diagram below). In the diagrams below, the points of the set are indicated by solid dots, the vertices of the convex hull by X&rsquo;s and the convex hull is drawn connecting the vertices.&nbsp;<\/p>\r\n\r\n<p>Note that not all points on the convex hull polygon are vertices.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/latice.png\" style=\"height:249px; width:584px\" \/><\/p>\r\n\r\n<p>The vertices of a lattice polygon are in standard order if:&nbsp;<\/p>\r\n\r\n<ol>\r\n\t<li>The first vertex is the one with the largest y value. If two vertices have the same y value, the one with the smaller x value is the first.&nbsp;<\/li>\r\n\t<li>Vertices are given in clockwise order around the polygon.&nbsp;<\/li>\r\n<\/ol>\r\n\r\n<p>Write a program that reads a set of lattice points and outputs the vertices of the convex hull of the points in standard order.&nbsp;<\/p>\r\n","input":"<p>The first line of input contains a single integer P, (1 &le; P &le; 1000), which is the number of data sets that follow. The first line of each data set contains the data set number, followed by a space, followed by a decimal integer giving the number of points N, (3 &le; N &le; 50), in the set. The remaining lines in the data set contain the points in the set, at most 5 points per line (the last line may have fewer). Each point consists of 2 space separated decimal integer values representing the x and y coordinates respectively.&nbsp;<\/p>\r\n","output":"<p>For each data set there are multiple lines of output. The first line contains a decimal integer giving the data set number followed by a single space, followed by a decimal integer giving the total number of vertices of the convex hull. The vertices of the convex hull follow, one per line in standard order. Each line contains the decimal integer x coordinate, a single space and the decimal integer y coordinate.&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

ACM-ICPC > Regionals > North America > Greater New York Region > 2009 Greater New York Programming Contest G번