answer:Given the point ( P ) inside triangle ( ABC ) satisfies[overrightarrow{PA} + 2 overrightarrow{PB} + 3 overrightarrow{PC} = mathbf{0},]we need to determine the ratio of the area of triangle ( ABC ) to the area of triangle ( APC ).We start by using the center of mass condition and vector properties. To solve this problem, let's place (A) at the origin (0, 0) of a coordinate system, and represent points (B(x_1, y_1)) and (C(x_2, y_2)). Let (P) be ((x, y)).The condition given by:[overrightarrow{PA} + 2 overrightarrow{PB} + 3 overrightarrow{PC} = mathbf{0}]translates to:[overrightarrow{PA} = P - A = (x, y) - (0,0) = (x, y)]Similarly,[overrightarrow{PB} = P - B = (x, y) - (x_1, y_1) = (x - x_1, y - y_1)]and[overrightarrow{PC} = P - C = (x, y) - (x_2, y_2) = (x - x_2, y - y_2)]Substituting these into the given equation:[(x, y) + 2(x - x_1, y - y_1) + 3(x - x_2, y - y_2) = (0, 0)]Separating the x and y components:[x + 2(x - x_1) + 3(x - x_2) = 0][y + 2(y - y_1) + 3(y - y_2) = 0]Simplifying, we get:[x + 2x - 2x_1 + 3x - 3x_2 = 0 implies 6x - 2x_1 - 3x_2 = 0 implies 6x = 2x_1 + 3x_2 implies x = frac{2x_1 + 3x_2}{6}]and[y + 2y - 2y_1 + 3y - 3y_2 = 0 implies 6y - 2y_1 - 3y_2 = 0 implies 6y = 2y_1 + 3y_2 implies y = frac{2y_1 + 3y_2}{6}]Thus, the coordinates of ( P ) are:[Pleft( frac{2x_1 + 3x_2}{6}, frac{2y_1 + 3y_2}{6} right)]To find the ratio of the areas, we realize that the vector sum equation implies a certain weighting of (P) within the triangle. Specifically, the coefficients (1, 2, 3) relate to the areas of smaller triangles. The weighted centroid theorem tells us that the area ratios are inversely proportional to these coefficients. Here the coefficients sum to (1+2+3 = 6).Let's calculate the area ratio using coding.Using determinants to calculate the ratios, we compute the determinants for area calculations. To get the final result, we may simplify the calculations through programming.```pythonimport sympy as sp# Define the symbolic variablesx1, y1, x2, y2 = sp.symbols('x1 y1 x2 y2')x = (2*x1 + 3*x2) / 6y = (2*y1 + 3*y2) / 6# Triangle verticesA = (0, 0)B = (x1, y1)C = (x2, y2)P = (x, y)# Area of triangle ABC using the determinant methodarea_ABC = sp.Abs((B[0] * C[1] + C[0] * A[1] + A[0] * B[1] - B[1] * C[0] - C[1] * A[0] - A[1] * B[0]) / 2)# Area of triangle APC using the determinant methodarea_APC = sp.Abs((A[0] * P[1] + P[0] * C[1] + C[0] * A[1] - A[1] * P[0] - P[1] * C[0] - C[1] * A[0]) / 2)# Simplify the areasarea_ABC = sp.simplify(area_ABC)area_APC = sp.simplify(area_APC)# The area ratio of triangle ABC to triangle APCarea_ratio = area_ABC / area_APCprint(area_ratio.simplify())``````output3```The ratio of the area of triangle ( ABC ) to the area of triangle ( APC ) is [boxed{3}]

answer:To solve the problem, we need to determine how many parts Liu Wei should process on average per day for the remaining days to complete a total of 190 parts in 10 days. Here are the steps to solve the problem:1. Calculate the total number of parts Liu Wei has already processed in the first 2 days.2. Calculate the number of parts remaining to be processed.3. Determine the number of days remaining.4. Calculate the average number of parts Liu Wei needs to process per day for the remaining days to meet the target.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variablestotal_parts = 190days_total = 10parts_first_two_days = 15 * 2days_first_two_days = 2days_remaining = days_total - days_first_two_days# Calculate the remaining parts to be processedparts_remaining = total_parts - parts_first_two_days# Calculate the average number of parts to be processed per day in the remaining daysaverage_parts_per_day = parts_remaining / days_remaining# Output the resultprint(average_parts_per_day)``````output20.0```The average number of parts Liu Wei needs to process per day for the remaining 8 days to meet the target of 190 parts in 10 days is (boxed{20}).

answer:To solve this problem, we need to find all real-valued functions ( f ) defined on the set of all non-zero real numbers that satisfy the given conditions. Let's break down the steps:1. Conditions: - ( f(1) = 1 ) - ( fleft( frac{1}{x+y} right) = fleft( frac{1}{x} right) + fleft( frac{1}{y} right) ) for all non-zero ( x, y ) such that ( x + y neq 0 ) - ( (x + y) f(x + y) = xy f(x) f(y) ) for all non-zero ( x, y ) such that ( x + y neq 0 )Let's use sympy to solve this step-by-step.```pythonimport sympy as sp# Define the function f and the variables x and yf = sp.Function('f')x, y = sp.symbols('x y')# Define the conditionscondition1 = f(1) - 1condition2 = f(1/(x+y)) - (f(1/x) + f(1/y))condition3 = (x + y) * f(x + y) - x*y*f(x)*f(y)# Solve the conditionssolution1 = sp.solve(condition1, f(1))solution2 = sp.solve(condition2, f(1/(x+y)))solution3 = sp.solve(condition3, f(x + y))print(solution1)print(solution2)print(solution3)``````output[1][f(1/x) + f(1/y)][x*y*f(x)*f(y)/(x + y)]```Based on the results of the calculations, we have the following insights:1. From ( f(1) = 1 ), we already know ( f(1) = 1 ).2. The equation ( fleft( frac{1}{x+y} right) = fleft( frac{1}{x} right) + fleft( frac{1}{y} right) ) suggests a functional form involving inverses.3. The equation ( (x + y) f(x + y) = x y f(x) f(y) ) gives us a multiplicative relation.Let's explore these insights further to deduce the form of ( f ).Given the second condition, let's consider ( f(x) = frac{1}{x} ):[ fleft( frac{1}{x+y} right) = fleft( frac{1}{x} right) + fleft( frac{1}{y} right) ]This becomes:[ frac{1}{frac{1}{x} + frac{1}{y}} = fleft( frac{1}{x} right) + fleft( frac{1}{y} right) ]which simplifies correctly.Next, let's check if this form satisfies the third condition:[ (x + y) f(x + y) = x y f(x) f(y) ]Substituting ( f(x) = frac{1}{x} ), we get:[ (x + y) frac{1}{x + y} = x y frac{1}{x} frac{1}{y} ][ 1 = 1 ]which is true.So the function ( f(x) = frac{1}{x} ) satisfies all the given conditions.Let's wrap this up in the final form and print the result.```pythonimport sympy as sp# Define the function fdef f(x): return 1 / x# Verify the conditionsx, y = sp.symbols('x y')condition1 = f(1) == 1condition2 = f(1/(x+y)) == f(1/x) + f(1/y)condition3 = (x + y) * f(x + y) == x * y * f(x) * f(y)print("Condition 1:", condition1)print("Condition 2:", condition2)print("Condition 3:", condition3)print("The function that satisfies all conditions is f(x) =", f(x))``````outputCondition 1: TrueCondition 2: TrueCondition 3: TrueThe function that satisfies all conditions is f(x) = 1/x```The function ( f(x) = frac{1}{x} ) satisfies all the given conditions. Therefore, the solution to the problem is:[boxed{f(x) = frac{1}{x}}]This function meets the criteria:1. ( f(1) = 1 )2. ( fleft( frac{1}{x+y} right) = fleft( frac{1}{x} right) + fleft( frac{1}{y} right) ) for all non-zero ( x, y ) such that ( x+y neq 0 )3. ( (x+y) f(x+y) = x y f(x) f(y) ) for all non-zero ( x, y ) such that ( x+y neq 0 )This completes the solution.

answer:To solve the problem, we need to find the number of travelers ( n ) that satisfies the given conditions:1. The number of travelers when divided by 15 leaves a remainder of 7.2. The number of travelers when divided by 25 leaves a remainder of 17 (since 25 - 8 = 17).3. The number of travelers is between 250 and 400.Let's use Python and sympy to solve this problem step-by-step:1. Define the constraints using modular arithmetic.2. Use sympy to solve the simultaneous congruences within the given range.Here's the Python code to achieve this:```pythonimport sympy as sp# Define the number of travelersn = sp.symbols('n', integer=True)# Define the constraintsconstraint1 = sp.Mod(n, 15) - 7constraint2 = sp.Mod(n, 25) - 17# Solve the congruencessolution = sp.solve([constraint1, constraint2], n)# Filter the solution to be within the range 250 to 400final_solution = [sol for sol in solution if 250 <= sol <= 400]# Print the final solutionprint(final_solution[0])``````outputTraceback (most recent call last): File " solution = sp.solve([constraint1, constraint2], n)NotImplementedError: could not solve Mod(n, 15) - 7```It looks like the `sp.solve` function is having trouble with the modular arithmetic directly. Instead, we can use the Chinese Remainder Theorem (CRT) to solve the system of congruences. We can write custom logic to solve the system within the given range.Here's the Python code to solve this problem using the Chinese Remainder Theorem:```pythonimport sympy as sp# Define the number of travelersn = sp.symbols('n', integer=True)# Constraints:# n ≡ 7 (mod 15)# n ≡ 17 (mod 25)# Find a solution using the Chinese Remainder Theorem directlyn_val = Nonefor x in range(250, 401): if x % 15 == 7 and x % 25 == 17: n_val = x breakprint(n_val)``````output292```The number of travelers on the liner that satisfies the given conditions is (boxed{292}).

answer:First, we need to determine the work rate of the initial 6 people. From the problem, we know that 6 people completed (frac{1}{3}) of the project in 35 days. Let's denote the total work required to complete the project as (W). Therefore, the work done in those 35 days is:[ text{Work done} = frac{1}{3} W ]We can calculate the work rate (work done per day) for those 6 people as:[ text{Work rate of 6 people} = frac{frac{1}{3} W}{35 text{ days}} = frac{W}{105 text{ days}} ]This means the work rate for 1 person is:[ text{Work rate of 1 person} = frac{W}{105 times 6 text{ days}} = frac{W}{630 text{ days}} ]Now, after 35 days, 6 more people joined. Therefore, the total number of people working on the project becomes 12. The new work rate with 12 people is:[ text{Work rate of 12 people} = 12 times frac{W}{630 text{ days}} = frac{12W}{630 text{ days}} = frac{2W}{105 text{ days}} = frac{2W}{105} ]Now, they need to complete the remaining (frac{2}{3}) of the project. We can find the time (in days) it takes for 12 people to complete the remaining project by using the formula:[ text{Remaining Work} = frac{2}{3} W ][ text{Time} = frac{text{Remaining Work}}{text{Work rate of 12 people}} ][ text{Time} = frac{frac{2}{3} W}{frac{2W}{105}} ]Let's compute this in Python to find the exact number of days.```python# Given datainitial_work_fraction = 1 / 3initial_days = 35initial_people = 6additional_people = 6total_people_after_joining = initial_people + additional_people# Total work (represented as W)total_work = 1 # We can assume total work W = 1 for simplicity# Work done by initial 6 people in 35 dayswork_done_initial = initial_work_fraction * total_work# Work rate of 6 peoplework_rate_6_people = work_done_initial / initial_days# Work rate of 1 personwork_rate_1_person = work_rate_6_people / initial_people# Work rate of 12 peoplework_rate_12_people = work_rate_1_person * total_people_after_joining# Remaining work to be doneremaining_work = total_work - work_done_initial# Time required to complete the remaining work with 12 peopletime_to_complete_remaining_work = remaining_work / work_rate_12_people# Total time to complete the projecttotal_time_to_complete_project = initial_days + time_to_complete_remaining_workprint(total_time_to_complete_project)``````output70.0```The total time it took to complete the project is (boxed{70}) days.

answer:To solve the problem, we need to find the absolute value of (|a+b| - |a-b|) given that (|a| = 3) and (|b| = 5).Here are the steps to solve the problem:1. Understand the problem: We are given (|a| = 3) and (|b| = 5). We need to compute (|a+b| - |a-b|) and then take its absolute value.2. Consider the possible values of (a) and (b): Since (|a| = 3), (a) can be either (3) or (-3). Similarly, since (|b| = 5), (b) can be either (5) or (-5).3. Evaluate (|a+b| - |a-b|) for all possible combinations of (a) and (b): - Case 1: (a = 3) and (b = 5) - Case 2: (a = 3) and (b = -5) - Case 3: (a = -3) and (b = 5) - Case 4: (a = -3) and (b = -5)4. Calculate the expression for each case and find the absolute value.Let's write the Python code to perform these calculations.```pythonimport sympy as sp# Possible values for a and ba_values = [3, -3]b_values = [5, -5]# Calculating |a+b| - |a-b| for all combinationsresults = []for a in a_values: for b in b_values: abs_a_plus_b = sp.Abs(a + b) abs_a_minus_b = sp.Abs(a - b) result = abs_a_plus_b - abs_a_minus_b results.append(result)# Taking the absolute value of the resultsabs_results = [sp.Abs(result) for result in results]# Find the unique absolute valuesunique_abs_results = set(abs_results)print(unique_abs_results)``````output{6}```The absolute value of (|a+b| - |a-b|) for all combinations of (a) and (b) where (|a| = 3) and (|b| = 5) is consistently (6).Therefore, the final answer is (boxed{6}).