#ifndef HIDDEN_IN_VS // 折りたたみ用 // 警告の抑制 #define _CRT_SECURE_NO_WARNINGS // ライブラリの読み込み #include using namespace std; // 型名の短縮 using ll = long long; using ull = unsigned long long; // -2^63 ~ 2^63 = 9 * 10^18(int は -2^31 ~ 2^31 = 2 * 10^9) using pii = pair; using pll = pair; using pil = pair; using pli = pair; using vi = vector; using vvi = vector; using vvvi = vector; using vvvvi = vector; using vl = vector; using vvl = vector; using vvvl = vector; using vvvvl = vector; using vb = vector; using vvb = vector; using vvvb = vector; using vc = vector; using vvc = vector; using vvvc = vector; using vd = vector; using vvd = vector; using vvvd = vector; template using priority_queue_rev = priority_queue, greater>; using Graph = vvi; // 定数の定義 const double PI = acos(-1); const vi DX = { 1, 0, -1, 0 }; // 4 近傍(下,右,上,左) const vi DY = { 0, 1, 0, -1 }; int INF = 1001001001; ll INFL = 4004004003104004004LL; // (int)INFL = 1010931620; // 入出力高速化 struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp; // 汎用マクロの定義 #define all(a) (a).begin(), (a).end() #define sz(x) ((int)(x).size()) #define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), x)) #define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), x)) #define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");} #define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 から n-1 まで昇順 #define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s から t まで昇順 #define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s から t まで降順 #define repe(v, a) for(const auto& v : (a)) // a の全要素(変更不可能) #define repea(v, a) for(auto& v : (a)) // a の全要素(変更可能) #define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d ビット全探索(昇順) #define repis(i, set) for(int i = lsb(set), bset##i = set; i >= 0; bset##i -= 1 << i, i = lsb(bset##i)) // set の全要素(昇順) #define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a の順列全て(昇順) #define smod(n, m) ((((n) % (m)) + (m)) % (m)) // 非負mod #define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去 #define EXIT(a) {cout << (a) << endl; exit(0);} // 強制終了 #define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) // 半開矩形内判定 // 汎用関数の定義 template inline ll powi(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; } template inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新(更新されたら true を返す) template inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新(更新されたら true を返す) template inline T get(T set, int i) { return (set >> i) & T(1); } // 演算子オーバーロード template inline istream& operator>>(istream& is, pair& p) { is >> p.first >> p.second; return is; } template inline istream& operator>>(istream& is, vector& v) { repea(x, v) is >> x; return is; } template inline vector& operator--(vector& v) { repea(x, v) --x; return v; } template inline vector& operator++(vector& v) { repea(x, v) ++x; return v; } #endif // 折りたたみ用 #if __has_include() #include using namespace atcoder; #ifdef _MSC_VER #include "localACL.hpp" #endif //using mint = modint1000000007; using mint = modint998244353; //using mint = modint; // mint::set_mod(m); namespace atcoder { inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; } inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; } } using vm = vector; using vvm = vector; using vvvm = vector; using vvvvm = vector; using pim = pair; #endif #ifdef _MSC_VER // 手元環境(Visual Studio) #include "local.hpp" #else // 提出用(gcc) inline int popcount(int n) { return __builtin_popcount(n); } inline int popcount(ll n) { return __builtin_popcountll(n); } inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : -1; } inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : -1; } inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; } inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; } #define dump(...) #define dumpel(v) #define dump_list(v) #define dump_mat(v) #define input_from_file(f) #define output_to_file(f) #define Assert(b) { if (!(b)) while (1) cout << "OLE"; } #endif //【局面のニム値】O(?)(遅いので実験用) /* * 初期局面 p から遷移可能な局面とそのニム値のリストを返す. * nxt(p, nps) を呼ぶと,p から遷移可能な局面のリストを nps に格納するものとする. */ template map calc_nimber(const T& p, function&)>& nxt) { map nim; // これをグローバル変数にすれば再利用可能 function calc_nimber = [&](const T& p) { if (nim.count(p)) return nim[p]; vector nps; nxt(p, nps); vi next_nimbers; repe(np, nps) { next_nimbers.push_back(calc_nimber(np)); } uniq(next_nimbers); int i = 0; while (i < sz(next_nimbers) && next_nimbers[i] == i) i++; nim[p] = i; return nim[p]; }; calc_nimber(p); return nim; /* nxt の定義の雛形 using T = vi; function&)> nxt = [&](const T& p, vector& nps) { }; */ } bool naive(int n, ll m, vl a) { using T = pair; function&)> nxt = [&](const T& p, vector& nps) { auto& [a, x] = p; rep(i, n) { repi(k, 1, a[i]) { if (k % m <= x) { auto na(a); na[i] -= k; sort(all(na)); nps.push_back({ na, k % m }); } } } }; sort(all(a)); auto res = calc_nimber({ a, m - 1 }, nxt); //for (auto [tmp, nimber] : res) { // auto [a, x] = tmp; // if (x != 0) continue; // ll a_xor_1 = 0; // repe(v, a) a_xor_1 ^= v / 1; // ll a_xor_2 = 0; // repe(v, a) a_xor_2 ^= v / 2; // ll a_xor_3 = 0; // repe(v, a) a_xor_3 ^= v / 3; // dump("(", a, x, ")", nimber, "a_xor:", a_xor_1, a_xor_2, a_xor_3); //} //for (auto [tmp, nimber] : res) { // auto [a, x] = tmp; // if (x != 1) continue; // ll a_xor_1 = 0; // repe(v, a) a_xor_1 ^= v / 1; // ll a_xor_2 = 0; // repe(v, a) a_xor_2 ^= v / 2; // ll a_xor_3 = 0; // repe(v, a) a_xor_3 ^= v / 3; // dump("(", a, x, ")", nimber, "a_xor:", a_xor_1, a_xor_2, a_xor_3); //} set> xor_to_nimber; for (auto [tmp, nimber] : res) { auto [a, x] = tmp; if (x != m - 1) continue; // if (nimber != 0) continue; ll a_xor_1 = 0; repe(v, a) a_xor_1 ^= v / 1; ll a_xor_2 = 0; repe(v, a) a_xor_2 ^= v / 2; ll a_xor_3 = 0; repe(v, a) a_xor_3 ^= v / 3; ll a_xor_4 = 0; repe(v, a) a_xor_4 ^= v / 4; ll a_xor_5 = 0; repe(v, a) a_xor_5 ^= v / 5; ll a_xor_6 = 0; repe(v, a) a_xor_6 ^= v / 6; if (a_xor_3 != 0) continue; // dump("(", a, x, ")", nimber, " a_xor:", a_xor_1, a_xor_2, a_xor_3, a_xor_4, a_xor_5, a_xor_6); dump("(", a, x, ")", nimber, " a_xor:", a_xor_1, a_xor_2, a_xor_3); xor_to_nimber.insert({a_xor_1, a_xor_2, a_xor_3, nimber > 0}); } dumpel(xor_to_nimber); return res[{a, m - 1}] > 0; } bool solve(int n, ll m, vl a) { ll q_xor = 0; rep(i, n) q_xor ^= a[i] / m; if (q_xor > 0) return 1; ll r_xor = 0; rep(i, n) r_xor ^= a[i] % m; return r_xor > 0; } void bug_find() { #ifdef _MSC_VER // 合わない入力例を見つける. mute_dump = true; mt19937_64 mt; mt.seed((int)time(NULL)); uniform_int_distribution rnd(0LL, 1LL << 60); rep(hoge, 100) { int n = rnd(mt) % 5 + 1; ll m = rnd(mt) % 5 + 1; vl a(n); rep(i, n) a[i] = rnd(mt) % 10 + 1; auto res_naive = naive(n, m, a); auto res_solve = solve(n, m, a); if (res_naive != res_solve) { cout << "----------error!----------" << endl; cout << "input:" << endl; cout << n << " " << m << endl; cout << a << endl; cout << "results:" << endl; cout << res_naive << endl; cout << res_solve << endl; cout << "--------------------------" << endl; } } mute_dump = false; exit(0); #endif } int main() { input_from_file("input.txt"); // output_to_file("output.txt"); bug_find(); int n; ll m; cin >> n >> m; vl a(n); cin >> a; dump(naive(n, m, a)); dump("----"); cout << (solve(n, m, a) ? "Alice" : "Bob") << endl; }