🧠:<think>Alright, so I have these two sub-problems to solve related to Martin Motnik's concert tour. Let me tackle them one by one.Starting with Sub-problem 1: Martin wants his tour to follow a Fibonacci sequence in both x and y coordinates. He starts at (x₁, y₁) and ends at (xₙ, yₙ). I need to find the coordinates of the 10th city. Hmm, okay. So, I think this means that each coordinate (x and y) is generated by the Fibonacci sequence. Let me recall how the Fibonacci sequence works. It starts with F(1) = 1, F(2) = 1, and each subsequent term is the sum of the two preceding ones. So, F(3) = 2, F(4) = 3, F(5) = 5, and so on.Wait, so does that mean each city's x and y coordinates are the Fibonacci numbers? Like, the first city is (F(1), F(1)), the second is (F(2), F(2)), and so on? Or is it that the path between cities follows the Fibonacci sequence? Hmm, the problem says the coordinates are derived from the Fibonacci sequence. Maybe each coordinate is a Fibonacci number. So, for the 10th city, both x and y would be F(10).Let me confirm. The Fibonacci sequence starting from F(1) = 1, F(2) = 1:F(1) = 1F(2) = 1F(3) = F(2) + F(1) = 1 + 1 = 2F(4) = F(3) + F(2) = 2 + 1 = 3F(5) = F(4) + F(3) = 3 + 2 = 5F(6) = 5 + 3 = 8F(7) = 8 + 5 = 13F(8) = 13 + 8 = 21F(9) = 21 + 13 = 34F(10) = 34 + 21 = 55So, F(10) is 55. Therefore, the 10th city would have coordinates (55, 55). That seems straightforward.Moving on to Sub-problem 2: Martin spends time in each city following T(i) = 2^(i-1), where i is the position in the tour. I need to calculate the total time up to the 10th city.So, T(1) = 2^(1-1) = 2^0 = 1T(2) = 2^(2-1) = 2^1 = 2T(3) = 2^(3-1) = 2^2 = 4And so on, up to T(10) = 2^(10-1) = 2^9 = 512This looks like a geometric series where each term is double the previous one. The total time is the sum from i=1 to i=10 of 2^(i-1). The formula for the sum of a geometric series is S_n = a*(r^n - 1)/(r - 1), where a is the first term, r is the common ratio, and n is the number of terms.Here, a = 1 (since T(1)=1), r = 2, and n = 10.So, S_10 = 1*(2^10 - 1)/(2 - 1) = (1024 - 1)/1 = 1023.Therefore, the total time spent is 1023 units.Wait, let me double-check the calculations. For the Fibonacci sequence, F(10) is indeed 55, so the coordinates are (55,55). For the total time, summing 2^0 to 2^9 is 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512. Adding these up:1 + 2 = 33 + 4 = 77 + 8 = 1515 + 16 = 3131 + 32 = 6363 + 64 = 127127 + 128 = 255255 + 256 = 511511 + 512 = 1023Yes, that adds up correctly. So, the total time is 1023.Final AnswerSub-problem 1: The coordinates of the 10th city are boxed{(55, 55)}.Sub-problem 2: The total time spent is boxed{1023}.</think>

🧠:<think>First, I'll tackle the first problem. The supplier needs to provide silk fabric for both chairs and window drapes. The fabric is 54 inches wide and costs 150 per yard. The chairs require 120 square feet of fabric, and the drapes require 80 square feet, totaling 200 square feet.Since fabric is sold by the yard, I need to convert the total area from square feet to square yards. There are 9 square feet in a square yard, so dividing 200 by 9 gives approximately 22.222 square yards. To avoid waste, I'll round up to 23 square yards.Next, I'll calculate the total cost by multiplying the number of yards by the cost per yard: 23 yards * 150 = 3,450.Now, for the second problem, the customer needs fabric for 12 windows, each requiring 3 yards of fabric. This totals 36 yards. The fabric's pattern repeats every 1.5 yards, so I need to ensure that each window's fabric starts at the same point in the pattern.To achieve perfect alignment, the total fabric length must be a multiple of the pattern repeat. Since 36 yards is already a multiple of 1.5 yards (36 / 1.5 = 24), no additional fabric is needed. Therefore, the supplier needs to provide exactly 36 yards of fabric.Finally, calculating the total cost: 36 yards * 150 = 5,400.</think>

🧠:<think>Alright, so I've got this problem about a software engineer working on a custom HashMap. It's divided into two parts. Let me try to figure out each step carefully.Starting with the first part: calculating the index using the custom hashing function. The function is given as h(x) = (a*x + b) mod p mod n. The constants are a=17, b=31, p=101, and n=20. The key x is 42.Okay, so first, I need to compute h(42). Let's break it down step by step.First, calculate a*x + b. That would be 17*42 + 31. Let me compute 17*42. Hmm, 17*40 is 680, and 17*2 is 34, so 680 + 34 = 714. Then add 31: 714 + 31 = 745.Now, take that result modulo p, which is 101. So 745 mod 101. To compute this, I need to find how many times 101 fits into 745 and what the remainder is.Let me calculate 101*7 = 707. Subtract that from 745: 745 - 707 = 38. So 745 mod 101 is 38.Next, take that result modulo n, which is 20. So 38 mod 20. 20*1=20, 38-20=18. So 38 mod 20 is 18.Therefore, the index for key x=42 is 18.Wait, let me double-check my calculations to make sure I didn't make a mistake.17*42: 17*40=680, 17*2=34, so 680+34=714. Correct.714 +31=745. Correct.745 divided by 101: 101*7=707, 745-707=38. Correct.38 mod 20: 20*1=20, 38-20=18. Correct.Okay, so the first part seems solid. The index is 18.Moving on to the second part: collision resolution using a secondary hash function h'(x) = (c*x + d) mod q. The constants here are c=13, d=7, q=97, and the key x is still 42.Since the key x=42 caused a collision at index 18, we need to compute the new index using h'(42).So, let's compute h'(42). First, calculate c*x + d. That's 13*42 +7.13*40=520, 13*2=26, so 520+26=546. Then add 7: 546 +7=553.Now, take that modulo q, which is 97. So 553 mod 97.To find this, let's see how many times 97 fits into 553.97*5=485. 553 - 485=68. So 553 mod 97 is 68.Therefore, the new index computed by the secondary hash function is 68.Wait, hold on. Let me verify that again.13*42: 13*40=520, 13*2=26, so 520+26=546. Correct.546 +7=553. Correct.553 divided by 97: 97*5=485, 553-485=68. Correct.So yes, 553 mod 97 is 68. Therefore, the secondary index is 68.But wait, in the context of collision resolution, does this mean that the new index is 68? Or is there another step?In some collision resolution strategies, like linear probing or quadratic probing, you might add the secondary hash value to the original index. But in this case, the problem states that the secondary hash function is used to resolve collisions, so I think it's just using h'(x) as the new index.But let me think again. The problem says: "determine the new index computed by the secondary hash function for this key." So it's just h'(x), which is 68.But wait, the array size is n=20. So 68 is larger than 20. How does that work? Because in the original hash function, we had mod n=20, but in the secondary hash function, it's mod q=97.Hmm, that seems odd because the secondary index might be outside the array bounds. Maybe I misunderstood the problem.Wait, perhaps the secondary hash function is used to compute the step size for probing, rather than the actual index. So in linear probing, you might add h'(x) each time, but in this case, the problem says it's a novel strategy.Wait, the problem says: "using a secondary hash function h'(x) to resolve collisions." So it might be that when a collision occurs at index h(x), the secondary function h'(x) is used to find the next index. But since h'(x) can be larger than n, perhaps we take h'(x) mod n?Wait, let me check the problem statement again.It says: "the secondary hash function is defined as h'(x) = (c x + d) mod q." So it's mod q, which is 97, not mod n. So the result is between 0 and 96.But the array size is 20, so 68 is larger than 20. So how is this used?Hmm, perhaps the secondary hash function is used to compute the step size, and then you add that to the original index modulo n. So the new index would be (h(x) + h'(x)) mod n.But the problem doesn't specify that. It just says "determine the new index computed by the secondary hash function for this key." So maybe it's just h'(x) mod n?Wait, no, because h'(x) is defined as mod q, which is 97, so h'(x) is 68. If we need an index in the array of size 20, perhaps we take 68 mod 20.68 divided by 20 is 3*20=60, remainder 8. So 68 mod 20 is 8.But the problem didn't specify that. It just said to compute h'(x). So I'm a bit confused.Wait, let me read the problem again."2. The novel collision resolution strategy involves using a secondary hash function h'(x) to resolve collisions. The secondary hash function is defined as h'(x) = (c x + d) mod q... determine the new index computed by the secondary hash function for this key."So it's just h'(x). So the new index is 68. But 68 is larger than n=20. So perhaps the array is larger? Or maybe the secondary hash function is used in a different way.Wait, maybe the array size is still n=20, but the secondary hash function is used to compute the step size for probing. So the new index would be (original index + h'(x)) mod n.So original index was 18, h'(x)=68. So 18 +68=86. 86 mod 20 is 6. So new index is 6.But the problem doesn't specify that. It just says "determine the new index computed by the secondary hash function for this key." So I think it's just h'(x)=68.But 68 is larger than n=20. So perhaps the array is actually size q=97? But in the first part, n=20.Wait, maybe the array size is still 20, but the secondary hash function is mod q=97, so the result is 68, but then we have to map it back to the array size. So 68 mod 20=8.But again, the problem doesn't specify that. It just says to compute h'(x). So I think the answer is 68.But let me think again. If the array size is 20, and the secondary hash function is mod 97, then the index computed by h'(x) is 68, but since the array is only size 20, we have to wrap around. So 68 mod 20=8.But the problem didn't specify that. It just says to compute h'(x). So maybe the answer is 68.Wait, but in the context of collision resolution, the secondary function is used to find the next position. So if the array is size 20, and h'(x)=68, then the next position would be (18 +68) mod 20= (86) mod20=6.But again, the problem doesn't specify that. It just says to compute h'(x). So I think the answer is 68.But to be safe, maybe I should compute both and see which makes sense.Wait, let me check the problem statement again."2. The novel collision resolution strategy involves using a secondary hash function h'(x) to resolve collisions. The secondary hash function is defined as h'(x) = (c x + d) mod q... determine the new index computed by the secondary hash function for this key."So it's just h'(x). So the answer is 68.But 68 is larger than n=20. So perhaps the array is actually size q=97? But in the first part, n=20.Wait, maybe the array size is still 20, but the secondary hash function is used to compute the step size. So the new index is (h(x) + h'(x)) mod n.So h(x)=18, h'(x)=68. 18+68=86. 86 mod20=6.But the problem didn't specify that. It just said to compute h'(x). So I think the answer is 68.But I'm a bit confused because 68 is larger than the array size. Maybe the problem expects the answer to be 68 regardless of the array size.Alternatively, perhaps the secondary hash function is used to compute the next index in a way that wraps around the array. So if the array is size 20, and the secondary hash is 68, then 68 mod20=8. So the new index is 8.But again, the problem didn't specify that. It just said to compute h'(x). So I think the answer is 68.Wait, let me think about how collision resolution typically works. In linear probing, you add 1 each time. In quadratic probing, you add i^2. In double hashing, you use a second hash function to determine the step size.So in double hashing, the step size is h'(x), and the new index is (h(x) + i*h'(x)) mod n, where i is the probe number.But in this case, the problem says it's a novel strategy, so maybe it's just using h'(x) as the new index. But if h'(x) is larger than n, then it's not a valid index.Alternatively, maybe h'(x) is computed mod n. So h'(x)=68 mod20=8.But the problem didn't specify that. It just said h'(x)= (c x + d) mod q.So I think the answer is 68.But I'm not entirely sure. Maybe I should go with 68.Wait, let me check the problem statement again."2. The novel collision resolution strategy involves using a secondary hash function h'(x) to resolve collisions. The secondary hash function is defined as h'(x) = (c x + d) mod q... determine the new index computed by the secondary hash function for this key."So it's just h'(x). So the answer is 68.But in the context of the array size being 20, 68 is not a valid index. So perhaps the problem expects the answer to be 68, assuming that the array size is q=97.But in the first part, n=20. So maybe the array size is still 20, and the secondary hash function is used to compute the step size.Wait, maybe the secondary hash function is used to compute the step size, so the new index is (h(x) + h'(x)) mod n.So h(x)=18, h'(x)=68. So 18+68=86. 86 mod20=6.But the problem didn't specify that. It just said to compute h'(x). So I think the answer is 68.But I'm still unsure. Maybe I should compute both and see which one makes sense.Alternatively, perhaps the secondary hash function is used to compute the new index directly, but since the array size is 20, we take h'(x) mod n.So 68 mod20=8.But again, the problem didn't specify that. It just said to compute h'(x).Hmm, I think I need to go with the problem's instructions. It says to compute h'(x), which is 68.So the answer is 68.But to be thorough, let me compute both possibilities.If h'(x)=68, and the array size is 20, then 68 mod20=8.Alternatively, if the array size is 97, then 68 is a valid index.But in the first part, n=20, so I think the array size is 20. Therefore, the secondary hash function's result should be mod20.So 68 mod20=8.But the problem didn't specify that. It just said to compute h'(x). So I think the answer is 68.Wait, but in the context of the problem, the secondary hash function is used to resolve collisions in the same array of size n=20. So it's likely that the secondary hash function's result is mod n.Therefore, h'(x)=68 mod20=8.But the problem didn't specify that. It just said h'(x)= (c x + d) mod q.So I think the answer is 68.But I'm still confused. Maybe I should look up how secondary hash functions are typically used.In double hashing, the secondary hash function is used to compute the step size, which is then added to the original index. The step size is usually computed as h'(x) mod (n-1) or something like that.But in this problem, it's a novel strategy, so maybe it's different.Wait, the problem says: "using a secondary hash function h'(x) to resolve collisions." So it's not necessarily double hashing. It could be that the secondary function is used to compute the next index directly.But if the array size is 20, and h'(x)=68, then 68 is not a valid index. So perhaps the secondary function is used in a way that wraps around the array.So 68 mod20=8.But again, the problem didn't specify that. It just said to compute h'(x).I think I need to stick with the problem's instructions. It says to compute h'(x), which is 68.Therefore, the new index is 68.But wait, in the context of the array size being 20, 68 is larger than 20, so it's not a valid index. So perhaps the problem expects the answer to be 68, assuming that the array size is 97, which is q.But in the first part, n=20, so the array size is 20. Therefore, the secondary hash function's result should be mod20.So 68 mod20=8.But the problem didn't specify that. It just said to compute h'(x).Hmm, this is a bit ambiguous. But I think the problem expects the answer to be 68, as per the definition of h'(x).So I'll go with 68.But just to be safe, I'll note that if the array size is 20, the index would be 8. But since the problem didn't specify, I think 68 is the answer.So, summarizing:1. The index for x=42 is 18.2. The new index using the secondary function is 68.But wait, let me check the calculations again.First part:h(x) = (17*42 +31) mod101 mod20.17*42=714, 714+31=745.745 mod101: 101*7=707, 745-707=38.38 mod20=18. Correct.Second part:h'(x)= (13*42 +7) mod97.13*42=546, 546+7=553.553 mod97: 97*5=485, 553-485=68. Correct.So yes, 68 is the answer.Therefore, the answers are 18 and 68.</think>

🧠:<think>First, I need to determine the dimensions of the community center. I know that the length ( L ) is 20 meters longer than the width ( W ), so I can express ( L ) as ( W + 20 ). The total area of the community center is given as 1800 square meters, which means that ( L times W = 1800 ). Substituting ( L ) with ( W + 20 ) in the area equation gives me a quadratic equation: ( W^2 + 20W - 1800 = 0 ). Solving this equation using the quadratic formula, I find that the width ( W ) is 30 meters, and consequently, the length ( L ) is 50 meters.Next, I need to calculate the width ( G ) of the garden surrounding the community center. The garden has a uniform width around the entire building, so the total area including the garden is ( (L + 2G) times (W + 2G) ). The area of the garden alone is the total area minus the area of the community center, which should equal 1800 square meters. Setting up the equation ( (50 + 2G)(30 + 2G) - 1800 = 1800 ) and simplifying it leads to a quadratic equation: ( 4G^2 + 160G - 1800 = 0 ). Solving this equation using the quadratic formula, I find that the width ( G ) is 7.5 meters.</think>