🧠:<think>Alright, so I've got this math problem here about optimizing data distribution across multiple servers. Let me try to wrap my head around it step by step.First, the problem is about minimizing total latency while making sure no server is overloaded. We have n servers, each with a maximum load capacity L_i. The total data to distribute is D gigabytes. The network is represented as a weighted graph where the edges have weights w_ij, which are the latencies in milliseconds between servers i and j.Okay, part 1 is to formulate an optimization problem. Hmm, optimization problems usually have an objective function and some constraints. So, the objective here is to minimize total latency. But how do we express that?I think the total latency would depend on how much data is being sent between the servers. If more data is sent between two servers, the latency would be higher, right? So, maybe the total latency is the sum over all pairs of servers of the product of the data sent between them and the latency weight. But wait, how is the data distributed? Each server has x_i amount of data, so the data sent from server i to j would depend on the distribution. Hmm, maybe it's not just x_i and x_j, but perhaps the flow between them.Wait, actually, in a network, the data distribution might involve routing, so the latency could be related to the amount of data passing through each edge. But I'm not entirely sure. Maybe it's simpler. If each server is handling x_i data, then the total latency could be the sum of x_i multiplied by some function of the connections. Or perhaps it's the sum over all edges of the latency weight times the data flowing through that edge.I think I need to model this properly. Let me think: each server has x_i data, so the amount of data that needs to be sent from server i to server j could be related to how much data is being handled by each. But without knowing the exact data flow, maybe we can model the latency as a function of the data each server is handling. Alternatively, perhaps the latency is a function of the total data in the network and the connections.Wait, maybe the total latency is the sum over all pairs of servers of the product of their data loads and the latency between them. So, the objective function would be something like the sum from i=1 to n, sum from j=1 to n of x_i * x_j * w_ij. That makes sense because if two servers have a lot of data, the latency between them would contribute more to the total.So, the objective function would be:Minimize Σ_{i=1 to n} Σ_{j=1 to n} x_i x_j w_ijAnd the constraints would be that the sum of all x_i equals D, because we have to distribute all D gigabytes. Also, each x_i must be less than or equal to L_i, since no server can exceed its maximum capacity. Additionally, x_i should be non-negative because you can't have negative data.So, putting it all together, the optimization problem is:Minimize Σ_{i=1}^n Σ_{j=1}^n x_i x_j w_{ij}Subject to:Σ_{i=1}^n x_i = Dx_i ≤ L_i for all i = 1, 2, ..., nx_i ≥ 0 for all i = 1, 2, ..., nOkay, that seems reasonable. Let me double-check. The objective is quadratic because it's x_i times x_j times the weight. So it's a quadratic optimization problem with linear constraints. That makes sense.Now, part 2 is to derive the optimal data distribution vector x given the adjacency matrix A and the maximum load capacities L_i. Hmm, how do we approach solving this quadratic optimization problem?Quadratic optimization problems can be tricky, but if the objective function is convex, we can use methods like Lagrange multipliers or quadratic programming. Let me see if the objective function is convex.The Hessian matrix of the objective function is 2A, where A is the adjacency matrix. Wait, actually, the Hessian would be 2 times the matrix of w_ij because the second derivative with respect to x_i and x_j is 2w_ij. So, if the adjacency matrix is positive semi-definite, the problem is convex, and we can find a global minimum.Assuming that the weights w_ij are non-negative, which they are since they represent latencies, the Hessian would be positive semi-definite. So, the problem is convex, and we can use convex optimization techniques.One approach is to use Lagrange multipliers. Let's set up the Lagrangian. Let me denote the Lagrange multipliers for the equality constraint as λ and for the inequality constraints as μ_i and ν_i for the upper and lower bounds, respectively. But since the constraints are x_i ≤ L_i and x_i ≥ 0, we can handle them using inequality multipliers.But maybe it's simpler to use the method of Lagrange multipliers for the equality constraint first and then check the inequality constraints.So, the Lagrangian would be:L = Σ_{i=1}^n Σ_{j=1}^n x_i x_j w_{ij} + λ (D - Σ_{i=1}^n x_i) + Σ_{i=1}^n μ_i (L_i - x_i) + Σ_{i=1}^n ν_i x_iWait, actually, the standard form for inequality constraints with Lagrangian multipliers is a bit different. The Lagrangian would include terms for the inequality constraints as well, but since we're dealing with both upper and lower bounds, it might get a bit complicated.Alternatively, since the problem is convex, we can use the KKT conditions. The KKT conditions state that at the optimal point, the gradient of the objective function plus the gradients of the constraints multiplied by their respective Lagrange multipliers equals zero.So, let's compute the gradient of the objective function. The derivative of L with respect to x_k is:dL/dx_k = 2 Σ_{j=1}^n x_j w_{kj} - λ - μ_k + ν_k = 0Wait, actually, the derivative of the quadratic term is 2 Σ_{j=1}^n x_j w_{kj} because the objective is Σ x_i x_j w_ij, so when you take the derivative with respect to x_k, you get 2 Σ x_j w_{kj}.Then, the derivative of the equality constraint is -λ, and the derivatives of the inequality constraints x_i ≤ L_i and x_i ≥ 0 are -μ_k and ν_k, respectively.So, the KKT conditions give us:2 Σ_{j=1}^n x_j w_{kj} - λ - μ_k + ν_k = 0 for each kAnd the complementary slackness conditions:μ_k (L_k - x_k) = 0ν_k x_k = 0Also, the primal feasibility conditions:Σ x_i = D0 ≤ x_k ≤ L_kAnd dual feasibility:μ_k ≥ 0, ν_k ≥ 0Hmm, this seems a bit involved. Maybe we can simplify it by assuming that the optimal solution doesn't hit the upper or lower bounds, meaning μ_k = 0 and ν_k = 0. Then, the gradient condition simplifies to:2 Σ_{j=1}^n x_j w_{kj} = λ for each kThis would mean that for each server k, the weighted sum of all x_j multiplied by the latency w_{kj} is equal to λ/2.But wait, this is a system of equations. Let me write it as:Σ_{j=1}^n w_{kj} x_j = λ / 2 for each kThis is a linear system where the matrix is W (the adjacency matrix) and the vector is x, and the right-hand side is λ/2 times a vector of ones.But wait, actually, the system is W x = (λ/2) e, where e is a vector of ones.But we also have the constraint that Σ x_i = D.So, we can write:W x = (λ/2) eande^T x = DThis is a system of n+1 equations with n+1 variables (x_1, ..., x_n, λ).But solving this system might not be straightforward because W could be singular or not invertible. However, if W is invertible, we can express x as:x = (λ/2) W^{-1} eThen, substituting into the constraint e^T x = D:e^T (λ/2) W^{-1} e = DSo,λ = 2 D / (e^T W^{-1} e)Then,x = (2 D / (e^T W^{-1} e)) * (1/2) W^{-1} e = D W^{-1} e / (e^T W^{-1} e)So, x = D W^{-1} e / (e^T W^{-1} e)But wait, this assumes that W is invertible and that the solution x doesn't violate the constraints x_i ≤ L_i and x_i ≥ 0.If the solution x obtained this way satisfies x_i ≤ L_i for all i, then it's the optimal solution. Otherwise, we have to consider the inequality constraints, which complicates things.So, in general, the optimal solution can be found by solving the system W x = (λ/2) e with the constraint Σ x_i = D, and then checking if x_i ≤ L_i for all i. If yes, that's the solution. If not, we have to adjust the variables by setting some x_i to their maximum L_i and solving the reduced system.This sounds like a typical quadratic programming problem where we can use an active set method or some other QP solver.Alternatively, if the problem is small, we can solve it using Lagrange multipliers and check the constraints. But for larger n, we'd need a more efficient algorithm.So, the general approach would be:1. Formulate the problem as a quadratic program with the objective function as Σ x_i x_j w_ij, subject to Σ x_i = D and 0 ≤ x_i ≤ L_i.2. Check if the unconstrained solution (ignoring the inequality constraints) satisfies the constraints. If yes, that's the optimal solution.3. If not, identify which constraints are binding (i.e., x_i = L_i or x_i = 0) and solve the reduced problem with those constraints fixed.4. Repeat until all constraints are satisfied.But this is a bit vague. Maybe a better approach is to use a quadratic programming algorithm that can handle the constraints directly.In summary, the optimal data distribution vector x can be found by solving the quadratic optimization problem with the given constraints, possibly using methods like Lagrange multipliers or quadratic programming, and checking for constraint satisfaction.I think that's the gist of it. Let me just recap:1. The optimization problem is to minimize the quadratic form Σ x_i x_j w_ij subject to Σ x_i = D and 0 ≤ x_i ≤ L_i.2. The solution involves solving a quadratic program, which can be done using methods like Lagrange multipliers, considering the KKT conditions, and checking for constraint violations.So, the final answer would involve setting up the quadratic program as described and solving it, possibly with the help of optimization software or algorithms.</think>

🧠:<think>Okay, so I have this problem about an indie band member trying to optimize their recording sessions. There are two parts: one about optimizing recording time and another about studio acoustics and eigenvalues. Let me tackle them one by one.Starting with the first part: Optimization of Recording Time. The band has a total time T hours and wants to record n tracks. Each track i has a time t_i, and the quality function Q is given by the sum of a_i times the square root of t_i. The goal is to maximize Q subject to the constraint that the sum of all t_i equals T. They want me to use the method of Lagrange multipliers.Alright, so I remember that Lagrange multipliers are used to find the extrema of a function subject to equality constraints. The general idea is to introduce a multiplier for the constraint and then take derivatives.Let me write down the problem formally.We need to maximize:Q(t₁, t₂, ..., tₙ) = Σ (a_i * sqrt(t_i)) from i=1 to nSubject to:Σ t_i = TSo, the Lagrangian function L would be the quality function minus λ times the constraint. So,L = Σ a_i sqrt(t_i) - λ (Σ t_i - T)To find the maximum, we take the partial derivatives of L with respect to each t_i and set them equal to zero.So, for each i, ∂L/∂t_i = (a_i)/(2 sqrt(t_i)) - λ = 0That gives us (a_i)/(2 sqrt(t_i)) = λ for each i.So, rearranging, sqrt(t_i) = a_i/(2λ), so t_i = (a_i)^2/(4λ²)Hmm, interesting. So each t_i is proportional to (a_i)^2. That makes sense because the more important a track is (higher a_i), the more time we should allocate to it.But we also have the constraint that Σ t_i = T. So, let's plug our expression for t_i into that.Σ t_i = Σ (a_i²)/(4λ²) = TSo, (Σ a_i²)/(4λ²) = TTherefore, 4λ² = (Σ a_i²)/TSo, λ² = (Σ a_i²)/(4T)Thus, λ = sqrt(Σ a_i²)/(2 sqrt(T))Wait, but λ is a Lagrange multiplier, so it can be positive or negative. But since we're dealing with time, which is positive, and a_i are positive constants, λ must be positive. So, we can take the positive square root.So, λ = sqrt(Σ a_i²)/(2 sqrt(T))Now, going back to t_i:t_i = (a_i²)/(4λ²) = (a_i²)/(4 * (Σ a_i²)/(4T)) ) = (a_i²)/( (Σ a_i²)/T ) = (a_i² * T)/Σ a_i²So, t_i = (T * a_i²)/Σ a_i²Therefore, each t_i is proportional to a_i squared. So, the time allocated to each track is proportional to the square of its importance.That seems logical because if a track is more important (higher a_i), we should spend more time on it, and the relationship is quadratic.Let me check if this makes sense. Suppose all a_i are equal, say a_i = a for all i. Then, t_i would be T/n for each track, which is uniform distribution. That makes sense because if all tracks are equally important, you'd spend equal time on each.Another test case: suppose one track is much more important, say a_1 is much larger than the others. Then, t_1 would be significantly larger than the others, which also makes sense.So, I think this solution is correct.Moving on to the second part: Studio Acoustics and Eigenvalues.The studio's acoustics are modeled by a symmetric matrix A. The overall acoustic quality is influenced by the eigenvalues of A, and the best quality corresponds to eigenvalues being as close to 1 as possible. I need to determine the conditions on the eigenvalues of A for the best recording quality and discuss the implications.Hmm, so eigenvalues close to 1. Since A is symmetric, it's diagonalizable, and all its eigenvalues are real.If the eigenvalues are close to 1, that suggests that the matrix A is close to the identity matrix. Because the identity matrix has all eigenvalues equal to 1.So, perhaps the condition is that all eigenvalues λ_i of A satisfy |λ_i - 1| ≤ ε for some small ε > 0.But maybe more precise conditions are needed.Alternatively, perhaps the matrix A should be such that it's a perturbation of the identity matrix. So, A = I + E, where E is a small perturbation matrix with small eigenvalues.But the problem says the eigenvalues should be as close to 1 as possible. So, maybe the best quality is when all eigenvalues are exactly 1, meaning A is the identity matrix. But in reality, it's probably not exactly the identity, but as close as possible.But what does this imply for the matrix A?If all eigenvalues are close to 1, then A is close to the identity matrix in some sense. Since A is symmetric, it can be written as A = Q D Q^T, where Q is orthogonal and D is diagonal with eigenvalues.If all eigenvalues are close to 1, then D is close to I, so A is close to Q I Q^T = Q Q^T = I. So, A is close to the identity matrix.But what does this mean for the studio acoustics? If A is close to the identity matrix, that might mean that the studio has neutral acoustics—neither too absorptive nor too reflective in any particular direction. Each frequency is treated similarly, leading to a balanced sound.Alternatively, if A is exactly the identity matrix, it might represent a perfectly neutral studio with no coloration, which is ideal for accurate recordings.But in practice, A might not be exactly the identity, but the eigenvalues should be as close to 1 as possible to minimize any distortions or colorations in the sound.So, the condition is that all eigenvalues of A are as close to 1 as possible. This would ensure that the acoustic properties are uniform across different frequencies or directions, leading to the best possible recording quality.In terms of implications, this means that the studio's absorption and reflection properties should be balanced. If some eigenvalues are much larger than 1, it might indicate excessive reflection in certain modes, leading to echoes or resonances. If some eigenvalues are much smaller than 1, it might indicate excessive absorption, leading to a dead or lifeless sound.Therefore, to ensure the best recording quality, the studio's acoustic matrix A should be designed such that all its eigenvalues are near 1, ensuring a balanced and neutral acoustic environment.Let me think if there's more to this. Since A is symmetric, it's associated with a quadratic form. The eigenvalues determine the maximum and minimum values of this quadratic form on the unit sphere. If all eigenvalues are close to 1, then the quadratic form is approximately the same in all directions, which is ideal for acoustics because it means the studio doesn't favor certain frequencies or directions over others.So, in summary, for the best recording quality, the eigenvalues of the acoustic matrix A should be as close to 1 as possible, implying that A is close to the identity matrix. This ensures a balanced and neutral acoustic environment, minimizing distortions and colorations.I think that covers both parts. The first part was about allocating recording time optimally using Lagrange multipliers, leading to each track's time being proportional to the square of its importance. The second part was about the eigenvalues of the acoustic matrix needing to be close to 1 for optimal sound quality, implying a balanced acoustic environment.Final Answer1. The optimal time allocation for each track is (boxed{t_i = dfrac{T a_i^2}{sum_{j=1}^n a_j^2}}).2. The best recording quality is achieved when all eigenvalues of matrix (A) are as close to 1 as possible, implying (A) is close to the identity matrix.</think>

🧠:<think>Okay, so I have two problems here related to modeling documentation quality and client satisfaction. Let me tackle them one by one.Starting with the first problem: It says that the quality of documentation D(t) follows a logistic growth model. The formula given is D(t) = L / (1 + e^{-k(t - t0)}). The parameters are L = 100, k = 0.1, and t0 = 10. I need to find the time t when the quality of documentation reaches 75% of its maximum level.Alright, so first, 75% of the maximum level L is 75% of 100, which is 75. So, I need to solve for t when D(t) = 75.So, plugging into the equation: 75 = 100 / (1 + e^{-0.1(t - 10)}).Let me write that down:75 = 100 / (1 + e^{-0.1(t - 10)})I need to solve for t. Let's rearrange this equation step by step.First, divide both sides by 100:75 / 100 = 1 / (1 + e^{-0.1(t - 10)})Which simplifies to:0.75 = 1 / (1 + e^{-0.1(t - 10)})Now, take the reciprocal of both sides:1 / 0.75 = 1 + e^{-0.1(t - 10)}Calculating 1 / 0.75: that's approximately 1.3333.So, 1.3333 = 1 + e^{-0.1(t - 10)}Subtract 1 from both sides:1.3333 - 1 = e^{-0.1(t - 10)}Which is:0.3333 = e^{-0.1(t - 10)}Now, take the natural logarithm of both sides to solve for the exponent:ln(0.3333) = -0.1(t - 10)Calculating ln(0.3333): I remember that ln(1/3) is approximately -1.0986.So, -1.0986 = -0.1(t - 10)Divide both sides by -0.1:(-1.0986) / (-0.1) = t - 10Which is:10.986 = t - 10Add 10 to both sides:t = 10 + 10.986t ≈ 20.986So, approximately 20.986 units of time. Since the problem doesn't specify the units, I'll assume it's just time units. Rounding to a reasonable decimal place, maybe two, so t ≈ 20.99.Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting from 75 = 100 / (1 + e^{-0.1(t - 10)}).Divide both sides by 100: 0.75 = 1 / (1 + e^{-0.1(t - 10)}).Reciprocal: 1 / 0.75 = 1.3333 = 1 + e^{-0.1(t - 10)}.Subtract 1: 0.3333 = e^{-0.1(t - 10)}.Take ln: ln(0.3333) ≈ -1.0986.So, -1.0986 = -0.1(t - 10).Multiply both sides by -1: 1.0986 = 0.1(t - 10).Divide by 0.1: 10.986 = t - 10.Add 10: t ≈ 20.986.Yes, that seems correct. So, t is approximately 20.99.Moving on to the second problem: The client satisfaction score S(D) is a quadratic function of the form S(D) = aD² + bD + c. We know that the maximum score is 90 when D is 80, and the score is 50 when D is 50. We need to find a, b, and c.Alright, so since it's a quadratic function and it has a maximum, the parabola opens downward, meaning a is negative.First, the vertex form of a quadratic is S(D) = a(D - h)² + k, where (h, k) is the vertex. Here, the maximum occurs at D = 80, S = 90. So, h = 80, k = 90.So, S(D) = a(D - 80)² + 90.We also know that when D = 50, S = 50. So, plugging that in:50 = a(50 - 80)² + 90Simplify (50 - 80): that's -30. Squared is 900.So, 50 = a*900 + 90Subtract 90 from both sides:50 - 90 = 900a-40 = 900aDivide both sides by 900:a = -40 / 900Simplify: divide numerator and denominator by 20: -2 / 45.So, a = -2/45.Now, let's write the quadratic in standard form.S(D) = (-2/45)(D - 80)² + 90.Expanding this:First, expand (D - 80)²: D² - 160D + 6400.Multiply by (-2/45):(-2/45)D² + (320/45)D - (12800/45) + 90.Simplify each term:-2/45 D² + (320/45)D - 12800/45 + 90.Convert 90 to 4050/45 to have a common denominator.So, S(D) = (-2/45)D² + (320/45)D - 12800/45 + 4050/45.Combine the constants:-12800 + 4050 = -8750.So, S(D) = (-2/45)D² + (320/45)D - 8750/45.Simplify fractions:-2/45 can stay as is.320/45 can be simplified: divide numerator and denominator by 5: 64/9.Wait, 320 divided by 5 is 64, and 45 divided by 5 is 9. So, 320/45 = 64/9.Similarly, -8750/45: Let's see, 8750 divided by 5 is 1750, 45 divided by 5 is 9. So, -1750/9.So, S(D) = (-2/45)D² + (64/9)D - 1750/9.But let me check if I did that correctly.Wait, when I expanded (-2/45)(D² - 160D + 6400), it should be:(-2/45)D² + (320/45)D - (12800/45). Then adding 90, which is 4050/45.So, combining constants: -12800/45 + 4050/45 = (-12800 + 4050)/45 = (-8750)/45.Yes, that's correct.So, S(D) = (-2/45)D² + (320/45)D - 8750/45.Simplify each term:-2/45 is already simplified.320/45: divide numerator and denominator by 5: 64/9.-8750/45: divide numerator and denominator by 5: -1750/9.So, S(D) = (-2/45)D² + (64/9)D - 1750/9.Alternatively, we can write all terms with denominator 45:-2/45 D² + (320/45)D - 8750/45.But to express a, b, c as constants, it's fine to leave them as fractions.So, a = -2/45, b = 64/9, c = -1750/9.Let me verify this with the given points.First, when D = 80, S should be 90.Plug D = 80 into S(D):S(80) = (-2/45)(80)^2 + (64/9)(80) - 1750/9.Calculate each term:(-2/45)(6400) = (-12800)/45 ≈ -284.444.(64/9)(80) = 5120/9 ≈ 568.888.-1750/9 ≈ -194.444.Adding them up: -284.444 + 568.888 - 194.444 ≈ (-284.444 - 194.444) + 568.888 ≈ (-478.888) + 568.888 ≈ 90. Correct.Now, when D = 50, S should be 50.S(50) = (-2/45)(2500) + (64/9)(50) - 1750/9.Calculate each term:(-2/45)(2500) = (-5000)/45 ≈ -111.111.(64/9)(50) = 3200/9 ≈ 355.555.-1750/9 ≈ -194.444.Adding them up: -111.111 + 355.555 - 194.444 ≈ (-111.111 - 194.444) + 355.555 ≈ (-305.555) + 355.555 ≈ 50. Correct.So, the constants are a = -2/45, b = 64/9, c = -1750/9.Alternatively, if we want to write them as decimals:a ≈ -0.0444, b ≈ 7.1111, c ≈ -194.444.But since the question doesn't specify, fractions are probably better.So, summarizing:1. t ≈ 20.992. a = -2/45, b = 64/9, c = -1750/9Final Answer1. The time ( t ) when the documentation quality reaches 75% of its maximum is boxed{20.99}.2. The constants are ( a = boxed{-dfrac{2}{45}} ), ( b = boxed{dfrac{64}{9}} ), and ( c = boxed{-dfrac{1750}{9}} ).</think>

🧠:<think>Okay, so I have this problem about a travel writer who's been to 12 countries. The first part is about finding the shortest possible route that visits each country exactly once and returns to the starting point. That sounds like the Traveling Salesman Problem (TSP). I remember TSP is a classic problem in graph theory where you have to find the shortest Hamiltonian circuit in a graph. Since the writer starts in Country A and needs to visit each country once before coming back, it's definitely a TSP.The problem mentions that each pair of countries has a direct flight connection, so the graph is a complete graph. The weights on the edges are the distances between countries, and it's symmetric, meaning the distance from A to B is the same as from B to A. Also, the triangle inequality holds, which means that for any three countries, the direct flight between two is never longer than going through the third. That's a useful property because it can sometimes help in finding approximate solutions, but since we're talking about the exact shortest route, we probably need an exact algorithm.But wait, 12 countries is a manageable number, but TSP is NP-hard, so exact solutions for larger numbers can be tough. However, 12 might still be feasible with dynamic programming or other exact methods. I think the Held-Karp algorithm is a dynamic programming approach for TSP, which can handle up to maybe 20 cities, so 12 should be okay.So, for part 1, the approach would be to model this as a TSP and apply an exact algorithm like Held-Karp to find the shortest Hamiltonian circuit. The distance matrix D is given, so we can use that to compute the distances between each pair of countries.Moving on to part 2. The writer wants to maximize their utility function U, which is the sum of weights multiplied by time spent in each country. The utility function is U = ∑ (wᵢ * tᵢ). The constraints are that the total time spent is less than or equal to 365 days, and the cost associated with each country is within a predefined budget.So, this sounds like a linear programming problem where we need to maximize U subject to constraints on time and cost. Let me break it down.First, the variables are the time spent in each country, t₁ to t₁₂. The objective is to maximize U, which is a linear combination of these variables with weights wᵢ. The constraints are:1. Total time: t₁ + t₂ + ... + t₁₂ ≤ 3652. For each country, the cost associated with spending tᵢ days is within the budget. I think this means that for each country i, the cost cᵢ * tᵢ ≤ budgetᵢ, where cᵢ is the cost per day in country i, and budgetᵢ is the maximum allowed cost for that country.Wait, the problem says "the cost associated with each country is within a predefined budget." So, maybe each country has its own budget, or there's a single budget for all. It's a bit unclear. If it's a single budget, then the total cost across all countries should be less than or equal to the budget. But since it says "each country," it might mean that for each country, the cost doesn't exceed its own budget. So, for each i, cᵢ * tᵢ ≤ budgetᵢ.Additionally, the time spent in each country can't be negative, so tᵢ ≥ 0 for all i.So, putting it all together, the linear programming problem would be:Maximize U = ∑ (wᵢ * tᵢ) for i = 1 to 12Subject to:1. ∑ tᵢ ≤ 3652. For each i, cᵢ * tᵢ ≤ budgetᵢ3. tᵢ ≥ 0 for all iThis is a standard linear program where we maximize a linear function subject to linear constraints. To solve this, we can use the simplex method or any linear programming solver.But wait, do we have the weights wᵢ? The problem says to formulate the optimization problem, so we don't need to solve it numerically, just set it up. So, we can define the decision variables, objective function, and constraints as above.I should also consider whether the time spent in each country is an integer or can be fractional. The problem doesn't specify, so I think it's safe to assume that tᵢ can be continuous variables, meaning the writer can spend a fraction of a day in a country if needed. If they had to be integers, it would become an integer linear program, but since it's not specified, I'll stick with continuous variables.So, in summary, for part 1, it's a TSP that can be solved with an exact algorithm given the size is manageable. For part 2, it's a linear program to maximize utility subject to time and cost constraints.Final Answer1. The shortest possible route is determined by solving the Traveling Salesman Problem, resulting in a Hamiltonian circuit with minimal total distance. The solution can be found using the Held-Karp algorithm.2. The optimization problem is formulated as a linear program:Maximize ( U = sum_{i=1}^{12} w_i t_i )Subject to:[sum_{i=1}^{12} t_i leq 365][c_i t_i leq text{budget}_i quad text{for each } i = 1, 2, ldots, 12][t_i geq 0 quad text{for each } i = 1, 2, ldots, 12]The final answers are:1. boxed{text{The shortest route is found using the Held-Karp algorithm.}}2. boxed{text{Maximize } U text{ subject to the given constraints.}}</think>