




from __future__ import annotations

from math import ceil, floor, sqrt






def  snake_case(    __magic_name__ = 2_00_00_00   )			->							int:





     '''simple docstring'''


     lowercase :       list[int]       =			[0]
     lowercase :       int

     for idx in range(1			,		ceil(sqrt(target * 2   ) * 1.1   )   ):
          triangle_numbers.append(triangle_numbers[-1] + idx   )

     # we want this to be as close as possible to target
     lowercase :       int       =			0
     # the area corresponding to the grid that gives the product closest to target
     lowercase :       int       =			0
     # an estimate of b, using the quadratic formula
     lowercase :       float
     # the largest integer less than b_estimate
     lowercase :       int
     # the largest integer less than b_estimate
     lowercase :       int
     # the triangle number corresponding to b_floor
     lowercase :       int
     # the triangle number corresponding to b_ceil
     lowercase :       int

     for idx_a, triangle_a in enumerate(triangle_numbers[1:]			,		1   ):
          lowercase :       Dict       =			(-1 + sqrt(1 + 8 * target / triangle_a   )) / 2
          lowercase :       int       =			floor(__magic_name__   )
          lowercase :       Union[str, Any]       =			ceil(__magic_name__   )
          lowercase :       Any       =			triangle_numbers[b_floor]
          lowercase :       Any       =			triangle_numbers[b_ceil]

          if abs(target - triangle_b_first_guess * triangle_a   ) < abs(
              target - best_product   ):
               lowercase :       Optional[int]       =			triangle_b_first_guess * triangle_a
               lowercase :       Union[str, Any]       =			idx_a * b_floor

          if abs(target - triangle_b_second_guess * triangle_a   ) < abs(
              target - best_product   ):
               lowercase :       Dict       =			triangle_b_second_guess * triangle_a
               lowercase :       Any       =			idx_a * b_ceil

     return area


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





import math






def  snake_case(    __magic_name__   )			->							bool:





     '''simple docstring'''


     lowercase :       Union[str, Any]       =			math.loga(math.sqrt(4 * positive_integer + 1   ) / 2 + 1 / 2   )

     return exponent == int(__magic_name__   )






def  snake_case(    __magic_name__ = 1 / 1_23_45   )			->							int:





     '''simple docstring'''


     lowercase :       Union[str, Any]       =			0
     lowercase :       str       =			0

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


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