;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;; Treap accessible by index (O(log(n))).
;; Virtually it works like std::set of C++ or TreeSet of Java.

;; Note:
;; - You shouldn't insert duplicate keys into a treap unless you know what you
;; are doing.
;; - You cannot rely on the side effect when you call any destructive operations
;; on a treap. Always use the returned value.
;; - An empty treap is NIL.

(defstruct (treap (:constructor %make-treap (key priority &key left right (count 1)))
                  (:copier nil)
                  (:conc-name %treap-))
  (key 0 :type fixnum)
  (priority 0 :type (integer 0 #.most-positive-fixnum))
  (count 0 :type (integer 0 #.most-positive-fixnum))
  (left nil :type (or null treap))
  (right nil :type (or null treap)))

(declaim (inline treap-count))
(defun treap-count (treap)
  "Returns the size of the (nullable) TREAP."
  (declare ((or null treap) treap))
  (if (null treap)
      0
      (%treap-count treap)))

(declaim (inline update-count))
(defun update-count (treap)
  (declare (treap treap))
  (setf (%treap-count treap)
        (+ 1
           (treap-count (%treap-left treap))
           (treap-count (%treap-right treap)))))

(declaim (inline treap-bisect-left)
         (ftype (function * (values (integer 0 #.most-positive-fixnum) t &optional)) treap-bisect-left))
(defun treap-bisect-left (value treap &key (order #'<))
  "Returns the smallest index and the corresponding key that satisfies
TREAP[index] >= VALUE. Returns the size of TREAP and VALUE if TREAP[size-1] <
VALUE."
  (declare (function order))
  (labels ((recur (count treap)
             (declare ((integer 0 #.most-positive-fixnum) count))
             (cond ((null treap) (values nil nil))
                   ((funcall order (%treap-key treap) value)
                    (recur count (%treap-right treap)))
                   (t (let ((left-count (- count (treap-count (%treap-right treap)) 1)))
                        (multiple-value-bind (idx key)
                            (recur left-count (%treap-left treap))
                          (if idx
                              (values idx key)
                              (values left-count (%treap-key treap)))))))))
    (declare (ftype (function * (values t t &optional)) recur))
    (multiple-value-bind (idx key)
        (recur (treap-count treap) treap)
      (if idx
          (values idx key)
          (values (treap-count treap) value)))))

(declaim (inline treap-split)
         (ftype (function * (values (or null treap) (or null treap) &optional)) treap-split))
(defun treap-split (key treap &key (order #'<))
  "Destructively splits the TREAP with reference to KEY and returns two treaps,
the smaller sub-treap (< KEY) and the larger one (>= KEY)."
  (declare (function order)
           ((or null treap) treap))
  (labels ((recur (treap)
             (cond ((null treap)
                    (values nil nil))
                   ((funcall order (%treap-key treap) key)
                    (multiple-value-bind (left right) (recur (%treap-right treap))
                      (setf (%treap-right treap) left)
                      (update-count treap)
                      (values treap right)))
                   (t
                    (multiple-value-bind (left right) (recur (%treap-left treap))
                      (setf (%treap-left treap) right)
                      (update-count treap)
                      (values left treap))))))
    (recur treap)))

(declaim (inline treap-insert))
(defun treap-insert (key treap &key (order #'<))
  "Destructively inserts KEY into TREAP and returns the resultant treap."
  (declare ((or null treap) treap)
           (function order))
  (let ((node (%make-treap key (random most-positive-fixnum))))
    (labels ((recur (treap)
               (declare (treap node))
               (cond ((null treap) node)
                     ((> (%treap-priority node) (%treap-priority treap))
                      (setf (values (%treap-left node) (%treap-right node))
                            (treap-split (%treap-key node) treap :order order))
                      (update-count node)
                      node)
                     (t
                      (if (funcall order (%treap-key node) (%treap-key treap))
                          (setf (%treap-left treap)
                                (recur (%treap-left treap)))
                          (setf (%treap-right treap)
                                (recur (%treap-right treap))))
                      (update-count treap)
                      treap))))
      (recur treap))))

(defun treap-map (function treap)
  "Successively applies FUNCTION to TREAP[0], ..., TREAP[SIZE-1]. FUNCTION must
take one argument."
  (declare (function function))
  (when treap
    (treap-map function (%treap-left treap))
    (funcall function (%treap-key treap))
    (treap-map function (%treap-right treap))))

(defmethod print-object ((object treap) stream)
  (print-unreadable-object (object stream :type t)
    (let ((init t))
      (treap-map (lambda (key)
                   (if init
                       (setq init nil)
                       (write-char #\  stream))
                   (write key :stream stream))
                 object))))

(define-condition invalid-treap-index-error (type-error)
  ((treap :initarg :treap :reader invalid-treap-index-error-treap)
   (index :initarg :index :reader invalid-treap-index-error-index))
  (:report
   (lambda (condition stream)
     (format stream "Invalid index ~W for treap ~W."
             (invalid-treap-index-error-index condition)
             (invalid-treap-index-error-treap condition)))))

(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))))))))

(declaim (inline bisect-left))
(defun bisect-left (target value &key (start 0) end (order #'<) (key #'identity))
  "TARGET := vector | function (taking an integer argument)
ORDER := strict order

Analogue of lower_bound of C++ or bisect_left of Python: Returns the smallest
index (or input) i that fulfills TARGET[i] >= VALUE, where '>=' is the
complement of ORDER. In other words, this function returns the leftmost index at
which VALUE can be inserted with keeping the order. Therefore, TARGET must be
monotonically non-decreasing with respect to ORDER.

This function returns END if VALUE exceeds TARGET[END-1]. Note that the range
[START, END) is half-open. END must be explicitly specified if TARGET is
function. KEY is applied to each element of TARGET before comparison."
  (declare (function key order)
           (integer start)
           ((or null integer) end))
  (macrolet
      ((body (accessor &optional declaration)
         `(progn
            (assert (<= start end))
            (if (= start end) end
                (labels
                    ((%bisect-left (left ok)
                       ;; TARGET[OK] >= VALUE always holds (assuming
                       ;; TARGET[END] = +infinity)
                       ,@(list declaration)
                       (let ((mid (ash (+ left ok) -1)))
                         (if (= mid left)
                             (if (funcall order (funcall key (,accessor target left)) value)
                                 ok
                                 left)
                             (if (funcall order (funcall key (,accessor target mid)) value)
                                 (%bisect-left mid ok)
                                 (%bisect-left left mid))))))
                  (%bisect-left start end))))))
    (etypecase target
      (vector
       (let ((end (or end (length target))))
         (body aref (declare ((integer 0 #.most-positive-fixnum) left ok)))))
      (function
       (assert end () "Requires END argument if TARGET is a function.")
       (body funcall (declare ((integer 0 #.most-positive-fixnum) left ok)))))))

(declaim (inline bisect-right))
(defun bisect-right (target value &key (start 0) end (order #'<) (key #'identity))
  "TARGET := vector | function (taking an integer argument)
ORDER := strict order

Analogue of upper_bound of C++ or bisect_right of Python: Returns the smallest
index (or input) i that fulfills TARGET[i] > VALUE. In other words, this
function returns the rightmost index at which VALUE can be inserted with keeping
the order. TARGET must be monotonically non-decreasing with respect to ORDER.

This function returns END if VALUE >= TARGET[END-1]. Note that the range [START,
END) is half-open. END must be explicitly specified if TARGET is function. KEY
is applied to each element of TARGET before comparison."
  (declare (function key order)
           (integer start)
           ((or null integer) end))
  (macrolet
      ((body (accessor &optional declaration)
         `(progn
            (assert (<= start end))
            (if (= start end)
                end
                (labels
                    ((%bisect-right (left ok)
                       ;; TARGET[OK] > VALUE always holds (assuming
                       ;; TARGET[END] = +infinity)
                       ,@(list declaration)
                       (let ((mid (ash (+ left ok) -1)))
                         (if (= mid left)
                             (if (funcall order value (funcall key (,accessor target left)))
                                 left
                                 ok)
                             (if (funcall order value (funcall key (,accessor target mid)))
                                 (%bisect-right left mid)
                                 (%bisect-right mid ok))))))

                  (%bisect-right start end))))))
    (etypecase target
      (vector
       (when (null end)
         (setf end (length target)))
       (body aref (declare ((integer 0 #.most-positive-fixnum) left ok))))
      (function
       (assert end () "Requires END argument if TARGET is a function.")
       (body funcall (declare ((integer 0 #.most-positive-fixnum) left ok)))))))

(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+ 998244353)

;; Body
;;;
;;; Arithmetic operations with static modulus
;;;

(defmacro define-mod-operations (&optional (divisor 1000000007))
  `(progn
     (defun mod* (&rest args)
       (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

     (sb-c:define-source-transform mod* (&rest args)
       (if (null args)
           1
           (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))

     (defun mod+ (&rest args)
       (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

     (sb-c:define-source-transform mod+ (&rest args)
       (if (null args)
           0
           (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))

     (define-modify-macro incfmod (delta divisor)
       (lambda (x y divisor) (mod (+ x y) divisor)))

     (define-modify-macro decfmod (delta divisor)
       (lambda (x y divisor) (mod (- x y) divisor)))))

(define-mod-operations 998244353)

(defun main ()
  (declare #.OPT
           (inline sort))
  (let* ((n (read))
         (xs (make-array n :element-type 'int32))
         (ys (make-array n :element-type 'int32))
         (powers (make-array 400000 :element-type 'uint32))
         (ord-xs (make-array n :element-type 'uint32))
         (res 0))
    (declare (uint32 n res)
             ((simple-array uint32 (*)) ord-xs))
    (setf (aref powers 0) 1)
    (loop for i from 1 below (length powers)
          do (setf (aref powers i)
                   (mod* 2 (aref powers (- i 1)))))
    (dotimes (i n)
      (setf (aref xs i) (read-fixnum)
            (aref ys i) (read-fixnum)))
    (dotimes (i n)
      (setf (aref ord-xs i) i))
    (setf ord-xs (sort ord-xs (lambda (i j) (< (aref xs i) (aref xs j)))))
    (incfmod res (mod* (- (aref powers n) 1) n) +mod+)
    ;; L R U D
    (dotimes (i n)
      (let ((lower i)
            (upper (- n i 1)))
        (decfmod res (* 2 (aref powers lower)) +mod+)
        (decfmod res (* 2 (aref powers upper)) +mod+)))
    ;; LU LD
    (let (treap)
      (dotimes (i n)
        (let* ((ord (aref ord-xs i))
               (y (aref ys ord))
               (ld (treap-bisect-left y treap))
               (lu (- (treap-count treap) ld)))
          (incfmod res (aref powers ld) +mod+)
          (incfmod res (aref powers lu) +mod+)
          (setq treap (treap-insert y treap)))))
    ;; RU RD
    (let (treap)
      (loop for i from (- n 1) downto 0
            do (let* ((ord (aref ord-xs i))
                      (y (aref ys ord))
                      (rd (treap-bisect-left y treap))
                      (ru (- (treap-count treap) rd)))
                 (incfmod res (aref powers rd) +mod+)
                 (incfmod res (aref powers ru) +mod+)
                 (setq treap (treap-insert y treap)))))
    (println res)))

#-swank (main)

Score : 600 points

Problem Statement

We have a set S of N points in a two-dimensional plane. The coordinates of the i-th point are (x_i, y_i). The N points have distinct x-coordinates and distinct y-coordinates.

For a non-empty subset T of S, let f(T) be the number of points contained in the smallest rectangle, whose sides are parallel to the coordinate axes, that contains all the points in T. More formally, we define f(T) as follows:

f(T) := (the number of integers i (1 \leq i \leq N) such that a \leq x_i \leq b and c \leq y_i \leq d, where a, b, c, and d are the minimum x-coordinate, the maximum x-coordinate, the minimum y-coordinate, and the maximum y-coordinate of the points in T)

Find the sum of f(T) over all non-empty subset T of S. Since it can be enormous, print the sum modulo 998244353.

Constraints

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

-10^9 \leq x_i, y_i \leq 10^9

x_i \neq x_j (i \neq j)

y_i \neq y_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the sum of f(T) over all non-empty subset T of S, modulo 998244353.

Sample Input 1

3
-1 3
2 1
3 -2

Sample Output 1

13

Let the first, second, and third points be P_1, P_2, and P_3, respectively. S = \{P_1, P_2, P_3\} has seven non-empty subsets, and f has the following values for each of them:

f(\{P_1\}) = 1

f(\{P_2\}) = 1

f(\{P_3\}) = 1

f(\{P_1, P_2\}) = 2

f(\{P_2, P_3\}) = 2

f(\{P_3, P_1\}) = 3

f(\{P_1, P_2, P_3\}) = 3

The sum of these is 13.

Sample Input 2

4
1 4
2 1
3 3
4 2

Sample Output 2

34

Sample Input 3

10
19 -11
-3 -12
5 3
3 -15
8 -14
-9 -20
10 -9
0 2
-7 17
6 -6

Sample Output 3

7222

Be sure to print the sum modulo 998244353.


(let* ((n (read))
       (m (read)))
  (princ (if (evenp (+ n m))
             (/ (+ n m) 2)
             "IMPOSSIBLE")))

Global OOD problem IDs are disjoint from train, validation, and in-distribution test problem IDs.

Score: 100 points

Problem Statement

We have two distinct integers A and B.

Print the integer K such that |A - K| = |B - K|.

If such an integer does not exist, print IMPOSSIBLE instead.

Constraints

All values in input are integers.

0 \leq A,\ B \leq 10^9

A and B are distinct.

Input

Input is given from Standard Input in the following format:

A B

Output

Print the integer K satisfying the condition.

If such an integer does not exist, print IMPOSSIBLE instead.

Sample Input 1

2 16

Sample Output 1

9

|2 - 9| = 7 and |16 - 9| = 7, so 9 satisfies the condition.

Sample Input 2

0 3

Sample Output 2

IMPOSSIBLE

No integer satisfies the condition.

Sample Input 3

998244353 99824435

Sample Output 3

549034394

Submitted source-code file size, in bytes.

Execution CPU time reported by official Project CodeNet metadata, in milliseconds.

Peak memory usage reported by official Project CodeNet metadata, in kilobytes.


(let* ((n (read))
       (p (make-array `(,n)))
       (q (make-array `(,n))))
  (loop :for i :from 0 :to (1- n)
        :for x := (read)
        :do (setf (aref p i) x)
        :do (setf (aref q i) x))
  (setf q (sort q #'<))
  (if (<= (loop :for i :from 0 :to (1- n)
                :if (/= (aref p i) (aref q i))
                :sum 1) 2)
      (format t "YES~%")
      (format t "NO~%")))

Score : 200 points

Problem Statement

We have a sequence p = {p_1,\ p_2,\ ...,\ p_N} which is a permutation of {1,\ 2,\ ...,\ N}.

You can perform the following operation at most once: choose integers i and j (1 \leq i < j \leq N), and swap p_i and p_j. Note that you can also choose not to perform it.

Print YES if you can sort p in ascending order in this way, and NO otherwise.

Constraints

All values in input are integers.

2 \leq N \leq 50

p is a permutation of {1,\ 2,\ ...,\ N}.

Input

Input is given from Standard Input in the following format:

N
p_1 p_2 ... p_N

Output

Print YES if you can sort p in ascending order in the way stated in the problem statement, and NO otherwise.

Sample Input 1

5
5 2 3 4 1

Sample Output 1

YES

You can sort p in ascending order by swapping p_1 and p_5.

Sample Input 2

5
2 4 3 5 1

Sample Output 2

NO

In this case, swapping any two elements does not sort p in ascending order.

Sample Input 3

7
1 2 3 4 5 6 7

Sample Output 3

YES

p is already sorted in ascending order, so no operation is needed.


(defun split (x str)
  (let ((pos (search x str))
        (size (length x)))
    (if pos
      (cons (subseq str 0 pos)
            (split x (subseq str (+ pos size))))
      (list str))))

(setq *n* (read))
(setq *ps* (mapcar #'parse-integer (split " " (read-line))))

(defun count-transposition (l i)
  (if (equal l nil)
    0
    (if (equal (car l) i)
      (count-transposition (cdr l) (1+ i))
      (1+ (count-transposition (cdr l) (1+ i))))))

(format t "~a~%"
 (if (<= (count-transposition *ps* 1) 2)
   "YES"
   "NO"))


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Rolling hash (62-bit)
;;;

;; Reference:
;; https://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
;; https://ei1333.github.io/luzhiled/snippets/string/rolling-hash.html

(defstruct (rhash (:constructor %make-rhash (mod1 base1 cumul1 powers1 mod2 base2 cumul2 powers2)))
  ;; lower 31-bit value
  (mod1 2147483647 :type (unsigned-byte 31))
  (base1 1059428526 :type (unsigned-byte 31))
  (cumul1 nil :type (simple-array (unsigned-byte 31) (*)))
  (powers1 nil :type (simple-array (unsigned-byte 31) (*)))
  ;; upper 31-bit value
  (mod2 2147483629 :type (unsigned-byte 31))
  (base2 2090066834 :type (unsigned-byte 31))
  (cumul2 nil :type (simple-array (unsigned-byte 31) (*)))
  (powers2 nil :type (simple-array (unsigned-byte 31) (*))))

;; This table consists of pairs of primes less than 2^31 and the random
;; primitive roots modulo them larger than 10^9. We randomly choose a pair and
;; adopt the prime as modulus and the primitive root as base.
(declaim ((simple-array (unsigned-byte 31) (100)) *moduli-table* *base-table*))
(defparameter *moduli-table*
  (make-array 100 :element-type '(unsigned-byte 31)
                  :initial-contents '(2147483647 2147483629 2147483587 2147483579 2147483563 2147483549 2147483543
 2147483497 2147483489 2147483477 2147483423 2147483399 2147483353 2147483323
 2147483269 2147483249 2147483237 2147483179 2147483171 2147483137 2147483123
 2147483077 2147483069 2147483059 2147483053 2147483033 2147483029 2147482951
 2147482949 2147482943 2147482937 2147482921 2147482877 2147482873 2147482867
 2147482859 2147482819 2147482817 2147482811 2147482801 2147482763 2147482739
 2147482697 2147482693 2147482681 2147482663 2147482661 2147482621 2147482591
 2147482583 2147482577 2147482507 2147482501 2147482481 2147482417 2147482409
 2147482367 2147482361 2147482349 2147482343 2147482327 2147482291 2147482273
 2147482237 2147482231 2147482223 2147482121 2147482093 2147482091 2147482081
 2147482063 2147482021 2147481997 2147481967 2147481949 2147481937 2147481907
 2147481901 2147481899 2147481893 2147481883 2147481863 2147481827 2147481811
 2147481797 2147481793 2147481673 2147481629 2147481571 2147481563 2147481529
 2147481509 2147481499 2147481491 2147481487 2147481373 2147481367 2147481359
 2147481353 2147481337)))
(defparameter *base-table*
  (make-array 100 :element-type '(unsigned-byte 31)
                  :initial-contents '(1059428526 2090066834 1772913519 1695158082 1516083910 1622025757 1248368302
 1894391153 2094976878 1193495823 1783230399 1520742486 1748395380 1703688443
 2138630366 1942049269 2066548889 1890950855 1480056952 1792721876 1092797280
 1204851872 1035383130 1002272185 1319736653 1980774767 1748793187 1866963602
 1200445534 1732959733 1214706585 1957228822 1479411729 1323155655 1052714514
 1989821027 1163834549 1095622874 2087901566 1670886084 1191975321 2091468260
 1429690292 1116037844 1420457779 1937649612 1552519679 1328604092 2090326292
 1397132095 1316705322 1664351025 1391513321 1851038917 1556301575 1928956735
 1764506480 1449537491 2119470570 1793768237 1831208371 1723755364 1643456516
 1993819805 1419297891 1755252963 1775153034 1388979165 2144586633 1501222238
 1872274033 1143076711 1229125474 1483974015 1997206147 1593231852 1632083893
 1601537043 2012194627 1299923971 1566635240 1814404069 1619988648 2072686565
 2014361572 1213868607 1166967329 1009325840 1306167671 1915239658 1223190075
 1821151471 2037700892 1646950698 1517859810 1099233635 1004913731 1653443892
 1782112665 1018916580)))

(defun %choose-moduli (mod1 mod2 base1 base2 rhash)
  "Chooses two appropriate pairs of moduli and bases."
  (declare ((or null (unsigned-byte 31)) mod1 mod2 base1 base2))
  (when rhash
    (return-from %choose-moduli
      (values (rhash-mod1 rhash)
              (rhash-mod2 rhash)
              (rhash-base1 rhash)
              (rhash-base2 rhash))))
  (let* ((rand1 (random (length *moduli-table*)))
         (rand2 (loop (let ((tmp (random (length *moduli-table*))))
                        (unless (= tmp rand1)
                          (return tmp))))))
    (if mod1
        (progn
          (assert (sb-int:positive-primep mod1))
          (setq base1 (or base1 (+ 1 (random (- mod1 1))))))
        (progn
          (setq mod1 (or mod1 (aref *moduli-table* rand1)))
          (if base1
              (assert (<= 1 base1 (- mod1 1)))
              (setq base1 (aref *base-table* rand1)))))
    (if mod2
        (progn
          (assert (sb-int:positive-primep mod2))
          (setq base2 (or base2 (+ 1 (random (- mod2 1))))))
        (progn
          (setq mod2 (or mod2 (aref *moduli-table* rand2)))
          (if base2
              (assert (<= 1 base2 (- mod2 1)))
              (setq base2 (aref *base-table* rand2))))))
  (values mod1 mod2 base1 base2))

(defun make-rhash (vector &key (key #'char-code) mod1 mod2 base1 base2 rhash)
  "Returns the table of rolling-hash of VECTOR modulo MOD1 and MOD2. KEY is
applied to each element of VECTOR prior to computing the hash value. If moduli
and bases are NIL, this function randomly chooses them. If RHASH is specified,
the same moduli and bases as RHASH is adopted.

MOD[1|2] := NIL | unsigned 31-bit prime number
BASE1 := NIL | 1 | 2 | ... | MOD1 - 1
BASE2 := NIL | 1 | 2 | ... | MOD2 - 1
KEY := FUNCTION returning FIXNUM
RHASH := NIL | RHASH"
  (declare (optimize (speed 3))
           (vector vector)
           ((or null (unsigned-byte 31)) mod1 mod2 base1 base2)
           (function key))
  (multiple-value-bind (mod1 mod2 base1 base2) (%choose-moduli mod1 mod2 base1 base2 rhash)
    (declare ((unsigned-byte 31) mod1 mod2 base1 base2))
    (let* ((size (length vector))
           (cumul1 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (powers1 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (cumul2 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (powers2 (make-array (+ 1 size) :element-type '(unsigned-byte 31))))
      (setf (aref powers1 0) 1
            (aref powers2 0) 1)
      (dotimes (i size)
        (setf (aref powers1 (+ i 1))
              (mod (* (aref powers1 i) base1) mod1)
              (aref powers2 (+ i 1))
              (mod (* (aref powers2 i) base2) mod2))
        (let ((sum1 (+ (mod (* base1 (aref cumul1 i)) mod1)
                       (mod (the fixnum (funcall key (aref vector i))) mod1)))
              (sum2 (+ (mod (* base2 (aref cumul2 i)) mod2)
                       (mod (the fixnum (funcall key (aref vector i))) mod2))))
          (setf (aref cumul1 (+ i 1)) (if (> sum1 mod1)
                                          (- sum1 mod1)
                                          sum1)
                (aref cumul2 (+ i 1)) (if (> sum2 mod2)
                                          (- sum2 mod2)
                                          sum2))))
      (%make-rhash mod1 base1 cumul1 powers1 mod2 base2 cumul2 powers2))))

(declaim (ftype (function * (values (unsigned-byte 62) &optional)) rhash-vector-hash)
         (inline rhash-vector-hash))
(defun rhash-vector-hash (rhash vector &key (key #'char-code))
  "Returns the hash code of VECTOR w.r.t. the moduli and bases of RHASH."
  (declare (optimize (speed 3))
           (vector vector)
           (function key))
  (let* ((mod1 (rhash-mod1 rhash))
         (mod2 (rhash-mod2 rhash))
         (base1 (rhash-base1 rhash))
         (base2 (rhash-base2 rhash))
         (size (length vector))
         (lower 0)
         (upper 0))
    (declare ((unsigned-byte 31) lower upper))
    (dotimes (i size)
      (setf lower (+ (mod (* base1 lower) mod1)
                     (mod (the fixnum (funcall key (aref vector i))) mod1)))
      (setf upper (+ (mod (* base2 upper) mod2)
                     (mod (the fixnum (funcall key (aref vector i))) mod2))))
    (dpb upper (byte 31 31) lower)))

(declaim (inline rhash-query)
         (ftype (function * (values (unsigned-byte 62) &optional)) rhash-query))
(defun rhash-query (rhash l r)
  "Returns the hash value of the interval [L, R)."
  (declare ((integer 0 #.most-positive-fixnum) l r))
  (assert (<= l r))
  (let ((cumul1 (rhash-cumul1 rhash))
        (powers1 (rhash-powers1 rhash))
        (mod1 (rhash-mod1 rhash))
        (cumul2 (rhash-cumul2 rhash))
        (powers2 (rhash-powers2 rhash))
        (mod2 (rhash-mod2 rhash)))
    (let ((lower (+ (aref cumul1 r)
                    (- mod1 (mod (* (aref cumul1 l) (aref powers1 (- r l))) mod1))))
          (upper (+ (aref cumul2 r)
                    (- mod2 (mod (* (aref cumul2 l) (aref powers2 (- r l))) mod2)))))
      (let ((lower (if (> lower mod1) (- lower mod1) lower))
            (upper (if (> upper mod2) (- upper mod2) upper)))
        (declare ((unsigned-byte 31) lower upper))
        (dpb upper (byte 31 31) lower)))))

(declaim (inline rhash-concat))
(defun rhash-concat (rhash hash1 hash2 length2)
  "Returns the hash value of the concatenated sequence.

HASH1 := hash value of the first sequence
HASH2 := hash value of the second sequence
LENGTH2 := length of the second sequence."
  (declare ((unsigned-byte 62) hash1 hash2)
           ((integer 0 #.most-positive-fixnum) length2))
  (let* ((hash1-lower (ldb (byte 31 0) hash1))
         (hash1-upper (ldb (byte 31 31) hash1))
         (hash2-lower (ldb (byte 31 0) hash2))
         (hash2-upper (ldb (byte 31 31) hash2))
         (mod1 (rhash-mod1 rhash))
         (mod2 (rhash-mod2 rhash))
         (res-lower (mod (+ hash2-lower
                            (mod (* hash1-lower
                                    (aref (rhash-powers1 rhash) length2))
                                 mod1))
                         mod1))
         (res-upper (mod (+ hash2-upper
                            (mod (* hash1-upper
                                    (aref (rhash-powers2 rhash) length2))
                                 mod2))
                         mod2)))
    (declare ((unsigned-byte 31) res-lower res-upper))
    (dpb res-upper (byte 31 31) res-lower)))

(declaim (ftype (function * (values (integer 0 #.most-positive-fixnum) &optional)) rhash-get-lcp))
(defun rhash-get-lcp (rhash1 start1 rhash2 start2)
  "Returns the length of the longest common prefix of two suffixes which begin
at START1 and START2."
  (declare (optimize (speed 3))
           ((integer 0 #.most-positive-fixnum) start1 start2))
  (assert (and (= (rhash-mod1 rhash1) (rhash-mod1 rhash2))
               (= (rhash-mod2 rhash1) (rhash-mod2 rhash2))))
  (assert (and (< start1 (length (rhash-cumul1 rhash1)))
               (< start2 (length (rhash-cumul1 rhash2)))))
  (let ((max-length (min (- (length (rhash-cumul1 rhash1)) start1 1)
                         (- (length (rhash-cumul1 rhash2)) start2 1))))
    (declare (optimize (safety 0)))
    (if (= (rhash-query rhash1 start1 (+ start1 max-length))
           (rhash-query rhash2 start2 (+ start2 max-length)))
        max-length
        (labels ((bisect (ok ng)
                   (declare ((integer 0 #.most-positive-fixnum) ok ng))
                   (if (<= (- ng ok) 1)
                       ok
                       (let ((mid (ash (+ ng ok) -1)))
                         (if (= (rhash-query rhash1 start1 (+ start1 mid))
                                (rhash-query rhash2 start2 (+ start2 mid)))
                             (bisect mid ng)
                             (bisect ok mid))))))
          (bisect 0 max-length)))))

(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 main ()
  (declare #.OPT)
  (let* ((ss (coerce (read-line) 'simple-base-string))
         (ts (coerce (read-line) 'simple-base-string))
         (slen (length ss))
         (tlen (length ts))
         (snum (max 3 (ceiling (* tlen 4) slen)))
         (ex-ss (make-string (* snum slen) :element-type 'base-char))
         (tnum (floor (* snum slen) tlen))
         (ex-ts (make-string (* tnum tlen) :element-type 'base-char)))
    (declare (uint31 slen tlen snum tnum))
    (dotimes (i slen)
      (dotimes (fac snum)
        (setf (aref ex-ss (+ i (* fac slen)))
              (aref ss i))))
    (dotimes (i tlen)
      (dotimes (fac tnum)
        (setf (aref ex-ts (+ i (* fac tlen)))
              (aref ts i))))
    (let* ((rhash (make-rhash ex-ss))
           (total-len (* snum slen))
           (thash (rhash-vector-hash rhash ts))
           (base 0))
      (block maybe-infinite
        (loop (unless (<= (+ base tlen) total-len)
                (return))
              (when (= thash (rhash-query rhash base (+ base tlen)))
                ;; possibly infinite
                (loop for i from base by tlen
                      while (<= (+ i tlen) total-len)
                      do (unless (= thash (rhash-query rhash i (+ i tlen)))
                           (return-from maybe-infinite))
                      finally (let* ((l base)
                                     (r i)
                                     (dif (+ l (- total-len r))))
                                (if (<= dif (* 2 (- tlen 1)))
                                    (progn (println -1)
                                           (return-from main))
                                    (return-from maybe-infinite)))))
              (incf base)))
      #>total-len
      ;; not infinite
      (let ((base 0)
            (res 0)
            (rhash2 (make-rhash ex-ts :rhash rhash)))
        (declare (uint31 res))
        (dotimes (pos total-len)
          (let ((length (rhash-get-lcp rhash pos rhash2 0)))
            (setf res (max res (floor length tlen)))))
        (println res)))))

#-swank (main)

Score : 600 points

Problem Statement

Given are two strings s and t consisting of lowercase English letters. Determine if the number of non-negative integers i satisfying the following condition is finite, and find the maximum value of such i if the number is finite.

There exists a non-negative integer j such that the concatenation of i copies of t is a substring of the concatenation of j copies of s.

Notes

A string a is a substring of another string b if and only if there exists an integer x (0 \leq x \leq |b| - |a|) such that, for any y (1 \leq y \leq |a|), a_y = b_{x+y} holds.

We assume that the concatenation of zero copies of any string is the empty string. From the definition above, the empty string is a substring of any string. Thus, for any two strings s and t, i = 0 satisfies the condition in the problem statement.

Constraints

1 \leq |s| \leq 5 \times 10^5

1 \leq |t| \leq 5 \times 10^5

s and t consist of lowercase English letters.

Input

Input is given from Standard Input in the following format:

s
t

Output

If the number of non-negative integers i satisfying the following condition is finite, print the maximum value of such i; if the number is infinite, print -1.

Sample Input 1

abcabab
ab

Sample Output 1

3

The concatenation of three copies of t, ababab, is a substring of the concatenation of two copies of s, abcabababcabab, so i = 3 satisfies the condition.

On the other hand, the concatenation of four copies of t, abababab, is not a substring of the concatenation of any number of copies of s, so i = 4 does not satisfy the condition.

Similarly, any integer greater than 4 does not satisfy the condition, either. Thus, the number of non-negative integers i satisfying the condition is finite, and the maximum value of such i is 3.

Sample Input 2

aa
aaaaaaa

Sample Output 2

-1

For any non-negative integer i, the concatenation of i copies of t is a substring of the concatenation of 4i copies of s. Thus, there are infinitely many non-negative integers i that satisfy the condition.

Sample Input 3

aba
baaab

Sample Output 3

0

As stated in Notes, i = 0 always satisfies the condition.


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Rolling hash (62-bit)
;;;

;; Reference:
;; https://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
;; https://ei1333.github.io/luzhiled/snippets/string/rolling-hash.html

(defstruct (rhash (:constructor %make-rhash (mod1 base1 cumul1 powers1 mod2 base2 cumul2 powers2)))
  ;; lower 31-bit value
  (mod1 2147483647 :type (unsigned-byte 31))
  (base1 1059428526 :type (unsigned-byte 31))
  (cumul1 nil :type (simple-array (unsigned-byte 31) (*)))
  (powers1 nil :type (simple-array (unsigned-byte 31) (*)))
  ;; upper 31-bit value
  (mod2 2147483629 :type (unsigned-byte 31))
  (base2 2090066834 :type (unsigned-byte 31))
  (cumul2 nil :type (simple-array (unsigned-byte 31) (*)))
  (powers2 nil :type (simple-array (unsigned-byte 31) (*))))

;; This table consists of pairs of primes less than 2^31 and the random
;; primitive roots modulo them larger than 10^9. We randomly choose a pair and
;; adopt the prime as modulus and the primitive root as base.
(declaim ((simple-array (unsigned-byte 31) (100)) *moduli-table* *base-table*))
(defparameter *moduli-table*
  (make-array 100 :element-type '(unsigned-byte 31)
                  :initial-contents '(2147483647 2147483629 2147483587 2147483579 2147483563 2147483549 2147483543
 2147483497 2147483489 2147483477 2147483423 2147483399 2147483353 2147483323
 2147483269 2147483249 2147483237 2147483179 2147483171 2147483137 2147483123
 2147483077 2147483069 2147483059 2147483053 2147483033 2147483029 2147482951
 2147482949 2147482943 2147482937 2147482921 2147482877 2147482873 2147482867
 2147482859 2147482819 2147482817 2147482811 2147482801 2147482763 2147482739
 2147482697 2147482693 2147482681 2147482663 2147482661 2147482621 2147482591
 2147482583 2147482577 2147482507 2147482501 2147482481 2147482417 2147482409
 2147482367 2147482361 2147482349 2147482343 2147482327 2147482291 2147482273
 2147482237 2147482231 2147482223 2147482121 2147482093 2147482091 2147482081
 2147482063 2147482021 2147481997 2147481967 2147481949 2147481937 2147481907
 2147481901 2147481899 2147481893 2147481883 2147481863 2147481827 2147481811
 2147481797 2147481793 2147481673 2147481629 2147481571 2147481563 2147481529
 2147481509 2147481499 2147481491 2147481487 2147481373 2147481367 2147481359
 2147481353 2147481337)))
(defparameter *base-table*
  (make-array 100 :element-type '(unsigned-byte 31)
                  :initial-contents '(1059428526 2090066834 1772913519 1695158082 1516083910 1622025757 1248368302
 1894391153 2094976878 1193495823 1783230399 1520742486 1748395380 1703688443
 2138630366 1942049269 2066548889 1890950855 1480056952 1792721876 1092797280
 1204851872 1035383130 1002272185 1319736653 1980774767 1748793187 1866963602
 1200445534 1732959733 1214706585 1957228822 1479411729 1323155655 1052714514
 1989821027 1163834549 1095622874 2087901566 1670886084 1191975321 2091468260
 1429690292 1116037844 1420457779 1937649612 1552519679 1328604092 2090326292
 1397132095 1316705322 1664351025 1391513321 1851038917 1556301575 1928956735
 1764506480 1449537491 2119470570 1793768237 1831208371 1723755364 1643456516
 1993819805 1419297891 1755252963 1775153034 1388979165 2144586633 1501222238
 1872274033 1143076711 1229125474 1483974015 1997206147 1593231852 1632083893
 1601537043 2012194627 1299923971 1566635240 1814404069 1619988648 2072686565
 2014361572 1213868607 1166967329 1009325840 1306167671 1915239658 1223190075
 1821151471 2037700892 1646950698 1517859810 1099233635 1004913731 1653443892
 1782112665 1018916580)))

(defun %choose-moduli (mod1 mod2 base1 base2 rhash)
  "Chooses two appropriate pairs of moduli and bases."
  (declare ((or null (unsigned-byte 31)) mod1 mod2 base1 base2))
  (when rhash
    (return-from %choose-moduli
      (values (rhash-mod1 rhash)
              (rhash-mod2 rhash)
              (rhash-base1 rhash)
              (rhash-base2 rhash))))
  (let* ((rand1 (random (length *moduli-table*)))
         (rand2 (loop (let ((tmp (random (length *moduli-table*))))
                        (unless (= tmp rand1)
                          (return tmp))))))
    (if mod1
        (progn
          (assert (sb-int:positive-primep mod1))
          (setq base1 (or base1 (+ 1 (random (- mod1 1))))))
        (progn
          (setq mod1 (or mod1 (aref *moduli-table* rand1)))
          (if base1
              (assert (<= 1 base1 (- mod1 1)))
              (setq base1 (aref *base-table* rand1)))))
    (if mod2
        (progn
          (assert (sb-int:positive-primep mod2))
          (setq base2 (or base2 (+ 1 (random (- mod2 1))))))
        (progn
          (setq mod2 (or mod2 (aref *moduli-table* rand2)))
          (if base2
              (assert (<= 1 base2 (- mod2 1)))
              (setq base2 (aref *base-table* rand2))))))
  (values mod1 mod2 base1 base2))

(defun make-rhash (vector &key (key #'char-code) mod1 mod2 base1 base2 rhash)
  "Returns the table of rolling-hash of VECTOR modulo MOD1 and MOD2. KEY is
applied to each element of VECTOR prior to computing the hash value. If moduli
and bases are NIL, this function randomly chooses them. If RHASH is specified,
the same moduli and bases as RHASH is adopted.

MOD[1|2] := NIL | unsigned 31-bit prime number
BASE1 := NIL | 1 | 2 | ... | MOD1 - 1
BASE2 := NIL | 1 | 2 | ... | MOD2 - 1
KEY := FUNCTION returning FIXNUM
RHASH := NIL | RHASH"
  (declare (optimize (speed 3))
           (vector vector)
           ((or null (unsigned-byte 31)) mod1 mod2 base1 base2)
           (function key))
  (multiple-value-bind (mod1 mod2 base1 base2) (%choose-moduli mod1 mod2 base1 base2 rhash)
    (declare ((unsigned-byte 31) mod1 mod2 base1 base2))
    (let* ((size (length vector))
           (cumul1 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (powers1 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (cumul2 (make-array (+ 1 size) :element-type '(unsigned-byte 31)))
           (powers2 (make-array (+ 1 size) :element-type '(unsigned-byte 31))))
      (setf (aref powers1 0) 1
            (aref powers2 0) 1)
      (dotimes (i size)
        (setf (aref powers1 (+ i 1))
              (mod (* (aref powers1 i) base1) mod1)
              (aref powers2 (+ i 1))
              (mod (* (aref powers2 i) base2) mod2))
        (let ((sum1 (+ (mod (* base1 (aref cumul1 i)) mod1)
                       (mod (the fixnum (funcall key (aref vector i))) mod1)))
              (sum2 (+ (mod (* base2 (aref cumul2 i)) mod2)
                       (mod (the fixnum (funcall key (aref vector i))) mod2))))
          (setf (aref cumul1 (+ i 1)) (if (> sum1 mod1)
                                          (- sum1 mod1)
                                          sum1)
                (aref cumul2 (+ i 1)) (if (> sum2 mod2)
                                          (- sum2 mod2)
                                          sum2))))
      (%make-rhash mod1 base1 cumul1 powers1 mod2 base2 cumul2 powers2))))

(declaim (ftype (function * (values (unsigned-byte 62) &optional)) rhash-vector-hash)
         (inline rhash-vector-hash))
(defun rhash-vector-hash (rhash vector &key (key #'char-code))
  "Returns the hash code of VECTOR w.r.t. the moduli and bases of RHASH."
  (declare (optimize (speed 3))
           (vector vector)
           (function key))
  (let* ((mod1 (rhash-mod1 rhash))
         (mod2 (rhash-mod2 rhash))
         (base1 (rhash-base1 rhash))
         (base2 (rhash-base2 rhash))
         (size (length vector))
         (lower 0)
         (upper 0))
    (declare ((unsigned-byte 31) lower upper))
    (dotimes (i size)
      (setf lower (+ (mod (* base1 lower) mod1)
                     (mod (the fixnum (funcall key (aref vector i))) mod1)))
      (setf upper (+ (mod (* base2 upper) mod2)
                     (mod (the fixnum (funcall key (aref vector i))) mod2))))
    (dpb upper (byte 31 31) lower)))

(declaim (inline rhash-query)
         (ftype (function * (values (unsigned-byte 62) &optional)) rhash-query))
(defun rhash-query (rhash l r)
  "Returns the hash value of the interval [L, R)."
  (declare ((integer 0 #.most-positive-fixnum) l r))
  (assert (<= l r))
  (let ((cumul1 (rhash-cumul1 rhash))
        (powers1 (rhash-powers1 rhash))
        (mod1 (rhash-mod1 rhash))
        (cumul2 (rhash-cumul2 rhash))
        (powers2 (rhash-powers2 rhash))
        (mod2 (rhash-mod2 rhash)))
    (let ((lower (+ (aref cumul1 r)
                    (- mod1 (mod (* (aref cumul1 l) (aref powers1 (- r l))) mod1))))
          (upper (+ (aref cumul2 r)
                    (- mod2 (mod (* (aref cumul2 l) (aref powers2 (- r l))) mod2)))))
      (let ((lower (if (> lower mod1) (- lower mod1) lower))
            (upper (if (> upper mod2) (- upper mod2) upper)))
        (declare ((unsigned-byte 31) lower upper))
        (dpb upper (byte 31 31) lower)))))

(declaim (inline rhash-concat))
(defun rhash-concat (rhash hash1 hash2 length2)
  "Returns the hash value of the concatenated sequence.

HASH1 := hash value of the first sequence
HASH2 := hash value of the second sequence
LENGTH2 := length of the second sequence."
  (declare ((unsigned-byte 62) hash1 hash2)
           ((integer 0 #.most-positive-fixnum) length2))
  (let* ((hash1-lower (ldb (byte 31 0) hash1))
         (hash1-upper (ldb (byte 31 31) hash1))
         (hash2-lower (ldb (byte 31 0) hash2))
         (hash2-upper (ldb (byte 31 31) hash2))
         (mod1 (rhash-mod1 rhash))
         (mod2 (rhash-mod2 rhash))
         (res-lower (mod (+ hash2-lower
                            (mod (* hash1-lower
                                    (aref (rhash-powers1 rhash) length2))
                                 mod1))
                         mod1))
         (res-upper (mod (+ hash2-upper
                            (mod (* hash1-upper
                                    (aref (rhash-powers2 rhash) length2))
                                 mod2))
                         mod2)))
    (declare ((unsigned-byte 31) res-lower res-upper))
    (dpb res-upper (byte 31 31) res-lower)))

(declaim (ftype (function * (values (integer 0 #.most-positive-fixnum) &optional)) rhash-get-lcp))
(defun rhash-get-lcp (rhash1 start1 rhash2 start2)
  "Returns the length of the longest common prefix of two suffixes which begin
at START1 and START2."
  (declare (optimize (speed 3))
           ((integer 0 #.most-positive-fixnum) start1 start2))
  (assert (and (= (rhash-mod1 rhash1) (rhash-mod1 rhash2))
               (= (rhash-mod2 rhash1) (rhash-mod2 rhash2))))
  (assert (and (< start1 (length (rhash-cumul1 rhash1)))
               (< start2 (length (rhash-cumul1 rhash2)))))
  (let ((max-length (min (- (length (rhash-cumul1 rhash1)) start1 1)
                         (- (length (rhash-cumul1 rhash2)) start2 1))))
    (declare (optimize (safety 0)))
    (if (= (rhash-query rhash1 start1 (+ start1 max-length))
           (rhash-query rhash2 start2 (+ start2 max-length)))
        max-length
        (labels ((bisect (ok ng)
                   (declare ((integer 0 #.most-positive-fixnum) ok ng))
                   (if (<= (- ng ok) 1)
                       ok
                       (let ((mid (ash (+ ng ok) -1)))
                         (if (= (rhash-query rhash1 start1 (+ start1 mid))
                                (rhash-query rhash2 start2 (+ start2 mid)))
                             (bisect mid ng)
                             (bisect ok mid))))))
          (bisect 0 max-length)))))

(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 main ()
  (let* ((ss (coerce (read-line) 'simple-base-string))
         (ts (coerce (read-line) 'simple-base-string))
         (slen (length ss))
         (tlen (length ts))
         (snum (max 3 (ceiling (* tlen 4) slen)))
         (ex-ss (make-string (* snum slen) :element-type 'base-char)))
    (declare (uint31 slen tlen snum))
    (dotimes (i slen)
      (dotimes (fac snum)
        (setf (aref ex-ss (+ i (* fac slen)))
              (aref ss i))))
    (let* ((rhash (make-rhash ex-ss))
           (total-len (* snum slen))
           (thash (rhash-vector-hash rhash ts))
           (base 0))
      (block maybe-infinite
        (loop (unless (<= (+ base tlen) total-len)
                (return))
              (when (= thash (rhash-query rhash base (+ base tlen)))
                ;; possibly infinite
                (loop for i from base by tlen
                      while (<= (+ i tlen) total-len)
                      do (unless (= thash (rhash-query rhash i (+ i tlen)))
                           (return-from maybe-infinite))
                      finally (let* ((l base)
                                     (r i)
                                     (dif (+ l (- total-len r))))
                                (if (<= dif (* 2 (- tlen 1)))
                                    (progn (println -1)
                                           (return-from main))
                                    (return-from maybe-infinite)))))
              (incf base)))
      #>total-len
      ;; not infinite
      (let ((base 0)
            (res 0))
        (loop
          (unless (<= (+ base tlen) total-len)
            (return))
          (if (= thash (rhash-query rhash base (+ base tlen)))
              (loop for i from base by tlen
                    do (unless (<= (+ i tlen) total-len)
                         (setf base i)
                         (setf res (max res (floor (- i base) tlen)))
                         (loop-finish))
                       (unless (= thash (rhash-query rhash i (+ i tlen)))
                         (setf res (max res (floor (- i base) tlen)))
                         (setf base i)
                         (loop-finish)))
              (incf base)))
        (println res)))))

#-swank (main)

;; For Test
#+swank
(defun io-equal (in-string out-string &optional (func #'main))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNC, and returns true if the
string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (and (> (length s) 0)
                      (eql (char s (- (length s) 1)) #\Linefeed))
                 s
                 (uiop:strcat s uiop:+lf+))))
    (equal (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 func)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "C:/msys64/usr/bin/cat.exe" '("/dev/clipboard") :output out)))

#+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*))
  (let ((*standard-output* out))
    (etypecase thing
      (null ; Runs #'MAIN with the string on clipboard
       (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 ; Runs #'MAIN with the string in a text file
       (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 "abcabab
ab
"
    "3
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "aa
aaaaaaa
"
    "-1
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "aba
baaab
"
    "0
")))


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(defmacro with-buffered-stdout (&body body)
  "Buffers all outputs to *STANDARD-OUTPUT* in BODY and flushes them to
*STANDARD-OUTPUT* after BODY has been done (without error). Note that only
BASE-CHAR is allowed."
  (let ((out (gensym)))
    `(let ((,out (make-string-output-stream :element-type 'base-char)))
       (let ((*standard-output* ,out))
         ,@body)
       (write-string (get-output-stream-string ,out)))))

(declaim (inline println-sequence))
(defun println-sequence (sequence &key (out *standard-output*) (key #'identity))
  (let ((init t))
    (sequence:dosequence (x sequence)
      (if init
          (setq init nil)
          (write-char #\  out))
      (princ (funcall key x) out))
    (terpri out)))

;;;
;;; Calculate a^n on any monoids in O(log(n)) time
;;;

(declaim (inline power))
(defun power (base exponent op identity)
  "OP := binary operation (on a monoid)
IDENTITY := identity element w.r.t. OP"
  (declare ((integer 0) exponent)
           (function op))
  (labels ((recur (x p)
             (declare ((integer 0 #.most-positive-fixnum) p))
             (cond ((= 0 p) identity)
                   ((evenp p) (recur (funcall op x x) (ash p -1)))
                   (t (nth-value 0 (funcall op x (recur x (- p 1)))))))
           (recur-big (x p)
             (declare ((integer 0) p))
             (cond ((zerop p) identity)
                   ((evenp p) (recur-big (funcall op x x) (ash p -1)))
                   (t (nth-value 0 (funcall op x (recur-big x (- p 1))))))))
    (typecase exponent
      (fixnum (recur base exponent))
      (otherwise (recur-big base exponent)))))

(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 main ()
  (declare #.OPT)
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'uint32 :initial-element 0))
         (last-table (make-array 200001 :element-type 'uint32 :initial-element 0))
         (nexts (make-array (+ n 1) :element-type 'uint32 :initial-element 0))
         (iden (make-array (+ n 1) :element-type 'uint32 :initial-element 0)))
    (declare (uint62 n k))
    (dotimes (i n) (setf (aref as i) (read-fixnum)))
    (dotimes (i (+ n 1)) (setf (aref iden i) i))
    (loop for i from (- (* 2 n) 1) downto 0
          do (when (< i n)
               (setf (aref nexts i) (+ 1 (aref last-table (aref as i)))))
             (setf (aref last-table (aref as (mod i n))) i))
    (let ((perm (copy-seq nexts)))
      (loop for i from (- n 1) downto 0
            do (when (< (aref perm i) n)
                 (setf (aref perm i) (aref perm (aref perm i)))))
      (dotimes (i n)
        (setf (aref perm i) (- (aref perm i) n)))
      (let* ((final-perm
               (power perm (- k 1)
                      (lambda (x y)
                        (declare ((simple-array uint32 (*)) x y))
                        (let ((new (make-array (+ n 1) :element-type 'uint32)))
                          (dotimes (i (+ n 1))
                            (setf (aref new i) (aref y (aref x i))))
                          new))
                      iden))
             (pos (aref final-perm 0))
             (res (make-array 0 :fill-pointer 0)))
        ;; #>final-perm
        (loop (when (= pos n)
                (with-buffered-stdout
                  (println-sequence res))
                (return-from main))
              (if (<= (aref nexts pos) n)
                  (setq pos (aref nexts pos))
                  (progn
                    (vector-push-extend (aref as pos) res)
                    (incf pos))))))))

#-swank (main)

Score : 700 points

Problem Statement

We have a sequence of N \times K integers: X=(X_0,X_1,\cdots,X_{N \times K-1}).
Its elements are represented by another sequence of N integers: A=(A_0,A_1,\cdots,A_{N-1}). For each pair i, j (0 \leq i \leq K-1,\ 0 \leq j \leq N-1), X_{i \times N + j}=A_j holds.

Snuke has an integer sequence s, which is initially empty.
For each i=0,1,2,\cdots,N \times K-1, in this order, he will perform the following operation:

If s does not contain X_i: add X_i to the end of s.

If s does contain X_i: repeatedly delete the element at the end of s until s no longer contains X_i. Note that, in this case, we do not add X_i to the end of s.

Find the elements of s after Snuke finished the operations.

Constraints

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

1 \leq K \leq 10^{12}

1 \leq A_i \leq 2 \times 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_0 A_1 \cdots A_{N-1}

Output

Print the elements of s after Snuke finished the operations, in order from beginning to end, with spaces in between.

Sample Input 1

3 2
1 2 3

Sample Output 1

2 3

In this case, X=(1,2,3,1,2,3).
We will perform the operations as follows:

i=0: s does not contain 1, so we add 1 to the end of s, resulting in s=(1).

i=1: s does not contain 2, so we add 2 to the end of s, resulting in s=(1,2).

i=2: s does not contain 3, so we add 3 to the end of s, resulting in s=(1,2,3).

i=3: s does contain 1, so we repeatedly delete the element at the end of s as long as s contains 1, which causes the following changes to s: (1,2,3)→(1,2)→(1)→().

i=4: s does not contain 2, so we add 2 to the end of s, resulting in s=(2).

i=5: s does not contain 3, so we add 3 to the end of s, resulting in s=(2,3).

Sample Input 2

5 10
1 2 3 2 3

Sample Output 2

3

Sample Input 3

6 1000000000000
1 1 2 2 3 3

Sample Output 3

s may be empty in the end.

Sample Input 4

11 97
3 1 4 1 5 9 2 6 5 3 5

Sample Output 4

9 2 6


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Calculate a^n on any monoids in O(log(n)) time
;;;

(declaim (inline power))
(defun power (base exponent op identity)
  "OP := binary operation (on a monoid)
IDENTITY := identity element w.r.t. OP"
  (declare ((integer 0) exponent)
           (function op))
  (labels ((recur (x p)
             (declare ((integer 0 #.most-positive-fixnum) p))
             (cond ((zerop p) identity)
                   ((evenp p) (recur (funcall op x x) (ash p -1)))
                   (t (nth-value 0 (funcall op x (recur x (- p 1)))))))
           (recur-big (x p)
             (declare ((integer 0) p))
             (cond ((zerop p) identity)
                   ((evenp p) (recur-big (funcall op x x) (ash p -1)))
                   (t (nth-value 0 (funcall op x (recur-big x (- p 1))))))))
    (typecase exponent
      (fixnum (recur base exponent))
      (otherwise (recur-big base exponent)))))

(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
                                  (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 main ()
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'uint32 :initial-element 0))
         (last-table (make-array 200001 :element-type 'uint32 :initial-element 0))
         (nexts (make-array n :element-type 'uint32 :initial-element 0))
         (iden (make-array n :element-type 'uint32 :initial-element 0)))
    (dotimes (i n)
      (setf (aref as i) (read-fixnum))
      (setf (aref iden i) i))
    (loop for i from (- (* 2 n) 1) downto 0
          do (when (< i n)
               (setf (aref nexts i) (+ 1 (aref last-table (aref as i)))))
             (setf (aref last-table (aref as (mod i n))) i))
    #>nexts
    (let ((perm (copy-seq nexts)))
      (loop for i from (- n 1) downto 0
            do (when (< (aref perm i) n)
                 (setf (aref perm i) (aref perm (aref perm i)))))
      (dotimes (i n)
        (setf (aref perm i) (mod (aref perm i) n)))
      (let* ((final-perm (power perm (- k 1)
                                (lambda (x y)
                                  (let ((new (make-array n :element-type 'uint32 :initial-element 0)))
                                    (dotimes (i n)
                                      (setf (aref new i) (aref y (aref x i))))
                                    new))
                                iden))
             (pos (aref final-perm 0))
             (res (make-array 0 :fill-pointer 0)))
        (loop (when (= pos n)
                (loop with init = t
                      for i below (length res)
                      do (if init
                             (setq init nil)
                             (write-char #\ ))
                         (princ (aref res i)))
                (terpri)
                (return-from main))
              (if (<= (aref nexts pos) n)
                  (setq pos (aref nexts pos))
                  (progn
                    (vector-push-extend (aref as pos) res)
                    (incf pos))))))))

#-swank (main)


(print (ceiling (/ (read) (1+ (* 2 (read))))))

Score : 100 points

Problem Statement

It is known that the area of a regular dodecagon inscribed in a circle of radius a is 3a^2.

Given an integer r, find the area of a regular dodecagon inscribed in a circle of radius r.

Constraints

1 \leq r \leq 100

r is an integer.

Input

Input is given from Standard Input in the following format:

r

Output

Print an integer representing the area of the regular dodecagon.

Sample Input 1

4

Sample Output 1

48

The area of the regular dodecagon is 3 \times 4^2 = 48.

Sample Input 2

15

Sample Output 2

675

Sample Input 3

80

Sample Output 3

19200


(let ((value (read)))
     (format t "~a" (* 3 value value)))


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(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 main ()
  (let* ((r (read)))
    (println (* 3 r r))))

#-swank (main)

;; For Test
#+swank
(defun io-equal (in-string out-string &optional (func #'main))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNC, and returns true if the
string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (and (> (length s) 0)
                      (eql (char s (- (length s) 1)) #\Linefeed))
                 s
                 (uiop:strcat s uiop:+lf+))))
    (equal (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 func)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "C:/msys64/usr/bin/cat.exe" '("/dev/clipboard") :output out)))

#+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*))
  (let ((*standard-output* out))
    (etypecase thing
      (null ; Runs #'MAIN with the string on clipboard
       (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 ; Runs #'MAIN with the string in a text file
       (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)))


(let ((n (read))
      (d (read)))
  (format t "~A~%" (ceiling (/ n (+ d d 1)))))

Score : 200 points

Problem Statement

There are N apple trees in a row. People say that one of them will bear golden apples.

We want to deploy some number of inspectors so that each of these trees will be inspected.

Each inspector will be deployed under one of the trees. For convenience, we will assign numbers from 1 through N to the trees. An inspector deployed under the i-th tree (1 \leq i \leq N) will inspect the trees with numbers between i-D and i+D (inclusive).

Find the minimum number of inspectors that we need to deploy to achieve the objective.

Constraints

All values in input are integers.

1 \leq N \leq 20

1 \leq D \leq 20

Input

Input is given from Standard Input in the following format:

N D

Output

Print the minimum number of inspectors that we need to deploy to achieve the objective.

Sample Input 1

6 2

Sample Output 1

2

We can achieve the objective by, for example, placing an inspector under Tree 3 and Tree 4.

Sample Input 2

14 3

Sample Output 2

2

Sample Input 3

20 4

Sample Output 3

3


;; -*- 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 (ql:quickload '(:cl-debug-print :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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Disjoint sparse table
;;;

;;; Reference:
;;; https://discuss.codechef.com/questions/117696/tutorial-disjoint-sparse-table
;;; http://noshi91.hatenablog.com/entry/2018/05/08/183946 (Japanese)
;;; http://drken1215.hatenablog.com/entry/2018/09/08/162600 (Japanese)

(declaim (inline make-disjoint-sparse-table))
(defun make-disjoint-sparse-table (vector binop)
  (let* ((n (length vector))
         (height (integer-length (- n 1)))
         (table (make-array (list height n) :element-type '(unsigned-byte 32))))
    (dotimes (j n)
      (setf (aref table 0 j) (aref vector j)))
    (do ((i 1 (+ i 1)))
        ((>= i height))
      (let* ((width/2 (ash i 1))
             (width (* width/2 2)))
        (do ((j 0 (+ j width)))
            ((>= j n))
          (let ((mid (min (+ j width/2) n)))
            ;; fill the first half
            (setf (aref table i (- mid 1))
                  (aref vector (- mid 1)))
            (do ((k (- mid 2) (- k 1)))
                ((< k j))
              (setf (aref table i k)
                    (funcall binop (aref vector k) (aref table i (+ k 1)))))
            (when (>= mid n)
              (return))
            ;; fill the second half
            (setf (aref table i mid)
                  (aref vector mid))
            (let ((end (min n (+ mid width/2))))
              (do ((k (+ mid 1) (+ k 1)))
                  ((>= k end))
                (setf (aref table i k)
                      (funcall binop (aref table i (- k 1)) (aref vector k)))))))))
    table))

(declaim (inline dst-query))
(defun dst-query (table binop left right)
  "Queries the interval [LEFT, RIGHT). Note that a null interval [x, x) is not
allowed as disjoint sparse table deals with a semigroup."
  (declare ((integer 0 #.most-positive-fixnum) left right)
           ((simple-array (unsigned-byte 32) (* *)) table))
  (assert (< left right))
  (setq right (- right 1)) ;; change to closed interval
  (if (= left right)
      (aref table 0 left)
      (let ((h (- (integer-length (logxor left right)) 1)))
        (funcall binop
                 (aref table h left)
                 (aref table h right)))))

(defmacro with-buffered-stdout (&body body)
  "Buffers all outputs to *STANDARD-OUTPUT* in BODY and flushes them to
*STANDARD-OUTPUT* after BODY has been done (without error). Note that only
BASE-CHAR is allowed."
  (let ((out (gensym)))
    `(let ((,out (make-string-output-stream :element-type 'base-char)))
       (let ((*standard-output* ,out))
         ,@body)
       (write-string (get-output-stream-string ,out)))))

(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 main ()
  (let* ((n (read))
         (as (make-array n :element-type 'uint32)))
    (dotimes (i n)
      (setf (aref as i) (read-fixnum)))
    (let ((table (make-disjoint-sparse-table as #'max)))
      (with-buffered-stdout
        (dotimes (i n)
          (println (max (if (zerop i)
                            0
                            (dst-query table #'max 0 i))
                        (if (= (+ i 1) n)
                            0
                            (dst-query table #'max (+ i 1) n)))))))))

#-swank (main)

Score : 300 points

Problem Statement

You are given a sequence of length N: A_1, A_2, ..., A_N.
For each integer i between 1 and N (inclusive), answer the following question:

Find the maximum value among the N-1 elements other than A_i in the sequence.

Constraints

2 \leq N \leq 200000

1 \leq A_i \leq 200000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1
:
A_N

Output

Print N lines. The i-th line (1 \leq i \leq N) should contain the maximum value among the N-1 elements other than A_i in the sequence.

Sample Input 1

3
1
4
3

Sample Output 1

4
3
4

The maximum value among the two elements other than A_1, that is, A_2 = 4 and A_3 = 3, is 4.

The maximum value among the two elements other than A_2, that is, A_1 = 1 and A_3 = 3, is 3.

The maximum value among the two elements other than A_3, that is, A_1 = 1 and A_2 = 4, is 4.

Sample Input 2

2
5
5

Sample Output 2

5
5


(defun c ()
  (let* ((n (read))
         (m (list (list -1 -1))))
    (dotimes (i n)
      (let ((x (read)))
        (cond ((> x (caar m))
               (push (list x i) m))
              ((> x (caadr m))
               (setf (cdr m) (list (list x i)))))))
    (dotimes (i n)
      (format t "~a~%"
              (if (= i (cadar m))
                  (caadr m)
                  (caar m))))))
(c)


(let ((n (read))
      (a (make-array 1 :element-type 'integer
                       :initial-element 0
                       :adjustable t
                       :fill-pointer 1))
      (ans (make-array 0 :element-type 'integer
                         :adjustable t
                         :fill-pointer 0))
      (ans-c 0))
     (dotimes (i n)
       (vector-push-extend (read) a))
     (loop for i from n downto 1
           do (loop for j from 2 upto n
                    while (<= (* i j) n)
                    do (setf (aref a i) (logxor (aref a i) (aref a (* i j)))))
              (if (= (aref a i) 1) (progn (incf ans-c)
                                          (vector-push-extend i ans))))
     (format t "~A~%" ans-c)
     (if (not (= 0 ans-c)) (format t "~{~A~%~}" (coerce ans 'list))))

Score : 400 points

Problem Statement

There are N empty boxes arranged in a row from left to right.
The integer i is written on the i-th box from the left (1 \leq i \leq N).

For each of these boxes, Snuke can choose either to put a ball in it or to put nothing in it.

We say a set of choices to put a ball or not in the boxes is good when the following condition is satisfied:

For every integer i between 1 and N (inclusive), the total number of balls contained in the boxes with multiples of i written on them is congruent to a_i modulo 2.

Does there exist a good set of choices? If the answer is yes, find one good set of choices.

Constraints

All values in input are integers.

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

a_i is 0 or 1.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

If a good set of choices does not exist, print -1.

If a good set of choices exists, print one such set of choices in the following format:

M
b_1 b_2 ... b_M

where M denotes the number of boxes that will contain a ball, and b_1,\ b_2,\ ...,\ b_M are the integers written on these boxes, in any order.

Sample Input 1

3
1 0 0

Sample Output 1

1
1

Consider putting a ball only in the box with 1 written on it.

There are three boxes with multiples of 1 written on them: the boxes with 1, 2, and 3. The total number of balls contained in these boxes is 1.

There is only one box with a multiple of 2 written on it: the box with 2. The total number of balls contained in these boxes is 0.

There is only one box with a multiple of 3 written on it: the box with 3. The total number of balls contained in these boxes is 0.

Thus, the condition is satisfied, so this set of choices is good.

Sample Input 2

5
0 0 0 0 0

Sample Output 2

0

Putting nothing in the boxes can be a good set of choices.


#-swank
(unless (member :child-sbcl *features*)
  (quit
   :unix-status
   (process-exit-code
    (run-program *runtime-pathname*
                 `("--control-stack-size" "128MB"
                   "--noinform" "--disable-ldb" "--lose-on-corruption" "--end-runtime-options"
                   "--eval" "(push :child-sbcl *features*)"
                   "--script" ,(namestring *load-pathname*))
                 :output t :error t :input t))))

(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
           #\# #\> (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))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (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 #\-))
                                  (setq 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))))))))

(in-package :cl-user)

(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 main ()
  (let* ((n (read-fixnum))
         (m (read-fixnum))
         (graph (make-array n :element-type 'list :initial-element nil))
         ;; 1: tmp. marked, 2: fixed
         (marked (make-array n :element-type 'uint8 :initial-element 0))
         (out (make-string-output-stream :element-type 'base-char)))
    (dotimes (i m)
      (let ((a (- (read-fixnum) 1))
            (b (- (read-fixnum) 1)))
        (push a (aref graph b))
        (push b (aref graph a))))
    (labels ((%print (a b)
               (format out "~D ~D~%" (+ a 1) (+ b 1)))
             (dfs (v)
               #>v
               (setf (aref marked v) 1)
               (let ((parity 0))
                 (dolist (child (aref graph v))
                   (cond ((= 1 (aref marked child)))
                         ((= 2 (aref marked child))
                          (%print v child)
                          (xorf parity 1))
                         ((zerop (dfs child))
                          (%print v child)
                          (xorf parity 1))
                         (t (%print child v))))
                 (dbg v parity)
                 (setf (aref marked v) 2)
                 parity)))
      (let ((parity (dfs 0)))
        (if (zerop parity)
            (write-string (get-output-stream-string out))
            (println -1))))))

#-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 4
1 2
2 3
3 4
4 1
"
    "1 2
1 4
3 2
3 4
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "5 5
1 2
2 3
3 4
2 5
4 5
"
    "-1
")))

Score : 700 points

Problem Statement

You are given a simple connected undirected graph with N vertices and M edges. The vertices are numbered 1 to N, and the i-th edge connects Vertex A_i and Vertex B_i.
Takahashi will assign one of the two possible directions to each of the edges in the graph to make a directed graph.
Determine if it is possible to make a directed graph with an even number of edges going out from every vertex. If the answer is yes, construct one such graph.

Notes

An undirected graph is said to be simple when it contains no self-loops or multiple edges.

Constraints

2 \leq N \leq 10^5

N-1 \leq M \leq 10^5

1 \leq A_i,B_i \leq N (1\leq i\leq M)

The given graph is simple and connected.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
:
A_M B_M

Output

If it is impossible to assign directions to satisfy the requirement, print -1.
Otherwise, print an assignment of directions that satisfies the requirement, in the following format:

C_1 D_1
:
C_M D_M

Here each pair (C_i, D_i) means that there is an edge directed from Vertex C_i to Vertex D_i. The edges may be printed in any order.

Sample Input 1

4 4
1 2
2 3
3 4
4 1

Sample Output 1

1 2
1 4
3 2
3 4

After this assignment of directions, Vertex 1 and 3 will each have two outgoing edges, and Vertex 2 and 4 will each have zero outgoing edges.

Sample Input 2

5 5
1 2
2 3
3 4
2 5
4 5

Sample Output 2

-1


(defun min-index(lst)
  (labels ((rec (lst i n minvalue)
				(if (null (cdr lst))
				  n
				  (if (< (car lst) minvalue)
					(rec (cdr lst) (1+ i) i (car lst))
					(rec (cdr lst) (1+ i) n minvalue)))))
	(rec (cdr lst) 1 1 (cadr lst))))
(defun skip-n(lst n)
  (let ((a (nth n lst)))
	(labels ((rec (lst i acc)
				  (if (null lst)
					(nreverse acc)
					(if (= i n)
					  (rec (cdr lst) (1+ i) acc)
					  (if (or (= (1+ i) n)
							  (= (1- i) n))
						(rec (cdr lst) (1+ i) (cons (+ a (car lst)) acc))
						(rec (cdr lst) (1+ i) (cons (car lst) acc)))))))
	  (rec lst 0 nil))))
(defun f(lst)
  (let ((llsstt (skip-n lst (min-index lst))))
	(if (null (cddr llsstt))
	  (+ (car llsstt) (cadr llsstt))
	  (f llsstt))))

(defun splitat (c line)
  (labels ((rec (line acc)
				(let ((pos (position-if (lambda(cc) (char= c cc)) line)))
				  (if pos
					  (rec (subseq line (1+ pos)) (cons (subseq line 0 pos) acc))
					(nreverse (cons line acc))))))
	(rec line nil)))

(let* ((n (parse-integer (read-line nil nil)))
	   (lst (mapcar #'parse-integer (splitat #\space (read-line nil nil)))))
  (format t "~A" (f lst)))

Score : 1000 points

Problem Statement

There is a stack of N cards, each of which has a non-negative integer written on it. The integer written on the i-th card from the top is A_i.

Snuke will repeat the following operation until two cards remain:

Choose three consecutive cards from the stack.

Eat the middle card of the three.

For each of the other two cards, replace the integer written on it by the sum of that integer and the integer written on the card eaten.

Return the two cards to the original position in the stack, without swapping them.

Find the minimum possible sum of the integers written on the last two cards remaining.

Constraints

2 \leq N \leq 18

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

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the minimum possible sum of the integers written on the last two cards remaining.

Sample Input 1

4
3 1 4 2

Sample Output 1

16

We can minimize the sum of the integers written on the last two cards remaining by doing as follows:

Initially, the integers written on the cards are 3, 1, 4, and 2 from top to bottom.

Choose the first, second, and third card from the top. Eat the second card with 1 written on it, add 1 to each of the other two cards, and return them to the original position in the stack. The integers written on the cards are now 4, 5, and 2 from top to bottom.

Choose the first, second, and third card from the top. Eat the second card with 5 written on it, add 5 to each of the other two cards, and return them to the original position in the stack. The integers written on the cards are now 9 and 7 from top to bottom.

The sum of the integers written on the last two cards remaining is 16.

Sample Input 2

6
5 2 4 1 6 9

Sample Output 2

51

Sample Input 3

10
3 1 4 1 5 9 2 6 5 3

Sample Output 3

115


Score : 100 points

Problem Statement

N of us are going on a trip, by train or taxi.

The train will cost each of us A yen (the currency of Japan).

The taxi will cost us a total of B yen.

How much is our minimum total travel expense?

Constraints

All values in input are integers.

1 \leq N \leq 20

1 \leq A \leq 50

1 \leq B \leq 50

Input

Input is given from Standard Input in the following format:

N A B

Output

Print an integer representing the minimum total travel expense.

Sample Input 1

4 2 9

Sample Output 1

8

The train will cost us 4 \times 2 = 8 yen, and the taxi will cost us 9 yen, so the minimum total travel expense is 8 yen.

Sample Input 2

4 2 7

Sample Output 2

7

Sample Input 3

4 2 8

Sample Output 3

8


(let* ((n (read))
       (d (read))
       (lst (loop :repeat n :collect(loop :repeat d :collect (read))))
       (ans '()))
  (mapcar (lambda (pos1)
            (mapcar (lambda (pos2)
                      (push (reduce #'+ (mapcar (lambda (p q) (expt (- p q) 2)) pos1 pos2)) ans)) lst)) lst)
  (princ (floor (length (remove-if-not (lambda (x) (= (expt (isqrt x) 2) x)) ans)) 2)))

Score : 200 points

Problem Statement

There are N points in a D-dimensional space.

The coordinates of the i-th point are (X_{i1}, X_{i2}, ..., X_{iD}).

The distance between two points with coordinates (y_1, y_2, ..., y_D) and (z_1, z_2, ..., z_D) is \sqrt{(y_1 - z_1)^2 + (y_2 - z_2)^2 + ... + (y_D - z_D)^2}.

How many pairs (i, j) (i < j) are there such that the distance between the i-th point and the j-th point is an integer?

Constraints

All values in input are integers.

2 \leq N \leq 10

1 \leq D \leq 10

-20 \leq X_{ij} \leq 20

No two given points have the same coordinates. That is, if i \neq j, there exists k such that X_{ik} \neq X_{jk}.

Input

Input is given from Standard Input in the following format:

N D
X_{11} X_{12} ... X_{1D}
X_{21} X_{22} ... X_{2D}
\vdots
X_{N1} X_{N2} ... X_{ND}

Output

Print the number of pairs (i, j) (i < j) such that the distance between the i-th point and the j-th point is an integer.

Sample Input 1

3 2
1 2
5 5
-2 8

Sample Output 1

1

The number of pairs with an integer distance is one, as follows:

The distance between the first point and the second point is \sqrt{|1-5|^2 + |2-5|^2} = 5, which is an integer.

The distance between the second point and the third point is \sqrt{|5-(-2)|^2 + |5-8|^2} = \sqrt{58}, which is not an integer.

The distance between the third point and the first point is \sqrt{|-2-1|^2+|8-2|^2} = 3\sqrt{5}, which is not an integer.

Sample Input 2

3 4
-3 7 8 2
-12 1 10 2
-2 8 9 3

Sample Output 2

2

Sample Input 3

5 1
1
2
3
4
5

Sample Output 3

10


(let* ((n (read))
    (p (make-array n))
    (ans 0))

    (loop for i below n do
        (setf (aref p i) (read))
    )

    (loop for i from 1 below (- n 1) do
        (let ((a (make-array 3)))
            (setf (aref a 0) (aref p (- i 1)))
            (setf (aref a 1) (aref p i))
            (setf (aref a 2) (aref p (+ i 1)))
            (setf a (sort a #'<))
            (if (= (aref p i) (aref a 1))
                (incf ans)
            )
        )
    )
    (princ ans)
)

Score : 200 points

Problem Statement

We have a permutation p = {p_1,\ p_2,\ ...,\ p_n} of {1,\ 2,\ ...,\ n}.

Print the number of elements p_i (1 < i < n) that satisfy the following condition:

p_i is the second smallest number among the three numbers p_{i - 1}, p_i, and p_{i + 1}.

Constraints

All values in input are integers.

3 \leq n \leq 20

p is a permutation of {1,\ 2,\ ...,\ n}.

Input

Input is given from Standard Input in the following format:

n
p_1 p_2 ... p_n

Output

Print the number of elements p_i (1 < i < n) that satisfy the condition.

Sample Input 1

5
1 3 5 4 2

Sample Output 1

2

p_2 = 3 is the second smallest number among p_1 = 1, p_2 = 3, and p_3 = 5. Also, p_4 = 4 is the second smallest number among p_3 = 5, p_4 = 4, and p_5 = 2. These two elements satisfy the condition.

Sample Input 2

9
9 6 3 2 5 8 7 4 1

Sample Output 2

5


(defun solve(a &optional (ans 0))
    (cond
      ((<= (length a) 2) ans)
      (t
       (when
	   (or (< (first a) (second a) (third a))
	       (> (first a) (second a) (third a)))
	 (incf ans))
       (solve (rest a) ans))))

(defun main()
  (let ((n (read))
	(a (read-from-string (concatenate 'string "(" (read-line) ")"))))
    (princ (solve a))
    (fresh-line)))

(main)


(defmacro defsolver (name vars &body body)
  `(defun ,name ()
     (let (,@(mapcar #'list
            vars
            (mapcar (constantly '(read))
                vars)))
       ,@body)))

(defsolver solution-b (n)
  (let ((l (loop for i below n collect (read))))
    (princ
     (loop for i from 1 below (- n 1)
	count (let ((ll
		     (sort (list (nth (1- i) l)
				 (nth i l)
				 (nth (1+ i) l))
			   #'<)))
		(= (nth i l)
		   (nth 1 ll)))))))

(solution-b)


(defun splitat (c line)
  (labels ((rec (line acc)
				(let ((pos (position-if (lambda(cc) (char= c cc)) line)))
				  (if pos
					  (rec (subseq line (1+ pos)) (cons (subseq line 0 pos) acc))
					(nreverse (cons line acc))))))
	(rec line nil)))
(defun f (ns line)
  (let* ((n (parse-integer ns))
		 (arr (map 'vector (lambda(a) (parse-integer a)) (splitat #\space line))))
	(labels ((rec (i acc)
				  (if (< i (- n 2))
					(progn
					  (let ((a0 (elt arr i))
							(a1 (elt arr (1+ i)))
							(a2 (elt arr (+ i 2))))
						(rec (1+ i) (if (or (and (<= a0 a1) (<= a1 a2))
											(and (>= a0 a1) (>= a1 a2))
											)(1+ acc) acc))))
					acc)))
	  (format t "~A" (rec 0 0)))))

(let (
	  (line0 (read-line nil nil))
	  (line1 (read-line nil nil))
	  )
  (f line0 line1))