Problem:
Score : 600 points

Problem Statement

There are N holes in a two-dimensional plane. The coordinates of the i-th hole are (x_i,y_i).

Let R=10^{10^{10^{10}}}. Ringo performs the following operation:

Randomly choose a point from the interior of a circle of radius R centered at the origin, and put Snuke there. Snuke will move to the hole with the smallest Euclidean distance from the point, and fall into that hole. If there are multiple such holes, the hole with the smallest index will be chosen.

For every i (1 \leq i \leq N), find the probability that Snuke falls into the i-th hole.

Here, the operation of randomly choosing a point from the interior of a circle of radius R is defined as follows:

Pick two real numbers x and y independently according to uniform distribution on [-R,R].

If x^2+y^2\leq R^2, the point (x,y) is chosen. Otherwise, repeat picking the real numbers x,y until the condition is met.

Constraints

2 \leq N \leq 100

|x_i|,|y_i| \leq 10^6(1\leq i\leq N)

All given points are pairwise distinct.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print N real numbers. The i-th real number must represent the probability that Snuke falls into the i-th hole.

The output will be judged correct when, for all output values, the absolute or relative error is at most 10^{-5}.

Sample Input 1

2
0 0
1 1

Sample Output 1

0.5
0.5

If Ringo put Snuke in the region x+y\leq 1, Snuke will fall into the first hole. The probability of this happening is very close to 0.5.
Otherwise, Snuke will fall into the second hole, the probability of which happening is also very close to 0.5.

Sample Input 2

5
0 0
2 8
4 5
2 6
3 10

Sample Output 2

0.43160120892732328768
0.03480224363653196956
0.13880483535586193855
0.00000000000000000000
0.39479171208028279727

Solution in Rust (1.15.1) (judge verdict: Accepted)

use self::mod_int::ModInt;
const MOD: usize = 998244353;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());
    let n: usize = sc.read();
    let mut points = vec![];
    for _ in 0..n {
        let x: f64 = sc.read();
        let y: f64 = sc.read();
        points.push(Point { x: x, y: y });
    }

    let indices = extract_convex_hull(&points, true);
    let num_convex = indices.len();

    // let mut max_x = std::f64::MIN;
    // let mut min_x = std::f64::MAX;
    // let mut max_y = std::f64::MIN;
    // let mut min_y = std::f64::MAX;
    // for i in 0..num_convex {
    //     max_x = max_x.max(points[indices[i]].x);
    //     max_y = max_y.max(points[indices[i]].y);
    //     min_x = min_x.min(points[indices[i]].x);
    //     min_y = min_y.min(points[indices[i]].y);
    // }
    //
    // let cx = (max_x + min_x) / 2.;
    // let cy = (max_y + min_y) / 2.;
    // for i in 0..num_convex {
    //     points[indices[i]].x += cx;
    //     points[indices[i]].y += cy;
    // }

    let mut ans = vec![0.0; n];
    for i in 0..num_convex {
        let prev = indices[(i + num_convex - 1) % num_convex];
        let next = indices[(i + 1) % num_convex];

        let v1 = points[prev];
        let v2 = points[indices[i]];
        let v3 = points[next];

        let v12 = Point {
            x: v1.x - v2.x,
            y: v1.y - v2.y,
        };
        let v23 = Point {
            x: v2.x - v3.x,
            y: v2.y - v3.y,
        };
        ans[indices[i]] = calc(
            Point {
                x: v12.y,
                y: -v12.x,
            },
            Point {
                x: v23.y,
                y: -v23.x,
            },
        );
    }

    for ans in ans.into_iter() {
        println!("{}", ans);
    }
}

fn calc(a: Point, b: Point) -> f64 {
    let cos =
        (a.x * b.x + a.y * b.y) / (a.x * a.x + a.y * a.y).sqrt() / (b.x * b.x + b.y * b.y).sqrt();
    cos.min(1.0).max(-1.0).acos() / std::f64::consts::PI / 2.0
}

pub fn extract_convex_hull(points: &Vec<Point>, contain_on_segment: bool) -> Vec<usize> {
    let n = points.len();
    if n <= 1 {
        return vec![0];
    }

    let mut ps: Vec<usize> = (0..n).collect();
    ps.sort_by(|&a, &b| {
        if points[a].x == points[b].x {
            points[a].y.partial_cmp(&points[b].y).unwrap()
        } else {
            points[a].x.partial_cmp(&points[b].x).unwrap()
        }
    });

    let mut qs: Vec<usize> = Vec::new();
    for &i in &ps {
        while qs.len() > 1 {
            let k = qs.len();
            let det = (points[qs[k - 1]] - points[qs[k - 2]]).det(&(points[i] - points[qs[k - 1]]));
            if det < 0.0 || (det <= 0.0 && !contain_on_segment) {
                qs.pop();
            } else {
                break;
            }
        }
        qs.push(i);
    }

    let t = qs.len();
    for i in (0..(n - 1)).rev() {
        let i = ps[i];
        while qs.len() > t {
            let k = qs.len();
            let det = (points[qs[k - 1]] - points[qs[k - 2]]).det(&(points[i] - points[qs[k - 1]]));
            if det < 0.0 || (det <= 0.0 && !contain_on_segment) {
                qs.pop();
            } else {
                break;
            }
        }
        qs.push(i);
    }

    qs.pop();
    return qs;
}

#[derive(Debug, Copy, Clone)]
pub struct Point {
    x: f64,
    y: f64,
}

impl std::ops::Sub for Point {
    type Output = Point;
    fn sub(self, other: Point) -> Point {
        Point {
            x: self.x - other.x,
            y: self.y - other.y,
        }
    }
}

impl Point {
    pub fn det(&self, other: &Point) -> f64 {
        self.x * other.y - self.y * other.x
    }
}
pub struct UnionFind {
    parent: Vec<usize>,
    sizes: Vec<usize>,
    size: usize,
}

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        UnionFind {
            parent: (0..n).map(|i| i).collect::<Vec<usize>>(),
            sizes: vec![1; n],
            size: n,
        }
    }

    pub fn find(&mut self, x: usize) -> usize {
        if x == self.parent[x] {
            x
        } else {
            let px = self.parent[x];
            self.parent[x] = self.find(px);
            self.parent[x]
        }
    }

    pub fn unite(&mut self, x: usize, y: usize) -> bool {
        let parent_x = self.find(x);
        let parent_y = self.find(y);
        if parent_x == parent_y {
            return false;
        }

        let (large, small) = if self.sizes[parent_x] < self.sizes[parent_y] {
            (parent_y, parent_x)
        } else {
            (parent_x, parent_y)
        };

        self.parent[small] = large;
        self.sizes[large] += self.sizes[small];
        self.sizes[small] = 0;
        self.size -= 1;
        return true;
    }
}

/// `FenwickTree` is a data structure that can efficiently update elements
/// and calculate prefix sums in a table of numbers.
/// [https://en.wikipedia.org/wiki/Fenwick_tree](https://en.wikipedia.org/wiki/Fenwick_tree)
pub struct FenwickTree<T> {
    n: usize,
    data: Vec<T>,
    init: T,
}

impl<T: Copy + std::ops::AddAssign + std::ops::Sub<Output = T>> FenwickTree<T> {
    /// Constructs a new `FenwickTree`. The size of `FenwickTree` should be specified by `size`.
    pub fn new(size: usize, init: T) -> FenwickTree<T> {
        FenwickTree {
            n: size + 1,
            data: vec![init; size + 1],
            init: init,
        }
    }

    pub fn add(&mut self, k: usize, value: T) {
        let mut x = k;
        while x < self.n {
            self.data[x] += value;
            x |= x + 1;
        }
    }

    /// Returns a sum of range `[l, r)`
    pub fn sum(&self, l: usize, r: usize) -> T {
        self.sum_one(r) - self.sum_one(l)
    }

    /// Returns a sum of range `[0, k)`
    pub fn sum_one(&self, k: usize) -> T {
        assert!(k < self.n, "k={} n={}", k, self.n);

        let mut result = self.init;
        let mut x = k as i32 - 1;
        while x >= 0 {
            result += self.data[x as usize];
            x = (x & (x + 1)) - 1;
        }

        result
    }
}
pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

    type Num = usize;

    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);

    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }

    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }

    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }

    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }

    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }

    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }

    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }

    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            self * ModInt(rhs).pow(MOD - 2)
        }
    }

    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self / rhs.0
        }
    }

    impl DivAssign<Num> for ModInt<Num> {
        fn div_assign(&mut self, rhs: Num) {
            *self = *self / rhs
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self / rhs
        }
    }

    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }
    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }

    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }

    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }

    impl ModInt<Num> {
        pub fn pow(self, e: Num) -> ModInt<Num> {
            let mut result = ModInt(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}