Problem:
Score : 700 points

Problem Statement

We will define the median of a sequence b of length M, as follows:

Let b' be the sequence obtained by sorting b in non-decreasing order. Then, the value of the (M / 2 + 1)-th element of b' is the median of b. Here, / is integer division, rounding down.

For example, the median of (10, 30, 20) is 20; the median of (10, 30, 20, 40) is 30; the median of (10, 10, 10, 20, 30) is 10.

Snuke comes up with the following problem.

You are given a sequence a of length N.
For each pair (l, r) (1 \leq l \leq r \leq N), let m_{l, r} be the median of the contiguous subsequence (a_l, a_{l + 1}, ..., a_r) of a.
We will list m_{l, r} for all pairs (l, r) to create a new sequence m.
Find the median of m.

Constraints

1 \leq N \leq 10^5

a_i is an integer.

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

Print the median of m.

Sample Input 1

3
10 30 20

Sample Output 1

30

The median of each contiguous subsequence of a is as follows:

The median of (10) is 10.

The median of (30) is 30.

The median of (20) is 20.

The median of (10, 30) is 30.

The median of (30, 20) is 30.

The median of (10, 30, 20) is 20.

Thus, m = (10, 30, 20, 30, 30, 20) and the median of m is 30.

Sample Input 2

1
10

Sample Output 2

10

Sample Input 3

10
5 9 5 9 8 9 3 5 4 3

Sample Output 3

8

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

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (progn (ql:quickload '(:cl-debug-print :fiveam))
                 (shadow :run)
                 (use-package :fiveam))
  #-swank (set-dispatch-macro-character #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil (values) t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
;;;
;;; ARRAY-ELEMENT-TYPE is not constant-folded on SBCL version earlier than
;;; 1.5.0. See
;;; https://github.com/sbcl/sbcl/commit/9f0d12e7ab961828931d01c0b2a76a5885ad35d2
;;;

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-c:deftransform array-element-type ((array))
    (let ((type (sb-c::lvar-type array)))
      (flet ((element-type (type)
               (and (sb-c::array-type-p type)
                    (sb-int:neq (sb-kernel::array-type-specialized-element-type type) sb-kernel:*wild-type*)
                    (sb-kernel:type-specifier (sb-kernel::array-type-specialized-element-type type)))))
        (cond ((let ((type (element-type type)))
                 (and type
                      `',type)))
              ((sb-kernel:union-type-p type)
               (let (result)
                 (loop for type in (sb-kernel:union-type-types type)
                       for et = (element-type type)
                       unless (and et
                                   (if result
                                       (equal result et)
                                       (setf result et)))
                       do (sb-c::give-up-ir1-transform))
                 `',result))
              ((sb-kernel:intersection-type-p type)
               (loop for type in (sb-kernel:intersection-type-types type)
                     for et = (element-type type)
                     when et
                     return `',et
                     finally (sb-c::give-up-ir1-transform)))
              (t
               (sb-c::give-up-ir1-transform)))))))

;;;
;;; Calculate inversion number by merge sort
;;;

(declaim (inline %merge-count))
(defun %merge-count (l mid r source-vec dest-vec predicate)
  (declare ((integer 0 #.array-total-size-limit) l mid r)
           (function predicate))
  (loop with count of-type (integer 0 #.most-positive-fixnum) = 0
        with i = l
        with j = mid
        for idx from l
        when (= i mid)
        do (loop for j from j below r
                 for idx from idx
                 do (setf (aref dest-vec idx)
                          (aref source-vec j))
                 finally (return-from %merge-count count))
        when (= j r)
        do (loop for i from i below mid
                 for idx from idx
                 do (setf (aref dest-vec idx)
                          (aref source-vec i))
                 finally (return-from %merge-count count))
        do (if (funcall predicate
                        (aref source-vec j)
                        (aref source-vec i))
               (setf (aref dest-vec idx) (aref source-vec j)
                     j (1+ j)
                     count (+ count (- mid i)))
               (setf (aref dest-vec idx) (aref source-vec i)
                     i (1+ i)))))

(defmacro with-fixnum+ (form)
  (let ((fixnum+ '(integer 0 #.most-positive-fixnum)))
    `(the ,fixnum+
          ,(reduce (lambda (f1 f2)`(,(car form)
                                   (the ,fixnum+ ,f1)
                                   (the ,fixnum+ ,f2)))
	           (cdr form)))))

(declaim (inline %calc-by-insertion-sort!))
(defun %calc-by-insertion-sort! (vec predicate l r)
  (declare (function predicate)
           ((integer 0 #.array-total-size-limit) l r))
  (loop with inv-count of-type (integer 0 #.most-positive-fixnum) = 0
        for end from (+ l 1) below r
        do (loop for i from end above l
                 while (funcall predicate (aref vec i) (aref vec (- i 1)))
                 do (rotatef (aref vec (- i 1)) (aref vec i))
                    (incf inv-count))
        finally (return inv-count)))

(declaim (inline calc-inversion-number!))
(defun calc-inversion-number! (vector predicate &key (start 0) end)
  "Calculates the inversion number of VECTOR w.r.t. the strict order
PREDICATE. This function sorts VECTOR as a side effect."
  (declare (vector vector)
           (function predicate))
  (let ((end (or end (length vector))))
    (declare ((integer 0 #.array-total-size-limit) start end))
    (assert (<= start end))
    (let ((buffer (make-array (length vector) :element-type (array-element-type vector))))
      (labels
          ((recurse (l r merge-to-vec1-p)
             (declare (optimize (safety 0))
                      ((integer 0 #.array-total-size-limit) l r))
             (cond ;; It is faster to use insertion sort. I don't adopt it
                   ;; by default, however, because that makes it hard to
                   ;; change the code to fit some special settings.
                   ((and (<= (- r l) 24) merge-to-vec1-p)
                    (%calc-by-insertion-sort! vector predicate l r))
                   (t
                    (let ((mid (floor (+ l r) 2)))
                      (with-fixnum+
                          (+ (recurse l mid (not merge-to-vec1-p))
                             (recurse mid r (not merge-to-vec1-p))
                             (if merge-to-vec1-p
                                 (%merge-count l mid r buffer vector predicate)
                                 (%merge-count l mid r vector buffer predicate)))))))))
        (recurse start end t)))))

;; Scheme-style named let
(defmacro nlet (name args &body body)
  (labels ((ensure-list (x) (if (listp x) x (list x))))
    (let ((args (mapcar #'ensure-list args)))
      `(labels ((,name ,(mapcar #'car args) ,@body))
         (,name ,@(mapcar #'cadr args))))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (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 feasible-p (as x)
  (declare #.OPT
           ((simple-array uint32 (*)) as)
           (uint32 x))
  (let* ((n (length as))
         (cumul (make-array (+ n 1) :element-type 'int32 :initial-element 0)))
    (dotimes (i n)
      (setf (aref cumul (+ i 1))
            (+ (aref cumul i) (if (>= (aref as i) x) 1 -1))))
    (let ((non-inversion-number (- (ash (* n (+ n 1)) -1)
                                   (calc-inversion-number! cumul #'<)))
          (threshold (ceiling (* n (+ n 1)) 4)))
      (>= non-inversion-number threshold))))

(defun main ()
  (let* ((n (read))
         (as (make-array n :element-type 'uint32)))
    (dotimes (i n) (setf (aref as i) (read-fixnum)))
    (nlet bisect ((ok 1) (ng 1000000001))
      (if (<= (- ng ok) 1)
          (println ok)
          (let ((mid (ash (+ ok ng) -1)))
            (if (feasible-p as mid)
                (bisect mid ng)
                (bisect ok mid)))))))

#-swank(main)