input: Write a Python function called `determine_flower_pairs` that simulates a game between Alice and Bob on a circular field with flowers. Alice plays first, and they alternately pick a flower in either clockwise or anti-clockwise direction. The function should count the number of valid (x, y) pairs, where x represents the flowers in the clockwise direction and y represents the flowers in the anti-clockwise direction, such that Alice wins the game. The parameters n and m represent the ranges [1, n] and [1, m] for x and y respectively. The function should return the count of these valid pairs.
  
  Examples:
  
  n = 3, m = 2
  3
  
  n = 1, m = 1
  0
  
  
  Constraints:
  1 <= n, m <= 10^5
output: def determine_flower_pairs(n, m):
      def gcd(a, b):
          while b:
              a, b = b, a % b
          return a
  
      def alice_wins(x, y):
          if x == y:
              return False
          g = gcd(x, y)
          if (x // g) % 2 == 1 and (y // g) % 2 == 1:
              return True
          return False
  
      count = 0
      for x in range(1, n + 1):
          for y in range(1, m + 1):
              if alice_wins(x, y):
                  count += 1
      return count