AtCoder Grand Contest 039 - Incenters

Given are N points on the circumference of a circle centered at (0,0) in an xy-plane. The coordinates of the i-th point are (\cos(\frac{2\pi T_i}{L}),\sin(\frac{2\pi T_i}{L})).

Three distinct points will be chosen uniformly at random from these N points. Find the expected x- and y-coordinates of the center of the circle inscribed in the triangle formed by the chosen points.

Constraints

* 3 \leq N \leq 3000
* N \leq L \leq 10^9
* 0 \leq T_i \leq L-1
* T_i<T_{i+1}
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N L
T_1
:
T_N


Output

Print the expected x- and y-coordinates of the center of the circle inscribed in the triangle formed by the chosen points. Your output will be considered correct when the absolute or relative error is at most 10^{-9}.

Examples

Input

3 4
0
1
3


Output

0.414213562373095 -0.000000000000000


Input

4 8
1
3
5
6


Output

-0.229401949926902 -0.153281482438188


Input

10 100
2
11
35
42
54
69
89
91
93
99


Output

0.352886583546338 -0.109065017701873

Source: atcoder | Difficulty: UNKNOWN_DIFFICULTY


Solutions:
from math import cos, sin, pi
from itertools import combinations
(N, L) = map(int, input().split())
T = [int(input()) for _ in range(N)]
PI2oL = 2 * pi / L
(csum, ssum) = (0, 0)
count = 0
for (i, j) in combinations(range(N), 2):
	count += 1
	(mid1, num2) = ((T[i] + T[j]) / 2, j - i - 1)
	(mid2, num1) = ((mid1 + L / 2) % L, N - num2 - 2)
	csum += cos(PI2oL * mid1) * num1 + cos(PI2oL * mid2) * num2
	ssum += sin(PI2oL * mid1) * num1 + sin(PI2oL * mid2) * num2
print(csum / count / (N - 2) * 3, ssum / count / (N - 2) * 3)


import math
(N, L) = list(map(int, input().split()))
T = [int(input()) for _ in range(N)]
(x, y) = (0, 0)
for i in range(N - 1):
	for j in range(i + 1, N):
		a = (T[i] + T[j]) / 2
		b = a + L / 2
		t = j - i - 1
		k = N - 2 - t
		x += k * math.cos(2 * math.pi * a / L)
		x += t * math.cos(2 * math.pi * b / L)
		y += k * math.sin(2 * math.pi * a / L)
		y += t * math.sin(2 * math.pi * b / L)
x /= N * (N - 1) * (N - 2) / 3 / 2
y /= N * (N - 1) * (N - 2) / 3 / 2
print(x, y)


from math import sin, cos, pi
(N, L) = map(int, input().split())
S = N * (N - 1) * (N - 2) / 6
T = [int(input()) * pi / L for _ in range(N)]
print(sum([sum([cos(T[i] + T[j]) * (N + 2 * (i - j)) for j in range(i + 1, N)]) for i in range(N)]) / S, sum([sum([sin(T[i] + T[j]) * (N + 2 * (i - j)) for j in range(i + 1, N)]) for i in range(N)]) / S)


import math
(N, L) = map(int, input().split())
T = [int(input()) for i in range(N)]
X = 0
Y = 0
for i in range(N):
	for j in range(i + 1, N):
		angle = math.pi * (T[j] + T[i]) / L
		X += math.cos(angle) * (N - 2 - (j - i - 1) * 2)
		Y += math.sin(angle) * (N - 2 - (j - i - 1) * 2)
A = N * (N - 1) * (N - 2) // 6
print(X / A, Y / A)


import sys
sys.setrecursionlimit(2147483647)
INF = float('inf')
MOD = 10 ** 9 + 7
input = lambda : sys.stdin.readline().rstrip()

def resolve():
	(n, l) = map(int, input().split())
	T = [int(input()) for _ in range(n)]
	import math, cmath
	pi = math.acos(-1)
	ans = 0
	for i in range(n):
		for j in range(i + 1, n):
			ans += cmath.rect(1, (T[i] + T[j]) * pi / l) * (n - (j - i + 1))
			ans += -cmath.rect(1, (T[i] + T[j]) * pi / l) * (j - i - 1)
	ans /= n * (n - 1) * (n - 2) / 6
	print(ans.real, ans.imag)
resolve()


import math
(N, L) = map(int, input().split())
T = [int(input()) for i in range(N)]
ANSX = ANSY = 0
for i in range(N - 1):
	for j in range(i + 1, N):
		ANSX += math.cos((T[i] + T[j]) * math.pi / L) * (N - (j - i + 1))
		ANSY += math.sin((T[i] + T[j]) * math.pi / L) * (N - (j - i + 1))
		ANSX += math.cos((T[i] + T[j]) * math.pi / L + math.pi) * (j - i - 1)
		ANSY += math.sin((T[i] + T[j]) * math.pi / L + math.pi) * (j - i - 1)
print(ANSX / (N * (N - 1) * (N - 2) // 6), ANSY / (N * (N - 1) * (N - 2) // 6))


import sys
from math import *
from itertools import *
input = sys.stdin.readline
(N, L) = list(map(int, input().split()))
T = [int(input().strip()) for i in range(N)]
T.sort()
pairN = factorial(N) / (factorial(N - 2) * factorial(2))
tripleN = factorial(N) / (factorial(N - 3) * factorial(3))
ans = [0, 0]
for (i0, i1) in combinations(range(N), 2):
	(t0, t1) = (T[i0], T[i1])
	tn = (t0 + t1) / 2
	to = tn + L / 2.0
	cn = (cos(2.0 * pi * tn / L), sin(2.0 * pi * tn / L))
	co = (cos(2.0 * pi * to / L), sin(2.0 * pi * to / L))
	onum = i1 - i0 - 1
	nnum = N - 2 - onum
	ans[0] += (cn[0] * nnum + co[0] * onum) / tripleN
	ans[1] += (cn[1] * nnum + co[1] * onum) / tripleN
print(' '.join(map(str, ans)))


from math import cos, sin, pi
(n, L) = map(int, input().split())
t = [int(input()) for _ in range(n)]
ans = [0, 0]
for i in range(n):
	for j in range(i + 1, n):
		k = n - 1 - j + i - (j - i - 1)
		ans[0] += cos(pi * (t[i] + t[j]) / L) * k
		ans[1] += sin(pi * (t[i] + t[j]) / L) * k
for i in range(2):
	ans[i] /= n * (n - 1) * (n - 2) / 6
print(*ans, sep=' ')


from math import sqrt, sin, cos, pi
import time

def main():
	(N, L) = map(int, input().split())
	t = [int(input()) for i in range(N)]
	comb = N * (N - 1) * (N - 2) / 6
	(ans_X, ans_Y) = (0, 0)
	for i in range(N - 1):
		for j in range(i + 1, N):
			theta = pi * (t[j] + t[i]) / L
			I_X = cos(theta) * (N - 2 * (j - i))
			I_Y = sin(theta) * (N - 2 * (j - i))
			ans_X += I_X / comb
			ans_Y += I_Y / comb
	print(ans_X, ans_Y)
main()