#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template vector sort_unique(vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template istream &operator>>(istream &is, vector &vec) { for (auto &v : vec) is >> v; return is; } template ostream &operator<<(ostream &os, const vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template istream &operator>>(istream &is, tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template ostream &operator<<(ostream &os, const tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } #endif template ostream &operator<<(ostream &os, const deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const pair &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; } template ostream &operator<<(ostream &os, const map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl #define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr) #else #define dbg(x) (x) #define dbgif(cond, x) 0 #endif // Subset sum (fast zeta transform) // Complexity: O(N 2^N) for array of size 2^N template void subset_sum(std::vector &f) { const int sz = f.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (S & (1 << d)) f[S] += f[S ^ (1 << d)]; } } // Inverse of subset sum (fast moebius transform) // Complexity: O(N 2^N) for array of size 2^N template void subset_sum_inv(std::vector &g) { const int sz = g.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (S & (1 << d)) g[S] -= g[S ^ (1 << d)]; } } // Superset sum / its inverse (fast zeta/moebius transform) // Complexity: O(N 2^N) for array of size 2^N template void superset_sum(std::vector &f) { const int sz = f.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (!(S & (1 << d))) f[S] += f[S | (1 << d)]; } } template void superset_sum_inv(std::vector &g) { const int sz = g.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (!(S & (1 << d))) g[S] -= g[S | (1 << d)]; } } // Subset convolution // h[S] = \sum_T f[T] * g[S - T] // Complexity: O(N^2 2^N) for arrays of size 2^N template std::vector subset_convolution(std::vector f, std::vector g) { const int sz = f.size(), m = __builtin_ctz(sz) + 1; assert(__builtin_popcount(sz) == 1 and f.size() == g.size()); std::vector> ff(m, std::vector(sz)), fg(m, std::vector(sz)); for (int i = 0; i < sz; i++) { int pc = __builtin_popcount(i); ff[pc][i] += f[i], fg[pc][i] += g[i]; } for (auto &vec : ff) subset_sum(vec); for (auto &vec : fg) subset_sum(vec); std::vector> fh(m, std::vector(sz)); for (int d = 0; d < m; d++) { for (int e = 0; d + e < m; e++) { for (int i = 0; i < sz; i++) fh[d + e][i] += ff[d][i] * fg[e][i]; } } for (auto &vec : fh) subset_sum_inv(vec); std::vector ret(sz); for (int i = 0; i < sz; i++) ret[i] = fh[__builtin_popcount(i)][i]; return ret; } int main() { int N; cin >> N; vector A(N); cin >> A; int sum = accumulate(ALL(A), 0); vector v(1 << N); REP(S, 1 << N) { int tot = 0; REP(d, N) if ((S >> d) & 1) tot += A[d]; if (tot * 3 == sum) v[S] = 1; } auto m = subset_convolution(v, subset_convolution(v, v)); cout << (m.back() ? "Yes" : "No") << '\n'; }