Problem:
Score : 600 points

Problem Statement

Takahashi wants to grill N pieces of meat on a grilling net, which can be seen as a two-dimensional plane. The coordinates of the i-th piece of meat are \left(x_i, y_i\right), and its hardness is c_i.

Takahashi can use one heat source to grill the meat. If he puts the heat source at coordinates \left(X, Y\right), where X and Y are real numbers, the i-th piece of meat will be ready to eat in c_i \times \sqrt{\left(X - x_i\right)^2 + \left(Y-y_i\right)^2} seconds.

Takahashi wants to eat K pieces of meat. Find the time required to have K or more pieces of meat ready if he put the heat source to minimize this time.

Constraints

All values in input are integers.

1 \leq N \leq 60

1 \leq K \leq N

-1000 \leq x_i , y_i \leq 1000

\left(x_i, y_i\right) \neq \left(x_j, y_j\right) \left(i \neq j \right)

1 \leq c_i \leq 100

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
\vdots
x_N y_N c_N

Output

Print the answer.

It will be considered correct if its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

4 3
-1 0 3
0 0 3
1 0 2
1 1 40

Sample Output 1

2.4

If we put the heat source at \left(-0.2, 0\right), the 1-st, 2-nd, and 3-rd pieces of meat will be ready to eat within 2.4 seconds. This is the optimal place to put the heat source.

Sample Input 2

10 5
-879 981 26
890 -406 81
512 859 97
362 -955 25
128 553 17
-885 763 2
449 310 57
-656 -204 11
-270 76 40
184 170 16

Sample Output 2

7411.2252

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>

#include <algorithm>
#include <bitset>
#include <complex>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <vector>

#include <cassert>
#include <functional>

typedef long long ll;
using namespace std;

#ifndef LOCAL
#define debug(x) ;
#else
#define debug(x) cerr << __LINE__ << " : " << #x << " = " << (x) << endl;

template<typename T1, typename T2>
ostream &operator<<(ostream &out, const pair<T1, T2> &p) {
  out << "{" << p.first << ", " << p.second << "}";
  return out;
}

template<typename T>
ostream &operator<<(ostream &out, const vector<T> &v) {
  out << '{';
  for (const T &item : v) out << item << ", ";
  out << "\b\b}";
  return out;
}
#endif

#define mod 1000000007 //1e9+7(prime number)
#define INF 1000000000 //1e9
#define LLINF 2000000000000000000LL //2e18
#define SIZE 200010

typedef long double P_type; //座標(integer or real)
typedef double G_real; //実数の戻り値(float or double or long double)
typedef complex<P_type> P;
const G_real P_eps = 1e-8; //整数の時はゼロ

namespace std {
  template<class T>
  bool operator<(const complex<T> &a, const complex<T> &b) {
    return abs(a.real() - b.real()) < P_eps ? a.imag() + P_eps < b.imag() : a.real() + P_eps < b.real();
  }
};

P rotate(P p, double theta) {
  return p * P(cos(theta), sin(theta));
}

//内積
P_type dot(P a, P b) {
  return (a * conj(b)).real();
}

//外積
P_type cross(P a, P b) {
  return (conj(a) * b).imag();
}

//反時計回り
int ccw(P a, P b, P c) {
  if (cross(b - a, c - a) > P_eps) return 1; //COUNTER_CLOCKWISE(center:a)
  if (cross(b - a, c - a) < -P_eps) return -1; //CLOCKWISE(center:a)
  if (dot(b - a, c - a) < -P_eps) return -2; //c -> a -> b
  if (dot(a - b, c - b) < -P_eps) return 2; //a -> b -> c
  return 0; //a -> c -> b
}

/*
// 偏角 [-pi, pi]
// complex<int|ll> 使用時の std::arg の代わり
G_real arg(P p) {
  return atan2(imag(p), real(p));
}
*/

/* ベクトルa, bの角度 */
// ({0, 1}, {1, 0}) => 90, ({0, 1}, {-1, -1}) => -135
G_real degree(P a, P b) {
  return atan2(cross(b, a), dot(b, a)) / M_PI * 180;
}

/* 線分abと点cの距離 */
G_real distanceSP(P a, P b, P c) {
  if (dot(b - a, c - a) < P_eps) return abs(c - a);
  if (dot(a - b, c - b) < P_eps) return abs(c - b);
  return abs(cross(b - a, c - a)) / abs(b - a);
}

/* 直線abと点cの距離 */
G_real distanceLP(P a, P b, P c) {
  return abs(cross(b - a, c - a)) / abs(b - a);
}

/* 円の点包含判定 */
bool isContainedCP(P c, P_type r, P p) {
  //return abs(c - p) < r - P_eps; //円周上を含まない
  return abs(c - p) <= r + P_eps; //円周上を含む
}

/* 円の円包含判定 (円1の内部に円2が存在するか) */
bool isContainedCC(P c1, P_type r1, P c2, P_type r2) {
  return abs(c1 - c2) + r2 < r1 - P_eps; //接する場合を含まない
  //return abs(c1 - c2) + r2 <= r1 + P_eps; //接する場合を含まない
}

/* 直線交差判定 */
bool isIntersectedLL(P a1, P a2, P b1, P b2) {
  return abs(cross(a1 - a2, b1 - b2)) > P_eps;
}

/* 線分交差判定 */
bool isIntersectedSS(P a1, P a2, P b1, P b2) {
  //線分a と 直線b
  int a = ccw(b1, b2, a1);
  int b = ccw(b1, b2, a2);

  //線分b と 直線a
  int c = ccw(a1, a2, b1);
  int d = ccw(a1, a2, b2);

  return a * b <= 0 && c * d <= 0; // T字を除く時は (** < 0)
}

/* 直線A線分B交差判定 */
bool isIntersectedLS(P a1, P a2, P b1, P b2) {
  int a = ccw(a1, a2, b1);
  int b = ccw(a1, a2, b2);

  // 直線上のとき a or b = 0 or -2 or 2
  return (a % 2) * (b % 2) <= 0; // T字を除く時は (** < 0)
}

/* 円交差判定 */
bool isIntersectedCC(P c1, G_real r1, P c2, P_type r2) {
  G_real dist = abs(c1 - c2);

  return abs(r1 - r2) <= dist + P_eps && dist - P_eps <= r1 + r2; //外接内接を含む
  //return abs(r1 - r2) < dist - P_eps && dist + P_eps < r1 + r2; //外接内接を除く
}

/* 円直線交差判定 */
bool isIntersectedCL(P c, G_real r, P a1, P a2) {
  return distanceLP(a1, a2, c) <= r + P_eps; //接する場合を含まない場合 < r - P_eps
}

/* 円線分交差判定 */
bool isIntersectedCS(P c, P_type r, P a1, P a2) {
  return (!isContainedCP(c, r, a1) || !isContainedCP(c, r, a2)) &&
         distanceLP(a1, a2, c) <= r + P_eps; //接する場合を含まない場合 < r - P_eps
}

/* 直線/線分交点 */
vector<P> getCrosspointLL(P a1, P a2, P b1, P b2) {
  //assert(isIntersectedLL(a1, a2, b1, b2));
  P a = a2 - a1;
  P b = b2 - b1;
  return {a1 + a * cross(b, b1 - a1) / cross(b, a)};
}

vector<P> getCrosspointSS(P a1, P a2, P b1, P b2) {
  //assert(isIntersectedSS(a1, a2, b1, b2));
  return getCrosspointLL(a1, a2, b1, b2);
}

/* 円交点 */
vector<P> getCrosspointCC(P c1, P_type r1, P c2, P_type r2) {
  //assert(isIntersectedCC(c1, r1, c2, r2));

  P_type dist = abs(c1 - c2);
  P_type a = acos((r1 * r1 + dist * dist - r2 * r2) / (2 * r1 * dist));
  return {c1 + polar(r1, arg(c2 - c1) + a), c1 + polar(r1, arg(c2 - c1) - a)};
}

/* 円直線交点 */
vector<P> getCrosspointCL(P c, P_type r, P a1, P a2) {
  if (!isIntersectedCL(c, r, a1, a2)) return {};

  P base1 = a2 - a1;
  P proj = a1 + base1 * dot(c - a1, base1) / norm(base1); //射影
  P e = (a2 - a1) / abs(a2 - a1);
  P base2 = sqrt(r * r - norm(proj - c));
  return {proj - e * base2, proj + e * base2};
}

/* 円線分交点 */
vector<P> getCrosspointCS(P c, P_type r, P a1, P a2) {
  if (!isIntersectedCS(c, r, a1, a2)) return {};

  vector<P> res;
  for (P p : getCrosspointCL(c, r, a1, a2))
    if (dot(a1 - p, a2 - p) <= P_eps) res.push_back(p);
  return res;
}

/* 多角形-点包含 */
bool isContainedPolyP(vector<P> &g, P p) {
  int n = g.size();
  bool f = false;
  for (int i = 0; i < n; i++) {
    P a = g[i] - p, b = g[(i + 1) % n] - p;
    if (abs(cross(a, b)) < P_eps && dot(a, b) < P_eps) return true; //辺上
    if (a.imag() > b.imag()) swap(a, b);
    if (a.imag() < P_eps && P_eps < b.imag() && cross(a, b) > P_eps) f = !f;
  }
  return f; //内部 or 外部
}

/* 射影(直線abとpからの垂線との交点) */
P getProject(P a, P b, P p) {
  P base = b - a;
  return a + base * dot(p - a, base) / norm(base);
}

/* 反射(直線abを対称軸としたpの線対称の位置) */
P getReflection(P a, P b, P p) {
  return p + (getProject(a, b, p) - p) * (P_type)2.0;
}

/* 多角形面積 */
G_real getAreaPoly(vector<P> &g) {
  P_type res = 0;
  for (int i = 0; i < (int)g.size(); i++)
    res += cross(g[i], g[(i + 1) % g.size()]);
  return abs(res / 2.0);
}

/* 円の共通面積 */
G_real getAreaCC(P c1, P_type r1, P c2, P_type r2) {
  G_real d = abs(c1 - c2);
  if (r1 + r2 <= d + P_eps) {
    return 0.0;
  } else if (d <= abs(r1 - r2) + P_eps) {
    G_real r = min(r1, r2);
    return r * r * M_PI;
  } else {
    G_real res = 0;
    for (int i = 0; i < 2; i++) {
      G_real rc = (G_real)(norm(c1 - c2) + r1 * r1 - r2 * r2) / (2 * d * r1);
      G_real theta = acos(rc) * 2;
      res += (theta - sin(theta)) * r1 * r1 / 2;
      swap(r1, r2);
      swap(c1, c2);
    }
    return res;
  }
}

/* 円-直線(中心との三角形) 共通面積 */
G_real getAreaCS(P c, P_type r, P a1, P a2) {
  if (min({abs(a1 - a2), abs(c - a1), abs(c - a2), r}) <= P_eps) return 0;

  G_real res = 0;
  vector<P> vec = {a1};
  for (auto p : getCrosspointCS(c, r, a1, a2)) vec.push_back(p);
  vec.push_back(a2);

  for (int i = 0; i + 1 < (int)vec.size(); i++) {
    P p1 = vec[i] - c, p2 = vec[i + 1] - c;
    if (max(abs(p1), abs(p2)) <= r + P_eps)
      res += cross(p1, p2) / 2.0;
    else
      res += r * r * arg(conj(p1) * p2) / 2.0;
  }

  return res;
}

/* 円-多角形共通面積 */
G_real getAreaPolyC(vector<P> &g, P c, P_type r) {
  G_real res = 0;
  for (int i = 0; i < (int)g.size(); i++)
    res += getAreaCS(c, r, g[i], g[(i + 1) % g.size()]);
  return abs(res);
}

/* 凸多角形の切断 */
// 直線Aで切断し、a1からa2方向に見たときに直線Aの左側にできる凸多角形を返す
vector<P> convexCut(const vector<P> &g, P a1, P a2) {
  vector<P> res;
  P prev = *g.rbegin();
  for (P now : g) {
    if (ccw(a1, a2, prev) != -1) res.push_back(prev);
    int c = ccw(a1, a2, prev) * ccw(a1, a2, now);
    if (-4 < c && c < 0) {
      res.push_back(getCrosspointLL(prev, now, a1, a2)[0]);
    }
    prev = now;
  }
  return res;
}

/* 凸多角形の直径(最遠頂点対間距離) */
// 反時計回りのみ
G_real convexDiameter(const vector<P> &g) {
  int N = g.size(), is = 0, js = 0;
  for (int i = 1; i < N; i++) {
    if (imag(g[i]) > imag(g[is])) is = i;
    if (imag(g[i]) < imag(g[js])) js = i;
  }

  G_real maxd = abs(g[is] - g[js]);

  int i, maxi, j, maxj;
  i = maxi = is;
  j = maxj = js;

  do {
    if (cross(g[(i + 1) % N] - g[i], g[(j + 1) % N] - g[j]) >= 0)
      j = (j + 1) % N;
    else
      i = (i + 1) % N;

    if (abs(g[i] - g[j]) > maxd) {
      maxd = abs(g[i] - g[j]);
      maxi = i;
      maxj = j;
    }
  } while (i != is || j != js);

  return maxd; /* farthest pair is {maxi, maxj}. */
}

int N, K;
int x[SIZE], y[SIZE], c[SIZE];
P p[SIZE];

bool check(double r) {
  vector<P> points;

  for (int i = 0; i < N; i++) {
    for (int j = 0; j < i; j++) {
      if (!isIntersectedCC(p[i], r / c[i], p[j], r / c[j])) continue;

      auto cps = getCrosspointCC(p[i], r / c[i], p[j], r / c[j]);
      for (auto cp : cps) points.push_back(cp);
    }
  }

  for (int i = 0; i < N; i++)
    points.push_back(p[i]);

  for (auto cp : points) {
    int counter = 0;
    for (int i = 0; i < N; i++) {
      counter += isContainedCP(p[i], r / c[i], cp);
    }
    if (counter >= K) return true;
  }

  return false;
}

int main() {
  scanf("%d%d", &N, &K);

  for (int i = 0; i < N; i++) {
    scanf("%d%d%d", x + i, y + i, c + i);
    p[i] = P(x[i], y[i]);
  }

  double l = 0, r = 1e5;

  for (int _ = 0; _ < 1000; _++) {
    double mid = (l + r) / 2;

    if (check(mid)) {
      r = mid;
    } else {
      l = mid;
    }
  }

  printf("%.10lf\n", l);

  return 0;
}