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문제

원자력 발전소는 현대 문명의 축복이자 저주이다. 원자력 발전소는 많은 위험성이 있지만, 전기를 만드는 가장 싼 방법이다. 이 문제에서는 거리가 멀지 않은 두 원자력 발전소에 어떤 상황에 대해서 생각해 본다.

그림 1. 두 원자력 발전소. (81, 49)와 (77, 33)에 있는 집은 매우 위험한 지역이다.

먼저, 땅이 모두 평평하고, 2차원 좌표 평면 위에 모든 집이 있다고 가정한다. 두 원자력 발전소가 각각 (ax, ay)와 (bx, by)에 있다고 가정한다. 발전소 (ax, ay)와 거리가 R1이내인 곳(R1인 곳도 포함)은 방사능 고위험 지역이다. 발전소 (bx, by)와 거리가 R2이내인 곳(R2인 곳도 포함)도 방사능 고위험 지역이다. 따라서, 원자력 발전소 1과 2의 관계자는 특별한 보호 장비를 방사능 고위험 지역에 있는 집에게 나누어 준다. 따라서, 두 원자력 발전소의 위험 지역에 모두 포함되는 집은 보호 장비를 두 개 받게 된다. 하지만, 이 보호 장비는 하나만 있어도 집을 안전하게 보호할 수 있다.

방사능 고위험 지역의 밖에 있는 집은 방사능 저위험 지역이다. 하지만, 저위험 지역에 있는 집은 보호 장비를 받지 못한다. 이 경우에 보호 장비를 두 개 가지고 있는 집에서 하나를 저위험 지역에 있는 집에게 주면, 저위험 지역도 보호 장비를 하나 가질 수 있다. 이렇게 보호 장비를 나누어주어도, 받지 못하는 집이 있을 수 있다.

집의 위치와 두 원자력 발전소의 위치, 그리고 가능한 R1, R2가 주어졌을 때, 각각의 경우에 대해서, 보호 장비를 가질 수 없는 집의 수를 구하는 프로그램을 작성하시오.

입력

입력은 최대 3개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스는 아래와 같이 이루어져 있다.

첫째 줄에는 집의 수 N이 주어진다. (0 < N ≤ 200000) 다음 N개 줄에는 집의 좌표 xi, yi가 주어진다. (0 ≤ xi, yi ≤ 20000) 두 집이 같은 곳에 있는 경우는 없다.

다음 줄에는 ax, ay, bx, by, q가 주어진다. (0 ≤ ax, ay, bx, by ≤ 20000, 0 < q ≤ 20000) ax, ay, bx, by는 원자력 발전소의 좌표이고, q는 가능한 R1, R2값의 수이다.

다음 q개 줄에는 R1, R2가 주어진다. (0 < R1, R2 ≤ 13000)

마지막 줄에는 0이 하나 주어진다.

출력

각 테스트 케이스는 q+1 줄을 출력해야 한다. 첫째 줄은 테스트 케이스 번호를 출력한다. 그 다음, 가능한 R1, R2값에 대해서, 보호 장비를 받을 수 없는 집의 수를 출력한다.

예제 입력 1

11
95 75
27 6
93 5
124 13
34 49
65 61
81 49
77 33
110 50
91 22
110 25
57 42 97 36 2
31 25
25 25
0

예제 출력 1

Case 1:
2
2
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(81, 49)\uc640 (77, 33)\uc5d0 \uc788\ub294 \uc9d1\uc740 \ub9e4\uc6b0 \uc704\ud5d8\ud55c \uc9c0\uc5ed\uc774\ub2e4.<\/p>\r\n\r\n<p>\uba3c\uc800, \ub545\uc774 \ubaa8\ub450 \ud3c9\ud3c9\ud558\uace0, 2\ucc28\uc6d0 \uc88c\ud45c \ud3c9\uba74 \uc704\uc5d0 \ubaa8\ub4e0 \uc9d1\uc774 \uc788\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4. \ub450 \uc6d0\uc790\ub825 \ubc1c\uc804\uc18c\uac00 \uac01\uac01 (ax, ay)\uc640 (bx, by)\uc5d0 \uc788\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4. \ubc1c\uc804\uc18c (ax, ay)\uc640 \uac70\ub9ac\uac00 R1\uc774\ub0b4\uc778 \uacf3(R1\uc778 \uacf3\ub3c4 \ud3ec\ud568)\uc740 \ubc29\uc0ac\ub2a5 \uace0\uc704\ud5d8 \uc9c0\uc5ed\uc774\ub2e4. \ubc1c\uc804\uc18c (bx, by)\uc640 \uac70\ub9ac\uac00 R2\uc774\ub0b4\uc778 \uacf3(R2\uc778 \uacf3\ub3c4 \ud3ec\ud568)\ub3c4 \ubc29\uc0ac\ub2a5 \uace0\uc704\ud5d8 \uc9c0\uc5ed\uc774\ub2e4. \ub530\ub77c\uc11c, \uc6d0\uc790\ub825 \ubc1c\uc804\uc18c 1\uacfc 2\uc758 \uad00\uacc4\uc790\ub294 \ud2b9\ubcc4\ud55c \ubcf4\ud638 \uc7a5\ube44\ub97c \ubc29\uc0ac\ub2a5 \uace0\uc704\ud5d8 \uc9c0\uc5ed\uc5d0 \uc788\ub294 \uc9d1\uc5d0\uac8c \ub098\ub204\uc5b4 \uc900\ub2e4. \ub530\ub77c\uc11c, \ub450 \uc6d0\uc790\ub825 \ubc1c\uc804\uc18c\uc758 \uc704\ud5d8 \uc9c0\uc5ed\uc5d0 \ubaa8\ub450 \ud3ec\ud568\ub418\ub294 \uc9d1\uc740 \ubcf4\ud638 \uc7a5\ube44\ub97c \ub450 \uac1c \ubc1b\uac8c \ub41c\ub2e4. \ud558\uc9c0\ub9cc, \uc774 \ubcf4\ud638 \uc7a5\ube44\ub294 \ud558\ub098\ub9cc \uc788\uc5b4\ub3c4 \uc9d1\uc744 \uc548\uc804\ud558\uac8c \ubcf4\ud638\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\ubc29\uc0ac\ub2a5 \uace0\uc704\ud5d8 \uc9c0\uc5ed\uc758 \ubc16\uc5d0 \uc788\ub294 \uc9d1\uc740 \ubc29\uc0ac\ub2a5 \uc800\uc704\ud5d8 \uc9c0\uc5ed\uc774\ub2e4. \ud558\uc9c0\ub9cc, \uc800\uc704\ud5d8 \uc9c0\uc5ed\uc5d0 \uc788\ub294 \uc9d1\uc740 \ubcf4\ud638 \uc7a5\ube44\ub97c \ubc1b\uc9c0 \ubabb\ud55c\ub2e4. \uc774 \uacbd\uc6b0\uc5d0 \ubcf4\ud638 \uc7a5\ube44\ub97c \ub450 \uac1c \uac00\uc9c0\uace0 \uc788\ub294 \uc9d1\uc5d0\uc11c \ud558\ub098\ub97c \uc800\uc704\ud5d8 \uc9c0\uc5ed\uc5d0 \uc788\ub294 \uc9d1\uc5d0\uac8c \uc8fc\uba74, \uc800\uc704\ud5d8 \uc9c0\uc5ed\ub3c4 \ubcf4\ud638 \uc7a5\ube44\ub97c \ud558\ub098 \uac00\uc9c8 \uc218 \uc788\ub2e4. \uc774\ub807\uac8c \ubcf4\ud638 \uc7a5\ube44\ub97c \ub098\ub204\uc5b4\uc8fc\uc5b4\ub3c4, \ubc1b\uc9c0 \ubabb\ud558\ub294 \uc9d1\uc774 \uc788\uc744 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc9d1\uc758 \uc704\uce58\uc640 \ub450 \uc6d0\uc790\ub825 \ubc1c\uc804\uc18c\uc758 \uc704\uce58, \uadf8\ub9ac\uace0 \uac00\ub2a5\ud55c R1, R2\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac01\uac01\uc758 \uacbd\uc6b0\uc5d0 \ub300\ud574\uc11c, \ubcf4\ud638 \uc7a5\ube44\ub97c \uac00\uc9c8 \uc218 \uc5c6\ub294 \uc9d1\uc758 \uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \ucd5c\ub300 3\uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \uc544\ub798\uc640 \uac19\uc774 \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \uc9d1\uc758 \uc218 N\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; N &le; 200000) \ub2e4\uc74c N\uac1c \uc904\uc5d0\ub294 \uc9d1\uc758 \uc88c\ud45c xi, yi\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &le; xi, yi &le; 20000) \ub450 \uc9d1\uc774 \uac19\uc740 \uacf3\uc5d0 \uc788\ub294 \uacbd\uc6b0\ub294 \uc5c6\ub2e4.<\/p>\r\n\r\n<p>\ub2e4\uc74c \uc904\uc5d0\ub294 ax, ay, bx, by, q\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &le; ax, ay, bx, by &le; 20000, 0 &lt; q &le; 20000) ax, ay, bx, by\ub294 \uc6d0\uc790\ub825 \ubc1c\uc804\uc18c\uc758 \uc88c\ud45c\uc774\uace0, q\ub294 \uac00\ub2a5\ud55c R1, R2\uac12\uc758 \uc218\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub2e4\uc74c q\uac1c \uc904\uc5d0\ub294 R1, R2\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; R1, R2 &le; 13000)<\/p>\r\n\r\n<p>\ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 0\uc774 \ud558\ub098 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 q+1 \uc904\uc744 \ucd9c\ub825\ud574\uc57c \ud55c\ub2e4. \uccab\uc9f8 \uc904\uc740 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \ubc88\ud638\ub97c \ucd9c\ub825\ud55c\ub2e4. \uadf8 \ub2e4\uc74c, \uac00\ub2a5\ud55c R1, R2\uac12\uc5d0 \ub300\ud574\uc11c, \ubcf4\ud638 \uc7a5\ube44\ub97c \ubc1b\uc744 \uc218 \uc5c6\ub294 \uc9d1\uc758 \uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"5631","problem_lang":"1","title":"Radiation","description":"<p>Nuclear power plants (NPP) are a blessing and curse of modern civilization. NPPs have some risks but still it is one of the cheapest ways to produce electricity in the developed world. In this problem we will discuss a situation related to two nuclear plants, which are not far away from each other.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/nppe.png\" style=\"height:376px; width:594px\" \/><\/p>\r\n\r\n<p>Figure 1: Two Nuclear Power Plants. Houses at (81, 49) and (77,33) are at high risk from both the plants.&nbsp;<\/p>\r\n\r\n<p>We will describe the entire scenario in a flat land, so two-dimensional Cartesian coordinate system is used to denote each location. Lets assume that the coordinate of the two nuclear power plants are (ax, ay) and (bx, by). Houses that are located within distance R1 (inclusive) of the power plant at (ax, ay) are under high risk of radiation. Similarly, houses that are located within distance R2 (inclusive) of the power plant at (bx, by) are under high risk of radiation. So the authorities of power plant 1 and power plant 2 distribute special protective equipments to the houses that are within radius (inclusive) R1 and R2 of the respective power plants. As a result each of the houses that are endangered by both the plants actually receive two sets of equipments to protect their house, however only one set is enough for full protection. Houses that are outside the high-risk area are under low risk of radiation but they do not receive any protective equipment due to budget constraints. However, each owner of the houses that have two sets of protective equipments gives away one set of equipment to the owner of a house that has none. Still,some houses in the low-risk area remain un-protected. Given the location of the houses and the values of ax, ay, bx, by and possible values of R1 and R2 your job is to find out the number of houses that are without protective equipments for each pair of values of R1 and R2.&nbsp;<\/p>\r\n","input":"<p>The input file contains at most 3 test cases. The description of each test case is given below:&nbsp;<\/p>\r\n\r\n<p>A test case starts with a line containing a positive integer N (0 &lt; N &le; 200000) that denotes the number of houses that are under either low risk or high risk of radiation. Each of the next N lines contains two integers xi, yi (0 &le; xi, yi &le; 20000) that denotes the coordinate of the i-th house. No two houses are at the same location. The next line contains five integers ax, ay, bx, by and q (0 &le; ax, ay, bx, by &le; 20000, 0 &lt;q &le; 20000). The meaning of ax, ay, bx and by are given in the problem statement. Here q denotes the total number of query. Each of the next q lines contains two integers, which denote the values of R1 and R2 (0 &lt; R1, R2 &le; 13000) respectively.<\/p>\r\n\r\n<p>A line containing a single zero terminates input. This line should not be processed.&nbsp;<\/p>\r\n","output":"<p>For each test case produce q+1 lines of output. The first line is the serial of output. For each query (given value of R1 and R2) determine how many houses in the low risk region remains without protective equipment. You may consider using faster IO as judge input file is large.&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]