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

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Numerics;
using System.Linq;
using System.Text;
using static System.Math;

class MainClass
{

    public static void Main(string[] args)
    {
        var _ = LnInts();
        var N = _[0];
        var K = _[1];
        var ps = LnInts();
        var Dice = new double[N];
        for (int i = 0; i < N; i++)
        {
            Dice[i] = (double)SumAtoB(1, ps[i]) / (double)ps[i];
        }
        var Sums = new double[N+1];
        for (int i = 0; i < N; i++)
        {
            Sums[i+1] = Sums[i] + Dice[i];
        }
        var left = 0;var right = K;
        var max = 0D;
        while (right <= N)
        {
            var sum = Sums[right] - Sums[left];
            max = sum > max ? sum : max;
            right++;
            left++;
        }
        print(max);
    }

public static long SumAtoB(long a,long b){var length = b - a + 1;return (a + b) * (length / 2) + (length % 2 == 1 ? (a + b) / 2 : 0);}
public static long PowLite(long n, long p, int mod = (int)1e9 + 7){long ans = 1;long now = n;for (; p > 0; p >>= 1){if ((p & 1) == 1){ans *= now;ans %= mod;}now *= now;now %= mod;}return ans;}
public static void print(object obj) { Console.WriteLine(obj); }
public static long Gcd(long a, long b) { if (a > b) { var r = a % b; if (r == 0) return b; else return Gcd(b, r); } else { var r = b % a; if (r == 0) return a; else return Gcd(a, r); } }
public static int Gcd(int a, int b) { if (a > b) { var r = a % b; if (r == 0) return b; else return Gcd(b, r); } else { var r = b % a; if (r == 0) return a; else return Gcd(a, r); } }
public static int[] LnInts() { var vs = Console.ReadLine().Trim().Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries); var ins = new int[vs.Length]; for (int i = 0; i < vs.Length; i++) ins[i] = int.Parse(vs[i]); return ins; }
public static int LnInt() { var vs = Console.ReadLine().Trim(); return int.Parse(vs); }
public static long[] LnLngs() { var vs = Console.ReadLine().Trim().Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries); var ins = new long[vs.Length]; for (int i = 0; i < vs.Length; i++) ins[i] = long.Parse(vs[i]); return ins; }
public static long LnLng() { var vs = Console.ReadLine().Trim(); return long.Parse(vs); }
public static string[] LnStrs() { var vs = Console.ReadLine().Trim().Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries); return vs; }
public static string LnStr() { var vs = Console.ReadLine().Trim(); return vs; }
public static List<T[]> AllPermutation<T>(params T[] array) where T : IComparable{var a = new List<T>(array).ToArray();var res = new List<T[]>();res.Add(new List<T>(a).ToArray());var n = a.Length;var next = true;while (next){next = false;int i;for (i = n - 2; i >= 0; i--){if (a[i].CompareTo(a[i + 1]) < 0) break;}if (i < 0) break;var j = n;do{j--;} while (a[i].CompareTo(a[j]) > 0);if (a[i].CompareTo(a[j]) < 0){var tmp = a[i];a[i] = a[j];a[j] = tmp;Array.Reverse(a, i + 1, n - i - 1);res.Add(new List<T>(a).ToArray());next = true;}}return res;}}