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 OCaml (4.02.3) (judge verdict: Accepted)

let () =
    let main () =
        let n = Scanf.scanf "%d" (fun n -> n) in
        let xy = Array.init n (fun _ -> Scanf.scanf " %d %d" (fun x y -> x, y)) in
        let gx, gy = Array.fold_left (fun (ax, ay) (x, y) -> ax + x, ay + y) (0, 0) xy in
        let gx = float gx /. float n in
        let gy = float gy /. float n in

        let ax = Array.init n (fun i -> xy.(i) |> fst |> float) in
        let ay = Array.init n (fun i -> xy.(i) |> snd |> float) in

        let rec loop_i i p x y last_dist =
            if i = 0 then sqrt last_dist else
                let rec loop_j j fx fy max_dist =
                    if j = n then loop_i (i - 1) (p *. 0.999) (x +. (fx -. x) *. p) (y +. (fy -. y) *. p) max_dist else
                        let d = (ax.(j) -. x) *. (ax.(j) -. x) +. (ay.(j) -. y) *. (ay.(j) -. y) in
                        if max_dist < d then loop_j (j + 1) ax.(j) ay.(j) d else
                            loop_j (j + 1) fx fy max_dist
                in
                loop_j 0 0.0 0.0 0.0
        in
        loop_i 30000 0.1 gx gy 0.0 |> Printf.printf "%.10f\n"
    in
    main ()