🧠:<think>Alright, so I have this problem about designing a culinary tour that maximizes the number of unique dishes. Let me try to break it down step by step.First, the problem is divided into two parts. The first part is about selecting exactly k restaurants out of n to maximize the total unique dishes. The second part introduces the complication that some dishes are shared between restaurants, so we have to adjust our selection to account for overlaps.Starting with the first part: I need to select k restaurants such that the sum of their unique dishes is maximized. Each restaurant has a different number of unique dishes, denoted as d_i for the i-th restaurant. So, the straightforward approach is to pick the top k restaurants with the highest d_i values. That makes sense because each restaurant contributes its own unique dishes, and since they are all different, adding more high-value restaurants should give the maximum total.But wait, is that always the case? What if some restaurants have overlapping dishes? Oh, but in the first part, I think we're assuming that all dishes are unique across all restaurants. So, no overlaps. Therefore, the total unique dishes would just be the sum of the d_i's of the selected k restaurants. So, the algorithm is simply to sort the restaurants in descending order of d_i and pick the top k. The mathematical formulation would be an optimization problem where we maximize the sum of d_i for the selected k restaurants.Let me write that down. Let’s define a binary variable x_i, where x_i = 1 if we select the i-th restaurant, and 0 otherwise. Then, the objective function is to maximize the sum over all i of d_i * x_i. The constraint is that the sum of x_i equals k, and each x_i is either 0 or 1. So, in mathematical terms:Maximize Σ (d_i * x_i) for i = 1 to nSubject to:Σ x_i = kx_i ∈ {0,1} for all iThat seems right. So, the first part is a straightforward maximum selection problem.Moving on to the second part. Now, we have a matrix A where A_ij = 1 if restaurants i and j share at least one dish. So, if we select both i and j, the total unique dishes won't just be d_i + d_j, but less because of the overlap. The problem now is to modify the algorithm to account for this overlap and determine how it affects the total unique dishes.Hmm, so this complicates things because now the total unique dishes depend not just on the sum of d_i's but also on the overlaps between the selected restaurants. So, the total unique dishes would be the sum of d_i's minus the overlaps. But how do we quantify the overlaps?Wait, the matrix A tells us whether two restaurants share at least one dish. But it doesn't tell us how many dishes they share. So, for the purpose of this problem, I think we can only know that if A_ij = 1, then there is at least one overlapping dish. But we don't know the exact number. So, how does this affect the total unique dishes?If two restaurants share at least one dish, then the total unique dishes from both would be less than the sum of their individual dishes. But since we don't know how many dishes they share, we can't compute the exact reduction. However, in the worst case, each pair of restaurants sharing a dish would reduce the total unique dishes by 1. But since we don't know the exact number, maybe we have to consider the minimum possible reduction or the maximum possible?Wait, the problem says "modify the algorithm from sub-problem 1 to account for this overlap." So, perhaps we need to adjust the selection process to minimize the overlaps, thereby maximizing the total unique dishes.But without knowing the exact number of overlapping dishes, it's tricky. Maybe we can model the problem by considering the overlaps as a penalty. So, when selecting restaurants, if two selected restaurants share a dish, we have to subtract 1 from the total. But again, without knowing how many dishes they share, we can't be precise.Alternatively, maybe the problem is expecting us to consider that each pair of restaurants with A_ij = 1 contributes a certain amount of overlap, but since we don't know the exact number, perhaps we have to make an assumption or find a way to model it.Wait, perhaps the problem is expecting us to consider the inclusion-exclusion principle. The total unique dishes would be the sum of d_i minus the sum of overlaps between each pair, plus the overlaps between each triple, and so on. But since we don't have information about how many dishes are shared between multiple restaurants, this might not be feasible.Alternatively, maybe the problem is simplified by assuming that each pair of restaurants shares exactly one dish. Then, the total unique dishes would be the sum of d_i minus the number of overlapping pairs. But the problem doesn't specify that, so I'm not sure.Alternatively, perhaps the problem is expecting us to realize that without knowing the exact overlaps, we can't compute the exact total unique dishes, but we can adjust our selection to minimize the number of overlaps, thereby maximizing the total unique dishes.So, in that case, the optimization problem would not only aim to maximize the sum of d_i but also to minimize the number of overlapping pairs among the selected restaurants.So, the objective function would be a combination of maximizing the sum of d_i and minimizing the number of overlapping pairs. But how do we combine these two objectives? Maybe we can assign a weight to the overlap penalty.Alternatively, perhaps the problem is expecting us to modify the selection process to avoid selecting restaurants that share dishes, but that might not always be possible if k is large.Wait, but the problem says "modify the algorithm from sub-problem 1 to account for this overlap." So, in the first problem, we just selected the top k restaurants by d_i. Now, we have to adjust that selection to account for overlaps.So, perhaps the algorithm needs to consider both the number of dishes and the overlaps. So, it's a trade-off between selecting a restaurant with a high d_i but which overlaps with many others, versus selecting a restaurant with a slightly lower d_i but which overlaps less.This sounds like a problem that can be modeled as a graph where nodes are restaurants and edges represent overlaps (A_ij = 1). Then, selecting k nodes such that the sum of their d_i is maximized, but also considering the edges (overlaps) to minimize the total overlaps.But since overlaps reduce the total unique dishes, perhaps we can model this as a graph and find a maximum weight independent set of size k, where the weight of each node is d_i, and edges represent overlaps. However, finding a maximum weight independent set is NP-hard, so for large n, this might not be feasible.Alternatively, maybe we can use a greedy approach. Start by selecting the restaurant with the highest d_i, then select the next restaurant with the highest d_i that doesn't overlap with the already selected ones, and so on. But this might not always give the optimal solution because sometimes selecting a slightly lower d_i restaurant might allow for more non-overlapping restaurants later.Alternatively, perhaps we can model this as a problem where the total unique dishes is the sum of d_i minus the number of overlapping pairs. So, the total unique dishes T = sum(d_i for selected) - sum(A_ij for all i < j selected). So, the problem becomes selecting k restaurants to maximize T.But since A_ij is 1 if they share at least one dish, the total unique dishes would be sum(d_i) - C, where C is the number of overlapping pairs. So, our objective is to maximize sum(d_i) - C.But how do we model this? It's a bit more complex because now the objective function depends on both the sum of d_i and the number of overlapping pairs.Alternatively, perhaps we can think of it as a modified sum where each overlapping pair reduces the total by 1. So, the total unique dishes is sum(d_i) - (number of overlapping pairs). Therefore, to maximize this, we need to maximize sum(d_i) while minimizing the number of overlapping pairs.So, the optimization problem becomes:Maximize [sum(d_i) - C]Where C is the number of overlapping pairs among the selected k restaurants.But how do we model this in an optimization framework? It's a bit more involved.Alternatively, perhaps we can model this as a quadratic optimization problem because the overlapping pairs contribute a quadratic term.Let me think. Let’s denote x_i as before, a binary variable indicating whether restaurant i is selected. Then, the total unique dishes would be sum(d_i x_i) - sum(A_ij x_i x_j for i < j). Because for each pair i < j, if both are selected (x_i x_j = 1), and A_ij = 1, then we subtract 1.So, the objective function becomes:Maximize Σ d_i x_i - Σ A_ij x_i x_jSubject to:Σ x_i = kx_i ∈ {0,1}This is a quadratic unconstrained binary optimization problem, which is more complex than the linear one in the first part.So, the mathematical formulation changes from a linear problem to a quadratic one because of the overlapping pairs.Therefore, the algorithm needs to account for both the individual contributions of each restaurant and the penalties from overlapping pairs.This makes the problem more challenging because quadratic terms are harder to handle, especially for large n. However, for small n, exact methods like integer programming could be used, while for larger n, heuristic or approximation algorithms might be necessary.So, in summary, the first part is a simple maximum selection problem, while the second part introduces overlaps, turning it into a quadratic optimization problem where we need to balance the sum of dishes against the number of overlapping pairs.I think that's the gist of it. Now, let me try to formalize this.</think>

🧠:<think>First, I'll tackle the first problem. The relative is traveling a total of 3,600 miles using a combination of biking and hitchhiking. Let's denote the distance the relative bikes as ( x ) miles. Consequently, the distance hitchhiked will be ( 3600 - x ) miles.The time spent biking can be calculated by dividing the distance by the biking speed: ( frac{x}{12} ) hours. Similarly, the time spent hitchhiking is ( frac{3600 - x}{45} ) hours. The total time for the journey is given as 100 hours, so I can set up the equation:[frac{x}{12} + frac{3600 - x}{45} = 100]To solve for ( x ), I'll find a common denominator for the fractions, which is 180. Multiplying each term by 180 to eliminate the denominators:[15x + 4(3600 - x) = 18000]Expanding and simplifying:[15x + 14400 - 4x = 18000][11x + 14400 = 18000][11x = 3600][x = frac{3600}{11} approx 327.27 text{ miles}]So, the relative bikes approximately 327.27 miles.Next, for the second problem, the couple and the relative are meeting every 600 miles. The couple is driving at 60 miles per hour, and the relative is biking at 12 miles per hour. The time it takes for each to reach the first checkpoint can be calculated by dividing the distance by their respective speeds:- Couple's time: ( frac{600}{60} = 10 ) hours- Relative's time: ( frac{600}{12} = 50 ) hoursSince the couple reaches the checkpoint much faster, they will have to wait for the relative. The time interval between meetings is determined by the relative's slower pace. After the first meeting, both the couple and the relative will continue to the next checkpoint, maintaining their speeds. Therefore, the time intervals between each checkpoint meeting will be consistent and equal to the relative's time to reach each checkpoint, which is 50 hours.</think>

🧠:<think>Okay, so I have this problem about a politician analyzing the impact of criminal justice reform on case resolution times. There are two parts: first, testing whether the reform has reduced the average resolution time, and second, constructing a confidence interval for the variance after the reform. Let me take this step by step.Starting with part 1: Testing the hypothesis that the reform has reduced the average resolution time. Before the reform, the average time was 180 days with a standard deviation of 30 days, and it followed a normal distribution. After the reform, they took a sample of 100 cases with a sample mean of 160 days and a sample variance of 400 days². They want to test this at a 5% significance level.Alright, so hypothesis testing. I remember that when testing means, we usually set up a null hypothesis and an alternative hypothesis. Since the politician wants to know if the reform reduced the average time, this is a one-tailed test. So, the null hypothesis (H0) would be that the mean is still 180 days, and the alternative hypothesis (H1) is that the mean is less than 180 days.H0: μ = 180 H1: μ < 180Given that the sample size is 100, which is pretty large, and they mentioned the sample follows a normal distribution, so we can use the z-test here. The formula for the z-test statistic is:z = (x̄ - μ) / (σ / sqrt(n))Wait, but hold on. The population standard deviation before the reform was 30 days, but after the reform, the variance is given as 400 days² in the sample. Hmm, so do we use the population standard deviation or the sample standard deviation for the test?Wait, in the problem statement, it says "the variance of the time to resolve cases after the reform is unknown." So, actually, for part 1, we don't know the population variance after the reform. Hmm, but in the first part, the question is about testing the mean, so if the population variance is unknown, we might need to use a t-test instead. But wait, the sample size is 100, which is large, so the z-test is still appropriate because the Central Limit Theorem tells us that the sampling distribution will be approximately normal.But wait, the variance given is 400, which is the sample variance. So, the sample standard deviation is sqrt(400) = 20 days. So, in this case, since the population variance is unknown, we use the sample standard deviation in the formula.So, the formula becomes:z = (x̄ - μ) / (s / sqrt(n))Where x̄ is 160, μ is 180, s is 20, and n is 100.Plugging in the numbers:z = (160 - 180) / (20 / sqrt(100)) z = (-20) / (20 / 10) z = (-20) / 2 z = -10Wait, that seems like a huge z-score. Let me double-check my calculations.x̄ = 160 μ = 180 s = sqrt(400) = 20 n = 100 Standard error (SE) = s / sqrt(n) = 20 / 10 = 2 z = (160 - 180) / 2 = (-20)/2 = -10Yeah, that's correct. A z-score of -10 is extremely low. The critical z-value for a one-tailed test at 5% significance level is -1.645. Since our calculated z-score is -10, which is much less than -1.645, we reject the null hypothesis.So, the conclusion is that there's significant evidence at the 5% level that the reform has reduced the average resolution time.Moving on to part 2: Constructing a 95% confidence interval for the population variance after the reform. We have the same sample data: n=100, sample variance s²=400.I remember that for confidence intervals of variance, we use the chi-square distribution. The formula is:[(n - 1)s² / χ²(α/2, n - 1), (n - 1)s² / χ²(1 - α/2, n - 1)]Where α is 0.05 for a 95% confidence interval, and degrees of freedom (df) is n - 1 = 99.So, first, we need the chi-square critical values for 99 degrees of freedom at α/2 = 0.025 and 1 - α/2 = 0.975.I don't remember the exact values, but I can look them up or approximate. However, since this is a thought process, I'll note that for large degrees of freedom, the chi-square distribution approximates a normal distribution. But for exact values, I might need to refer to a table or use a calculator.But since I don't have a table here, I can recall that for df=99, the critical values are approximately:χ²(0.025, 99) ≈ 73.361 χ²(0.975, 99) ≈ 128.419Wait, is that right? Let me think. For df=100, the critical values are around 74.22 and 129.58. So, for df=99, it's slightly less. Maybe 73.36 and 128.42.Assuming those values, let's compute the confidence interval.Lower limit: (99 * 400) / 128.419 Upper limit: (99 * 400) / 73.361Calculating the numerator first: 99 * 400 = 39,600Lower limit: 39,600 / 128.419 ≈ 308.3 Upper limit: 39,600 / 73.361 ≈ 540.0So, the 95% confidence interval for the population variance is approximately (308.3, 540.0) days².Now, comparing this to the variance before the reform. Before the reform, the standard deviation was 30 days, so variance was 900 days². The confidence interval after the reform is (308.3, 540.0), which is much lower than 900. So, the variance has decreased, meaning the variability of case resolution times has reduced.Wait a second, but hold on. The sample variance is 400, which is less than 900, but the confidence interval is (308.3, 540.0). So, the entire interval is below 900, which suggests that the population variance after the reform is significantly lower than before.Therefore, the variability has decreased, which is a positive outcome as it indicates more consistent case resolution times.Let me just recap to make sure I didn't make any mistakes. For the hypothesis test, we used the z-test because the sample size is large, even though the population variance was unknown, we used the sample variance. The z-score was extremely low, leading us to reject the null hypothesis. For the confidence interval, using the chi-square distribution, we found that the variance is significantly lower than before, so the variability has decreased.I think that's solid. I don't see any errors in the calculations or reasoning.Final Answer1. The hypothesis test leads to rejecting the null hypothesis, indicating the reform has reduced the average resolution time. The result is boxed{z = -10}.2. The 95% confidence interval for the population variance is approximately (308.3, 540.0) days², indicating reduced variability. The interval is boxed{(308.3, 540.0)}.</think>

🧠:<think>Okay, so I have two problems to solve here. The first one is about finding the coefficients of a quadratic function that models tuition fees based on the rank of universities. The second problem is about calculating the growth rate of an exponential function for salary increase and then determining the salary after 10 years. Let me tackle them one by one.Starting with the first problem. I need to find the quadratic function ( T(x) = ax^2 + bx + c ) where ( x ) is the rank of the university, and ( T(x) ) is the tuition fee in thousands of dollars. I have three data points: University 1 (rank 1, fee 60k), University 2 (rank 2, fee 50k), and University 5 (rank 5, fee 75k). So, these points should satisfy the quadratic equation.Let me write down the equations based on these points.For University 1 (x=1, T=60):( a(1)^2 + b(1) + c = 60 )Simplifying: ( a + b + c = 60 ) -- Equation 1For University 2 (x=2, T=50):( a(2)^2 + b(2) + c = 50 )Simplifying: ( 4a + 2b + c = 50 ) -- Equation 2For University 5 (x=5, T=75):( a(5)^2 + b(5) + c = 75 )Simplifying: ( 25a + 5b + c = 75 ) -- Equation 3Now, I have a system of three equations:1. ( a + b + c = 60 )2. ( 4a + 2b + c = 50 )3. ( 25a + 5b + c = 75 )I need to solve for a, b, and c. Let me subtract Equation 1 from Equation 2 to eliminate c.Equation 2 - Equation 1:( (4a + 2b + c) - (a + b + c) = 50 - 60 )Simplify:( 3a + b = -10 ) -- Equation 4Similarly, subtract Equation 2 from Equation 3:Equation 3 - Equation 2:( (25a + 5b + c) - (4a + 2b + c) = 75 - 50 )Simplify:( 21a + 3b = 25 ) -- Equation 5Now, I have two equations:4. ( 3a + b = -10 )5. ( 21a + 3b = 25 )Let me solve Equation 4 for b:From Equation 4: ( b = -10 - 3a )Now, substitute this into Equation 5:( 21a + 3(-10 - 3a) = 25 )Simplify:( 21a - 30 - 9a = 25 )Combine like terms:( 12a - 30 = 25 )Add 30 to both sides:( 12a = 55 )Divide by 12:( a = 55/12 )Hmm, that's approximately 4.5833. Let me keep it as a fraction for accuracy.So, ( a = 55/12 ). Now, substitute back into Equation 4 to find b.( 3*(55/12) + b = -10 )Simplify:( 165/12 + b = -10 )Convert 165/12 to decimal: 13.75So, 13.75 + b = -10Subtract 13.75:( b = -10 - 13.75 = -23.75 )Which is -95/4 as a fraction.Wait, let me check that again. 165 divided by 12 is indeed 13.75. So, 13.75 + b = -10, so b = -23.75.Alternatively, in fractions: 165/12 is 55/4, so:( 55/4 + b = -10 )Convert -10 to -40/4:( 55/4 + b = -40/4 )Subtract 55/4:( b = -40/4 - 55/4 = -95/4 )Yes, that's correct. So, b = -95/4.Now, go back to Equation 1 to find c.Equation 1: ( a + b + c = 60 )Substitute a and b:( 55/12 + (-95/4) + c = 60 )First, let me convert all terms to twelfths to add them up.55/12 is already in twelfths.-95/4 is equal to -285/12.So:55/12 - 285/12 + c = 60Combine the fractions:(55 - 285)/12 + c = 60(-230)/12 + c = 60Simplify -230/12: that's -115/6 ≈ -19.1667So:-115/6 + c = 60Add 115/6 to both sides:c = 60 + 115/6Convert 60 to sixths: 60 = 360/6So:c = 360/6 + 115/6 = 475/6 ≈ 79.1667So, summarizing:a = 55/12 ≈ 4.5833b = -95/4 = -23.75c = 475/6 ≈ 79.1667Let me check these values with the original equations to make sure.First, Equation 1: a + b + c55/12 - 95/4 + 475/6Convert all to twelfths:55/12 - 285/12 + 950/12(55 - 285 + 950)/12 = (55 + 665)/12 = 720/12 = 60. Correct.Equation 2: 4a + 2b + c4*(55/12) + 2*(-95/4) + 475/6Simplify:220/12 - 190/4 + 475/6Convert to twelfths:220/12 - 570/12 + 950/12(220 - 570 + 950)/12 = (220 + 380)/12 = 600/12 = 50. Correct.Equation 3: 25a + 5b + c25*(55/12) + 5*(-95/4) + 475/6Simplify:1375/12 - 475/4 + 475/6Convert to twelfths:1375/12 - 1425/12 + 950/12(1375 - 1425 + 950)/12 = (1375 + 525)/12 = 1900/12 ≈ 158.3333Wait, that's not 75. Hmm, that's a problem.Wait, 1900/12 is approximately 158.3333, but Equation 3 should equal 75.Wait, so that means I made a mistake in my calculations.Let me check Equation 3 again.25a + 5b + c25*(55/12) + 5*(-95/4) + 475/6Calculate each term:25*(55/12) = (25*55)/12 = 1375/12 ≈ 114.58335*(-95/4) = (-475)/4 ≈ -118.75475/6 ≈ 79.1667Now, add them up:114.5833 - 118.75 + 79.1667 ≈ (114.5833 + 79.1667) - 118.75 ≈ 193.75 - 118.75 = 75. Correct.Wait, so in fractions:1375/12 - 475/4 + 475/6Convert all to twelfths:1375/12 - (475*3)/12 + (475*2)/12Which is:1375/12 - 1425/12 + 950/12Now, 1375 - 1425 = -50, so -50 + 950 = 900900/12 = 75. Correct.Wait, earlier I thought 1375 - 1425 + 950 was 1900, but that's incorrect. It's actually 1375 - 1425 = -50, then -50 + 950 = 900. So, 900/12 = 75. Correct.So, my initial calculation was right, but I made a mistake when adding 1375 - 1425 + 950. I thought it was 1900, but it's actually 900. So, all equations are satisfied.Therefore, the coefficients are:a = 55/12b = -95/4c = 475/6Alternatively, in decimal form:a ≈ 4.5833b ≈ -23.75c ≈ 79.1667So, that's the quadratic function.Now, moving on to the second problem. Alex expects a salary increase modeled by an exponential growth function ( S(t) = S_0 e^{kt} ). The initial salary ( S_0 ) is 50,000, and the salary doubles in 5 years. I need to find the growth rate ( k ) and then determine the salary 10 years post-graduation.First, let's note that the salary doubles in 5 years. So, at t=5, S(5) = 2*S_0 = 2*50,000 = 100,000.So, plug into the equation:( 100,000 = 50,000 e^{5k} )Divide both sides by 50,000:( 2 = e^{5k} )Take the natural logarithm of both sides:( ln(2) = 5k )Therefore, ( k = ln(2)/5 )Compute the value:( ln(2) ≈ 0.6931 )So, ( k ≈ 0.6931 / 5 ≈ 0.1386 ) per year.So, the growth rate k is approximately 0.1386 or 13.86% per year.Now, to find the salary after 10 years, plug t=10 into the equation:( S(10) = 50,000 e^{0.1386*10} )Simplify the exponent:0.1386*10 = 1.386So, ( S(10) = 50,000 e^{1.386} )Compute ( e^{1.386} ). Since ( ln(4) ≈ 1.386 ), so ( e^{1.386} ≈ 4 ).Therefore, ( S(10) ≈ 50,000 * 4 = 200,000 ).Alternatively, using a calculator:( e^{1.386} ≈ 4.000 ), so yes, it's exactly 4 because 1.386 is ln(4). So, S(10) is exactly 200,000.Wait, let me verify:Since k = ln(2)/5, then after 10 years, the exponent is 10*(ln(2)/5) = 2 ln(2) = ln(4). So, e^{ln(4)} = 4. Therefore, S(10) = 50,000 * 4 = 200,000.Yes, that's correct.So, the growth rate k is ln(2)/5, which is approximately 0.1386, and the salary after 10 years is 200,000.Let me recap:1. Quadratic function coefficients: a = 55/12 ≈ 4.5833 b = -95/4 ≈ -23.75 c = 475/6 ≈ 79.16672. Exponential growth: Growth rate k = ln(2)/5 ≈ 0.1386 Salary after 10 years = 200,000I think that's all. Let me just double-check the quadratic function with the given points to ensure there are no calculation errors.For University 1 (x=1):T(1) = (55/12)(1)^2 + (-95/4)(1) + 475/6= 55/12 - 95/4 + 475/6Convert to twelfths:55/12 - 285/12 + 950/12= (55 - 285 + 950)/12= 720/12 = 60. Correct.For University 2 (x=2):T(2) = (55/12)(4) + (-95/4)(2) + 475/6= 220/12 - 190/4 + 475/6Convert to twelfths:220/12 - 570/12 + 950/12= (220 - 570 + 950)/12= 600/12 = 50. Correct.For University 5 (x=5):T(5) = (55/12)(25) + (-95/4)(5) + 475/6= 1375/12 - 475/4 + 475/6Convert to twelfths:1375/12 - 1425/12 + 950/12= (1375 - 1425 + 950)/12= 900/12 = 75. Correct.All points satisfy the quadratic function, so my coefficients are correct.For the exponential growth, since the salary doubles every 5 years, after 10 years it should quadruple, which is exactly what we got: 200,000. So, that's consistent.Therefore, I'm confident with my solutions.Final Answer1. The coefficients are ( a = boxed{dfrac{55}{12}} ), ( b = boxed{-dfrac{95}{4}} ), and ( c = boxed{dfrac{475}{6}} ).2. The growth rate ( k ) is ( boxed{dfrac{ln 2}{5}} ) and Alex's salary 10 years post-graduation is ( boxed{200000} ) dollars.</think>