answer：First, I need to identify the structure of the three-digit number. Since the hundreds digit is 4 and the units digit is also 4, the number can be represented as 4x4, where x is the tens digit.Next, I recall that a number is divisible by 3 if the sum of its digits is divisible by 3. Therefore, I calculate the sum of the known digits: 4 (hundreds place) + 4 (units place) = 8.Now, I need to find the value of x such that the total sum, 8 + x, is divisible by 3. I will test each possible value of x from 0 to 9:- If x = 0, the sum is 8 + 0 = 8 (not divisible by 3)- If x = 1, the sum is 8 + 1 = 9 (divisible by 3)- If x = 2, the sum is 8 + 2 = 10 (not divisible by 3)- If x = 3, the sum is 8 + 3 = 11 (not divisible by 3)- If x = 4, the sum is 8 + 4 = 12 (divisible by 3)- If x = 5, the sum is 8 + 5 = 13 (not divisible by 3)- If x = 6, the sum is 8 + 6 = 14 (not divisible by 3)- If x = 7, the sum is 8 + 7 = 15 (divisible by 3)- If x = 8, the sum is 8 + 8 = 16 (not divisible by 3)- If x = 9, the sum is 8 + 9 = 17 (not divisible by 3)From this analysis, the valid values for x are 1, 4, and 7. Therefore, the three-digit numbers that satisfy the conditions are 414, 444, and 474.

answer：Okay, so I'm trying to figure out how many people have reappeared in total after the magician's performances this year. Let me start by understanding the problem step by step.First, the magician makes a certain number of audience members disappear and then reappear. The number of people he makes disappear depends on the size of the audience, specifically one person for every 50 people present. So, if there are 50 people, one disappears; if there are 100 people, two disappear, and so on.Now, the problem mentions that the audience sizes vary from 50 to 500 people, and there are 100 performances. I think I need to find the average number of people who disappear per performance and then apply the probabilities of different reappearance scenarios.Let me calculate the average audience size first. Since the audience varies from 50 to 500, the average would be (50 + 500) / 2 = 275 people. So, on average, there are 275 people per performance.Given that one person disappears for every 50 people, the number of people who disappear per performance would be 275 / 50 = 5.5. But since we can't have half a person, I think we should round this down to 5 people disappearing per performance. That seems reasonable.Next, the problem states that one-tenth of the time, one of the audience members never reappears. That means in 10% of the performances, one person doesn't come back. Since there are 100 performances, this happens 10 times. So, 10 people never reappear in total.Then, one-fifth of the time, twice the number of people reappear instead of the original group. That's 20% of the performances, which is 20 times. If 5 people disappear, then twice that number would be 10 people reappearing. So, in these 20 performances, there are 10 extra people reappearing each time, totaling 20 * 10 = 200 extra people.Additionally, one-twentieth of the time, the magician accidentally makes three times the number of people reappear. That's 5% of the performances, which is 5 times. If 5 people disappear, three times that number would be 15 people reappearing. So, in these 5 performances, there are 10 extra people reappearing each time (since 15 - 5 = 10), totaling 5 * 10 = 50 extra people.Now, let's calculate the total number of people who reappear. The base number is 5 people per performance for 100 performances, which is 5 * 100 = 500 people. Then, we subtract the 10 people who never reappeared: 500 - 10 = 490 people. Next, we add the 200 extra people from the twice reappearance scenario: 490 + 200 = 690 people. Finally, we add the 50 extra people from the three times reappearance scenario: 690 + 50 = 740 people.Wait, that doesn't match the previous calculation of 640. Did I make a mistake somewhere? Let me check.In the initial calculation, I considered the extra people as 100 and 50, but now I'm calculating 200 and 50. Hmm, maybe I misapplied the probabilities. Let me recast the problem.For each performance, the number of people who disappear is 5. Normally, all 5 reappear. But:- 10% of the time, one person doesn't reappear, so only 4 reappear.- 20% of the time, twice the number reappear, so 10 reappear.- 5% of the time, three times the number reappear, so 15 reappear.- The remaining 65% of the time, all 5 reappear.So, let's calculate the expected number of people reappearing per performance.- 10% of the time: 4 people- 20% of the time: 10 people- 5% of the time: 15 people- 65% of the time: 5 peopleCalculating the expected value:(0.10 * 4) + (0.20 * 10) + (0.05 * 15) + (0.65 * 5) = 0.4 + 2 + 0.75 + 3.25 = 6.4 people per performance.Over 100 performances, the total number of people reappearing would be 6.4 * 100 = 640 people.Okay, that makes sense now. I think my initial mistake was in how I added the extra people, not considering the probabilities correctly. The expected value approach gives a more accurate result.

answer：First, I need to determine the total number of possible outcomes when both spinners are spun. Spinner C has 4 sections, and Spinner D has 3 sections, so there are 4 multiplied by 3, which equals 12 total possible outcomes.Next, I will list all possible sums of the numbers from each spinner:- When Spinner C lands on 1: 1 + 2 = 3, 1 + 4 = 5, 1 + 6 = 7- When Spinner C lands on 3: 3 + 2 = 5, 3 + 4 = 7, 3 + 6 = 9- When Spinner C lands on 5: 5 + 2 = 7, 5 + 4 = 9, 5 + 6 = 11- When Spinner C lands on 7: 7 + 2 = 9, 7 + 4 = 11, 7 + 6 = 13Now, I will identify which of these sums are divisible by 3:- 3 is divisible by 3- 9 is divisible by 3- 9 is divisible by 3There are 3 favorable outcomes where the sum is divisible by 3.Finally, to find the probability, I will divide the number of favorable outcomes by the total number of possible outcomes: 3 divided by 12 equals 1/4.

answer：Okay, so I have this problem about a function f(x) = A sin²(ωx + φ). The problem has two parts: first, I need to find φ, and second, I need to calculate the sum of f(1) + f(2) + ... + f(2017). Let me try to figure this out step by step.Starting with part (1), finding φ. The function is given as f(x) = A sin²(ωx + φ). They tell me that the maximum value of y = f(x) is 2. Since sin² of anything is always between 0 and 1, the maximum value of f(x) would be A. So, if the maximum is 2, that means A must be 2. Okay, so A = 2.Next, they mention the distance between adjacent symmetry axes of its graph is 2. Hmm, I need to recall what symmetry axes mean for a sine squared function. The sine function has symmetry about its peaks and troughs, but since it's squared, the graph of sin² is always non-negative and has a different kind of symmetry. The graph of sin² is symmetric about its maxima and minima. So, the distance between these symmetry axes would be half the period, right? Because the period of sin² is π, but wait, let me think.Actually, the standard sin²(x) has a period of π because sin²(x) = (1 - cos(2x))/2, which has a period of π. So, if the distance between adjacent symmetry axes is 2, that would mean the period is 4? Wait, no. If the distance between symmetry axes is 2, then the period should be 4 because the symmetry axes are spaced by half the period. Wait, no, actually, in the standard sin²(x), the distance between symmetry axes is π/2, right? Because the maxima are at π/2, 3π/2, etc., so the distance between them is π.Wait, maybe I'm getting confused. Let me think again. For sin²(x), the graph has maxima at x = π/2, 3π/2, 5π/2, etc., and minima at x = 0, π, 2π, etc. So, the distance between adjacent maxima is π, and the distance between a maximum and the next minimum is π/2. So, if the distance between adjacent symmetry axes is 2, that would correspond to the distance between a maximum and the next minimum, which is half the period. So, if that distance is 2, then the full period would be 4. Therefore, the period of the function is 4.But wait, the function is sin²(ωx + φ). The period of sin²(θ) is π, so when we have sin²(ωx + φ), the period becomes π / ω. So, if the period is 4, then π / ω = 4, which means ω = π / 4. Hmm, but wait, earlier I thought the distance between symmetry axes was 2, which might be half the period. Let me double-check.Wait, the distance between adjacent symmetry axes is 2. For sin²(x), the symmetry axes are at the maxima and minima. So, the distance between a maximum and the next minimum is π/2, and the distance between two maxima is π. So, if the distance between adjacent symmetry axes is 2, that would mean that the distance between a maximum and the next minimum is 2, which would be half the period. So, the full period would be 4. Therefore, π / ω = 4, so ω = π / 4.Wait, but in the problem statement, they mention the distance between adjacent symmetry axes is 2. So, if the distance between symmetry axes is 2, that would mean that the period is 4, as I thought. So, ω = π / 4.But wait, let me think again. Maybe the distance between symmetry axes is the distance between two consecutive points where the function is symmetric. For sin²(x), the function is symmetric about every maximum and every minimum. So, the distance between a maximum and the next minimum is π/2, and the distance between two maxima is π. So, if the distance between adjacent symmetry axes is 2, that would mean that the distance between a maximum and the next minimum is 2, so the period would be 4. Therefore, ω = π / 4.But wait, let me confirm. If the period is 4, then the function repeats every 4 units. So, the distance between two consecutive maxima would be 4, but in reality, for sin²(x), the maxima are spaced by π. So, maybe I'm mixing something up.Alternatively, maybe the distance between symmetry axes is the distance between two consecutive points where the function is symmetric, which could be the distance between two consecutive maxima or minima. If the function is sin²(ωx + φ), then the maxima occur where sin(ωx + φ) = ±1, so sin² is 1. The minima occur where sin(ωx + φ) = 0, so sin² is 0.So, the distance between a maximum and the next minimum would be half the period, and the distance between two maxima would be the full period. So, if the distance between adjacent symmetry axes is 2, that could mean the distance between a maximum and the next minimum is 2, which would make the period 4. Alternatively, if the distance between two maxima is 2, then the period would be 2.Wait, but in the problem statement, it says "the distance between adjacent symmetry axes of its graph is 2". So, symmetry axes are the lines where the function is symmetric. For sin²(x), the function is symmetric about each maximum and each minimum. So, the distance between two consecutive symmetry axes would be half the period. Because between a maximum and the next minimum is half the period, and that's where the symmetry axes are.So, if the distance between adjacent symmetry axes is 2, that means half the period is 2, so the full period is 4. Therefore, the period of the function is 4. Since the period of sin²(ωx + φ) is π / ω, we have π / ω = 4, so ω = π / 4.Wait, but let me check that again. The period of sin²(θ) is π, so when we have sin²(ωx + φ), the period becomes π / ω. So, if the period is 4, then π / ω = 4, so ω = π / 4. Yes, that seems correct.So, now we have A = 2, ω = π / 4. Now, the function passes through the point (1, 2). So, when x = 1, f(x) = 2. Let's plug that into the function:f(1) = 2 sin²(ω * 1 + φ) = 2So, 2 sin²(π/4 * 1 + φ) = 2Divide both sides by 2: sin²(π/4 + φ) = 1Take square root: sin(π/4 + φ) = ±1So, π/4 + φ = π/2 + kπ, where k is an integer.Solving for φ: φ = π/2 - π/4 + kπ = π/4 + kπBut we are given that 0 < φ < π/2. So, let's find k such that φ is in that interval.If k = 0: φ = π/4, which is approximately 0.785, which is less than π/2 (≈1.5708). So, that works.If k = 1: φ = π/4 + π = 5π/4, which is greater than π/2, so that's out.If k = -1: φ = π/4 - π = -3π/4, which is negative, so that's out.Therefore, φ = π/4.Wait, but let me double-check. If φ = π/4, then f(1) = 2 sin²(π/4 + π/4) = 2 sin²(π/2) = 2*(1)^2 = 2, which matches the point (1,2). So, that seems correct.So, for part (1), φ = π/4.Wait, but in the initial problem statement, the function is f(x) = A sin²(ωx + φ), and they mention 0 < φ < π/2. So, φ = π/4 is within that range. So, that's good.Now, moving on to part (2), calculating f(1) + f(2) + ... + f(2017). Hmm, that's a lot of terms. But maybe there's a pattern or periodicity we can exploit.We know that the function f(x) = 2 sin²(π/4 x + π/4). Let's see if we can simplify this function.Recall that sin²(θ) = (1 - cos(2θ))/2. So, f(x) = 2 * [ (1 - cos(2*(π/4 x + π/4)) ) / 2 ] = 1 - cos(π/2 x + π/2).So, f(x) = 1 - cos(π/2 x + π/2).Let me see if I can simplify cos(π/2 x + π/2). Using the cosine addition formula: cos(A + B) = cos A cos B - sin A sin B.So, cos(π/2 x + π/2) = cos(π/2 x) cos(π/2) - sin(π/2 x) sin(π/2).But cos(π/2) = 0 and sin(π/2) = 1. So, this simplifies to:cos(π/2 x + π/2) = 0 - sin(π/2 x) * 1 = -sin(π/2 x).Therefore, f(x) = 1 - (-sin(π/2 x)) = 1 + sin(π/2 x).Wait, that's interesting. So, f(x) = 1 + sin(π/2 x).Wait, let me double-check that:f(x) = 2 sin²(π/4 x + π/4) = 2 * [ (1 - cos(π/2 x + π/2)) / 2 ] = 1 - cos(π/2 x + π/2).Then, cos(π/2 x + π/2) = -sin(π/2 x), so f(x) = 1 - (-sin(π/2 x)) = 1 + sin(π/2 x). Yes, that's correct.So, f(x) = 1 + sin(π/2 x).Now, let's analyze this function. The function sin(π/2 x) has a period of 4, because the period of sin(kx) is 2π / k, so here k = π/2, so period is 2π / (π/2) = 4. So, f(x) has a period of 4.Therefore, f(x) is periodic with period 4. So, every 4 terms, the function repeats its values.So, to compute the sum from x=1 to x=2017, we can find how many complete periods are in 2017 terms and then compute the sum accordingly.First, let's find the values of f(x) for x = 1, 2, 3, 4, and see the pattern.Compute f(1): 1 + sin(π/2 * 1) = 1 + sin(π/2) = 1 + 1 = 2f(2): 1 + sin(π/2 * 2) = 1 + sin(π) = 1 + 0 = 1f(3): 1 + sin(π/2 * 3) = 1 + sin(3π/2) = 1 + (-1) = 0f(4): 1 + sin(π/2 * 4) = 1 + sin(2π) = 1 + 0 = 1f(5): 1 + sin(π/2 * 5) = 1 + sin(5π/2) = 1 + 1 = 2So, the pattern repeats every 4 terms: 2, 1, 0, 1, 2, 1, 0, 1, ...So, the sum of one period (4 terms) is 2 + 1 + 0 + 1 = 4.Now, let's see how many complete periods are in 2017 terms.2017 divided by 4 is 504 with a remainder of 1, because 504 * 4 = 2016, so 2017 = 504 * 4 + 1.Therefore, there are 504 complete periods and one extra term.Each complete period sums to 4, so 504 periods sum to 504 * 4 = 2016.Now, the extra term is the first term of the next period, which is f(2017). Since 2017 = 4 * 504 + 1, so f(2017) = f(1) = 2.Therefore, the total sum is 2016 + 2 = 2018.Wait, but let me double-check the calculations.Number of terms: 2017.Number of complete periods: 2017 // 4 = 504, remainder 1.Sum of complete periods: 504 * 4 = 2016.Sum of remaining terms: f(2017) = f(1) = 2.Total sum: 2016 + 2 = 2018.Yes, that seems correct.Alternatively, let's think about the function f(x) = 1 + sin(π/2 x). The average value over a period is (sum of one period) / period length = 4 / 4 = 1. So, over 2017 terms, the sum would be approximately 2017 * 1 = 2017, but since we have an extra term which is 2, the sum is 2017 + 1 = 2018. Wait, that doesn't quite make sense, but actually, since the average is 1, and we have 2017 terms, the sum is 2017 * 1 = 2017, but we have an extra 1 because the first term is 2 instead of 1. Hmm, maybe that's not the right way to think about it.Alternatively, since the function is periodic with period 4, and the sum over each period is 4, and we have 504 periods (2016 terms) summing to 2016, plus the last term f(2017) = 2, so total sum is 2016 + 2 = 2018.Yes, that seems correct.So, to summarize:(1) φ = π/4.(2) The sum f(1) + f(2) + ... + f(2017) = 2018.Wait, but let me double-check the function f(x) again. When I simplified f(x) = 2 sin²(π/4 x + π/4) to 1 + sin(π/2 x), is that correct?Let me verify:f(x) = 2 sin²(π/4 x + π/4) = 2 * [ (1 - cos(2*(π/4 x + π/4)) ) / 2 ] = 1 - cos(π/2 x + π/2).Then, cos(π/2 x + π/2) = cos(π/2 x) cos(π/2) - sin(π/2 x) sin(π/2) = 0 - sin(π/2 x) = -sin(π/2 x).So, f(x) = 1 - (-sin(π/2 x)) = 1 + sin(π/2 x). Yes, that's correct.So, f(x) = 1 + sin(π/2 x). Therefore, the values are as I calculated: 2, 1, 0, 1, 2, 1, 0, 1, etc.So, the sum over each period is 4, and with 504 periods and one extra term, the total sum is 2018.Therefore, the answers are:(1) φ = π/4.(2) The sum is 2018.

answer：First, I analyze the first balance, which shows that one triangle is equivalent in weight to two squares:[triangle = 2square]Next, I look at the second balance, which indicates that two triangles are equivalent in weight to three circles:[2triangle = 3bigcirc]To find out how many circles are needed to balance four squares, I express the weight of four squares in terms of triangles:[4square = 2triangle]Then, I convert the triangles to circles using the second balance:[2triangle = 3bigcirc]Therefore, four squares are equivalent in weight to three circles:[4 square = 3 bigcirc]Thus, it would take three circles to balance four squares.

answer：Okay, so I have this problem here: "A person crosses a 1,500-meter-long street in 3 minutes. What is their speed in kilometers per hour?" Hmm, speed, distance, and time. I remember that speed is basically how fast someone is going, right? So, speed equals distance divided by time. But wait, the units here are a bit tricky. The distance is in meters, and the time is in minutes, but they want the speed in kilometers per hour. I need to make sure all the units match before I can calculate the speed properly.First, let's think about the distance. It's 1,500 meters. I know that 1 kilometer is 1,000 meters, so 1,500 meters must be 1.5 kilometers. That seems straightforward enough. So, the distance is 1.5 kilometers.Next, the time is given as 3 minutes. But speed is usually measured in hours, especially when talking about kilometers per hour. So, I need to convert 3 minutes into hours. There are 60 minutes in an hour, right? So, 3 minutes is 3 divided by 60 hours. Let me calculate that: 3 divided by 60 is 0.05 hours. Okay, so the time is 0.05 hours.Now, I can plug these values into the speed formula. Speed equals distance divided by time. So, that would be 1.5 kilometers divided by 0.05 hours. Let me do that division: 1.5 divided by 0.05. Hmm, dividing by 0.05 is the same as multiplying by 20, because 1 divided by 0.05 is 20. So, 1.5 times 20 is 30. So, the speed is 30 kilometers per hour.Wait, does that make sense? Let me double-check my calculations. 1,500 meters is indeed 1.5 kilometers. 3 minutes is 0.05 hours. Dividing 1.5 by 0.05 gives 30. Yeah, that seems right. So, the person is moving at 30 kilometers per hour.I guess another way to think about it is to first find the speed in meters per minute and then convert that to kilometers per hour. Let's try that approach to verify. If the person crosses 1,500 meters in 3 minutes, then their speed is 1,500 meters divided by 3 minutes, which is 500 meters per minute. Now, to convert meters per minute to kilometers per hour, I need to consider that there are 60 minutes in an hour and 1,000 meters in a kilometer. So, 500 meters per minute times 60 minutes per hour gives 30,000 meters per hour. Then, dividing by 1,000 to convert meters to kilometers, we get 30 kilometers per hour. Yep, same result.So, both methods give me the same answer, which is reassuring. I think I've got it right. The person's speed is 30 kilometers per hour.