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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <bitset>
using namespace std;

int main(){
  int N;
  cin >> N;
  vector<int> A(N);
  for(int i = 0; i < N; ++i) cin >> A[i];
  bitset<4000001> bit = 0;
  bit[0] = 1;
  int s = 0;
  for(int i = 0; i < N; ++i){
    bit |= (bit<<A[i]);
    s += A[i];
  }
  for(int i = (s+1)/2; i <= s; ++i){
    if(bit[i]){
      cout << i << endl;
      return 0;
    }
  }
}