G.Finding the Radius for an Inserted Circle 2017 ACM-ICPC 亚洲区（南宁赛区）网络赛

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G.Finding the Radius for an Inserted Circle 2017 ACM-ICPC 亚洲区（南宁赛区）网络赛
地址：https://nanti.jisuanke.com/t/17314

题目：

Three circles $C_{a}Ca, C_{b}Cb, and C_{c}Cc, all with radius RR and tangent to each other, are located in two-dimensional space as shown in Figure 11. A smaller circle C_{1}C1 with radius R_{1}R1 (R_{1}<RR1<R) is then inserted into the blank area bounded by C_{a}Ca, C_{b}Cb, and C_{c}Cc so that C_{1}C1 is tangent to the three outer circles, C_{a}Ca, C_{b}Cb, and C_{c}Cc. Now, we keep inserting a number of smaller and smaller circles C_{k} (2 leq k leq N)Ck (2≤k≤N) with the corresponding radius R_{k}Rk into the blank area bounded by C_{a}Ca, C_{c}Cc and C_{k-1}Ck−1 (2 leq k leq N)(2≤k≤N), so that every time when the insertion occurs, the inserted circle C_{k}Ck is always tangent to the three outer circles C_{a}Ca, C_{c}Cc and C_{k-1}Ck−1, as shown in Figure 11$

Figure 1.

(Left) Inserting a smaller circle $C_{1}C1 into a blank area bounded by the circle C_{a}Ca, C_{b}Cb and C_{c}Cc.$

(Right) An enlarged view of inserting a smaller and smaller circle $C_{k}Ck into a blank area bounded by C_{a}Ca, C_{c}Cc and C_{k-1}Ck−1 (2 leq k leq N2≤k≤N), so that the inserted circle C_{k}Ck is always tangent to the three outer circles, C_{a}Ca, C_{c}Cc, and C_{k-1}Ck−1.$

Now, given the parameters $R_{k}Rk, i.e., the radius of the k-thk−th inserted circle. Please note that since the value of R_kRk may not be an integer, you only need to report the integer part of R_{k}Rk. For example, if you find that R_{k}Rk = 1259.89981259.8998 for some kk, then the answer you should report is 12591259.$

Another example, if $R_{k}Rk = 39.102939.1029 for some kk, then the answer you should report is 3939.$

Assume that the total number of the inserted circles is no more than

$10^{4} leq R leq 10^{7}104≤R≤107.$

Input Format

Contains

Line

Line

Line

Line

Line

Line

Output Format

Contains

Line $R_{k}Rk ----------------output for the value of kk and R_{k}Rk at the test case #11, each of which should be separated by a blank.$

Line $R_{k}Rk ----------------output for kk and the value of R_{k}Rk at the test case # ii, each of which should be separated by a blank.$

Line $R_{k}Rk ----------------output for kk and the value ofR_{k}Rk at the test case # ll, each of which should be separated by a blank.$
样例输入

1 152973.6 1 -1

样例输出

1 23665
题目来源

2017 ACM-ICPC 亚洲区（南宁赛区）网络赛

思路：

　　圆的反演。

　　很容易想到把上面两大圆的切点作为反演中心，这样会得到下图。

　　绿色的是反演前的圆，黄色的是反演后的图形，两个大圆成了平行直线，下面的大圆成了直线间的小圆，后面添加的圆都在这个小圆的下面。

　　所以求出小圆的圆心的y即可，然后反演回去可以得到半径。
1 #include <bits/stdc++.h> 2 3 using namespace std; 4 5 #define MP make_pair 6 #define PB push_back 7 typedef long long LL; 8 typedef pair<int,int> PII; 9 const double eps=1e-8; 10 const double PI=acos(-1.0); 11 const int K=1e6+7; 12 const int mod=1e9+7; 13 14 15 int main(void) 16 { 17 double r,x,y,ls,dis,ans[11]; 18 int t; 19 cin>>t>>r; 20 x=0.5*r,ls=-0.5*sqrt(3.0)*r; 21 dis=x*x+ls*ls; 22 ls=ls/dis; 23 r=1.0/(2*r); 24 for(int i=1;i<=10;i++) 25 { 26 y=ls-r*2; 27 ans[i]=0.5*(1.0/(y-r)-1.0/(y+r)); 28 ls=y; 29 } 30 for(int i=1,k;i<=t;i++) 31 scanf("%d",&k),printf("%d %d ",k,(int)ans[k]); 32 return 0; 33 }
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原文地址：https://www.cnblogs.com/weeping/p/7624767.html

G.Finding the Radius for an Inserted Circle 2017 ACM-ICPC 亚洲区（南宁赛区）网络赛

Input Format

Output Format

样例输入

样例输出

题目来源