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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Runtime.CompilerServices;
using System.Text;

namespace AtCoder
{
    static class Program_D
    {
        class Solver
        {
            
            public void Solve()
            {
                var nk = Ints();
                var n = nk[0];
                var k = nk[1];
                int[] p = Ints();
                float[] exp = new float[p.Length];
                for(int i=0;i<p.Length;++i)
                {
                    exp[i] = (float)((1 + p[i]) * p[i]) / (2 * p[i]);
                    //Console.WriteLine(string.Format("{0} : {1} : {2}",i,p[i],exp[i]));
                }

                float expSum=0;
                // setup
                for(int i =0; i < k; ++i)
                {
                    expSum += exp[i];
                }
                var expMax = expSum;
                // iteration
                for(int i=k; i <n;++i)
                {
                    var diff = exp[i] - exp[i - k];
                    expSum = expSum + diff;
                    //Console.WriteLine($"exp[{i}] - exp[{i - k}] = {diff}, expSum = {expSum}");
                    expMax = Math.Max(expMax, expSum);

                }
                Console.WriteLine(expMax);
            }



            // ----------------------------------------------------------------------------

            public int Int()
            {
                return int.Parse(Console.ReadLine());
            }
            public int[] Ints()
            {
                return Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();
            }
            public long[] Longs()
            {
                return Console.ReadLine().Split(' ').Select(s => long.Parse(s)).ToArray();
            }
            public int[][] MultiInts(int num)
            {
                var result = new int[num][];
                for (int i = 0; i < num; ++i)
                {
                    result[i] = Ints();
                }
                return result;
            }
            public T[] MultiItems<T>(int num, Func<string, T> parser)
            {
                var result = new T[num];
                for (int i = 0; i < num; ++i)
                {
                    result[i] = parser(Console.ReadLine());
                }
                return result;
            }
        }



        static void Main(string[] args)
        {
            new Solver().Solve();
        }
    }
    struct ModInt
    {
        const int MOD = 1000000007;
        long Data;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static implicit operator long(ModInt modInt) => modInt.Data;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static implicit operator ModInt(long val) => new ModInt(val);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator +(ModInt a, ModInt b)
        {
            long res = a.Data + b.Data;
            return new ModInt() { Data = res >= MOD ? res - MOD : res };
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator +(ModInt a, long b) => a.Data + b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator -(ModInt a, ModInt b)
        {
            long res = a.Data - b.Data;
            return new ModInt() { Data = res < 0 ? res + MOD : res };
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator -(ModInt a, long b) => a.Data - b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % MOD };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator /(ModInt a, ModInt b) => a.Data * GetInverse(b);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string ToString() => Data.ToString();

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long GetInverse(long a)
        {
            long div;
            long p = MOD;
            long x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return x2;
                div = a / p;
                x1 -= x2 * div;
                y1 -= y2 * div;
                a %= p;
                if (a == 1) return x1;
                div = p / a;
                x2 -= x1 * div;
                y2 -= y1 * div;
                p %= a;
            }
        }
    }

}