Problem:
Score : 600 points

Problem Statement

Given are N points (x_i, y_i) in a two-dimensional plane.

Find the minimum radius of a circle such that all the points are inside or on it.

Constraints

2 \leq N \leq 50

0 \leq x_i \leq 1000

0 \leq y_i \leq 1000

The given N points are all different.

The values in input are all integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the minimum radius of a circle such that all the N points are inside or on it.

Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

2
0 0
1 0

Sample Output 1

0.500000000000000000

Both points are contained in the circle centered at (0.5,0) with a radius of 0.5.

Sample Input 2

3
0 0
0 1
1 0

Sample Output 2

0.707106781186497524

Sample Input 3

10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9

Sample Output 3

6.726812023536805158

If the absolute or relative error from our answer is at most 10^{-6}, the output will be considered correct.

Solution in PyPy3 (2.4.0) (judge verdict: Wrong Answer)

import sys
input=sys.stdin.readline

def main():
    n=int(input())
    XY=[list(map(int,input().split())) for _ in range(n)]
    
    if n==2:
        return (dist2(XY[0][0],XY[0][1],XY[1][0],XY[1][1])**0.5)/2
    
    ans=float('inf')
    for i in range(n-2):
        for j in range(i+1,n-1):
            for k in range(j+1,n):
                x1,y1=XY[i]
                x2,y2=XY[j]
                x3,y3=XY[k]
                r2,x,y=circle(x1,y1,x2,y2,x3,y3)
                if r2==-1:
                    d1,d2,d3=dist2(x1,y1,x2,y2),dist2(x2,y2,x3,y3),dist2(x3,y3,x1,y1)
                    if d1>=d2:
                        if d1>=d3:
                            x,y=(x1+x2)/2,(y1+y2)/2
                            r2=dist2(x,y,x1,y1)
                        else:
                            x,y=(x1+x3)/2,(y1+y3)/2
                            r2=dist2(x,y,x1,y1)
                    else:
                        if d2>=d3:
                            x,y=(x2+x3)/2,(y2+y3)/2
                            r2=dist2(x,y,x2,y2)
                        else:
                            x,y=(x1+x3)/2,(y1+y3)/2
                            r2=dist2(x,y,x1,y1)
                for x4,y4 in XY:
                    if dist2(x,y,x4,y4)>r2:
                        break
                else:
                    ans=min(ans,r2)
    return ans**0.5
                
def dist2(x1,y1,x2,y2):
    return (x2-x1)**2+(y2-y1)**2

def circle(x1,y1,x2,y2,x3,y3):
    #length of each side to the second power and area of the triangle
    a2=(x2-x3)**2+(y2-y3)**2 
    b2=(x1-x3)**2+(y1-y3)**2
    c2=(x2-x1)**2+(y2-y1)**2
    area=((x2-x1)*(y3-y1)-(x3-x1)*(y2-y1))

    #not on the same line please
    if area==0:
        return -1,0,0
    
    #circumcenter coordinate and radius
    x=(a2*(b2+c2-a2)*x1+b2*(c2+a2-b2)*x2+c2*(a2+b2-c2)*x3)/(4*area*area)
    y=(a2*(b2+c2-a2)*y1+b2*(c2+a2-b2)*y2+c2*(a2+b2-c2)*y3)/(4*area*area) 
    r2=(x1-x)**2+(y1-y)**2
    
    #join to return
    return r2,x,y
    
if __name__=='__main__':
    print(main())