"""simple docstring"""






import math


def      UpperCAmelCase__			(							lowerCAmelCase__	:int						)					->       bool:
      '''simple docstring'''


      lowercase								=      math.loga(math.sqrt(4 * positive_integer + 1						) / 2 + 1 / 2						)

      return exponent == int(lowerCAmelCase__						)


def      UpperCAmelCase__			(							lowerCAmelCase__	:float = 1 / 1_2_3_4_5						)					->       int:
      '''simple docstring'''


      lowercase								=      0
      lowercase								=      0

      lowercase								=      3
      while True:
            lowercase								=      (integer**2 - 1) / 4
            # if candidate is an integer, then there is a partition for k
            if partition_candidate == int(lowerCAmelCase__						):
                  lowercase								=      int(lowerCAmelCase__						)
                  total_partitions += 1
                  if check_partition_perfect(lowerCAmelCase__						):
                        perfect_partitions += 1
            if perfect_partitions > 0:
                  if perfect_partitions / total_partitions < max_proportion:
                        return int(lowerCAmelCase__						)
            integer += 1


if __name__ == "__main__":
       print(F"""{solution() = }""")




"""simple docstring"""






from math import pow, sqrt


def      UpperCAmelCase__			(							*lowerCAmelCase__	:float						)					->       bool:
      '''simple docstring'''


      lowercase								=      len(lowerCAmelCase__						) > 0 and all(value > 0.0 for value in values						)
      return result


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError("""Input Error: Molar mass values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(effusion_rate * sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(effusion_rate / sqrt(molar_mass_a / molar_mass_a						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(molar_mass / pow(effusion_rate_a / effusion_rate_a   ,						2						)   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )






def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float						)					->       float | ValueError:
      '''simple docstring'''


      return (
          round(pow(effusion_rate_a / effusion_rate_a   ,						2						) / molar_mass   ,						6						)
          if validate(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)
          else ValueError(
              """Input Error: Molar mass and effusion rate values must greater than 0."""						)
      )



