#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template IStream &operator>>(IStream &is, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif template struct Point2d { static T_P EPS; static void set_eps(T_P e) { EPS = e; } T_P x, y; Point2d() : x(0), y(0) {} Point2d(T_P x, T_P y) : x(x), y(y) {} Point2d(const std::pair &p) : x(p.first), y(p.second) {} Point2d(const std::complex &p) : x(p.real()), y(p.imag()) {} std::complex to_complex() const noexcept { return {x, y}; } Point2d operator+(const Point2d &p) const noexcept { return Point2d(x + p.x, y + p.y); } Point2d operator-(const Point2d &p) const noexcept { return Point2d(x - p.x, y - p.y); } Point2d operator*(const Point2d &p) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x * p.x - y * p.y, x * p.y + y * p.x); } Point2d operator*(T_P d) const noexcept { return Point2d(x * d, y * d); } Point2d operator/(T_P d) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x / d, y / d); } Point2d inv() const { static_assert(std::is_floating_point::value == true); return conj() / norm2(); } Point2d operator/(const Point2d &p) const { return (*this) * p.inv(); } bool operator<(const Point2d &r) const noexcept { return x != r.x ? x < r.x : y < r.y; } bool operator==(const Point2d &r) const noexcept { return x == r.x and y == r.y; } bool operator!=(const Point2d &r) const noexcept { return !((*this) == r); } T_P dot(Point2d p) const noexcept { return x * p.x + y * p.y; } T_P det(Point2d p) const noexcept { return x * p.y - y * p.x; } T_P absdet(Point2d p) const noexcept { return std::abs(det(p)); } T_P norm() const noexcept { static_assert(std::is_floating_point::value == true); return std::sqrt(x * x + y * y); } T_P norm2() const noexcept { return x * x + y * y; } T_P arg() const noexcept { return std::atan2(y, x); } // rotate point/vector by rad Point2d rotate(T_P rad) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x * std::cos(rad) - y * std::sin(rad), x * std::sin(rad) + y * std::cos(rad)); } Point2d normalized() const { static_assert(std::is_floating_point::value == true); return (*this) / this->norm(); } Point2d conj() const noexcept { return Point2d(x, -y); } template friend IStream &operator>>(IStream &is, Point2d &p) { T_P x, y; is >> x >> y; p = Point2d(x, y); return is; } template friend OStream &operator<<(OStream &os, const Point2d &p) { return os << '(' << p.x << ',' << p.y << ')'; } }; template <> double Point2d::EPS = 1e-9; template <> long double Point2d::EPS = 1e-12; template <> long long Point2d::EPS = 0; template int ccw(const Point2d &a, const Point2d &b, const Point2d &c) { // a->b->cの曲がり方 Point2d v1 = b - a; Point2d v2 = c - a; if (v1.det(v2) > Point2d::EPS) return 1; // 左折 if (v1.det(v2) < -Point2d::EPS) return -1; // 右折 if (v1.dot(v2) < -Point2d::EPS) return 2; // c-a-b if (v1.norm2() < v2.norm2()) return -2; // a-b-c return 0; // a-c-b } // Convex hull (凸包) // return: IDs of vertices used for convex hull, counterclockwise // include_boundary: If true, interior angle pi is allowed template std::vector convex_hull(const std::vector> &ps, bool include_boundary = false) { int n = ps.size(); if (n <= 1) return std::vector(n, 0); std::vector, int>> points(n); for (size_t i = 0; i < ps.size(); i++) points[i] = std::make_pair(ps[i], i); std::sort(points.begin(), points.end()); int k = 0; std::vector, int>> qs(2 * n); auto ccw_check = [&](int c) { return include_boundary ? (c == -1) : (c <= 0); }; for (int i = 0; i < n; i++) { while (k > 1 and ccw_check(ccw(qs[k - 2].first, qs[k - 1].first, points[i].first))) k--; qs[k++] = points[i]; } for (int i = n - 2, t = k; i >= 0; i--) { while (k > t and ccw_check(ccw(qs[k - 2].first, qs[k - 1].first, points[i].first))) k--; qs[k++] = points[i]; } std::vector ret(k - 1); for (int i = 0; i < k - 1; i++) ret[i] = qs[i].second; return ret; } // Point on grid, sortable by its argument struct Point { constexpr static double eps = 1e-2; long long X, Y; double theta; Point() = default; Point(long long x, long long y) : X(x), Y(y), theta(std::atan2(y, x)) {} bool operator<(const Point &r) const { double b = theta - r.theta; return std::abs(b) > eps ? (b < 0) : (X * r.Y > r.X * Y); } bool operator==(const Point &r) const { return std::abs(theta - r.theta) < eps and X * r.Y == r.X * Y; } void rotate_pi() { theta += M_PI; X *= -1; Y *= -1; } }; int main() { const int N = 1000000; // const int N = 1000; cout << N << endl; const lint R = 1000000000; lint sum = 0, n = 0; vector vs; for (lint y = 1; n * 4 <= 1000000; ++y) { FOR(e, 1, y) { if (std::gcd(e, y - e) == 1) vs.emplace_back(e, y - e), sum += y, ++n; } } dbg(make_tuple(n, sum, vs.size())); vs.resize(N / 4); sort(vs.begin(), vs.end()); // dbg(vs.front().X); // dbg(vs.front().Y); // dbg(vs.back().X); // dbg(vs.back().Y); const double pi = acos(-1); const double r = pi * 2 / N; vector> xys; { lint x = 0, y = -R; REP(_, 4) { for (auto &p : vs) { xys.emplace_back(x, y); x += p.X; y += p.Y; swap(p.X, p.Y); p.X *= -1; // p.rotate_pi(); } } } // REP(t, N) { // double theta = r * t; // int x = roundl(cos(theta) * R); // int y = roundl(sin(theta) * R); // assert(abs(x) <= R); // assert(abs(y) <= R); // cout << x << ' ' << y << '\n'; // xys.emplace_back(x, y); // } REP(i, N) { int j = (i + 1) % N; int k = (i + 2) % N; int c = ccw(xys.at(i), xys.at(j), xys.at(k)); if (c != 1) { dbg(make_tuple(i, c, xys.at(i), xys.at(j), xys.at(k))); exit(1); } } dbg(xys.size()); dbg(convex_hull(xys, false).size()); for (auto p : xys) { assert(abs(p.x) <= R); assert(abs(p.y) <= R); cout << p.x << ' ' << p.y << '\n'; } cout.flush(); }