🧠:<think>Alright, so I have this problem about a Sharia law scholar analyzing intellectual property rights, specifically dealing with a patented algorithm in financial technology. The problem has two parts: solving a differential equation for revenue and then figuring out the maximum allowable total royalty distributed over the first year. Let me try to break this down step by step.Starting with part 1: The revenue ( R(t) ) is given by the differential equation ( frac{dR}{dt} = kR(t) ln(R(t)) ), with the initial condition ( R(0) = R_0 ). I need to solve this differential equation to find ( R(t) ).Hmm, this looks like a separable equation. So, I should be able to rewrite it so that all terms involving ( R ) are on one side and all terms involving ( t ) are on the other. Let me try that.So, starting with:[frac{dR}{dt} = kR ln(R)]I can rewrite this as:[frac{dR}{R ln(R)} = k dt]Okay, now I need to integrate both sides. The left side is with respect to ( R ), and the right side is with respect to ( t ).Let me focus on the left integral first: ( int frac{1}{R ln(R)} dR ). Hmm, I think a substitution might help here. Let me set ( u = ln(R) ). Then, ( du = frac{1}{R} dR ). So, substituting, the integral becomes:[int frac{1}{u} du]Which is straightforward. The integral of ( 1/u ) is ( ln|u| + C ), so substituting back:[ln|ln(R)| + C]So, putting it all together, the integral of the left side is ( ln(ln(R)) ) (assuming ( ln(R) ) is positive, which makes sense for revenue). The integral of the right side is ( int k dt = kt + C ).Therefore, combining both sides:[ln(ln(R)) = kt + C]Now, I need to solve for ( R ). Let me exponentiate both sides to get rid of the natural log:[ln(R) = e^{kt + C} = e^{kt} cdot e^C]Let me denote ( e^C ) as another constant, say ( C_1 ), since ( C ) is just a constant of integration. So:[ln(R) = C_1 e^{kt}]Now, exponentiate both sides again to solve for ( R ):[R = e^{C_1 e^{kt}}]Hmm, that seems a bit complicated, but let's see if we can find ( C_1 ) using the initial condition ( R(0) = R_0 ).At ( t = 0 ):[R(0) = e^{C_1 e^{0}} = e^{C_1} = R_0]So, ( e^{C_1} = R_0 ), which implies that ( C_1 = ln(R_0) ).Substituting back into the equation for ( R(t) ):[R(t) = e^{ln(R_0) e^{kt}} = e^{ln(R_0)}^{e^{kt}} = R_0^{e^{kt}}]Wait, is that right? Let me double-check the exponentiation. So, ( e^{C_1 e^{kt}} ) becomes ( e^{ln(R_0) e^{kt}} ). Since ( e^{ln(R_0)} = R_0 ), then ( e^{ln(R_0) e^{kt}} = R_0^{e^{kt}} ). Yeah, that seems correct.So, the solution is ( R(t) = R_0^{e^{kt}} ). Hmm, that seems a bit non-intuitive, but let's see if it makes sense.Let me test it with ( t = 0 ): ( R(0) = R_0^{e^{0}} = R_0^1 = R_0 ), which matches the initial condition. Good.Also, let's plug it back into the differential equation to verify.Compute ( frac{dR}{dt} ):First, ( R(t) = R_0^{e^{kt}} ). Let me take the natural log to make differentiation easier.Let ( ln(R) = e^{kt} ln(R_0) ).Differentiating both sides with respect to ( t ):( frac{1}{R} frac{dR}{dt} = k e^{kt} ln(R_0) ).Multiply both sides by ( R ):( frac{dR}{dt} = R cdot k e^{kt} ln(R_0) ).But ( R = R_0^{e^{kt}} ), so ( ln(R) = e^{kt} ln(R_0) ), which means ( ln(R) = ln(R_0^{e^{kt}}) ). Therefore, ( ln(R) = e^{kt} ln(R_0) ).So, substituting back into the expression for ( frac{dR}{dt} ):( frac{dR}{dt} = R cdot k ln(R) ).Which is exactly the original differential equation. So, that checks out. Okay, so part 1 is solved: ( R(t) = R_0^{e^{kt}} ).Moving on to part 2: The scholar needs to ensure that the total royalties distributed over the first year (from ( t = 0 ) to ( t = 1 )) do not exceed 50% of the revenue generated during that period. So, I need to calculate the maximum allowable total royalty ( T ) in terms of ( R_0 ) and ( k ).First, let's understand what is being asked. The total revenue generated over the first year is the integral of ( R(t) ) from 0 to 1. Then, 50% of that total revenue is the maximum royalty that can be distributed. So, ( T = 0.5 times ) (total revenue over [0,1]).Therefore, I need to compute:[T = 0.5 times int_{0}^{1} R(t) dt = 0.5 times int_{0}^{1} R_0^{e^{kt}} dt]So, my task is to compute this integral and express ( T ) in terms of ( R_0 ) and ( k ).Hmm, integrating ( R_0^{e^{kt}} ) with respect to ( t ) from 0 to 1. That seems a bit tricky. Let me think about substitution.Let me denote ( u = e^{kt} ). Then, ( du/dt = k e^{kt} ), so ( du = k e^{kt} dt ), which implies ( dt = du/(k e^{kt}) = du/(k u) ).But let's see if that helps. Let me write the integral:[int R_0^{e^{kt}} dt = int R_0^{u} cdot frac{du}{k u}]So, substituting ( u = e^{kt} ), the integral becomes:[frac{1}{k} int frac{R_0^{u}}{u} du]Hmm, that doesn't look straightforward. The integral of ( R_0^{u}/u ) du is not an elementary function, as far as I know. Maybe I need to consider a different substitution or perhaps express it in terms of a special function.Wait, let's think about the integral ( int frac{R_0^{u}}{u} du ). Let me rewrite ( R_0^{u} ) as ( e^{u ln(R_0)} ). So, the integral becomes:[int frac{e^{u ln(R_0)}}{u} du]Which is similar to the exponential integral function, denoted as ( text{Ei}(x) ). The exponential integral is defined as:[text{Ei}(x) = - int_{-x}^{infty} frac{e^{-t}}{t} dt]But for positive real numbers, it can also be expressed as:[text{Ei}(x) = gamma + ln(x) + int_{0}^{x} frac{e^{t} - 1}{t} dt]Where ( gamma ) is the Euler-Mascheroni constant. However, I'm not sure if that's helpful here. Alternatively, perhaps I can express the integral in terms of the exponential integral function.Given that:[int frac{e^{a u}}{u} du = text{Ei}(a u) + C]Where ( a ) is a constant. So, in our case, ( a = ln(R_0) ). Therefore, the integral becomes:[int frac{e^{u ln(R_0)}}{u} du = text{Ei}(u ln(R_0)) + C]Therefore, going back to our substitution, the integral ( int R_0^{e^{kt}} dt ) becomes:[frac{1}{k} left[ text{Ei}(u ln(R_0)) right]_{u = e^{0}}^{u = e^{k cdot 1}} = frac{1}{k} left[ text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0)) right]]So, putting it all together, the total revenue ( int_{0}^{1} R(t) dt ) is:[frac{1}{k} left[ text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0)) right]]Therefore, the maximum allowable total royalty ( T ) is 50% of this, so:[T = 0.5 times frac{1}{k} left[ text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0)) right]]Hmm, that's an expression in terms of the exponential integral function. However, the problem asks to express ( T ) in terms of ( R_0 ) and ( k ). The exponential integral is a special function, so unless there's a way to simplify this further or express it in terms of elementary functions, this might be as far as we can go.Wait, let me think again. Maybe I made a substitution that complicates things. Let me try a different substitution.Let me consider ( v = kt ). Then, ( dv = k dt ), so ( dt = dv/k ). But I don't know if that helps because the exponent is ( e^{kt} ), which would be ( e^{v} ). So, the integral becomes:[int R_0^{e^{v}} cdot frac{dv}{k}]Which is the same as before, leading us back to the exponential integral. So, perhaps there's no way around it, and the integral indeed requires the exponential integral function.Alternatively, maybe we can express the integral in terms of the original variables without substitution. Let me see.Wait, another thought: perhaps express the integral in terms of the logarithm of R(t). Since ( R(t) = R_0^{e^{kt}} ), then ( ln(R(t)) = e^{kt} ln(R_0) ). Let me denote ( y = ln(R(t)) ), so ( y = e^{kt} ln(R_0) ). Then, ( dy/dt = k e^{kt} ln(R_0) ), which is ( dy = k e^{kt} ln(R_0) dt ). Hmm, not sure if that helps.Alternatively, perhaps integrating ( R(t) ) directly. Let me write ( R(t) = e^{ln(R_0) e^{kt}} ). So, the integral becomes:[int_{0}^{1} e^{ln(R_0) e^{kt}} dt]Let me make a substitution: let ( u = e^{kt} ). Then, ( du = k e^{kt} dt ), so ( dt = du/(k u) ). When ( t = 0 ), ( u = 1 ); when ( t = 1 ), ( u = e^{k} ). So, the integral becomes:[int_{1}^{e^{k}} e^{ln(R_0) u} cdot frac{du}{k u} = frac{1}{k} int_{1}^{e^{k}} frac{e^{ln(R_0) u}}{u} du]Which is the same as before. So, this substitution doesn't seem to help in terms of simplifying the integral into elementary functions.Therefore, it seems that the integral cannot be expressed in terms of elementary functions and requires the exponential integral function. So, the expression for ( T ) is:[T = frac{1}{2k} left[ text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0)) right]]But the problem says to express ( T ) in terms of ( R_0 ) and ( k ). Since the exponential integral is a known function, perhaps this is acceptable. However, if the problem expects an expression without special functions, maybe I need to reconsider.Wait, another approach: perhaps approximate the integral numerically? But the problem doesn't specify that; it just asks to express ( T ) in terms of ( R_0 ) and ( k ). So, unless there's a way to express it in terms of elementary functions, which I don't see, I think the answer has to involve the exponential integral.Alternatively, maybe I made a mistake in solving the differential equation. Let me double-check part 1 again.Starting with ( frac{dR}{dt} = k R ln(R) ). Separating variables:( frac{dR}{R ln(R)} = k dt ). Integrating both sides:Left side: ( int frac{1}{R ln(R)} dR ). Let ( u = ln(R) ), ( du = 1/R dR ). So, integral becomes ( int frac{1}{u} du = ln|u| + C = ln|ln(R)| + C ).Right side: ( int k dt = kt + C ).So, ( ln(ln(R)) = kt + C ). Exponentiating both sides:( ln(R) = e^{kt + C} = e^{kt} e^{C} ). Let ( e^{C} = C_1 ), so ( ln(R) = C_1 e^{kt} ). Exponentiating again:( R = e^{C_1 e^{kt}} ). Using initial condition ( R(0) = R_0 ):( R_0 = e^{C_1 e^{0}} = e^{C_1} ), so ( C_1 = ln(R_0) ). Therefore, ( R(t) = e^{ln(R_0) e^{kt}} = R_0^{e^{kt}} ). Yep, that's correct.So, part 1 is correct, and the integral in part 2 indeed leads to the exponential integral function. Therefore, unless there's a different interpretation of the problem, I think the answer must involve ( text{Ei} ).Alternatively, maybe the problem expects a different approach. Let me think about whether the revenue function can be expressed differently or if there's a substitution that can make the integral solvable in terms of elementary functions.Wait, another idea: perhaps express ( R(t) ) in terms of its logarithm and then integrate. Let me denote ( y(t) = ln(R(t)) ). Then, ( y(t) = ln(R(t)) = e^{kt} ln(R_0) ). So, ( R(t) = e^{y(t)} ).But integrating ( R(t) ) from 0 to 1 is ( int_{0}^{1} e^{y(t)} dt ). Since ( y(t) = e^{kt} ln(R_0) ), it's ( int_{0}^{1} e^{e^{kt} ln(R_0)} dt ). Which is the same as before, leading to the same integral involving the exponential integral.So, I don't think that helps. Maybe another substitution? Let me try ( z = e^{kt} ). Then, ( dz = k e^{kt} dt ), so ( dt = dz/(k z) ). When ( t = 0 ), ( z = 1 ); when ( t = 1 ), ( z = e^{k} ). So, the integral becomes:[int_{1}^{e^{k}} e^{ln(R_0) z} cdot frac{dz}{k z} = frac{1}{k} int_{1}^{e^{k}} frac{R_0^{z}}{z} dz]Which is again the same integral as before. So, no progress there.Alternatively, perhaps express ( R_0^{z} ) as ( e^{z ln(R_0)} ), so the integral becomes:[frac{1}{k} int_{1}^{e^{k}} frac{e^{z ln(R_0)}}{z} dz = frac{1}{k} int_{1}^{e^{k}} frac{e^{a z}}{z} dz]Where ( a = ln(R_0) ). This is the definition of the exponential integral function:[text{Ei}(x) = - int_{-x}^{infty} frac{e^{-t}}{t} dt]But for positive ( x ), it can also be expressed as:[text{Ei}(x) = gamma + ln(x) + int_{0}^{x} frac{e^{t} - 1}{t} dt]However, our integral is from 1 to ( e^{k} ), so we can express it as:[int_{1}^{e^{k}} frac{e^{a z}}{z} dz = text{Ei}(a e^{k}) - text{Ei}(a)]Where ( a = ln(R_0) ). Therefore, the integral becomes:[frac{1}{k} [text{Ei}(a e^{k}) - text{Ei}(a)] = frac{1}{k} [text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0))]]So, that's consistent with what I had earlier. Therefore, the total revenue is:[frac{1}{k} [text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0))]]And the maximum royalty ( T ) is half of that:[T = frac{1}{2k} [text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0))]]So, unless there's a way to simplify this expression further or express it in terms of more elementary functions, this is the answer.Wait, another thought: perhaps if ( R_0 = 1 ), the integral simplifies? But the problem doesn't specify ( R_0 ), so I think we have to keep it general.Alternatively, maybe the problem expects an expression in terms of the original variables without referencing special functions. But I don't see a way to do that, given the form of ( R(t) ).Alternatively, perhaps I made a mistake in the substitution or in the integration step. Let me check again.Starting with ( R(t) = R_0^{e^{kt}} ). So, ( R(t) = e^{e^{kt} ln(R_0)} ). Therefore, integrating ( R(t) ) from 0 to 1:[int_{0}^{1} e^{e^{kt} ln(R_0)} dt]Let me make a substitution: let ( u = e^{kt} ). Then, ( du = k e^{kt} dt ), so ( dt = du/(k u) ). When ( t = 0 ), ( u = 1 ); when ( t = 1 ), ( u = e^{k} ). So, the integral becomes:[int_{1}^{e^{k}} e^{u ln(R_0)} cdot frac{du}{k u} = frac{1}{k} int_{1}^{e^{k}} frac{R_0^{u}}{u} du]Which is the same as before. So, no mistake there.Alternatively, perhaps the problem expects a different interpretation of the differential equation. Let me double-check the original equation:( frac{dR}{dt} = k R ln(R) ). Yes, that's what was given.Wait, another idea: perhaps the problem is expecting a solution in terms of the Lambert W function? But I don't see how that would apply here, since the integral leads to the exponential integral, not the Lambert W.Alternatively, maybe the problem is expecting a series expansion for the integral? That could be a possibility, but it would complicate things further and wouldn't give a closed-form expression.Alternatively, perhaps the problem is expecting a different approach to the differential equation. Let me think again.Wait, the differential equation is ( frac{dR}{dt} = k R ln(R) ). Let me consider if this is a Bernoulli equation or something else. But no, it's separable, as I did before.Alternatively, perhaps I can write it as ( frac{dR}{R ln(R)} = k dt ), which is what I did, leading to the integral involving ( ln(ln(R)) ).So, I think my solution to part 1 is correct, and part 2 indeed requires the exponential integral function.Therefore, the maximum allowable total royalty ( T ) is:[T = frac{1}{2k} left[ text{Ei}(e^{k} ln(R_0)) - text{Ei}(ln(R_0)) right]]Expressed in terms of ( R_0 ) and ( k ), using the exponential integral function.Alternatively, if the problem expects a different form, perhaps expressing it in terms of the original differential equation's solution. But I don't see a way to avoid the exponential integral here.So, I think this is the answer. It might be a bit advanced, involving special functions, but given the nature of the integral, I don't see another way around it.</think>

🧠:<think>Alright, so I have these two sub-problems to solve related to Alex's slow travel trip. Let me tackle them one by one.Starting with Sub-problem 1: Alex is in Rome for 30 days, walking a random distance each day, which is normally distributed with a mean of 5 km and a standard deviation of 1.5 km. I need to find the probability that the total distance walked exceeds 160 km. Hmm, okay.First, I remember that when dealing with sums of independent random variables, the Central Limit Theorem (CLT) can be applied, especially when the sample size is large enough. Here, we have 30 days, which is a decent sample size, so CLT should be applicable.Each day's walk is a normal variable, so the sum of these should also be normal, but I think the CLT is more about when the underlying distribution isn't normal, but in this case, since each D_i is normal, the sum will also be normal. So maybe I don't even need CLT here, but the problem mentions using CLT, so perhaps they want me to treat it as such.Anyway, let's proceed. The total distance walked over 30 days is the sum of 30 independent normal variables. The mean of the sum will be 30 times the mean of each day, and the variance will be 30 times the variance of each day.So, mean total distance, μ_total = 30 * 5 = 150 km.Variance of each day is (1.5)^2 = 2.25. So, variance of the total distance is 30 * 2.25 = 67.5. Therefore, the standard deviation of the total distance is sqrt(67.5). Let me calculate that.sqrt(67.5) is sqrt(67.5) ≈ 8.2158 km.So, the total distance walked, let's call it S, is normally distributed with μ = 150 and σ ≈ 8.2158.We need P(S > 160). To find this probability, we can standardize it.Z = (160 - 150) / 8.2158 ≈ 10 / 8.2158 ≈ 1.217.Now, we need the probability that Z > 1.217. Looking at standard normal tables, the probability that Z < 1.217 is approximately 0.8888. Therefore, the probability that Z > 1.217 is 1 - 0.8888 = 0.1112, or about 11.12%.Wait, let me double-check the Z-score calculation. 160 - 150 is 10, divided by 8.2158, which is indeed approximately 1.217. And yes, looking up 1.217 in the Z-table, it's about 0.8888. So, yeah, 1 - 0.8888 is 0.1112. So, approximately 11.12% chance.Alternatively, using a calculator, the exact value might be slightly different, but 11.12% is a reasonable estimate.Moving on to Sub-problem 2: Alex is in Kyoto for 30 days, writing blogs each day. The time per blog, T_i, follows an exponential distribution with a mean of 2 hours. We need the probability that the total time spent writing exceeds 70 hours.Okay, exponential distributions are memoryless, and the sum of exponential variables follows a gamma distribution. But since we have 30 variables, which is a large number, maybe we can apply the Central Limit Theorem here as well to approximate the sum with a normal distribution.Each T_i has a mean of 2 hours. So, the total time, let's call it S, will have a mean of 30 * 2 = 60 hours.The variance of an exponential distribution is equal to the square of its mean. So, variance of each T_i is (2)^2 = 4. Therefore, the variance of the total time S is 30 * 4 = 120. So, the standard deviation is sqrt(120) ≈ 10.954 hours.So, S is approximately normally distributed with μ = 60 and σ ≈ 10.954.We need P(S > 70). Again, standardize this.Z = (70 - 60) / 10.954 ≈ 10 / 10.954 ≈ 0.913.Looking up Z = 0.913 in the standard normal table, the probability that Z < 0.913 is approximately 0.8190. Therefore, the probability that Z > 0.913 is 1 - 0.8190 = 0.1810, or about 18.10%.Wait, let me verify the Z-score: 70 - 60 is 10, divided by 10.954 is approximately 0.913. Yes, that seems right. And the corresponding probability, 0.8190, so 1 - 0.8190 is 0.1810. So, approximately 18.1%.Alternatively, since the exponential distribution is skewed, the normal approximation might not be perfect, but with 30 days, it should be decent.Alternatively, if we wanted to be more precise, we could use the exact gamma distribution. The sum of n exponential variables with rate λ is a gamma distribution with shape n and rate λ. Here, each T_i has mean 2, so λ = 1/2.So, S ~ Gamma(30, 1/2). The CDF of the gamma distribution can be used to find P(S > 70). However, calculating this exactly might be more involved, but perhaps using software or tables.But since the problem doesn't specify, and given that n=30 is reasonably large, the normal approximation should suffice.So, summarizing:Sub-problem 1: Probability ≈ 11.12%Sub-problem 2: Probability ≈ 18.10%I think that's it.Final AnswerSub-problem 1: The probability is boxed{0.1112}.Sub-problem 2: The probability is boxed{0.1810}.</think>

🧠:<think>Alright, so I have this problem about designing a store layout with artisan paper. The store is a perfect square with each side being 20 meters. The designer wants to put a circular display stand in the center with a radius of 5 meters. Then, around this circle, they want to arrange some rectangular tables. Each table has a length-to-width ratio of 2:1, and the total area of these tables should be exactly half of the remaining store area after subtracting the circular display. Also, the number of tables has to be an integer. Okay, let me break this down into two sub-problems as given.Sub-problem 1: Calculate the area available for the rest of the store layout (excluding the circular display area).First, I need to find the total area of the store. Since it's a perfect square with side length 20 meters, the area is side squared. So, 20 meters times 20 meters.Total area = 20 m * 20 m = 400 m².Next, I need to calculate the area of the circular display. The radius is given as 5 meters. The area of a circle is π times radius squared.Circular area = π * (5 m)² = π * 25 m² ≈ 78.54 m².Now, subtract the circular area from the total area to find the remaining area available for the rest of the layout.Remaining area = Total area - Circular area= 400 m² - 78.54 m²≈ 321.46 m².Wait, but the problem says the total area occupied by the tables should be exactly half of the remaining store area. So, I think I need to compute that for Sub-problem 2.But first, let me just confirm my calculations for Sub-problem 1.Total area: 20*20=400. Correct.Circular area: π*5²=25π≈78.54. Correct.Remaining area: 400-78.54≈321.46. Hmm, but maybe I should keep it exact instead of using an approximate value for π. So, 25π is exact, so remaining area is 400 - 25π.Yes, that's better. So, I can write the remaining area as 400 - 25π square meters.So, Sub-problem 1 answer is 400 - 25π m².Sub-problem 2: Determine the dimensions of each table, assuming there are an integer number of tables.Alright, so the total area occupied by the tables should be half of the remaining store area. So, half of (400 - 25π).Total table area = (1/2)*(400 - 25π) = 200 - (25/2)π.Let me compute that numerically to get a sense.25/2 is 12.5, so 12.5π ≈ 39.27.Therefore, total table area ≈ 200 - 39.27 ≈ 160.73 m².So, the total area of all the tables combined is approximately 160.73 m².Each table has a length-to-width ratio of 2:1. So, if I let the width be 'w', then the length is '2w'. Therefore, the area of one table is length times width, which is 2w * w = 2w².Let me denote the number of tables as 'n', which is an integer. So, the total area of all tables is n * 2w².We know that total table area is 200 - (25/2)π, so:n * 2w² = 200 - (25/2)π.I need to find 'w' and 'n' such that this equation holds, with 'n' being an integer.But wait, I don't know 'n' yet. So, perhaps I can express 'w' in terms of 'n' or vice versa.Let me rearrange the equation:2w² = (200 - (25/2)π) / nSo,w² = (200 - (25/2)π) / (2n)Therefore,w = sqrt[(200 - (25/2)π) / (2n)]Hmm, but this seems a bit abstract. Maybe I can plug in the approximate value for π to get a numerical value for the total table area.Total table area ≈ 160.73 m².So, 2w² * n ≈ 160.73.Therefore, w² ≈ 160.73 / (2n) = 80.365 / nSo, w ≈ sqrt(80.365 / n)But since the store is 20x20 meters, the tables have to fit around the circular display without overlapping. So, the dimensions of the tables must be such that they can fit in the remaining space.Wait, perhaps I need to consider the arrangement of the tables around the circular area. Since the circular area is in the center, the tables are arranged around it. So, the circular area has a radius of 5 meters, so the diameter is 10 meters. The store is 20x20, so from the center to the wall is 10 meters in all directions.Therefore, the tables must be arranged in the annular region between the circle and the walls.But how exactly? Are the tables placed along the walls or around the circle?I think the problem doesn't specify, so maybe I can assume that the tables are placed in such a way that they don't overlap with the circle, but their exact arrangement isn't specified beyond that.So, perhaps the main constraints are:1. Each table has a 2:1 ratio.2. Total area of tables is 200 - (25/2)π ≈ 160.73 m².3. Number of tables is an integer.4. The tables must fit within the remaining area without overlapping the circle.But without knowing the exact arrangement, it's a bit tricky. Maybe I can assume that the tables are placed along the walls, but given the store is a square, perhaps they can be arranged in a symmetrical fashion around the circle.Alternatively, maybe the store is divided into sections, each containing a table.But perhaps another approach is needed.Wait, maybe I can consider the maximum possible size of the tables given the space.Since the circular area is 5 meters radius, the distance from the center to the wall is 10 meters. So, the maximum distance from the center to the edge is 10 meters.If the tables are placed around the circle, their placement must be such that they don't encroach into the circular area.So, the tables must be placed at least 5 meters away from the center in all directions.But the store is 20x20, so the tables can be placed along the walls or in the corners, but their placement must not overlap with the circle.Alternatively, perhaps the tables are arranged in a square pattern around the circle, each placed in a quadrant.But without more specifics, perhaps I can proceed with the given information.So, total table area is 200 - (25/2)π.Let me compute that exactly:200 - (25/2)π = 200 - 12.5π.So, total table area = 200 - 12.5π m².Each table's area is 2w², so total area is n*2w² = 200 - 12.5π.So, 2w² = (200 - 12.5π)/n.Thus, w² = (200 - 12.5π)/(2n).Therefore, w = sqrt[(200 - 12.5π)/(2n)].But since w must be a positive real number, and n must be an integer, we can look for integer values of n such that the resulting w is a reasonable dimension for a table, and that the tables can fit in the store without overlapping the circle.But perhaps I can find n such that (200 - 12.5π) is divisible by 2n, or at least gives a perfect square when divided by 2n, so that w is a whole number.Alternatively, maybe n is a factor of (200 - 12.5π). But since 200 - 12.5π is approximately 160.73, which is not an integer, perhaps we can look for n such that 160.73/(2n) is a perfect square.But this might be complicated. Alternatively, maybe I can assume that the tables are arranged in a way that their placement is symmetrical, so n could be 4, 8, 12, etc., depending on the store's layout.Alternatively, perhaps the number of tables is such that their dimensions fit neatly into the remaining space.Wait, another approach: since the store is 20x20, and the circle is in the center with radius 5, the remaining area is a square annulus around the circle.But the exact arrangement of the tables isn't specified, so perhaps the tables can be placed in any configuration as long as they don't overlap the circle.But given that, perhaps the maximum possible size of the tables is constrained by the distance from the circle to the walls.Since the circle has a radius of 5, the distance from the circle to the wall is 10 - 5 = 5 meters in all directions.So, the tables must be placed within a 5-meter border around the circle.But if the tables are placed in this border, their dimensions can't exceed 5 meters in any direction from the circle.But since the tables have a 2:1 ratio, perhaps their length is 2w and width is w.So, if the tables are placed along the walls, their width could be up to 5 meters, but that might not make sense because the tables need to fit without overlapping.Alternatively, perhaps the tables are placed in the corners, each in a quadrant.But without more information, maybe I can proceed by assuming that the number of tables is such that their total area is 200 - 12.5π, and each table has area 2w².So, let's compute 200 - 12.5π numerically:12.5π ≈ 39.27So, 200 - 39.27 ≈ 160.73 m².So, total table area ≈ 160.73 m².Each table's area is 2w², so 2w² * n = 160.73.Thus, w² = 160.73 / (2n) ≈ 80.365 / n.So, w ≈ sqrt(80.365 / n).We need w to be a positive real number, and n must be an integer.But since the tables must fit within the store without overlapping the circle, their dimensions must be such that they can be placed in the remaining space.Given that the remaining space is a square annulus around the circle, the maximum dimension of a table in any direction can't exceed 5 meters from the circle, but that's just a rough estimate.Alternatively, perhaps the tables are placed along the walls, so their length can be up to 20 meters, but that seems too large.Wait, but the tables have a 2:1 ratio, so if the length is 2w, and the width is w, perhaps the tables can be placed along the walls with their length parallel to the walls.But given that the store is 20x20, and the circle is in the center, the tables can be placed along each wall, but their placement must not overlap with the circle.So, perhaps the tables are placed in the corners, each in a quadrant, but that might not be the case.Alternatively, perhaps the tables are arranged in a circular pattern around the central circle, but that might complicate things.Wait, maybe I can think about the maximum possible size of a table that can fit in the remaining space.Since the circle has a radius of 5, the distance from the center to the wall is 10 meters. So, the maximum distance from the circle to the wall is 5 meters.Therefore, any table placed in the remaining area must be within 5 meters of the circle's edge.But if the table is placed against the wall, its distance from the center would be 10 meters, but the circle is only 5 meters radius, so the table can be placed up to 5 meters away from the circle.Wait, no, the distance from the center to the wall is 10 meters, so the table can be placed anywhere in the square, as long as it doesn't overlap with the circle.So, perhaps the tables can be placed in such a way that their edges are at least 5 meters away from the center.But since the tables have a 2:1 ratio, perhaps their placement is such that their longer side is along the wall.But without more specifics, maybe I can proceed by assuming that the tables are placed in a way that their dimensions fit into the remaining space, and that the number of tables is such that the total area is 160.73 m².So, let's try to find integer values of n such that w is a reasonable number.Let me compute 80.365 / n for various integer n and see if the square root is a nice number.Let's try n=4:w² ≈ 80.365 / 4 ≈ 20.091w ≈ sqrt(20.091) ≈ 4.48 metersSo, width ≈ 4.48 m, length ≈ 8.96 m.But 8.96 meters is quite long. The store is 20 meters, so placing 4 tables each 8.96 meters long might be possible, but perhaps it's too long.Alternatively, n=8:w² ≈ 80.365 / 8 ≈ 10.0456w ≈ sqrt(10.0456) ≈ 3.17 metersSo, width ≈ 3.17 m, length ≈ 6.34 m.That seems more manageable.n=10:w² ≈ 80.365 / 10 ≈ 8.0365w ≈ sqrt(8.0365) ≈ 2.836 metersSo, width ≈ 2.84 m, length ≈ 5.68 m.n=16:w² ≈ 80.365 / 16 ≈ 5.0228w ≈ sqrt(5.0228) ≈ 2.24 metersSo, width ≈ 2.24 m, length ≈ 4.48 m.n=20:w² ≈ 80.365 / 20 ≈ 4.01825w ≈ sqrt(4.01825) ≈ 2.0045 metersSo, width ≈ 2.0045 m, length ≈ 4.009 m.Hmm, that's very close to 2 meters width and 4 meters length.Wait, 2 meters width and 4 meters length would give an area of 8 m² per table.So, with n=20, total area would be 20*8=160 m², which is very close to 160.73 m².So, perhaps n=20 is the integer number of tables, each with dimensions 2 meters by 4 meters.But let me check:If each table is 2m x 4m, area is 8 m².20 tables would occupy 20*8=160 m².But the required total area is 200 - 12.5π ≈ 160.73 m².So, 160 m² is slightly less than required.Alternatively, maybe n=20 is acceptable, considering rounding.But perhaps the exact value is 200 - 12.5π, which is approximately 160.73, so 20 tables of 8 m² each give 160 m², which is about 0.73 m² less.Alternatively, maybe n=19 or n=21.Let me check n=19:w² ≈ 80.365 / 19 ≈ 4.2297w ≈ sqrt(4.2297) ≈ 2.057 metersSo, width ≈ 2.057 m, length ≈ 4.114 m.Area per table ≈ 2.057*4.114 ≈ 8.47 m².Total area ≈ 19*8.47 ≈ 160.93 m², which is very close to 160.73.Similarly, n=21:w² ≈ 80.365 / 21 ≈ 3.827w ≈ sqrt(3.827) ≈ 1.956 metersSo, width ≈ 1.956 m, length ≈ 3.912 m.Area per table ≈ 1.956*3.912 ≈ 7.66 m².Total area ≈ 21*7.66 ≈ 160.86 m², which is also close.But since the problem says the total area should be exactly half of the remaining store area, which is 200 - 12.5π, which is an exact value, perhaps we need to find n such that 2w² * n = 200 - 12.5π exactly.So, 2w² * n = 200 - 12.5π.But 200 - 12.5π is approximately 160.73, but exactly, it's 200 - (25/2)π.So, perhaps we can express w in terms of n:w = sqrt[(200 - (25/2)π)/(2n)].But since w must be a real number, and n must be an integer, perhaps we can find n such that (200 - (25/2)π) is divisible by 2n, but that's unlikely because π is irrational.Alternatively, perhaps the problem expects us to use the approximate value of π as 3.14, so 25/2 π ≈ 39.25, so total table area ≈ 160.75 m².Then, 160.75 / 2 = 80.375 per table if n=2, but that's not helpful.Wait, no, 160.75 is the total area, so 2w² * n = 160.75.So, 2w² = 160.75 / n.Thus, w² = 80.375 / n.So, w = sqrt(80.375 / n).Looking for integer n such that 80.375 / n is a perfect square.But 80.375 is 80 + 0.375 = 80 + 3/8 = 640/8 + 3/8 = 643/8.So, 643/8 divided by n should be a perfect square.But 643 is a prime number, I think, so 643/8 is not a perfect square.Therefore, perhaps the problem expects us to use an approximate value and find n such that 80.375 / n is close to a perfect square.Looking back, when n=20, 80.375 /20 ≈4.01875, which is close to 4, which is a perfect square.So, w ≈2 meters, length≈4 meters.Thus, perhaps the intended answer is 2 meters by 4 meters, with n=20 tables.But let me check:20 tables, each 2x4, area=8 m², total=160 m².But the required total area is 200 - 12.5π ≈160.73 m².So, 160 m² is slightly less, but perhaps the problem expects us to round.Alternatively, maybe n=19 or 21.But 20 is a nice number, and 2x4 is a standard ratio.Alternatively, perhaps the problem expects an exact answer in terms of π.Wait, let me try to express the total table area as 200 - (25/2)π.So, 2w² * n = 200 - (25/2)π.We can write this as:n = (200 - (25/2)π) / (2w²).But since n must be an integer, perhaps we can choose w such that (200 - (25/2)π) is divisible by 2w².But without knowing w, it's difficult.Alternatively, perhaps the problem expects us to assume that the tables are arranged in a way that their dimensions fit into the remaining space, and that the number of tables is such that the area is exactly half.But perhaps I'm overcomplicating.Let me think differently.The remaining area after the circle is 400 -25π.Half of that is 200 -12.5π.Each table has area 2w².So, total area is n*2w² =200 -12.5π.So, 2w² = (200 -12.5π)/n.We can write this as:w² = (200 -12.5π)/(2n).So, w = sqrt[(200 -12.5π)/(2n)].But since w must be a real number, and n must be an integer, perhaps we can find n such that (200 -12.5π)/(2n) is a perfect square.But 200 -12.5π is approximately 160.73, so 160.73/(2n) must be a perfect square.Looking for integer n such that 160.73/(2n) is a perfect square.Let me try n=20:160.73/(2*20)=160.73/40≈4.018, which is close to 4, which is 2².So, w≈2 meters.Thus, each table is 2 meters in width and 4 meters in length.So, n=20 tables, each 2x4 meters.That seems to fit.But let me check the exact calculation:If w=2 meters, then each table's area is 2*2²=8 m².20 tables would be 160 m².But the exact total area needed is 200 -12.5π≈160.73 m².So, 160 m² is slightly less, but perhaps the problem expects us to round.Alternatively, maybe the problem expects us to use exact values.Wait, 200 -12.5π is exactly 200 - (25/2)π.So, 2w² *n =200 - (25/2)π.If we take n=20, then 2w²= (200 - (25/2)π)/20=10 - (25/40)π=10 - (5/8)π.So, w²=5 - (5/16)π.Thus, w= sqrt(5 - (5/16)π).But that's not a nice number, so perhaps n=20 is not the exact answer.Alternatively, maybe n=16.Then, 2w²= (200 -12.5π)/16≈160.73/16≈10.0456.So, w²≈5.0228, w≈2.24 meters.But that's not a nice number either.Alternatively, maybe n=8.2w²=160.73/8≈20.091.w²≈10.0456, w≈3.17 meters.Still not a nice number.Alternatively, maybe n=4.2w²≈40.1825.w²≈20.091, w≈4.48 meters.Hmm.Alternatively, maybe n=5.2w²≈32.146.w²≈16.073, w≈4.009 meters.So, width≈4.009 m, length≈8.018 m.But that's a very long table.Alternatively, maybe n=10.2w²≈16.073.w²≈8.0365, w≈2.836 meters.So, width≈2.84 m, length≈5.68 m.Hmm.But none of these give a nice integer or simple fractional dimensions.Wait, perhaps the problem expects us to use the exact value of π and find n such that (200 - (25/2)π) is divisible by 2w², but that seems complicated.Alternatively, perhaps the problem expects us to assume that the tables are arranged in such a way that their dimensions are 2 meters by 4 meters, with n=20, even though it's slightly less than the required area.Alternatively, maybe the problem expects us to use the exact value and express the dimensions in terms of π.But that seems unlikely.Alternatively, perhaps the problem expects us to use the approximate value of π as 3.14, so 25/2 π≈39.25.Thus, total table area≈160.75 m².So, 2w² *n=160.75.Looking for integer n such that 160.75/(2n) is a perfect square.160.75=643/4.So, 643/4 divided by 2n=643/(8n).So, 643/(8n) must be a perfect square.But 643 is a prime number, so 8n must divide 643, which is not possible since 643 is prime and 8n would have to be a factor of 643, which is only 1 and 643.Thus, n would have to be 643/8, which is not an integer.Therefore, perhaps the problem expects us to use an approximate value and choose n=20, giving w≈2 meters.Alternatively, perhaps the problem expects us to express the dimensions in terms of π.But that seems complicated.Alternatively, maybe the problem expects us to find the dimensions such that the total area is exactly 200 -12.5π, and n is an integer, but without specifying the exact arrangement, it's difficult.Alternatively, perhaps the problem expects us to assume that the tables are arranged in a way that their placement is such that their length is along the wall, and their width is away from the wall, so that the width is limited by the distance from the wall to the circle.Since the distance from the center to the wall is 10 meters, and the circle has a radius of 5 meters, the distance from the circle to the wall is 5 meters.So, if the tables are placed with their width away from the wall, their width can't exceed 5 meters, otherwise they would overlap with the circle.But since the tables have a 2:1 ratio, if their width is 5 meters, their length would be 10 meters.But 10 meters is half the length of the store, which is 20 meters, so placing a table of 10x5 meters along the wall would fit.But let's see:If each table is 10 meters long and 5 meters wide, area=50 m².Total table area=50n.We need 50n=200 -12.5π≈160.73.So, n≈160.73/50≈3.214.But n must be an integer, so n=3 tables would give 150 m², which is less than required, and n=4 tables would give 200 m², which is more than required.But 200 m² is more than the required 160.73 m², so that's not acceptable.Alternatively, maybe the tables are placed with their length along the wall, but their width is less than 5 meters.So, let's say the width is 'w', then length is '2w'.The area per table is 2w².Total area=2w²n=200 -12.5π.Also, since the width 'w' can't exceed 5 meters, as the distance from the wall to the circle is 5 meters.So, w≤5 meters.Thus, 2w²n=200 -12.5π.Let me solve for n:n=(200 -12.5π)/(2w²).Since w≤5, let's see what happens when w=5:n=(200 -12.5π)/(2*25)= (200 -12.5π)/50≈(200 -39.27)/50≈160.73/50≈3.214.So, n≈3.214, which is not an integer.If w=4 meters:n=(200 -12.5π)/(2*16)= (200 -39.27)/32≈160.73/32≈5.022.So, n≈5.022, which is close to 5.So, n=5 tables, each 4x8 meters.Total area=5*32=160 m², which is slightly less than 160.73 m².Alternatively, w=4.1 meters:n=(200 -12.5π)/(2*(4.1)^2)= (200 -39.27)/(2*16.81)=160.73/33.62≈4.78.So, n≈4.78, which is not an integer.Alternatively, w=4.2 meters:n=(200 -12.5π)/(2*(4.2)^2)=160.73/(2*17.64)=160.73/35.28≈4.556.Still not an integer.Alternatively, w=4.5 meters:n=(200 -12.5π)/(2*(4.5)^2)=160.73/(2*20.25)=160.73/40.5≈3.968.Close to 4.So, n=4 tables, each 4.5x9 meters.But 9 meters is quite long, and the store is 20 meters, so placing 4 tables each 9 meters long along the wall would require 4*9=36 meters, which is more than the 20 meters available.Thus, that's not possible.Alternatively, perhaps the tables are placed in the corners, each in a quadrant.So, each quadrant is 10x10 meters.But the circle is in the center, so each quadrant is a 10x10 square with a quarter-circle of radius 5 meters removed.Thus, the area available in each quadrant is 100 - (25π)/4≈100 -19.635≈80.365 m².So, if we place one table in each quadrant, each table would have an area of 80.365 m².But each table has a 2:1 ratio, so area=2w²=80.365.Thus, w²=40.1825, so w≈6.34 meters.But the quadrant is 10x10, so a table of 6.34x12.68 meters would not fit in the 10x10 quadrant.Thus, that's not possible.Alternatively, maybe the tables are placed along the walls, but not in the corners.So, for example, along the top wall, we can place a table of length 'l' and width 'w', with l=2w.The table must be placed such that it doesn't overlap with the circle.The distance from the center to the table must be at least 5 meters.But the table is placed along the wall, so the distance from the center to the table is 10 meters minus the distance from the wall to the table.Wait, no, the distance from the center to the table is the distance from the center to the edge of the table.If the table is placed along the wall, its distance from the center is 10 meters minus the width of the table.Wait, no, the table is placed along the wall, so its distance from the center is determined by its position.Wait, perhaps it's better to model the store as a coordinate system, with the center at (0,0), and the walls at x=±10 and y=±10.The circle is centered at (0,0) with radius 5.A table placed along the top wall would have its bottom edge at y=10 - w, where w is the width of the table.But the table must not overlap with the circle, so the distance from the center to the table must be at least 5 meters.The distance from the center to the table is the distance from (0,0) to the table's edge.If the table is placed along the top wall, its bottom edge is at y=10 - w.The distance from the center to this edge is 10 - w.This distance must be ≥5 meters, so 10 - w ≥5 ⇒ w ≤5 meters.Thus, the width of the table can be at most 5 meters.Given that, and the ratio of 2:1, the length would be 10 meters.But the store is 20 meters long, so placing a table of 10 meters along the wall would fit, but we can only place two such tables along the wall without overlapping.But let's see:If each table is 10 meters long and 5 meters wide, area=50 m².If we place two tables along the top and bottom walls, that's 100 m².Similarly, two tables along the left and right walls, another 100 m².But that would give a total of 200 m², which is more than the required 160.73 m².Alternatively, maybe we can place fewer tables.But let's see:If each table is 10 meters long and 5 meters wide, area=50 m².If we place 3 tables, total area=150 m², which is less than required.If we place 4 tables, total area=200 m², which is more than required.But we need exactly 160.73 m².Alternatively, maybe the tables are smaller.If the width is less than 5 meters, say w meters, then length=2w meters.The area per table=2w².The distance from the center to the table is 10 - w ≥5 ⇒ w ≤5.So, w can be up to 5 meters.But if we place multiple tables along the walls, their total area would be n*2w²=160.73.But we also need to ensure that the tables fit along the walls without overlapping.Each wall is 20 meters long, but the tables are placed along the walls, so the length of the table along the wall is 2w.But wait, no, the length of the table is 2w, and the width is w, perpendicular to the wall.So, if the table is placed along the wall, its length is along the wall, so 2w meters, and its width is w meters away from the wall.Thus, the number of tables that can fit along one wall is 20 / (2w).But since we have four walls, total number of tables would be 4*(20 / (2w))=40/w.But the total area would be n*2w²= (40/w)*2w²=80w.We need 80w=160.73 ⇒ w≈2.009 meters.So, w≈2 meters, length≈4 meters.Thus, each table is 2x4 meters, and number of tables=40/w=40/2=20 tables.So, 20 tables, each 2x4 meters, placed along the walls, 5 tables per wall (since 20/4=5), each table taking 4 meters of wall space (since length=4 meters), so 5 tables per wall would take 5*4=20 meters, which fits exactly.Thus, this seems to fit.So, the dimensions of each table are 2 meters by 4 meters.Therefore, the answer is width=2 meters, length=4 meters.</think>

🧠:<think>Okay, so I have this problem about a biotech entrepreneur developing a new therapy based on DNA manipulation. The stability of a protein P is modeled by a function S, which depends on the concentrations of various amino acids. The function S is given as a quadratic form involving both individual amino acid concentrations squared and their pairwise products. The first part asks for the conditions on the coefficients a_i and b_ij such that S has a local maximum at a certain point (x1*, x2*, ..., xn*). Hmm, so I remember that for functions of multiple variables, to find maxima or minima, we need to look at the critical points where the gradient is zero and then check the second derivative, which in this case would be the Hessian matrix. Since S is a quadratic function, it's a paraboloid, and whether it has a maximum or minimum depends on the definiteness of the Hessian. For a local maximum, the Hessian should be negative definite. So I need to figure out what conditions on a_i and b_ij make the Hessian negative definite.The function S is:S = sum_{i=1}^n a_i x_i^2 + sum_{1 <= i < j <=n} b_{ij} x_i x_jSo, to write this in matrix form, it would be S = (1/2) x^T Q x, where Q is a symmetric matrix with a_i on the diagonal and b_ij/2 on the off-diagonal entries. Wait, actually, the standard quadratic form is (1/2) x^T Q x when you have cross terms. So maybe the function is written without the 1/2 factor, so the Hessian would be 2Q? Or maybe just Q? Let me think.Actually, the gradient of S would be the vector of partial derivatives. For each x_k, the partial derivative is 2 a_k x_k + sum_{j ≠ k} b_{kj} x_j. So the Hessian matrix H would have entries H_ik = 2 a_i if i = k, and H_ij = b_ij if i ≠ j. So H is a symmetric matrix with 2 a_i on the diagonal and b_ij on the off-diagonal.For S to have a local maximum at some point, the Hessian must be negative definite at that point. So the conditions would be that all the leading principal minors of H have alternating signs starting with negative. That is, the first leading minor (the top-left 1x1 matrix) is negative, the next 2x2 determinant is positive, the next 3x3 determinant is negative, and so on, with the last one being (-1)^n times positive.But since the Hessian is constant (because S is quadratic), the definiteness is the same everywhere. So if H is negative definite, then S has a maximum at every critical point. So the conditions are that H is negative definite.Alternatively, since H is symmetric, it's negative definite if and only if all its eigenvalues are negative. But checking eigenvalues might be complicated. Instead, the leading principal minors condition is more straightforward but might be tedious for large n.Alternatively, since H is a symmetric matrix, another way to check negative definiteness is that all the diagonal entries are negative, and the matrix is diagonally dominant with negative diagonal entries. Wait, no, diagonal dominance is for positive definiteness. For negative definiteness, all diagonal entries must be negative, and the matrix must be diagonally dominant in the negative sense, meaning that for each row, the absolute value of the diagonal entry is greater than the sum of the absolute values of the other entries in that row.But I'm not sure if that's a sufficient condition. Maybe it's better to stick with the leading principal minors.So, for the Hessian H, which is a symmetric matrix with 2a_i on the diagonal and b_ij on the off-diagonal, the conditions for negative definiteness are:1. The first leading principal minor (the (1,1) entry) is negative: 2a_1 < 0 => a_1 < 0.2. The second leading principal minor is the determinant of the top-left 2x2 matrix:| 2a1 b12 || b12 2a2 |Determinant is (2a1)(2a2) - b12^2. For negative definiteness, this determinant should be positive. So (2a1)(2a2) - b12^2 > 0.Similarly, for the third leading minor, the determinant of the top-left 3x3 matrix should be negative, and so on, alternating signs.This pattern continues, with each leading principal minor of order k having a sign of (-1)^k.So in general, for all k from 1 to n, the k-th leading principal minor should have sign (-1)^k.Therefore, the conditions on the coefficients a_i and b_ij are that the Hessian matrix H, with diagonal entries 2a_i and off-diagonal entries b_ij, is negative definite. This requires that all leading principal minors of H alternate in sign starting with negative.That's the first part.For the second part, we have a constraint that the sum of concentrations is constant: sum x_i = C. We need to maximize S subject to this constraint using Lagrange multipliers.So, set up the Lagrangian:L = S - λ (sum x_i - C)Which is:L = sum a_i x_i^2 + sum_{i<j} b_ij x_i x_j - λ (sum x_i - C)Take partial derivatives with respect to each x_i and set them to zero.Partial derivative of L with respect to x_k:2 a_k x_k + sum_{j ≠ k} b_kj x_j - λ = 0So, for each k, we have:2 a_k x_k + sum_{j ≠ k} b_kj x_j = λThis can be written in matrix form as H x = λ e, where H is the Hessian matrix (as before), x is the vector of concentrations, and e is the vector of ones.So, H x = λ e.Additionally, we have the constraint sum x_i = C.So, we have a system of equations:H x = λ esum x_i = CTo solve for x, we can write this as a linear system. Let me denote H as the matrix with 2a_i on the diagonal and b_ij on the off-diagonal.So, the system is:2a1 x1 + b12 x2 + b13 x3 + ... + b1n xn = λb21 x1 + 2a2 x2 + b23 x3 + ... + b2n xn = λ...bn1 x1 + bn2 x2 + bn3 x3 + ... + 2an xn = λAnd sum x_i = C.This is a system of n+1 equations with n+1 variables (x1, x2, ..., xn, λ).To solve this, we can subtract the equations from each other to eliminate λ.For example, subtract the first equation from the second:(2a2 - 2a1) x2 + (b23 - b13) x3 + ... + (b2n - b1n) xn - (b21 - b11) x1 = 0Wait, actually, since H is symmetric, b_ij = b_ji, so the coefficients would be symmetric.But this might get complicated. Alternatively, we can write the system as (H - λ I) x = 0, but with the additional constraint sum x_i = C.Wait, no, because H x = λ e, so (H - λ I) x = -λ e + λ I x = ?Wait, maybe it's better to think of it as H x = λ e, and sum x_i = C.Let me denote e as the vector of ones. Then, sum x_i = e^T x = C.So, we have H x = λ e and e^T x = C.We can solve for x in terms of λ from the first equation: x = λ H^{-1} e.Then substitute into the constraint: e^T x = λ e^T H^{-1} e = C.So, λ = C / (e^T H^{-1} e).Therefore, x = (C / (e^T H^{-1} e)) H^{-1} e.So, the concentrations x_i are proportional to the components of H^{-1} e, scaled by C divided by the sum of the components of H^{-1} e.Alternatively, x = (C / (1^T H^{-1} 1)) H^{-1} 1, where 1 is the vector of ones.This gives the optimal concentrations.But this is quite abstract. Maybe there's a more explicit way.Alternatively, since H is symmetric, we can write the system as H x = λ e, and e^T x = C.Let me denote the vector x as x = [x1, x2, ..., xn]^T.Then, H x = λ e implies that each row of H times x equals λ.So, for each i, sum_j H_ij x_j = λ.But H_ij = 2a_i if i=j, and b_ij otherwise.So, for each i:2a_i x_i + sum_{j ≠ i} b_ij x_j = λ.This is a system of linear equations.We can write this as:(2a1) x1 + b12 x2 + b13 x3 + ... + b1n xn = λb21 x1 + (2a2) x2 + b23 x3 + ... + b2n xn = λ...bn1 x1 + bn2 x2 + bn3 x3 + ... + (2an) xn = λAnd sum x_i = C.This is a linear system with n+1 equations.To solve this, we can subtract the first equation from all others to eliminate λ.For example, subtract the first equation from the second:(2a2 - 2a1) x2 + (b23 - b13) x3 + ... + (b2n - b1n) xn - (b21 - b11) x1 = 0But since H is symmetric, b_ij = b_ji, so b21 = b12, etc.This might not simplify much, but perhaps we can write it in terms of the differences.Alternatively, we can express the system as:H x = λ ee^T x = CLet me denote the vector e as [1,1,...,1]^T.Then, from H x = λ e, we can write x = λ H^{-1} e.Substitute into e^T x = C:e^T (λ H^{-1} e) = C => λ (e^T H^{-1} e) = C => λ = C / (e^T H^{-1} e).Therefore, x = (C / (e^T H^{-1} e)) H^{-1} e.So, the optimal concentrations are x = (C / (1^T H^{-1} 1)) H^{-1} 1.This is the solution.But to express it more explicitly, we might need to know the specific form of H, but since H is given in terms of a_i and b_ij, we can't simplify further without more information.So, in conclusion, the optimal concentrations are proportional to the components of H^{-1} e, scaled by C divided by the sum of the components of H^{-1} e.Alternatively, if we let v = H^{-1} e, then x_i = (C / sum_j v_j) v_i.So, each x_i is proportional to the corresponding component of H^{-1} e.Therefore, the concentrations that maximize S under the constraint sum x_i = C are given by x = (C / (1^T H^{-1} 1)) H^{-1} 1.I think that's the answer.</think>