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: Wrong Answer)


#include <bits/stdc++.h>

using namespace std;

/* g++ -g -fsanitize=address -lasan -std=c++11 -Dfoo_ */

/* freopen("input.txt", "rt", stdin); */
/* freopen("output.txt", "wt", stdout); */

#define ALL(c)          (c).begin(), (c).end()
#define ALLR(c)         (c).rbegin(), (c).rend()
#define FOR(i,a,b)      for (int i=(a); i < (b); ++i)
#define FORR(i,a,b)     for (int i=(a); i > (b); --i)
#define FOR_ALL(i,c)    for (__typeof((c).begin()) i=(c).begin();   \
                             i != (c).end(); ++i)
#define FOR_ALLR(i,c)   for (__typeof((c).rbegin()) i=(c).rbegin(); \
                             i != (c).rend(); ++i)
#define SZ(array)       (sizeof(array)/sizeof(array[0]))
#define lc(x)           (x<<1)     /* 2*x */
#define rc(x)           (x<<1 | 1) /* 2*x+1 */
#define lowbit(x)       (x & (-x)) /* 0b10100 -> 0b100 */

typedef long long       LL;
typedef map<int,int>    MII;
typedef pair<int,int>   PII;
typedef set<int>        SI;
typedef vector<bool>    VB;
typedef vector<double>  VD;
typedef vector<int>     VI;
typedef vector<string>  VS;

/* check if a key is in container C */
template <class C>
inline bool in_(const typename C::key_type& k, const C& A)
{ return A.find(k) != A.end(); }
inline bool in_(const string& s, const string& S)
{ return S.find(s) != string::npos; }

const int maxy = 1e3+5;
vector<pair<double,double>> A;

double dist(const double x, const double y)
{
    const int n = A.size();
    double ans = 0;
    FOR(i,0,n) {
        const double X = A[i].first;
        const double Y = A[i].second;
        const double dx = x-X;
        const double dy = y-Y;
        const double d = sqrt(dx*dx + dy*dy);
        ans = max(ans,d);
    }
    return ans;
}

int main()
{
#ifdef foo_
    freopen("foo", "rt", stdin);
#endif
    ios::sync_with_stdio(false);
    int n;
    while (cin >> n) {
        A.resize(n);
        FOR(i,0,n)
            cin >> A[i].first >> A[i].second;
        sort(ALL(A));
        const double x = (A.back().first + A[0].first) / 2;
        double lft = 0, rht = maxy;
        FOR(_,0,300) {
            const double m1 = lft+(rht-lft)/3;
            const double m2 = rht-(rht-lft)/3;
            const double v1 = dist(x,m1);
            const double v2 = dist(x,m2);
            if (v1 > v2)
                lft = m1;
            else
                rht = m2;
        }
        cout.precision(9);
        cout << fixed << dist(x,lft) << endl;
    }
    return 0;
}