結果
| 問題 |
No.132 点と平面との距離
|
| コンテスト | |
| ユーザー |
 T101010101
|
| 提出日時 | 2024-11-22 19:34:27 |
| 言語 | C++17(gcc12) (gcc 12.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 27 ms / 5,000 ms |
| コード長 | 37,730 bytes |
| コンパイル時間 | 19,182 ms |
| コンパイル使用メモリ | 380,080 KB |
| 最終ジャッジ日時 | 2025-02-25 05:52:58 |
|
ジャッジサーバーID （参考情報） |
judge4 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| other | AC * 3 |
ソースコード
#pragma region Macros
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,fma,mmx,abm,bmi,bmi2,popcnt,lzcnt")
#pragma GCC target("avx2") // CF, CodeChef, HOJ ではコメントアウト
#include <bits/extc++.h>
// #include <atcoder/all>
// using namespace atcoder;
using namespace std;
using namespace __gnu_pbds;
// #include <boost/multiprecision/cpp_dec_float.hpp>
// #include <boost/multiprecision/cpp_int.hpp>
// namespace mp = boost::multiprecision;
// using Bint = mp::cpp_int;
// using Bdouble = mp::number<mp::cpp_dec_float<256>>;
// Bdouble Beps = 0.00000000000000000000000000000001; // 1e-32
// const bool equals(Bdouble a, Bdouble b) { return mp::fabs(a - b) < Beps; }
#define pb emplace_back
#define int ll
#define endl '\n'
// #define sqrt __builtin_sqrtl
// #define cbrt __builtin_cbrtl
// #define hypot __builtin_hypotl
using ll = long long;
using ld = long double;
const ld PI = acosl(-1);
const int INF = 1 << 30;
const ll INFL = 1LL << 61;
const int MOD = 998244353;
// const int MOD = 1000000007;
const ld EPS = 1e-10;
const bool equals(ld a, ld b) { return fabs((a) - (b)) < EPS; }
const vector<int> dx = {0, 1, 0, -1, 1, 1, -1, -1, 0}; // → ↓ ← ↑ ↘ ↙ ↖ ↗ 自
const vector<int> dy = {1, 0, -1, 0, 1, -1, -1, 1, 0};
#define EC int
struct Edge {
int from, to;
EC cost;
Edge() {}
// Edge() : from(-1), to(-1), cost(-1) {}
Edge(int to, EC cost) : to(to), cost(cost) {}
Edge(int from, int to, EC cost) : from(from), to(to), cost(cost) {}
bool operator ==(const Edge &e) {
return this->from == e.from && this->to == e.to && this->cost == e.cost;
}
bool operator !=(const Edge &e) {
return this->from != e.from or this->to != e.to or this->cost != e.cost;
}
bool operator <(const Edge &e) { return this->cost < e.cost; }
bool operator >(const Edge &e) { return this->cost > e.cost; }
};
chrono::system_clock::time_point start;
__attribute__((constructor))
void constructor() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
cout << fixed << setprecision(10);
start = chrono::system_clock::now();
}
random_device seed_gen;
mt19937_64 rng(seed_gen());
uniform_int_distribution<int> dist_x(0, 1e9);
struct RNG {
unsigned Int() {
return dist_x(rng);
}
unsigned Int(unsigned l, unsigned r) {
return dist_x(rng) % (r - l + 1) + l;
}
ld Double() {
return ld(dist_x(rng)) / 1e9;
}
} rnd;
namespace bit_function {
using i64 = ll;
// using i64 = uint64_t;
// bit演算, x==0の場合は例外処理した方がよさそう. 区間は [l, r)
i64 lrmask(int l, int r) { return (1LL << r) - (1LL << l); }
i64 sub_bit(i64 x, int l, int r) { i64 b = x & ((1LL << r) - (1LL << l)); return b >> l; } // r溢れ可
i64 bit_width(i64 x) { return 64 - __builtin_clzll(x) + (x == 0); }
i64 popcount(i64 x) { return __builtin_popcountll(x); }
i64 popcount(i64 x, int l, int r) { return __builtin_popcountll(sub_bit(x, l, r)); }
i64 unpopcount(i64 x) { return bit_width(x) - __builtin_popcountll(x); } // 最上位bitより下のみ
i64 unpopcount(i64 x, int l, int r) { return r - l - __builtin_popcountll(sub_bit(x, l, r)); } // 最上位bitより上も含まれうる
bool is_pow2(i64 x) { return __builtin_popcountll(x) == 1; } // xが負のときは常にfalse
bool is_pow4(i64 x) { return __builtin_popcountll(x) == 1 && __builtin_ctz(x) % 2 == 0; }
//bool is_pow4(ll x) { return __builtin_popcountll(x) == 1 && (x&0x55555555); }
int top_bit(i64 x) { return 63 - __builtin_clzll(x);} // 2^kの位 (x > 0)
int bot_bit(i64 x) { return __builtin_ctzll(x);} // 2^kの位 (x > 0)
int next_bit(i64 x, int k) { // upper_bound
x >>= (k + 1);
int pos = k + 1;
while (x > 0) {
if (x & 1) return pos;
x >>= 1;
pos++;
}
return -1;
}
int prev_bit(i64 x, int k) {
// k = min(k, bit_width(x)); ?
int pos = 0;
while (x > 0 && pos < k) {
if (x & 1) {
if (pos < k) return pos;
}
x >>= 1;
pos++;
}
return -1;
}
int kth_bit(i64 x, int k) { // kは1-indexed
int pos = 0, cnt = 0;
while (x > 0) {
if (x & 1) {
cnt++;
if (cnt == k) return pos;
}
x >>= 1;
pos++;
}
return -1;
}
i64 msb(i64 x) { if (x == 0) return 0; return 1LL << (63 - __builtin_clzll(x)); } // mask
i64 lsb(i64 x) { return (x & -x); } // mask
int countl_zero(i64 x) { return __builtin_clzll(x); }
int countl_one(i64 x) { // countl_oneと定義が異なるので注意
i64 ret = 0, k = 63 - __builtin_clzll(x);
while (k != -1 && (x & (1LL << k))) { k--; ret++; }
return ret;
}
int countr_zero(i64 x) { return __builtin_ctzll(x); } // x=0のとき64
int countr_one(i64 x) { int ret = 0; while (x & 1) { x >>= 1; ret++; } return ret; }
// int countr_one(ll x) { return __builtin_popcount(x ^ (x & -~x));
i64 l_one(i64 x) { // 最上位で連なってる1のmask
if (x == 0) return 0;
i64 ret = 0, k = 63 - __builtin_clzll(x);
while (k != -1 && (x & (1LL << k))) { ret += 1LL << k; k--; }
return ret;
}
int floor_log2(i64 x) { return 63 - __builtin_clzll(x); } // top_bit
int ceil_log2(i64 x) { return 64 - __builtin_clzll(x - 1); }
i64 bit_floor(i64 x) { if (x == 0) return 0; return 1LL << (63 - __builtin_clzll(x)); } // msb
i64 bit_ceil(i64 x) { if (x == 0) return 0; return 1LL << (64 - __builtin_clzll(x - 1)); }
i64 rotl(i64 x, int k) { // 有効bit内でrotate. オーバーフロー注意
i64 w = bit_width(x); k %= w;
return ((x << k) | (x >> (w - k))) & ((1LL << w) - 1);
}
// i64 rotl(i64 x, i64 l, i64 m, i64 r) {}
i64 rotr(i64 x, int k) {
i64 w = bit_width(x); k %= w;
return ((x >> k) | (x << (w - k))) & ((1LL << w) - 1);
}
// i64 rotr(i64 x, i64 l, i64 m, i64 r) {}
i64 bit_reverse(i64 x) { // 有効bit内で左右反転
i64 r = 0, w = bit_width(x);
for (i64 i = 0; i < w; i++) r |= ((x >> i) & 1) << (w - i - 1);
return r;
}
// i64 bit_reverse(i64 x, int l, int r) {}
bool is_palindrome(i64 x) { return x == bit_reverse(x); }
bool is_palindrome(i64 x, int l, int r) { i64 b = sub_bit(x, l, r); return b == bit_reverse(b); }
i64 concat(i64 a, i64 b) { return (a << bit_width(b)) | b; } // オーバーフロー注意
i64 erase(i64 x, int l, int r) { return x >> r << l | x & ((1LL << l) - 1); } // [l, r) をカット
i64 hamming(i64 a, i64 b) { return __builtin_popcountll(a ^ b); }
i64 hamming(i64 a, i64 b, int l, int r) { return __builtin_popcountll(sub_bit(a, l, r) ^ sub_bit(b, l, r)); }
i64 compcount(i64 x) { return (__builtin_popcountll(x ^ (x >> 1)) + (x & 1)) / 2; }
i64 compcount2(i64 x) { return compcount(x & (x >> 1)); } // 長さ2以上の連結成分の個数
i64 adjacount(i64 x) { return __builtin_popcountll(x & (x >> 1)); } // 隣接する1のペアの個数
i64 next_combination(i64 x) {
i64 t = x | (x - 1); return (t + 1) | (((~t & -~t) - 1) >> (__builtin_ctzll(x) + 1));
}
} using namespace bit_function;
namespace util_function {
namespace Std = std;
__int128_t POW(__int128_t x, int n) {
__int128_t ret = 1;
assert(n >= 0);
if (x == 1 or n == 0) ret = 1;
else if (x == -1 && n % 2 == 0) ret = 1;
else if (x == -1) ret = -1;
else if (n % 2 == 0) {
// assert(x < INFL);
ret = POW(x * x, n / 2);
} else {
// assert(x < INFL);
ret = x * POW(x, n - 1);
}
return ret;
}
int per(int x, int y) { // x = qy + r (0 <= r < y) を満たすq
assert(y != 0);
if (x >= 0 && y > 0) return x / y;
if (x >= 0 && y < 0) return x / y - (x % y < 0);
if (x < 0 && y < 0) return x / y + (x % y < 0);
return x / y - (x % y < 0); // (x < 0 && y > 0)
}
int mod(int x, int y) { // x = qy + r (0 <= r < y) を満たすr
assert(y != 0);
return x - y * per(x, y);
} // https://yukicoder.me/problems/no/2781
int floor(int x, int y) { // (ld)x / y 以下の最大の整数
assert(y != 0);
if (y < 0) x = -x, y = -y;
return x >= 0 ? x / y : (x + 1) / y - 1;
}
int ceil(int x, int y) { // (ld)x / y 以上の最小の整数
assert(y != 0);
if (y < 0) x = -x, y = -y;
return x > 0 ? (x - 1) / y + 1 : x / y;
}
int round(int x, int y) { // (ld)x / y を小数第1位について四捨五入
assert(y != 0);
return (x * 2 + y) / (y * 2);
}
int round(int x, int y, int k) { // (ld)x / y を10^kの位に関して四捨五入
assert(y != 0 && k >= 0);
if (k == 0) return (x * 2 + y) / (y * 2);
x /= y * POW(10, k - 1);
if (x % 10 >= 5) return (x + 10 - x % 10) * POW(10, k - 1);
return x * POW(10, k - 1);
}
int round2(int x, int y) { // 五捨五超入 // 未verify
assert(y != 0);
if (y < 0) y = -y, x = -x;
int z = x / y;
if ((z * 2 + 1) * y <= y * 2) z++;
return z;
}
ld round(ld x, int k) { // xを10^kの位に関して四捨五入.
// x += EPS;
ld d = pow(10, -k);
return Std::round(x * d) / d;
}
ld floor(ld x, int k) { // xを10^kの位に関してflooring
// x += EPS;
ld d = pow(10, -k);
return Std::floor(x * d) / d; // 未verify
}
ld ceil(ld x, int k) { // xを10^kの位に関してceiling
// x -= EPS;
ld d = pow(10, -k);
return Std::ceil(x * d) / d; // 未verify
}
// int kth(int x, int y, int k) { // x / yの10^kの位の桁
// }
int floor(ld x, ld y) { // 誤差対策TODO
assert(!equals(y, 0));
return Std::floor(x / y);
// floor(x) = ceil(x - 1) という話も
}
int ceil(ld x, ld y) { // 誤差対策TODO // ceil(p/q) = -floor(-(p/q))らしい
assert(!equals(y, 0));
return Std::ceil(x / y);
// ceil(x) = floor(x + 1)
}
int perl(ld x, ld y) { // x = qy + r (0 <= r < y, qは整数) を満たす q
// 未verify. 誤差対策TODO. EPS外してもいいかも。
assert(!equals(y, 0));
if (x >= 0 && y > 0) return Std::floor(x / y)+EPS;
if (x >= 0 && y < 0) return -Std::floor(x / fabs(y));
if (x < 0 && y < 0) return Std::floor(x / y) + (x - Std::floor(x/y)*y < -EPS);
return Std::floor(x / y) - (x - Std::floor(x/y)*y < -EPS); // (x < 0 && y > 0)
}
ld modl(ld x, ld y) { // x = qy + r (0 <= r < y, qは整数) を満たす r
// 未verify. 誤差対策TODO. -0.0が返りうる。
assert(!equals(y, 0));
if (x >= 0) return x - fabs(y)*fabs(per(x, y));
return x - fabs(y)*floor(x, fabs(y));
}
int seisuu(ld x) { return (int)x; } // 整数部分. 誤差対策TODO
int modf(ld x) {
if (x < 0) return ceill(x);
else return floorl(x);
}
// 正なら+EPS, 負なら-EPSしてから、文字列に直して小数点以下を捨てる?
int seisuu(int x, int y) {
assert(y != 0);
return x / y;
}
int seisuu(ld x, ld y) { // 誤差対策TODO
assert(!equals(y, 0));
return (int)(x / y);
}
int floor_log(int base, int x) {
assert(base >= 2);
int ret = 0, now = 1;
while (now <= x) {
now *= base;
if (now <= x) ret++;
}
return ret;
}
int ceil_log(int base, int x) {
assert(base >= 2);
int ret = 0, now = 1;
while (now < x) {
now *= base;
ret++;
}
return ret;
}
template <class T> pair<T, T> max(const pair<T, T> &a, const pair<T, T> &b) {
if (a.first > b.first or a.first == b.first && a.second > b.second) return a;
return b;
}
template <class T> pair<T, T> min(const pair<T, T> &a, const pair<T, T> &b) {
if (a.first < b.first or a.first == b.first && a.second < b.second) return a;
return b;
}
template <class T> bool chmax(T &a, const T &b) {
if (a < b) { a = b; return true; } return false;
}
template <class T> bool chmin(T &a, const T &b) {
if (a > b) { a = b; return true; } return false;
}
template <class T> bool chmax(pair<T, T> &a, const pair<T, T> &b) {
if (a.first < b.first or a.first == b.first && a.second < b.second) { a = b; return true; }
return false;
}
template <class T> bool chmin(pair<T, T> &a, const pair<T, T> &b) {
if (a.first > b.first or a.first == b.first && a.second > b.second) { a = b; return true; }
return false;
}
template <class T> T mid(T a, T b, T c) { // 誤差対策TODO
return a + b + c - Std::max({a, b, c}) - Std::min({a, b, c});
}
template <typename T, typename... Args>
void Sort(T& a, T& b, T& c, Args&... args) {
vector<T> vec = {a, b, c, args...};
sort(vec.begin(), vec.end());
auto it = vec.begin();
a = *it++; b = *it++; c = *it++;
int dummy[] = { (args = *it++, 0)... };
static_cast<void>(dummy);
}
template <typename T, typename... Args>
void Sortr(T& a, T& b, T& c, Args&... args) {
vector<T> vec = {a, b, c, args...};
sort(vec.rbegin(), vec.rend());
auto it = vec.begin();
a = *it++; b = *it++; c = *it++;
int dummy[] = { (args = *it++, 0)... };
static_cast<void>(dummy);
}
template <class T>
void sort(vector<T> &A, vector<T> &B) {
vector<pair<T, T>> P(A.size());
for (int i = 0; i < A.size(); i++) P[i] = {A[i], B[i]};
sort(P.begin(), P.end());
for (int i = 0; i < A.size(); i++) A[i] = P[i].first, B[i] = P[i].second;
}
istream &operator >>(istream &is, __int128_t& x) {
string S; is >> S;
__int128_t ret = 0;
int f = 1;
if (S[0] == '-') f = -1;
for (int i = 0; i < S.length(); i++)
if ('0' <= S[i] && S[i] <= '9')
ret = ret * 10 + S[i] - '0';
x = ret * f;
return (is);
}
ostream &operator <<(ostream &os, __int128_t x) {
ostream::sentry s(os);
if (s) {
__uint128_t tmp = x < 0 ? -x : x;
char buffer[128]; char *d = end(buffer);
do {
--d; *d = "0123456789"[tmp % 10]; tmp /= 10;
} while (tmp != 0);
if (x < 0) { --d; *d = '-'; }
int len = end(buffer) - d;
if (os.rdbuf()->sputn(d, len) != len) os.setstate(ios_base::badbit);
}
return os;
}
__int128_t sto128(const string &S) {
__int128_t ret = 0; int f = 1;
if (S[0] == '-') f = -1;
for (int i = 0; i < S.length(); i++)
if ('0' <= S[i] && S[i] <= '9') ret = ret * 10 + S[i] - '0';
return ret * f;
}
__int128_t gcd(__int128_t a, __int128_t b) { return b ? gcd(b, a % b) : a; }
__int128_t lcm(__int128_t a, __int128_t b) {
return a / gcd(a, b) * b;
// lcmが__int128_tに収まる必要あり
}
string to_string(double x, int k) { // 小数第k+1を四捨五入して小数第k位までを出力
// 切り捨てがほしい場合は to_string(x, k+1) として pop_back() すればよい?
ostringstream os;
os << fixed << setprecision(k) << x;
return os.str();
}
string to_string(__int128_t x) {
string ret = "";
if (x < 0) { ret += "-"; x *= -1; }
while (x) { ret += (char)('0' + x % 10); x /= 10; }
reverse(ret.begin(), ret.end());
return ret;
}
string to_string(char c) { string s = ""; s += c; return s; }
} using namespace util_function;
struct custom_hash {
static uint64_t splitmix64(uint64_t x) {
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);
}
};
template<class T> size_t HashCombine(const size_t seed,const T &v) {
return seed^(hash<T>()(v)+0x9e3779b9+(seed<<6)+(seed>>2));
}
template<class T,class S> struct hash<pair<T,S>>{
size_t operator()(const pair<T,S> &keyval) const noexcept {
return HashCombine(hash<T>()(keyval.first), keyval.second);
}
};
template<class T> struct hash<vector<T>>{
size_t operator()(const vector<T> &keyval) const noexcept {
size_t s=0;
for (auto&& v: keyval) s=HashCombine(s,v);
return s;
}
};
template<int N> struct HashTupleCore{
template<class Tuple> size_t operator()(const Tuple &keyval) const noexcept{
size_t s=HashTupleCore<N-1>()(keyval);
return HashCombine(s,get<N-1>(keyval));
}
};
template <> struct HashTupleCore<0>{
template<class Tuple> size_t operator()(const Tuple &keyval) const noexcept{ return 0; }
};
template<class... Args> struct hash<tuple<Args...>>{
size_t operator()(const tuple<Args...> &keyval) const noexcept {
return HashTupleCore<tuple_size<tuple<Args...>>::value>()(keyval);
}
};
template<typename T>
class Compress {
public:
int sz = 0;
vector<T> uniqV;
Compress() {}
template<typename... Vecs>
Compress(const Vecs&... vecs) {
(uniqV.insert(uniqV.end(), vecs.begin(), vecs.end()), ...);
sort(uniqV.begin(), uniqV.end());
uniqV.erase(unique(uniqV.begin(), uniqV.end()), uniqV.end());
sz = uniqV.size();
}
vector<int> zip(const vector<T> &V) {
vector<int> ret(V.size());
for (int i = 0; i < V.size(); i++) {
ret[i] = encode(V[i]);
}
return ret;
}
vector<T> unzip(const vector<int> &V) {
vector<T> ret(V.size());
for (int i = 0; i < V.size(); i++) {
ret[i] = decode(V[i]);
}
return ret;
}
int size() { return sz; }
int encode(T x) {
auto it = lower_bound(uniqV.begin(), uniqV.end(), x);
return it - uniqV.begin();
}
T decode(int x) {
if (x < 0 or x >= uniqV.size()) return -1; // xが範囲外の場合
return uniqV[x];
}
};
class UnionFind {
public:
UnionFind() = default;
UnionFind(int N) : par(N), sz(N, 1) {
iota(par.begin(), par.end(), 0);
}
int root(int x) {
if (par[x] == x) return x;
return (par[x] = root(par[x]));
}
bool unite(int x, int y) {
int rx = root(x);
int ry = root(y);
if (rx == ry) return false;
if (sz[rx] < sz[ry]) swap(rx, ry);
sz[rx] += sz[ry];
par[ry] = rx;
return true;
}
bool issame(int x, int y) { return (root(x) == root(y)); }
int size(int x) { return sz[root(x)]; }
vector<vector<int>> groups(int N) {
vector<vector<int>> G(N);
for (int x = 0; x < N; x++) {
G[root(x)].push_back(x);
}
G.erase( remove_if(G.begin(), G.end(),
[&](const vector<int>& V) { return V.empty(); }), G.end());
return G;
}
private:
vector<int> par, sz;
};
template<typename T> struct BIT {
int N; // 要素数
vector<T> bit[2]; // データの格納先
BIT(int N_, int x = 0) {
N = N_ + 1;
bit[0].assign(N, 0); bit[1].assign(N, 0);
if (x != 0) {
for (int i = 0; i < N; i++) add(i, x);
}
}
BIT(const vector<T> &A) {
N = A.size() + 1;
bit[0].assign(N, 0); bit[1].assign(N, 0);
for (int i = 0; i < (int)A.size(); i++) add(i, A[i]);
}
void add_sub(int p, int i, T x) {
while (i < N) { bit[p][i] += x; i += (i & -i); }
}
void add(int l, int r, T x) {
add_sub(0, l + 1, -x * l); add_sub(0, r + 1, x * r);
add_sub(1, l + 1, x); add_sub(1, r + 1, -x);
}
void add(int i, T x) { add(i, i + 1, x); }
T sum_sub(int p, int i) {
T ret = 0;
while (i > 0) { ret += bit[p][i]; i -= (i & -i); }
return ret;
}
T sum(int i) { return sum_sub(0, i) + sum_sub(1, i) * i; }
T sum(int l, int r) { return sum(r) - sum(l); }
T get(int i) { return sum(i, i + 1); }
void set(int i, T x) { T s = get(i); add(i, -s + x); }
};
template<int mod> class Modint {
public:
int val = 0;
Modint(int x = 0) { while (x < 0) x += mod; val = x % mod; }
Modint(const Modint &r) { val = r.val; }
Modint operator -() { return Modint(-val); } // 単項
Modint operator +(const Modint &r) { return Modint(*this) += r; }
Modint operator +(const int &q) { Modint r(q); return Modint(*this) += r; }
Modint operator -(const Modint &r) { return Modint(*this) -= r; }
Modint operator -(const int &q) { Modint r(q); return Modint(*this) -= r; }
Modint operator *(const Modint &r) { return Modint(*this) *= r; }
Modint operator *(const int &q) { Modint r(q); return Modint(*this) *= r; }
Modint operator /(const Modint &r) { return Modint(*this) /= r; }
Modint operator /(const int &q) { Modint r(q); return Modint(*this) /= r; }
Modint& operator ++() { val++; if (val >= mod) val -= mod; return *this; } // 前置
Modint operator ++(signed) { ++*this; return *this; } // 後置
Modint& operator --() { val--; if (val < 0) val += mod; return *this; }
Modint operator --(signed) { --*this; return *this; }
Modint &operator +=(const Modint &r) { val += r.val; if (val >= mod) val -= mod; return *this; }
Modint &operator +=(const int &q) { Modint r(q); val += r.val; if (val >= mod) val -= mod; return *this; }
Modint &operator -=(const Modint &r) { if (val < r.val) val += mod; val -= r.val; return *this; }
Modint &operator -=(const int &q) { Modint r(q); if (val < r.val) val += mod; val -= r.val; return *this; }
Modint &operator *=(const Modint &r) { val = val * r.val % mod; return *this; }
Modint &operator *=(const int &q) { Modint r(q); val = val * r.val % mod; return *this; }
Modint &operator /=(const Modint &r) {
int a = r.val, b = mod, u = 1, v = 0;
while (b) {int t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v);}
val = val * u % mod; if (val < 0) val += mod;
return *this;
}
Modint &operator /=(const int &q) {
Modint r(q); int a = r.val, b = mod, u = 1, v = 0;
while (b) {int t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v);}
val = val * u % mod; if (val < 0) val += mod;
return *this;
}
bool operator ==(const Modint& r) { return this -> val == r.val; }
bool operator <(const Modint& r) { return this -> val < r.val; }
bool operator >(const Modint& r) { return this -> val > r.val; }
bool operator !=(const Modint& r) { return this -> val != r.val; }
friend istream &operator >>(istream &is, Modint& x) {
int t; is >> t; x = t; return (is);
}
friend ostream &operator <<(ostream &os, const Modint& x) {
return os << x.val;
}
};
using mint = Modint<MOD>;
mint modpow(const mint &x, int n) {
if (n < 0) return (mint)1 / modpow(x, -n); // 未verify
assert(n >= 0);
if (n == 0) return 1;
mint t = modpow(x, n / 2);
t = t * t;
if (n & 1) t = t * x;
return t;
}
int modpow(__int128_t x, int n, int mod) {
if (n == 0 && mod == 1) return 0;
assert(n >= 0 && mod > 0); // TODO: n <= -1
__int128_t ret = 1;
while (n > 0) {
if (n % 2 == 1) ret = ret * x % mod;
x = x * x % mod;
n /= 2;
}
return ret;
}
// int modinv(__int128_t x, int mod) { //
// assert(mod > 0);
// // assert(x > 0);
// if (x == 1 or x == 0) return 1;
// return mod - modinv(mod % x, mod) * (mod / x) % mod;
// }
vector<mint> _fac, _finv, _inv;
void COMinit(int N) {
_fac.resize(N + 1); _finv.resize(N + 1); _inv.resize(N + 1);
_fac[0] = _fac[1] = 1; _finv[0] = _finv[1] = 1; _inv[1] = 1;
for (int i = 2; i <= N; i++) {
_fac[i] = _fac[i-1] * mint(i);
_inv[i] = -_inv[MOD % i] * mint(MOD / i);
_finv[i] = _finv[i - 1] * _inv[i];
}
}
mint FAC(int N) {
if (N < 0) return 0; return _fac[N];
}
mint FACinv(int N) {
if (N < 0) return 0; return _finv[N];
}
mint COM(int N, int K) {
if (N < K) return 0; if (N < 0 or K < 0) return 0;
return _fac[N] * _finv[K] * _finv[N - K];
}
mint COMinv(int N, int K) {
if (N < K) return 0; if (N < 0 or K < 0) return 0;
return _finv[N] * _fac[K] * _fac[N - K];
}
mint MCOM(const vector<int> &V) {
int N = 0;
for (int i = 0; i < V.size(); i++) N += V[i];
mint ret = _fac[N];
for (int i = 0; i < V.size(); i++) ret *= _finv[V[i]];
return ret;
}
mint PERM(int N, int K) {
if (N < K) return 0; if (N < 0 or K < 0) return 0;
return _fac[N] * _finv[N - K];
}
mint NHK(int N, int K) { // initのサイズに注意
if (N == 0 && K == 0) return 1;
return COM(N + K - 1, K);
}
#pragma endregion
struct Point3d {
double x, y, z;
Point3d() {}
Point3d(double x, double y, double z) : x(x), y(y), z(z) {}
Point3d operator+(const Point3d &p) { return Point3d(x+p.x, y+p.y, z+p.z); }
Point3d operator-(const Point3d &p) { return Point3d(x-p.x, y-p.y, z-p.z); }
Point3d operator*(const double &k) { return Point3d(x*k, y*k, z*k); }
Point3d operator/(const double &k) { return Point3d(x/k, y/k, z/k); }
friend istream& operator>>(istream &is, Point3d &p) {
is >> p.x >> p.y >> p.z; return is;
}
friend ostream& operator<<(ostream& os, Point3d &p) {
os << p.x << " " << p.y << " " << p.z; return os;
}
bool operator==(const Point3d &p) const {
return equals(x, p.x) && equals(y, p.y) && equals(z, p.z);
}
bool operator<(const Point3d &p) const {
if (!equals(x, p.x)) return x < p.x;
if (!equals(y, p.y)) return y < p.y;
if (!equals(z, p.z)) return z < p.z;
return false;
}
};
using Vector3d = Point3d;
int sign(double x) { return x < -EPS ? -1 : x > EPS; } // -1(負)/0/1(正)
double norm(Point3d p) { return p.x*p.x + p.y*p.y + p.z*p.z; }
double abs(Point3d p) { return sqrt(norm(p)); }
double dist(Point3d a, Point3d b) { return sqrt(norm(a - b)); }
double dot(Point3d a, Point3d b) { return a.x*b.x + a.y*b.y + a.z*b.z; }
Vector3d cross(Point3d a, Point3d b) {
return Vector3d(a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x);
} // https://yukicoder.me/problems/no/2765
Vector3d normalize(Vector3d v) { // 長さを1に正規化(単位ベクトル)
assert(abs(v) > EPS);
return v / abs(v);
}
double rad_to_deg(double rad) { return (rad * 180. / PI); }
double deg_to_rad(double deg) { return (deg * PI / 180.); }
double normalize(double rad) { // to [-PI, PI)
double ret = fmod(rad + PI, PI*2.);
if (ret < 0) ret += PI;
else ret -= PI;
return ret;
} // maroon's library, Contest 2: ptKU Contest 1, ptTZ Summer 2022 Day 4
double angle(Point3d a, Point3d b) {
double costheta = dot(a, b) / norm(a) / norm(b);
return acos(max<double>(-1., min<double>(1., costheta)));
}
double small_angle(Point3d a, Point3d b) {
return acos(min(fabs(dot(a, b)) / abs(a) / abs(b), (double)1.));
}
bool is_lattice(Point3d p) { // 未
return equals(abs(p.x - round(p.x)), 0.) && equals(abs(p.y - round(p.y)), 0.) && equals(abs(p.z - round(p.z)), 0.);
}
bool is_lattice(double x, double y, double z) { // 未
return equals(abs(x - round(x)), 0.) && equals(abs(y - round(y)), 0.) && equals(abs(z - round(z)), 0.);
}
bool is_parallel(Vector3d v1, Vector3d v2) {
if (equals(v1.x*v2.y, v1.y*v2.x) &&
equals(v1.y*v2.z, v1.z*v2.y) &&
equals(v1.z*v2.x, v1.x*v2.z)) return true;
return false;
}
bool is_orthogonal(Vector3d v1, Vector3d v2) {
if (equals(v1.x*v2.x + v1.y*v2.y + v1.z*v2.z, 0.)) return true;
return false;
}
struct Line3d {
Point3d o, d; // p = o + k * d (k is a real parameter)
Line3d() {}
Line3d(Point3d p1, Point3d p2) : d(p2 - p1), o(p1) {}
Line3d(double a, double b, double c, double e) { // ax + by + cz + e = 0. 未
if (equals(a, 0)) o = Point3d{0, -e / b, 0}, d = Point3d{1, -a / b, 0};
else if (equals(b, 0)) o = Point3d{-e / a, 0, 0}, d = Point3d{-b / a, 1, 0};
else o = Point3d{0, 0, -e / c}, d = Point3d{1, 0, -a / c};
}
double dist2(Point3d p) { return dot(cross(d, p - o), cross(d, p - o)) / dot(d, d); }
double dist(Point3d p) { return sqrt(dist2(p)); }
bool cmp_proj(Point3d p, Point3d q) { return dot(d, p) < dot(d, q); }
friend istream& operator>>(istream &is, Line3d &L) {
Point3d p1, p2;
is >> p1 >> p2;
L = Line3d(p1, p2);
return is;
}
// bool operator==(const Line3d &L) const {}
};
struct Segment3d {
Point3d p1, p2;
Segment3d() {}
Segment3d(Point3d p1, Point3d p2) : p1(p1), p2(p2) {}
double length() { return dist(p1, p2); }
friend istream& operator>>(istream &is, Segment3d &S) {
is >> S.p1 >> S.p2;
return is;
}
bool operator==(const Segment3d &S) const {
return (p1 == S.p1 && p2 == S.p2) or (p1 == S.p2 && p2 == S.p1);
}
};
bool on_line(Line3d L, Point3d p) {
return equals(abs(cross(L.o - p, L.o + L.d - p)), 0);
// return is_parallel(L.o - p, L.o + L.d - p);
}
bool on_segment(Segment3d S, Point3d p) {
if (!on_line(Line3d{S.p1, S.p2}, p)) return false;
return equals(abs(S.p2 - S.p1), abs(p - S.p1) + abs(p - S.p2));
}
// small angle between direction vectors of two lines
double angle(Line3d L1, Line3d L2) {
return small_angle(L1.d, L2.d);
}
Point3d projection(Line3d L, Point3d p) {
Vector3d base = L.d;
double t = dot(p - L.o, base) / norm(base);
return L.o + base * t;
}
// Point3d projection(Segment3d S, Point3d p) { // 未. closet_point
// Vector base = S.p2 - S.p1;
// double r = dot(p - S.p1, base) / base.norm();
// Point3d proj = S.p1 + base * r;
// if (r < 0.) return S.p1;
// if (r > 1.) return S.p2;
// return proj;
// }
Point3d reflection(Line3d L, Point3d p) {
return p + (projection(L, p) - p) * 2.;
}
bool is_parallel(Line3d L1, Line3d L2) { return is_parallel(L1.d, L2.d); }
bool is_parallel(Line3d L, Segment3d S) { return is_parallel(L.d, S.p2 - S.p1); }
bool is_parallel(Segment3d S1, Segment3d S2) { return is_parallel(S1.p2 - S1.p1, S2.p2 - S2.p1); }
bool is_orthogonal(Line3d L1, Line3d L2) { return is_orthogonal(L1.d, L2.d); }
bool is_orthogonal(Line3d L, Segment3d S) { return is_orthogonal(L.d, S.p2 - S.p1); }
bool is_orthogonal(Segment3d S1, Segment3d S2) { return is_orthogonal(S1.p2 - S1.p1, S2.p2 - S2.p1); }
// bool intersect_LL(Line3d L1, Line3d L2) {
// return !is_parallel(L1, L2) && (同一平面上);
// }
// bool intersect_LS(Line3d L, Segment3d S) { // 2次元のもの
// return (cross(L.d, S.p1 - L.p1) * cross(L.d, S.d) < EPS);
// }
// bool intersect_SS(Segment3d S1, Segment3d S2) {
// }
// vector<Point3d> cross_point_LL(Line3d L1, Line3d L2) {}
// vector<Point3d> cross_point_LS(Line3d L, Segment3d S) {}
// vector<Point3d> cross_point_SS(Segment3d S1, Segment3d S2) {}
double dist_Lp(Line3d L, Point3d p) {
return abs(cross(L.d, p - L.o)) / abs(L.d);
}
double dist_Sp(Segment3d S, Point3d p) {
Point3d r = projection(Line3d{S.p1, S.p2}, p);
if (on_segment(S, r)) return abs(p - r);
return min(abs(S.p1 - p), abs(S.p2 - p));
}
pair<Point3d, Point3d> closest_pair(Line3d L1, Line3d L2) {
Point3d d1 = normalize(L1.d);
Point3d d2 = normalize(L2.d);
// P(x) = L1.o + d1 * x, Q(y) = L2.o + d2 * y. min |P(x) - Q(y)| is the distance of the lines L1, L2.
double p = dot(L2.d, d1), q = dot(L2.d, d2), c = dot(d1, d2);
// x - cy = p
// cx - y = q
double x, y;
if (equals(abs(c), 1.)) { // parallel <=> d1 // d2
x = p, y = 0;
} else {
double den = 1 - c*c;
x = (p - c*q) / den, y = (c*p - q) / den;
}
return {L1.o + d1*x, L2.o + d2*y};
} // ライブラリにほかの実装あり
// pair<Point3d, Point3d> closest_pair(Segment3d L1, Segment3d L2) {}
double dist_LL(Line3d L1, Line3d L2) {
auto [p, q] = closest_pair(L1, L2);
return dist(p, q);
}
double dist_LS(Line3d L, Segment3d S) {
auto [p, q] = closest_pair(L, Line3d{S.p1, S.p2});
return on_segment(S, q) ? dist(p, q) : min(dist_Lp(L, S.p1), dist_Lp(L, S.p2));
}
double dist_SS(Segment3d S1, Segment3d S2) {
auto [p, q] = closest_pair(Line3d{S1.p1, S1.p2}, Line3d{S2.p1, S2.p2});
return on_segment(S1, p) && on_segment(S2, q) ?
dist(p, q) :
min({dist_Sp(S1, S2.p1), dist_Sp(S1, S2.p2), dist_Sp(S2, S1.p1), dist_Sp(S2, S1.p2)});
}
struct Plane {
Point3d n; // 法線ベクトル
double d; // dot(n, p)
Plane() {}
Plane(Point3d n, double d) : n(n), d(d) {}
Plane(Point3d n, Point3d p) : n(n), d(dot(n, p)) {}
Plane(Point3d p, Point3d q, Point3d r) : Plane(cross(q - p, r - p), p) {} // non-collinear points P,Q,R
// 正:nと同じ側, 0:面上, 負:nと反対側
double side(Point3d p) { return dot(n, p) - d; }
// tだけ平行移動
Plane translate(Point3d t) { return {n, d + dot(n, t)}; }
// nの方向にdだけ平行移動
Plane shiftUp(double dist) { return {n, d + dist * abs(n)}; }
// 平面上の異なる点を2つ返す
pair<Point3d, Point3d> get_two_points_on_plane() {
assert(sign(n.x) != 0 or sign(n.y) != 0 or sign(n.z) != 0);
if (sign(n.x) == 0 && sign(n.y) == 0) return {Point3d(1, 0, d/n.z), Point3d(0, 1, d/n.z)};
if (sign(n.y) == 0 && sign(n.z) == 0) return {Point3d(d/n.x, 1, 0), Point3d(d/n.x, 0, 1)};
if (sign(n.z) == 0 && sign(n.x) == 0) return {Point3d(1, d/n.y, 0), Point3d(0, d/n.y, 1)};
if (sign(n.x) == 0) return {Point3d(1, d/n.y, 0), Point3d(0, 0, d/n.z)};
if (sign(n.y) == 0) return {Point3d(0, 1, d/n.z), Point3d(d/n.x, 0, 0)};
if (sign(n.z) == 0) return {Point3d(d/n.x, 0, 1), Point3d(0, d/n.y, 0)};
if (sign(d)!=0) return {Point3d(d/n.x, 0, 0), Point3d(0, d/n.y, 0)};
return {Point3d(n.y, -n.x, 0), Point3d(-n.y, n.x, 0)};
}
};
Point3d projection(Plane P, Point3d p) {
Point3d a = P.n * P.d;
return p - (P.n * dot(p - a, P.n));
}
Point3d reflection(Plane P, Point3d p) {
return p + (projection(P, p) - p) * 2.;
}
bool is_parallel(Plane P, Line3d L) { return is_orthogonal(P.n, L.d); }
bool is_parallel(Plane P, Segment3d S) { return is_orthogonal(P.n, S.p1 - S.p2); }
bool is_orthogonal(Plane P, Line3d L) { return is_parallel(P.n, L.d); }
bool is_orthogonal(Plane P, Segment3d S) { return is_parallel(P.n, S.p1 - S.p2); }
bool is_parallel(Plane P1, Plane P2) { // (0, 0, 0)か
return abs(cross(P1.n, P2.n)) < EPS;
}
bool is_orthogonal(Plane P1, Plane P2) { return sign(dot(P1.n, P2.n)) == 0; }
double angle(Plane P, Line3d l) {
return PI / 2. - acos(min(fabs(dot(P.n, l.d)) / abs(P.n) / abs(l.d), 1.));
}
bool intersect_PL(Plane P, Line3d L) { return !is_parallel(P, L); }
bool intersect_PS(Plane P, Segment3d S) {
Point3d a = P.n * P.d;
double b = dot(a - S.p1, P.n);
double c = dot(a - S.p2, P.n);
if (b > c) swap(b, c);
if (b < EPS && c > -EPS) return true;
return false;
}
bool intersect_PP(Plane P1, Plane P2) { return !is_parallel(P1, P2); }
Point3d cross_point_PL(Plane P, Line3d L) {
return L.o - L.d * P.side(L.o) / dot(L.d, P.n);
}
Point3d cross_point_PS(Plane P, Segment3d S) {
Point3d a = P.n * P.d;
double dot_p0a = fabs(dot(S.p1 - a, P.n));
double dot_p1a = fabs(dot(S.p2 - a, P.n));
if (equals(dot_p0a + dot_p1a, 0)) return S.p1;
return S.p1 + (S.p2 - S.p1) * (dot_p0a / (dot_p0a + dot_p1a));
}
Line3d cross_line_PP(Plane p1, Plane p2) { // p1とp2は同一でない
Point3d d = cross(p1.n, p2.n);
Point3d o = cross(p2.n * p1.d - p1.n * p2.d, d) / dot(d, d);
return {o, d};
}
double dist_Pp(Plane P, Point3d p) { return fabs(P.side(p)) / abs(P.n); }
// https://yukicoder.me/problems/no/132
// double dist_Pp(Plane P, Point3d p) {
// Point3d a = P.n * P.d; // 平面上の適当な点をつくる
// return abs(dot(p - a, P.n));
// } // https://yukicoder.me/problems/no/132
double dist_PL(Plane P, Line3d L) { return is_parallel(P, L) ? dist_Pp(P, L.o) : 0; }
double dist_PS(Plane P, Segment3d S) {
Point3d ha = projection(P, S.p1), hb = projection(P, S.p2);
double ipa = dot(S.p1 - ha, S.p2 - S.p1), ipb = dot(S.p2 - hb, S.p1 - S.p2);
return sign(ipa) < 0 && sign(ipb) < 0 ? 0 : min(dist_Pp(P, S.p1), dist_Pp(P, S.p2));
}
double dist_PP(Plane P1, Plane P2) { // 未
if (!is_parallel(P1, P2)) return 0;
return abs(P1.d - P2.d) / abs(P1.n);
}
using P3db = pair<Point3d, bool>;
double area(Point3d a, Point3d b, Point3d c) {
return abs(cross(b - a, c - a)) / 2.;
}
signed main() {
int N;
cin >> N;
Point3d p;
cin >> p;
vector<Point3d> P(N);
for (int i = 0; i < N; i++) {
cin >> P[i];
}
double ans = 0;
for (int i = 0; i < N; i++) {
for (int j = i + 1; j < N; j++) {
for (int k = j + 1; k < N; k++) {
Plane f(P[i], P[j], P[k]);
ans += dist_Pp(f, p);
}
}
}
cout << ans << endl;
}