



"""simple docstring"""






from itertools import combinations, product
def      A      (			_A,     _A,     _A,     _A							):




					"""simple docstring"""
					if xa == ya == 0 or xa == ya == 0:
										return False
					snake_case_						:Any								=						xa * xa + ya * ya
					snake_case_						:str								=						xa * xa + ya * ya
					snake_case_						:Any								=						(xa - xa) * (xa - xa) + (ya - ya) * (ya - ya)
					return (
					    a_square + b_square == c_square
					    or a_square + c_square == b_square
					    or b_square + c_square == a_square
					)
def      A      (			_A = 50							):




					"""simple docstring"""
					return sum(
					    1
					    for pta, pta in combinations(product(range(limit + 1							),     repeat=2							),     2							)
					    if is_right(*_A,     *_A							)							)


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






"""simple docstring"""






from typing import Any

import numpy as np

class 	__lowerCAmelCase							:


					'''simple docstring'''


					def __init__(      self    ,       a    ,       a   ):




										"""simple docstring"""





										if np.any(tableau[:, -1]    ,       where=tableau[:, -1] < 0   ):
															raise ValueError("RHS must be > 0"   )

										snake_case_						:List[Any]								=						tableau
										snake_case_	,  snake_case_						:Dict								=						tableau.shape

										# Number of decision variables x1, x2, x3...
										snake_case_						:Union[str, Any]								=						n_vars

										# Number of artificial variables to be minimised
										snake_case_						:Tuple								=						len(np.where(tableau[self.n_vars : -1] == -1   )[0]   )

										# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
										snake_case_						:List[str]								=						(self.n_art_vars > 0) + 1

										# Number of slack variables added to make inequalities into equalities
										snake_case_						:Optional[int]								=						self.n_rows - self.n_stages

										# Objectives for each stage
										snake_case_						:Union[str, Any]								=						["max"]

										# In two stage simplex, first minimise then maximise
										if self.n_art_vars:
															self.objectives.append("min"   )

										snake_case_						:Dict								=						[""]

										# Index of current pivot row and column
										snake_case_						:List[str]								=						None
										snake_case_						:List[Any]								=						None

										# Does objective row only contain (non)-negative values?
										snake_case_						:Union[str, Any]								=						False







					@staticmethod
					def 		_a (      *a   ):




										"""simple docstring"""





										if len(a   ) != 3:
															raise ValueError("Must provide n_vars, n_slack, and n_art_vars"   )

										if not all(x >= 0 and isinstance(a    ,       a   ) for x in args   ):
															raise ValueError("All arguments must be non-negative integers"   )

										# decision | slack | artificial
										snake_case_						:Tuple								=						["x", "s", "a"]
										snake_case_						:Dict								=						[]
										for i in range(3   ):
															for j in range(args[i]   ):
																				titles.append(string_starts[i] + str(j + 1   )   )
										titles.append("RHS"   )
										return titles







					def 		_a (      self    ,       a   ):




										"""simple docstring"""





										snake_case_						:str								=						self.objectives[-1]

										# Find entries of highest magnitude in objective rows
										snake_case_						:Dict								=						(objective == "min") - (objective == "max")
										snake_case_						:Optional[Any]								=						np.argmax(sign * tableau[0, : self.n_vars]   )

										# Choice is only valid if below 0 for maximise, and above for minimise
										if sign * self.tableau[0, col_idx] <= 0:
															snake_case_						:Dict								=						True
															return 0, 0

										# Pivot row is chosen as having the lowest quotient when elements of
										# the pivot column divide the right-hand side

										# Slice excluding the objective rows
										snake_case_						:str								=						slice(self.n_stages    ,       self.n_rows   )

										# RHS
										snake_case_						:Optional[Any]								=						tableau[s, -1]

										# Elements of pivot column within slice
										snake_case_						:Union[str, Any]								=						tableau[s, col_idx]

										# Array filled with nans
										snake_case_						:Optional[Any]								=						np.full(self.n_rows - self.n_stages    ,       np.nan   )

										# If element in pivot column is greater than zeron_stages, return
										# quotient or nan otherwise
										snake_case_						:Dict								=						np.divide(a    ,       a    ,       out=a    ,       where=divisor > 0   )

										# Arg of minimum quotient excluding the nan values. n_stages is added
										# to compensate for earlier exclusion of objective columns
										snake_case_						:List[Any]								=						np.nanargmin(a   ) + self.n_stages
										return row_idx, col_idx







					def 		_a (      self    ,       a    ,       a    ,       a   ):




										"""simple docstring"""





										snake_case_						:Tuple								=						tableau[row_idx].copy()

										snake_case_						:Any								=						piv_row[col_idx]

										# Entry becomes 1
										piv_row *= 1 / piv_val

										# Variable in pivot column becomes basic, ie the only non-zero entry
										for idx, coeff in enumerate(tableau[:, col_idx]   ):
															tableau[idx] += -coeff * piv_row
										snake_case_						:str								=						piv_row
										return tableau







					def 		_a (      self    ,       a   ):




										"""simple docstring"""





										self.objectives.pop()

										if not self.objectives:
															return tableau

										# Slice containing ids for artificial columns
										snake_case_						:str								=						slice(-self.n_art_vars - 1    ,       -1   )

										# Delete the artificial variable columns
										snake_case_						:Optional[int]								=						np.delete(a    ,       a    ,       axis=1   )

										# Delete the objective row of the first stage
										snake_case_						:List[str]								=						np.delete(a    ,       0    ,       axis=0   )

										snake_case_						:Optional[Any]								=						1
										self.n_rows -= 1
										snake_case_						:List[Any]								=						0
										snake_case_						:Dict								=						False
										return tableau







					def 		_a (      self   ):




										"""simple docstring"""





										for _ in range(1_00   ):
															# Completion of each stage removes an objective. If both stages
															# are complete, then no objectives are left
															if not self.objectives:
																				snake_case_						:str								=						self.generate_col_titles(
																				    self.n_vars    ,       self.n_slack    ,       self.n_art_vars   )

																				# Find the values of each variable at optimal solution
																				return self.interpret_tableau(self.tableau    ,       self.col_titles   )

															snake_case_	,  snake_case_						:Union[str, Any]								=						self.find_pivot(self.tableau   )

															# If there are no more negative values in objective row
															if self.stop_iter:
																				# Delete artificial variable columns and rows. Update attributes
																				snake_case_						:str								=						self.change_stage(self.tableau   )
															else:
																				snake_case_						:str								=						self.pivot(self.tableau    ,       a    ,       a   )
										return {}







					def 		_a (      self    ,       a    ,       a   ):




										"""simple docstring"""





										snake_case_						:Any								=						{"P": abs(tableau[0, -1]   )}

										for i in range(self.n_vars   ):
															# Gives ids of nonzero entries in the ith column
															snake_case_						:Dict								=						np.nonzero(tableau[:, i]   )
															snake_case_						:Tuple								=						len(nonzero[0]   )

															# First entry in the nonzero ids
															snake_case_						:Any								=						nonzero[0][0]
															snake_case_						:int								=						tableau[nonzero_rowidx, i]

															# If there is only one nonzero value in column, which is one
															if n_nonzero == nonzero_val == 1:
																				snake_case_						:Tuple								=						tableau[nonzero_rowidx, -1]
																				snake_case_						:Tuple								=						rhs_val

        # Check for basic variables
										for title in col_titles:
															# Don't add RHS or slack variables to output dict
															if title[0] not in "R-s-a":
																				output_dict.setdefault(a    ,       0   )
										return output_dict


if __name__ == "__main__":
				import doctest

				doctest.testmod()

