Problem:
Score : 700 points

Problem Statement

You are given N integers A_1, A_2, ..., A_N.

Consider the sums of all non-empty subsequences of A. There are 2^N - 1 such sums, an odd number.

Let the list of these sums in non-decreasing order be S_1, S_2, ..., S_{2^N - 1}.

Find the median of this list, S_{2^{N-1}}.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq 2000

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the median of the sorted list of the sums of all non-empty subsequences of A.

Sample Input 1

3
1 2 1

Sample Output 1

2

In this case, S = (1, 1, 2, 2, 3, 3, 4). Its median is S_4 = 2.

Sample Input 2

1
58

Sample Output 2

58

In this case, S = (58).

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Diagnostics;
class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	
	public void Solve(){
		
		uint x = 123456789;
		uint y = 362436069;
		uint z = 521288629;
		uint w = 88675123; 
		uint t;
		
		z ^= 7777777;
		z ^= z >> 21; z ^= z << 3; z ^= z >> 4;
		
		long[] cnt = new long[2000 * 2000 + 1];
		var sw = new Stopwatch();
		sw.Start();
		long cc = 0;
		long tt = sw.ElapsedMilliseconds;
		while(true){
			cc++;
			if(cc % 100 == 0){
				if(sw.ElapsedMilliseconds - tt > 1800) break;
			}
			
			int tot = 0;
			for(int i=0;i<N;i+=30){
				t = x ^ (x << 11);
				x = y; y = z; z = w;
				w = (w ^ (w >> 19)) ^ (t ^ (t >> 8));
				for(int j=0;j<30;j++){
					if(i * 30 + j >= N) break;
					if(((w >> j) & 1) == 1) tot += A[i * 30 + j];
				}
			}
			if(tot == 0) continue;
			cnt[tot]++;
		}
		
		long total = cnt.Sum();
//for(int i=1;i<=4;i++) Console.Write("{0} ",cnt[i]);Console.WriteLine();		
		
		total /= 2;
		for(int i=0;i<cnt.Length;i++){
			total -= cnt[i];
			if(total <= 0){
				Console.WriteLine(i);
				return;
			}
		}
		
	}
	int N;
	int[] A;

	public Sol(){
		N = ri();
		A = ria();
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}