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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define rep(X,N) for(ll X = 0; X < (N); X++)
#define PI (acos(-1.0))
#define pback push_back
#define mpair make_pair
#define MODN 1000000007
#define ALL(V) (V).begin(),(V).end()

typedef long long ll;

using namespace std;

int main(){

    int n;
    cin >> n;

    typedef pair<double, double> Point;

    vector<Point> v;

    rep(i,n){
        Point p;
        cin >> p.first >> p.second;

        v.pback(p);
    }

    double down = 0.0;
    double up = 1000.0;

    double tmpd;

    while(up - down > 0.0000001){

        //cerr << down  << " " << up << endl;

        double yfirst = (down * 2 + up) / 3;
        double ysecond = (down + up * 2) / 3;

        double left = 0.0;
        double right = 1000.0;

        //yがyfirstの時のxの値を求める
        while(right - left > 0.0000001){

            //cerr << left << " " << right << endl;

            double xfirst = (left * 2 + right) / 3;
            double xsecond = (left + right * 2) / 3;

            //cerr << xfirst << " " << xsecond << endl;

            double maxd_first = 0.0;
            double maxd_second = 0.0;

            rep(i,n){
                double tmp1 = sqrt(pow(v[i].first - xfirst, 2.0) + pow(v[i].second - yfirst, 2.0));
                double tmp2 = sqrt(pow(v[i].first - xsecond, 2.0) + pow(v[i].second - yfirst, 2.0));

                maxd_first = max(maxd_first, tmp1);
                maxd_second = max(maxd_second, tmp2);
            }

            if(maxd_first <= maxd_second){
                right = xsecond;
            }else{
                left = xfirst;
            }
        }

        double yfirst_x = (left + right) / 2;

        left = 0.0;
        right = 1000.0;

        //yがysecondの時のxの値を求める
        while(right - left > 0.0000001){

            double xfirst = (left * 2 + right) / 3;
            double xsecond = (left + right * 2) / 3;

            double maxd_first = 0.0;
            double maxd_second = 0.0;

            rep(i,n){
                double tmp1 = sqrt(pow(v[i].first - xfirst, 2.0) + pow(v[i].second - ysecond, 2.0));
                double tmp2 = sqrt(pow(v[i].first - xsecond, 2.0) + pow(v[i].second - ysecond, 2.0));

                maxd_first = max(maxd_first, tmp1);
                maxd_second = max(maxd_second, tmp2);
            }

            if(maxd_first <= maxd_second){
                right = xsecond;
            }else{
                left = xfirst;
            }
        }

        double ysecond_x = (left + right) / 2;

        double maxd1 = 0.0;
        double maxd2 = 0.0;

        rep(i,n){
            double tmp1 = sqrt(pow(v[i].first - yfirst_x, 2.0) + pow(v[i].second - yfirst, 2.0));
            double tmp2 = sqrt(pow(v[i].first - ysecond_x, 2.0) + pow(v[i].second - ysecond, 2.0));

            maxd1 = max(maxd1, tmp1);
            maxd2 = max(maxd2, tmp2);
        }

        if(maxd1 <= maxd2){
            up = ysecond;
        }else{
            down = yfirst;
        }
    }

    double y = (up + down) / 2;

    double left = 0.0;
    double right = 1000.0;

    while(right - left > 0.0000001){

        double xfirst = (left * 2 + right) / 3;
        double xsecond = (left + right * 2) / 3;

        double maxd_first = 0.0;
        double maxd_second = 0.0;

        rep(i,n){
            double tmp1 = sqrt(pow(v[i].first - xfirst, 2.0) + pow(v[i].second - y, 2.0));
            double tmp2 = sqrt(pow(v[i].first - xsecond, 2.0) + pow(v[i].second - y, 2.0));

            maxd_first = max(maxd_first, tmp1);
            maxd_second = max(maxd_second, tmp2);
        }

        if(maxd_first <= maxd_second){
            right = xsecond;
        }else{
            left = xfirst;
        }
    }

    double x = (left + right) / 2;

    double ans = 0.0;

    rep(i,n){
        double tmp = sqrt(pow(v[i].first - x, 2.0) + pow(v[i].second - y, 2.0));

        ans = max(ans, tmp);
    }

    cout << setprecision(14) << ans << endl;

    return 0;
}