


from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray





def     _UpperCamelCase     (a__   :float   ,			a__   :float   ,			a__   :bool = False       ):






							"""simple docstring"""





							if radian_mode:
														return [magnitude * cos(lowerCamelCase_       ), magnitude * sin(lowerCamelCase_       )]
							return [magnitude * cos(radians(lowerCamelCase_       )       ), magnitude * sin(radians(lowerCamelCase_       )       )]





def     _UpperCamelCase     (a__   :NDArray[floataa]   ,			a__   :NDArray[floataa]   ,			a__   :float = 10**-1       ):






							"""simple docstring"""





							UpperCamelCase__				     =			cross(lowerCamelCase_   ,			lowerCamelCase_       )
							UpperCamelCase__				     =			sum(lowerCamelCase_       )
							return abs(lowerCamelCase_       ) < eps


if __name__ == "__main__":
					# Test to check if it works
					UpperCamelCase__											= array(
					    [
					        polar_force(7_18.4, 180 - 30),
					        polar_force(8_79.54, 45),
					        polar_force(100, -90),
					    ]
					)

					UpperCamelCase__											= array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem 1 in image_data/2D_problems.jpg
					UpperCamelCase__											= array(
					    [
					        polar_force(30 * 9.81, 15),
					        polar_force(215, 180 - 45),
					        polar_force(264, 90 - 30),
					    ]
					)

					UpperCamelCase__											= array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem in image_data/2D_problems_1.jpg
					UpperCamelCase__											= array([[0, -2_000], [0, -1_200], [0, 15_600], [0, -12_400]])

					UpperCamelCase__											= array([[0, 0], [6, 0], [10, 0], [12, 0]])

					assert in_static_equilibrium(forces, location)

					import doctest

					doctest.testmod()











from math import ceil, sqrt





def     _UpperCamelCase     (a__   :int = 100_0000       ):






							"""simple docstring"""





							UpperCamelCase__				     =			0

							for outer_width in range(3   ,			(limit // 4) + 2       ):
														if outer_width**2 > limit:
																					UpperCamelCase__				     =			max(ceil(sqrt(outer_width**2 - limit       )       )   ,			1       )
														else:
																					UpperCamelCase__				     =			1
														if (outer_width - hole_width_lower_bound) % 2:
																					hole_width_lower_bound += 1

														answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

							return answer


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




