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 Rust (1.15.1) (judge verdict: Wrong Answer)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[macro_export]
macro_rules ! chmax { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: max ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! chmin { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: min ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! max { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: max ( $ x , max ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! min { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: min ( $ x , min ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! dvec { ( $ t : expr ; $ len : expr ) => { vec ! [ $ t ; $ len ] } ; ( $ t : expr ; $ len : expr , $ ( $ rest : expr ) ,* ) => { vec ! [ dvec ! ( $ t ; $ ( $ rest ) ,* ) ; $ len ] } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let out = stdout();
    let mut out = BufWriter::new(out.lock());
    input!{
        n:usize,
        xy:[(f64,f64);n],
    }
    let mut ps = vec![];
    for i in 0..n {
        let (x,y) = xy[i];
        ps.push(Vector2D(x,y));
    }

    let mut ans = std::f64::MAX;

    // 2点選んで真ん中を中心にとる
    for i in 0..n-1 {
        for j in i+1..n {
            let center = (ps[i]+ps[j]) / 2.;
            let mut maxv = 0.;
            for k in 0..n {
                let L = (ps[k]-center).len();
                if L > maxv {
                    maxv = L;
                }
            }
            if maxv < ans {
                ans = maxv;
            }
        }
    }

    // 3点とって外接円の中心を中心にする
    for a in 0..n-2 {
        for b in a+1..n-1 {
            for c in b+1..n {

                // // 面積があることをチェック
                // let A = (ps[a]-ps[b]).len();
                // let B = (ps[b]-ps[c]).len();
                // let C = (ps[c]-ps[a]).len();
                // let eps = 0.0000000001;
                // if (A+B-C) < eps { continue; }
                // if (B+C-A) < eps { continue; }
                // if (C+A-B) < eps { continue; }

                let circle = Circle::outer_circle_stable(ps[a], ps[b], ps[c]);
                let center = circle.center;

                let mut maxv = 0.;
                for k in 0..n {
                    let L = (ps[k]-center).len();
                    if L > maxv {
                        maxv = L;
                    }
                }
                if maxv < ans {
                    ans = maxv;
                }
            }
        }
    }

    println!("{}", ans);
}
#[derive(PartialEq, PartialOrd)]
#[doc = " Implement Eq and Ord for a type which has only PartialEq and PartialOrd."]
#[doc = " It is useful when sorting a Vec of f64"]
pub struct Total<T>(pub T);
impl<T: PartialEq> Eq for Total<T> {}
impl<T: PartialOrd> Ord for Total<T> {
    fn cmp(&self, other: &Total<T>) -> Ordering {
        self.0.partial_cmp(&other.0).unwrap()
    }
}

#[derive(Debug, Clone, Copy, PartialEq)]
#[allow(dead_code)]
pub struct Vector2D(f64, f64);

impl Vector2D {
    pub fn add(a: f64, b: f64) -> f64 {
        let c = a + b;
        if c.abs() < 1e-10 {
            0.0
        } else {
            c
        }
    }
    pub fn dot(self, other: Vector2D) -> f64 {
        Self::add(self.0 * other.0, self.1 * other.1)
    }
    pub fn det(self, other: Vector2D) -> f64 {
        Self::add(self.0 * other.1, -self.1 * other.0)
    }
    pub fn len(&self) -> f64 {
        f64::sqrt((self.0).powi(2) + (self.1).powi(2))
    }
    pub fn unit(self) -> Vector2D {
        let l = self.len();
        Vector2D(self.0 / l, self.1 / l)
    }
    #[doc = "orthogonal vector"]
    pub fn normal(self) -> Vector2D {
        Vector2D(self.1, -self.0)
    }
    #[doc = "bisection of the angle"]
    pub fn bisect(a: Vector2D, b: Vector2D) -> Vector2D {
        (a.unit() + b.unit()).unit()
    }
}

impl std::ops::Add for Vector2D {
    type Output = Vector2D;
    fn add(self, rhs: Vector2D) -> Self::Output {
        Vector2D(Vector2D::add(self.0, rhs.0), Vector2D::add(self.1, rhs.1))
    }
}

impl std::ops::Sub for Vector2D {
    type Output = Vector2D;
    fn sub(self, rhs: Vector2D) -> Self::Output {
        Vector2D(Vector2D::add(self.0, -rhs.0), Vector2D::add(self.1, -rhs.1))
    }
}

impl std::ops::Mul<f64> for Vector2D {
    type Output = Vector2D;
    fn mul(self, rhs: f64) -> Self::Output {
        Vector2D(rhs * self.0, rhs * self.1)
    }
}

impl std::ops::Div<f64> for Vector2D {
    type Output = Vector2D;
    fn div(self, rhs: f64) -> Self::Output {
        Vector2D(self.0 / rhs, self.1 / rhs)
    }
}

pub struct Circle {
    center: Vector2D,
    radius: f64,
}

impl Circle {
    pub fn inner_circle(a: Vector2D, b: Vector2D, c: Vector2D) -> Circle {
        let a_bisect = Line2D {
            p: a,
            d: Vector2D::bisect(a-b, a-c),
        };
        let b_bisect = Line2D {
            p: b,
            d: Vector2D::bisect(b-a, b-c),
        };

        let center = Line2D::intersection(a_bisect, b_bisect);
        let ab = Line2D {
            p: a,
            d: b-a,
        };
        let radius = ab.distance(center);
        Circle {
            center: center,
            radius: radius,
        }
    }
    pub fn outer_circle(a: Vector2D, b: Vector2D, c: Vector2D) -> Circle {
        let ubn = Line2D {
             p: (a+b) / 2.,
             d: (a-b).normal()
        };
        let vbn = Line2D {
            p: (b+c) / 2.,
            d: (b-c).normal(),
        };

        let center = Line2D::intersection(ubn, vbn);
        let radius = (a - center).len();

        Circle {
            center: center,
            radius: radius,
        }
    }
    pub fn outer_circle_stable(x: Vector2D, y: Vector2D, z: Vector2D) -> Circle {
        let a = (y-z).len();
        let a2 = a*a;
        let b = (x-z).len();
        let b2 = b*b;
        let c = (x-y).len();
        let c2 = c*c;
        let X = x*(a2*(b2+c2-a2)) + y*(b2*(c2+a2-b2)) + z*(c2*(a2+b2-c2));
        let Y = a2*(b2+c2-a2) + b2*(c2+a2-b2) + c2*(a2+b2-c2);
        let center = X / Y;
        let radius = (x-center).len();
        Circle {
            center: center,
            radius: radius
        }
    }
}

#[derive(Clone, Copy, Debug)]
pub struct Line2D {
    p: Vector2D,
    d: Vector2D,
}
impl Line2D {
    pub fn intersection(a: Line2D, b: Line2D) -> Vector2D {
        let n = b.d.normal();
        let x = n.dot(b.p - a.p) / n.dot(a.d);
        a.p + a.d * x
    }
    pub fn distance(self, a: Vector2D) -> f64 {
        let perpendicular = Line2D {
            p: a,
            d: self.d.unit(),
        };
        let q = Self::intersection(self, perpendicular);
        (a-q).len()
    }
}