🧠:<think>Alright, so I've got this problem about tracking a cult leader's movements using complex functions. It's divided into two sub-problems. Let me try to tackle them one by one.Starting with Sub-problem 1: The cult leader's position is given by the function ( f(t) = e^{i(omega t + phi)} ). They mention that the movements repeat every ( T ) hours, so I need to find ( omega ) in terms of ( T ).Hmm, okay. So, the function ( f(t) ) is a complex exponential function, which in the complex plane represents a point moving around the unit circle. The parameter ( omega ) is the angular frequency, and ( phi ) is the initial phase. Since the movement repeats every ( T ) hours, that means the function is periodic with period ( T ).I remember that for a function like ( e^{i(omega t + phi)} ), the period ( T ) is related to the angular frequency ( omega ) by the formula ( T = frac{2pi}{omega} ). Is that right? Let me think. The exponential function completes one full cycle when the exponent increases by ( 2pi ). So, if ( omega t ) increases by ( 2pi ), that should correspond to one period ( T ).So, setting ( omega T = 2pi ), which gives ( omega = frac{2pi}{T} ). Yeah, that seems correct. So, the angular frequency ( omega ) is ( 2pi ) divided by the period ( T ).Okay, so Sub-problem 1 seems straightforward. I just need to express ( omega ) as ( frac{2pi}{T} ).Moving on to Sub-problem 2: The investigator has data points ( (t_1, f(t_1)), (t_2, f(t_2)), ldots, (t_n, f(t_n)) ) and wants to predict the cult leader's location at any future time ( t ). Specifically, they want to find the position at ( t = T + frac{T}{4} ).Given that ( f(t) = e^{i(omega t + phi)} ), and we already found ( omega = frac{2pi}{T} ), we can write ( f(t) = e^{ileft(frac{2pi}{T} t + phiright)} ).But wait, the problem says to derive a general expression for the cult leader's location at any future time ( t ) using the given data points. So, maybe it's not just plugging into the formula, but perhaps we need to determine ( phi ) as well?Since ( f(t) ) is given for multiple times ( t_k ), we can use these data points to solve for ( phi ). Let's see. If we have multiple points, we can set up equations and solve for ( phi ). However, since the function is periodic, the phase ( phi ) can be determined if we know the position at a specific time.But actually, since ( f(t) ) is a complex exponential, it's fully determined by ( omega ) and ( phi ). We already know ( omega ) from Sub-problem 1, so if we have at least one data point, we can solve for ( phi ). If we have multiple data points, they should all be consistent with the same ( phi ).So, let's suppose we have one data point ( (t_1, f(t_1)) ). Then, ( f(t_1) = e^{ileft(frac{2pi}{T} t_1 + phiright)} ). Taking the natural logarithm of both sides, we get:( ln f(t_1) = ileft(frac{2pi}{T} t_1 + phiright) ).But wait, ( f(t_1) ) is a complex number on the unit circle, so its logarithm is multi-valued. However, since we're dealing with the principal value, we can write:( ln f(t_1) = ileft(frac{2pi}{T} t_1 + phi - 2pi kright) ), where ( k ) is an integer.But in practice, since we can choose ( phi ) modulo ( 2pi ), we can set ( k = 0 ) for simplicity. Therefore, ( phi = arg(f(t_1)) - frac{2pi}{T} t_1 ).So, if we have multiple data points, we can compute ( phi ) for each and ensure they are consistent. If they aren't, that might indicate some error in the data or the model.Once we have ( phi ), we can write the general expression for ( f(t) ) as:( f(t) = e^{ileft(frac{2pi}{T} t + phiright)} ).Now, to find the position at ( t = T + frac{T}{4} ), let's plug that into the function.First, compute ( frac{2pi}{T} times left(T + frac{T}{4}right) ):( frac{2pi}{T} times T + frac{2pi}{T} times frac{T}{4} = 2pi + frac{pi}{2} ).So, the exponent becomes ( 2pi + frac{pi}{2} + phi ).But since ( e^{itheta} ) is periodic with period ( 2pi ), adding ( 2pi ) doesn't change the value. Therefore, ( e^{i(2pi + frac{pi}{2} + phi)} = e^{i(frac{pi}{2} + phi)} ).So, the position at ( t = T + frac{T}{4} ) is ( e^{i(frac{pi}{2} + phi)} ).Alternatively, if we consider that ( t = T + frac{T}{4} ) is just ( frac{5T}{4} ), which is ( T ) plus a quarter period. Since the function repeats every ( T ), the position at ( t = frac{5T}{4} ) is the same as at ( t = frac{T}{4} ).So, ( fleft(frac{5T}{4}right) = fleft(frac{T}{4}right) = e^{ileft(frac{2pi}{T} times frac{T}{4} + phiright)} = e^{ileft(frac{pi}{2} + phiright)} ).Therefore, the position is ( e^{i(frac{pi}{2} + phi)} ).But wait, do we need to express this in terms of the data points? The problem says to derive a general expression using the given data points. So, perhaps we need to express ( phi ) in terms of the data.Suppose we have a data point ( (t_k, f(t_k)) ). Then, as I mentioned earlier, ( phi = arg(f(t_k)) - frac{2pi}{T} t_k ).Therefore, the general expression for ( f(t) ) is:( f(t) = e^{ileft(frac{2pi}{T} t + arg(f(t_k)) - frac{2pi}{T} t_kright)} ).Simplifying, this becomes:( f(t) = e^{ileft(frac{2pi}{T}(t - t_k) + arg(f(t_k))right)} ).So, that's the general expression. Now, to find the position at ( t = T + frac{T}{4} ), we can plug that into the expression:( fleft(T + frac{T}{4}right) = e^{ileft(frac{2pi}{T}left(T + frac{T}{4} - t_kright) + arg(f(t_k))right)} ).Simplifying inside the exponent:( frac{2pi}{T} times left(T + frac{T}{4} - t_kright) = 2pi + frac{pi}{2} - frac{2pi}{T} t_k ).So, the exponent becomes:( 2pi + frac{pi}{2} - frac{2pi}{T} t_k + arg(f(t_k)) ).Again, since ( e^{itheta} ) is periodic, the ( 2pi ) term can be dropped, so we have:( e^{ileft(frac{pi}{2} - frac{2pi}{T} t_k + arg(f(t_k))right)} ).But notice that ( - frac{2pi}{T} t_k + arg(f(t_k)) ) is just ( phi ), as we defined earlier. So, this simplifies back to ( e^{i(frac{pi}{2} + phi)} ).Therefore, regardless of the data point used, the position at ( t = T + frac{T}{4} ) is ( e^{i(frac{pi}{2} + phi)} ).Alternatively, if we think geometrically, moving a quarter period ahead from any point would rotate the position by ( frac{pi}{2} ) radians (90 degrees) in the complex plane. So, if the current position is ( e^{iphi} ), after a quarter period, it becomes ( e^{i(phi + frac{pi}{2})} ).So, that makes sense.In summary, for Sub-problem 1, ( omega = frac{2pi}{T} ). For Sub-problem 2, the general expression is ( f(t) = e^{ileft(frac{2pi}{T} t + phiright)} ), and the position at ( t = T + frac{T}{4} ) is ( e^{i(frac{pi}{2} + phi)} ).But wait, the problem says to derive the expression using the given data points. So, maybe I should express ( phi ) in terms of the data points.Suppose we have multiple data points, but since the function is deterministic, all data points should agree on ( phi ). So, using any one data point, say ( t_1 ), we can write:( phi = arg(f(t_1)) - frac{2pi}{T} t_1 ).Therefore, plugging back into the general expression, we get:( f(t) = e^{ileft(frac{2pi}{T} t + arg(f(t_1)) - frac{2pi}{T} t_1right)} = e^{ileft(frac{2pi}{T}(t - t_1) + arg(f(t_1))right)} ).So, that's the expression derived from the data point ( (t_1, f(t_1)) ).Therefore, the position at ( t = T + frac{T}{4} ) is:( fleft(T + frac{T}{4}right) = e^{ileft(frac{2pi}{T}left(T + frac{T}{4} - t_1right) + arg(f(t_1))right)} ).Simplifying:( frac{2pi}{T} times left(T + frac{T}{4} - t_1right) = 2pi + frac{pi}{2} - frac{2pi}{T} t_1 ).So, the exponent is ( 2pi + frac{pi}{2} - frac{2pi}{T} t_1 + arg(f(t_1)) ).Again, ( 2pi ) can be dropped, so we have:( e^{ileft(frac{pi}{2} - frac{2pi}{T} t_1 + arg(f(t_1))right)} ).But ( - frac{2pi}{T} t_1 + arg(f(t_1)) = phi ), so it's ( e^{i(frac{pi}{2} + phi)} ).Alternatively, if we use another data point ( t_2 ), we'd get the same result because ( phi ) is consistent across all data points.Therefore, regardless of which data point we use, the position at ( t = T + frac{T}{4} ) is ( e^{i(frac{pi}{2} + phi)} ).So, to wrap up:Sub-problem 1: ( omega = frac{2pi}{T} ).Sub-problem 2: The general expression is ( f(t) = e^{ileft(frac{2pi}{T} t + phiright)} ), and the position at ( t = T + frac{T}{4} ) is ( e^{i(frac{pi}{2} + phi)} ).But the problem asks to determine the cult leader's position at ( t = T + frac{T}{4} ). Since ( phi ) is determined from the data, we can express it as ( e^{i(frac{pi}{2} + phi)} ), but perhaps we can express it in terms of the data points without referencing ( phi ).Alternatively, if we consider that ( t = T + frac{T}{4} ) is equivalent to ( t = frac{5T}{4} ), which is ( frac{T}{4} ) after a full period. So, the position is the same as at ( t = frac{T}{4} ).But without knowing ( phi ), we can't simplify it further. So, the answer is ( e^{i(frac{pi}{2} + phi)} ).Alternatively, if we have a specific data point, say ( t_1 = 0 ), then ( f(0) = e^{iphi} ), so ( phi = arg(f(0)) ). Then, the position at ( t = T + frac{T}{4} ) would be ( e^{i(frac{pi}{2} + arg(f(0)))} ).But since the problem doesn't specify particular data points, we can't compute a numerical value, only express it in terms of ( phi ).So, I think that's as far as we can go. The key takeaway is that the position at ( t = T + frac{T}{4} ) is a rotation of ( frac{pi}{2} ) radians from the initial phase ( phi ).Final AnswerSub-problem 1: (boxed{omega = dfrac{2pi}{T}})Sub-problem 2: The cult leader's position at ( t = T + dfrac{T}{4} ) is (boxed{e^{ileft(dfrac{pi}{2} + phiright)}}).</think>

🧠:<think>Okay, so I'm trying to solve these two sub-problems related to optimizing a graphics rendering engine. Let me take them one at a time.Starting with Sub-problem 1. The rendering engine uses a parallel processing algorithm with a time complexity of ( T(n) = n log_2(n) ). We need to find the maximum allowable time complexity constant ( k ) such that ( T(n) = k cdot n log_2(n) leq 10 ) seconds for an image with ( n = 8 times 10^6 ) pixels.Alright, so the formula is ( T(n) = k cdot n log_2(n) ), and we want this to be less than or equal to 10 seconds. So, plugging in the values, we have:( 10 geq k cdot 8 times 10^6 cdot log_2(8 times 10^6) )First, I need to calculate ( log_2(8 times 10^6) ). Let me compute that.I know that ( 8 times 10^6 = 8,000,000 ). To find ( log_2(8,000,000) ), I can use logarithm properties or approximate it.I remember that ( log_2(10^6) ) is approximately ( log_2(1,000,000) ). Since ( 2^{20} ) is about a million (1,048,576), so ( log_2(10^6) approx 19.93 ). Therefore, ( log_2(8 times 10^6) = log_2(8) + log_2(10^6) ).( log_2(8) = 3 ), so adding that to 19.93 gives approximately 22.93.So, ( log_2(8,000,000) approx 22.93 ).Now, plugging back into the equation:( 10 geq k cdot 8 times 10^6 cdot 22.93 )Let me compute ( 8 times 10^6 times 22.93 ):First, 8,000,000 multiplied by 22.93.Let me compute 8,000,000 * 20 = 160,000,0008,000,000 * 2.93 = ?Compute 8,000,000 * 2 = 16,000,0008,000,000 * 0.93 = 7,440,000So, 16,000,000 + 7,440,000 = 23,440,000Therefore, total is 160,000,000 + 23,440,000 = 183,440,000So, 8,000,000 * 22.93 ≈ 183,440,000Therefore, the equation becomes:( 10 geq k cdot 183,440,000 )To solve for ( k ):( k leq frac{10}{183,440,000} )Calculating that:10 divided by 183,440,000 is approximately 5.454 x 10^-8.So, ( k approx 5.454 times 10^{-8} )Wait, let me double-check my calculations.First, ( log_2(8,000,000) ). Since 8,000,000 is 8 * 10^6, which is 2^3 * (10^6). As I did before, ( log_2(10^6) ) is approximately 19.93, so total is 3 + 19.93 = 22.93. That seems right.Then, 8,000,000 * 22.93. Let me compute this more accurately.22.93 * 8,000,000.22 * 8,000,000 = 176,000,0000.93 * 8,000,000 = 7,440,000So, 176,000,000 + 7,440,000 = 183,440,000. Correct.So, 10 / 183,440,000 = 10 / 1.8344 x 10^8 ≈ 5.454 x 10^-8.So, ( k approx 5.454 times 10^{-8} ).But let me express this in a more precise way.Alternatively, maybe I can compute ( log_2(8,000,000) ) more accurately.We know that ( log_2(8,000,000) = log_2(8 times 10^6) = log_2(8) + log_2(10^6) = 3 + log_2(10^6) ).We can compute ( log_2(10^6) ) as ( ln(10^6)/ln(2) ).Compute ( ln(10^6) = 6 ln(10) ≈ 6 * 2.302585 ≈ 13.81551 ).Then, ( log_2(10^6) = 13.81551 / 0.693147 ≈ 19.93157 ).So, ( log_2(8,000,000) = 3 + 19.93157 ≈ 22.93157 ).So, more accurately, it's approximately 22.93157.Then, 8,000,000 * 22.93157.Compute 8,000,000 * 22 = 176,000,0008,000,000 * 0.93157 = ?Compute 8,000,000 * 0.9 = 7,200,0008,000,000 * 0.03157 = 252,560So, 7,200,000 + 252,560 = 7,452,560Therefore, total is 176,000,000 + 7,452,560 = 183,452,560So, 8,000,000 * 22.93157 ≈ 183,452,560Therefore, 10 / 183,452,560 ≈ 5.454 x 10^-8So, ( k approx 5.454 times 10^{-8} )But perhaps we can write it as ( k leq frac{10}{8 times 10^6 times log_2(8 times 10^6)} )Alternatively, maybe we can express it in terms of exact logarithms, but I think the approximate value is sufficient here.So, the maximum allowable time complexity constant ( k ) is approximately ( 5.454 times 10^{-8} ).Moving on to Sub-problem 2.The deep learning model has a computational complexity ( C(m) = m^2 + 5m ), where ( m ) is the number of parameters in millions. The computational budget is 400 million operations, and the model can utilize up to 16 million parameters. We need to find the maximum ( m ) such that ( C(m) leq 400 ).Wait, hold on. The computational complexity is given in terms of operations, right? So, ( C(m) ) is in operations, and the budget is 400 million operations.But ( m ) is in millions of parameters. So, ( m ) is already in millions. So, if ( m = 16 ), that's 16 million parameters.But the computational complexity is ( m^2 + 5m ). So, plugging in ( m = 16 ), we get ( 16^2 + 5*16 = 256 + 80 = 336 ). So, 336 million operations.But the budget is 400 million operations, so 336 is under 400. So, maybe we can go higher than 16 million parameters?Wait, but the problem says the model can utilize up to 16 million parameters. So, does that mean the maximum ( m ) is 16? Or is it that the model can use up to 16 million, but we need to find the maximum ( m ) without exceeding 400 million operations?Wait, let me read the problem again."Given that the deep learning model can utilize up to 16 million parameters, what is the maximum number of parameters ( m ) that can be used without exceeding a computational budget of 400 million operations?"So, the model can use up to 16 million parameters, but we need to find the maximum ( m ) (in millions) such that ( C(m) = m^2 + 5m leq 400 ).So, ( m ) is in millions, so ( m ) can be up to 16, but we need to find the maximum ( m ) where ( m^2 + 5m leq 400 ).So, solving the quadratic inequality ( m^2 + 5m - 400 leq 0 ).Let me solve ( m^2 + 5m - 400 = 0 ).Using quadratic formula:( m = frac{-5 pm sqrt{25 + 1600}}{2} = frac{-5 pm sqrt{1625}}{2} )Compute ( sqrt{1625} ). Since 40^2 = 1600, so sqrt(1625) is 40.311 approx.So, ( m = frac{-5 + 40.311}{2} ≈ frac{35.311}{2} ≈ 17.655 )The other root is negative, so we discard it.So, the critical point is at approximately 17.655 million parameters.But since the model can only utilize up to 16 million parameters, the maximum ( m ) is 16 million.Wait, but hold on. If ( m = 17.655 ) would give exactly 400 million operations, but since the model can only go up to 16 million, then 16 million is the maximum.But wait, let's compute ( C(16) ):( 16^2 + 5*16 = 256 + 80 = 336 ). So, 336 million operations, which is under 400.So, perhaps we can actually go beyond 16 million? But the problem says the model can utilize up to 16 million parameters. So, maybe 16 million is the hard limit, regardless of the computational budget.But the question is asking: "what is the maximum number of parameters ( m ) that can be used without exceeding a computational budget of 400 million operations?"So, it's possible that even though the model can go up to 16 million, the computational budget allows for more, but since the model's limit is 16 million, that's the maximum.But wait, maybe I misread. Let me check."Given that the deep learning model can utilize up to 16 million parameters, what is the maximum number of parameters ( m ) that can be used without exceeding a computational budget of 400 million operations?"So, the model can use up to 16 million, but the computational budget is 400 million. So, if using 16 million parameters only uses 336 million operations, which is under 400, then perhaps we can use more parameters? But the model can't go beyond 16 million. So, the maximum is 16 million.Alternatively, maybe the model can use more than 16 million, but the problem states it can utilize up to 16 million. So, perhaps 16 million is the upper limit.Wait, but let's think again. If the computational budget is 400 million, and the model can use up to 16 million parameters, but using 16 million only uses 336 million operations, then perhaps the model could actually use more parameters without exceeding the budget? But the model's limit is 16 million, so we can't go beyond that. So, the maximum ( m ) is 16 million.But wait, maybe the question is implying that the model can use up to 16 million, but perhaps the computational budget is a separate constraint. So, if the model can use up to 16 million, but the computational budget is 400 million, which allows for more parameters, but since the model can't go beyond 16 million, then 16 million is the answer.Alternatively, maybe the model can use more than 16 million if the computational budget allows, but the problem says it can utilize up to 16 million, so perhaps 16 million is the maximum regardless.Wait, let me think differently. Maybe the model can use more than 16 million, but the problem says it can utilize up to 16 million. So, perhaps 16 million is the maximum, regardless of the computational budget.But in that case, the computational budget is not a limiting factor here because 16 million parameters only use 336 million operations, which is under 400 million.So, the maximum number of parameters is 16 million.But wait, let me check the quadratic solution again.We have ( m^2 + 5m leq 400 ).Solving ( m^2 + 5m - 400 = 0 ), we get ( m approx 17.655 ).So, if the model could use up to 17.655 million parameters, it would exactly use 400 million operations. But since the model is limited to 16 million, the maximum ( m ) is 16 million.Therefore, the answer is 16 million parameters.But wait, let me confirm by plugging in ( m = 17.655 ):( C(17.655) = (17.655)^2 + 5*(17.655) ≈ 311.7 + 88.275 ≈ 400 ). So, yes, that's correct.But since the model can only go up to 16 million, which is less than 17.655, the maximum ( m ) is 16 million.Alternatively, if the model wasn't limited to 16 million, then the maximum ( m ) would be approximately 17.655 million. But given the constraint, it's 16 million.So, the answer is 16 million parameters.Wait, but let me think again. The problem says "the model can utilize up to 16 million parameters." So, does that mean 16 million is the maximum it can handle, regardless of the computational budget? Or is 16 million just a limit, but the computational budget might allow for more?I think it's the former. The model's capacity is limited to 16 million parameters, so even if the computational budget allows for more, the model can't use more than 16 million. Therefore, the maximum ( m ) is 16 million.But just to be thorough, let's compute ( C(16) = 16^2 + 5*16 = 256 + 80 = 336 ) million operations, which is under 400. So, the computational budget isn't the limiting factor here; the model's parameter limit is.Therefore, the maximum number of parameters ( m ) is 16 million.Wait, but the problem says "the model can utilize up to 16 million parameters," so maybe it's implying that 16 million is the maximum, so we can't go beyond that. So, the answer is 16 million.Alternatively, if the model could use more, but the problem states it can only go up to 16 million, then 16 million is the answer.So, summarizing:Sub-problem 1: ( k approx 5.454 times 10^{-8} )Sub-problem 2: ( m = 16 ) million parameters.But let me express the first answer more precisely.Earlier, I calculated ( k approx 5.454 times 10^{-8} ). Let me compute it more accurately.We had ( 8,000,000 * 22.93157 ≈ 183,452,560 )So, ( k = 10 / 183,452,560 ≈ 5.454 x 10^{-8} )But let me compute 10 / 183,452,560.10 divided by 183,452,560 is approximately 5.454 x 10^-8.Expressed as a decimal, that's 0.00000005454.But perhaps we can write it as ( k leq frac{10}{8 times 10^6 times log_2(8 times 10^6)} ), but I think the numerical value is acceptable.Alternatively, we can write it as ( k leq frac{10}{8 times 10^6 times 22.93157} approx 5.454 times 10^{-8} )So, that's the value.Therefore, the final answers are:Sub-problem 1: ( k approx 5.454 times 10^{-8} )Sub-problem 2: ( m = 16 ) million parameters.But wait, in Sub-problem 2, the quadratic solution suggests that ( m ) can be up to approximately 17.655 million, but since the model is limited to 16 million, the answer is 16 million.Yes, that makes sense.So, to recap:1. For the rendering time, the maximum constant ( k ) is approximately ( 5.454 times 10^{-8} ).2. For the deep learning model, the maximum number of parameters ( m ) is 16 million.I think that's it.</think>

🧠:<think>Alright, so I've got this problem about a FIFA content creator who wants to make a video series. Each video focuses on a combination of 4 skill moves out of 12 distinct ones. The first part is asking how many unique videos the creator can make. Hmm, okay, so this sounds like a combinatorics problem. I remember that combinations are used when the order doesn't matter, which is the case here because the skill moves in a video don't have a specific sequence, right?So, the formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items we're choosing. In this case, n is 12 and k is 4. Let me plug those numbers in.C(12, 4) = 12! / (4! * (12 - 4)!) = 12! / (4! * 8!). Calculating that, I know that 12! is 12 × 11 × 10 × 9 × 8!, so the 8! in the numerator and denominator will cancel out. That leaves us with (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1). Let me compute that step by step.First, multiply the numerator: 12 × 11 is 132, 132 × 10 is 1320, and 1320 × 9 is 11880. Now the denominator: 4 × 3 is 12, 12 × 2 is 24, and 24 × 1 is 24. So now, divide 11880 by 24. Let's see, 24 × 495 is 11880 because 24 × 500 is 12000, which is 120 more, so subtract 5 × 24 which is 120. So 500 - 5 = 495. Therefore, the number of unique videos is 495.Okay, that seems straightforward. So the first answer is 495 unique videos.Moving on to the second part. The creator wants each skill move to be featured in exactly 10 different videos. So each of the 12 skill moves should appear in 10 videos. Now, given that there are 495 videos, is this possible?Hmm, let's think about this. Each video uses 4 skill moves, so each video contributes to the count of 4 skill moves. So the total number of skill move appearances across all videos is 495 videos × 4 skill moves per video = 1980 skill move appearances.On the other hand, if each of the 12 skill moves is featured in exactly 10 videos, then the total number of skill move appearances is 12 skill moves × 10 videos each = 120 skill move appearances.Wait, that doesn't add up. 1980 vs. 120? That's a huge discrepancy. So clearly, 120 is much less than 1980. That means it's not possible for each skill move to only appear in 10 videos if we have 495 videos each featuring 4 skill moves. Because the total number of appearances required is way higher.So, the creator's goal isn't feasible with the current setup. Therefore, adjustments need to be made. Let me think about what adjustments can be made.One approach is to either reduce the number of videos or increase the number of skill moves. Alternatively, the creator could change the number of skill moves per video. Let's explore these options.First, let's see if we can adjust the number of skill moves each video features. Suppose instead of 4 skill moves per video, the creator uses fewer or more. Let's denote the number of skill moves per video as k. The total number of skill move appearances would then be C(12, k) × k. We want this to equal 12 × 10 = 120.So, C(12, k) × k = 120. Let's solve for k.We can write this as (12! / (k! * (12 - k)!)) × k = 120.Simplify that: (12! / ((k - 1)! * (12 - k)!)) = 120.So, 12! / ((k - 1)! * (12 - k)!) = 120.Let me compute this for different k values.Starting with k=1: C(12,1) ×1 =12×1=12≠120.k=2: C(12,2)=66, 66×2=132≠120.k=3: C(12,3)=220, 220×3=660≠120.k=4: 495×4=1980≠120.k=5: C(12,5)=792, 792×5=3960≠120.Wait, as k increases, the total number of skill move appearances increases as well. So, actually, the total number of skill move appearances is increasing with k, so it's not possible to get 120 with k>1.Wait, but when k=1, it's 12, which is less than 120. So, perhaps if we have k=2, we get 132, which is more than 120. So, maybe the creator can't have each skill move in exactly 10 videos if they stick to combinations of 4 or more. Alternatively, maybe the creator can't have each skill move in exactly 10 videos if they stick to 12 skill moves.Alternatively, maybe the creator can adjust the number of skill moves. Let's see.Suppose instead of 12 skill moves, the creator uses a different number, say m. Then, the total number of skill move appearances is C(m, k) × k. We want this to equal m × 10, since each of the m skill moves appears in 10 videos.So, C(m, k) × k = 10m.Which simplifies to (m! / (k! (m - k)!)) × k = 10m.Simplify: (m! / ((k - 1)! (m - k)!)) = 10m.Which is equal to m × (m - 1)! / ((k - 1)! (m - k)!)) = 10m.Cancel m from both sides: (m - 1)! / ((k - 1)! (m - k)!)) = 10.So, C(m - 1, k - 1) = 10.So, we need to find integers m and k such that C(m - 1, k - 1) = 10.Looking for combinations that equal 10.C(10,1)=10, so m -1=10, k -1=1 => m=11, k=2.Alternatively, C(5,2)=10, so m -1=5, k -1=2 => m=6, k=3.Or C(10,9)=10, but that would mean m=11, k=10, which seems impractical because k=10 would mean each video has 10 skill moves, which is almost all of them, so not useful.Similarly, C(5,3)=10, so m -1=5, k -1=3 => m=6, k=4.So, possible solutions are:1. m=11, k=2: So 11 skill moves, each video has 2 skill moves, each skill move appears in 10 videos.2. m=6, k=3: 6 skill moves, each video has 3 skill moves, each skill move appears in 10 videos.3. m=6, k=4: 6 skill moves, each video has 4 skill moves, each skill move appears in 10 videos.Wait, but let's check if these actually work.First, m=11, k=2:Total videos: C(11,2)=55.Total skill move appearances: 55×2=110.Each skill move appears in 10 videos: 11×10=110. So yes, that works.Similarly, m=6, k=3:Total videos: C(6,3)=20.Total skill move appearances: 20×3=60.Each skill move appears in 10 videos: 6×10=60. So that works too.Similarly, m=6, k=4:Total videos: C(6,4)=15.Total skill move appearances:15×4=60.Each skill move appears in 10 videos:6×10=60. So that also works.Alternatively, m=10, k= something?Wait, C(m -1, k -1)=10.If m=10, then C(9, k -1)=10. But C(9,1)=9, C(9,2)=36, so no.Similarly, m=7: C(6, k -1)=10. C(6,3)=20, which is too high. C(6,2)=15, still not 10. So m=7 doesn't work.Similarly, m=5: C(4, k -1)=10. C(4,3)=4, C(4,2)=6, so no.So the only possible m are 11 and 6.Therefore, the creator can either reduce the number of skill moves to 6 and have each video feature 3 or 4 skill moves, or reduce to 11 skill moves and have each video feature 2 skill moves.Alternatively, if the creator doesn't want to change the number of skill moves per video, which is 4, then perhaps they need to adjust the number of skill moves or the number of videos.Wait, let's think differently. Suppose the creator wants to keep 4 skill moves per video, but adjust the total number of videos so that each skill move appears in 10 videos.So, total skill move appearances would be 12×10=120.Each video contributes 4 skill moves, so number of videos needed is 120 / 4 = 30 videos.But the number of unique combinations of 4 skill moves from 12 is 495, which is way more than 30. So, the creator can't have each skill move in exactly 10 videos if they use all possible combinations. Instead, they can only use 30 videos, each with 4 unique skill moves, such that each skill move appears in exactly 10 videos.But 30 videos is much less than 495, so the creator would have to limit themselves to only 30 videos instead of using all possible combinations.Alternatively, if the creator wants to use all 495 videos, then each skill move would have to appear in more than 10 videos. Let's calculate how many times each skill move appears in total.Total skill move appearances: 495 × 4 = 1980.Number of skill moves:12.So, each skill move appears in 1980 / 12 = 165 videos.So, each skill move would appear in 165 videos, which is way more than 10. So, if the creator wants each skill move in exactly 10 videos, they can't use all 495 combinations. They have to limit the number of videos.Alternatively, if they don't mind each skill move appearing more than 10 times, then they can use all 495 videos. But since the creator wants exactly 10, they need to adjust.So, the conclusion is that it's not possible with 12 skill moves and 495 videos. The creator needs to either reduce the number of skill moves or reduce the number of videos.If they reduce the number of skill moves to 6, as we saw earlier, they can have each skill move in 10 videos with either 20 or 15 videos. Or if they reduce to 11 skill moves, they can have 55 videos with 2 skill moves each.Alternatively, if they keep 12 skill moves, they need to have only 30 videos, each with 4 skill moves, to have each skill move in 10 videos.So, the creator has a few options:1. Keep 12 skill moves, make 30 videos, each with 4 skill moves, ensuring each skill move is in 10 videos.2. Reduce to 11 skill moves, make 55 videos, each with 2 skill moves, each skill move in 10 videos.3. Reduce to 6 skill moves, make 20 videos, each with 3 skill moves, each skill move in 10 videos.Or another option is to have 15 videos with 4 skill moves each, but that would require only 6 skill moves.So, depending on what the creator prefers—whether they want more videos, fewer skill moves, or fewer videos—the adjustments can be made accordingly.I think the key takeaway is that with 12 skill moves and 4 per video, the total number of required appearances is too high for each skill move to only be in 10 videos. So, either the number of skill moves needs to decrease, the number of videos needs to decrease, or the number of skill moves per video needs to change.Another angle is to think about this as a design of experiments or a block design problem, where each skill move is a treatment and each video is a block. The creator wants a balanced incomplete block design (BIBD) where each treatment appears in r blocks, each block contains k treatments, and every pair of treatments appears in λ blocks. But in this case, the creator only specifies that each treatment appears in r=10 blocks, but doesn't specify anything about pairs. So, it's a more general question of whether a BIBD exists with these parameters.But maybe that's overcomplicating it. The main point is that the total number of appearances must match on both sides: total videos × skill moves per video = total skill moves × appearances per skill move.So, 495 videos × 4 = 1980, which must equal 12 × 10 = 120. Since 1980 ≠ 120, it's impossible. Therefore, the creator needs to adjust either the number of skill moves, the number of videos, or the number of skill moves per video to make the totals match.So, summarizing:1. The number of unique videos is 495.2. It's not possible to have each skill move in exactly 10 videos with 12 skill moves and 495 videos. The creator needs to adjust either the number of skill moves, the number of videos, or the number of skill moves per video. For example, reducing to 6 skill moves and making 20 videos with 3 skill moves each would work, as 6×10=60 total appearances, and 20×3=60.Final Answer1. The number of unique videos is boxed{495}.2. It is not possible with the given parameters. Adjustments such as reducing the number of skill moves or videos are necessary. For example, using 6 skill moves and 20 videos with 3 skill moves each would make it feasible.</think>

🧠:<think>Okay, so I have this problem about integrating smart technology into public safety, which sounds pretty cool. The politician is proposing a system that uses advanced algorithms to optimize where emergency response units are deployed in a city modeled as an n x n grid. Each block has a probability p_ij of an emergency happening there. The first part asks me to formulate an optimization problem to maximize the expected coverage, given a budget B and deployment costs C_ij for each block. Hmm, okay. So, I think this is a linear programming problem because we're dealing with maximizing a linear objective function subject to linear constraints.Let me break it down. The expected coverage is the sum of the probabilities p_ij for each block where we deploy a unit. So, if x_ij is a binary variable (0 or 1) indicating whether we deploy a unit to block (i,j), then the expected coverage is the sum over all i and j of p_ij * x_ij. The goal is to maximize this sum. But we have a constraint on the budget. The total cost of deploying units can't exceed B. The cost for each block is C_ij times x_ij, so the total cost is the sum over all i and j of C_ij * x_ij, and this has to be less than or equal to B.So putting it together, the optimization problem should be:Maximize Σ (from i=1 to n) Σ (from j=1 to n) p_ij * x_ijSubject to:Σ (from i=1 to n) Σ (from j=1 to n) C_ij * x_ij ≤ BAnd x_ij is binary (0 or 1) for each i, j.Wait, is that all? I think so. It's a binary integer linear programming problem because the variables x_ij are binary. So, the problem is to choose which blocks to deploy units to, such that the total cost doesn't exceed the budget, and we maximize the expected coverage.Now, moving on to the second part. The algorithm's runtime is influenced by the complexity of calculating the expected coverage. Each block's expected coverage calculation involves a matrix multiplication with time complexity O(n^3). I need to find the overall time complexity of the algorithm in terms of n.Hmm, okay. So, if each block's calculation is O(n^3), and there are n x n blocks, then naively, the total time complexity would be O(n^3 * n^2) = O(n^5). But that seems really high. Maybe I'm misunderstanding.Wait, the problem says the expected coverage calculation for each block involves a matrix multiplication with time complexity O(n^3). So, does that mean for each block, we have to perform a matrix multiplication that takes O(n^3) time? If so, then for each block, it's O(n^3), and there are n^2 blocks, so the total complexity would be O(n^5). But that seems excessive. Maybe the matrix multiplication isn't per block, but for the entire grid?Wait, let me read it again. "If the expected coverage calculation for each block involves a matrix multiplication operation with time complexity O(n^3), determine the overall time complexity of the algorithm in terms of n."Hmm, so for each block, calculating its expected coverage involves a matrix multiplication that is O(n^3). So, if we have n^2 blocks, each requiring O(n^3) operations, then the total time complexity would be O(n^5). But that seems too high, maybe I'm misinterpreting.Alternatively, perhaps the matrix multiplication is part of the algorithm, not per block. Maybe the algorithm uses matrix multiplication in some step, and that step has O(n^3) complexity. But the question says "for each block", so I think it's per block. So, each block's coverage calculation is O(n^3), so n^2 blocks would make it O(n^5). But wait, maybe the matrix multiplication is not per block but for the entire grid. Let me think. If the expected coverage is calculated by multiplying matrices, perhaps the grid is represented as a matrix, and some operation is done on it. For example, if the coverage is calculated using some transformation matrix, then multiplying two n x n matrices would take O(n^3) time. So, if the algorithm does this once, the complexity is O(n^3). But the question says "for each block", so maybe it's not the case.Alternatively, maybe for each block, the algorithm does a matrix multiplication that is O(n^3). So, for each of the n^2 blocks, it's O(n^3), leading to O(n^5). But that seems very high. Maybe the question is referring to the calculation of the expected coverage in total, not per block. Let me read it again."If the expected coverage calculation for each block involves a matrix multiplication operation with time complexity O(n^3), determine the overall time complexity of the algorithm in terms of n."Hmm, so for each block, the calculation is O(n^3). So, if we have n^2 blocks, each with O(n^3) operations, then the total is O(n^5). But that seems too much. Maybe the matrix multiplication is a one-time operation, not per block. But the wording says "for each block", so I think it's per block.Alternatively, perhaps the matrix multiplication is part of the algorithm, not per block. Maybe the algorithm uses matrix operations to compute the expected coverage, and that operation is O(n^3). So, if the algorithm's runtime is dominated by this matrix multiplication, then the overall time complexity is O(n^3). But the question says "for each block", so I'm confused.Wait, maybe the expected coverage isn't just the sum of p_ij x_ij, but something more complicated that involves matrix multiplication. For example, if the coverage depends on some interaction between blocks, modeled by a matrix, then computing the coverage might involve multiplying matrices, each of size n x n, leading to O(n^3) time. But then, how does that relate to each block?Alternatively, perhaps the coverage calculation for each block requires a matrix multiplication, so for each block, you have to do O(n^3) work. So, for n^2 blocks, it's O(n^5). But that seems impractical. Maybe the question is assuming that the matrix multiplication is done once, not per block, so the overall complexity is O(n^3). Wait, perhaps the expected coverage is computed as a matrix product, say, P * X, where P is the probability matrix and X is the deployment matrix. Then, the multiplication would be O(n^3). But if the algorithm needs to compute this for each block, maybe it's not the case.I'm getting a bit stuck here. Let me try to parse the question again: "the expected coverage calculation for each block involves a matrix multiplication operation with time complexity O(n^3)". So, for each block, when calculating its expected coverage, you have to do a matrix multiplication that takes O(n^3) time. So, if you have n^2 blocks, each needing O(n^3) operations, then the total time is O(n^5). But that seems too high, so maybe I'm misinterpreting.Alternatively, maybe the matrix multiplication is part of the algorithm, not per block. For example, the algorithm might need to compute something like the expected coverage across the entire grid, which involves a matrix multiplication of two n x n matrices, taking O(n^3) time. Then, the overall time complexity would be O(n^3). But the question says "for each block", so I'm not sure.Wait, maybe the expected coverage is computed using a matrix that represents the probabilities and another that represents the deployment, and their product gives the coverage. So, if you have to compute this product, it's O(n^3). But then, if the algorithm needs to compute this multiple times, say, in each iteration of an optimization process, then the complexity could be higher. But the question doesn't specify that; it just says the calculation for each block involves a matrix multiplication with O(n^3) time.I think I need to go with the straightforward interpretation: for each block, calculating its expected coverage involves a matrix multiplication that takes O(n^3) time. Therefore, for n^2 blocks, the total time complexity is O(n^5). But that seems really high, so maybe the question is referring to the entire calculation, not per block. Alternatively, perhaps the matrix multiplication is a one-time operation, not per block.Wait, another thought: maybe the expected coverage isn't just the sum of p_ij x_ij, but involves some transformation. For example, if the coverage depends on neighboring blocks or something, modeled by a matrix, then computing the coverage might involve a matrix multiplication. So, if you have to compute this for each block, it's O(n^3) per block, leading to O(n^5). But that seems too much.Alternatively, maybe the matrix multiplication is part of the algorithm's process, not per block. For example, the algorithm might precompute some matrix that helps in calculating the coverage, and that precomputation is O(n^3). Then, the overall complexity is O(n^3). But the question says "for each block", so I'm not sure.I think I need to make a decision here. Given the wording, I think it's saying that for each block, the expected coverage calculation involves a matrix multiplication with O(n^3) time. So, for each block, it's O(n^3), and there are n^2 blocks, so total time is O(n^5). But that seems too high, so maybe the question is referring to the entire coverage calculation, not per block. If it's the entire coverage, then it's O(n^3). Wait, the question says "the expected coverage calculation for each block involves a matrix multiplication operation with time complexity O(n^3)". So, for each individual block, the calculation is O(n^3). So, if you have n^2 blocks, each needing O(n^3) operations, the total is O(n^5). But that seems too much. Maybe the question is referring to the entire calculation, not per block. Alternatively, perhaps the matrix multiplication is a one-time operation, and the per-block calculation is something else. Maybe the matrix multiplication is used to precompute something, and then the per-block calculation is O(1). But the question says "for each block", so I think it's per block.Wait, another angle: perhaps the expected coverage isn't just the sum of p_ij x_ij, but involves some more complex calculation, like considering the influence of neighboring blocks, which might require matrix operations. So, for each block, calculating its contribution to the coverage might involve multiplying matrices, leading to O(n^3) per block. But that still leads to O(n^5).Alternatively, maybe the matrix multiplication is part of the algorithm's process, not per block. For example, if the algorithm uses a matrix to represent the grid and performs operations on it, the time complexity is O(n^3). So, the overall algorithm's time complexity is O(n^3). But the question says "for each block", so I'm confused.I think I need to stick with the straightforward interpretation. If each block's expected coverage calculation involves a matrix multiplication of O(n^3), then for n^2 blocks, it's O(n^5). But that seems too high, so maybe the question is referring to the entire coverage calculation, not per block. If the entire coverage calculation is O(n^3), then the overall time complexity is O(n^3).Wait, the question says "the expected coverage calculation for each block involves a matrix multiplication operation with time complexity O(n^3)". So, for each block, it's O(n^3). So, total is O(n^5). But that seems too high. Maybe the question is referring to the entire calculation, not per block. Alternatively, perhaps the matrix multiplication is a one-time operation, not per block.Wait, maybe the matrix multiplication is done once for the entire grid, so it's O(n^3), and then the rest of the algorithm is linear or something. But the question says "for each block", so I think it's per block.Alternatively, maybe the matrix multiplication is part of the algorithm's process, not per block. For example, the algorithm might need to compute some matrix product that represents the coverage, and that product is O(n^3). So, the overall time complexity is O(n^3). But the question says "for each block", so I'm not sure.I think I need to go with the straightforward interpretation: for each block, the expected coverage calculation involves a matrix multiplication of O(n^3), so for n^2 blocks, it's O(n^5). But that seems too high, so maybe the question is referring to the entire calculation, not per block. If it's the entire calculation, then it's O(n^3).Wait, another thought: perhaps the expected coverage is computed as a matrix product, say, P * X, where P is n x n and X is n x n, so the multiplication is O(n^3). So, the entire coverage calculation is O(n^3). Then, the algorithm's time complexity is O(n^3). But the question says "for each block", so maybe it's not.I think I'm overcomplicating this. Let me try to answer based on the straightforward interpretation.For part 1, the optimization problem is a binary integer linear program:Maximize ΣΣ p_ij x_ijSubject to ΣΣ C_ij x_ij ≤ Bx_ij ∈ {0,1}For part 2, if each block's coverage calculation involves O(n^3) operations, and there are n^2 blocks, then the total time complexity is O(n^5). But that seems too high, so maybe it's O(n^3). I'm not sure, but I'll go with O(n^5) based on the wording.Wait, but maybe the matrix multiplication is not per block but for the entire grid. So, the expected coverage is calculated once, involving a matrix multiplication of O(n^3). So, the overall time complexity is O(n^3). That makes more sense. So, the algorithm's runtime is dominated by the matrix multiplication, which is O(n^3). Therefore, the overall time complexity is O(n^3).I think that's more plausible. So, the answer for part 2 is O(n^3).But I'm still a bit confused because the question says "for each block". Maybe it's a translation issue. If it's for each block, then O(n^5), but if it's for the entire calculation, then O(n^3). Given that matrix multiplication is typically a single operation, not per element, I think it's the latter. So, the overall time complexity is O(n^3).</think>