🧠:<think>Okay, so I have this problem about digital advertising campaigns using machine learning to optimize ad placements. It's split into two parts, and I need to solve both. Let me start with the first one.Problem 1: Maximizing CTRThe model's predicted CTR is given by the function:[ P(t, d, h) = frac{e^{alpha t + beta d + gamma h}}{1 + e^{alpha t + beta d + gamma h}} ]We have parameters α = 0.05, β = 1.2, γ = 0.8. We need to find the time of day t that maximizes P(t, 0.5, 0.7). So, d is 0.5 and h is 0.7.First, let me write out the function with the given values:[ P(t) = frac{e^{0.05t + 1.2*0.5 + 0.8*0.7}}{1 + e^{0.05t + 1.2*0.5 + 0.8*0.7}} ]Let me compute the constants first:1.2 * 0.5 = 0.60.8 * 0.7 = 0.56So, the exponent becomes:0.05t + 0.6 + 0.56 = 0.05t + 1.16So, the function simplifies to:[ P(t) = frac{e^{0.05t + 1.16}}{1 + e^{0.05t + 1.16}} ]Hmm, this is a logistic function. It's an S-shaped curve that increases with t. Since the exponent is linear in t, the function P(t) will increase as t increases. But wait, does it have a maximum?Wait, actually, as t approaches infinity, the exponent goes to infinity, so e^{0.05t + 1.16} dominates, so P(t) approaches 1. Similarly, as t approaches negative infinity, the exponent approaches negative infinity, so e^{...} approaches 0, so P(t) approaches 0. So, the function is monotonically increasing with t. Therefore, it doesn't have a maximum at a finite t; it just asymptotically approaches 1 as t increases.But that can't be right because time of day is limited. t is in hours, so t is between 0 and 24, right? So, within this range, the function is increasing, so the maximum CTR occurs at t = 24.Wait, but maybe I'm missing something. Let me double-check the function. It's a logistic function, which is indeed increasing everywhere. So, within the domain of t from 0 to 24, the maximum CTR is at t=24.But wait, is t allowed to be beyond 24? The problem says t represents the time of day in hours, so I think t is between 0 and 24. So, the maximum occurs at t=24.But let me think again. Maybe I made a mistake in interpreting the function. Let me take the derivative of P(t) with respect to t and see where it's zero.The derivative of P(t) is:P'(t) = [e^{0.05t + 1.16} * 0.05 * (1 + e^{0.05t + 1.16}) - e^{0.05t + 1.16} * e^{0.05t + 1.16} * 0.05] / (1 + e^{0.05t + 1.16})^2Simplify numerator:0.05 e^{0.05t + 1.16} (1 + e^{0.05t + 1.16} - e^{0.05t + 1.16}) ) = 0.05 e^{0.05t + 1.16}So, P'(t) = 0.05 e^{0.05t + 1.16} / (1 + e^{0.05t + 1.16})^2Which is always positive because all terms are positive. So, P(t) is strictly increasing in t. Therefore, the maximum occurs at the maximum t, which is 24.But wait, let me check the problem statement again. It says "the time of day t (in hours)", so t is between 0 and 24. So, the maximum CTR is at t=24.But wait, in reality, CTR might peak at certain times, but according to this model, it's always increasing. So, the answer is t=24.Wait, but maybe I should consider that t is in a 24-hour period, so t=24 is the same as t=0. So, perhaps the maximum is at t=24, but since it's the same as t=0, maybe the maximum is at t=24, but in terms of time of day, it's midnight. Hmm, but that might not make sense in real life, but according to the model, it's increasing.Alternatively, maybe I made a mistake in the exponent. Let me check:The exponent is αt + βd + γh. So, with α=0.05, which is positive, so as t increases, the exponent increases, so P(t) increases.Therefore, the maximum CTR is at t=24.Wait, but let me compute P(t) at t=24:Compute exponent: 0.05*24 + 1.16 = 1.2 + 1.16 = 2.36So, e^{2.36} ≈ e^2 is about 7.389, e^0.36 ≈ 1.433, so e^2.36 ≈ 7.389 * 1.433 ≈ 10.56So, P(24) ≈ 10.56 / (1 + 10.56) ≈ 10.56 / 11.56 ≈ 0.914Similarly, at t=0:Exponent: 0 + 1.16 = 1.16e^1.16 ≈ 3.19So, P(0) ≈ 3.19 / (1 + 3.19) ≈ 3.19 / 4.19 ≈ 0.76So, yes, it increases from ~0.76 to ~0.914 as t goes from 0 to 24.Therefore, the maximum CTR is at t=24.But wait, in reality, t=24 is the same as t=0, so maybe the model is assuming t is in a 24-hour period, so t=24 is the same as t=0. So, perhaps the maximum is at t=24, but in terms of time of day, it's midnight. Hmm, but that might not make sense in real life, but according to the model, it's increasing.Alternatively, maybe the model is designed such that t is a continuous variable beyond 24, but in reality, t is cyclic. But the problem doesn't specify that, so I think we can assume t is between 0 and 24, and the maximum is at t=24.Wait, but let me think again. Maybe I should consider that t is in a 24-hour period, so t=24 is the same as t=0, so the maximum is at t=24, but in terms of time of day, it's midnight. Hmm, but that might not make sense in real life, but according to the model, it's increasing.Alternatively, maybe I should consider that t is a continuous variable beyond 24, but the problem says t is time of day in hours, so I think it's between 0 and 24.Therefore, the answer is t=24.Wait, but let me check the derivative again. Since P'(t) is always positive, the function is increasing, so the maximum is at t=24.Problem 2: Minimizing Cost with Quadratic FunctionThe advertising budget follows a quadratic cost function:C(x) = a x² + b x + c, where a=0.0005, b=0.3, c=100.We need to achieve a target number of clicks T=500. Each impression has the maximum predicted CTR from Problem 1, which we found to be P_max ≈ 0.914.Wait, but actually, in Problem 1, we found that P(t) is maximized at t=24, with P(24) ≈ 0.914. So, each impression has a CTR of 0.914.Therefore, the number of clicks is x * P_max = x * 0.914.We need x * 0.914 ≥ 500.So, x ≥ 500 / 0.914 ≈ 547.08.Since x must be an integer, x ≥ 548.But we need to minimize the cost C(x) = 0.0005 x² + 0.3 x + 100, subject to x ≥ 548.But wait, the cost function is quadratic, which is convex, so the minimum occurs at the vertex. However, since we have a constraint x ≥ 548, the minimum cost under the constraint will be at x=548.Wait, but let me think again. The cost function is C(x) = 0.0005 x² + 0.3 x + 100.The vertex of this parabola is at x = -b/(2a) = -0.3/(2*0.0005) = -0.3 / 0.001 = -300.But since x cannot be negative, the minimum of the cost function is at x=0, but we have a constraint x ≥ 548. Therefore, the minimum cost under the constraint is at x=548.Wait, but let me verify. The cost function is increasing for x > -b/(2a). Since -b/(2a) is negative, the function is increasing for all x > 0. Therefore, the cost increases as x increases. So, to minimize the cost while satisfying x ≥ 548, we should choose the smallest x that satisfies the constraint, which is x=548.But wait, let me compute the cost at x=548 and see.C(548) = 0.0005*(548)^2 + 0.3*548 + 100Compute 548²: 548*548. Let's compute 500²=250000, 48²=2304, and cross term 2*500*48=48000. So, (500+48)^2=500² + 2*500*48 +48²=250000 + 48000 + 2304=250000+48000=298000+2304=300,304.So, 0.0005*300304=0.0005*300304=150.1520.3*548=164.4So, total C(548)=150.152 + 164.4 + 100=150.152+164.4=314.552+100=414.552So, approximately 414.55.But wait, is this the minimum? Since the cost function is increasing for x>0, yes, the minimum cost under x≥548 is at x=548.Alternatively, if we could choose x less than 548, but we can't because we need at least 500 clicks. So, x must be at least 548.Wait, but let me check: x=548 gives clicks=548*0.914≈548*0.914.Compute 548*0.9=493.2, 548*0.014≈7.672, so total≈493.2+7.672≈500.872, which is just above 500. So, x=548 is sufficient.Therefore, the number of impressions to purchase is 548.But wait, let me check if x=547 would give enough clicks.547*0.914≈547*0.9=492.3, 547*0.014≈7.658, total≈492.3+7.658≈499.958≈500. So, almost 500, but slightly less. Since we need at least 500, x=547 gives approximately 499.96, which is just below 500. Therefore, x=548 is the minimum number of impressions needed.Therefore, the answer is x=548.Wait, but let me think again. The problem says "minimize the cost while achieving at least T=500 clicks." So, we need to find the smallest x such that x*P_max ≥500, and then compute C(x). Since the cost function is increasing, the minimal cost is at the minimal x that satisfies the constraint.Therefore, x=548.But let me compute x more precisely.We have x ≥ 500 / 0.914.Compute 500 / 0.914:0.914 * 547 = ?0.914*500=4570.914*47=42.958Total=457+42.958=499.958≈500.So, x=547 gives approximately 499.96 clicks, which is just below 500. Therefore, x=548 is needed.Therefore, the number of impressions is 548.But wait, let me compute 548*0.914 precisely.548 * 0.914:First, 500*0.914=45748*0.914=48*0.9 +48*0.014=43.2 +0.672=43.872Total=457+43.872=500.872So, 548 impressions give 500.872 clicks, which is above 500.Therefore, x=548 is the minimal number of impressions needed.So, the answer is x=548.But wait, let me think again. The cost function is C(x)=0.0005x² +0.3x +100.Is there a way to express x in terms of T and P_max?Yes, x= T / P_max.But since P_max is 0.914, x=500 /0.914≈547.08, so x=548.Therefore, the minimal x is 548.But wait, let me think about the cost function. Since it's quadratic, maybe we can set up an equation where the derivative equals zero, but with the constraint x≥548.But as we saw earlier, the cost function is increasing for x>0, so the minimal cost under x≥548 is at x=548.Therefore, the answer is x=548.Final Answer1. The time of day that maximizes the predicted CTR is boxed{24} hours.2. The number of impressions to purchase is boxed{548}.</think>

🧠:<think>Okay, so I have this problem where a student created a website using a variation of RSA encryption. The public key is n = 10403, which is the product of two primes p and q that are very close to each other, specifically consecutive primes. I need to find p and q. Then, I also have to compute Euler's totient function φ(n) and find the smallest suitable exponent e that's coprime to φ(n) and between 1 and φ(n).Starting with the first part: finding p and q. Since p and q are consecutive primes and their product is 10403, I can think of this as a factoring problem. I remember that in RSA, n is the product of two large primes, and if they are close to each other, it might be easier to factor n by checking primes around the square root of n.Let me calculate the square root of 10403 to get an approximate idea of where p and q might be. The square root of 10000 is 100, so sqrt(10403) should be a bit more than 100. Let me compute it more accurately.Calculating sqrt(10403):I know that 100^2 = 10000, so 102^2 = 10404. Wait, that's really close. 102 squared is 10404, which is just 1 more than 10403. So that means 102^2 = 10404, so 102^2 -1 = 10403. Hmm, interesting.So, 10403 = 102^2 - 1^2 = (102 - 1)(102 + 1) = 101 * 103. Wait, that's factoring using the difference of squares. So, 10403 = 101 * 103.Let me check if 101 and 103 are primes. 101 is a well-known prime number. 103 is also a prime number because it's not divisible by 2, 3, 5, 7, or 11. Let me confirm:103 divided by 2 is 51.5, not integer.103 divided by 3 is about 34.333, not integer.103 divided by 5 is 20.6, nope.103 divided by 7 is approximately 14.714, not integer.103 divided by 11 is about 9.36, still not integer.Since the square root of 103 is around 10.14, I only need to check primes up to 11, which I did, and none divide 103. So, 103 is prime.Similarly, 101 is prime. So, p and q are 101 and 103. Since they are consecutive primes, that makes sense.So, part 1 is solved: p = 101 and q = 103.Moving on to part 2: computing φ(n) and finding the smallest suitable e.Euler's totient function φ(n) for n = p * q, where p and q are primes, is given by φ(n) = (p - 1)(q - 1). So, let me compute that.First, p = 101, so p - 1 = 100.q = 103, so q - 1 = 102.Therefore, φ(n) = 100 * 102.Calculating 100 * 102: 100 * 100 = 10000, plus 100 * 2 = 200, so total is 10200.So, φ(n) = 10200.Now, we need to find the smallest e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1.In RSA, the exponent e is typically chosen as a small prime number, often 3, 5, 17, etc., to make encryption faster. But it must be coprime with φ(n).Let me check the smallest possible e starting from 2 upwards.e must be greater than 1, so starting with e = 2.Check gcd(2, 10200). Since 10200 is even, gcd(2, 10200) = 2 ≠ 1. So, e=2 is invalid.Next, e=3.Compute gcd(3, 10200). 10200 divided by 3 is 3400, so 3 is a divisor. Therefore, gcd(3, 10200)=3≠1. So, e=3 is invalid.Next, e=4.gcd(4, 10200). Since 10200 is divisible by 4 (as 100 is divisible by 4), gcd(4,10200)=4≠1. So, e=4 invalid.e=5.Check gcd(5, 10200). 10200 ends with a 0, so divisible by 5. Therefore, gcd(5,10200)=5≠1. So, e=5 invalid.e=6.gcd(6,10200). 10200 is divisible by both 2 and 3, so 6 divides 10200. gcd=6≠1. Invalid.e=7.Check gcd(7,10200). Let's see if 7 divides 10200.Divide 10200 by 7: 7*1457=10199, which is 1 less than 10200. So, 10200 = 7*1457 +1. Therefore, 7 does not divide 10200. So, gcd(7,10200)=1.Therefore, e=7 is a valid exponent.But wait, is 7 the smallest e? Let's check e=7 is the first one after 2,3,4,5,6 that works.Wait, hold on, e=2,3,4,5,6 all don't work, so e=7 is the next candidate. So, e=7 is the smallest suitable exponent.But wait, hold on, is 7 the smallest? Let me double-check.Wait, e=7 is the first after 6, but is there a smaller e? Since e must be greater than 1, so starting from 2, which didn't work, then 3,4,5,6 didn't work, so 7 is indeed the next.But just to make sure, let me check e=7.Compute gcd(7,10200). Since 10200 divided by 7 is approximately 1457.142..., which is not an integer, so 7 does not divide 10200. Therefore, gcd(7,10200)=1. So, yes, e=7 is coprime.Therefore, the smallest suitable e is 7.Wait, but hold on, in RSA, e is often chosen as 65537 for security reasons, but since the question asks for the smallest e, 7 is correct.But let me just check if I did everything correctly.Wait, φ(n)=10200. So, the factors of 10200 are 2^3 * 3 * 5^2 * 17.Wait, 10200: Let's factorize 10200.10200 divided by 100 is 102, so 10200 = 100 * 102 = (2^2 * 5^2) * (2 * 3 * 17) = 2^3 * 3 * 5^2 * 17.So, prime factors of φ(n)=10200 are 2,3,5,17.Therefore, e must not share any of these factors. So, e must be coprime with 2,3,5,17.So, e must be an integer greater than 1, less than 10200, and not divisible by 2,3,5,17.So, starting from e=2, which is divisible by 2, so no.e=3: divisible by 3, no.e=4: divisible by 2, no.e=5: divisible by 5, no.e=6: divisible by 2 and 3, no.e=7: not divisible by 2,3,5,17. So, yes, 7 is coprime.Therefore, e=7 is the smallest suitable exponent.Hence, the answers are p=101, q=103, φ(n)=10200, and e=7.Final Answer1. The prime numbers are (boxed{101}) and (boxed{103}).2. The Euler's totient function (phi(n)) is (boxed{10200}) and the smallest suitable exponent (e) is (boxed{7}).</think>

🧠:<think>Alright, so I have these two graph theory problems related to analyzing graphic novels. Let me try to work through them step by step.Starting with the first problem: I need to prove that in a strongly connected directed graph ( G ) with ( n ) vertices, the sum of the in-degrees equals the sum of the out-degrees and both are equal to the number of edges in ( G ).Hmm, okay. I remember that in any directed graph, the sum of all in-degrees equals the sum of all out-degrees. Is that always true? Let me think. Each edge contributes +1 to the out-degree of its starting vertex and +1 to the in-degree of its ending vertex. So, if I add up all the out-degrees, I'm essentially counting each edge once. Similarly, adding up all the in-degrees also counts each edge once. Therefore, the total sum of in-degrees must equal the total sum of out-degrees, and both must equal the number of edges in the graph. But wait, the problem mentions that ( G ) is strongly connected. Does that affect this result? I don't think so. The property that the sum of in-degrees equals the sum of out-degrees is true for any directed graph, regardless of whether it's strongly connected or not. So, even if the graph wasn't strongly connected, this equality would still hold. Therefore, the fact that ( G ) is strongly connected might be a red herring here. Maybe it's just emphasizing that we're dealing with a directed graph, but the proof doesn't actually require the graph to be strongly connected.So, to summarize my thoughts: Each edge contributes to both an out-degree and an in-degree, so when you sum all out-degrees, you're counting each edge once, same with in-degrees. Therefore, the sums must be equal and both equal to the number of edges. So, I think that's the proof.Moving on to the second problem: The scholar wants to identify a Hamiltonian cycle in a directed graph ( G ) that is Eulerian and has ( n ) vertices. I need to determine under what conditions a Hamiltonian cycle can be guaranteed.First, let me recall what an Eulerian graph is. In a directed graph, an Eulerian circuit exists if and only if every vertex has equal in-degree and out-degree, and the graph is strongly connected. So, since ( G ) is Eulerian, it must satisfy these conditions: it's strongly connected, and for every vertex, the in-degree equals the out-degree.Now, a Hamiltonian cycle is a cycle that visits every vertex exactly once and returns to the starting vertex. So, in a directed graph, a Hamiltonian cycle is a directed cycle that includes all vertices.I need to figure out if an Eulerian directed graph necessarily has a Hamiltonian cycle. Or, if not, under what additional conditions can we guarantee it.I remember that not all Eulerian graphs are Hamiltonian. For example, consider a graph with two vertices where each vertex has a self-loop. This graph is Eulerian because each vertex has equal in-degree and out-degree (both are 1), and it's strongly connected. However, it doesn't have a Hamiltonian cycle because a Hamiltonian cycle would require visiting each vertex exactly once, but with two vertices, you can't form a cycle that includes both without repeating edges, but since each has a self-loop, the cycle would just loop on each vertex individually, which isn't a single cycle covering both.Wait, actually, in that case, each vertex has a self-loop, so the graph is Eulerian, but there's no Hamiltonian cycle because a Hamiltonian cycle would need to traverse both vertices in a single cycle, which isn't possible since each vertex only has a loop. So, that's a counterexample where an Eulerian graph doesn't have a Hamiltonian cycle.But maybe if the graph is strongly connected and satisfies some other condition, like being regular or something else, then it might have a Hamiltonian cycle. Alternatively, perhaps if the graph is also 2-connected or has some other property.Wait, but in the problem statement, the graph is already Eulerian, which implies it's strongly connected. So, maybe if the graph is also 2-connected or has some other property, we can guarantee a Hamiltonian cycle.Alternatively, perhaps if the graph is a complete directed graph, meaning every pair of vertices has edges in both directions, then it's Hamiltonian. But that's a stronger condition than being Eulerian.Wait, let me think of another example. Suppose I have a directed graph with three vertices: A, B, C. Each vertex has an out-degree of 2 and in-degree of 2. So, each vertex has edges to the other two vertices. This graph is Eulerian because in-degree equals out-degree for each vertex, and it's strongly connected. Does it have a Hamiltonian cycle?Yes, in this case, you can have a cycle like A -> B -> C -> A, which is a Hamiltonian cycle. So, in this case, the Eulerian graph does have a Hamiltonian cycle.But in the two-vertex example, it didn't. So, maybe the number of vertices matters? Or perhaps the minimum degree?Wait, in the two-vertex case, each vertex has a degree of 1, but in the three-vertex case, each vertex has a higher degree.Wait, perhaps if the graph is strongly connected and has minimum out-degree at least n/2, then it's Hamiltonian. I think there's a theorem related to that.Yes, I recall that in a strongly connected directed graph, if for every vertex, the out-degree is at least n/2, then the graph is Hamiltonian. This is similar to Dirac's theorem for undirected graphs, which states that if every vertex has degree at least n/2, then the graph is Hamiltonian.But in our case, the graph is Eulerian, which requires that in-degree equals out-degree for each vertex, but doesn't necessarily specify the minimum degree. So, unless the graph also satisfies that each vertex has out-degree at least n/2, we can't guarantee a Hamiltonian cycle.Alternatively, if the graph is Eulerian and also satisfies some other condition, like being regular of degree at least 2, maybe that's sufficient. But I'm not sure.Wait, let's think about another example. Suppose we have four vertices: A, B, C, D. Each vertex has in-degree and out-degree 2. So, it's Eulerian. Now, can we construct such a graph without a Hamiltonian cycle?Yes, for example, partition the graph into two separate cycles: A -> B -> A and C -> D -> C. This graph is Eulerian because each vertex has equal in-degree and out-degree, and it's strongly connected within each cycle, but overall, it's not strongly connected because there's no path from A to C or vice versa. Wait, no, actually, in this case, the graph isn't strongly connected because you can't get from one cycle to the other. So, that's not a valid example because the graph must be strongly connected to be Eulerian.Wait, right, in a directed graph, being Eulerian requires strong connectivity. So, my previous example isn't Eulerian because it's not strongly connected. So, maybe in a strongly connected Eulerian graph, does it necessarily have a Hamiltonian cycle?Wait, let me think of a strongly connected Eulerian graph that isn't Hamiltonian. Maybe a graph with four vertices where each vertex has in-degree and out-degree 2, but arranged in such a way that there's no Hamiltonian cycle.Let me try to construct such a graph. Let's have vertices A, B, C, D. Let's have edges: A -> B, A -> C; B -> C, B -> D; C -> D, C -> A; D -> A, D -> B. Now, let's check the degrees: Each vertex has out-degree 2 and in-degree 2. So, it's Eulerian and strongly connected.Does it have a Hamiltonian cycle? Let's see. Starting at A: A -> B -> C -> D -> A. That's a cycle covering all four vertices. So, that's a Hamiltonian cycle. Hmm, so in this case, it does have a Hamiltonian cycle.Wait, maybe I need a different construction. Let me try another graph. Let's have A -> B, B -> C, C -> D, D -> A, forming a 4-cycle. Then, add another edge from each vertex to another vertex, say A -> C, B -> D, C -> A, D -> B. Now, each vertex has out-degree 2 and in-degree 2. Is this graph strongly connected? Yes, because you can go from any vertex to any other vertex through the additional edges.Does it have a Hamiltonian cycle? Let's see. The original 4-cycle is a Hamiltonian cycle. So, yes, it does have one.Hmm, maybe it's difficult to construct a strongly connected Eulerian graph without a Hamiltonian cycle. Maybe such graphs are actually Hamiltonian. But I'm not sure. I need to think of a theorem or a result.Wait, I recall that in a strongly connected directed graph, if it's also regular (i.e., all vertices have the same in-degree and out-degree), then it might have a Hamiltonian cycle, but I'm not sure.Alternatively, maybe if the graph is Eulerian and has a certain property, like being 2-connected or having high enough minimum degree, then it's Hamiltonian.Wait, another thought: In an Eulerian graph, the number of edges is equal to the sum of the degrees divided by 2, but in the directed case, it's just the sum of the out-degrees, which equals the number of edges.But I'm not sure how that helps.Wait, maybe I should look for a theorem. I think there's a theorem by Ghouila-Houri which states that if a strongly connected directed graph has ( n geq 2 ) vertices and every vertex has out-degree at least ( n/2 ), then the graph is Hamiltonian.So, in our case, if the Eulerian graph also satisfies that every vertex has out-degree at least ( n/2 ), then it's Hamiltonian. But in the two-vertex example, each vertex has out-degree 1, which is equal to ( 2/2 = 1 ), so it satisfies the condition, but it doesn't have a Hamiltonian cycle. Wait, that contradicts the theorem.Wait, no, in the two-vertex example, each vertex has out-degree 1, which is equal to ( n/2 = 1 ). But the graph doesn't have a Hamiltonian cycle because it's just two self-loops. So, maybe the theorem requires strict inequality or something else.Wait, let me check. Ghouila-Houri's theorem states that if a strongly connected directed graph has ( n geq 2 ) vertices and every vertex has out-degree at least ( n/2 ), then the graph is Hamiltonian. But in the two-vertex case, each vertex has out-degree 1, which is exactly ( n/2 ), but the graph isn't Hamiltonian. So, maybe the theorem requires the out-degree to be strictly greater than ( n/2 ).Wait, actually, in the two-vertex case, if each vertex has out-degree 1, the graph is two self-loops, which isn't strongly connected in the sense of having a single cycle, but actually, it's two separate cycles. Wait, no, in a directed graph, two vertices each with a self-loop are strongly connected because you can go from each vertex to itself, but you can't go from one to the other. Wait, actually, no, in that case, the graph isn't strongly connected because there's no path from A to B or B to A. So, that graph isn't strongly connected, hence it's not Eulerian.Wait, hold on. If each vertex has a self-loop, the graph is strongly connected only if you can go from any vertex to any other vertex. But with just self-loops, you can't go from A to B or B to A. So, that graph isn't strongly connected, hence it's not Eulerian. So, my previous example was incorrect because it wasn't strongly connected.So, perhaps in a strongly connected Eulerian graph, if every vertex has out-degree at least ( n/2 ), then it's Hamiltonian. But in the two-vertex case, if you have a strongly connected graph, each vertex must have an edge to the other vertex. So, in that case, each vertex has out-degree 1, which is equal to ( n/2 = 1 ), and the graph is strongly connected. Does it have a Hamiltonian cycle? Yes, because you can go A -> B -> A, which is a cycle covering both vertices. So, in that case, it does have a Hamiltonian cycle.Wait, so in the two-vertex case, if the graph is strongly connected and Eulerian, it does have a Hamiltonian cycle. So, maybe Ghouila-Houri's theorem applies here. So, if a strongly connected directed graph has every vertex with out-degree at least ( n/2 ), then it's Hamiltonian.But in our case, the graph is Eulerian, which requires that in-degree equals out-degree for each vertex, but doesn't specify the minimum degree. So, unless the graph also satisfies that each vertex has out-degree at least ( n/2 ), we can't guarantee a Hamiltonian cycle.But wait, in the three-vertex example I had earlier, each vertex had out-degree 2, which is greater than ( 3/2 = 1.5 ), so it satisfied the condition and had a Hamiltonian cycle.So, perhaps the condition is that if the Eulerian graph is also such that every vertex has out-degree at least ( n/2 ), then it's Hamiltonian.But the problem is asking under what conditions can the scholar guarantee the existence of a Hamiltonian cycle, given that the graph is Eulerian and has ( n ) vertices.So, maybe the answer is that if the graph is Eulerian and every vertex has out-degree at least ( n/2 ), then it's Hamiltonian. But I need to verify this.Alternatively, maybe being Eulerian and strongly connected is sufficient, but I don't think so because I can think of a graph that is strongly connected and Eulerian but not Hamiltonian. Wait, but earlier I thought that in a strongly connected Eulerian graph, you can have a Hamiltonian cycle, but maybe not always.Wait, let me think of a graph with four vertices. Each vertex has out-degree 2 and in-degree 2. Let's arrange the edges such that there's a cycle of length 3 and another edge. Wait, but in a directed graph, if it's strongly connected and Eulerian, it must have a single cycle that covers all edges, but not necessarily all vertices.Wait, no, an Eulerian circuit covers all edges, but a Hamiltonian cycle covers all vertices. So, they are different concepts.Wait, perhaps if the graph is Eulerian and also satisfies that the number of edges is at least ( n ), but that's always true because in a strongly connected directed graph, you need at least ( n ) edges to form a cycle.Wait, I'm getting confused. Let me try to think differently.I think the key here is that being Eulerian doesn't necessarily imply being Hamiltonian, but under certain additional conditions, like high minimum degree, it might.So, perhaps the answer is that if the Eulerian graph is also such that every vertex has out-degree at least ( n/2 ), then it's Hamiltonian. Otherwise, it's not guaranteed.Alternatively, maybe the graph being Eulerian and having a certain connectivity, like being 2-connected, ensures a Hamiltonian cycle, but I'm not sure.Wait, another thought: In an Eulerian graph, since every vertex has equal in-degree and out-degree, and it's strongly connected, it must have at least one cycle that covers all edges, but not necessarily all vertices. So, the existence of a Hamiltonian cycle is a separate condition.Therefore, the scholar cannot guarantee a Hamiltonian cycle just from the graph being Eulerian. They need additional conditions, such as the graph satisfying Ghouila-Houri's theorem, i.e., every vertex has out-degree at least ( n/2 ).So, to answer the second question: The scholar can guarantee the existence of a Hamiltonian cycle if, in addition to being Eulerian, every vertex has an out-degree of at least ( n/2 ). Otherwise, it's not guaranteed.But wait, let me check if this is correct. Ghouila-Houri's theorem states that a strongly connected directed graph with ( n geq 2 ) vertices is Hamiltonian if every vertex has out-degree at least ( n/2 ). Since our graph is Eulerian, it is strongly connected, so if it also satisfies the minimum out-degree condition, then it's Hamiltonian.Therefore, the conditions are: ( G ) is Eulerian (hence strongly connected and in-degree equals out-degree for all vertices) and every vertex has out-degree at least ( n/2 ). Then, ( G ) has a Hamiltonian cycle.So, summarizing my thoughts:1. For the first problem, the sum of in-degrees equals the sum of out-degrees and both equal the number of edges, regardless of strong connectivity.2. For the second problem, the graph being Eulerian alone doesn't guarantee a Hamiltonian cycle, but if it also satisfies that every vertex has out-degree at least ( n/2 ), then a Hamiltonian cycle exists.But wait, in the two-vertex example, each vertex has out-degree 1, which is equal to ( n/2 = 1 ), and the graph is strongly connected and Eulerian, and it does have a Hamiltonian cycle (A -> B -> A). So, in that case, it works.Another example: three vertices, each with out-degree 2, which is greater than ( 3/2 = 1.5 ). It has a Hamiltonian cycle.So, I think the condition is that if the Eulerian graph has every vertex with out-degree at least ( n/2 ), then it's Hamiltonian.Therefore, the answer to the second question is that the graph must have every vertex with out-degree at least ( n/2 ) in addition to being Eulerian to guarantee a Hamiltonian cycle.But wait, let me think if there's a case where the graph is Eulerian, every vertex has out-degree at least ( n/2 ), but it's not Hamiltonian. I don't think so, because Ghouila-Houri's theorem says that if a strongly connected directed graph has every vertex with out-degree at least ( n/2 ), then it's Hamiltonian. Since Eulerian graphs are strongly connected, the condition is sufficient.Therefore, the conditions are: ( G ) is Eulerian (so it's strongly connected and in-degree equals out-degree for all vertices) and every vertex has out-degree at least ( n/2 ). Then, ( G ) has a Hamiltonian cycle.So, to wrap up:1. The sum of in-degrees equals the sum of out-degrees and both equal the number of edges.2. The graph must be Eulerian and have every vertex with out-degree at least ( n/2 ) to guarantee a Hamiltonian cycle.</think>

🧠:<think>Okay, so I have two problems to solve here about transformations on 3D models. Let me take them one at a time.Problem 1: Transforming an EllipsoidAlright, the first problem is about an ellipsoid. The equation given is (frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2} = 1), with semi-principal axes (a = 3), (b = 2), and (c = 1). The designer wants to apply a non-uniform scaling, doubling the x-axis and tripling the z-axis. I need to find the new equation and the new volume.First, scaling transformations. Scaling affects each axis differently here. If we double the x-axis, that means the new semi-principal axis along x will be (2a). Similarly, tripling the z-axis means the new semi-principal axis along z will be (3c). The y-axis isn't scaled, so it remains (b).So, substituting the given values:Original semi-axes:- (a = 3)- (b = 2)- (c = 1)After scaling:- New (a' = 2a = 2*3 = 6)- New (b' = b = 2)- New (c' = 3c = 3*1 = 3)So the equation of the transformed ellipsoid should be (frac{x^2}{(6)^2} + frac{y^2}{(2)^2} + frac{z^2}{(3)^2} = 1), which simplifies to (frac{x^2}{36} + frac{y^2}{4} + frac{z^2}{9} = 1).Now, for the volume. The volume of an ellipsoid is given by (frac{4}{3}pi a b c). So, the original volume is (frac{4}{3}pi * 3 * 2 * 1 = frac{4}{3}pi * 6 = 8pi).After scaling, the new volume will be scaled by the product of the scaling factors. The scaling factors are 2 along x, 1 along y, and 3 along z. So the volume scaling factor is (2 * 1 * 3 = 6). Therefore, the new volume is (8pi * 6 = 48pi).Wait, hold on. Let me verify that. The original volume is (frac{4}{3}pi * 3 * 2 * 1 = 8pi). After scaling, the new semi-axes are 6, 2, 3. So the new volume is (frac{4}{3}pi * 6 * 2 * 3). Calculating that: 6*2=12, 12*3=36, so (frac{4}{3}pi * 36 = 48pi). Yep, that matches. So the volume is indeed 48π.Problem 2: Rotating a CylinderThe second problem involves rotating a cylinder. The cylinder is initially aligned along the z-axis with equation (x^2 + y^2 = r^2) and height (h = 10). The designer rotates it 45 degrees around the x-axis, then 60 degrees around the y-axis. I need to find the equation of the cylinder after these transformations.Hmm, rotations in 3D can be tricky. I remember that rotating an object involves using rotation matrices. The order of rotations matters, so first rotating around x, then around y.Let me recall the rotation matrices. A rotation around the x-axis by θ is given by:[R_x(theta) = begin{pmatrix}1 & 0 & 0 0 & costheta & -sintheta 0 & sintheta & costhetaend{pmatrix}]And a rotation around the y-axis by φ is:[R_y(phi) = begin{pmatrix}cosphi & 0 & sinphi 0 & 1 & 0 -sinphi & 0 & cosphiend{pmatrix}]Since the cylinder is first rotated around x by 45 degrees, then around y by 60 degrees, the total transformation is (R_y(60^circ) cdot R_x(45^circ)). So the combined rotation matrix is the product of these two.Let me compute this step by step.First, convert degrees to radians because when implementing, but since we're just writing the equation, maybe we can keep it in degrees for the equation.But actually, in the rotation matrices, the angles are in radians, but since we're just going to write the sine and cosine terms, perhaps we can keep them as functions of 45° and 60°.So, let me compute (R_x(45^circ)) and (R_y(60^circ)):First, (R_x(45^circ)):[R_x = begin{pmatrix}1 & 0 & 0 0 & cos45^circ & -sin45^circ 0 & sin45^circ & cos45^circend{pmatrix}]We know that (cos45^circ = sin45^circ = frac{sqrt{2}}{2}), so:[R_x = begin{pmatrix}1 & 0 & 0 0 & frac{sqrt{2}}{2} & -frac{sqrt{2}}{2} 0 & frac{sqrt{2}}{2} & frac{sqrt{2}}{2}end{pmatrix}]Next, (R_y(60^circ)):[R_y = begin{pmatrix}cos60^circ & 0 & sin60^circ 0 & 1 & 0 -sin60^circ & 0 & cos60^circend{pmatrix}]We know that (cos60^circ = frac{1}{2}), (sin60^circ = frac{sqrt{3}}{2}), so:[R_y = begin{pmatrix}frac{1}{2} & 0 & frac{sqrt{3}}{2} 0 & 1 & 0 -frac{sqrt{3}}{2} & 0 & frac{1}{2}end{pmatrix}]Now, the combined rotation is (R = R_y cdot R_x). Let's compute that matrix multiplication.Multiplying (R_y) and (R_x):First row of R_y times each column of R_x:First element: (frac{1}{2}*1 + 0*0 + frac{sqrt{3}}{2}*0 = frac{1}{2})Second element: (frac{1}{2}*0 + 0*frac{sqrt{2}}{2} + frac{sqrt{3}}{2}*0 = 0)Third element: (frac{1}{2}*0 + 0*(-frac{sqrt{2}}{2}) + frac{sqrt{3}}{2}*frac{sqrt{2}}{2} = frac{sqrt{6}}{4})Wait, hold on. Wait, actually, the multiplication is row by column. Let me write it out properly.Wait, no, actually, the combined rotation is (R = R_y cdot R_x), so each point is first transformed by (R_x), then by (R_y). So the transformation is (R_y(R_x(v))).But when we write the matrix multiplication, it's (R = R_y cdot R_x), so the first transformation is (R_x), then (R_y).So, to compute (R = R_y R_x), let's compute each element:First row of R:- First element: (1st row of R_y) • (1st column of R_x) = (frac{1}{2}*1 + 0*0 + frac{sqrt{3}}{2}*0 = frac{1}{2})- Second element: (1st row of R_y) • (2nd column of R_x) = (frac{1}{2}*0 + 0*frac{sqrt{2}}{2} + frac{sqrt{3}}{2}*0 = 0)- Third element: (1st row of R_y) • (3rd column of R_x) = (frac{1}{2}*0 + 0*(-frac{sqrt{2}}{2}) + frac{sqrt{3}}{2}*frac{sqrt{2}}{2} = frac{sqrt{6}}{4})Second row of R:- First element: (2nd row of R_y) • (1st column of R_x) = (0*1 + 1*0 + 0*0 = 0)- Second element: (2nd row of R_y) • (2nd column of R_x) = (0*0 + 1*frac{sqrt{2}}{2} + 0*0 = frac{sqrt{2}}{2})- Third element: (2nd row of R_y) • (3rd column of R_x) = (0*0 + 1*(-frac{sqrt{2}}{2}) + 0*frac{sqrt{2}}{2} = -frac{sqrt{2}}{2})Third row of R:- First element: (3rd row of R_y) • (1st column of R_x) = (-frac{sqrt{3}}{2}*1 + 0*0 + frac{1}{2}*0 = -frac{sqrt{3}}{2})- Second element: (3rd row of R_y) • (2nd column of R_x) = (-frac{sqrt{3}}{2}*0 + 0*frac{sqrt{2}}{2} + frac{1}{2}*0 = 0)- Third element: (3rd row of R_y) • (3rd column of R_x) = (-frac{sqrt{3}}{2}*0 + 0*(-frac{sqrt{2}}{2}) + frac{1}{2}*frac{sqrt{2}}{2} = frac{sqrt{2}}{4})So putting it all together, the combined rotation matrix R is:[R = begin{pmatrix}frac{1}{2} & 0 & frac{sqrt{6}}{4} 0 & frac{sqrt{2}}{2} & -frac{sqrt{2}}{2} -frac{sqrt{3}}{2} & 0 & frac{sqrt{2}}{4}end{pmatrix}]Wait, let me double-check the third element of the third row. It was (frac{1}{2}*frac{sqrt{2}}{2} = frac{sqrt{2}}{4}), yes.So now, the cylinder is being rotated by this matrix R. So to find the equation of the transformed cylinder, we need to express the original cylinder in terms of the transformed coordinates.The original cylinder is (x^2 + y^2 = r^2), with z ranging from 0 to 10 (assuming it's along the positive z-axis). But since the height is 10, maybe it's from -5 to 5? Wait, the problem doesn't specify, but I think it's from 0 to 10. But actually, in the equation, it's just (x^2 + y^2 = r^2), so z can be anything, but in the context, it's a cylinder of height 10. So perhaps z ranges from 0 to 10.But for the equation, we can consider the entire cylinder, so z can be any real number, but in the design, it's limited to height 10. But for the equation, we just need the surface.So, to find the equation after rotation, we can express the original coordinates (x, y, z) in terms of the transformed coordinates (x', y', z'). Wait, actually, when we apply a rotation, the new coordinates (x', y', z') are related to the original coordinates (x, y, z) by the rotation matrix.So, the transformation is:[begin{pmatrix}x' y' z'end{pmatrix}= R cdotbegin{pmatrix}x y zend{pmatrix}]But to find the equation in terms of x', y', z', we need to express x, y, z in terms of x', y', z'. That is, we need the inverse of R.But since R is a rotation matrix, its inverse is its transpose. So, (R^{-1} = R^T).So, let me compute (R^T):[R^T = begin{pmatrix}frac{1}{2} & 0 & -frac{sqrt{3}}{2} 0 & frac{sqrt{2}}{2} & 0 frac{sqrt{6}}{4} & -frac{sqrt{2}}{2} & frac{sqrt{2}}{4}end{pmatrix}]So, the original coordinates (x, y, z) can be expressed as:[x = frac{1}{2}x' - frac{sqrt{3}}{2}z'][y = frac{sqrt{2}}{2}y'][z = frac{sqrt{6}}{4}x' - frac{sqrt{2}}{2}y' + frac{sqrt{2}}{4}z']Wait, let me write that correctly.From (R^T):First row: x = (1/2)x' + 0*y' - (√3/2)z'Second row: y = 0*x' + (√2/2)y' + 0*z'Third row: z = (√6/4)x' + (-√2/2)y' + (√2/4)z'So, substituting back into the original cylinder equation (x^2 + y^2 = r^2):[left( frac{1}{2}x' - frac{sqrt{3}}{2}z' right)^2 + left( frac{sqrt{2}}{2}y' right)^2 = r^2]Let me expand this:First term: (left( frac{1}{2}x' - frac{sqrt{3}}{2}z' right)^2 = frac{1}{4}x'^2 - frac{sqrt{3}}{2}x'z' + frac{3}{4}z'^2)Second term: (left( frac{sqrt{2}}{2}y' right)^2 = frac{2}{4}y'^2 = frac{1}{2}y'^2)So, combining:[frac{1}{4}x'^2 - frac{sqrt{3}}{2}x'z' + frac{3}{4}z'^2 + frac{1}{2}y'^2 = r^2]To simplify, multiply both sides by 4 to eliminate denominators:[x'^2 - 2sqrt{3}x'z' + 3z'^2 + 2y'^2 = 4r^2]So, the equation of the transformed cylinder is:[x'^2 + 2y'^2 + 3z'^2 - 2sqrt{3}x'z' = 4r^2]Hmm, that seems a bit complicated. Let me check my steps.Wait, perhaps I made a mistake in the substitution. Let me go back.Original equation: (x^2 + y^2 = r^2).Expressed in terms of x', y', z':x = (1/2)x' - (√3/2)z'y = (√2/2)y'So, substituting:[(1/2)x' - (√3/2)z']² + [(√2/2)y']² = r²Expanding:(1/4)x'² - (√3/2)x'z' + (3/4)z'² + (2/4)y'² = r²Simplify:(1/4)x'² - (√3/2)x'z' + (3/4)z'² + (1/2)y'² = r²Multiply both sides by 4:x'² - 2√3 x'z' + 3z'² + 2y'² = 4r²Yes, that's correct. So the equation is quadratic in x', y', z', which makes sense because a rotated cylinder is a quadric surface.Alternatively, we can write it as:x² + 2y² + 3z² - 2√3 xz = 4r²But in terms of the original coordinates, it's in terms of x', y', z', so perhaps we can leave it as is.Wait, but in the problem statement, it just says "derive the equation for the boundary of the cylinder after these transformations in the xyz-coordinate system." So, maybe we can write it in terms of x, y, z, but actually, the primed coordinates are the transformed ones, so perhaps we should express it in terms of the original x, y, z.Wait, no. The equation after transformation is in terms of the new coordinates, but the problem says "in the xyz-coordinate system," which is the original coordinate system. Wait, that's confusing.Wait, actually, when we apply a transformation, the object is transformed in space, so the equation in the original coordinate system would involve the inverse transformation. Hmm, maybe I need to think differently.Alternatively, perhaps it's better to consider that after rotation, the cylinder is described by the transformed coordinates, so the equation in the original coordinate system is as above.But I'm a bit confused. Let me think.When we rotate the cylinder, the points on the cylinder are transformed by the rotation matrix. So, if a point (x, y, z) is on the original cylinder, then after rotation, it becomes (x', y', z') = R*(x, y, z). So, to find the equation of the rotated cylinder in the original coordinate system, we need to express the condition that (x', y', z') lies on the original cylinder.Wait, no. Wait, actually, the equation of the cylinder in the original coordinate system is (x^2 + y^2 = r^2). After rotation, the cylinder is transformed, so the equation in the original coordinate system would be the set of points (x, y, z) such that when transformed by R^{-1}, they lie on the original cylinder.So, if (x, y, z) is a point on the transformed cylinder, then R^{-1}*(x, y, z) lies on the original cylinder. So, let me denote (x', y', z') = R^{-1}*(x, y, z). Then, (x')² + (y')² = r².So, substituting:x' = (1/2)x - (√3/2)zy' = (√2/2)yz' = (√6/4)x - (√2/2)y + (√2/4)zWait, no, actually, R^{-1} is R^T, so:x' = (1/2)x + 0*y - (√3/2)zy' = 0*x + (√2/2)y + 0*zz' = (√6/4)x + (-√2/2)y + (√2/4)zSo, the condition is:(x')² + (y')² = r²Which is:[(1/2)x - (√3/2)z]^2 + [(√2/2)y]^2 = r²Which is the same equation as before:(1/4)x² - (√3/2)xz + (3/4)z² + (1/2)y² = r²Multiply by 4:x² - 2√3 xz + 3z² + 2y² = 4r²So, in the original coordinate system, the equation of the transformed cylinder is:x² + 2y² + 3z² - 2√3 xz = 4r²But wait, the original cylinder was (x^2 + y^2 = r^2). So, in the transformed coordinates, it's this quadratic equation.But the problem says "derive the equation for the boundary of the cylinder after these transformations in the xyz-coordinate system." So, yes, this is the equation in the original xyz system.But wait, the cylinder also has a height. The original cylinder has height h=10, so z ranges from 0 to 10. After rotation, the height is still 10, but the orientation has changed. However, the equation we derived is for the surface of the cylinder, regardless of z. So, perhaps we need to also consider the bounds on z.But the problem just asks for the equation of the boundary, so I think the quadratic equation is sufficient.So, summarizing, after rotating the cylinder first 45 degrees around x, then 60 degrees around y, the equation of the cylinder in the original coordinate system is:x² + 2y² + 3z² - 2√3 xz = 4r²But wait, the original cylinder equation was (x^2 + y^2 = r^2). So, in the transformed coordinates, it's this quadratic equation.Alternatively, if we want to write it in terms of the original coordinates, it's the same as above.I think that's the answer.Final Answer1. The equation of the transformed ellipsoid is (boxed{frac{x^2}{36} + frac{y^2}{4} + frac{z^2}{9} = 1}) and its new volume is (boxed{48pi}).2. The equation of the transformed cylinder is (boxed{x^2 + 2y^2 + 3z^2 - 2sqrt{3}xz = 4r^2}).</think>