answer:To find all functions ( f: mathbb{R} rightarrow mathbb{R} ) that satisfy the given functional equation[ y^{2} f(x) + x^{2} f(y) + xy = xy f(x + y) + x^{2} + y^{2} ]for all real numbers ( x ) and ( y ), we can proceed as follows:1. Substitution and Simplification: Substitute specific values for ( x ) and ( y ) to derive equations that can help identify the function ( f ).2. General Solution: Combine the results from the substitutions to find a general form for ( f(x) ).Let's start the process by plugging in various values for ( x ) and ( y ):a. Substitute ( y = 0 ):[ 0^{2} f(x) + x^{2} f(0) + 0 = 0cdot x f(x + 0) + x^{2} + 0^{2} ]This simplifies to:[ x^{2} f(0) = x^{2} ]Since this must hold for all ( x neq 0 ):[ f(0) = 1 ]b. Substitute ( x = 0 ):[ y^{2} f(0) + 0^{2} f(y) + 0 = 0cdot y f(0 + y) + 0^{2} + y^{2} ]This simplifies to:[ y^{2} cdot 1 = y^{2} ]which is always true and yields no new information.c. Substitute ( x = y ):[ y^{2} f(y) + y^{2} f(y) + y^{2} = y^{2} f(2y) + y^{2} + y^{2} ]Simplifying:[ 2y^{2} f(y) + y^{2} = y^{2} f(2y) + 2y^{2} ][ 2y^{2} f(y) + y^{2} = y^{2} f(2y) + 2y^{2} ][ y^{2} (2f(y) - f(2y)) = y^{2} ]For ( y neq 0 ):[ 2f(y) - f(2y) = 1 ]We now have two key pieces of information:1. ( f(0) = 1 )2. ( 2f(y) - f(2y) = 1 )Let's use these to find the general form of ( f(x) ).Now, we can use the sympy library to solve this system:```pythonimport sympy as sp# Define the function f(x)x, y = sp.symbols('x y')f = sp.Function('f')# Define the equations based on our substitutionseq1 = sp.Eq(f(0), 1)eq2 = sp.Eq(2*f(y) - f(2*y), 1)# Solve the system of equationssol = sp.solve([eq1, eq2], f(y), f(2*y))print(sol)``````output[]```The empty list of solutions indicates that sympy couldn't directly find a general solution using the given constraints. We need to manually analyze the functional equation further to determine ( f(x) ). From our derived equation:[ 2f(y) - f(2y) = 1 ]Let's manipulate this equation to find ( f(y) ).1. Let ( y = 1 ):[ 2f(1) - f(2) = 1 ]2. Let ( y = 2 ):[ 2f(2) - f(4) = 1 ]By continuing this pattern, we notice that ( f(2y) = 2f(y) - 1 ). This suggests a recursive pattern. Let's assume a linear form for ( f(x) ) and see if it satisfies the functional equation.Assume ( f(x) = ax + b ).Substitute ( f(x) ) into the equation ( 2f(y) - f(2y) = 1 ):[ 2(ay + b) - (a(2y) + b) = 1 ][ 2ay + 2b - 2ay - b = 1 ][ b = 1 ]So, ( f(x) = ax + 1 ). Now, substitute ( f(x) = ax + 1 ) back into the original functional equation to determine ( a ):[ y^2 (ax + 1) + x^2 (ay + 1) + xy = xy(a(x + y) + 1) + x^2 + y^2 ]Simplify and collect like terms:[ y^2 ax + y^2 + x^2 ay + x^2 + xy = axy(x + y) + xy + x^2 + y^2 ][ y^2 ax + x^2 ay + xy = axy(x + y) + xy ]This simplifies to:[ y^2 ax + x^2 ay = axy(x + y) ]Both sides are equal, confirming that our assumed form ( f(x) = ax + 1 ) is correct. To further verify, note that substituting ( f(x) = x + 1 ) into the functional equation holds true.Thus, the general solution is:[ f(x) = x + 1 ]Let's format the final result:[ boxed{f(x) = x + 1} ]

answer:To determine the total number of buttons Sally needs to sew all the shirts, we can follow these steps:1. Calculate the total number of shirts Sally has sewn from Monday to Wednesday.2. Calculate the total number of buttons by multiplying the total number of shirts by the number of buttons per shirt.Let's write the Python code to perform these calculations.```python# Number of shirts sewed on each dayshirts_monday = 4shirts_tuesday = 3shirts_wednesday = 2# Number of buttons per shirtbuttons_per_shirt = 5# Total number of shirtstotal_shirts = shirts_monday + shirts_tuesday + shirts_wednesday# Total number of buttons neededtotal_buttons = total_shirts * buttons_per_shirtprint(total_buttons)``````output45```The total number of buttons Sally needs to sew all the shirts is (boxed{45}).

answer:To find the intercept of the line ( l ) on the ( x )-axis, we need to determine the point where the line crosses the ( x )-axis. This means we need to find the ( x )-coordinate when the ( y )-coordinate is zero.Given:- The line passes through the point ((-3, 4)).- The direction vector of the line is ((1, -2)).To solve this problem, we will:1. Write the parametric equations of the line.2. Set the ( y )-coordinate to 0 to find the corresponding ( x )-coordinate.The parametric equations for the line can be written as:[ x = -3 + t ][ y = 4 - 2t ]We need to find the value of ( t ) when ( y = 0 ):[ 0 = 4 - 2t ][ 2t = 4 ][ t = 2 ]Now substitute ( t = 2 ) back into the ( x ) equation to find the intercept on the ( x )-axis:[ x = -3 + 2 times 1 ][ x = -3 + 2 ][ x = -1 ]Thus, the intercept of the line on the ( x )-axis is (-1).Let's verify this using Python and Sympy.```pythonimport sympy as sp# Define the parameter tt = sp.symbols('t')# Parametric equations of the linex = -3 + ty = 4 - 2*t# Find the value of t when y = 0t_value = sp.solve(y, t)[0]# Substitute t_value in the x equationx_intercept = x.subs(t, t_value)print(x_intercept)``````output-1```The intercept of the line on the ( x )-axis is (boxed{-1}).

answer:To find the area of (triangle PQS), we start with the given information and proceed step by step.Given:- (PQRS) is a trapezoid with area (12).- (RS = 2 times PQ).We need to find the area of (triangle PQS). Step-by-Step Approach1. Understanding the Trapezoid: - Let (PQ = a). - Hence, (RS = 2a). - The height (distance between the parallel sides (PQ) and (RS)) of the trapezoid can be denoted as (h).2. Area of the Trapezoid: The area (A) of a trapezoid with parallel sides (a) and (b) and height (h) is given by: [ A = frac{1}{2} times (a + b) times h ] Here, (PQ = a) and (RS = 2a), so: [ 12 = frac{1}{2} times (a + 2a) times h Rightarrow 12 = frac{1}{2} times 3a times h Rightarrow 12 = frac{3ah}{2} ]3. Solving for Height: [ 12 = frac{3ah}{2} Rightarrow 24 = 3ah Rightarrow h = frac{24}{3a} = frac{8}{a} ]4. Area of (triangle PQS): (triangle PQS) shares a height (h) with the trapezoid where (PQ) is the base of the triangle. The area of a triangle is given by: [ text{Area of } triangle PQS = frac{1}{2} times text{base} times text{height} = frac{1}{2} times a times h ] Substituting (h = frac{8}{a}): [ text{Area of } triangle PQS = frac{1}{2} times a times frac{8}{a} = frac{1}{2} times 8 = 4 ]Let's verify this reasoning using Python code to confirm the calculations.```python# Given area of the trapezoidarea_trapezoid = 12# Let PQ = a, then RS = 2a# Using the formula for the area of a trapezoid: A = 1/2 * (a + b) * h# Here, a = PQ and b = RS (which is given as 2a)# So area = 1/2 * (a + 2a) * h = 3ah / 2# 12 = 3ah / 2 => h = 24 / 3a = 8 / a# Let's verify with symbolic calculationfrom sympy import symbols, Eq, solve# Define symbolsa, h = symbols('a h')# Equation for area of the trapezoideq1 = Eq((3 * a * h) / 2, area_trapezoid)# Solve for h in terms of aheight = solve(eq1, h)[0]# Area of triangle PQS# Area = 1/2 * base * heightarea_triangle = (1/2) * a * height# Simplify to final areaprint(area_triangle.simplify())``````output4.00000000000000```Let's try solving this problem through symbolic calculations step by step.1. Set up the variables and equation for the height: - Given area of trapezoid: [ text{Area} = 12 ] - For the trapezoid (PQRS): [ 12 = frac{1}{2} times (PQ + RS) times h ] - We know (PQ = a) and (RS = 2a), so: [ 12 = frac{1}{2} times (a + 2a) times h = frac{1}{2} times 3a times h ]2. Solving for height: [ 12 = frac{3ah}{2} implies 24 = 3ah implies h = frac{24}{3a} = frac{8}{a} ]3. Calculate the area of (triangle PQS): ( text{Area of } triangle PQS = frac{1}{2} times PQ times h = frac{1}{2} times a times frac{8}{a} = frac{1}{2} times 8 = 4 )Finally, the area of (triangle PQS) is:[boxed{4}]

answer:To solve the problem, let's break it down into manageable parts: Part 1: Find the Analytical Expression of the ParabolaWe know the parabola intersects the (x)-axis at (A(-1,0)) and (B(3,0)), and passes through point (C(0,3)).1. Form of the Parabola: A parabola that intersects the (x)-axis at points (x_1) and (x_2) can be written as: [ y = a(x - x_1)(x - x_2) ] Given (x_1 = -1) and (x_2 = 3), the parabola is: [ y = a(x + 1)(x - 3) ]2. Determine the Coefficient (a): Since the parabola passes through (C(0, 3)), we can substitute (x = 0) and (y = 3) to find the value of (a): [ 3 = a(0 + 1)(0 - 3) ] [ 3 = -3a implies a = -1 ]Thus, the analytical expression of the parabola is:[y = -(x + 1)(x - 3)] Part 2: Find the Area of (triangle ABM)1. Find the Vertex (M) of the Parabola: The vertex form of the parabola is: [ y = -1(x^2 - 2x - 3) implies y = -x^2 + 2x + 3 ] The (x)-coordinate of the vertex of the parabola (y = ax^2 + bx + c) is given by: [ x = -frac{b}{2a} ] Here, (a = -1) and (b = 2): [ x = -frac{2}{2(-1)} = 1 ] Substituting (x = 1) into the parabola equation to find (y): [ y = -1(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4 ] So, the vertex (M(1, 4)).2. Calculate the Area of (triangle ABM): The vertices of (triangle ABM) are (A(-1, 0)), (B(3, 0)), and (M(1, 4)). The area of a triangle with vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is given by: [ text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right| ] Substituting the coordinates of (A), (B), and (M): [ text{Area} = frac{1}{2} left| (-1)(0 - 4) + 3(4 - 0) + 1(0 - 0) right| ] [ text{Area} = frac{1}{2} left| 4 + 12 + 0 right| = frac{1}{2} times 16 = 8 ] Python Code:Now let's implement the above steps in Python to ensure the results are accurate.```pythonimport sympy as sp# Define the variablesx, y = sp.symbols('x y')# Given pointsA = (-1, 0)B = (3, 0)C = (0, 3)# Find the equation of the parabolaa = sp.symbols('a')parabola_eq = a * (x + 1) * (x - 3)a_value = sp.solve(parabola_eq.subs(x, 0) - 3, a)[0]parabola_eq = parabola_eq.subs(a, a_value)# Find the vertex Mvertex_x = -sp.Rational(2, 2 * -1)vertex_y = parabola_eq.subs(x, vertex_x)vertex = (vertex_x, vertex_y)# Calculate the area of triangle ABMarea = sp.Rational(1, 2) * abs(A[0] * (B[1] - vertex[1]) + B[0] * (vertex[1] - A[1]) + vertex[0] * (A[1] - B[1]))print(area)``````output8```The area of (triangle ABM) is (boxed{8}).

answer:To solve the given sum (S), we need to compute the following expression and iterate through it:[S = sum_{k=1}^{2007} sqrt{1 + frac{1}{k^2} + frac{1}{(k+1)^2}}]Once we have (S), we need to then compute (lfloor S^2 rfloor).First, we will write a Python code to compute (S). Here is the step-by-step plan:1. Calculate each term in the sum ( sqrt{1 + frac{1}{k^2} + frac{1}{(k+1)^2}} ) for ( k = 1 ) to ( k = 2007 ).2. Sum up all these terms to get (S).3. Compute (S^2).4. Use the floor function to get (lfloor S^2 rfloor).Let's execute this plan using Python.```pythonimport math# Initialize the sum SS = 0# Calculate the sum of the seriesfor k in range(1, 2008): term = math.sqrt(1 + 1 / (k2) + 1 / ((k+1)2)) S += term# Compute S^2S_squared = S2# Compute the floor of S^2result = math.floor(S_squared)print(result)``````output4032062```The computed value of (lfloor S^2 rfloor ) is (boxed{4032062}).