Problem:
Score: 400 points

Problem Statement

We have N integers A_1, A_2, ..., A_N.

There are \frac{N(N-1)}{2} ways to choose two of them and form a pair. If we compute the product of each of those pairs and sort the results in ascending order, what will be the K-th number in that list?

Constraints

All values in input are integers.

2 \leq N \leq 2 \times 10^5

1 \leq K \leq \frac{N(N-1)}{2}

-10^9 \leq A_i \leq 10^9\ (1 \leq i \leq N)

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \dots A_N

Output

Print the answer.

Sample Input 1

4 3
3 3 -4 -2

Sample Output 1

-6

There are six ways to form a pair. The products of those pairs are 9, -12, -6, -12, -6, 8.

Sorting those numbers in ascending order, we have -12, -12, -6, -6, 8, 9. The third number in this list is -6.

Sample Input 2

10 40
5 4 3 2 -1 0 0 0 0 0

Sample Output 2

6

Sample Input 3

30 413
-170202098 -268409015 537203564 983211703 21608710 -443999067 -937727165 -97596546 -372334013 398994917 -972141167 798607104 -949068442 -959948616 37909651 0 886627544 -20098238 0 -948955241 0 -214720580 277222296 -18897162 834475626 0 -425610555 110117526 663621752 0

Sample Output 3

448283280358331064

Solution in Common Lisp (SBCL 2.0.3) (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
(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (declare #.OPT)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10))
                                         result))))
              (return (if minus (- result) result))))))))

(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 sort< (as)
  (declare #.OPT
           (inline sort)
           ((simple-array int32 (*)) as))
  (sort as #'<))

(defun sort> (as)
  (declare #.OPT
           (inline sort)
           ((simple-array int32 (*)) as))
  (sort as #'>))

(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'int32))
         (as+ (make-array n :fill-pointer 0 :element-type 'int32))
         (as- (make-array n :fill-pointer 0 :element-type 'int32)))
    (declare (uint31 n k))
    (dotimes (i n)
      (let ((a (read-fixnum)))
        (setf (aref as i) a)
        (vector-push a (if (>= a 0) as+ as-))))
    (let* ((as+< (sort< (coerce as+ '(simple-array int32 (*)))))
           (as+> (sort> (coerce as+ '(simple-array int32 (*)))))
           (as-< (sort< (coerce as- '(simple-array int32 (*)))))
           (as-> (sort> (coerce as- '(simple-array int32 (*))))))
      (declare ((simple-array int32 (*)) as+< as+> as-< as->))
      (labels ((calc-count (threshold)
                 (declare (fixnum threshold))
                 (let ((res 0))
                   (declare (uint62 res))
                   (when (>= threshold 0)
                     (let ((end 0))
                       (sb-int:dovector (a as+>)
                         (loop until (or (= end (length as+<))
                                         (< threshold (* a (aref as+< end))))
                               do (incf end)
                               finally (incf res end)))))
                   (if (>= threshold 0)
                       (incf res (the uint62 (* (length as+<) (length as-<))))
                       (let ((end 0))
                         (sb-int:dovector (a as+<)
                           (loop until (or (= end (length as-<))
                                           (< threshold (* a (aref as-< end))))
                                 do (incf end)
                                 finally (incf res end)))))
                   (if (>= threshold 0)
                       (incf res (the uint62 (* (length as+<) (length as-<))))
                       (let ((end 0))
                         (sb-int:dovector (a as->)
                           (loop until (or (= end (length as+>))
                                           (< threshold (* a (aref as+> end))))
                                 do (incf end)
                                 finally (incf res end)))))
                   (when (>= threshold 0)
                     (let ((end 0))
                       (sb-int:dovector (a as-<)
                         (loop until (or (= end (length as->))
                                         (< threshold (* a (aref as-> end))))
                               do (incf end)
                               finally (incf res end)))))
                   (sb-int:dovector (a as)
                     (when (<= (* a a) threshold)
                       (decf res)))
                   (setq res (ash res -1))
                   res)))
        (sb-int:named-let bisect ((ng #.(- (expt 10 18))) (ok #.(+ 1 (expt 10 18))))
          (declare (fixnum ng ok))
          (if (<= (- ok ng) 1)
              (println ok)
              (let ((mid (floor (+ ng ok) 2)))
                (if (>= (calc-count mid) k)
                    (bisect ng mid)
                    (bisect mid ok)))))))))

#-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
3 3 -4 -2
"
    "-6
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "10 40
5 4 3 2 -1 0 0 0 0 0
"
    "6
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "30 413
-170202098 -268409015 537203564 983211703 21608710 -443999067 -937727165 -97596546 -372334013 398994917 -972141167 798607104 -949068442 -959948616 37909651 0 886627544 -20098238 0 -948955241 0 -214720580 277222296 -18897162 834475626 0 -425610555 110117526 663621752 0
"
    "448283280358331064
")))