#line 1 "test/yukicoder/4_modular_subset_sum.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/4" #line 1 "my_template.hpp" #include using namespace std; using ll = long long; using pi = pair; using vi = vector; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vec(type, name, ...) vector name(__VA_ARGS__) #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll i = 0; i < ll(a); ++i) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define FOR4_R(i, a, b, c) for (ll i = (b)-1; i >= ll(a); i -= (c)) #define overload4(a, b, c, d, e, ...) e #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) \ overload4(__VA_ARGS__, FOR4_R, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) for (ll t = s; t >= 0; t = (t == 0 ? -1 : (t - 1) & s)) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll template T SUM(vector &A) { T sum = T(0); for (auto &&a: A) sum += a; return sum; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()) int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T ceil(T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } template pair divmod(T x, U y) { T q = floor(x, y); return {q, x - q * y}; } ll binary_search(function check, ll ok, ll ng) { assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; if (check(x)) ok = x; else ng = x; } return ok; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } vi s_to_vi(const string &S, char first_char) { vi A(S.size()); FOR(i, S.size()) { A[i] = S[i] - first_char; } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } template vc bincount(const vc &A, int size) { vc C(size); for (auto &&x: A) { ++C[x]; } return C; } template vector argsort(const vector &A) { // stable vector ids(A.size()); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return A[i] < A[j] || (A[i] == A[j] && i < j); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { int n = len(A); assert(len(I) == n); vc B(n); FOR(i, n) B[i] = A[I[i]]; return B; } #line 1 "other/io.hpp" // based on yosupo's fastio #include namespace detail { template std::true_type check_value(int); template std::false_type check_value(long); } // namespace detail template struct is_modint : decltype(detail::check_value(0)) {}; template using is_modint_t = enable_if_t::value>; template using is_not_modint_t = enable_if_t::value>; struct Scanner { FILE *fp; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) { bool sep = false; for (size_t i = st; i < ed; i++) { if (isspace(line[i])) { sep = true; break; } } if (!sep) reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); } if (neg) ref = -ref; return true; } template * = nullptr> bool read_single(T &ref) { long long val = 0; bool f = read_single(val); ref = T(val); return f; } bool read_single(double &ref) { string s; if (!read_single(s)) return false; ref = std::stod(s); return true; } bool read_single(char &ref) { string s; if (!read_single(s) || s.size() != 1) return false; ref = s[0]; return true; } template bool read_single(vector &ref) { for (auto &d: ref) { if (!read_single(d)) return false; } return true; } template bool read_single(pair &p) { return (read_single(p.first) && read_single(p.second)); } template bool read_single(tuple &p) { return (read_single(get<0>(p)) && read_single(get<1>(p)) && read_single(get<2>(p))); } template bool read_single(tuple &p) { return (read_single(get<0>(p)) && read_single(get<1>(p)) && read_single(get<2>(p)) && read_single(get<3>(p))); } void read() {} template void read(H &h, T &... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE *fp) : fp(fp) {} }; struct Printer { Printer(FILE *_fp) : fp(_fp) {} ~Printer() { flush(); } static constexpr size_t SIZE = 1 << 15; FILE *fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write(const char &val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write('0'); return; } if (val < 0) { write('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char(0x30 | (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write(const string &s) { for (char c: s) write(c); } void write(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write(s[i]); } void write(const double &x) { ostringstream oss; oss << setprecision(15) << x; string s = oss.str(); write(s); } void write(const long double &x) { ostringstream oss; oss << setprecision(15) << x; string s = oss.str(); write(s); } template * = nullptr> void write(T &ref) { write(ref.val); } template void write(const vector &val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } template void write(const pair &val) { write(val.first); write(' '); write(val.second); } template void write(const tuple &val) { auto &[a, b, c] = val; write(a), write(' '), write(b), write(' '), write(c); } template void write(const tuple &val) { auto &[a, b, c, d] = val; write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d); } template void write(const tuple &val) { auto &[a, b, c, d, e] = val; write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e); } template void write(const tuple &val) { auto &[a, b, c, d, e, f] = val; write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e), write(' '), write(f); } template void write(const array &val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } void write(i128 val) { string s; bool negative = 0; if(val < 0){ negative = 1; val = -val; } while (val) { s += '0' + int(val % 10); val /= 10; } if(negative) s += "-"; reverse(all(s)); if (len(s) == 0) s = "0"; write(s); } }; Scanner scanner = Scanner(stdin); Printer printer = Printer(stdout); void flush() { printer.flush(); } void print() { printer.write('\n'); } template void print(Head &&head, Tail &&... tail) { printer.write(head); if (sizeof...(Tail)) printer.write(' '); print(forward(tail)...); } void read() {} template void read(Head &head, Tail &... tail) { scanner.read(head); read(tail...); } #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 1 "other/random.hpp" struct RandomNumberGenerator { mt19937 mt; RandomNumberGenerator() : mt(chrono::steady_clock::now().time_since_epoch().count()) {} ll operator()(ll a, ll b) { // [a, b) uniform_int_distribution dist(a, b - 1); return dist(mt); } ll operator()(ll b) { // [0, b) return (*this)(0, b); } }; #line 2 "mod/modint61.hpp" struct modint61 { static constexpr bool is_modint = true; static constexpr ll mod = (1LL << 61) - 1; ll val; constexpr modint61(const ll val = 0) : val(val) {} bool operator<(const modint61 &other) const { return val < other.val; } // To use std::map bool operator==(const modint61 &p) const { return val == p.val; } bool operator!=(const modint61 &p) const { return val != p.val; } modint61 &operator+=(const modint61 &p) { if ((val += p.val) >= mod) val -= mod; return *this; } modint61 &operator-=(const modint61 &p) { if ((val += mod - p.val) >= mod) val -= mod; return *this; } modint61 &operator*=(const modint61 &p) { ll a = val, b = p.val; const ll MASK30 = (1LL << 30) - 1; const ll MASK31 = (1LL << 31) - 1; const ll MASK61 = (1LL << 61) - 1; ll au = a >> 31, ad = a & MASK31; ll bu = b >> 31, bd = b & MASK31; ll x = ad * bu + au * bd; ll xu = x >> 30, xd = x & MASK30; x = au * bu * 2 + xu + (xd << 31) + ad * bd; xu = x >> 61, xd = x & MASK61; x = xu + xd; if (x >= MASK61) x -= MASK61; val = x; return *this; } modint61 &operator/=(const modint61 &p) { *this *= p.inverse(); return *this; } modint61 operator+(const modint61 &p) const { return modint61(*this) += p; } modint61 operator-(const modint61 &p) const { return modint61(*this) -= p; } modint61 operator*(const modint61 &p) const { return modint61(*this) *= p; } modint61 operator/(const modint61 &p) const { return modint61(*this) /= p; } modint61 inverse() const { ll a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return modint61(u); } modint61 pow(int64_t n) const { modint61 ret(1), mul(val); while (n > 0) { if (n & 1) ret = ret * mul; mul = mul * mul; n >>= 1; } return ret; } static constexpr ll get_mod() { return mod; } }; #line 3 "mod/modular_subset_sum.hpp" // Faster Deterministic Modular Subset Sum. arXiv preprint arXiv:2012.06062. // modular subset sum のための、シフト付きセグ木 // shift には 2^(N-k) 時間かかる struct ShiftTree { using M61 = modint61; int delta; int N, n; M61 base; vc dat; vc base_pow; ShiftTree(int N, ll base) : delta(0), N(N), n(topbit(N)), base(base) { assert(N == (1 << n)); dat.assign(2 * N, 0); base_pow.assign(n, 1); base_pow[n - 1] = base; FOR_R(i, n - 1) base_pow[i] = base_pow[i + 1] * base_pow[i + 1]; } inline int skew(int k) { return (delta >> (n - k)) & 1; } inline int left(int k, int i) { int mask = (1 << (k + 1)) - 1; return ((2 * i + 0 - skew(k + 1)) & mask) + (1 << (k + 1)); } inline int right(int k, int i) { int mask = (1 << (k + 1)) - 1; return ((2 * i + 1 - skew(k + 1)) & mask) + (1 << (k + 1)); } inline int parent(int k, int i) { int mask = (1 << k) - 1; return (((i + skew(k)) & mask) + (1 << k)) / 2; } inline void update(int k, int i) { M61 b = base_pow[k]; dat[i] = b * dat[left(k, i)] + dat[right(k, i)]; } inline void set(int i, ll x) { i = (i + N - delta) % N + N; dat[i] = x; int k = n; while (i != 1) { i = parent(k, i); --k; update(k, i); } } void shift(int k) { k %= N; if (k < 0) k += N; if (k == 0) return; int j = lowbit(k); delta = (delta + k) % N; FOR_R(k, n - j) { FOR3(i, 1 << k, 2 << k) update(k, i); } } // [a,b) における difference の列挙。output sensitive。 // T のノード i、Q のノード j が (x,y) を指すとする。 static void find_differences(vc& res, ShiftTree& T, ShiftTree& Q, int a, int b, int k, int i, int j, int x, int y) { if (T.dat[i] == Q.dat[j]) return; if (max(a, x) >= min(b, y)) return; if (y == x + 1) { res.eb(x); return; } int z = (x + y) / 2; find_differences(res, T, Q, a, b, k + 1, T.left(k, i), Q.left(k, j), x, z); find_differences(res, T, Q, a, b, k + 1, T.right(k, i), Q.right(k, j), z, y); } static vc diff(ShiftTree& T, ShiftTree& Q, int a, int b) { assert(T.N == Q.N); vc res; find_differences(res, T, Q, a, b, 0, 1, 1, 0, T.N); return res; } }; /* 計算量:(|vals| + mod) * log(mod) ・can(x) または [x] で bool を返す。 ・restore(x) で復元。 コンストラクタには、(mod, vals) をわたす */ template struct Modular_Subset_Sum { int mod; vc& vals; vc par; Modular_Subset_Sum(int mod, vc& vals) : mod(mod), vals(vals) { par.assign(mod, -1); RandomNumberGenerator RNG; const ll base = RNG(0, (1LL << 61) - 1); int k = 1; while ((1 << k) < 2 * mod) ++k; int L = 1 << k; assert(L >= 2 * mod); ShiftTree T1(L, base); ShiftTree T2(L, base); T1.set(0, 1); T2.set(0, 1); T2.set(L - mod, 1); auto bit_rev = [&](int i) -> int { int x = 0; FOR(k) { x = 2 * x + (i & 1); i >>= 1; } return x; }; vc IDS(L); FOR(i, len(vals)) { IDS[vals[i]].eb(i); } FOR(i, 1, L) { int x = bit_rev(i); if (len(IDS[x]) == 0) continue; T2.shift(x - T2.delta); for (auto&& idx: IDS[x]) { auto diff = ShiftTree::diff(T1, T2, 0, mod); for (auto&& d: diff) { if (can(d)) continue; par[d] = idx; T1.set(d, 1); T2.set((d + x) % L, 1); T2.set((L + d + x - mod) % L, 1); } } } } bool can(int x) { return (x == 0 || par[x] != -1); } bool operator[](int x) { return can(x); } vc restore(int x) { vc res; while (x) { int i = par[x]; res.eb(i); x -= vals[i]; if (x < 0) x += mod; } reverse(all(res)); return res; } }; #line 5 "test/yukicoder/4_modular_subset_sum.test.cpp" void solve() { LL(N); VEC(int, A, N); ll S = SUM(A); Modular_Subset_Sum MSS(S + 1, A); if (MSS[S / 2] && S % 2 == 0) { print("possible"); } else { print("impossible"); } } signed main() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << setprecision(15); ll T = 1; // LL(T); FOR(_, T) solve(); return 0; }