Problem:
Score : 600 points

Problem Statement

You are given an integer sequence of length N, a = {a_1, a_2, …, a_N}, and an integer K.

a has N(N+1)/2 non-empty contiguous subsequences, {a_l, a_{l+1}, …, a_r} (1 ≤ l ≤ r ≤ N). Among them, how many have an arithmetic mean that is greater than or equal to K?

Constraints

All input values are integers.

1 ≤ N ≤ 2 \times 10^5

1 ≤ K ≤ 10^9

1 ≤ a_i ≤ 10^9

Input

Input is given from Standard Input in the following format:

N K
a_1
a_2
:
a_N

Output

Print the number of the non-empty contiguous subsequences with an arithmetic mean that is greater than or equal to K.

Sample Input 1

3 6
7
5
7

Sample Output 1

5

All the non-empty contiguous subsequences of a are listed below:

{a_1} = {7}

{a_1, a_2} = {7, 5}

{a_1, a_2, a_3} = {7, 5, 7}

{a_2} = {5}

{a_2, a_3} = {5, 7}

{a_3} = {7}

Their means are 7, 6, 19/3, 5, 6 and 7, respectively, and five among them are 6 or greater. Note that {a_1} and {a_3} are indistinguishable by the values of their elements, but we count them individually.

Sample Input 2

1 2
1

Sample Output 2

0

Sample Input 3

7 26
10
20
30
40
30
20
10

Sample Output 3

13

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

(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 (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
;; 1D BIT
(defmacro define-bitree (name &key (operator '#'+) (identity 0) sum-type (order '#'>))
  "OPERATOR := binary operator (on a commutative monoid)
IDENTITY := object (identity element of the monoid)
ORDER := nil | strict comparison operator on the monoid
SUM-TYPE := nil | type specifier

Defines no structure; BIT is just a vector. Defines the three function:
<NAME>-UPDATE!, <NAME>-SUM and COERCE-TO-<NAME>!. In addition this macro defines
the bisection function <NAME>-BISECT-LEFT if ORDER is specified. (Note that the
last function works only when the sequence of sums (VECTOR[0],
VECTOR[0]+VECTOR[1], ...) is monotonous.)

SUM-TYPE is used only for the type declaration: each sum
VECTOR[i]+VECTOR[i+1]...+VECTOR[i+k] is declared to be this type. (The
element-type of vector itself doesn't need to be SUM-TYPE.)"
  (let* ((name (string name))
         (fname-update (intern (format nil "~A-UPDATE!" name)))
         (fname-sum (intern (format nil "~A-SUM" name)))
         (fname-coerce (intern (format nil "COERCE-TO-~A!" name)))
         (fname-bisect (intern (format nil "~A-BISECT-LEFT" name))))
    `(progn
       (declaim (inline ,fname-update))
       (defun ,fname-update (bitree index delta)
         "Destructively increments the vector: vector[INDEX] = vector[INDEX] +
DELTA"
         (let ((len (length bitree)))
           (do ((i index (logior i (+ i 1))))
               ((>= i len) bitree)
             (declare ((integer 0 #.most-positive-fixnum) i))
             (setf (aref bitree i)
                   (funcall ,operator (aref bitree i) delta)))))

       (declaim (inline ,fname-sum))
       (defun ,fname-sum (bitree end)
         "Returns the sum of prefix: vector[0] + ... + vector[END-1]."
         (declare ((integer 0 #.most-positive-fixnum) end))
         (let ((res ,identity))
           ,@(when sum-type `((declare (type ,sum-type res))))
           (do ((i (- end 1) (- (logand i (+ i 1)) 1)))
               ((< i 0) res)
             (declare ((integer -1 #.most-positive-fixnum) i))
             (setf res (funcall ,operator res (aref bitree i))))))

       (declaim (inline ,fname-coerce))
       (defun ,fname-coerce (vector)
         "Destructively constructs BIT from VECTOR."
         (loop with len = (length vector)
               for i below len
               for dest-i = (logior i (+ i 1))
               when (< dest-i len)
               do (setf (aref vector dest-i)
                        (funcall ,operator (aref vector dest-i) (aref vector i)))
               finally (return vector)))

       ,@(when order
           `((declaim (inline ,fname-bisect))
             (defun ,fname-bisect (bitree value)
               "Returns the smallest index that fulfills VECTOR[0]+ ... +
VECTOR[index-1] >= VALUE. Returns the length of VECTOR if VECTOR[0]+
... +VECTOR[END-1] > VALUE."
               (declare (vector bitree))
               (if (not (funcall ,order value ,identity))
                   0
                   (let ((len (length bitree))
                         (index+1 0)
                         (cumul ,identity))
                     (declare ((integer 0 #.most-positive-fixnum) index+1)
                              ,@(when sum-type
                                  `((type ,sum-type cumul))))
                     (do ((delta (ash 1 (- (integer-length len) 1))
                                 (ash delta -1)))
                         ((zerop delta) index+1)
                       (declare ((integer 0 #.most-positive-fixnum) delta))
                       (let ((next-index (+ index+1 delta -1)))
                         (when (< next-index len)
                           (let ((next-cumul (funcall ,operator cumul (aref bitree next-index))))
                             ,@(when sum-type
                                 `((declare (type ,sum-type next-cumul))))
                             (when (funcall ,order value next-cumul)
                               (setf cumul next-cumul)
                               (incf index+1 delta))))))))))))))

(define-bitree bitree
  :operator #'+
  :identity 0
  :sum-type fixnum)

(declaim (inline make-reverse-lookup-table))
(defun make-reverse-lookup-table (vector &key (test #'eql))
  "Assigns each value of the (sorted) VECTOR of length n to the integers 0, ...,
n-1."
  (let ((table (make-hash-table :test test :size (length vector))))
    (dotimes (i (length vector) table)
      (setf (gethash (aref vector i) table) i))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (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
                                  ;; (return-from read-fixnum 0)
                                  (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) (the fixnum (* result 10))))
              (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)

(defmacro println (obj &optional (stream '*standard-output*))
  `(let ((*read-default-float-format* 'double-float))
     (prog1 (princ ,obj ,stream) (terpri ,stream))))

(defconstant +mod+ 1000000007)

;; Body

(defun main ()
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'int32))
         (cumul (make-array (1+ n) :element-type 'fixnum :initial-element 0))
         (bitree (make-array (1+ n) :element-type 'fixnum :initial-element 0)))
    (declare (uint31 n k))
    (dotimes (i n)
      (setf (aref as i) (- (read-fixnum) k))
      (setf (aref cumul (1+ i)) (+ (aref cumul i) (aref as i))))
    (let ((table (make-reverse-lookup-table (sort (copy-seq cumul) #'<))))
      (println
       (loop for i to n
             sum (bitree-sum bitree (1+ (the uint31 (gethash (aref cumul i) table))))
             of-type fixnum
             do (bitree-update! bitree (gethash (aref cumul i) table) 1))))))

#-swank(main)