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.IO;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using static Ha2ne2Util.Cin;
using static Ha2ne2Util.Util;
using static System.Console;
using static System.Math;
using System.Collections;
using Pair = System.Tuple<long, long>;
using Ha2ne2Util;

/// <summary>
/// ABC154
/// 
/// </summary>
namespace ABC154
{
    public class Program
    {
        public static void Main(string[] args)
        {
            // 1回目 R981
            // A(); 5m
            // B(); 6m
            // C(); 8m
            // D(); 32m
            // E(); だめだった
            // F();

            // 2回目
            // A(); 3m
            // B(); 4m
            // C(); 6m
            D();
            // E();
            // F();
        }
        public static void F()
        {

        }

        public static void E()
        {

        }

        public static void D()
        {
            long N = rl;
            long K = rl;
            long[] A = rla;
            double[] B = new double[N];
            for (int i = 0; i < N; i++)
            {
                B[i] = (A[i] + 1) / 2.0;
            }

            double[] acc = new double[N+1];
            for (int i = 0; i < N; i++)
            {
                acc[i + 1] = acc[i] + B[i];
            }

            double ans = 0;
            for (int i = 0; i + K <= N; i++)
            {
                double res = acc[i + K] - acc[i];
                ans = Max(ans, res);
            }

            Console.WriteLine(ans);
        }
        public static void C()
        {
            long N = rl;
            long[] A = rla;
            if (A.Length == A.Distinct().Count())
            {
                Console.WriteLine("YES");
            }
            else
            {
                Console.WriteLine("NO");
            }
        }

        public static void B()
        {
            string s = rs;
            Console.WriteLine(new string('x', s.Length));
        }

        public static void A()
        {
            string s = rs;
            string t = rs;
            long a = rl;
            long b = rl;
            string u = rs;

            if (s == u)
            {                
                Console.WriteLine((a-1) + " " + b);
            }
            else
            {
                Console.WriteLine(a + " " + (b-1));
            }
        }

        //public static void F()
        //{

        //}

        //public static void E()
        //{
        //    string N = rs;
        //    long K = rl;
        //    long len = N.Length;
        //    long[,] dp = new long[len + 1, 5];
        //    dp[0, 0] = 1;
        //    long[,] dp2 = new long[len + 1, 5];
        //    dp2[0, 0] = 0;

        //    for (int i = 0; i < len; i++)
        //    {
        //        int c = N[i] - '0';

        //        // 未満テーブルから未満テーブルへの遷移
        //        for (int j = 0; j < 4; j++)
        //        {
        //            dp2[i + 1, j] += dp2[i, j];
        //            dp2[i + 1, j + 1] += dp2[i, j] * 9;
        //        }

        //        if (c != 0)
        //        {
        //            for (int j = 0; j < 4; j++)
        //            {
        //                // 同値テーブルの更新（1が右下に動く）
        //                dp[i + 1, j + 1] = dp[i, j];

        //                // 同値テーブルから未満テーブルへの遷移
        //                dp2[i + 1, j] += dp[i, j];
        //                dp2[i + 1, j + 1] += dp[i, j] * (c - 1);
        //            }
        //        }
        //        else
        //        {
        //            // 同値テーブルの更新（1が右に動く）
        //            for (int j = 0; j < 4; j++)
        //            {
        //                dp[i + 1, j] = dp[i, j];
        //            }
        //        }
        //    }

        //    long ans = dp2[len,K] + dp[len,K];
        //    Console.WriteLine(ans);
        //}

        //public static void D()
        //{
        //    long N = rl;
        //    long K = rl;
        //    long[] A = rla;

        //    long[] range = new long[1001];
        //    for (int i = 0; i <= 1000; i++)
        //    {
        //        range[i] = i+1;
        //    }

        //    long[] acc = new long[1002];
        //    for (int i = 0; i < 1001; i++)
        //    {
        //        acc[i + 1] = acc[i] + range[i];
        //    }

        //    double[] A2 = new double[N];
        //    for (int i = 0; i < N; i++)
        //    {
        //        A2[i] = acc[A[i]] / (double)A[i];
        //    }

        //    double[] A2acc = new double[N + 1];
        //    for (int i = 0; i < N; i++)
        //    {
        //        A2acc[i + 1] = A2[i] + A2acc[i];
        //    }

        //    double ans = 0;
        //    for (int i = 0; i + K <= N; i++)
        //    {
        //        ans = Max(ans, A2acc[i + K] - A2acc[i]);
        //    }

        //    Console.WriteLine(ans);
        //}

        //public static void C()
        //{
        //    long n = rl;
        //    long[] a = rla;
        //    if(a.Count() == a.Distinct().Count())
        //    {
        //        Console.WriteLine("YES");
        //    }
        //    else
        //    {
        //        Console.WriteLine("NO");
        //    }
        //}

        //public static void B()
        //{
        //    string s = rs;
        //    string ans = new string('x', s.Length);
        //    Console.WriteLine(ans);
        //}

        //public static void A()
        //{            
        //    string s = rs;
        //    string t = rs;
        //    long sx = rl;
        //    long tx = rl;
        //    string u = rs;
        //    if(s == u)
        //    {
        //        Console.WriteLine((sx-1).ToString() + " " + tx);
        //    }
        //    else
        //    {
        //        Console.WriteLine(sx.ToString() + " " + (tx - 1).ToString());
        //    }
        //}
    }
}

namespace Ha2ne2Util
{
    public static class Cin
    {
        public static long rl => ReadLong();
        public static long[] rla => ReadLongArray();
        public static double rd => ReadDouble();
        public static double[] rda => ReadDoubleArray();
        public static string rs => ReadString();

        private static string[] Tokens { get; set; }

        private static int Pointer { get; set; }

        public static string Next()
        {
            if (Tokens == null || Tokens.Length <= Pointer)
            {
                Tokens = Console.ReadLine().Split(' ');
                Pointer = 0;
            }

            return Tokens[Pointer++];
        }

        public static int ReadInt()
        {
            return int.Parse(Next());
        }

        public static long ReadLong()
        {
            return long.Parse(Next());
        }

        public static double ReadDouble()
        {
            return double.Parse(Next());
        }

        public static string ReadString()
        {
            return Next();
        }

        public static int[] ReadIntArray()
        {
            Tokens = null;
            return Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
        }

        public static long[] ReadLongArray()
        {
            Tokens = null;
            return Array.ConvertAll(Console.ReadLine().Split(' '), long.Parse);
        }

        public static double[] ReadDoubleArray()
        {
            Tokens = null;
            return Array.ConvertAll(Console.ReadLine().Split(' '), double.Parse);
        }

        public static void ReadCols(out long[] a, out long[] b, long N)
        {
            a = new long[N];
            b = new long[N];
            for (int i = 0; i < N; i++)
            {
                a[i] = ReadLong();
                b[i] = ReadLong();
            }
        }

        public static void ReadCols(out long[] a, out long[] b, out long[] c, long N)
        {
            a = new long[N];
            b = new long[N];
            c = new long[N];
            for (int i = 0; i < N; i++)
            {
                a[i] = ReadLong();
                b[i] = ReadLong();
                c[i] = ReadLong();
            }
        }

        public static int[][] ReadIntTable(int n)
        {
            Tokens = null;
            int[][] ret = new int[n][];
            for (int i = 0; i < n; i++)
            {
                ret[i] = ReadIntArray();
            }

            return ret;
        }

        public static long[][] ReadLongTable(long n)
        {
            Tokens = null;
            long[][] ret = new long[n][];
            for (long i = 0; i < n; i++)
            {
                ret[i] = ReadLongArray();
            }

            return ret;
        }
    }

    /// <summary>
    /// ユーティリティー
    /// </summary>
    public static class Util
    {
        // 10^17
        public static readonly long INF = (long)1e17;

        // 10^9 + 7
        public readonly static long MOD = (long)1e9 + 7;

        /// <summary>
        /// Console.WriteLineの自動フラッシュをしないようにする
        /// </summary>
        public static void DontAutoFlush()
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
        }

        public static void Flush()
        {
            Console.Out.Flush();
        }

        public static IEnumerable<int> Range(int end)
        {
            for (int i = 0; i < end; i++)
            {
                yield return i;
            }
        }

        public static IEnumerable<int> Range(int from, int end)
        {
            for (int i = from; i < end; i++)
            {
                yield return i;
            }
        }

        public static IEnumerable<int> Range(int from, int end, int step)
        {
            for (int i = from; i < end; i += step)
            {
                yield return i;
            }
        }

        /// <summary>
        /// ソートをして結果を返します。
        /// 破壊的関数です。
        /// </summary>
        /// <typeparam name="T"></typeparam>
        /// <param name="array"></param>
        /// <returns></returns>
        public static T[] Sort<T>(T[] array)
        {
            Array.Sort<T>(array);
            return array;
        }

        /// <summary>
        /// 降順ソートをして結果を返します。
        /// 破壊的関数です。
        /// </summary>
        /// <typeparam name="T"></typeparam>
        /// <param name="array"></param>
        /// <returns></returns>
        public static T[] SortDecend<T>(T[] array)
        {
            Array.Sort<T>(array);
            Array.Reverse(array);
            return array;
        }

        /// <summary>
        /// 引数aと引数bの値を入れ替えます
        /// </summary>
        /// <typeparam name="T"></typeparam>
        /// <param name="a"></param>
        /// <param name="b"></param>
        public static void Swap<T>(ref T a, ref T b)
        {
            T _a = a;
            a = b;
            b = _a;
        }

        /// <summary>
        /// 先頭と末尾に要素を増やした新しい配列を返します。
        /// </summary>
        /// <typeparam name="T"></typeparam>
        /// <param name="array"></param>
        /// <returns></returns>
        public static T[] AddHeadAndTail<T>(T[] array)
        {
            int len = array.Length;
            T[] res = new T[len + 2];
            Array.Copy(array, 0, res, 1, len);
            return res;
        }

        /// <summary>
        /// 昇順ソート済みの配列を2分探索します。
        /// 要素が見つからなかった場合はnより大きい数値の中で最小の数値のインデックスを返す。
        /// </summary>
        /// <param name="array"></param>
        /// <param name="n"></param>
        /// <returns></returns>
        public static int BinarySearch(long[] array, long n)
        {
            int left = 0;
            int right = array.Length - 1;

            while (left <= right)
            {
                int mid = (right - left) / 2 + left;

                if (array[mid] == n)
                {
                    return mid;
                }
                else if (n < array[mid])
                {
                    right = mid - 1;
                }
                else
                {
                    left = mid + 1;
                }
            }

            return left;
        }

        public static int GCD(int a, int b)
        {
            return (b == 0) ? a : GCD(b, a % b);
        }

        /// <summary>
        /// nの約数の配列を返します。
        /// 配列は昇順ソートされています。
        /// </summary>
        /// <param name="n"></param>
        /// <returns></returns>
        public static int[] GetDivisor(int n)
        {
            double sqrt = Math.Sqrt(n);

            List<int> divisor = new List<int>();

            for (int i = 1; i <= sqrt; i++)
            {
                if (n % i == 0)
                {
                    divisor.Add(i);
                    int d = n / i;
                    if (d != i)
                    {
                        divisor.Add(d);
                    }
                }
            }

            return Sort(divisor.ToArray());
        }

        public static void Deconstruct<T>(this T[] items, out T t0)
        {
            t0 = items.Length > 0 ? items[0] : default(T);
        }

        public static void Deconstruct<T>(this T[] items, out T t0, out T t1)
        {
            if (items.Length < 2)
                throw new ArgumentException();

            t0 = items[0];
            t1 = items[1];
        }

        public static void Deconstruct<T>(this T[] items, out T t0, out T t1, out T t2)
        {
            if (items.Length < 3)
                throw new ArgumentException();

            t0 = items[0];
            t1 = items[1];
            t2 = items[2];
        }

        public static void Deconstruct<T>(this T[] items, out T t0, out T t1, out T t2, out T t3)
        {
            if (items.Length < 4)
                throw new ArgumentException();

            t0 = items[0];
            t1 = items[1];
            t2 = items[2];
            t3 = items[3];
        }

        public static void ChMax(ref long a, long b)
        {
            if (a < b) a = b;
        }

        public static void ChMin(ref long a, long b)
        {
            if (a > b) a = b;
        }

        public static long ModAdd(long a, long b)
        {
            long res = a + b;
            if (res >= MOD)
                return res % MOD;
            return res;
        }

        public static long ModMul(long a, long b)
        {
            long res = a * b;
            if (res >= MOD)
                return res % MOD;
            return res;
        }

        public static void ChModAdd(ref long a, long b)
        {
            a += b;
            if (a >= MOD)
                a %= MOD;
        }

        public static void ChModMul(ref long a, long b)
        {
            a *= b;
            if (a >= MOD)
                a %= MOD;
        }

        public static T MinBy<T>(this IEnumerable<T> source, Func<T, long> conv)
        {
            T min = source.First();
            long minValue = long.MaxValue;
            foreach (T x in source)
            {
                long n = conv(x);
                if (n < minValue)
                {
                    min = x;
                }
            }

            return min;
        }

        public static T MaxBy<T>(this IEnumerable<T> source, Func<T, long> conv)
        {
            T min = source.First();
            long maxValue = long.MinValue;
            foreach (T x in source)
            {
                long n = conv(x);
                if (maxValue < n)
                {
                    min = x;
                }
            }

            return min;
        }

        public static void FillArray<T>(T[] array, T value)
        {
            int max = array.Length;
            for (int i = 0; i < max; i++)
            {
                array[i] = value;
            }
        }

        public static void FillArray<T>(T[,] array, T value)
        {
            int max0 = array.GetLength(0);
            int max1 = array.GetLength(1);

            for (int i = 0; i < max0; i++)
            {
                for (int j = 0; j < max1; j++)
                {
                    array[i, j] = value;
                }
            }
        }

        public static void FillArray<T>(T[,,] array, T value)
        {
            int max0 = array.GetLength(0);
            int max1 = array.GetLength(1);
            int max2 = array.GetLength(2);

            for (int i = 0; i < max0; i++)
            {
                for (int j = 0; j < max1; j++)
                {
                    for (int k = 0; k < max2; k++)
                    {
                        array[i, j, k] = value;
                    }
                }
            }
        }

        /// <summary>
        /// めぐる式二分探索
        /// okにはpredを満たすindexを、ngにはpredを満たさないindexを指定します。
        /// 引数がng＜okの場合、最小のokとなるindexをokに代入します。(lower_bound）
        /// 引数がok＜ngの場合、最大のokとなるindexをokに代入します。(upper_bound）
        /// </summary>
        /// <typeparam name="T"></typeparam>
        /// <param name="array"></param>
        /// <param name="okIndex"></param>
        /// <param name="ngIndex"></param>
        /// <param name="pred"></param>
        public static void MeguruBinarySearch<T>(
            T[] array,
            ref long okIndex,
            ref long ngIndex,
            Predicate<T> pred)
        {
            while (Math.Abs(okIndex - ngIndex) > 1)
            {
                long mid = (okIndex + ngIndex) / 2;
                if (pred(array[mid]))
                {
                    okIndex = mid;
                }
                else
                {
                    ngIndex = mid;
                }
            }
        }
    }

    public class HashMap<K, V> : Dictionary<K, V>
    {
        private V DefaltValue;
        public HashMap() { }
        public HashMap(V defaultValue)
        {
            DefaltValue = defaultValue;
        }
        new public V this[K i]
        {
            get
            {
                V v;
                return TryGetValue(i, out v) ? v : base[i] = DefaltValue;
            }
            set { base[i] = value; }
        }
    }

    /// <summary>
    /// UnionFindTree
    /// </summary>
    public class UnionFindTree
    {
        public int[] Node;

        public UnionFindTree(int N)
        {
            Node = new int[N];
            for (int i = 0; i < N; i++)
            {
                Node[i] = -1;
            }
        }

        public bool IsSameGroup(int x, int y)
        {
            return GetRoot(x) == GetRoot(y);
        }

        public bool Merge(int x, int y)
        {
            int xr = GetRoot(x);
            int yr = GetRoot(y);

            if (xr == yr)
                return false;

            // xが、大きなグループを示すようにSwapする（値が小さい方が大きなグループ）
            if (Node[xr] > Node[yr])
                Swap(ref xr, ref yr);

            // グループの数を合算する
            Node[xr] += Node[yr];

            // 根を張り替える
            Node[yr] = xr;
            return true;
        }

        public int Size(int x)
        {
            return -Node[GetRoot(x)];
        }

        public int GetRoot(int n)
        {
            if (Node[n] < 0)
            {
                return n;
            }
            else
            {
                // 根を張りなおす。
                Node[n] = GetRoot(Node[n]);
                return Node[n];
            }
        }
    }

    /// <summary>
    /// PriorityQueue
    /// </summary>
    /// <typeparam name="T"></typeparam>
    public class PriorityQueue<T>
    {
        private Comparison<T> _comparison = null;
        private int _type = 0;

        private T[] _heap;
        private int _sz = 0;

        private int _count = 0;

        /// <summary>
        /// Priority queue
        /// </summary>
        /// <param name="maxSize">max size</param>
        /// <param name="type">0: asc, 1:desc</param>
        public PriorityQueue(int maxSize, int type = 0)
        {
            _heap = new T[maxSize];
            _type = type;
        }

        public PriorityQueue(int maxSize, Comparison<T> comparison)
        {
            _heap = new T[maxSize];
            _comparison = comparison;
        }

        private int Compare(T x, T y)
        {
            return _comparison(x, y);
            //if (_comparison != null) return _comparison(x, y);
            //return _type == 0 ? x.CompareTo(y) : y.CompareTo(x);
        }

        public void Push(T x)
        {
            _count++;

            //node number
            var i = _sz++;

            while (i > 0)
            {
                //parent node number
                var p = (i - 1) / 2;

                if (Compare(_heap[p], x) <= 0) break;

                _heap[i] = _heap[p];
                i = p;
            }

            _heap[i] = x;
        }

        public T Pop()
        {
            _count--;

            T ret = _heap[0];
            T x = _heap[--_sz];

            int i = 0;
            while (i * 2 + 1 < _sz)
            {
                //children
                int a = i * 2 + 1;
                int b = i * 2 + 2;

                if (b < _sz && Compare(_heap[b], _heap[a]) < 0) a = b;

                if (Compare(_heap[a], x) >= 0) break;

                _heap[i] = _heap[a];
                i = a;
            }

            _heap[i] = x;

            return ret;
        }

        public int Count()
        {
            return _count;
        }

        public T Peek()
        {
            return _heap[0];
        }

        public bool Contains(T x)
        {
            for (int i = 0; i < _sz; i++) if (x.Equals(_heap[i])) return true;
            return false;
        }

        public void Clear()
        {
            while (this.Count() > 0) this.Pop();
        }

        public IEnumerator<T> GetEnumerator()
        {
            var ret = new List<T>();

            while (this.Count() > 0)
            {
                ret.Add(this.Pop());
            }

            foreach (var r in ret)
            {
                this.Push(r);
                yield return r;
            }
        }

        public T[] ToArray()
        {
            T[] array = new T[_sz];
            int i = 0;

            foreach (var r in this)
            {
                array[i++] = r;
            }

            return array;
        }
    }
}