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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(macrolet ((def (name fname)
             `(define-modify-macro ,name (new-value) ,fname)))
  (def minf min)
  (def maxf max)
  (def mulf *)
  (def divf /)
  (def iorf logior)
  (def xorf logxor)
  (def andf logand))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(defun calc (n k xs ys cs)
  (when ))

(defun main ()
  (declare #.OPT
           (inline sort))
  (let* ((n (read))
         (k (read))
         (xs (make-array n :element-type 'double-float))
         (ys (make-array n :element-type 'double-float))
         (cs (make-array n :element-type 'double-float)))
    (declare (uint8 n k))
    (dotimes (i n)
      (setf (aref xs i) (float (read) 1d0)
            (aref ys i) (float (read) 1d0)
            (aref cs i) (float (read) 1d0)))
    (let ((tmp (make-array n :element-type 'double-float)))
      (declare ((simple-array double-float (*)) tmp))
      (labels ((calc (pivot-x pivot-y)
                 (declare (double-float pivot-x pivot-y))
                 (dotimes (i n)
                   (let ((x (aref xs i))
                         (y (aref ys i))
                         (c (aref cs i)))
                     (setf (aref tmp i)
                           (* c (sqrt (+ (expt (- x pivot-x) 2)
                                         (expt (- y pivot-y) 2)))))))
                 (setq tmp (sort tmp #'<))
                 (aref tmp (- k 1)))
               (%%minimize (pivot-x pivot-y rad div)
                 (declare (double-float pivot-x pivot-y rad)
                          (uint32 div))
                 (let ((delta (/ rad div))
                       (res most-positive-double-float))
                   (declare (double-float delta res))
                   (loop for x from (- pivot-x rad) to (+ pivot-x rad) by delta
                         do (loop for y from (- pivot-y rad) to (+ pivot-y rad) by delta
                                  do (minf res (calc x y))))
                   res))
               (%minimize (x1 y1 x2 y2 rate div)
                 (declare (double-float x1 y1 x2 y2 rate))
                 (let* ((xmid (* 0.5d0 (+ x1 x2)))
                        (ymid (* 0.5d0 (+ y1 y2)))
                        (delta (* 0.5d0 (max (abs (* 0.5d0 (- xmid x1)))
                                             (abs (* 0.5d0 (- ymid y1)))))))
                   (let* ((min00 (%%minimize (* 0.5d0 (+ x1 xmid))
                                             (* 0.5d0 (+ y1 ymid))
                                             (* 1.5d0 delta) div))
                          (min01 (%%minimize (* 0.5d0 (+ xmid x2))
                                             (* 0.5d0 (+ y1 ymid))
                                             (* 1.5d0 delta) div))
                          (min10 (%%minimize (* 0.5d0 (+ x1 xmid))
                                             (* 0.5d0 (+ ymid y2))
                                             (* 1.5d0 delta) div))
                          (min11 (%%minimize (* 0.5d0 (+ xmid x2))
                                             (* 0.5d0 (+ ymid y2))
                                             (* 1.5d0 delta) div))
                          (min (min min00 min01 min10 min11))
                          (d (* rate delta)))
                     (cond ((< delta 1d-10) min)
                           ((= min min00) (%minimize (- x1 d) (- y1 d) (+ xmid d) (+ ymid d) rate div))
                           ((= min min01) (%minimize (- xmid d) (- y1 d) (+ x2 d) (+ ymid d) rate div))
                           ((= min min10) (%minimize (- x1 d) (- ymid d) (+ xmid d) (+ y2 d) rate div))
                           ((= min min11) (%minimize (- xmid d) (- ymid d) (+ x2 d) (+ y2 d) rate div))
                           (t (error "Huh?")))))))
        (println (min (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.75d0 20)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.5d0 10)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.25d0 10)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.35d0 10)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.1d0 10)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.6d0 10)
                      (%minimize -1101.1d0 -1050.9d0 1040.3d0 1031.6d0 0.8d0 10)))))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "4 3
-1 0 3
0 0 3
1 0 2
1 1 40
"
    "2.4
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "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
"
    "7411.2252
")))