/** * date : 2023-09-12 15:41:26 * author : Nyaan */ #define NDEBUG using namespace std; // intrinstic #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 #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // utility namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template using V = vector; template using VV = vector>; using vi = vector; using vl = vector; using vd = V; using vs = V; using vvi = vector>; using vvl = vector>; template using minpq = priority_queue, greater>; template struct P : pair { template P(Args... args) : pair(args...) {} using pair::first; using pair::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P; using pi = P; using vp = V; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template int sz(const T &t) { return t.size(); } template inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline T Max(const vector &v) { return *max_element(begin(v), end(v)); } template inline T Min(const vector &v) { return *min_element(begin(v), end(v)); } template inline long long Sum(const vector &v) { return accumulate(begin(v), end(v), 0LL); } template int lb(const vector &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template int ub(const vector &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template pair mkp(const T &t, const U &u) { return make_pair(t, u); } template vector mkrui(const vector &v, bool rev = false) { vector ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template vector mkuni(const vector &v) { vector ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template vector mkord(int N, F f) { vector ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template vector mkinv(vector &v) { int max_val = *max_element(begin(v), end(v)); vector inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector mkiota(int n) { vector ret(n); iota(begin(ret), end(ret), 0); return ret; } template T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template bool nxp(vector &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template vector> product(const vector &a) { vector> ret; vector v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I, const function &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I) { return Power(a, n, I, function{[](T &) -> void {}}); } } // namespace Nyaan // bit operation namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return _mm_popcnt_u64(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template inline int gbit(const T &a, int i) { return (a >> i) & 1; } template inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan // inout namespace Nyaan { template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan // debug #ifdef NyaanDebug #define trc(...) (void(0)) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) (void(0)) #else #define trc2(...) (void(0)) #endif // macro #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } // using namespace std; using namespace std; namespace internal { template using is_broadly_integral = typename conditional_t || is_same_v || is_same_v, true_type, false_type>::type; template using is_broadly_signed = typename conditional_t || is_same_v, true_type, false_type>::type; template using is_broadly_unsigned = typename conditional_t || is_same_v, true_type, false_type>::type; #define ENABLE_VALUE(x) \ template \ constexpr bool x##_v = x::value; ENABLE_VALUE(is_broadly_integral); ENABLE_VALUE(is_broadly_signed); ENABLE_VALUE(is_broadly_unsigned); #undef ENABLE_VALUE #define ENABLE_HAS_TYPE(var) \ template \ struct has_##var : false_type {}; \ template \ struct has_##var> : true_type {}; \ template \ constexpr auto has_##var##_v = has_##var::value; #define ENABLE_HAS_VAR(var) \ template \ struct has_##var : false_type {}; \ template \ struct has_##var> : true_type {}; \ template \ constexpr auto has_##var##_v = has_##var::value; } // namespace internal using namespace std; namespace BinaryGCDImpl { using u64 = unsigned long long; using i8 = char; u64 binary_gcd(u64 a, u64 b) { if (a == 0 || b == 0) return a + b; i8 n = __builtin_ctzll(a); i8 m = __builtin_ctzll(b); a >>= n; b >>= m; n = min(n, m); while (a != b) { u64 d = a - b; i8 s = __builtin_ctzll(d); bool f = a > b; b = f ? b : a; a = (f ? d : -d) >> s; } return a << n; } using u128 = __uint128_t; // a > 0 int ctz128(u128 a) { u64 lo = a & u64(-1); return lo ? __builtin_ctzll(lo) : 64 + __builtin_ctzll(a >> 64); } u128 binary_gcd128(u128 a, u128 b) { if (a == 0 || b == 0) return a + b; i8 n = ctz128(a); i8 m = ctz128(b); a >>= n; b >>= m; n = min(n, m); while (a != b) { u128 d = a - b; i8 s = ctz128(d); bool f = a > b; b = f ? b : a; a = (f ? d : -d) >> s; } return a << n; } } // namespace BinaryGCDImpl long long binary_gcd(long long a, long long b) { return BinaryGCDImpl::binary_gcd(abs(a), abs(b)); } __int128_t binary_gcd128(__int128_t a, __int128_t b) { if (a < 0) a = -a; if (b < 0) b = -b; return BinaryGCDImpl::binary_gcd128(a, b); } /** * @brief binary GCD */ // T : 値, U : 比較用 template struct RationalBase { using R = RationalBase; using Key = T; T x, y; RationalBase() : x(0), y(1) {} template RationalBase(const T1& _x) : RationalBase(_x, T1{1}) {} template RationalBase(const T1& _x, const T2& _y) : x(_x), y(_y) { assert(y != 0); if (y == -1) x = -x, y = -y; if (y != 1) { T g; if constexpr (internal::is_broadly_integral_v) { if constexpr (sizeof(T) == 16) { g = binary_gcd128(x, y); } else { g = binary_gcd(x, y); } } else { g = gcd(x, y); } if (g != 0) x /= g, y /= g; if (y < 0) x = -x, y = -y; } } // y = 0 の代入も認める static R raw(T _x, T _y) { R r; r.x = _x, r.y = _y; return r; } friend R operator+(const R& l, const R& r) { if (l.y == r.y) return R{l.x + r.x, l.y}; return R{l.x * r.y + l.y * r.x, l.y * r.y}; } friend R operator-(const R& l, const R& r) { if (l.y == r.y) return R{l.x - r.x, l.y}; return R{l.x * r.y - l.y * r.x, l.y * r.y}; } friend R operator*(const R& l, const R& r) { return R{l.x * r.x, l.y * r.y}; } friend R operator/(const R& l, const R& r) { return R{l.x * r.y, l.y * r.x}; } R& operator+=(const R& r) { return (*this) = (*this) + r; } R& operator-=(const R& r) { return (*this) = (*this) - r; } R& operator*=(const R& r) { return (*this) = (*this) * r; } R& operator/=(const R& r) { return (*this) = (*this) / r; } R operator-() const { return raw(-x, y); } R inverse() const { assert(x != 0); R r = raw(y, x); if (r.y < 0) r.x = -r.x, r.y = -r.y; return r; } R pow(long long p) const { R res{1}, base{*this}; while (p) { if (p & 1) res *= base; base *= base; p >>= 1; } return res; } friend bool operator==(const R& l, const R& r) { return l.x == r.x && l.y == r.y; }; friend bool operator!=(const R& l, const R& r) { return l.x != r.x || l.y != r.y; }; friend bool operator<(const R& l, const R& r) { return U{l.x} * r.y < U{l.y} * r.x; }; friend bool operator<=(const R& l, const R& r) { return l < r || l == r; } friend bool operator>(const R& l, const R& r) { return U{l.x} * r.y > U{l.y} * r.x; }; friend bool operator>=(const R& l, const R& r) { return l > r || l == r; } friend ostream& operator<<(ostream& os, const R& r) { os << r.x; if (r.x != 0 && r.y != 1) os << "/" << r.y; return os; } // T にキャストされるので T が bigint の場合は to_ll も要る T to_mint(T mod) const { assert(mod != 0); T a = y, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b); swap(u -= t * v, v); } return U((u % mod + mod) % mod) * x % mod; } }; using Rational = RationalBase; // enumerate x : x \subset b vector enumerate_subset(int b) { vector res; for (int i = b; i >= 0; --i) res.push_back(i &= b); return res; }; // enumerate x : x \in {n} and x \superset b vector enumerate_superset(int b, int n) { vector res; for (int i = b; i < (1 << n); i = (i + 1) | b) res.push_back(i); return res; } /** * @brief 下位集合/上位集合の列挙 */ using namespace Nyaan; using R = Rational; void q() { inl(N); vl a(N); in(a); V> dp(PW(N)); rep(i, N) dp[PW(i)].insert(a[i]); rep(i, PW(N)) { if (popcnt(i) <= 1) continue; each(j, enumerate_subset(i)) { each(x, dp[i - j]) { each(y, dp[j]) { if (x == y) die("YES"); if (i != MSK(N)) { dp[i].insert(x + y); dp[i].insert(x - y); dp[i].insert(x * y); dp[i].insert(x / y); } } } } } out("NO"); } void Nyaan::solve() { int t = 1; // in(t); while (t--) q(); }