;;middle-of-3 : number number number -> number
;;to produces the number that would be in between
;;the other two if they were listed in increasing order
;;example:(middle-of-3 5 3 7) should be 5s
(define (middle-of-3 x y z)
  (cond
    [(< x y) ;for the case when x<y; then, if x<z, compare y & z; else, x must be the middle
     (cond
       [(< x z) ;for the case when x<z
        (cond
          [(< y z) y] ;if y<z, then the middle must be y
          [else    z])] ;else, the middle must be z
       [else
        x])];if x>=z, the middle must be x
    [else ;for the case when x>=y; then, if x<z, x must be the meddle; else, compare y & z
     (cond
       [(< x z) ;if x<z, the middle must be x
        x]
       [else
        (cond
          [(< y z) z] ;if y<z, z must be the meddle
          [else    y])])])) ;if y>=z, y must be the meddle

;;interval-proper-subset? : number number number number -> boolean
;;to determine whether set of points in the first interval is a
;;subset of the set of points contained in the second interval,
;;but not equal to it.
;;example: (interval-proper-subset? 3 4 2 5) should be ture, but
;;         (interval-proper-subset? 3 4 3 4) should be false
(define (interval-proper-subset? left1 right1 left2 right2)
  (and
   (and                ;interval1 must be a subset of interval2
    (>= left1 left2)
    (<= right1 right2))
   (not                ;interval1 must not equal to interval2
    (and
     (= left1 left2)
     (= right1 right2)))))

;;is-leap-year : number -> number
;;to determine whether the year is leap; if leap, then return 1; else return 0
(define (is-leap-year year)
  (cond
    [(= (remainder year 400) 0) ;when year is divisible by 400 the year is leap
     1]
    [(and ;when year is divisible by 4 but not the century year (-00) the year is leap
     (= (remainder year 4) 0)
     (not
      (= (remainder year 100) 0)))
     1]
    [else ;else year is not leap year
     0]))
 
;;day-within-year : symbol number number -> number
;;to calculate the day of the year that the date falls
;;example: (day-within-year 'Sep 24 2000) = 268
(define (day-within-year month day year)
 (cond
   [(symbol=? month 'Jan)
    day]
   [(symbol=? month 'Feb)
    (+ 31 day)]
   [(symbol=? month 'Mar)
    (+ 
     (+
      (+ 31 28)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Apr)
    (+ 
     (+
      (+ 31 28 31)
      day)
     (is-leap-year year))]
   [(symbol=? month 'May)
    (+ 
     (+
      (+ 31 28 31 30)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Jun)
    (+ 
     (+
      (+ 31 28 31 30 31)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Jul)
    (+ 
     (+
      (+ 31 28 31 30 31 30)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Aug)
    (+ 
     (+
      (+ 31 28 31 30 31 30 31)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Sep)
    (+ 
     (+
      (+ 31 28 31 30 31 30 31 31)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Oct)
    (+ 
     (+
      (+ 31 28 31 30 31 30 31 31 30)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Nov)
    (+ 
     (+
      (+ 31 28 31 30 31 30 31 31 30 31)
      day)
     (is-leap-year year))]
   [(symbol=? month 'Dec)
    (+ 
     (+
      (+ 31 28 31 30 31 30 31 31 30 31 30)
      day)
     (is-leap-year year))]))