#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 #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 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; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::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, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::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_CYAN = "\033[1;36m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif map, bool> memo; bool Bruteforce(vector A) { if (A.size() <= 1) return true; sort(ALL(A)); if (memo.contains(A)) return memo.at(A); bool found = false; REP(i, A.size()) REP(j, i) { const lint sum = A.at(i) + A.at(j); if (binary_search(ALL(A), sum)) continue; vector nxt = A; nxt.erase(nxt.begin() + i); nxt.erase(nxt.begin() + j); const bool b = Bruteforce(nxt); found |= b; } return memo[A] = found; } // https://hitonanode.github.io/cplib-cpp/random/rand_nondeterministic.hpp #include #include struct rand_int_ { using lint = long long; std::mt19937 mt; rand_int_() : mt(42) {} // rand_int_() : mt(std::chrono::steady_clock::now().time_since_epoch().count()) {} lint operator()(lint x) { return this->operator()(0, x); } // [0, x) lint operator()(lint l, lint r) { std::uniform_int_distribution d(l, r - 1); return d(mt); } } rnd; bool Solve(const vector &A) { const int n = A.size(); const int n0 = count(ALL(A), 0LL); if (n0 >= 2) return false; if (n % 2 == 0 and n0) return false; if (n % 2 and n0) { int nl = 0, nr = 0; map mp; for (lint a : A) { if (a > 0) mp[a].first++, nl++; if (a < 0) mp[-a].second++, nr++; } if (mp.size() == 1) { for (auto [_, fs] : mp) { auto [l, r] = fs; if (l % 2) return false; // if (l > nr - r) return false; // if (r > nl - l) return false; } } if (mp.size() == 2) { auto [b1, lr1] = *mp.begin(); auto [b2, lr2] = *(next(mp.begin())); if (b1 * 2 != b2) return true; auto [l1, r1] = lr1; auto [l2, r2] = lr2; if ((l1 + l2) % 2 == 0) return true; if (l1 <= l2) return true; if (r1 <= r2) return true; if (!l1) return true; if (!r1) return true; if (!l1 and !r2) return true; if (!r1 and !l2) return true; if (!r1 and !r2) return true; if (!l1 and !l2) return true; return false; } if (mp.size() == 3) { auto [b1, lr1] = *mp.begin(); auto [b2, lr2] = *next(mp.begin()); auto [b3, lr3] = *next(next(mp.begin())); if (b1 * 2 != b2) return true; if (b1 * 3 != b3) return true; auto [l1, r1] = lr1; auto [l2, r2] = lr2; auto [l3, r3] = lr3; // if (l3 and r3) return true; // return false; if ((l1 + l2 + l3) % 2 == 0) return true; if (l1 <= l2 + l3) return true; if (r1 <= r2 + r3) return true; if (l2 % 2 == 0 and l3) return true; if (l3 > 1) return true; if (r3 > 1) return true; // if (l2 ) if (r2 % 2 == 0 and r3) return true; // if (l1 == 0 and r1 == 0) return true; return false; } } return true; } int main() { for (long long ntry = 1;; ntry++) { break; // const int n = rnd(1, 12); const int n = rnd(1, 8) * 2 + 1; vector a(n); // for (auto &x : a) x = rnd(-10, 11); for (auto &x : a) { while (x == 0) x = rnd(-3, 4); } a.at(0) = 0; sort(ALL(a)); const auto sol = Solve(a); const auto bru = Bruteforce(a); if (sol != bru) { dbg(make_tuple(bru, sol, a)); // dbg(sol); // dbg(bru); // exit(1); } if (__builtin_popcountll(ntry) == 1) { dbg(ntry); ; } } int T; cin >> T; while (T--) { int N; cin >> N; vector A(N); cin >> A; sort(ALL(A)); const auto ret = Solve(A); cout << (ret ? "Yes" : "No") << '\n'; } }