Problem:


Score : 400 points

Problem Statement

We have N dice arranged in a line from left to right. The i-th die from the left shows p_i numbers from 1 to p_i with equal probability when thrown.

We will choose K adjacent dice, throw each of them independently, and compute the sum of the numbers shown. Find the maximum possible value of the expected value of this sum.

Constraints

1 ≤ K ≤ N ≤ 200000

1 ≤ p_i ≤ 1000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
p_1 ... p_N

Output

Print the maximum possible value of the expected value of the sum of the numbers shown.

Your output will be considered correct when its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

5 3
1 2 2 4 5

Sample Output 1

7.000000000000

When we throw the third, fourth, and fifth dice from the left, the expected value of the sum of the numbers shown is 7. This is the maximum value we can achieve.

Sample Input 2

4 1
6 6 6 6

Sample Output 2

3.500000000000

Regardless of which die we choose, the expected value of the number shown is 3.5.

Sample Input 3

10 4
17 13 13 12 15 20 10 13 17 11

Sample Output 3

32.000000000000



Sample input:
5 3
1 2 2 4 5


Sample output:
7.000000000000

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
P = list(map(int, input().split()))

summation = [0] * 1001
expected = [0] * 1001
for i in range(1000):
    summation[i + 1] = summation[i] + (i + 1)

for i in range(1, 1001):
    expected[i] = summation[i] / i

S = [0] * (N + 1)
for i in range(N):
    S[i + 1] = S[i] + expected[P[i]]

ans = 0
for i in range(0, N - K + 1):
    tmp = S[i + K] - S[i]
    ans = max(ans, tmp)

print(ans)

Submission status: Accepted

Solution code:
N,K = map(int,input().split())
p = list(map(int,input().split()))
e = []
for i in range(N):
    x = p[i] * (p[i]+1) / (2 * p[i])
    e.append(x)
#print(e)
ans = 0
tmp = 0
for i in range(K):
    tmp += e[i]
if tmp > ans:
    ans = tmp
for i in range(N-K):
    #print(tmp)
    tmp -= e[i]
    tmp += e[i+K]
    if tmp > ans:
        ans = tmp
print(ans)

Submission status: Accepted

Solution code:
n,k = [int(x) for x in input().split()]
p = [(int(x)+1)/2.0 for x in input().split()]
m = 0
t = 0

for i in range(k):
    t += p[i]
m = t
for i in range(k,n):
    t =t + p[i] - p[i-k]
    m = max(m,t)

print(m)

Submission status: Accepted

Solution code:
n, k = [ int(v) for v in input().split() ]
dice = [ int(v) for v in input().split() ]
dice = [ (1+v)/2 for v in dice ]

s = sum(dice[:k])
dice2 = [s]

for i in range(n - k):
    s += dice[i+k] - dice[i]
    dice2.append(s)
print(max(dice2))

Submission status: Accepted

Solution code:
N,K = map(int, input().strip().split())
P = list(map(int, input().strip().split()))
P = [(p+1)/2 for p in P]

ans = sum(P[0:K])
now = ans
for i in range(1,N-K+1):
  now = now - P[i-1] + P[i+K-1]
  ans = max(ans, now)
print(ans)

Submission status: Accepted

Solution code:
N,K=map(int,input().split())
P=list(map(int,input().split()))
a=N-K
D=0
for i in range(K):
  b=(1+P[i])*P[i]/2
  D+=b/P[i]
ans=D
for i in range(a):
  b=(1+P[i])*P[i]/2
  c=(1+P[i+K])*P[i+K]/2
  D+=c/P[i+K]
  D-=b/P[i]
  ans=max(ans,D)
print(ans)

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
p = list(map(int, input().split()))

e = []
for v in p:
    if v % 2 == 0:
        e.append(((v+1)*(v//2)) / v)
    else:
        e.append(((v+1)*(v//2) + (v+1) // 2) / v)

e_sum = [0]

for i in range(len(e)):
    e_sum.append(e_sum[i] + e[i])

ans = 0

for i in range(k, len(e_sum)):
    ans = max(ans, e_sum[i] - e_sum[i-k])

print(ans)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=list(map(int,input().split()))

s=sum(p[:k])
ma = s
for i in range(n-k):
  s = s+p[k+i]-p[i]
  ma = max(ma,s)
print((ma+k)/2)

Submission status: Accepted

Solution code:
# coding: utf-8
N, K = map(int, input().split())
P = list(map(int, input().split()))
# P.sort(reverse=True)
l = []
for i in range(N):
    l.append((P[i]+1) / 2)
ans = 0
tmp = sum(l[:K-1])
p = l[0]
idx = 0
for i in range(K-1, N):
    tmp += l[i]
    # print(tmp)
    ans = max(ans, tmp)
    p = l[idx]
    tmp -= p
    idx += 1
print(ans)

Submission status: Accepted

Solution code:
a,b = map(int,input().split())
p = list(map(int,input().split()))
ans = sum(p[:b])
count = ans
for i in range(1,a-b+1):
  count = count-p[i-1]+p[i+b-1]
  if count > ans:
    ans = count
print((ans+b)/2)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
p = list(map(int, input().split()))

S = sum(p[0:K])
max_S = S
for i in range(N - K):
    S += p[i + K] - p[i]
    max_S = max(S, max_S)
print((max_S + K) / 2)

Submission status: Accepted

Solution code:
# -*- coding: utf-8 -*-
import bisect
import heapq
import math
import random
import sys
from collections import Counter, defaultdict, deque
from decimal import ROUND_CEILING, ROUND_HALF_UP, Decimal
from functools import lru_cache, reduce
from itertools import combinations, combinations_with_replacement, product, permutations
from operator import add, mul, sub

sys.setrecursionlimit(100000)
input = sys.stdin.readline
INF = 2**62-1

def read_int():
    return int(input())


def read_int_n():
    return list(map(int, input().split()))


def read_float():
    return float(input())


def read_float_n():
    return list(map(float, input().split()))


def read_str():
    return input().strip()


def read_str_n():
    return list(map(str, input().split()))


def error_print(*args):
    print(*args, file=sys.stderr)


def mt(f):
    import time

    def wrap(*args, **kwargs):
        s = time.time()
        ret = f(*args, **kwargs)
        e = time.time()

        error_print(e - s, 'sec')
        return ret

    return wrap


@mt
def slv(N, K, A):
    B = deque(A[:K])
    S = sum(A[:K])
    ms = S
    mi = 0
    for i in range(K, N):
        S -= B.popleft()
        S += A[i]
        B.append(A[i])
        if S > ms:
            ms = S
            mi = i - K + 1

    ans = 0
    for i in range(mi, mi+K):
        ans += (A[i] + 1) / 2
    return ans


def main():
    N, K = read_int_n()
    A = read_int_n()
    print(slv(N, K, A))


if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
p = list(map(int, input().split()))
p_ave = [(x+1)/2 for x in p]

ans = sum(p_ave[:K])
temp, tempmax = 0, 0

for i in range(N-K):
    temp += p_ave[i+K] - p_ave[i]
    if temp > tempmax:
        tempmax = temp

print(ans + tempmax)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
P=list(map(int,input().split()))
now=sum(P[:k])
m=now
for i in range(n-k):
    now+=P[k+i]
    now-=P[i]
    m=max(m,now)
E=(m+k)/2
print(E)

Submission status: Accepted

Solution code:
N, K, *P = map(int, open(0).read().split())

A = [0]
for i in range(1, 1001):
    A.append(A[-1] + i)
B = [0] + [A[i] / i for i in range(1, 1001)]

Q = [B[p] for p in P]
ans = [sum(Q[:K])]
for i in range(N - K):
    ans.append(ans[-1] - Q[i] + Q[i + K])
    
print(max(ans))

Submission status: Accepted

Solution code:
N,K = list(map(int, input().split()))
P = list(map(int, input().split()))

cumsum = [0] * (N+1)

for i in range(N):
  cumsum[i+1] = cumsum[i] + P[i]
  
ans = -99
for i in range(N-K+1):
  ans = max(cumsum[i+K] - cumsum[i], ans)
  
print((ans + K)/2)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
P=list(map(int,input().split()))
R=[0]*(n-k+1)
R[0]=sum(P[:k])
for i in range(n-k):
    R[i+1]=R[i]-P[i]+P[i+k]
print((max(R)+k)/2)

Submission status: Accepted

Solution code:
#coding:utf-8

def main():
    n, k = map(int, input().split())
    nums = list(map(int, input().split()))
    
    for i in range(len(nums)):
        nums[i] = (nums[i] + 1) / 2
    
    sumv = sum(nums[:k])
    maxv = sumv
    for i in range(len(nums) - k):
        f = nums[i]
        l = nums[i + k]
        sumv = sumv - f + l
        if sumv > maxv:
            maxv = sumv
    print(maxv)
main()

Submission status: Accepted

Solution code:
import numpy as np
N, K = map(int, input().split())
plist = list(map(int, input().split()))
kitaiti = np.array([(0.5*(p+1)) for p in plist])
ans = 0
val = sum(kitaiti[:K])
ans = max(ans, val)
for i in range(1, N-K+1):
    # print(i-1, i+K)
    val -= kitaiti[i-1]
    val += kitaiti[i+K-1]
    ans = max(ans, val)
print(ans)

Submission status: Accepted

Solution code:
N,K = map(int,input().split())
P = list(map(int,input().split()))
cnt = 0
for i in range(K):
    cnt += P[i]
cmax = cnt
for i in range(K,N):
    cnt = cnt+P[i]-P[i-K]
    cmax = max(cmax,cnt)
print((K+cmax)/2)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
p = list(map(int, input().split()))

l = 0
r = K

maxl = l
maxr = r

s = sum(p[l:r])
maxs = s

for i in range(N-K):
    s -= p[i]
    s += p[i+K]
    if s > maxs:
        maxl = i+1
        maxr = i+K+1
        maxs = s

res = 0
for i in range(maxl, maxr):
    res += p[i]/2+0.5

print(res)

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
p = list(map(int, input().split()))

def cumsum(s):
    n = len(s)
    cs = [0] * (n+1)
    for i in range(n):
        cs[i+1] = cs[i] + s[i]
    return cs

pc = cumsum(p)
nc = cumsum(range(1, 1001))
mx = 0
s = 0

for i in range(n-k+1):
    sm = pc[i+k] - pc[i]
    if mx < sm:
        s = i
        mx = sm

ans = 0
for i in range(k):
    x = p[s+i]
    ans += nc[x] / x

print(ans)

Submission status: Accepted

Solution code:
a = [int(i) for i in input().split()]
b = [(int(i)+1)/2 for i in input().split()]

he = sum(b[:a[1]])
maxer = he
for i in range(a[0] - a[1]):
    he -= b[i]
    he += b[i + a[1]]
    maxer = max(he, maxer)

print(maxer)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
p = list(map(int, input().split()))

A = [(p[i]+1)/2 for i in range(N)]

B = [0]
for i in range(N):
  B.append(B[-1]+A[i])

ans = 0
for i in range(K, N+1):
  ans = max(ans, B[i]-B[i-K])

print(ans)

Submission status: Accepted

Solution code:
N,K = map(int,input().split())
p = list(map(int,input().split()))
ex = [p[i] + 1 for i in range(N)]
total = [0]
for i in range(N):
	total.append(total[-1] + ex[i])
m = [total[i] - total[i-K] for i in range(K, N+1)]
print(max(m)/2)

Submission status: Accepted

Solution code:
import sys
def input():
    return sys.stdin.readline()[:-1]
N,K=map(int,input().split())
p=list(map(int, input().split()))
t=sum(p[:K])
M=(t+K)/2
for i in range(N-K):
    t-=p[i]
    t+=p[i+K]
    M=max(M,(t+K)/2)
print(M)

Submission status: Accepted

Solution code:
n,k = map(int,input().split())
p = list(map(int,input().split()))

tmp = sum(p[0:k])
#print(tmp)
ans = tmp

for i in range(k,n):
   tmp -= p[i-k]
   tmp += p[i]
   ans = max(tmp,ans)

print(0.5*k+ans/2)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=[int(i) for i in input().split()]
ans=chk=0
w=[0]*(n+1)
for i in range(n):
    w[i+1]=w[i]+p[i]
for i in range(k,n+1):
    chk=max(chk,w[i]-w[i-k])
print((chk+k)/2)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
P = list(map(int, input().split()))
P = list(map(lambda x : (x + 1) / 2, P))
cur = sum(P[0 : K])
ans = cur

for i in range(K, N):
	cur -= P[i - K]
	cur += P[i]
	ans = max(ans, cur)
	
print(ans)

Submission status: Accepted

Solution code:
N,K=map(int,input().split())
p=list(map(int,input().split()))
ans=sum(p[:K])
cnt=sum(p[:K])
for i in range(N-K):
    cnt+=p[K+i]
    cnt-=p[i]
    if cnt>ans:
        ans=cnt
print(K+(ans-K)/2)

Submission status: Accepted

Solution code:
N,K=map(int,input().split())
p=list(map(int,input().split()))

x=[0]*N
y=[0]*(N+1-K)

for i in range(N):
  x[i]=0.5*(p[i]+1)
for i in range(K):
  y[0]+=x[i]
for i in range(N+1-K):
  if i!=0:
    y[i]=y[i-1]-x[i-1]+x[i-1+K]
print(max(y))

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
p = list(map(int, input().split()))
a = []
for i in p:
    a.append((1 + i) / 2)

buf=0
for i in range(k):
    buf+=a[i]

ans = buf

for i in range(k, n):
    buf += a[i]
    buf -= a[i - k]
    ans = max(ans, buf)

print(ans)

Submission status: Accepted

Solution code:


def main():
    N, K = [int(s) for s in input().split()]
    p = [int(s) for s in input().split()]
    s = sum(p[:K])
    res = s
    for i in range(1,N):
        if i+K > N:
            break
        s = s + p[i+K-1]
        s = s - p[i-1]
        res = max(s,res)
    print((res+K)/2)

main()

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
P = list(map(int, input().split()))
S = [0] * (n+1)
for i in range(n):
    S[i+1] = S[i] + (P[i] + 1) / 2

ans = 0
for i in range(k,n+1):
    ans = max(ans, S[i] - S[i-k])
print(ans)

Submission status: Accepted

Solution code:
''' بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ '''
#codeforces
gi = lambda : list(map(int,input().split()))
n, k = gi()
a = gi()
ans = 0
temp = 0
for j in range(k):
	ans += (a[j] * (a[j] + 1)) / (2 * a[j])
temp = ans
for j in range(k, n):
	temp -= (a[j - k] * (a[j - k] + 1)) / (2 * a[j - k])
	temp += (a[j] * (a[j] + 1)) / (2 * a[j])
	ans = max(ans, temp)
print(ans)

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
p = list(map(int, input().split()))
kp = []
for i in range(n):
  kp.append((p[i]+1)/2)
sumkp = []
sumkpi = 0
for i in range(n):
  sumkpi += kp[i]
  sumkp.append(sumkpi)
maxans = -1
maxans = sumkp[k-1]
for i in range(1, n-k+1):
  maxans = max(maxans, sumkp[i+k-1] - sumkp[i-1])
print(maxans)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
A = [int(x) for x in input().split()]
ans_kouho = sum(A[:K])
now_kouho = sum(A[:K])
for i in range(K, N):
    # print(ans_kouho)
    now_kouho = now_kouho-A[i-K]+A[i]
    ans_kouho = max(ans_kouho, now_kouho)
    # print(ans_kouho)
ans = (ans_kouho+K)/2
print(ans)

Submission status: Accepted

Solution code:




N, K = map(int, input().split())

P = list(map(int, input().split()))

dice = [0] * 1001
for i in range(1,1001):
    dice[i] = dice[i-1] + i

ans = 0
tmp = 0
for i in range(N):
    if i < K-1:
        tmp += dice[P[i]] / P[i]
        #print(i, tmp)
    elif i == K-1:
        tmp += dice[P[i]] / P[i]
        #print(i, tmp)
        ans = max(ans, tmp)
    else:
        tmp += dice[P[i]] / P[i]
        tmp -= dice[P[i-K]] / P[i-K]
        #print(i, tmp)
        ans = max(ans, tmp)
        
        

print(ans)

Submission status: Accepted

Solution code:
n, k = map(int, input().split())
p = list(map(int, input().split()))
q = [0]
acc = 0
for i in p:
    acc += (1+i)/2
    q.append(acc)

ma = 0
for i in range(n-k+1):
    ma = max(q[i+k] - q[i], ma)
print(ma)

Submission status: Accepted

Solution code:
def main(sample_file = ''):

    """ convenient functions
    # for i, a in enumerate(iterable)
    # q, mod = divmod(a, b)
    # divmod(x, y) returns the tuple (x//y, x%y)
    # Higher-order function: reduce(operator.mul, xyz_count, 1)
    # manage median(s) using two heapq https://atcoder.jp/contests/abc127/tasks/abc127_f
    """

    """convenient decorator
    # @functools.lru_cache():
    # to facilitate use of recursive function
        # ex:
        # from functools import lru_cache
        # import sys
        # sys.setrecursionlimit(10**9)
        # @lru_cache(maxsize=None)
        # def fib(n):
        #     if n < 2:
        #         return n
        #     return fib(n-1) + fib(n-2)
        # print(fib(1000))
    """
    
    # import numpy as np
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product # https://docs.python.org/ja/3/library/itertools.html
    # accumulate() returns iterator! to get list: list(accumulate())
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  # prime factor
        b, e = 2, 0  # base, exponent
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):   
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        # TODO: How should I do when n - r is negative?
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict # https://docs.python.org/ja/3/library/collections.html#collections.deque
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest # https://docs.python.org/ja/3/library/heapq.html
    from copy import deepcopy, copy # https://docs.python.org/ja/3/library/copy.html
    import operator
    from operator import itemgetter #sort 
    # ex1: List.sort(key=itemgetter(1))
    # ex2: sorted(tuples, key=itemgetter(1,2))
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd # Deprecated since version 3.5: Use math.gcd() instead.
    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)
    
    # first create factorial_list
    # fac_list = mod_factorial_list(n)
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    modpow = lambda a, n, p = MOD: pow(a, n, p) # Recursive function in python is slow!
    def modinv(a, p = MOD):
        # evaluate reciprocal using Fermat's little theorem:
        # a**(p-1) is identical to 1 (mod p) when a and p is coprime
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        # fac_list = modfactorial_list(100)
        # print(modcomb(100, 5, modfactorial_list(100)))
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something
        >>  else:
        >>      do something
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    # Graph: https://en.wikipedia.org/wiki/Directed_graph
    # Bellman-Ford: O(|V||E|). Use this if there exists an edge with negative length in the graph
    # After N steps, the shortest path has converded if there doesn't exist an cycle of edges with negative
    # Watch out: d[N] == d[2*N] doesn't necessarily mean the graph doesn't have negative cycle
    # ref: https://www.youtube.com/watch?v=1Z6ofKN03_Y

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    """ initialize variables and set inputs
    # initialize variables
        # to initialize list, use [0] * n
        # to initialize two dimentional array:
            # ex) [[0] * N for _ in range(N)]
            # ex2) dp = [[0] * (N+1) for _ in range(W*2)]
    # set inputs
        # put inputs between specific values (almost as quickly)
        # ex) S = [-INF] + [int(r()) for _ in range(A)] + [INF]
        # open(0).read() is sometimes useful:
        # ex) n, m, *x = map(int, open(0).read().split())
        #     min(x[::2]) - max(x[1::2])
        # ex2) *x, = map(int, open(0).read().split())
        #     don't forget to add comma after *x if only one variable is used
    # preprocessing
        # transpose = [x for x in zip(*data)]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [(1, 4, 7), (2, 5, 8), (3, 6, 9)]
        # flat = [flatten for inner in data for flatten in inner]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [1, 2, 3, 4, 5, 6, 7, 8, 9]
    # calculate and output
        # output pattern
        # ex1) print(*l) => when l = [2, 5, 6], printed 2 5 6
    """

    # functions to read input
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    r_float = lambda: float(r())
    R = lambda: list(map(int, r().split()))
    R_str = lambda: r().split()
    R_map = lambda: map(int, r().split())
    R_float = lambda: list(map(float, r().split()))
    R_tuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    # single int: int(r())
    # single string: r()
    # single float: float(r())
    # line int: R()
    # line string: r().split()
    # line (str, int, int): [j if i == 0 else int(j) for i, j in enumerate(r().split())]
    # lines int: [R() for _ in range(n)]
    """

    # for test
    if sample_file:
        sys.stdin = open(sample_file)
        
    # ----------------------------------
    # main

    # S, T = R_str()
    # A, B = R()
    # U = r()
    # if S == U:
    #     print(A-1,B)
    # else:
    #     print(A, B-1)
    
    # S = r()
    # print('x'*len(S))

    # N = r_int()
    # A = R()
    # A.sort()
    # temp = -1
    # for i in A:
    #     if temp == i:
    #         print('NO')
    #         return
    #     temp = i
    # print('YES')

    N, K = R()
    P = R()
    Q = [(1+p)/2 for p in P]
    now = sum(Q[:K])
    ans = now
    for i in range(0,N-K):
        ans = max(ans, now)
        now = now - Q[i] + Q[i+K]
    else:
        ans = max(ans, now)
    print(ans)
    # to finish code, use return instead of exit()
    # end of main
    # ----------------------------------

    """memo: how to use defaultdict of list
    # initialize
    Dic = defaultdict(list)
    # append / extend
    Dic[x].append(y)
    # three methods for loop: keys(), values(), items()
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    # to make binary digits text
    >>> a = 5
    >>> bin_str_a = format(a, '#06b')
    >>> print(bin_str_a)
    0b0101 # first 2 strings (='0b') indicates it is binary
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
def inpl():
    return list(map(int, input().split()))


N, K = inpl()
P = inpl()

cumP = [0]
for p in P:
    cumP.append(cumP[-1] + p)

max_s = 0
for i in range(N - K + 1):
    max_s = max(max_s, cumP[i + K] - cumP[i])

print((max_s + K) / 2)

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
p = list(map(int, input().split()))

p_K = []
p_K.append(sum(p[0:K]))

for i in range(N-K):
    p_K.append(p_K[i] - p[i] + p[i+K])

print((max(p_K) + (K)) / 2)

Submission status: Accepted

Solution code:
def get_E(p):
    return sum(range(1, p + 1)) / p


N, K = map(int, input().split())
P = [int(x) for x in input().split()]
E = [get_E(p) for p in P]

left = 0
cur = 0
res = 0
for right in range(N):
    cur += E[right]
    while right - left + 1 > K:
        cur -= E[left]
        left += 1
    if right - left + 1 == K:
        res = max(res, cur)
print(res)