#include using namespace std; #define rep(i, n) for (int i = 0; i < n; i++) #define rep2(i, x, n) for (int i = x; i <= n; i++) #define rep3(i, x, n) for (int i = x; i >= n; i--) #define each(e, v) for (auto &e : v) #define pb push_back #define eb emplace_back #define all(x) x.begin(), x.end() #define rall(x) x.rbegin(), x.rend() #define sz(x) (int)x.size() using ll = long long; using pii = pair; using pil = pair; using pli = pair; using pll = pair; template bool chmax(T &x, const T &y) { return (x < y) ? (x = y, true) : false; } template bool chmin(T &x, const T &y) { return (x > y) ? (x = y, true) : false; } template int flg(T x, int i) { return (x >> i) & 1; } template void print(const vector &v, T x = 0) { int n = v.size(); for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' '); if (v.empty()) cout << '\n'; } template void printn(const vector &v, T x = 0) { int n = v.size(); for (int i = 0; i < n; i++) cout << v[i] + x << '\n'; } template int lb(const vector &v, T x) { return lower_bound(begin(v), end(v), x) - begin(v); } template int ub(const vector &v, T x) { return upper_bound(begin(v), end(v), x) - begin(v); } template void rearrange(vector &v) { sort(begin(v), end(v)); v.erase(unique(begin(v), end(v)), end(v)); } template vector id_sort(const vector &v, bool greater = false) { int n = v.size(); vector ret(n); iota(begin(ret), end(ret), 0); sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; }); return ret; } template pair operator+(const pair &p, const pair &q) { return make_pair(p.first + q.first, p.second + q.second); } template pair operator-(const pair &p, const pair &q) { return make_pair(p.first - q.first, p.second - q.second); } template istream &operator>>(istream &is, pair &p) { S a; T b; is >> a >> b; p = make_pair(a, b); return is; } template ostream &operator<<(ostream &os, const pair &p) { return os << p.first << ' ' << p.second; } struct io_setup { io_setup() { ios_base::sync_with_stdio(false); cin.tie(NULL); cout << fixed << setprecision(15); } } io_setup; const int inf = (1 << 30) - 1; const ll INF = (1LL << 60) - 1; // const int MOD = 1000000007; const int MOD = 998244353; template struct Mod_Int { int x; Mod_Int() : x(0) {} Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {} static int get_mod() { return mod; } Mod_Int &operator+=(const Mod_Int &p) { if ((x += p.x) >= mod) x -= mod; return *this; } Mod_Int &operator-=(const Mod_Int &p) { if ((x += mod - p.x) >= mod) x -= mod; return *this; } Mod_Int &operator*=(const Mod_Int &p) { x = (int)(1LL * x * p.x % mod); return *this; } Mod_Int &operator/=(const Mod_Int &p) { *this *= p.inverse(); return *this; } Mod_Int &operator++() { return *this += Mod_Int(1); } Mod_Int operator++(int) { Mod_Int tmp = *this; ++*this; return tmp; } Mod_Int &operator--() { return *this -= Mod_Int(1); } Mod_Int operator--(int) { Mod_Int tmp = *this; --*this; return tmp; } Mod_Int operator-() const { return Mod_Int(-x); } Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; } Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; } Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; } Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; } bool operator==(const Mod_Int &p) const { return x == p.x; } bool operator!=(const Mod_Int &p) const { return x != p.x; } Mod_Int inverse() const { assert(*this != Mod_Int(0)); return pow(mod - 2); } Mod_Int pow(long long k) const { Mod_Int now = *this, ret = 1; for (; k > 0; k >>= 1, now *= now) { if (k & 1) ret *= now; } return ret; } friend ostream &operator<<(ostream &os, const Mod_Int &p) { return os << p.x; } friend istream &operator>>(istream &is, Mod_Int &p) { long long a; is >> a; p = Mod_Int(a); return is; } }; using mint = Mod_Int; template struct Number_Theoretic_Transform { static int max_base; static T root; static vector r, ir; Number_Theoretic_Transform() {} static void init() { if (!r.empty()) return; int mod = T::get_mod(); int tmp = mod - 1; root = 2; while (root.pow(tmp >> 1) == 1) root++; max_base = 0; while (tmp % 2 == 0) tmp >>= 1, max_base++; r.resize(max_base), ir.resize(max_base); for (int i = 0; i < max_base; i++) { r[i] = -root.pow((mod - 1) >> (i + 2)); // r[i] := 1 の 2^(i+2) 乗根 ir[i] = r[i].inverse(); // ir[i] := 1/r[i] } } static void ntt(vector &a) { init(); int n = a.size(); assert((n & (n - 1)) == 0); assert(n <= (1 << max_base)); for (int k = n; k >>= 1;) { T w = 1; for (int s = 0, t = 0; s < n; s += 2 * k) { for (int i = s, j = s + k; i < s + k; i++, j++) { T x = a[i], y = w * a[j]; a[i] = x + y, a[j] = x - y; } w *= r[__builtin_ctz(++t)]; } } } static void intt(vector &a) { init(); int n = a.size(); assert((n & (n - 1)) == 0); assert(n <= (1 << max_base)); for (int k = 1; k < n; k <<= 1) { T w = 1; for (int s = 0, t = 0; s < n; s += 2 * k) { for (int i = s, j = s + k; i < s + k; i++, j++) { T x = a[i], y = a[j]; a[i] = x + y, a[j] = w * (x - y); } w *= ir[__builtin_ctz(++t)]; } } T inv = T(n).inverse(); for (auto &e : a) e *= inv; } static vector convolve(vector a, vector b) { if (a.empty() || b.empty()) return {}; int k = (int)a.size() + (int)b.size() - 1, n = 1; while (n < k) n <<= 1; a.resize(n), b.resize(n); ntt(a), ntt(b); for (int i = 0; i < n; i++) a[i] *= b[i]; intt(a), a.resize(k); return a; } }; template int Number_Theoretic_Transform::max_base = 0; template T Number_Theoretic_Transform::root = T(); template vector Number_Theoretic_Transform::r = vector(); template vector Number_Theoretic_Transform::ir = vector(); using NTT = Number_Theoretic_Transform; template struct Formal_Power_Series : vector { using NTT_ = Number_Theoretic_Transform; using vector::vector; Formal_Power_Series(const vector &v) : vector(v) {} Formal_Power_Series pre(int n) const { return Formal_Power_Series(begin(*this), begin(*this) + min((int)this->size(), n)); } Formal_Power_Series rev(int deg = -1) const { Formal_Power_Series ret = *this; if (deg != -1) ret.resize(deg, T(0)); reverse(begin(ret), end(ret)); return ret; } void normalize() { while (!this->empty() && this->back() == 0) this->pop_back(); } Formal_Power_Series operator-() const { Formal_Power_Series ret = *this; for (int i = 0; i < (int)ret.size(); i++) ret[i] = -ret[i]; return ret; } Formal_Power_Series &operator+=(const T &x) { if (this->empty()) this->resize(1); (*this)[0] += x; return *this; } Formal_Power_Series &operator+=(const Formal_Power_Series &v) { if (v.size() > this->size()) this->resize(v.size()); for (int i = 0; i < (int)v.size(); i++) (*this)[i] += v[i]; this->normalize(); return *this; } Formal_Power_Series &operator-=(const T &x) { if (this->empty()) this->resize(1); *this[0] -= x; return *this; } Formal_Power_Series &operator-=(const Formal_Power_Series &v) { if (v.size() > this->size()) this->resize(v.size()); for (int i = 0; i < (int)v.size(); i++) (*this)[i] -= v[i]; this->normalize(); return *this; } Formal_Power_Series &operator*=(const T &x) { for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= x; return *this; } Formal_Power_Series &operator*=(const Formal_Power_Series &v) { if (this->empty() || empty(v)) { this->clear(); return *this; } return *this = NTT_::convolve(*this, v); } Formal_Power_Series &operator/=(const T &x) { assert(x != 0); T inv = x.inverse(); for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= inv; return *this; } Formal_Power_Series &operator/=(const Formal_Power_Series &v) { if (v.size() > this->size()) { this->clear(); return *this; } int n = this->size() - (int)v.size() + 1; return *this = (rev().pre(n) * v.rev().inv(n)).pre(n).rev(n); } Formal_Power_Series &operator%=(const Formal_Power_Series &v) { return *this -= (*this / v) * v; } Formal_Power_Series &operator<<=(int x) { Formal_Power_Series ret(x, 0); ret.insert(end(ret), begin(*this), end(*this)); return *this = ret; } Formal_Power_Series &operator>>=(int x) { Formal_Power_Series ret; ret.insert(end(ret), begin(*this) + x, end(*this)); return *this = ret; } Formal_Power_Series operator+(const T &x) const { return Formal_Power_Series(*this) += x; } Formal_Power_Series operator+(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) += v; } Formal_Power_Series operator-(const T &x) const { return Formal_Power_Series(*this) -= x; } Formal_Power_Series operator-(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) -= v; } Formal_Power_Series operator*(const T &x) const { return Formal_Power_Series(*this) *= x; } Formal_Power_Series operator*(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) *= v; } Formal_Power_Series operator/(const T &x) const { return Formal_Power_Series(*this) /= x; } Formal_Power_Series operator/(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) /= v; } Formal_Power_Series operator%(const Formal_Power_Series &v) const { return Formal_Power_Series(*this) %= v; } Formal_Power_Series operator<<(int x) const { return Formal_Power_Series(*this) <<= x; } Formal_Power_Series operator>>(int x) const { return Formal_Power_Series(*this) >>= x; } T val(const T &x) const { T ret = 0; for (int i = (int)this->size() - 1; i >= 0; i--) ret *= x, ret += (*this)[i]; return ret; } Formal_Power_Series diff() const { // df/dx int n = this->size(); Formal_Power_Series ret(n - 1); for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * i; return ret; } Formal_Power_Series integral() const { // ∫f(x)dx int n = this->size(); Formal_Power_Series ret(n + 1); for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / (i + 1); return ret; } Formal_Power_Series inv(int deg) const { // 1/f(x) (f[0] != 0) assert((*this)[0] != T(0)); Formal_Power_Series ret(1, (*this)[0].inverse()); for (int i = 1; i < deg; i <<= 1) { Formal_Power_Series f = pre(2 * i), g = ret; f.resize(2 * i), g.resize(2 * i); NTT_::ntt(f), NTT_::ntt(g); Formal_Power_Series h(2 * i); for (int j = 0; j < 2 * i; j++) h[j] = f[j] * g[j]; NTT_::intt(h); for (int j = 0; j < i; j++) h[j] = 0; NTT_::ntt(h); for (int j = 0; j < 2 * i; j++) h[j] *= g[j]; NTT_::intt(h); for (int j = 0; j < i; j++) h[j] = 0; ret -= h; } ret.resize(deg); return ret; } Formal_Power_Series inv() const { return inv(this->size()); } Formal_Power_Series log(int deg) const { // log(f(x)) (f[0] = 1) assert((*this)[0] == 1); Formal_Power_Series ret = (diff() * inv(deg)).pre(deg - 1).integral(); ret.resize(deg); return ret; } Formal_Power_Series log() const { return log(this->size()); } Formal_Power_Series exp(int deg) const { // exp(f(x)) (f[0] = 0) assert((*this)[0] == 0); Formal_Power_Series inv; inv.reserve(deg + 1); inv.push_back(0), inv.push_back(1); auto inplace_integral = [&](Formal_Power_Series &F) -> void { int n = F.size(); int mod = T::get_mod(); while ((int)inv.size() <= n) { int i = inv.size(); inv.push_back((-inv[mod % i]) * (mod / i)); } F.insert(begin(F), 0); for (int i = 1; i <= n; i++) F[i] *= inv[i]; }; auto inplace_diff = [](Formal_Power_Series &F) -> void { if (F.empty()) return; F.erase(begin(F)); T coeff = 1, one = 1; for (int i = 0; i < (int)F.size(); i++) { F[i] *= coeff; coeff += one; } }; Formal_Power_Series ret{1, this->size() > 1 ? (*this)[1] : 0}, c{1}, z1, z2{1, 1}; for (int m = 2; m < deg; m *= 2) { auto y = ret; y.resize(2 * m); NTT_::ntt(y); z1 = z2; Formal_Power_Series z(m); for (int i = 0; i < m; i++) z[i] = y[i] * z1[i]; NTT_::intt(z); fill(begin(z), begin(z) + m / 2, 0); NTT_::ntt(z); for (int i = 0; i < m; i++) z[i] *= -z1[i]; NTT_::intt(z); c.insert(end(c), begin(z) + m / 2, end(z)); z2 = c, z2.resize(2 * m); NTT_::ntt(z2); Formal_Power_Series x(begin(*this), begin(*this) + min((int)this->size(), m)); inplace_diff(x); x.push_back(0); NTT_::ntt(x); for (int i = 0; i < m; i++) x[i] *= y[i]; NTT_::intt(x); x -= ret.diff(), x.resize(2 * m); for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = 0; NTT_::ntt(x); for (int i = 0; i < 2 * m; i++) x[i] *= z2[i]; NTT_::intt(x); x.pop_back(); inplace_integral(x); for (int i = m; i < min((int)this->size(), 2 * m); i++) x[i] += (*this)[i]; fill(begin(x), begin(x) + m, 0); NTT_::ntt(x); for (int i = 0; i < 2 * m; i++) x[i] *= y[i]; NTT_::intt(x); ret.insert(end(ret), begin(x) + m, end(x)); } ret.resize(deg); return ret; } Formal_Power_Series exp() const { return exp(this->size()); } Formal_Power_Series pow(long long k, int deg) const { // f(x)^k int n = this->size(); for (int i = 0; i < n; i++) { if ((*this)[i] == 0) continue; T rev = (*this)[i].inverse(); Formal_Power_Series C(*this * rev), D(n - i, 0); for (int j = i; j < n; j++) D[j - i] = C[j]; D = (D.log() * k).exp() * ((*this)[i].pow(k)); Formal_Power_Series E(deg, 0); if (i > 0 && k > deg / i) return E; long long S = i * k; for (int j = 0; j + S < deg && j < D.size(); j++) E[j + S] = D[j]; E.resize(deg); return E; } return Formal_Power_Series(deg, 0); } Formal_Power_Series pow(long long k) const { return pow(k, this->size()); } Formal_Power_Series Taylor_shift(T c) const { // f(x+c) int n = this->size(); vector ifac(n, 1); Formal_Power_Series f(n), g(n); for (int i = 0; i < n; i++) { f[n - 1 - i] = (*this)[i] * ifac[n - 1]; if (i < n - 1) ifac[n - 1] *= i + 1; } ifac[n - 1] = ifac[n - 1].inverse(); for (int i = n - 1; i > 0; i--) ifac[i - 1] = ifac[i] * i; T pw = 1; for (int i = 0; i < n; i++) { g[i] = pw * ifac[i]; pw *= c; } f *= g; Formal_Power_Series b(n); for (int i = 0; i < n; i++) b[i] = f[n - 1 - i] * ifac[i]; return b; } }; using fps = Formal_Power_Series; template vector> subproduct_tree(const vector &xs) { int n = xs.size(); int k = 1; while (k < n) k <<= 1; vector> g(2 * k, {1}); for (int i = 0; i < n; i++) g[k + i] = {-xs[i], 1}; for (int i = k - 1; i > 0; i--) g[i] = g[2 * i] * g[2 * i + 1]; return g; } template Formal_Power_Series polynomial_interpolation(const vector &xs, const vector &ys) { int n = xs.size(); assert(ys.size() == n); vector> g = subproduct_tree(xs); int k = g.size() / 2; vector> f(2 * k); f[1] = g[1].diff(); for (int i = 2; i < k + n; i++) f[i] = f[i / 2] % g[i]; for (int i = 0; i < n; i++) f[k + i][0] = ys[i] / f[k + i][0]; for (int i = k - 1; i > 0; i--) f[i] = f[2 * i] * g[2 * i + 1] + f[2 * i + 1] * g[2 * i]; f[1].resize(n); return f[1]; } int main() { int N; mint X; cin >> N >> X; vector x(N), y(N); rep(i, N) cin >> x[i] >> y[i]; rep(i, N) { if (X == x[i]) { vector xs, ys; rep(j, N) { if (j != i) xs.eb(x[j]), ys.eb(y[j]); } fps f = polynomial_interpolation(xs, ys); mint pw = 1, ans = 0; rep(i, sz(f)) { ans += pw * f[i]; pw *= X; } ans += y[i] * (N - 1); cout << ans << '\n'; return 0; } } vector iv(N); rep(i, N) iv[i] = (X - x[i]).inverse(); mint ans = 0; rep(i, N) { mint L = 1, R = 1; rep(j, i) L *= x[i] - x[j]; rep2(j, i + 1, N - 1) R *= x[i] - x[j]; L = L.inverse(), R = R.inverse(); rep(j, i) L *= X - x[j]; rep2(j, i + 1, N - 1) R *= X - x[j]; rep(j, i) { mint tmp = y[i] * L * (x[i] - x[j]) * iv[j] * R; ans += tmp; } rep2(j, i + 1, N - 1) { mint tmp = y[i] * L * R * (x[i] - x[j]) * iv[j]; ans += tmp; } } cout << ans << '\n'; }