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



Sample input:
2
0 0
1 1


Sample output:
0.5
0.5

Submission status: Time Limit Exceeded

Solution code:
import math
N = int(input())
x = [0 for i in range(N)]
y = [0 for i in range(N)]
for i in range(N):
    x[i], y[i] = map(int, input().split())


def mindis(t):
    j = 0
    for i in range(N):
        if x[i]*math.cos(t)+y[i]*math.sin(t)<x[j]*math.cos(t)+y[j]*math.sin(t):
            j = i
    return j


r = [0 for i in range(N)]
M = 250000
for i in range(M):
    k = mindis(2*math.pi*i/M)
    r[k] += 1
S = sum(r)
for i in range(N):
    print(r[i]/S)

Submission status: Time Limit Exceeded

Solution code:
import math
N = int(input())
x = [0 for i in range(N)]
y = [0 for i in range(N)]
for i in range(N):
    x[i], y[i] = map(int, input().split())


def mindis(p, q):
    j = 0
    for i in range(N):
        if (x[i]-p)**2+(y[i]-q)**2<(x[j]-p)**2+(y[j]-q)**2:
            j = i
    return j


r = [0 for i in range(N)]
M = 19*10**5
for i in range(M):
    c = math.cos(2*math.pi*i/M)*10**10
    s = math.sin(2*math.pi*i/M)*10**10
    k = mindis(c, s)
    r[k] += 1
S = sum(r)
for i in range(N):
    print(r[i]/S)

Submission status: Time Limit Exceeded

Solution code:
import math
N = int(input())
x = [0 for i in range(N)]
y = [0 for i in range(N)]
for i in range(N):
    x[i], y[i] = map(int, input().split())


def mindis(p, q):
    j = 0
    for i in range(N):
        if (x[i]-p)**2+(y[i]-q)**2<(x[j]-p)**2+(y[j]-q)**2:
            j = i
    return j


r = [0 for i in range(N)]
M = 18*10**5
for i in range(M):
    c = math.cos(2*math.pi*i/M)*10**10
    s = math.sin(2*math.pi*i/M)*10**10
    k = mindis(c, s)
    r[k] += 1
S = sum(r)
for i in range(N):
    print(r[i]/S)

Submission status: Time Limit Exceeded

Solution code:
import math
N = int(input())
x = [0 for i in range(N)]
y = [0 for i in range(N)]
for i in range(N):
    x[i], y[i] = map(int, input().split())


def mindis(p, q):
    j = 0
    for i in range(N):
        if (x[i]-p)**2+(y[i]-q)**2<(x[j]-p)**2+(y[j]-q)**2:
            j = i
    return j


r = [0 for i in range(N)]
M = 20*10**5
for i in range(M):
    c = math.cos(2*math.pi*i/M)*10**10
    s = math.sin(2*math.pi*i/M)*10**10
    k = mindis(c, s)
    r[k] += 1
S = sum(r)
for i in range(N):
    print(r[i]/S)

Submission status: Time Limit Exceeded

Solution code:
import math
N = int(input())
x = [0 for i in range(N)]
y = [0 for i in range(N)]
for i in range(N):
    x[i], y[i] = map(int, input().split())


def mindis(p, q):
    j = 0
    for i in range(N):
        if (x[i]-p)**2+(y[i]-q)**2<(x[j]-p)**2+(y[j]-q)**2:
            j = i
    return j


r = [0 for i in range(N)]
M = 25*10**5
for i in range(M):
    c = math.cos(2*math.pi*i/M)*10**10
    s = math.sin(2*math.pi*i/M)*10**10
    k = mindis(c, s)
    r[k] += 1
S = sum(r)
for i in range(N):
    print(r[i]/S)

Submission status: Time Limit Exceeded

Solution code:
import random

def distance(p1, p2):
     return pow(p1[0] - p2[0], 2) + pow(p1[1] - p2[1], 2)
     
lim=1000
#lim=pow(pow(10,10), 10)

sample_num=1000000
samples=[]

for i in range(sample_num):
#    samples.append([random.uniform(-lim, lim), random.uniform(-lim, lim)])
    samples.append([random.randint(-lim, lim), random.randint(-lim, lim)])

N=int(raw_input())

set_points=[]
point_score=[]

for i in range(N):
    point = map(int,raw_input().split())
    set_points.append(point)
    point_score.append(0)

#print set_points
#print samples

for sample in samples:
    p = 0
    d = distance(set_points[p], sample)
    for i in range(1, N):
        tmp_d = distance(set_points[i], sample)
        if d < tmp_d:
            p = i
            d = tmp_d
    point_score[p] += 1
#    print point_score

for score in point_score:
    print (score + 0.00) / sample_num

Submission status: Time Limit Exceeded

Solution code:
import math
def solve():
    N = int(input())
    clist = []
    dist = {}
    for i in range(N):
        x,y = map(int,input().split())
        clist.append((x,y))
        dist[i] = 0
    num = 100000
    calc(dist,clist,num)
    for i in range (N):
        print(str(dist[i]))

def dotprod(theta,x,y):
    return math.cos(theta)*x + math.sin(theta)*y

def prefered(theta,li):
    i = 0
    max = -10000000000
    maxi = -1
    for (x,y) in li:
        cand = dotprod(theta,x,y) 
        if cand > max:
            max = cand
            maxi = i
        i += 1
    return maxi

def calc(d,li,number):
    for k in range(number):
        theta = 2 * math.pi * (k/number)
        d[prefered(theta,li)] += 1

    for key in d:
        d[key] = d[key]/number

solve()

Submission status: Memory Limit Exceeded

Solution code:
import math
import os
import sys

import numpy as np

if os.getenv("LOCAL"):
    sys.stdin = open("_in.txt", "r")

sys.setrecursionlimit(10 ** 9)
INF = float("inf")
IINF = 10 ** 18
MOD = 10 ** 9 + 7
# MOD = 998244353


N = int(sys.stdin.buffer.readline())
X, Y = list(zip(*[map(int, sys.stdin.buffer.readline().split()) for _ in range(N)]))

X = np.array(X, dtype=np.float128)
Y = np.array(Y, dtype=np.float128)

COUNT = 100000

# 角度を決めると確率も決まる
# 端っこにある2つが50%ずつ
xs = 10 ** 8
ys = np.tan(np.linspace(0, math.pi, COUNT)) * xs
d = (X[:, None] - xs) ** 2 + (Y[:, None] - ys[None, :]) ** 2
counts = np.bincount(d.argmin(axis=0), minlength=N) + np.bincount(d.argmax(axis=0), minlength=N)
ans = counts / counts.sum()
print(*ans, sep='\n')

Submission status: Memory Limit Exceeded

Solution code:
from collections import Counter

import numpy as np


def solve(n, pts):
    ans = [0] * n

    div = 200000
    a = np.linspace(0, 2 * np.pi, div, endpoint=False)
    b = np.dot(pts, np.array([np.cos(a), np.sin(a)]))
    c = np.argmax(b, axis=0)
    for i, v in Counter(c).items():
        ans[i] = v / div
    return ans


n = int(input())
pts = [list(map(int, input().split())) for _ in range(n)]
print('\n'.join(map(str, solve(n, pts))))

Submission status: Memory Limit Exceeded

Solution code:
from collections import Counter
import numpy as np

p = 2.5e5
n = int(input())
pts = [list(map(int, input().split())) for _ in range(n)]
ans = np.zeros(n)

a = np.linspace(0, 2 * np.pi, p, endpoint=False)

for i, v in Counter(np.argmax(np.dot(pts, [np.sin(a), np.cos(a)]), axis=0)).items():
  ans[i] += v / p

print('\n'.join(map(str, ans)))

Submission status: Memory Limit Exceeded

Solution code:
from collections import Counter

import numpy as np


def solve(n, pts):
    ans = [0] * n

    div = 200000
    a = np.linspace(0, 2 * np.pi, div)
    b = np.dot(pts, [np.sin(a), np.cos(a)])
    c = np.argmax(b, axis=0)
    for i, v in Counter(c).items():
        ans[i] = v / div
    return ans


n = int(input())
pts = [list(map(int, input().split())) for _ in range(n)]
print('\n'.join(map(str, solve(n, pts))))

Submission status: Accepted

Solution code:
import math

N = int(input())
xy = [list(map(int, input().split())) for _ in range(N)]

for i in range(N):
  x, y = xy[i]
  lmin = 0
  lmax = 180
  rmin = 180
  rmax = 360
  for j in range(N):
    if i == j: continue
    p, q = xy[j][0] - x, xy[j][1] - y
    lt = math.degrees(math.atan2(q, p)) % 360
    rt = (lt + 180) % 360
    if lt > rt:
      rt, lt = lt, rt
      lmax = min(lmax, lt)
      rmin = max(rmin, rt)
    else:
      lmin = max(lmin, lt)
      rmax = min(rmax, rt)
  ans = 0
  if lmin < lmax:
    ans += (lmax - lmin) / 360
  if rmin < rmax:
    ans += (rmax - rmin) / 360
  print(ans)

Submission status: Accepted

Solution code:

"""

https://atcoder.jp/contests/agc021/tasks/agc021_b
偏角ソートそしてimos法

"""

import math
N = int(input())

xy = []
for i in range(N):
    x,y = map(int,input().split())
    xy.append((x,y))

start = []
end = []
pi = math.pi
now = [0] * N

for i in range(N):
    for j in range(i):

        x,y = xy[i]
        X,Y = xy[j]

        #xが等しい時→角度は0とpiになる
        if x == X:
            if y > Y:
                start.append( (0,i) )
                end.append( ( pi , i) )
                start.append( ( pi,j) )
                end.append( (2 * pi , j) )
            else:
                start.append( (0,j) )
                end.append( ( pi , j) )
                start.append( ( pi,i) )
                end.append( (2 * pi , i) )
        elif y == Y: #yが等しい時→角度は1/2piと3/2pi
            if x > X:
                start.append( ( pi/2   ,j) )
                end.append(   ( 3*pi/2 ,j) )
                start.append( ( 3*pi/2 ,i) )
                end.append(   ( pi/2 ,i) )
                now[i] += 1
            else:
                start.append( ( pi/2   ,i) )
                end.append(   ( 3*pi/2 ,i) )
                start.append( ( 3*pi/2 ,j) )
                end.append(   ( pi/2 ,j) )
                now[j] += 1
        else: #一般の場合(面倒くさい)
            
            k = -1 / ((Y-y)/(X-x)) #二分線の傾き
            mx = (x+X)/2
            my = (y+Y)/2
            # y-my = k(x-mx) が方程式→ y=k(x-mx)+my
            #print (k)

            if k > 0:
                #上側にあればdeg～deg+pi
                deg = math.atan(k)
                if y > k*(x-mx)+my:
                    start.append( (deg   ,i) )
                    end.append  ( (deg+pi,i) )
                    start.append( (deg+pi  ,j))
                    end.append  ( (deg,j))
                    now[j] += 1
                else:
                    start.append( (deg   ,j) )
                    end.append  ( (deg+pi,j) )
                    start.append( (deg+pi  ,i))
                    end.append  ( (deg,i))
                    now[i] += 1
            else:
                #下側にあればdeg+pi～deg+2pi
                deg = math.atan(k)
                if y < k*(x-mx)+my:
                    start.append( (deg+pi,i) )
                    end.append  ( (deg+2*pi,i) )
                    start.append( (deg+2*pi,j))
                    end.append  ( (deg+pi,j))
                    now[j] += 1
                else:
                    start.append( (deg+pi,j) )
                    end.append  ( (deg+2*pi,j))
                    start.append( (deg+2*pi,i))
                    end.append  ( (deg+pi,i))
                    now[i] += 1
            

ans = [0] * N
start.sort()
end.sort()
si = 0
ei = 0
start.append((float("inf"),0))
end.append((float("inf"),0))

fmax = [None] * N
#print (start)
#print (end)
#print (now)

for i in range(N):
    if now[i] == N-1:
        fmax[i] = 0

for i in range(len(start)+len(end)-2):
    
    if start[si][0] < end[ei][0]:
        d,ind = start[si]
        si += 1
        now[ind] += 1
        if now[ind] == N-1:
            fmax[ind] = d
    else:
        d,ind = end[ei]
        ei += 1
        if now[ind] == N-1:
            #print (fmax)
            ans[ind] += (d-fmax[ind])/(2*pi)
        now[ind] -= 1

for i in range(N):
    if now[i] == N-1:
        ans[i] += 1 - fmax[i]/(2*pi)

for i in ans:
    print (i)

Submission status: Accepted

Solution code:
import math
from copy import deepcopy
import sys
input = sys.stdin.readline

class Vec:
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __str__(self):
        return '({}, {})'.format(self.x, self.y)

    def __add__(self, other):
        return Vec(self.x+other.x, self.y+other.y)

    def __sub__(self, other):
        return Vec(self.x-other.x, self.y-other.y)

    def __mul__(self, k):
        return Vec(self.x*k, self.y*k)

    def __truediv__(self, k):
        return Vec(self.x/k, self.y/k)

    def __neg__(self):
        return Vec(-self.x, -self.y)

    def __eq__(self, other):
        return self.x == other.x and self.y == other.y

    def __ne__(self, other):
        return not (self == other)

    def abs(self):
        return (self.x**2 + self.y**2)**.5

    def dot(self, other):
        return self.x*other.x + self.y*other.y

    def cross(self, other):
        return self.x*other.y - self.y*other.x

    def rot(self, ang):
        c = math.cos(ang)
        s = math.sin(ang)
        return Vec(c*self.x-s*self.y, s*self.x+c*self.y)

    def arg(self):
        return math.atan2(self.y, self.x)


def convex_hull(points, eps=1e-12):
    n = len(points)
    points.sort(key=lambda p: (p.x, p.y))
    k = 0  # # of vertices in the convex hull
    ch = [None] * (2*n)
    # bottom
    for i in range(n):
        while k > 1 and (ch[k-1]-ch[k-2]).cross(points[i]-ch[k-1]) < eps:
            k -= 1
        ch[k] = points[i]
        k += 1
    t = k
    # top
    for i in range(0, n-1)[::-1]:
        while k > t and (ch[k-1]-ch[k-2]).cross(points[i]-ch[k-1]) < eps:
            k -= 1
        ch[k] = points[i]
        k += 1
    return ch[:k-1]


N = int(input())
points = [Vec(*map(int, input().split())) for _ in range(N)]
tmp = deepcopy(points)
ch = convex_hull(tmp)
ch.sort(key=lambda v: v.arg())
ans = [0] * N
for i in range(len(ch)):
    p0 = ch[(i-1) % len(ch)]
    p1 = ch[i]
    p2 = ch[(i+1) % len(ch)]
    a = (p2 - p1).arg() - (p1 - p0).arg()
    if a < 0:
        a += 2 * math.pi
    for j in range(N):
        if p1 == points[j]:
            ans[j] = a / (2 * math.pi)
print(*ans, sep='\n')

Submission status: Accepted

Solution code:
from math import acos
pi = 3.14159265358979323846264338
N = int(input())
A = [[0, 0, 0, _] for _ in range(N)]
flg = [False for _ in range(N)]
for i in range(N):
    A[i][1], A[i][2] = map(int, input().split())
    A[i][0] = A[i][1] * A[i][1] + A[i][2] * A[i][2]
X = [[A[i][1], A[i][2]] for i in range(N)]
A.sort(reverse=True)
stx, sty = A[0][1], A[0][2]
px, py = stx, sty
flg[A[0][3]] = True
prev_point = A[0][3]
convex_hull = [A[0][3]]
while True:
    min_prod2 = [[1, 1], [N+1, N+1], 1]
    for i in range(N):
        for j in range(N):
            if i != prev_point and j != prev_point:
                ffflg = True
                dx1 = X[i][0] - px
                dx2 = X[j][0] - px
                dy1 = X[i][1] - py
                dy2 = X[j][1] - py
                inner_prod = dx1 * dx2 + dy1 * dy2
                norm1 = dx1 * dx1 + dy1 * dy1
                norm2 = dx2 * dx2 + dy2 * dy2
                sign = (inner_prod > 0) - (inner_prod < 0)
                if sign < min_prod2[2]:
                    min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                    min_prod2[1] = [i, j]
                    min_prod2[2] = sign
                elif sign == min_prod2[2]:
                    if sign == 1:
                        if min_prod2[0][0] * norm1 * norm2 > inner_prod * inner_prod * min_prod2[0][1]:
                            min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                            min_prod2[1] = [i, j]
                        elif min_prod2[0][0] * norm1 * norm2 == inner_prod * inner_prod * min_prod2[0][1]:
                            if min_prod2[0][1] > norm1 * norm2 or min_prod2[1] == [N+1, N+1]:
                                min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                                min_prod2[1] = [i, j]
                    elif sign == 0:
                        if min_prod2[0][1] > norm1 * norm2 or min_prod2[1] == [N+1, N+1]:
                            min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                            min_prod2[1] = [i, j]
                    else:
                        if min_prod2[0][0] * norm1 * norm2 < inner_prod * inner_prod * min_prod2[0][1]:
                            min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                            min_prod2[1] = [i, j]
                        elif min_prod2[0][0] * norm1 * norm2 == inner_prod * inner_prod * min_prod2[0][1]:
                            if min_prod2[0][1] > norm1 * norm2 or min_prod2[1] == [N+1, N+1]:
                                min_prod2[0] = [inner_prod*inner_prod, norm1*norm2]
                                min_prod2[1] = [i, j]
    if flg[min_prod2[1][0]] and flg[min_prod2[1][1]]:
        break
    else:
        if flg[min_prod2[1][0]] and not flg[min_prod2[1][1]]:
            prev_point = min_prod2[1][1]
        elif not flg[min_prod2[1][0]] and flg[min_prod2[1][1]]:
            prev_point = min_prod2[1][0]
        elif not flg[min_prod2[1][0]] and not flg[min_prod2[1][1]]:
            prev_point = min_prod2[1][0]
        convex_hull.append(prev_point)
        flg[prev_point] = True
        px, py = X[prev_point][0], X[prev_point][1]
M = len(convex_hull)
prob = [0 for _ in range(N)]
for i in range(M):
    center = convex_hull[i]
    ii, jj = convex_hull[(i-1)%M], convex_hull[(i+1)%M]
    dx1 = X[ii][0] - X[center][0]
    dx2 = X[jj][0] - X[center][0]
    dy1 = X[ii][1] - X[center][1]
    dy2 = X[jj][1] - X[center][1]
    ip = dx1 * dx2 + dy1 * dy2
    n1 = dx1 * dx1 + dy1 * dy1
    n2 = dx2 * dx2 + dy2 * dy2
    n1 = n1**0.5
    n2 = n2**0.5
    prob[center] = 0.5 - (acos((ip/n1)/n2))/(2*pi)
for i in range(N):
    print(prob[i])

Submission status: Accepted

Solution code:
import sys
def I(): return int(sys.stdin.readline().rstrip())
def MI(): return map(int,sys.stdin.readline().rstrip().split())

N = I()
A = []
for i in range(N):
    x,y = MI()
    A.append((x,y,i))

if N == 2:
    print(0.5)
    print(0.5)
    exit()


# グラハムスキャン(2次元の場合使用可能)

def area(X,Y,Z):  # 三角形の符号付き面積の2倍
    return (Y[0]-X[0])*(Z[1]-X[1])-(Y[1]-X[1])*(Z[0]-X[0])

from operator import itemgetter

def ConvexHull(A):  # Aの凸包を形成する点のリストを返す
    A = sorted(A,key=itemgetter(0,1))  # 適宜変更
    res = []
    N = len(A)
    for i in range(N):  # 下側
        a = A[i]
        while len(res) > 1 and area(res[-2],res[-1],a) <= 0:  # 適宜'>,<,>='に変更
            res.pop()
        res.append(a)
    r = len(res)
    for i in range(N-2,-1,-1):  # 上側
        a = A[i]
        while len(res) > r and area(res[-2],res[-1],a) <= 0:  # 適宜'>,<,>='に変更
            res.pop()
        res.append(a)
    return res


B = ConvexHull(A)
del B[-1]
ANS = [0]*N

from math import acos,pi

for j in range(len(B)):
    x,y,i = B[j]
    if j == 0:
        x1,y1,i1 = B[-1]
        x2,y2,i2 = B[j+1]
    elif j == len(B)-1:
        x1,y1,i1 = B[j-1]
        x2,y2,i2 = B[0]
    else:
        x1,y1,i1 = B[j-1]
        x2,y2,i2 = B[j+1]
    ANS[i] = (pi - acos(((x1-x)*(x2-x)+(y1-y)*(y2-y))/((((x1-x)**2+(y1-y)**2)*((x2-x)**2+(y2-y)**2))**.5)))/(2*pi)

print(*ANS,sep='\n')

Submission status: Accepted

Solution code:
"""
@ 凸包をつくる
    > 凸包内部の点に属する領域は、半径Rの円からするともはやないものとして考えて良い
    > というかこの多角形自体点として考えていい！
@ 凸包の頂点vについて
    > vの両隣との傾きだけあればよい
    > その間の角度 / 360が答え
"""

from math import gcd, atan, pi, degrees

def cross3(a, b, c):
    return (b[0]-a[0])*(c[1]-a[1]) - (b[1]-a[1])*(c[0]-a[0])

def convex_hull(ps):
    qs = []
    N = len(ps)
    for p in ps:
        # 一直線上で高々2点にする場合は ">=" にする
        while len(qs) > 1 and cross3(qs[-1], qs[-2], p) > 0:
            qs.pop()
        qs.append(p)
    t = len(qs)
    for i in range(N-2, -1, -1):
        p = ps[i]
        while len(qs) > t and cross3(qs[-1], qs[-2], p) > 0:
            qs.pop()
        qs.append(p)
    return qs

def tup_to_key(tup):
    x, y = tup
    x += 10 ** 7
    y += 10 ** 7
    x *= 10 ** 8
    return x + y

def key_to_tup(key):
    x, y = divmod(key, 10 ** 8)
    x -= 10 ** 7
    y -= 10 ** 7
    return x, y

def angle(x1, y1, x2, y2):
    if y1 == y2:
        return pi / 2
    return atan(-(x1 - x2) / (y1 - y2))

def on_line(ps):
    st = set()
    for p, q in zip(ps, ps[1:]):
        px, py = p
        qx, qy = q
        dx, dy = qx - px, qy - py
        g = gcd(dx, dy)
        dx //= g
        dy //= g
        st.add(tup_to_key((dx, dy)))
    return len(st) == 1

if __name__ == "__main__":
    N = int(input())
    if N == 2:
        print(0.5)
        print(0.5)
        exit(0)

    res = dict()
    points = [list(map(int,input().split())) for _ in range(N)]

    # ps = sorted(points)
    # for a, b in zip(ps, ps[1:]):
    #     x1, y1 = a
    #     x2, y2 = b
    #     x, y = x2 - x1, y2 - y1
    #     print(x, y)

    if on_line(sorted(points)):
        mn = min(points)
        mx = max(points)
        res[tup_to_key(mn)] = 0.5
        res[tup_to_key(mx)] = 0.5
    else:
        ps = convex_hull(sorted(points))[:-1]
        sz = len(ps)
        for i, (x, y) in enumerate(ps):
            key = tup_to_key((x, y))
            lx, ly = ps[i - 1]
            rx, ry = ps[(i + 1) % sz]
            la = angle(x, y, lx, ly)
            ra = angle(x, y, rx, ry)
            a = ra - la
            # a = min(a, 2 * pi - a)
            # print(x, y)
            # print(degrees(la), degrees(ra))
            if a < 0:
                a += pi
            # print(la, ra)
            res[key] = a / (2 * pi)

    for tup in points:
        key = tup_to_key(tup)
        if key in res:
            print(res[key])
        else:
            print(0)

Submission status: Accepted

Solution code:
import sys
input = lambda : sys.stdin.readline().rstrip()
sys.setrecursionlimit(max(1000, 10**9))
write = lambda x: sys.stdout.write(x+"\n")


n = int(input())
xy = [None]*n
for i in range(n):
    xy[i] = tuple(map(int, input().split()))
import cmath
ans = [0]*n
ps = [[] for _ in range(n)]
for i in range(n):
    for j in range(n):
        if i==j:
            continue
        v = (xy[i][0]-xy[j][0]) + (xy[i][1]-xy[j][1])*1j
        ps[i].append(cmath.polar(v)[1])
pi = cmath.pi
def sub(p):
    """p: 偏角のリスト
    """
    l = [[-pi, pi]]
    for pp in p:
        lb = pp-pi/2
        ub = pp+pi/2
        if lb<-pi:
            ll = [[lb+2*pi, pi], [-pi, ub]]
        elif ub>pi:
            ll = [[lb, pi], [-pi, ub-2*pi]]
        else:
            ll = [[lb,ub]]
        nl = []
        for i,item1 in enumerate(l):
            for j,item2 in enumerate(ll):
                item = [max(item1[0], item2[0]), min(item1[1], item2[1])]
                if item[0]<item[1]:
                    nl.append(item)
        l = nl
    return l
for i,p in enumerate(ps):
    lus = sub(p)
    for lu in lus:
        if lu[0]>lu[1]:
            ans[i] += 0
        else:
            ans[i] += (lu[1]-lu[0]) / (2*pi)
write("\n".join(map(str, ans)))

Submission status: Accepted

Solution code:
import copy
import numpy as np

def det(p, q):
    return p[0]*q[1] - p[1]*q[0]
def sub(p, q):
    return (p[0]-q[0], p[1]-q[1])
def get_convex_hull(points):
    n = len(points)
    points.sort()
    size_convex_hull = 0
    ch = []
    for i in range(n):
        while size_convex_hull > 1:
            v_cur = sub(ch[-1], ch[-2])
            v_new = sub(points[i], ch[-2])
            if det(v_cur, v_new) > 0:
                break
            size_convex_hull -= 1
            ch.pop()
        ch.append(points[i])
        size_convex_hull += 1

    t = size_convex_hull
    for i in range(n-2, -1, -1):
        while size_convex_hull > t:
            v_cur = sub(ch[-1], ch[-2])
            v_new = sub(points[i], ch[-2])
            if det(v_cur, v_new) > 0:
                break
            size_convex_hull -= 1
            ch.pop()
        ch.append(points[i])
        size_convex_hull += 1

    return ch[:-1]

N,*xyf = map(int, open(0).read().split())
xy = [(xyf[i*2],xyf[i*2+1]) for i in range(N)]
xyc = copy.deepcopy(xy)
ans = [0] * (N)
ch = get_convex_hull(xyc)
chs = set(ch)
l = len(ch)
for i in range(N):
    if xy[i] in chs:
        ind = ch.index(xy[i])
        a = (ch[ind-1],ch[ind])
        b = (ch[ind],ch[(ind+1)%l])
        u = np.array([a[0][0]-a[1][0],a[1][1]-a[0][1]])
        v = np.array([b[0][0]-b[1][0],b[1][1]-b[0][1]])
        inn = np.inner(u,v)
        nor = np.linalg.norm(u) * np.linalg.norm(v)
        cos = inn / nor
        ans[i] = (np.arccos(np.clip(cos, -1.0, 1.0))/(2*np.pi))
for x in ans:
    print(x)

Submission status: Accepted

Solution code:
def examA():
    N = I()
    ans = 0
    print(ans)
    return

# 参考 https://atcoder.jp/contests/agc021/submissions/8392122
def examB():
    def norm2(vec):
        return math.sqrt(vec[0] ** 2 + vec[1] ** 2)
    # any 2 points must have different position.
    def ConvexHull(point_list):
        pos2idx = {point_list[i]: i for i in range(len(point_list))}
        y_val = defaultdict(list)
        x_list = sorted(list(set([p[0] for p in point_list])))
        for x, y in point_list:
            y_val[x].append(y)

        upper = [(x_list[0], max(y_val[x_list[0]]))]
        lower = [(x_list[0], min(y_val[x_list[0]]))]
        prev = float('inf')
        for xi in x_list[1:]:
            x0, y0 = upper[-1]
            x1, y1 = xi, max(y_val[xi])
            if (y1 - y0) / (x1 - x0) < prev:
                upper.append((x1, y1))
                prev = (y1 - y0) / (x1 - x0)
            else:
                while True:
                    x0, y0 = upper[-1]
                    if len(upper) == 1:
                        upper.append((x1, y1))
                        break
                    x00, y00 = upper[-2]
                    if (y1 - y0) / (x1 - x0) > (y1 - y00) / (x1 - x00):
                        upper.pop()
                    else:
                        prev = (y1 - y0) / (x1 - x0)
                        upper.append((x1, y1))
                        break

        prev = -float('inf')
        for xi in x_list[1:]:
            x0, y0 = lower[-1]
            x1, y1 = xi, min(y_val[xi])
            if (y1 - y0) / (x1 - x0) > prev:
                lower.append((x1, y1))
                prev = (y1 - y0) / (x1 - x0)
            else:
                while True:
                    x0, y0 = lower[-1]
                    if len(lower) == 1:
                        lower.append((x1, y1))
                        break
                    x00, y00 = lower[-2]
                    if (y1 - y0) / (x1 - x0) < (y1 - y00) / (x1 - x00):
                        lower.pop()
                    else:
                        prev = (y1 - y0) / (x1 - x0)
                        lower.append((x1, y1))
                        break

        # return upper, lower
        # return [pos2idx[xy] for xy in upper], [pos2idx[xy] for xy in lower]

        upper_idx, lower_idx = [pos2idx[xy] for xy in upper], [pos2idx[xy] for xy in lower]
        if upper_idx[-1] == lower_idx[-1]:
            upper_idx.pop()
        CH_idx = upper_idx
        CH_idx.extend(reversed(lower_idx))
        if CH_idx[0] == CH_idx[-1] and len(CH_idx) > 1:
            CH_idx.pop()
        return CH_idx

    N = I()
    P = [[]for _ in range(N)]
    D = defaultdict(int)
    for i in range(N):
        x,y = LI()
        P[i] = (x,y)
        D[(x,y)] = i
    C = ConvexHull(P)
    ans = [0]*N
    if len(C)==2:
        for c in C:
            ans[c] = 0.5
        for v in ans:
            print(v)
        return
    #print(C)
    for i,c in enumerate(C):
        s, t, u = C[i - 1], C[i], C[(i + 1) % len(C)]
        x0, y0 = P[s]
        x1, y1 = P[t]
        x2, y2 = P[u]
        vec0 = (y0 - y1, x1 - x0)
        vec1 = (y1 - y2, x2 - x1)
        ans[t] =  math.acos((vec0[0] * vec1[0] + vec0[1] * vec1[1]) / (norm2(vec0) * norm2(vec1))) / (2 * math.pi)
    for v in ans:
        print(v)
    return

def examC():
    ans = 0
    print(ans)
    return

def examD():
    S = SI()
    N = len(S)
    K = I()
    dp = [[[0 for _ in range(K+1)]for _ in range(N)]for _ in range(N)]
    for i in range(N):
        for j in range(K+1):
            dp[i][i][j] = 1
    for i in range(2,N+1):
        for k in range(K+1):
            for l,r in enumerate(range(i-1,N)):
                if S[l]==S[r]:
                    dp[l][r][k] = dp[l+1][r-1][k] + 2
                else:
                    if k>0:
                        dp[l][r][k] = max(dp[l+1][r][k],dp[l][r-1][k],dp[l+1][r-1][k-1]+2)
                    else:
                        dp[l][r][k] = max(dp[l + 1][r][k], dp[l][r - 1][k])
    ans = max(dp[0][-1])
    print(ans)
    return

def examE():
    ans = 0
    print(ans)
    return

def examF():
    ans = 0
    print(ans)
    return

def test():
    i = I()
    li = LI()
    lsi = LSI()
    si = LS()
    print(i)
    print(li)
    print(lsi)
    print(si)
    return

from decimal import Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = 10**(-12)
alphabet = [chr(ord('a') + i) for i in range(26)]

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examB()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""