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: Accepted)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
static class Program{
	const int mod=(int)1e9+7;
		static void Main(){
		Sc sc=new Sc();
		int n=sc.I;
		int[] a=sc.Ia;
		int sum=a.Sum();
		int p=sum-(sum+1)/2;
		Bitset bs=new Bitset(p+1);
		bs[0]=1;
		for(int i = 0;i<n;i++) {bs|=bs<<a[i];}
		while(bs[p]==0){p--;}
		Console.WriteLine("{0}",sum-p);
	}
}
public class Bitset{
	public int n,l;
	public ulong[] he;
	public Bitset(int n){
		this.n=n;
		l=(n>>6)+1;
		he=new ulong[l];
	}
	public byte this[int i]{
		set{he[i>>6]=(he[i>>6]&(ulong.MaxValue^(1UL<<i)))|(1UL<<i);}
		get{return (byte)((he[i>>6]>>i)&1);}
	}
	public static Bitset operator&(Bitset a,Bitset b){
		Bitset c=new Bitset(a.n);
		int d=Math.Min(a.l,b.l);
		for(int i=0;i<d;i++){c.he[i]=a.he[i]&b.he[i];}
        return c;
	}
	public static Bitset operator|(Bitset a,Bitset b){
		Bitset c;
		if(a.l<b.l){c=a;a=b;b=c;}
		c=new Bitset(a.n);
		int i=0;
		for(;i<b.l;i++){c.he[i]=a.he[i]|b.he[i];}
		for(;i<a.l;i++){c.he[i]=a.he[i];}
        return c;
	}
	public static Bitset operator^(Bitset a,Bitset b){
		Bitset c;
		if(a.l<b.l){c=a;a=b;b=c;}
		c=new Bitset(a.n);
		int i=0;
		for(;i<b.l;i++){c.he[i]=a.he[i]^b.he[i];}
		for(;i<a.l;i++){c.he[i]=a.he[i];}
        return c;
	}
	public static Bitset operator<<(Bitset a,int b){
		Bitset c=new Bitset(a.n);
		if(b>a.l<<6){return c;}
		int d=(b+63)>>6;
		if(b%64==0){for(int i=a.l-1;i>=d;i--){c.he[i]=a.he[i-d];}}
		else{
			for(int i=a.l-1;i>=d;i--){c.he[i]=(a.he[i-d+1]<<b)|(a.he[i-d]>>(64-b));}
			c.he[d-1]=a.he[0]<<b;
		}
        return c;
	}
	public static Bitset operator>>(Bitset a,int b){
		Bitset c=new Bitset(a.n);
		if(b>a.l<<6){return c;}
		int d=(b+63)>>6;
		if(b%64==0){for(int i=0;i+d<c.l;i++){c.he[i]=a.he[i+d];}}
		else{
			for(int i=0;i+d<c.l;i++){c.he[i]=(a.he[i+d-1]>>b)|(a.he[i+d]<<(64-b));}
			c.he[c.l-d]=a.he[c.l-1]>>b;
		}
        return c;
	}
}

public class Sc{
	private string[] a;
	private int n=-1,m=0;
	public int I{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return int.Parse(a[n]);}}
	public long L{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return long.Parse(a[n]);}}
	public double D{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return double.Parse(a[n]);}}
	public string S{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return a[n];}}
	public int[] Ia{get{if(n!=-1&&n!=a.Length-1){m=n;n=-1;return Array.ConvertAll(a.Skip(m+1).ToArray(),int.Parse);}return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()).Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
}