#line 1 "/home/maspy/compro/library/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__)))) #define FOR_(n) for (ll _ = 0; (_) < (ll)(n); ++(_)) #define FOR(i, n) for (ll i = 0; (i) < (ll)(n); ++(i)) #define FOR3(i, m, n) for (ll i = (m); (i) < (ll)(n); ++(i)) #define FOR_R(i, n) for (ll i = (ll)(n)-1; (i) >= 0; --(i)) #define FOR3_R(i, m, n) for (ll i = (ll)(n)-1; (i) >= (ll)(m); --(i)) #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 : 31 - __builtin_clz(x)); } int lowbit(u32 x) { return (x==0 ? -1 : 31 - __builtin_clz(x)); } int lowbit(ll x) { return (x==0 ? -1 : 63 - __builtin_clzll(x)); } int lowbit(u64 x) { return (x==0 ? -1 : 63 - __builtin_clzll(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(string S, char first_char = 'a') { 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(vc &A, int size) { vc C(size); for (auto &&x: A) { ++C[x]; } return C; } template vector argsort(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; } #line 1 "/home/maspy/compro/library/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 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; while (val) { s += '0' + int(val % 10); val /= 10; } 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 2 "/home/maspy/compro/library/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 int get_mod() { return mod; } }; #line 3 "/home/maspy/compro/library/alg/monoid_rollinghash.hpp" struct Monoid_Rolling_Hash { using value_type = pair; // pow of base, val using X = value_type; static X op(X x, X y) { return {x.fi * y.fi, x.se * y.fi + y.se}; } static constexpr X unit = X(1, 0); static constexpr bool commute = false; }; #line 1 "/home/maspy/compro/library/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 "/home/maspy/compro/library/ds/segtree.hpp" template struct SegTree { using X = typename Monoid::value_type; using value_type = X; vc dat; int n, log, size; SegTree() : SegTree(0) {} SegTree(int n) : SegTree(vc(n, Monoid::unit)) {} SegTree(vc v) : n(len(v)) { log = 1; while ((1 << log) < n) ++log; size = 1 << log; dat.assign(size << 1, Monoid::unit); FOR(i, n) dat[size + i] = v[i]; FOR3_R(i, 1, size) update(i); } X operator[](int i) { return dat[size + i]; } void update(int i) { dat[i] = Monoid::op(dat[2 * i], dat[2 * i + 1]); } void set(int i, X x) { assert(i < n); dat[i += size] = x; while (i >>= 1) update(i); } X prod(int L, int R) { assert(L <= R); assert(R <= n); X vl = Monoid::unit, vr = Monoid::unit; L += size, R += size; while (L < R) { if (L & 1) vl = Monoid::op(vl, dat[L++]); if (R & 1) vr = Monoid::op(dat[--R], vr); L >>= 1, R >>= 1; } return Monoid::op(vl, vr); } X prod_all() { return dat[1];} template int max_right(F &check, int L) { assert(0 <= L && L <= n && check(Monoid::unit)); if (L == n) return n; L += size; X sm = Monoid::unit; do { while (L % 2 == 0) L >>= 1; if (!check(Monoid::op(sm, dat[L]))) { while (L < size) { L = 2 * L; if (check(Monoid::op(sm, dat[L]))) { sm = Monoid::op(sm, dat[L]); L++; } } return L - size; } sm = Monoid::op(sm, dat[L]); L++; } while ((L & -L) != L); return n; } template int min_left(F &check, int R) { assert(0 <= R && R <= n && check(Monoid::unit)); if (R == 0) return 0; R += size; X sm = Monoid::unit; do { --R; while (R > 1 && (R % 2)) R >>= 1; if (!check(Monoid::op(dat[R], sm))) { while (R < size) { R = 2 * R + 1; if (check(Monoid::op(dat[R], sm))) { sm = Monoid::op(dat[R], sm); R--; } } return R + 1 - size; } sm = Monoid::op(dat[R], sm); } while ((R & -R) != R); return 0; } void debug() { print("segtree", dat); } }; #line 4 "/home/maspy/compro/library/mod/modular_subset_sum.hpp" /* (|vals| + mod) * log^2(mod) verify: https://codeforces.com/gym/103428/problem/C */ vc modular_subset_sum(int mod, vc vals) { using Mono = Monoid_Rolling_Hash; RandomNumberGenerator RNG; const ll base = RNG(0, (1LL << 61) - 1); vc A(mod + mod); ll cnt = 0; vc> seg_raw(mod + mod); FOR(i, mod + mod) seg_raw[i] = {base, 0}; SegTree seg(seg_raw); auto add = [&](ll x) -> void { ++cnt; A[x] = A[x + mod] = 1; seg.set(x, {base, 1}); seg.set(x + mod, {base, 1}); }; add(0); for (auto&& val: vals) { val %= mod; if (cnt == mod) break; vc LR; LR.eb(0, mod); vi ADD; while (len(LR)) { auto [L, R] = LR.back(); LR.pop_back(); if (L == R) continue; modint61 x1 = seg.prod(L, R).se; modint61 x2 = seg.prod(mod + L - val, mod + R - val).se; if (x1 == x2) continue; if (R == L + 1) { // 対称差 L が見つかった if (!A[L]) ADD.eb(L); continue; } ll M = (L + R) / 2; LR.eb(L, M); LR.eb(M, R); } for (auto&& a: ADD) add(a); } A.resize(mod); return A; } #line 4 "main.cpp" void solve() { LL(N); VEC(int, A, N); ll S = SUM(A); auto can = modular_subset_sum(S + 1, A); if (can[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; }