


"""simple docstring"""







import math




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					return exponent == int(lowerCAmelCase_   )




def    UpperCAmelCase__       (lowerCAmelCase_ = 1 / 1_2345   ):




					'''simple docstring'''


					__SCREAMING_SNAKE_CASE						   =				0
					__SCREAMING_SNAKE_CASE						   =				0

					__SCREAMING_SNAKE_CASE						   =				3
					while True:
										__SCREAMING_SNAKE_CASE						   =				(integer**2 - 1) / 4
										# if candidate is an integer, then there is a partition for k
										if partition_candidate == int(lowerCAmelCase_   ):
															__SCREAMING_SNAKE_CASE						   =				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 collections import defaultdict
from math import ceil, sqrt




def    UpperCAmelCase__       (lowerCAmelCase_ = 100_0000  ,						lowerCAmelCase_ = 10   ):




					'''simple docstring'''


					__SCREAMING_SNAKE_CASE						   =				defaultdict(lowerCAmelCase_   )

					for outer_width in range(3  ,						(t_limit // 4) + 2   ):
										if outer_width * outer_width > t_limit:
															__SCREAMING_SNAKE_CASE						   =				max(
															    ceil(sqrt(outer_width * outer_width - t_limit   )   )  ,						1   )
										else:
															__SCREAMING_SNAKE_CASE						   =				1

										hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

										for hole_width in range(lowerCAmelCase_  ,						outer_width - 1  ,						2   ):
															count[outer_width * outer_width - hole_width * hole_width] += 1

					return sum(1 for n in count.values() if 1 <= n <= 10   )


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