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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;
using System.Numerics;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var k = ni();
        var x = new int[n];
        var y = new int[n];
        var c = new int[n];

        for (int i = 0; i < n; i++) {
            x[i] = ni();
            y[i] = ni();
            c[i] = ni();
        }

        var p = x.Zip(y, (_x, _y) => new Complex(_x, _y)).ToArray();

        double lower = 0, upper = 10000000;

        for (int _ = 0; _ < 150; _++) {
            var mid = (lower + upper) / 2.0;
            bool ok = false;

            // time = ci * distance 
            // distance = time / ci
            var points = new List<Complex>();

            points.AddRange(p);

            for (int i = 0; i < n; i++) {
                for (int j = 0; j < i; j++) {
                    points.AddRange(CrossCircles(p[i], mid / c[i], p[j], mid / c[j]));
                }
            }

            foreach (var cand in points) {
                var covered = 0;
                for (int i = 0; i < n; i++) {
                    var distance = Complex.Abs(cand - p[i]);
                    if (distance <= mid / c[i] + Epsilon) {
                        covered++;
                    }
                }
                if (covered >= k) {
                    ok = true; break;
                }
            }

            if (ok) {
                upper = mid;
            } else {
                lower = mid;
            }
        }

        cout.WriteLine("{0:F15}", lower);
    }

    const double Epsilon = 1e-9;

    public static Complex[] CrossCircles(Complex c1, double r1, Complex c2, double r2) {
        double d = (c1 - c2).Magnitude;
        if (r1 + r2 < d - Epsilon || d < Math.Abs(r2 - r1) - Epsilon) {
            return new Complex[0];
        }

        double l = 0.5 * ((r1 * r1 - r2 * r2) / d + d);
        double h = Math.Sqrt(r1 * r1 - l * l);
        return new[] {
                new Complex(l, +h) * (c2 - c1) / d + c1,
                new Complex(l, -h) * (c2 - c1) / d + c1,
            };
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new InvalidOperationException();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}