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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define fi first
#define se second
#define mp make_pair
#define pb push_back
#define ss(x) (int) x.size()
#define cat(x) cerr << #x << " = " << x << endl

using ll = long long;

using namespace std;

const int N = 4000005;

int n, s, a;
bitset <N> B;

int main() {
	B[0] = 1;
	scanf ("%d", &n);
	for (int i = 1; i <= n; ++i) {
		scanf ("%d", &a);
		s += a;
		B |= (B << a);
	}
	for (int i = (s + 1) / 2; true; ++i)
		if ((int) B[i] == 1) 
			return printf ("%d\n", i), 0;
	return 0;
}