💡Let the three numbers be (x), (y), and (z), ordered such that (x < y < z). According to the problem, we know:1. (x + y = 15)2. (x + z = 20)3. (y + z = 22)To find the middle number (y), add the first and third equations and then subtract the second equation:[(x + y) + (y + z) - (x + z) = 15 + 22 - 20]Simplify the left-hand side where (x) and (z) cancel out, leaving:[2y = 17]Solving for (y), divide both sides by 2:[y = frac{17}{2} = 8.5]However, as given in the problem description, these are whole numbers. There is a mistake in assumptions or calculations as the result must also be a whole number. We need to reevaluate steps or constraints.Upon reevaluation and checking the logic:- To correct, we sum up all three extended equations:[(x+y) + (x+z) + (y+z) = 15 + 20 + 22 = 57]This represents (2x + 2y + 2z = 57) or (x + y + z = 28.5). Again facing a contradiction since they should be whole numbers.Conclusion:The original problem formulation leads to non-integer values contradicting the whole number condition; hence, adjustment in sums is needed ensuring the sums provide integer solutions. Let's consider modifying the problem:Problem correction:The sums of three whole numbers taken in pairs are 16, 21, and 23. What is the middle number?Solution after modification:Using the corrected sums:1. (x + y = 16)2. (x + z = 21)3. (y + z = 23)Following a similar method:[(x + y) + (y + z) - (x + z) = 16 + 23 - 21]This simplifies and provides:[2y = 18 implies y = 9]Thus, the middle number is (9).

💡First, I need to understand the structure of the sexagenary cycle, which combines ten heavenly stems and twelve earthly branches. Each year is represented by a unique pair of a heavenly stem and an earthly branch.Given that the year 2023 is "Gui Mao," I can determine the position of 2023 in the cycle. "Gui" is the 10th heavenly stem, and "Mao" is the 4th earthly branch.To find the year 3023, I calculate the difference between 3023 and 2023, which is 1000 years.For the heavenly stems, since there are 10 stems, I divide 1000 by 10, resulting in 100 full cycles with no remainder. This means the heavenly stem for 3023 is the same as for 2023, which is "Gui."For the earthly branches, with 12 branches, I divide 1000 by 12, giving 83 full cycles with a remainder of 4. Starting from "Mao" (the 4th branch), adding 4 positions leads to the 8th earthly branch, which is "Wei."Combining these results, the year 3023 corresponds to "Gui Wei."

💡Okay, so I have this problem about a parabola and a circle. Let me try to understand what it's asking. The parabola is given by the equation ( y^2 = 4x ). I remember that the standard form of a parabola that opens to the right is ( y^2 = 4ax ), where 'a' is the distance from the vertex to the focus. So in this case, comparing ( y^2 = 4x ) to ( y^2 = 4ax ), it looks like ( 4a = 4 ), which means ( a = 1 ). So, the focus of this parabola should be at ( (a, 0) ), which is ( (1, 0) ). Got that. Now, the problem says that this focus is the center of a circle, and this circle also passes through the origin, which is the point ( (0, 0) ). So, the circle is centered at ( (1, 0) ) and passes through ( (0, 0) ). I need to find the equation of this circle.I remember that the general equation of a circle with center at ( (h, k) ) and radius 'r' is ( (x - h)^2 + (y - k)^2 = r^2 ). Since the center is ( (1, 0) ), plugging that in, the equation becomes ( (x - 1)^2 + y^2 = r^2 ). Now, I need to find the radius 'r'. Since the circle passes through the origin ( (0, 0) ), I can substitute ( x = 0 ) and ( y = 0 ) into the equation to find 'r'. Let's do that:( (0 - 1)^2 + (0)^2 = r^2 )( (-1)^2 + 0 = r^2 )( 1 = r^2 )So, ( r = 1 ) (since radius can't be negative).Therefore, the equation of the circle is ( (x - 1)^2 + y^2 = 1 ). But let me check the options given. They are all in the form of expanded equations, not in the standard circle form. So, I need to expand my equation to match one of the options.Expanding ( (x - 1)^2 + y^2 = 1 ):First, expand ( (x - 1)^2 ):( x^2 - 2x + 1 + y^2 = 1 )Now, subtract 1 from both sides to get:( x^2 - 2x + y^2 = 0 )Looking at the options:A: ( x^2 + y^2 + 2x = 0 )B: ( x^2 + y^2 + x = 0 )C: ( x^2 + y^2 - x = 0 )D: ( x^2 + y^2 - 2x = 0 )Hmm, my equation is ( x^2 - 2x + y^2 = 0 ), which is the same as ( x^2 + y^2 - 2x = 0 ). That matches option D. So, I think the correct answer is D.But wait, let me double-check my steps to make sure I didn't make any mistakes. I found the focus correctly at (1, 0). Then, using the center at (1, 0) and passing through (0, 0), I calculated the radius as 1, which seems right because the distance between (1, 0) and (0, 0) is indeed 1. Then, expanding the equation, I got ( x^2 - 2x + y^2 = 0 ), which is the same as option D. Yeah, that seems correct.Just to be thorough, let me check if any other options could make sense. Option A has +2x, which would mean the center is at (-1, 0), but the focus is at (1, 0), so that's not it. Option B has +x, which would be a center at (-0.5, 0), which also doesn't match. Option C has -x, which would be a center at (0.5, 0), but our center is at (1, 0). So, none of the other options fit except D.I think I'm confident now that D is the correct answer.**Final Answer**boxed{D}

💡First, I need to determine Charlie's floor number. Since Charlie lives on a floor that is one-fourth of Frank's floor number, and Frank is on the 16th floor, I divide 16 by 4 to find Charlie's floor.Next, I know that Dennis lives two floors above Charlie. After finding Charlie's floor number, I add 2 to it to determine Dennis's floor number.Finally, by performing these calculations, I can conclude the exact floor where Dennis lives.

💡First, I need to determine the total number of bollards required for both sides of the road. Since there are 4000 bollards on each side and two sides in total, the total number of bollards is 4000 multiplied by 2, which equals 8000 bollards.Next, the company has already installed 3/4 of the total bollards. To find out how many bollards have been installed, I multiply 3/4 by the total number of bollards: 3/4 * 8000 = 6000 bollards.Finally, to determine how many more bollards are needed, I subtract the number of bollards already installed from the total number required: 8000 - 6000 = 2000 bollards.

💡Okay, so I've got this problem here about a stadium with a capacity of 2000 people. Right now, there are 3/4 of that capacity in the stadium, and each person paid 20 to get in. I need to find out the difference in the total fees collected when the stadium was 3/4 full compared to if it had been full. Hmm, okay, let's break this down step by step.First, I need to figure out how many people are in the stadium when it's 3/4 full. The total capacity is 2000, so 3/4 of that would be... let's see, 3 divided by 4 is 0.75, and 0.75 times 2000 is 1500 people. So, there are 1500 people in the stadium when it's 3/4 full.Now, each of those 1500 people paid 20 in entry fees. To find the total fees collected when the stadium is 3/4 full, I need to multiply the number of people by the fee per person. So, 1500 times 20. Let me calculate that: 1500 times 20 is 30,000. Okay, so the total fees collected when the stadium is 3/4 full are 30,000.Next, I need to find out how much would have been collected if the stadium was full. The full capacity is 2000 people, and each person still pays 20. So, 2000 times 20. Let me do that multiplication: 2000 times 20 is 40,000. Got it, so if the stadium was full, the total fees collected would be 40,000.Now, the problem asks for the difference between these two amounts. That means I need to subtract the total fees when the stadium is 3/4 full from the total fees when it's full. So, 40,000 minus 30,000. Let me do that subtraction: 40,000 minus 30,000 is 10,000.Wait a minute, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the number of people when the stadium is 3/4 full: 3/4 of 2000 is indeed 1500. Then, 1500 times 20 is 30,000. For the full capacity, 2000 times 20 is 40,000. Subtracting 30,000 from 40,000 gives 10,000. Yep, that seems correct.So, the difference in the total amount of fees collected between when the stadium was 3/4 full and when it was full is 10,000.