#ifdef ONLINE_JUDGE #pragma GCC optimize("Ofast,unroll-loops") #pragma GCC target("avx2,bmi,bmi2,lzcnt,popcnt") #endif #include #include #include #include #include #include #include using namespace std; using namespace __gnu_cxx; using namespace __gnu_pbds; template using pbds_set = tree, rb_tree_tag,tree_order_statistics_node_update>; using Trie = trie, pat_trie_tag, trie_prefix_search_node_update>; // template using heapq = __gnu_pbds::priority_queue, pairing_heap_tag>; template using heapq = std::priority_queue, greater>; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; using u128 = __uint128_t; using f128 = __float128; using ld = long double; using ui = unsigned int; using ull = unsigned long long; using pii = pair; using pll = pair; using pdd = pair; using vi = vector; using vvi = vector>; using vll = vector; using vvll = vector>; using vpii = vector; using vpll = vector; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = std::priority_queue; template using pqg = std::priority_queue, greater>; #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 lb lower_bound #define ub upper_bound #define pb push_back #define pf push_front #define eb emplace_back #define fi first #define se second #define overload4(_1, _2, _3, _4, name, ...) name #define overload3(_1, _2, _3, name, ...) name #define rep1(n) for(ll _ = 0; _ < n; ++_) #define rep2(i, n) for(ll i = 0; i < n; ++i) #define rep3(i, a, b) for(ll i = a; i < b; ++i) #define rep4(i, a, b, c) for(int i = a; i < b; i += c) #define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1) (__VA_ARGS__) #define rrep1(n) for(ll i = n; i--; ) #define rrep2(i, n) for(ll i = n; i--; ) #define rrep3(i, a, b) for(ll i = a; i > b; i--) #define rrep4(i, a, b, c) for(ll i = a; i > b; i -= c) #define rrep(...) overload4(__VA_ARGS__, rrep4, rrep3, rrep2, rrep1) (__VA_ARGS__) #define each1(i, a) for(auto&& i : a) #define each2(x, y, a) for(auto&& [x, y] : a) #define each3(x, y, z, a) for(auto&& [x, y, z] : a) #define each(...) overload4(__VA_ARGS__, each3, each2, each1) (__VA_ARGS__) #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1) (__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R) (__VA_ARGS__) #define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define len(x) (int)x.size() #define elif else if #define all1(i) begin(i), end(i) #define all2(i, a) begin(i), begin(i) + a #define all3(i, a, b) begin(i) + a, begin(i) + b #define all(...) overload3(__VA_ARGS__, all3, all2, all1) (__VA_ARGS__) #define rall1(i) rbegin(i), rend(i) #define rall2(i, a) rbegin(i), rbegin(i) + a #define rall3(i, a, b) rbegin(i) + a, rbegin(i) + b #define rall(...) overload3(__VA_ARGS__, rall3, rall2, rall1) (__VA_ARGS__) #define mst(x, a) memset(x, a, sizeof(x)) #define bitcnt(x) (__builtin_popcountll(x)) #define endl "\n" #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()), x.shrink_to_fit() #define SORT(a) sort(all(a)) #define REV(a) reverse(all(a)) 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 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template auto max(const T& a){ return *max_element(all(a)); } template auto min(const T& a){ return *min_element(all(a)); } 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}; } template T SUM(const vector &A) { T sum = 0; for (auto &&a: A) sum += a; return sum; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); for (int i = 0; i < N; i++) B[i + 1] = B[i] + A[i]; if (off == 0) B.erase(B.begin()); return B; } template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { assert(!que.empty()); T a = que.top(); que.pop(); return a; } template T POP(vc &que) { assert(!que.empty()); T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { while (iter--) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } 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); } mt19937 rng( chrono::steady_clock::now().time_since_epoch().count() ); #define Ran(a, b) rng() % ( (b) - (a) + 1 ) + (a) struct custom_hash { static uint64_t splitmix64(uint64_t x) { // http://xorshift.di.unimi.it/splitmix64.c x += 0x9e3779b97f4a7c15; x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9; x = (x ^ (x >> 27)) * 0x94d049bb133111eb; return x ^ (x >> 31); } size_t operator()(uint64_t x) const { static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count(); return splitmix64(x + FIXED_RANDOM); } size_t operator()(pair x) const { static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count(); return splitmix64(x.first + FIXED_RANDOM) ^ (splitmix64(x.second + FIXED_RANDOM) >> 1); } }; #define FASTIO #include // https://judge.yosupo.jp/submission/21623 namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; struct Pre { char num[10000][4]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i][j] = n % 10 | '0'; n /= 10; } } } } constexpr pre; inline void load() { memcpy(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(u32 &x) { rd_integer(x); } void rd(u64 &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } void rd(f128 &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d: x) rd(d); } template void rd(vc &x) { for (auto &d: x) rd(d); } void read() {} template void read(H &h, T &... t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c: s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(u32 x) { wt_integer(x); } void wt(u64 x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } void wt(f128 x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ read(__VA_ARGS__) #define U64(...) \ u64 __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); } const i128 ONE = 1; template auto print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") -> decltype(fastio::wt(*v.begin())) { for (auto it = v.begin(); it != v.end();) { fastio::wt(*it); if (++it != v.end()) fastio::wt(sep); } fastio::wt(end); } ll gcd(ll x, ll y) { if(!x) return y; if(!y) return x; int t = __builtin_ctzll(x | y); x >>= __builtin_ctzll(x); do { y >>= __builtin_ctzll(y); if (x > y) swap(x, y); y -= x; } while (y); return x << t; } ll lcm(ll x, ll y) { return x * y / gcd(x, y); } ll exgcd(ll a, ll b, ll &x, ll &y) { if(!b) return x = 1, y = 0, a; ll d = exgcd(b, a % b, x, y); ll t = x; x = y; y = t - a / b * x; return d; } ll max(ll x, ll y) { return x > y ? x : y; } ll min(ll x, ll y) { return x < y ? x : y; } ll Mod(ll x, int mod) { return (x % mod + mod) % mod; } ll pow(ll x, ll y, ll mod){ ll res = 1, cur = x; while (y) { if (y & 1) res = res * cur % mod; cur = ONE * cur * cur % mod; y >>= 1; } return res % mod; } ll probabilityMod(ll x, ll y, ll mod) { return x * pow(y, mod-2, mod) % mod; } vvi getGraph(int n, int m, bool directed = false) { vvi res(n); rep(_, 0, m) { INT(u, v); u--, v--; res[u].emplace_back(v); if(!directed) res[v].emplace_back(u); } return res; } vector getWeightedGraph(int n, int m, bool directed = false) { vector res(n); rep(_, 0, m) { INT(u, v, w); u--, v--; res[u].emplace_back(v, w); if(!directed) res[v].emplace_back(u, w); } return res; } template auto ndvector(size_t n, Args &&...args) { if constexpr (sizeof...(args) == 1) { return vector(n, args...); } else { return vector(n, ndvector(args...)); } } const ll LINF = 0x1fffffffffffffff; const ll MINF = 0x7fffffffffff; const int INF = 0x3fffffff; const int MOD = 1000000007; const int MODD = 998244353; const int N = 1e6 + 10; void solve() { INT(R, K); const ld pi = acos(-1); ld area = pi * R * R; ld s = pi * R * R / (K + 1); rep(i, 1, K + 1) { ld t = s * i; auto check = [&] (ld x) -> bool { ld y = pi - 2 * x; return area * y / (2 * pi) - R * R * sin(y) / 2 >= t; }; ld x = binary_search_real(check, -pi / 2, pi / 2); ld y = pi - 2 * x; print(R * sin(x)); } } signed main() { int T = 1; // read(T); while (T--) { solve(); } return 0; }