Problem:
Score : 400 points

Problem Statement

One day, Niwango-kun, an employee of Dwango Co., Ltd., found an integer sequence (a_1, ..., a_N) of length N.
He is interested in properties of the sequence a.

For a nonempty contiguous subsequence a_l, ..., a_r (1 \leq l \leq r \leq N) of the sequence a, its beauty is defined as a_l + ... + a_r. Niwango-kun wants to know the maximum possible value of the bitwise AND of the beauties of K nonempty contiguous subsequences among all N(N+1)/2 nonempty contiguous subsequences. (Subsequences may share elements.)

Find the maximum possible value for him.

Constraints

2 \leq N \leq 1000

1 \leq a_i \leq 10^9

1 \leq K \leq N(N+1)/2

All numbers given in input are integers

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 ... a_N

Output

Print the answer.

Sample Input 1

4 2
2 5 2 5

Sample Output 1

12

There are 10 nonempty contiguous subsequences of a. Let us enumerate them:

contiguous subsequences starting from the first element: \{2\}, \{2, 5\}, \{2, 5, 2\}, \{2, 5, 2, 5\}

contiguous subsequences starting from the second element: \{5\}, \{5, 2\}, \{5, 2, 5\}

contiguous subsequences starting from the third element: \{2\}, \{2, 5\}

contiguous subsequences starting from the fourth element: \{5\}

(Note that even if the elements of subsequences are equal, subsequences that have different starting indices are considered to be different.)

The maximum possible bitwise AND of the beauties of two different contiguous subsequences is 12.
This can be achieved by choosing \{5, 2, 5\} (with beauty 12) and \{2, 5, 2, 5\} (with beauty 14).

Sample Input 2

8 4
9 1 8 2 7 5 6 4

Sample Output 2

32

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.Serialization.Formatters;

public static class Ex
{
    public static bool IsNullOrEmpty(this string s) { return string.IsNullOrEmpty(s); }
    public static List<string> FastSort(this List<string> s) { s.Sort(StringComparer.OrdinalIgnoreCase); return s.ToList(); }
    public static void yesno(this bool b) { Console.WriteLine(b ? "yes" : "no"); }
    public static void YesNo(this bool b) { Console.WriteLine(b ? "Yes" : "No"); }
    public static void YESNO(this bool b) { Console.WriteLine(b ? "YES" : "NO"); }
    public static int PopCount(this int bits)
    {
        bits = (bits & 0x55555555) + (bits >> 1 & 0x55555555);
        bits = (bits & 0x33333333) + (bits >> 2 & 0x33333333);
        bits = (bits & 0x0f0f0f0f) + (bits >> 4 & 0x0f0f0f0f);
        bits = (bits & 0x00ff00ff) + (bits >> 8 & 0x00ff00ff);
        return (bits & 0x0000ffff) + (bits >> 16 & 0x0000ffff);
    }
}



partial class Program
{
    public string GetStr() { return Console.ReadLine().Trim(); }
    public char GetChar() { return Console.ReadLine().Trim()[0]; }
    public int GetInt() { return int.Parse(Console.ReadLine().Trim()); }
    public long GetLong() { return long.Parse(Console.ReadLine().Trim()); }
    public double GetDouble() { return double.Parse(Console.ReadLine().Trim()); }
    public string[] GetStrArray() { return Console.ReadLine().Trim().Split(' '); }
    public string[][] GetStrArray(int N)
    {
        var res = new string[N][];
        for (int i = 0; i < N; i++) res[i] = Console.ReadLine().Trim().Split(' ');
        return res;
    }
    public int[] GetIntArray() { return Console.ReadLine().Trim().Split(' ').Select(int.Parse).ToArray(); }
    public int[][] GetIntArray(int N)
    {
        var res = new int[N][];
        for (int i = 0; i < N; i++) res[i] = Console.ReadLine().Trim().Split(' ').Select(int.Parse).ToArray();
        return res;
    }
    public long[] GetLongArray() { return Console.ReadLine().Trim().Split(' ').Select(long.Parse).ToArray(); }
    public long[][] GetLongArray(int N)
    {
        var res = new long[N][];
        for (int i = 0; i < N; i++) res[i] = Console.ReadLine().Trim().Split(' ').Select(long.Parse).ToArray();
        return res;
    }
    public char[] GetCharArray() { return Console.ReadLine().Trim().Split(' ').Select(char.Parse).ToArray(); }
    public double[] GetDoubleArray() { return Console.ReadLine().Trim().Split(' ').Select(double.Parse).ToArray(); }
    public double[][] GetDoubleArray(int N)
    {
        var res = new double[N][];
        for (int i = 0; i < N; i++) res[i] = Console.ReadLine().Trim().Split(' ').Select(double.Parse).ToArray();
        return res;
    }
    public char[][] GetGrid(int H)
    {
        var res = new char[H][];
        for (int i = 0; i < H; i++) res[i] = Console.ReadLine().Trim().ToCharArray();
        return res;
    }
    public T[] CreateArray<T>(int N, T value)
    {
        var res = new T[N];
        for (int i = 0; i < N; i++) res[i] = value;
        return res;
    }
    public T[][] CreateArray<T>(int H, int W, T value)
    {
        var res = new T[H][];
        for (int i = 0; i < H; i++)
        {
            res[i] = new T[W];
            for (int j = 0; j < W; j++) res[i][j] = value;
        }
        return res;
    }
    public T[][][] CreateArray<T>(int H, int W, int R, T value)
    {
        var res = new T[H][][];
        for (int i = 0; i < H; i++)
        {
            res[i] = new T[W][];
            for (int j = 0; j < W; j++)
            {
                res[i][j] = new T[R];
                for (int k = 0; k < R; k++) res[i][j][k] = value;
            }
        }
        return res;
    }

    Dictionary<int, List<int>> GetUnweightedAdjacencyList(int N, int M, bool isDirected, bool isNode_0indexed)
    {
        var dic = new Dictionary<int, List<int>>();
        foreach (var e in Enumerable.Range(0, N)) { dic.Add(e, new List<int>()); }
        for (int i = 0; i < M; i++)
        {
            var input = GetIntArray();
            var a = isNode_0indexed ? input[0] : input[0] - 1;
            var b = isNode_0indexed ? input[1] : input[1] - 1;
            dic[a].Add(b);
            if (isDirected == false) dic[b].Add(a);
        }
        return dic;
    }
    Dictionary<int, List<Edge>> GetWeightedAdjacencyList(int N, int M, bool isDirected, bool isNode_0indexed)
    {
        var dic = new Dictionary<int, List<Edge>>();
        foreach (var e in Enumerable.Range(0, N)) { dic.Add(e, new List<Edge>()); }
        for (int i = 0; i < M; i++)
        {
            var input = GetIntArray();
            var a = isNode_0indexed ? input[0] : input[0] - 1;
            var b = isNode_0indexed ? input[1] : input[1] - 1;
            var c = input[2];
            dic[a].Add(new Edge(b, c));
            if (isDirected == false) dic[b].Add(new Edge(a, c));
        }
        return dic;
    }

    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>() ? ct<T, int>(int.Parse(s))
                       : eq<T, long>() ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>() ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    void Multi<T>(out T a) => a = cv<T>(GetStr());
    void Multi<T, U>(out T a, out U b)
    {
        var ar = GetStrArray(); a = cv<T>(ar[0]); b = cv<U>(ar[1]);
    }
    void Multi<T, U, V>(out T a, out U b, out V c)
    {
        var ar = GetStrArray(); a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]);
    }
    void Multi<T, U, V, W>(out T a, out U b, out V c, out W d)
    {
        var ar = GetStrArray(); a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]);
    }
    void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
    {
        var ar = GetStrArray(); a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]);
    }
    void Multi<T, U, V, W, X, Y>(out T a, out U b, out V c, out W d, out X e, out Y f)
    {
        var ar = GetStrArray(); a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); f = cv<Y>(ar[5]);
    }

    int[] dx = new int[] { 1, 0, -1, 0, 1, 1, -1, -1 };
    int[] dy = new int[] { 0, 1, 0, -1, 1, -1, 1, -1 };
}

class Pair : IComparable<Pair>
{
    public Pair(int node, long cost)
    {
        Node = node; Cost = cost;
    }
    public int Node { get; set; }
    public long Cost { get; set; }
    public int CompareTo(Pair p)
    {
        return p.Cost > Cost ? 1 : -1;
    }
}

class Edge
{
    public Edge(int to, long length)
    {
        To = to;
        Length = length;
    }
    public int To { get; set; }
    public long Length { get; set; }
}

class Comparable : IComparable<Comparable>
{
    public Comparable(int x, int id)
    {
        X = x;
        Id = id;
    }

    public int X;
    public int Id;
    public int CompareTo(Comparable c)
    {
        return X >= c.X ? -1 : 1;
    }
}

class Comparer<T> : IComparer<T>
{
    public int Compare(T t1, T t2)
    {
        return 1;
    }
}




partial class Program
{
    static void Main()
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Program().Solve();
        Console.Out.Flush();
        Console.Read();
    }

    public void Solve()
    {
        int N, K;
        Multi(out N, out K);
        var A = GetLongArray();
        var cum = new long[N + 1];
        for (int i = 0; i < N; i++) cum[i + 1] = cum[i] + A[i];
        var set = new HashSet<long>();
        for (int i = 0; i < N; i++)
        {
            for (int j = i + 1; j <= N; j++) set.Add(cum[j] - cum[i]);
        }

        if (set.Count <= K)
        {
            long ans = 0;
            foreach (var s in set) ans &= s;
            Console.WriteLine(ans);
            return;
        }

        var top = (int)Math.Floor(Math.Log(set.OrderByDescending(o => o).Skip(K - 1).First(), 2));
        long sum = 0;
        var numbers = set.ToArray();
        for (int i = top; i>= 0; i--)
        {
            if (numbers.Count(s => s / (1 << i) % 2 == 1) >= K)
            {
                numbers = numbers.Where(s => s / (1 << i) % 2 == 1).ToArray();
                sum += (1 << i);
            }
        }
        Console.WriteLine(sum);
    }
}