Problem p00678 (C++) — buggy submission, verdict: Time Limit Exceeded

#include "bits/stdc++.h"
using namespace std;
#ifdef _DEBUG
#include "dump.hpp"
#else
#define dump(...)
#endif

// #define int long long
#define rep(i, a, b) for (int i = (a); i < (b); i++)
#define rrep(i, a, b) for (int i = (b)-1; i >= (a); i--)
#define all(c) begin(c), end(c)
const int INF =
    sizeof(int) == sizeof(long long) ? 0x3f3f3f3f3f3f3f3fLL : 0x3f3f3f3f;
const int MOD = (int)(1e9) + 7;
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 (b < a) {
    a = b;
    return true;
  }
  return false;
}

// #define double long double
const double EPS = 1e-8;
const double PI = acos(-1);
#define equals(a, b) (fabs((a) - (b)) < EPS)
#define next(P, i) P[(i + 1) % P.size()]
#define prev(P, i) P[(i + P.size() - 1) % P.size()]

struct Point {
  double x, y;
  Point() {}
  Point(double x, double y) : x(x), y(y) {}

  Point &operator+=(const Point &p) {
    x += p.x;
    y += p.y;
    return *this;
  }
  Point &operator-=(const Point &p) {
    x -= p.x;
    y -= p.y;
    return *this;
  }
  Point &operator*=(double a) {
    x *= a;
    y *= a;
    return *this;
  }
  Point &operator/=(double a) {
    x /= a;
    y /= a;
    return *this;
  }

  double abs() { return sqrt(norm()); }
  double norm() { return x * x + y * y; }
};
Point operator+(const Point &p1, const Point &p2) { return Point(p1) += p2; }
Point operator-(const Point &p1, const Point &p2) { return Point(p1) -= p2; }
Point operator*(const Point &p, double a) { return Point(p) *= a; }
Point operator/(const Point &p, double a) { return Point(p) /= a; }
bool operator==(const Point &p1, const Point &p2) {
  return equals(p1.x, p2.x) && equals(p1.y, p2.y);
}
bool operator<(const Point &p1, const Point &p2) {
  // return p1.y != p2.y ? p1.y < p2.y : p1.x < p2.x; //y?????? -> x??????
  return p1.x != p2.x ? p1.x < p2.x : p1.y < p2.y; // x?????? -> y??????
}
bool operator>(const Point &p1, const Point &p2) { return p2 < p1; }
inline istream &operator>>(istream &is, Point &p) {
  double x, y;
  is >> x >> y;
  p = Point(x, y);
  return is;
}
inline ostream &operator<<(ostream &os, const Point &p) {
  os << p.x << ' ' << p.y;
  return os;
}

struct Vector : public Point {
  using Point::Point;
  Vector() {}
  Vector(const Point &P) {
    x = P.x;
    y = P.y;
  }
  Vector rotate(double rad) {
    return Vector(x * cos(rad) - y * sin(rad), x * sin(rad) + y * cos(rad));
  }
  Vector unit() { return *this / abs(); }
};

//?????? dot product
double dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
//?????? cross product ?????§???????????£????????????
double cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }

struct Line {
  Point p1, p2;
  Line() {}
  Line(Point p1, Point p2) : p1(p1), p2(p2) {}
};

struct Segment : public Line {
  using Line::Line;
  Segment() {}
  Segment(const Line &L) {
    p1 = L.p1;
    p2 = L.p2;
  }
  Vector vec() { return p2 - p1; }
};

struct Circle {
  Point c;  // center
  double r; // radius
  Circle() {}
  Circle(Point c, double r) : c(c), r(r) {}
};
using Polygon = vector<Point>;

// degree to radian
double rad(double deg) { return PI * deg / 180; }
// radian to degree
double deg(double rad) { return rad / PI * 180; }

//????§? argument
double arg(Vector p) { return atan2(p.y, p.x); }

//?\\???¢??? polar form
Vector polar(double r, double a) { return Point(cos(a) * r, sin(a) * r); }

// 2??????????????????????§????
double angle(Vector a, Vector b) {
  double lena = a.abs(), lenb = b.abs();
  if (lena == 0 || lenb == 0)
    return 0; //?§£??????
  double costheta = dot(a, b) / (lena * lenb);
  if (equals(costheta, 1))
    costheta = 1; //????????????
  return acos(costheta);
}

bool inrange(Point p, double x1, double y1, double x2, double y2) {
  return x1 <= p.x && p.x <= x2 && y1 <= p.y && p.y <= y2;
}

//??´?????????
bool is_orthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }
bool is_orthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

//????????????
bool is_parallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }
bool is_parallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

//?°???±
Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / base.norm();
  return s.p1 + base * r;
}
//????°?
Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

//??????(p0,p1)????????????p2???????????¢???
enum {
  ONLINE_FRONT = -2,
  CLOCKWISE,
  ON_SEGMENT,
  COUNTER_CLOCKWISE,
  ONLINE_BACK
};
int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0, b = p2 - p0;
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK; // p2 p0 p1
  if (a.norm() < b.norm())
    return ONLINE_FRONT; // p0 p1 p2
  return ON_SEGMENT;
}
int ccw(Vector a, Vector b) {
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK; // p2 p0 p1
  if (a.norm() < b.norm())
    return ONLINE_FRONT; // p0 p1 p2
  return ON_SEGMENT;
}

//??´?????¨??´??????????????????
bool intersect(Segment a, Segment b) {
  Point p1 = a.p1, p2 = a.p2, p3 = b.p1, p4 = b.p2;
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}
//(?????????????¶????)

// 2??????????????¢
double get_distance(Point a, Point b) { return (a - b).abs(); }

//??´?????¨???????????¢
double get_distance(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / (l.p2 - l.p1).abs());
}

//????????¨???????????¢
double get_distance(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return (p - s.p1).abs();
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return (p - s.p2).abs();
  return get_distance(Line(s), p);
}

//????????¨??????????????¢
double get_distance(Segment s1, Segment s2) {
  if (intersect(s1, s2))
    return 0.0;
  return min(min(get_distance(s1, s2.p1), get_distance(s1, s2.p2)),
             min(get_distance(s2, s1.p1), get_distance(s2, s1.p2)));
}

//?????¨??´??????????????????
bool intersect(Circle c, Line l) { return get_distance(l, c.c) <= c.r + EPS; }

//?????¨?????????????????? ??±?????\\????????°
int intersect(Circle c1, Circle c2) {
  double d = get_distance(c1.c, c2.c);
  if (d > c1.r + c2.r)
    return 4;
  if (d == c1.r + c2.r)
    return 3;
  if (d + c1.r == c2.r || d + c2.r == c1.r)
    return 1;
  if (d + c1.r < c2.r || d + c2.r < c1.r)
    return 0;
  return 2;
}

//????????¨???????????????
Point get_cross_point(Segment a, Segment b) {
  assert(intersect(a, b));
  Vector base = b.p2 - b.p1;
  double area1 = abs(cross(base, a.p1 - b.p1));
  double area2 = abs(cross(base, a.p2 - b.p1));
  double t = area1 / (area1 + area2);
  return a.p1 + (a.p2 - a.p1) * t;
}

//?????¨??´????????????
pair<Point, Point> get_cross_points(Circle c, Line l) {
  assert(intersect(c, l));
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / (l.p2 - l.p1).abs();
  double base = sqrt(c.r * c.r - (pr - c.c).norm());
  return make_pair(pr + e * base, pr - e * base);
}

//?????¨????????????
pair<Point, Point> get_cross_points(Circle c1, Circle c2) {
  int m = intersect(c1, c2);
  assert(m != 4 && m != 0);
  double d = (c1.c - c2.c).abs();
  double a = acos((c1.r * c1.r - c2.r * c2.r + d * d) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
}

//????????????
enum { OUT = 0, ON, IN };
int contains(const Polygon &pl, Point p) {
  int n = pl.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = pl[i] - p, b = pl[(i + 1) % n] - p;
    if (abs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return ON;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return (x ? IN : OUT);
}

int contains(Circle c, Point p) {
  double d = get_distance(c.c, p);
  if (equals(d, c.r))
    return ON;
  if (d < c.r)
    return IN;
  return OUT;
}

//????§???¢?????¢???
double area(const Polygon &p) {
  double a = 0;
  for (size_t i = 0; i < p.size(); i++)
    a += cross(p[i], p[(i + 1) % p.size()]);
  return fabs(a / 2.0);
}

//?????§????????????????¨??????????
bool is_convex(Polygon g) {
  for (size_t i = 0; i < g.size(); i++)
    if (ccw(g[i], g[(i + 1) % g.size()], g[(i + 2) % g.size()]) == CLOCKWISE)
      return false;
  return true;
}

//??????
// Graham scan https://en.wikipedia.org/wiki/Graham_scan
//???????????????????????????
Polygon convex_hull(Polygon P) {
  sort(P.begin(), P.end());
  Polygon up;
  for (Point &p : P) {
    while (up.size() > 1 &&
           ccw(up[up.size() - 2], up[up.size() - 1], p) != CLOCKWISE)
      up.pop_back();
    up.emplace_back(p);
  }
  Polygon down;
  for (Point &p : P) {
    while (down.size() > 1 && ccw(down[down.size() - 2], down[down.size() - 1],
                                  p) != COUNTER_CLOCKWISE)
      down.pop_back();
    down.emplace_back(p);
  }
  reverse(up.begin(), up.end()); //???????¨??????????
  down.insert(down.end(), up.begin() + 1, up.end() - 1);
  return down;
}

//??????
// Graham scan https://en.wikipedia.org/wiki/Graham_scan
//?????????????????????
Polygon convex_hull_with_points_online(Polygon P) {
  sort(P.begin(), P.end());
  Polygon up;
  for (Point &p : P) {
    int _ccw;
    while (up.size() > 1 &&
           (_ccw = ccw(up[up.size() - 2], up[up.size() - 1], p)) != CLOCKWISE &&
           _ccw != ONLINE_FRONT)
      up.pop_back();
    up.emplace_back(p);
  }
  Polygon down;
  for (Point &p : P) {
    int _ccw;
    while (down.size() > 1 &&
           (_ccw = ccw(down[down.size() - 2], down[down.size() - 1], p)) !=
               COUNTER_CLOCKWISE &&
           _ccw != ONLINE_FRONT)
      down.pop_back();
    down.emplace_back(p);
  }
  reverse(up.begin(), up.end()); //???????¨??????????
  down.insert(down.end(), up.begin() + 1, up.end() - 1);
  return down;
}

//???????§???¢??????????????????????????¢
// calipers https://en.wikipedia.org/wiki/Rotating_calipers
double diameter(Polygon P) {
  P = convex_hull(P);
  auto mima = minmax_element(P.begin(), P.end());
  int I = mima.first - P.begin();
  int J = mima.second - P.begin();
  double maxd = get_distance(P[I], P[J]);

  int maxi, maxj, i, j;
  i = maxi = I;
  j = maxj = J;
  do {
    if (ccw(next(P, i) - P[i], next(P, j) - P[j]) == COUNTER_CLOCKWISE)
      j = (j + 1) % P.size();
    else
      i = (i + 1) % P.size();
    if (maxd < get_distance(P[i], P[j])) {
      maxd = get_distance(P[i], P[j]);
      maxi = i, maxj = j;
    }
  } while (!(i == I && j == J));
  return maxd;
}

//????§???¢???(0,0)???????????¨???????????¢
Polygon rotate(const Polygon &P, double rad) {
  Polygon ret;
  for (auto &p : P)
    ret.emplace_back(p.x * cos(rad) - p.y * sin(rad),
                     p.x * sin(rad) + p.y * cos(rad));
  return ret;
}

// Heron's formula
double area(double a, double b, double c) {
  double s = (a + b + c) / 2;
  return sqrt(s * (s - a) * (s - b) * (s - c));
}

//????§???¢?????????
Point center(const Polygon &P) {
  Point ret(0, 0);
  for (auto &p : P)
    ret = ret + p;
  ret = ret / P.size();
  return ret;
}

//?????´????????????
Line get_bisection(const Point &p1, const Point &p2) {
  Circle c1(p1, INF), c2(p2, INF); // INF ?????¨???????????????
  auto ps = get_cross_points(c1, c2);
  return Line(ps.first, ps.second);
}

// 3??????????????¢??????????????? (3???????????¨????????????????????????)
Point get_center(const Point &p1, const Point &p2, const Point &p3) {
  Line l1 = get_bisection(p1, p2), l2 = get_bisection(p2, p3);
  return get_cross_point(l1, l2);
}

//???p????????????c?????\\?????????????????\\???
pair<Point, Point> get_tangent(const Circle &c, const Point &p) {
  double d = get_distance(c.c, p);
  Circle c2((c.c + p) / 2, d / 2);
  return get_cross_points(c, c2);
}

signed main() {
  cin.tie(0);
  ios::sync_with_stdio(false);
  cout << fixed << setprecision(10);
  for (int N; cin >> N && N;) {
    Polygon p(N);
    vector<double> v(N);
    rep(i, 0, N) { cin >> p[i] >> v[i]; }
    Point g = center(p);
    vector<double> cost(N);
    double move = 0.5, y = 0.99;
    while (move > 1e-9) {
      rep(i, 0, N) cost[i] = get_distance(g, p[i]) / v[i];
      int maxidx = max_element(all(cost)) - cost.begin();
      g += (p[maxidx] - g) * move;
      move *= y;
    }
    double ans = 0;
    rep(i, 0, N) chmax(ans, get_distance(g, p[i]) / v[i]);
    dump(g);
    cout << ans << endl;
  }
  return 0;
}

Error: TLE

Fixed submission:

#include "bits/stdc++.h"
using namespace std;
#ifdef _DEBUG
#include "dump.hpp"
#else
#define dump(...)
#endif

// #define int long long
#define rep(i, a, b) for (int i = (a); i < (b); i++)
#define rrep(i, a, b) for (int i = (b)-1; i >= (a); i--)
#define all(c) begin(c), end(c)
const int INF =
    sizeof(int) == sizeof(long long) ? 0x3f3f3f3f3f3f3f3fLL : 0x3f3f3f3f;
const int MOD = (int)(1e9) + 7;
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 (b < a) {
    a = b;
    return true;
  }
  return false;
}

// #define double long double
const double EPS = 1e-8;
const double PI = acos(-1);
#define equals(a, b) (fabs((a) - (b)) < EPS)
#define next(P, i) P[(i + 1) % P.size()]
#define prev(P, i) P[(i + P.size() - 1) % P.size()]

struct Point {
  double x, y;
  Point() {}
  Point(double x, double y) : x(x), y(y) {}

  Point &operator+=(const Point &p) {
    x += p.x;
    y += p.y;
    return *this;
  }
  Point &operator-=(const Point &p) {
    x -= p.x;
    y -= p.y;
    return *this;
  }
  Point &operator*=(double a) {
    x *= a;
    y *= a;
    return *this;
  }
  Point &operator/=(double a) {
    x /= a;
    y /= a;
    return *this;
  }

  double abs() { return sqrt(norm()); }
  double norm() { return x * x + y * y; }
};
Point operator+(const Point &p1, const Point &p2) { return Point(p1) += p2; }
Point operator-(const Point &p1, const Point &p2) { return Point(p1) -= p2; }
Point operator*(const Point &p, double a) { return Point(p) *= a; }
Point operator/(const Point &p, double a) { return Point(p) /= a; }
bool operator==(const Point &p1, const Point &p2) {
  return equals(p1.x, p2.x) && equals(p1.y, p2.y);
}
bool operator<(const Point &p1, const Point &p2) {
  // return p1.y != p2.y ? p1.y < p2.y : p1.x < p2.x; //y?????? -> x??????
  return p1.x != p2.x ? p1.x < p2.x : p1.y < p2.y; // x?????? -> y??????
}
bool operator>(const Point &p1, const Point &p2) { return p2 < p1; }
inline istream &operator>>(istream &is, Point &p) {
  double x, y;
  is >> x >> y;
  p = Point(x, y);
  return is;
}
inline ostream &operator<<(ostream &os, const Point &p) {
  os << p.x << ' ' << p.y;
  return os;
}

struct Vector : public Point {
  using Point::Point;
  Vector() {}
  Vector(const Point &P) {
    x = P.x;
    y = P.y;
  }
  Vector rotate(double rad) {
    return Vector(x * cos(rad) - y * sin(rad), x * sin(rad) + y * cos(rad));
  }
  Vector unit() { return *this / abs(); }
};

//?????? dot product
double dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
//?????? cross product ?????§???????????£????????????
double cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }

struct Line {
  Point p1, p2;
  Line() {}
  Line(Point p1, Point p2) : p1(p1), p2(p2) {}
};

struct Segment : public Line {
  using Line::Line;
  Segment() {}
  Segment(const Line &L) {
    p1 = L.p1;
    p2 = L.p2;
  }
  Vector vec() { return p2 - p1; }
};

struct Circle {
  Point c;  // center
  double r; // radius
  Circle() {}
  Circle(Point c, double r) : c(c), r(r) {}
};
using Polygon = vector<Point>;

// degree to radian
double rad(double deg) { return PI * deg / 180; }
// radian to degree
double deg(double rad) { return rad / PI * 180; }

//????§? argument
double arg(Vector p) { return atan2(p.y, p.x); }

//?\\???¢??? polar form
Vector polar(double r, double a) { return Point(cos(a) * r, sin(a) * r); }

// 2??????????????????????§????
double angle(Vector a, Vector b) {
  double lena = a.abs(), lenb = b.abs();
  if (lena == 0 || lenb == 0)
    return 0; //?§£??????
  double costheta = dot(a, b) / (lena * lenb);
  if (equals(costheta, 1))
    costheta = 1; //????????????
  return acos(costheta);
}

bool inrange(Point p, double x1, double y1, double x2, double y2) {
  return x1 <= p.x && p.x <= x2 && y1 <= p.y && p.y <= y2;
}

//??´?????????
bool is_orthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }
bool is_orthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

//????????????
bool is_parallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }
bool is_parallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

//?°???±
Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / base.norm();
  return s.p1 + base * r;
}
//????°?
Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

//??????(p0,p1)????????????p2???????????¢???
enum {
  ONLINE_FRONT = -2,
  CLOCKWISE,
  ON_SEGMENT,
  COUNTER_CLOCKWISE,
  ONLINE_BACK
};
int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0, b = p2 - p0;
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK; // p2 p0 p1
  if (a.norm() < b.norm())
    return ONLINE_FRONT; // p0 p1 p2
  return ON_SEGMENT;
}
int ccw(Vector a, Vector b) {
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK; // p2 p0 p1
  if (a.norm() < b.norm())
    return ONLINE_FRONT; // p0 p1 p2
  return ON_SEGMENT;
}

//??´?????¨??´??????????????????
bool intersect(Segment a, Segment b) {
  Point p1 = a.p1, p2 = a.p2, p3 = b.p1, p4 = b.p2;
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}
//(?????????????¶????)

// 2??????????????¢
double get_distance(Point a, Point b) { return (a - b).abs(); }

//??´?????¨???????????¢
double get_distance(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / (l.p2 - l.p1).abs());
}

//????????¨???????????¢
double get_distance(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return (p - s.p1).abs();
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return (p - s.p2).abs();
  return get_distance(Line(s), p);
}

//????????¨??????????????¢
double get_distance(Segment s1, Segment s2) {
  if (intersect(s1, s2))
    return 0.0;
  return min(min(get_distance(s1, s2.p1), get_distance(s1, s2.p2)),
             min(get_distance(s2, s1.p1), get_distance(s2, s1.p2)));
}

//?????¨??´??????????????????
bool intersect(Circle c, Line l) { return get_distance(l, c.c) <= c.r + EPS; }

//?????¨?????????????????? ??±?????\\????????°
int intersect(Circle c1, Circle c2) {
  double d = get_distance(c1.c, c2.c);
  if (d > c1.r + c2.r)
    return 4;
  if (d == c1.r + c2.r)
    return 3;
  if (d + c1.r == c2.r || d + c2.r == c1.r)
    return 1;
  if (d + c1.r < c2.r || d + c2.r < c1.r)
    return 0;
  return 2;
}

//????????¨???????????????
Point get_cross_point(Segment a, Segment b) {
  assert(intersect(a, b));
  Vector base = b.p2 - b.p1;
  double area1 = abs(cross(base, a.p1 - b.p1));
  double area2 = abs(cross(base, a.p2 - b.p1));
  double t = area1 / (area1 + area2);
  return a.p1 + (a.p2 - a.p1) * t;
}

//?????¨??´????????????
pair<Point, Point> get_cross_points(Circle c, Line l) {
  assert(intersect(c, l));
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / (l.p2 - l.p1).abs();
  double base = sqrt(c.r * c.r - (pr - c.c).norm());
  return make_pair(pr + e * base, pr - e * base);
}

//?????¨????????????
pair<Point, Point> get_cross_points(Circle c1, Circle c2) {
  int m = intersect(c1, c2);
  assert(m != 4 && m != 0);
  double d = (c1.c - c2.c).abs();
  double a = acos((c1.r * c1.r - c2.r * c2.r + d * d) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
}

//????????????
enum { OUT = 0, ON, IN };
int contains(const Polygon &pl, Point p) {
  int n = pl.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = pl[i] - p, b = pl[(i + 1) % n] - p;
    if (abs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return ON;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return (x ? IN : OUT);
}

int contains(Circle c, Point p) {
  double d = get_distance(c.c, p);
  if (equals(d, c.r))
    return ON;
  if (d < c.r)
    return IN;
  return OUT;
}

//????§???¢?????¢???
double area(const Polygon &p) {
  double a = 0;
  for (size_t i = 0; i < p.size(); i++)
    a += cross(p[i], p[(i + 1) % p.size()]);
  return fabs(a / 2.0);
}

//?????§????????????????¨??????????
bool is_convex(Polygon g) {
  for (size_t i = 0; i < g.size(); i++)
    if (ccw(g[i], g[(i + 1) % g.size()], g[(i + 2) % g.size()]) == CLOCKWISE)
      return false;
  return true;
}

//??????
// Graham scan https://en.wikipedia.org/wiki/Graham_scan
//???????????????????????????
Polygon convex_hull(Polygon P) {
  sort(P.begin(), P.end());
  Polygon up;
  for (Point &p : P) {
    while (up.size() > 1 &&
           ccw(up[up.size() - 2], up[up.size() - 1], p) != CLOCKWISE)
      up.pop_back();
    up.emplace_back(p);
  }
  Polygon down;
  for (Point &p : P) {
    while (down.size() > 1 && ccw(down[down.size() - 2], down[down.size() - 1],
                                  p) != COUNTER_CLOCKWISE)
      down.pop_back();
    down.emplace_back(p);
  }
  reverse(up.begin(), up.end()); //???????¨??????????
  down.insert(down.end(), up.begin() + 1, up.end() - 1);
  return down;
}

//??????
// Graham scan https://en.wikipedia.org/wiki/Graham_scan
//?????????????????????
Polygon convex_hull_with_points_online(Polygon P) {
  sort(P.begin(), P.end());
  Polygon up;
  for (Point &p : P) {
    int _ccw;
    while (up.size() > 1 &&
           (_ccw = ccw(up[up.size() - 2], up[up.size() - 1], p)) != CLOCKWISE &&
           _ccw != ONLINE_FRONT)
      up.pop_back();
    up.emplace_back(p);
  }
  Polygon down;
  for (Point &p : P) {
    int _ccw;
    while (down.size() > 1 &&
           (_ccw = ccw(down[down.size() - 2], down[down.size() - 1], p)) !=
               COUNTER_CLOCKWISE &&
           _ccw != ONLINE_FRONT)
      down.pop_back();
    down.emplace_back(p);
  }
  reverse(up.begin(), up.end()); //???????¨??????????
  down.insert(down.end(), up.begin() + 1, up.end() - 1);
  return down;
}

//???????§???¢??????????????????????????¢
// calipers https://en.wikipedia.org/wiki/Rotating_calipers
double diameter(Polygon P) {
  P = convex_hull(P);
  auto mima = minmax_element(P.begin(), P.end());
  int I = mima.first - P.begin();
  int J = mima.second - P.begin();
  double maxd = get_distance(P[I], P[J]);

  int maxi, maxj, i, j;
  i = maxi = I;
  j = maxj = J;
  do {
    if (ccw(next(P, i) - P[i], next(P, j) - P[j]) == COUNTER_CLOCKWISE)
      j = (j + 1) % P.size();
    else
      i = (i + 1) % P.size();
    if (maxd < get_distance(P[i], P[j])) {
      maxd = get_distance(P[i], P[j]);
      maxi = i, maxj = j;
    }
  } while (!(i == I && j == J));
  return maxd;
}

//????§???¢???(0,0)???????????¨???????????¢
Polygon rotate(const Polygon &P, double rad) {
  Polygon ret;
  for (auto &p : P)
    ret.emplace_back(p.x * cos(rad) - p.y * sin(rad),
                     p.x * sin(rad) + p.y * cos(rad));
  return ret;
}

// Heron's formula
double area(double a, double b, double c) {
  double s = (a + b + c) / 2;
  return sqrt(s * (s - a) * (s - b) * (s - c));
}

//????§???¢?????????
Point center(const Polygon &P) {
  Point ret(0, 0);
  for (auto &p : P)
    ret = ret + p;
  ret = ret / P.size();
  return ret;
}

//?????´????????????
Line get_bisection(const Point &p1, const Point &p2) {
  Circle c1(p1, INF), c2(p2, INF); // INF ?????¨???????????????
  auto ps = get_cross_points(c1, c2);
  return Line(ps.first, ps.second);
}

// 3??????????????¢??????????????? (3???????????¨????????????????????????)
Point get_center(const Point &p1, const Point &p2, const Point &p3) {
  Line l1 = get_bisection(p1, p2), l2 = get_bisection(p2, p3);
  return get_cross_point(l1, l2);
}

//???p????????????c?????\\?????????????????\\???
pair<Point, Point> get_tangent(const Circle &c, const Point &p) {
  double d = get_distance(c.c, p);
  Circle c2((c.c + p) / 2, d / 2);
  return get_cross_points(c, c2);
}

signed main() {
  cin.tie(0);
  ios::sync_with_stdio(false);
  cout << fixed << setprecision(10);
  for (int N; cin >> N && N;) {
    Polygon p(N);
    vector<double> v(N);
    rep(i, 0, N) { cin >> p[i] >> v[i]; }
    Point g = center(p);
    vector<double> cost(N), cost2(N);
    rep(k, 0, 1000) {
      double x = 1, y = 0.98;
      rep(i, 0, N) cost[i] = get_distance(g, p[i]) / v[i];
      int maxidx = max_element(all(cost)) - cost.begin();
      Point ng;
      while (true) {
        ng = g + (p[maxidx] - g) * x / 2;
        rep(i, 0, N) cost2[i] = get_distance(ng, p[i]) / v[i];
        int maxidx2 = max_element(all(cost2)) - cost2.begin();
        if (cost[maxidx] >= cost2[maxidx2])
          break;
        x *= y;
      }
      g = ng;
    }
    double ans = 0;
    rep(i, 0, N) chmax(ans, get_distance(g, p[i]) / v[i]);
    dump(g);
    cout << ans << endl;
  }
  return 0;
}