// competitive-verifier: PROBLEM #include #include #include #include namespace internal { // @param m `1 <= m` // @return x mod m constexpr std::int64_t safe_mod(std::int64_t x, std::int64_t m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; std::uint64_t im; // @param m `1 <= m` explicit barrett(unsigned int m) : _m(m), im((std::uint64_t)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { std::uint64_t z = a; z *= b; std::uint64_t x = (std::uint64_t)(((__uint128_t)(z)*im) >> 64); std::uint64_t y = x * _m; return (unsigned int)(z - y + (z < y ? _m : 0)); } }; struct montgomery { std::uint64_t _m; std::uint64_t im; std::uint64_t r2; // @param m `1 <= m` explicit constexpr montgomery(std::uint64_t m) : _m(m), im(m), r2(-__uint128_t(m) % m) { for (int i = 0; i < 5; ++i) im = im * (2 - _m * im); im = -im; } // @return m constexpr std::uint64_t umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` constexpr std::uint64_t mul(std::uint64_t a, std::uint64_t b) const { return mr(mr(a, b), r2); } constexpr std::uint64_t exp(std::uint64_t a, std::uint64_t b) const { std::uint64_t res = 1, p = mr(a, r2); while (b) { if (b & 1) res = mr(res, p); p = mr(p, p); b >>= 1; } return res; } constexpr bool same_pow(std::uint64_t x, int s, std::uint64_t n) const { x = mr(x, r2), n = mr(n, r2); for (int r = 0; r < s; r++) { if (x == n) return true; x = mr(x, x); } return false; } private: constexpr std::uint64_t mr(std::uint64_t x) const { return ((__uint128_t)(x * im) * _m + x) >> 64; } constexpr std::uint64_t mr(std::uint64_t a, std::uint64_t b) const { __uint128_t t = (__uint128_t)a * b; std::uint64_t inc = std::uint64_t(t) != 0; std::uint64_t x = t >> 64, y = ((__uint128_t)(a * b * im) * _m) >> 64; unsigned long long z = 0; bool f = __builtin_uaddll_overflow(x, y, &z); z += inc; return f ? z - _m : z; } }; constexpr bool is_SPRP32(std::uint32_t n, std::uint32_t a) { std::uint32_t d = n - 1, s = 0; while ((d & 1) == 0) ++s, d >>= 1; std::uint64_t cur = 1, pw = d; while (pw) { if (pw & 1) cur = (cur * a) % n; a = (std::uint64_t)a * a % n; pw >>= 1; } if (cur == 1) return true; for (std::uint32_t r = 0; r < s; r++) { if (cur == n - 1) return true; cur = cur * cur % n; } return false; } // given 2 <= n,a < 2^64, a prime, check whether n is a-SPRP constexpr bool is_SPRP64(const montgomery &m, std::uint64_t a) { auto n = m.umod(); if (n == a) return true; if (n % a == 0) return false; std::uint64_t d = n - 1; int s = 0; while ((d & 1) == 0) ++s, d >>= 1; std::uint64_t cur = m.exp(a, d); if (cur == 1) return true; return m.same_pow(cur, s, n - 1); } constexpr bool is_prime_constexpr(std::uint64_t x) { if (x == 2 || x == 3 || x == 5 || x == 7) return true; if (x % 2 == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false; if (x < 121) return (x > 1); montgomery m(x); constexpr std::uint64_t bases[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022}; for (auto a : bases) { if (!is_SPRP64(m, a)) return false; } return true; } constexpr bool is_prime_constexpr(std::int64_t x) { if (x < 0) return false; return is_prime_constexpr(std::uint64_t(x)); } constexpr bool is_prime_constexpr(std::uint32_t x) { if (x == 2 || x == 3 || x == 5 || x == 7) return true; if (x % 2 == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false; if (x < 121) return (x > 1); std::uint64_t h = x; h = ((h >> 16) ^ h) * 0x45d9f3b; h = ((h >> 16) ^ h) * 0x45d9f3b; h = ((h >> 16) ^ h) & 255; constexpr uint16_t bases[] = { 15591, 2018, 166, 7429, 8064, 16045, 10503, 4399, 1949, 1295, 2776, 3620, 560, 3128, 5212, 2657, 2300, 2021, 4652, 1471, 9336, 4018, 2398, 20462, 10277, 8028, 2213, 6219, 620, 3763, 4852, 5012, 3185, 1333, 6227, 5298, 1074, 2391, 5113, 7061, 803, 1269, 3875, 422, 751, 580, 4729, 10239, 746, 2951, 556, 2206, 3778, 481, 1522, 3476, 481, 2487, 3266, 5633, 488, 3373, 6441, 3344, 17, 15105, 1490, 4154, 2036, 1882, 1813, 467, 3307, 14042, 6371, 658, 1005, 903, 737, 1887, 7447, 1888, 2848, 1784, 7559, 3400, 951, 13969, 4304, 177, 41, 19875, 3110, 13221, 8726, 571, 7043, 6943, 1199, 352, 6435, 165, 1169, 3315, 978, 233, 3003, 2562, 2994, 10587, 10030, 2377, 1902, 5354, 4447, 1555, 263, 27027, 2283, 305, 669, 1912, 601, 6186, 429, 1930, 14873, 1784, 1661, 524, 3577, 236, 2360, 6146, 2850, 55637, 1753, 4178, 8466, 222, 2579, 2743, 2031, 2226, 2276, 374, 2132, 813, 23788, 1610, 4422, 5159, 1725, 3597, 3366, 14336, 579, 165, 1375, 10018, 12616, 9816, 1371, 536, 1867, 10864, 857, 2206, 5788, 434, 8085, 17618, 727, 3639, 1595, 4944, 2129, 2029, 8195, 8344, 6232, 9183, 8126, 1870, 3296, 7455, 8947, 25017, 541, 19115, 368, 566, 5674, 411, 522, 1027, 8215, 2050, 6544, 10049, 614, 774, 2333, 3007, 35201, 4706, 1152, 1785, 1028, 1540, 3743, 493, 4474, 2521, 26845, 8354, 864, 18915, 5465, 2447, 42, 4511, 1660, 166, 1249, 6259, 2553, 304, 272, 7286, 73, 6554, 899, 2816, 5197, 13330, 7054, 2818, 3199, 811, 922, 350, 7514, 4452, 3449, 2663, 4708, 418, 1621, 1171, 3471, 88, 11345, 412, 1559, 194}; return is_SPRP32(x, bases[h]); } // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr std::int64_t pow_mod_constexpr(std::int64_t x, std::int64_t n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); std::uint64_t r = 1; std::uint64_t y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; std::int64_t d = n - 1; while (d % 2 == 0) d /= 2; constexpr std::int64_t bases[3] = {2, 7, 61}; for (std::int64_t a : bases) { std::int64_t t = d; std::int64_t y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair inv_gcd(std::int64_t a, std::int64_t b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; std::int64_t s = b, t = a; std::int64_t m0 = 0, m1 = 1; while (t) { std::int64_t u = s / t; s -= t * u; m0 -= m1 * u; auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (std::int64_t)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal #include #include namespace internal { template using is_signed_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using is_unsigned_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using make_unsigned_int128 = typename std::conditional::value, __uint128_t, unsigned __int128>; template using is_integral = typename std::conditional::value || is_signed_int128::value || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using is_signed_int = typename std::conditional<(is_integral::value && std::is_signed::value) || is_signed_int128::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional<(is_integral::value && std::is_unsigned::value) || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional< is_signed_int128::value, make_unsigned_int128, typename std::conditional::value, std::make_unsigned, std::common_type>::type>::type; template using is_signed_int_t = std::enable_if_t::value>; template using is_unsigned_int_t = std::enable_if_t::value>; template using to_unsigned_t = typename to_unsigned::type; } // namespace internal namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; } // namespace internal template * = nullptr> struct static_modint : internal::static_modint_base { using mint = static_modint; public: static constexpr int mod() { return m; } static constexpr mint raw(int v) { mint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} template * = nullptr> constexpr static_modint(T v) : _v(0) { std::int64_t x = (std::int64_t)(v % (std::int64_t)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template * = nullptr> constexpr static_modint(T v) : _v(0) { _v = (unsigned int)(v % umod()); } constexpr unsigned int val() const { return _v; } constexpr mint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr mint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr mint operator++(int) { mint result = *this; ++*this; return result; } constexpr mint operator--(int) { mint result = *this; --*this; return result; } constexpr mint &operator+=(const mint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr mint &operator-=(const mint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr mint &operator*=(const mint &rhs) { std::uint64_t z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } constexpr mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); } constexpr mint operator+() const { return *this; } constexpr mint operator-() const { return mint() - *this; } constexpr mint pow(std::int64_t n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } constexpr mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend constexpr mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; } friend constexpr mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; } friend constexpr mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; } friend constexpr mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; } friend constexpr bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; } friend constexpr bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; } friend std::istream &operator>>(std::istream &is, mint &rhs) { std::int64_t t; is >> t; rhs = mint(t); return is; } friend constexpr std::ostream &operator<<(std::ostream &os, const mint &rhs) { return os << rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime; }; template struct dynamic_modint : internal::modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template * = nullptr> dynamic_modint(T v) { std::int64_t x = (std::int64_t)(v % (std::int64_t)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template * = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } unsigned int val() const { return _v; } mint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint &operator+=(const mint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint &operator-=(const mint &rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint &operator*=(const mint &rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(std::int64_t n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; } friend std::istream &operator>>(std::istream &is, mint &rhs) { std::int64_t t; is >> t; rhs = mint(t); return is; } friend constexpr std::ostream &operator<<(std::ostream &os, const mint &rhs) { return os << rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); } }; template internal::barrett dynamic_modint::bt(998244353); using modint998 = static_modint<998244353>; using modint107 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template using is_static_modint = std::is_base_of; template using is_static_modint_t = std::enable_if_t::value>; template struct is_dynamic_modint : public std::false_type {}; template struct is_dynamic_modint> : public std::true_type {}; template using is_dynamic_modint_t = std::enable_if_t::value>; } // namespace internal template * = nullptr> bool has_sqrt_mod(mint x) { return x == 0 || x.pow(mint::mod() / 2) == 1; } template * = nullptr> mint sqrt_mod(mint x) { const int p = mint::mod(); if (x == 0 || x == 1) return x; if (p % 4 == 3) return x.pow(p / 4 + 1); int q = p - 1, s = 0; while (~q & 1) q >>= 1, ++s; mint z(1); while (has_sqrt_mod(z)) ++z; mint c = z.pow(q); mint t = x.pow(q); mint r = x.pow(q / 2 + 1); while (t != 1) { int m = 0; mint u = t; while (u != 1) ++m, u *= u; while (s != m) { --s; if (s == m) r *= c; c *= c; } t *= c; } return r; } #ifdef ATCODER #pragma GCC target("sse4.2,avx512f,avx512dq,avx512ifma,avx512cd,avx512bw,avx512vl,bmi2") #endif #pragma GCC optimize("Ofast,fast-math,unroll-all-loops") #include #ifndef ATCODER #pragma GCC target("sse4.2,avx2,bmi2") #endif template constexpr bool chmax(T &a, const U &b) { return a < (T)b ? a = (T)b, true : false; } template constexpr bool chmin(T &a, const U &b) { return (T)b < a ? a = (T)b, true : false; } constexpr std::int64_t INF = 1000000000000000003; constexpr int Inf = 1000000003; constexpr double EPS = 1e-7; constexpr double PI = 3.14159265358979323846; #define FOR(i, m, n) for (int i = (m); i < int(n); ++i) #define FORR(i, m, n) for (int i = (m)-1; i >= int(n); --i) #define FORL(i, m, n) for (int64_t i = (m); i < int64_t(n); ++i) #define rep(i, n) FOR (i, 0, n) #define repn(i, n) FOR (i, 1, n + 1) #define repr(i, n) FORR (i, n, 0) #define repnr(i, n) FORR (i, n + 1, 1) #define all(s) (s).begin(), (s).end() struct Sonic { Sonic() { std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout << std::fixed << std::setprecision(20); } constexpr void operator()() const {} } sonic; using namespace std; using ll = std::int64_t; using ld = long double; template std::istream &operator>>(std::istream &is, std::pair &p) { return is >> p.first >> p.second; } template std::istream &operator>>(std::istream &is, std::vector &v) { for (T &i : v) is >> i; return is; } template std::ostream &operator<<(std::ostream &os, const std::pair &p) { return os << '(' << p.first << ',' << p.second << ')'; } template std::ostream &operator<<(std::ostream &os, const std::vector &v) { for (auto it = v.begin(); it != v.end(); ++it) os << (it == v.begin() ? "" : " ") << *it; return os; } template void co(Head &&head, Tail &&...tail) { if constexpr (sizeof...(tail) == 0) std::cout << head << '\n'; else std::cout << head << ' ', co(std::forward(tail)...); } template void ce(Head &&head, Tail &&...tail) { if constexpr (sizeof...(tail) == 0) std::cerr << head << '\n'; else std::cerr << head << ' ', ce(std::forward(tail)...); } void Yes(bool is_correct = true) { std::cout << (is_correct ? "Yes\n" : "No\n"); } void No(bool is_not_correct = true) { Yes(!is_not_correct); } void YES(bool is_correct = true) { std::cout << (is_correct ? "YES\n" : "NO\n"); } void NO(bool is_not_correct = true) { YES(!is_not_correct); } void Takahashi(bool is_correct = true) { std::cout << (is_correct ? "Takahashi" : "Aoki") << '\n'; } void Aoki(bool is_not_correct = true) { Takahashi(!is_not_correct); } using Mint = modint; int main(void) { int p, r; cin >> p >> r; Mint::set_mod(p); int q; cin >> q; while (q--) { ll a, b, c; cin >> a >> b >> c; ll d = b * b - 4 * a * c; Mint x = Mint(-b) / 2 / a; if (d == 0) { co(x); } else if (has_sqrt_mod(d)) { co(x - sqrt_mod(d) / 2 / a, x + sqrt_mod(d) / 2 / a); } else { co(-1); } } return 0; }