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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <array>
#include <vector>
#include <map>
#include <unordered_map>
#include <set>
#include <unordered_set>
#include <algorithm>
#include <math.h>
#include <string>
#include <climits>
#include <assert.h>
#include <iomanip>
#include <bitset>
#include <queue>
#include <deque>
#include <stack>
#include <functional>
#include <fstream>
#include <random>

#define LEN(x) (long long)(x.size())
#define FOR(i, a, n) for(int i=(a);i<(n); ++i)
#define FOE(i, a) for(auto i : a)
#define ALL(c) (c).begin(), (c).end()
#define RALL(c) (c).rbegin(), (c).rend()
#define SUM(x) std::accumulate(ALL(x), 0LL)
#define MIN(v) *std::min_element(v.begin(), v.end())
#define MAX(v) *std::max_element(v.begin(), v.end())
#define EXIST(v, x) (std::find(v.begin(), v.end(), x) != v.end())
#define BIT_COUNT(bit) (__builtin_popcount(bit))

typedef long long LL;
template<typename T> std::vector<T> make_v(size_t a){return std::vector<T>(a);}
template<typename T,typename... Ts> auto make_v(size_t a, Ts... ts){ return std::vector<decltype(make_v<T>(ts...))>(a,make_v<T>(ts...));}    // C++14
template<typename T,typename V> typename std::enable_if<std::is_class<T>::value==0>::type fill_v(T &t,const V &v){t=v;}
template<typename T,typename V> typename std::enable_if<std::is_class<T>::value!=0>::type fill_v(T &t,const V &v){for(auto &e:t) fill_v(e,v);}
template<class T> inline T ceil(T a, T b) { return (a + b - 1) / b; }
void print() { std::cout << std::endl; }
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) { std::cout << head; if (sizeof...(tail) != 0) {std::cout << " ";} print(std::forward<Tail>(tail)...); }
template <class T> void print(std::vector<T> &v) {for (auto& a : v) { std::cout << a; if (&a != &v.back()) {std::cout << " ";} }std::cout << std::endl;}
template <class T> void print(std::vector<std::vector<T>> &vv) { for (auto& v : vv) { print(v); }}
void debug() { std::cerr << std::endl; }
template <class Head, class... Tail> void debug(Head&& head, Tail&&... tail) { std::cerr << head; if (sizeof...(tail) != 0) {std::cerr << " ";} print(std::forward<Tail>(tail)...); }
template <class T> void debug(std::vector<T> &v) {for (auto& a : v) { std::cerr << a; if (&a != &v.back()) {std::cerr << " ";} }std::cerr << std::endl;}
template <class T> void debug(std::vector<std::vector<T>> &vv) { for (auto& v : vv) { print(v); }}
inline bool inside(long long y, long long x, long long H, long long W) {return 0 <= y and y < H and 0 <= x and x < W; }
template<class T> inline double euclidean_distance(T y1, T x1, T y2, T x2) { return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)); }
template<class T> inline double manhattan_distance(T y1, T x1, T y2, T x2) { return abs(x1 - x2) + abs(y1 - y2); }
template<typename T> T &chmin(T &a, const T &b) { return a = std::min(a, b); }
template<typename T> T &chmax(T &a, const T &b) { return a = std::max(a, b); }
bool is_bit_on(const unsigned long long bit, const unsigned int i) { return (bit >> i) & 1u; }
unsigned long long bit_set(const unsigned long long bit, const unsigned int i, const unsigned int b) {
    assert(b == 0 or b == 1);
    if (b == 0) { return bit & ~(1ull << i); }
    else        {return bit | (1ull << i); }
}

// 初項s交差d長さnの数列の和
long long sum_of_arithmetic_progression(long long s, long long d, long long n) {
    return n * (2 * s + (n - 1) * d) / 2;
}

// xが2の階乗かどうか判定
bool is_power_of_two(long long x) {
    return !(x & (x - 1));
}

long long gcd(long long a, long long b) {
    if (b == 0) { return a; }
    return gcd(b, a % b);
}

long long gcd(std::vector<long long> &v) {
    long long ans = v[0];
    for (int i = 1; i < (int) v.size(); ++i) {
        ans = gcd(ans, v[i]);
    }
    return ans;
}

long long lcm(long long a, long long b) {
    long long g = gcd(a, b);
    return a / g * b;
}

const int INF = 1u << 30u;  // 1,073,741,824
const long long LINF = 1ull << 60u;
const double EPS = 1e-9;
const long double PI = acos(-1.0);
const std::vector<int> dy2 = {0, 1}, dx2 = {1, 0};
const std::vector<int> dy4 = {0, 1, 0, -1}, dx4 = {1, 0, -1, 0};
const std::vector<int> dy8 = {0, -1, 0, 1, 1, -1, -1, 1}, dx8 = {1, 0, -1, 0, 1, 1, -1, -1};

using namespace std;


class Vector2D {
public:
    // 平面ベクトル
    double x, y;

    Vector2D() {}

    Vector2D(double x, double y) : x(x), y(y) {}

    // ベクトルの和
    Vector2D operator+(const Vector2D p) { return Vector2D(x + p.x, y + p.y); }
    // ベクトルの差
    Vector2D operator-(const Vector2D p) { return Vector2D(x - p.x, y - p.y); }

    // ベクトルのスカラー倍
    Vector2D operator*(const double d) { return Vector2D(d * x, d * y); }
    Vector2D operator/(const double d) { return Vector2D(x / d, y / d); }

    // ベクトルの長さ
    double norm() { return sqrt(x * x + y * y); }

    // ベクトルの回転
    Vector2D rotate(double theta) {
        return Vector2D(x * cos(theta) - y * sin(theta), x * sin(theta) + y * cos(theta));
    }
};

class Circle {
public:
    Vector2D center;    // 中心
    double radius;      // 半径

    Circle() {}
    Circle(Vector2D center, double radius) : center(center), radius(radius) {}
};

vector<Vector2D> getIntersections(Circle C1, Circle C2) {
    double d = (C2.center - C1.center).norm();
    double rc = (C1.radius * C1.radius + d * d - C2.radius * C2.radius) / (2 * d);
    double rs = sqrt(C1.radius * C1.radius - rc * rc);

    Vector2D e1 = (C2.center - C1.center) / d;
    Vector2D e2 = e1.rotate(M_PI / 2);
    Vector2D e3 = e1.rotate(-M_PI / 2);
    Vector2D A = C1.center + e1 * rc + e2 * rs;
    Vector2D B = C1.center + e1 * rc + e3 * rs;

    return {A, B};
}


// 半径r，中心(y1, x1)の円に点(y2, x2)が含まれるか
bool inside_circle(Circle c, Vector2D v) {
    auto x1 = c.center.x;
    auto y1 = c.center.y;
    auto r = c.radius;
    auto x2 = v.x;
    auto y2 = v.y;

    return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2) <= r * r + EPS;  // 線も含む
}

LL K;
vector<tuple<double, double, double>> meat;

bool ok(double time) {
    int N = LEN(meat);
    auto cs = make_v<Circle>(0);
    auto p = make_v<Vector2D>(0);
    FOR(i, 0, N) {
        double x1, y1, c1;
        tie(x1, y1, c1) = meat[i];

        Circle circle1(Vector2D(x1, y1), time / c1);
        cs.emplace_back(circle1);
        p.emplace_back(circle1.center);

        FOR(j, i + 1, N) {
            double x2, y2, c2;
            tie(x2, y2, c2) = meat[j];

            Circle circle2(Vector2D(x2, y2), time / c2);

            auto a = getIntersections(circle1, circle2);

            FOR(k, 0, LEN(a)) {
                p.emplace_back(a[k]);
            }
        }
    }

    FOR(i, 0, LEN(p)) {
        int count = 0;
        FOE(c, cs) {
            count += inside_circle(c, p[i]);
        }
        if (count >= K) {
            return true;
        }
    }

    return false;
}



int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);

    LL N;
    cin >> N >> K;
    meat = make_v<tuple<double, double, double>>(N);

    FOR(i, 0, N) {
        double X, Y, C;
        cin >> X >> Y >> C;
        meat[i] = make_tuple(X, Y, C);
    }


    double low = 0;
    double high = 10000000;
    FOR(i, 0, 1000) {
        double middle = (low + high) / 2.0;
        if (ok(middle)) {
            high = middle;
        }
        else {
            low = middle;
        }
    }
    printf("%.10f\n", high);

    return 0;
}