answer:Given:1. Angle ( C ) is double the measure of angle ( B )2. The measure of angle ( A ) is what we need to find in triangle ( ABC )3. Angle ( B ) is given as ( 21^circ )Since the sum of the angles in a triangle is always ( 180^circ ):[A + B + C = 180^circ]Given ( angle B = 21^circ ):[C = 2B]Substitute ( angle B ) into the equation:[C = 2 times 21^circ = 42^circ]Now substitute the known values of ( angle B ) and ( angle C ) into the sum of angles equation:[A + 21^circ + 42^circ = 180^circ]Now, solve for ( angle A ):[A + 63^circ = 180^circ][A = 180^circ - 63^circ][A = 117^circ]So, the measure of angle ( A ) is ( 117^circ ).Let's confirm this result using Python as well.```python# Given angle Bangle_B = 21# Angle C is double angle Bangle_C = 2 * angle_B# Sum of angles in a triangle is 180 degreessum_of_angles = 180# Calculate angle Aangle_A = sum_of_angles - (angle_B + angle_C)print(angle_A)``````output117```The measure of angle ( A ) in triangle ( ABC ) is (boxed{117^circ}).

answer:To solve this problem, we need to translate the given conditions into equations and then solve those equations. Let's denote the three-digit number as ( overline{abc} ), where ( a, b, ) and ( c ) are its digits.1. Digit Sum Constraint: [ a + b + c = 16 ]2. Condition on Swapping Hundreds and Tens: Swapping the hundreds and tens digits results in the number ( overline{bac} ). - The original number can be expressed as ( 100a + 10b + c ). - The new number can be expressed as ( 100b + 10a + c ). - According to the problem, the number decreases by 360: [ 100b + 10a + c = 100a + 10b + c - 360 ] - Simplifying this equation: [ 90b - 90a = -360 ] [ b - a = -4 ] [ a = b + 4 ]3. Condition on Swapping Tens and Units: Swapping the tens and units digits results in the number ( overline{acb} ). - The new number can be expressed as ( 100a + 10c + b ). - According to the problem, the number increases by 54: [ 100a + 10c + b = 100a + 10b + c + 54 ] - Simplifying this equation: [ 10c + b = 10b + c + 54 ] [ 9c - 9b = 54 ] [ c - b = 6 ] [ c = b + 6 ]We now have three equations:1. ( a + b + c = 16 )2. ( a = b + 4 )3. ( c = b + 6 )We can substitute equations (2) and (3) into equation (1) to find the values of ( a ), ( b ), and ( c ).Let's solve these equations step by step using Python.```pythonfrom sympy import symbols, Eq, solve# Define the symbolsa, b, c = symbols('a b c')# Define the equations based on the problem statementeq1 = Eq(a + b + c, 16)eq2 = Eq(a, b + 4)eq3 = Eq(c, b + 6)# Solve the equationssolution = solve((eq1, eq2, eq3), (a, b, c))print(solution)``````output{a: 6, b: 2, c: 8}```The solution to the equations is:[ a = 6, ][ b = 2, ][ c = 8. ]This means the three-digit number is ( overline{628} ).Let's verify the conditions to ensure the accuracy:1. Digit Sum: ( 6 + 2 + 8 = 16 ).2. Swapping hundreds and tens: ( overline{628} ) becomes ( overline{268} ). - The decrease in number: ( 628 - 268 = 360 ).3. Swapping tens and units: ( overline{628} ) becomes ( overline{682} ). - The increase in number: ( 682 - 628 = 54 ).All conditions are satisfied. Thus, the three-digit number is:[ boxed{628} ]

answer:A palindrome is a number that reads the same backward as forward. For instance, 121 is a palindrome. To solve the problem, we need to identify all 3-digit numbers that are squares and check if they are palindromes.Firstly, 3-digit numbers range from 100 to 999. The smallest integer whose square is at least 100 is lceil sqrt{100} rceil = 10, and the largest integer whose square is at most 999 is lfloor sqrt{999} rfloor approx 31. We will square each integer from 10 to 31, check if the result is a 3-digit number, and if it is a palindrome.Let's write the Python code to do the same:```pythondef is_palindrome(num): return str(num) == str(num)[::-1]palindromic_squares = []for i in range(10, 32): # From 10 to 31 inclusive square = i 2 if 100 <= square <= 999: # Ensure it is a 3-digit number if is_palindrome(square): palindromic_squares.append(square)print((palindromic_squares, len(palindromic_squares)))``````output([121, 484, 676], 3)```The 3-digit squares that are palindromes are 121, 484, and 676. Therefore, the total number of 3-digit palindrome squares is:[boxed{3}]

answer:To solve this problem, we need to determine the length of the curve traced by the center of the coin when it rolls around the quadrilateral. Let's break down the problem into manageable parts:1. Understand the Problem: - The coin rolls around the quadrilateral with a circumference of 20 cm. - The coin itself has a circumference of about 5 cm, implying a diameter of approximately (5 / pi) cm. - As the coin makes a complete rotation around the quadrilateral, the center of the coin will trace a curve.2. Concept of the Curve Traced by the Center: - When a circular coin rolls without slipping, the center of the coin traces a curve called a "parallel curve" or "offset curve" around the boundary of the shape. - The length of this traced curve is equal to the boundary length of the quadrilateral plus an additional length due to the rolling coin.3. Calculate the Length of the Curve: - The center of the coin is always at a distance of the coin's radius ((5 / (2pi)) cm) from the boundary of the quadrilateral. - The length of the curve traced by the center is the original circumference of the quadrilateral plus an additional length equal to the circumference of the coin.Given these steps, let's write Python code to find the length of the curve traced by the center of the coin.```pythonimport sympy as sp# Define the circumferencesquadrilateral_circumference = 20 # cmcoin_circumference = 5 # cm# The diameter of the coincoin_diameter = coin_circumference / sp.pi# The length of the curve traced by the center of the coin# It will be the quadrilateral's circumference plus an additional length due to the rolling of the coinlength_of_traced_curve = quadrilateral_circumference + coin_circumferenceprint(length_of_traced_curve)``````output25```The length of the curve traced by the center of the coin when it rolls around the quadrilateral is (boxed{25}) cm.

answer:Let's define the variables:- (x_p) and (y_p) represent the number of sets containing 3 buns and 5 buns respectively brought by Petya.- (x_v) and (y_v) represent the number of sets containing 3 buns and 5 buns respectively brought by Vasya.Given the problem:1. Each set contains either 3 or 5 buns.2. Petya brought a total of 25 buns, so (3x_p + 5y_p = 25).3. Vasya brought a total of 35 buns, so (3x_v + 5y_v = 35).4. The number of sets brought by Petya and Vasya is equal, i.e., (N = x_p + y_p = x_v + y_v).We need to find (N).We can solve this system of equations by iterating over possible values of (x_p) and (x_v), then finding the corresponding (y_p) and (y_v).Let's implement this in Python.```pythondef find_number_of_sets(total_buns_petya, total_buns_vasya): for x_p in range(total_buns_petya // 3 + 1): for y_p in range(total_buns_petya // 5 + 1): if 3 * x_p + 5 * y_p == total_buns_petya: N_p = x_p + y_p for x_v in range(total_buns_vasya // 3 + 1): for y_v in range(total_buns_vasya // 5 + 1): if 3 * x_v + 5 * y_v == total_buns_vasya: N_v = x_v + y_v if N_p == N_v: # Check if the number of sets is equal return N_p return None# Provided datatotal_buns_petya = 25total_buns_vasya = 35# Finding the equal number of setsN = find_number_of_sets(total_buns_petya, total_buns_vasya)print(N)``````output7```The number of sets (N) brought by each boy is (boxed{7}).

answer:To solve the problem, we need to find the value of the expression (2023(a + b) + 3|m| - 2xy) given that:1. (a) and (b) are opposite numbers.2. (x) and (y) are reciprocals.3. (m) is the largest negative integer.Let's break down the problem:1. Opposite Numbers: If (a) and (b) are opposite numbers, then (a = -b). Hence, (a + b = 0).2. Reciprocals: If (x) and (y) are reciprocals, then (x = frac{1}{y}) and (y = frac{1}{x}). Hence, (xy = 1).3. Largest Negative Integer: The largest negative integer is (-1). So, (|m| = 1).Given these properties, we can rewrite the expression as:[2023(a + b) + 3|m| - 2xy = 2023 cdot 0 + 3 cdot 1 - 2 cdot 1 = 0 + 3 - 2 = 1]Let's implement this in Python to verify our reasoning:```pythonimport sympy as sp# Define the variablesa, b, x, y = sp.symbols('a b x y')# Given conditions# a and b are opposite numbers -> a = -b# x and y are reciprocals -> xy = 1# m is the largest negative integer -> m = -1# Define the expressionexpression = 2023*(a + b) + 3*abs(-1) - 2*(x * y)# Substitute the given conditions into the expressionexpression_substituted = expression.subs({a: -b, x*y: 1})# Simplify the expressionresult = sp.simplify(expression_substituted)print(result)``````output1```The value of the expression (2023(a + b) + 3|m| - 2xy) under the given conditions is (boxed{1}).