💡First, I need to determine the total monthly salary of the 20 employees. Since the average salary is Rs. 1200, the total salary is 20 multiplied by Rs. 1200, which equals Rs. 24,000.Next, when the manager's salary is added, the average salary increases by Rs. 100, making the new average Rs. 1300. Now there are 21 people in the organization, so the new total monthly salary for all 21 people is 21 multiplied by Rs. 1300, which equals Rs. 27,300.Finally, to find the manager's salary, I subtract the original total salary from the new total salary: Rs. 27,300 minus Rs. 24,000 equals Rs. 3,300. Therefore, the manager's monthly salary is Rs. 3,300.

💡Alright, so I'm trying to figure out how much tax Paphnutiy has to pay if he applies the tax for self-employed citizens, specifically the tax on professional income. Okay, first off, I need to understand what exactly this tax entails. From what I remember, self-employed individuals usually have different tax rules compared to regular employees. They might have to pay both income tax and possibly some other taxes like social security or Medicare, depending on the country.But the problem mentions "tax on professional income," so I think it's focusing on the income tax part. I guess I need to know the tax rate for self-employed individuals. I'm not entirely sure, but I think it's higher than the regular income tax rate because self-employed people don't have taxes withheld from their paychecks like regular employees do. Maybe it's around 15-20%? I'm not certain, though.Wait, the problem doesn't specify the tax rate. Hmm, that's a bit of a problem. Without knowing the tax rate, I can't calculate the exact amount. Maybe I need to assume a standard rate or look it up. Since I'm just trying to figure this out, I'll assume a tax rate of 15% for self-employed individuals. I think that's a common rate in some places.Okay, so if Paphnutiy is self-employed and earns, let's say, 50,000 a year from his professional income, how much tax would he have to pay? Using the 15% rate I assumed, it would be 0.15 times 50,000, which is 7,500. So, he'd have to pay 7,500 in taxes.But wait, I'm not sure if that's all. Maybe there are deductions or credits he can claim to reduce his taxable income. For example, if he has business expenses related to his profession, he might be able to subtract those from his income before calculating the tax. Let's say he has 10,000 in business expenses. Then his taxable income would be 50,000 minus 10,000, which is 40,000. Applying the 15% tax rate to 40,000 would be 6,000 in taxes. That's better for him.But I'm not entirely sure about the rules for deductions. Maybe there are limits on what kinds of expenses he can deduct. For example, can he deduct a portion of his home as a home office? Or can he deduct health insurance premiums? I think some of these deductions are allowed, but I'm not certain about the specifics.Also, I wonder if there are any additional taxes or fees he has to pay besides the income tax. For instance, in some countries, self-employed individuals have to pay self-employment tax, which covers Social Security and Medicare. That could add another 15.3% or so to his tax bill. If that's the case, his total tax would be higher.Let's say the self-employment tax is 15.3%. So, if his taxable income is 40,000, his self-employment tax would be 6,120. Adding that to the 6,000 income tax, his total tax would be 12,120. That's a significant amount.But I'm not sure if the problem is asking for just the income tax or the total tax including self-employment tax. It says "tax on professional income," which might just refer to the income tax part. But to be safe, maybe I should mention both possibilities.Another thing to consider is if there are any tax credits or incentives for self-employed individuals. For example, some places offer tax credits for health insurance or retirement contributions. If Paphnutiy contributes to a retirement plan, he might be able to reduce his taxable income further.Let's say he contributes 5,000 to a retirement plan. That would lower his taxable income to 35,000. Then, his income tax would be 15% of 35,000, which is 5,250. Plus, if there's a self-employment tax of 15.3%, that would be 5,355. So, his total tax would be 5,250 plus 5,355, totaling 10,605.This is getting a bit complicated. I think I need to break it down step by step to make sure I'm not missing anything.First, determine the taxable income:- Gross income: 50,000- Business expenses: 10,000- Retirement contributions: 5,000- Taxable income: 50,000 - 10,000 - 5,000 = 35,000Next, calculate the income tax:- Tax rate: 15%- Income tax: 0.15 * 35,000 = 5,250Then, calculate the self-employment tax:- Self-employment tax rate: 15.3%- Self-employment tax: 0.153 * 35,000 = 5,355Finally, add both taxes together:- Total tax: 5,250 + 5,355 = 10,605So, Paphnutiy would have to pay a total of 10,605 in taxes.But I'm still not sure if all these deductions and taxes apply. Maybe in the problem's context, it's simpler, and we're only supposed to consider the income tax on his professional income without the self-employment tax. If that's the case, then it's just 5,250.Alternatively, if the problem assumes a different tax rate or different deductions, the amount would change accordingly. I think I need to clarify the assumptions or see if there are any additional details provided in the problem that I missed.Also, I wonder if there are any thresholds or brackets that affect the tax calculation. For example, maybe the tax rate increases for higher income levels. If Paphnutiy's income is above a certain threshold, he might be taxed at a higher rate. But since I don't have that information, I'll stick with the 15% rate for simplicity.In summary, to calculate the tax amount for Paphnutiy as a self-employed citizen, I need to:1. Determine his gross professional income.2. Subtract any allowable business expenses and deductions.3. Apply the appropriate tax rate to the taxable income.4. Consider any additional taxes or fees, like self-employment tax, if applicable.Without specific numbers or more details, this is as far as I can go. But I think following these steps should give a reasonable estimate of the tax amount.

💡Let's denote the present age of Ramesh as ( R ) and the present age of Mahesh as ( M ).1. **Difference in Present Ages:** [ M - R = 7.5 quad text{(1)} ]2. **Ratio of Ages in 10 Years:** [ frac{R + 10}{M + 10} = frac{2}{3} quad text{(2)} ]3. **Cross-Multiplying Equation (2):** [ 3(R + 10) = 2(M + 10) ] [ 3R + 30 = 2M + 20 ] [ 2M - 3R = 10 quad text{(3)} ]4. **Solving Equations (1) and (3):** - Multiply Equation (1) by 2: [ 2M - 2R = 15 quad text{(4)} ] - Subtract Equation (4) from Equation (3): [ (2M - 3R) - (2M - 2R) = 10 - 15 ] [ -R = -5 ] [ R = 5 ] - Substitute ( R = 5 ) into Equation (1): [ M - 5 = 7.5 ] [ M = 12.5 ]5. **Ratio of Present Ages:** [ frac{R}{M} = frac{5}{12.5} = frac{2}{5} ]

💡Okay, so I have this problem where I need to find the sum of the squares of some constants in parametric equations. The line segment connects two points, (-3, 10) and (4, 16). The parametric equations are given as x = at + b and y = ct + d, where t ranges from 0 to 1. When t is 0, the point is (-3, 10), and when t is 1, it should be (4, 16). I need to find a² + b² + c² + d².Alright, let me start by understanding what parametric equations are. They are equations where the coordinates are expressed in terms of a parameter, which in this case is t. So, as t changes from 0 to 1, the point (x, y) moves from (-3, 10) to (4, 16).Given that t = 0 corresponds to (-3, 10), I can plug t = 0 into the equations. For x, that would be x = a*0 + b = b. Similarly, for y, it's y = c*0 + d = d. So, when t = 0, x = b and y = d. But we know that at t = 0, the point is (-3, 10). Therefore, b must be -3 and d must be 10. So, I've found b and d.Now, moving on to when t = 1. Plugging t = 1 into the parametric equations, we get x = a*1 + b = a + b and y = c*1 + d = c + d. We know that when t = 1, the point is (4, 16). So, setting up the equations:a + b = 4c + d = 16But I already know that b = -3 and d = 10, so I can substitute those values into these equations.Starting with the first equation: a + (-3) = 4. Solving for a, I add 3 to both sides: a = 4 + 3 = 7. So, a is 7.Next, the second equation: c + 10 = 16. Solving for c, I subtract 10 from both sides: c = 16 - 10 = 6. So, c is 6.Now, I have all the constants: a = 7, b = -3, c = 6, d = 10. The problem asks for the sum of their squares, which is a² + b² + c² + d².Calculating each term:a² = 7² = 49b² = (-3)² = 9c² = 6² = 36d² = 10² = 100Adding them all together: 49 + 9 + 36 + 100.Let me compute that step by step:49 + 9 = 5858 + 36 = 9494 + 100 = 194So, the sum of the squares is 194.Wait, let me double-check my calculations to make sure I didn't make a mistake. Sometimes when adding numbers, it's easy to slip up.Starting again:a² = 7² = 49b² = (-3)² = 9c² = 6² = 36d² = 10² = 100Adding them up:49 + 9 = 5858 + 36 = 9494 + 100 = 194Yes, that seems correct. So, the final answer is 194.Just to make sure I didn't make any mistakes in finding a and c, let me verify.We had:At t = 0: x = b = -3, y = d = 10.At t = 1: x = a + b = 4, so a = 4 - b = 4 - (-3) = 7.Similarly, y = c + d = 16, so c = 16 - d = 16 - 10 = 6.Yes, that's correct. So, a = 7 and c = 6 are correct.Therefore, the calculations for the squares are correct as well.I think I've got it right. The sum is 194.**Final Answer**boxed{194}

💡Okay, so I have this problem here where I need to find the minimum value of |PM|, where P is a point satisfying certain inequalities and M is a point on a circle. Let me try to break this down step by step.First, let's understand the constraints for point P(x, y). The problem says:cosθ ≤ x ≤ 3cosθ,sinθ ≤ y ≤ 3sinθ,for some real number θ. Hmm, okay. So for any angle θ, the x-coordinate of P is between cosθ and 3cosθ, and the y-coordinate is between sinθ and 3sinθ. That seems a bit abstract. Maybe I can visualize this region.If I think about θ varying over all real numbers, cosθ and sinθ will oscillate between -1 and 1. But since x is between cosθ and 3cosθ, and y is between sinθ and 3sinθ, the point P(x, y) must lie within some sort of region that depends on θ. Wait, maybe I can square the inequalities to see if that helps.So, squaring the inequalities:cos²θ ≤ x² ≤ 9cos²θ,sin²θ ≤ y² ≤ 9sin²θ.Adding these together:cos²θ + sin²θ ≤ x² + y² ≤ 9cos²θ + 9sin²θ.But cos²θ + sin²θ is 1, so:1 ≤ x² + y² ≤ 9.So, the point P(x, y) lies in the region between two circles centered at the origin with radii 1 and 3. But wait, are there any restrictions on x and y? Let me check the original inequalities again.The original inequalities are cosθ ≤ x ≤ 3cosθ and sinθ ≤ y ≤ 3sinθ. So, depending on θ, cosθ and sinθ can be positive or negative. For example, if θ is 0, cosθ is 1 and sinθ is 0, so x is between 1 and 3, and y is between 0 and 0, so y must be 0. Similarly, if θ is π/2, cosθ is 0 and sinθ is 1, so x is between 0 and 0 (so x must be 0) and y is between 1 and 3.Wait a minute, so depending on θ, the point P(x, y) can be in different quadrants. But when I squared the inequalities, I assumed x and y are positive, but actually, they can be negative as well. Hmm, so maybe the region isn't just in the first quadrant? Or is it?Wait, let me think. If θ is in the first quadrant, say between 0 and π/2, then cosθ and sinθ are positive, so x and y are positive. If θ is in the second quadrant, between π/2 and π, cosθ is negative and sinθ is positive, so x is between negative and positive values, but y is still positive. Similarly, for θ in the third and fourth quadrants, x and y can be negative or positive.But when I squared the inequalities, I lost the sign information. So, actually, the region defined by 1 ≤ x² + y² ≤ 9 is correct, but it's not restricted to any particular quadrant. So, P(x, y) can be anywhere in the annulus between the circles of radius 1 and 3 centered at the origin.But wait, is that the case? Let me check with θ = π. Then cosθ = -1, sinθ = 0, so x is between -1 and -3, and y is between 0 and 0, so y must be 0. Similarly, θ = 3π/2, cosθ = 0, sinθ = -1, so x is between 0 and 0, y is between -1 and -3.So, actually, for any θ, the point P(x, y) is constrained to a rectangle in the plane, but as θ varies, these rectangles sweep across the plane, covering the annulus between radius 1 and 3. So, the region is indeed the annulus 1 ≤ x² + y² ≤ 9.Okay, so now I know that P(x, y) is somewhere in this annulus. Now, point M(x, y) is on the circle (x + 5)² + (y + 5)² = 1. So, M is on a circle centered at (-5, -5) with radius 1.I need to find the minimum value of |PM|, which is the distance between point P and point M. So, essentially, I need to find the closest distance between any point P in the annulus 1 ≤ x² + y² ≤ 9 and any point M on the circle centered at (-5, -5) with radius 1.To find the minimum distance, I can think of it as the distance between the two regions: the annulus and the circle. The minimum distance between two regions is the minimum distance between any two points in those regions.But how do I compute that? Maybe I can find the distance between the centers of the two regions and then subtract the radii?Wait, the annulus is centered at the origin, and the circle is centered at (-5, -5). So, the distance between the centers is sqrt[(-5 - 0)² + (-5 - 0)²] = sqrt[25 + 25] = sqrt[50] = 5√2.Now, the annulus has an inner radius of 1 and an outer radius of 3. The circle has a radius of 1. So, the closest point from the annulus to the circle would be along the line connecting the centers, right?So, the distance between the centers is 5√2. The annulus extends from radius 1 to 3, so the closest point on the annulus to the center of the circle (-5, -5) would be along the line connecting (0,0) to (-5, -5), at a distance of 1 from the origin. Similarly, the closest point on the circle to the annulus would be along the same line, at a distance of 5√2 - 1 from the origin.Wait, no. Let me think again. The distance between the centers is 5√2. The annulus is from radius 1 to 3, so the closest point on the annulus to the center of the circle (-5, -5) is along the line connecting (0,0) to (-5, -5), at a distance of 1 from the origin. Similarly, the closest point on the circle to the annulus is along the same line, at a distance of 5√2 - 1 from the origin.But wait, the circle is centered at (-5, -5) with radius 1, so the closest point on the circle to the origin would be along the line connecting (0,0) to (-5, -5), at a distance of 5√2 - 1 from the origin. Similarly, the closest point on the annulus to (-5, -5) is along the same line, at a distance of 1 from the origin.Therefore, the minimum distance between the annulus and the circle is the distance between these two closest points, which would be 5√2 - 1 - 1 = 5√2 - 2? Wait, that doesn't seem right.Wait, no. The distance between the two closest points is the distance between the centers minus the sum of their radii? Or is it the distance between the centers minus the difference of their radii?Wait, let's clarify. The distance between the centers is 5√2. The annulus is from radius 1 to 3, so the closest point on the annulus to the circle is at radius 1, and the closest point on the circle is at radius 1 from (-5, -5). So, the distance between these two points would be 5√2 - 1 - 1 = 5√2 - 2.But wait, that would be if both points are along the same line. Let me visualize this.Imagine two circles: one centered at the origin with radius 1, and another centered at (-5, -5) with radius 1. The distance between their centers is 5√2. The closest distance between any two points on these circles would be 5√2 - 1 - 1 = 5√2 - 2.But in our case, the annulus is not just the inner circle, but the region between radius 1 and 3. So, the closest point on the annulus to the circle is still on the inner circle (radius 1), because that's closer to the circle centered at (-5, -5). So, the minimum distance between the annulus and the circle is 5√2 - 2.Wait, but looking at the answer choices, I don't see 5√2 - 2. The options are 2√2 -1, 4√2 -1, 6√2 -1, and √61 -1.Hmm, so maybe my approach is wrong. Let me think again.Alternatively, maybe I should parametrize point P and point M and find the minimum distance.Let me denote P as (x, y) in the annulus 1 ≤ x² + y² ≤ 9, and M as (a, b) on the circle (a + 5)² + (b + 5)² = 1.I need to minimize |PM| = sqrt[(x - a)² + (y - b)²].To minimize this, I can consider the distance squared: (x - a)² + (y - b)².Since both P and M are variables, this is a bit tricky. Maybe I can fix M and find the closest P, then minimize over M, or vice versa.Alternatively, perhaps I can use the fact that the minimum distance between two sets is the minimum distance between any two points in the sets. So, the minimum distance between the annulus and the circle is the minimum distance between any point in the annulus and any point on the circle.Given that, the minimum distance would be the distance between the two centers minus the sum of the maximum extent of the annulus and the radius of the circle.Wait, the annulus extends from radius 1 to 3, so the farthest point from the origin in the annulus is 3 units away. The circle is centered at (-5, -5) with radius 1, so the closest point on the circle to the origin is 5√2 - 1 units away.Wait, so the closest point on the circle to the origin is 5√2 - 1, and the closest point on the annulus to the origin is 1. So, the distance between these two points would be (5√2 - 1) - 1 = 5√2 - 2.But again, that's not one of the answer choices. Maybe I'm missing something.Wait, perhaps I should consider the distance between the origin and the circle. The distance from the origin to the center of the circle is 5√2, and the circle has radius 1, so the closest point on the circle to the origin is 5√2 - 1. Similarly, the closest point on the annulus to the origin is 1. Therefore, the minimum distance between the annulus and the circle is 5√2 - 1 - 1 = 5√2 - 2.But again, 5√2 - 2 is approximately 5*1.414 - 2 ≈ 7.07 - 2 = 5.07, which is not among the options. The options are 2√2 -1 ≈ 2.828 -1 ≈ 1.828, 4√2 -1 ≈ 5.656 -1 ≈ 4.656, 6√2 -1 ≈ 8.485 -1 ≈ 7.485, and √61 -1 ≈ 7.81 -1 ≈ 6.81.Hmm, none of these match 5√2 - 2 ≈ 5.07. So, maybe my initial approach is wrong.Wait, perhaps I should consider the point P not just on the inner circle, but anywhere in the annulus. Maybe the closest point P is not on the inner circle, but somewhere else.Alternatively, maybe I can parametrize P in terms of θ. Since P is defined in terms of θ, maybe I can express P as (3cosθ, 3sinθ) or something like that. Wait, no, because P is between cosθ and 3cosθ, and between sinθ and 3sinθ. So, P can be anywhere in the rectangle defined by those bounds for a given θ.Wait, but θ is a parameter that can vary. So, for each θ, P is in a certain rectangle, and as θ varies, these rectangles sweep across the plane, covering the annulus.But maybe instead of thinking about θ, I can think of P as any point in the annulus. So, perhaps the minimum distance from P to M is the minimum distance from the annulus to the circle.But I'm still stuck on how to compute that.Wait, another approach: the minimum distance between two sets is the minimum of the distances between all pairs of points in the sets. So, to find the minimum |PM|, I need to find the minimum of sqrt[(x - a)^2 + (y - b)^2] where (x, y) is in the annulus and (a, b) is on the circle.Alternatively, since both sets are compact, the minimum is achieved at some specific points.Maybe I can use optimization techniques. Let me set up the problem.Let me denote P = (x, y) and M = (a, b). We have:1 ≤ x² + y² ≤ 9,(a + 5)^2 + (b + 5)^2 = 1.We need to minimize sqrt[(x - a)^2 + (y - b)^2].To simplify, let's minimize the square of the distance: (x - a)^2 + (y - b)^2.We can use Lagrange multipliers to find the minimum, considering the constraints.But this might get complicated because we have two variables (x, y) and (a, b) with their own constraints.Alternatively, perhaps I can fix P and find the closest M, then minimize over P.Given a fixed P, the closest M to P is the point on the circle (a + 5)^2 + (b + 5)^2 = 1 that is closest to P. The distance from P to M would then be the distance from P to the center of the circle minus the radius.Wait, that's a good point. For a fixed P, the closest distance to the circle is |PM| = |P - (-5, -5)| - 1.So, the distance from P to M is at least |P - (-5, -5)| - 1.Therefore, the minimum distance |PM| over all M on the circle is |P - (-5, -5)| - 1.Therefore, to minimize |PM|, we need to minimize |P - (-5, -5)| - 1, which is equivalent to minimizing |P - (-5, -5)| and then subtracting 1.So, the problem reduces to finding the minimum distance from P to (-5, -5), where P is in the annulus 1 ≤ x² + y² ≤ 9, and then subtracting 1.So, what's the minimum distance from P to (-5, -5) where P is in the annulus?The distance from P to (-5, -5) is sqrt[(x + 5)^2 + (y + 5)^2]. We need to minimize this over all P such that 1 ≤ x² + y² ≤ 9.This is equivalent to finding the point P in the annulus that is closest to (-5, -5).The closest point in the annulus to (-5, -5) would lie along the line connecting the origin to (-5, -5), because that's the direction where the annulus is closest to (-5, -5).So, the line from the origin to (-5, -5) is the line y = x, but in the negative direction. So, the point on the annulus closest to (-5, -5) would be in the direction of (-5, -5), scaled to the inner radius of the annulus.Wait, the inner radius is 1, so the point would be (-5/√50, -5/√50) scaled by 1, because the direction vector is (-5, -5), which has length sqrt(25 + 25) = sqrt(50) = 5√2.So, the unit vector in the direction of (-5, -5) is (-5/5√2, -5/5√2) = (-1/√2, -1/√2).Therefore, the point on the inner circle (radius 1) in that direction is (-1/√2, -1/√2).So, the distance from this point to (-5, -5) is sqrt[(-5 + 1/√2)^2 + (-5 + 1/√2)^2].Let me compute that:First, (-5 + 1/√2)^2 = (5 - 1/√2)^2 = 25 - 10/√2 + 1/2.Similarly for the y-coordinate.So, total distance squared is 2*(25 - 10/√2 + 1/2) = 2*(25.5 - 10/√2) = 51 - 20/√2.Simplify 20/√2 = 10√2, so distance squared is 51 - 10√2.Therefore, the distance is sqrt(51 - 10√2).Wait, but that seems complicated. Maybe I made a mistake.Alternatively, perhaps I can compute the distance from (-5, -5) to the origin, which is 5√2, and then subtract the radius of the inner circle, which is 1. So, the minimum distance from (-5, -5) to the annulus is 5√2 - 1.Wait, that makes sense because the closest point on the annulus to (-5, -5) is along the line connecting the origin to (-5, -5), at a distance of 1 from the origin. Therefore, the distance from (-5, -5) to that point is 5√2 - 1.Therefore, the minimum distance |PM| is (5√2 - 1) - 1 = 5√2 - 2.But again, that's not one of the answer choices. Wait, let me check the answer choices again:(A) 2√2 -1 ≈ 1.828(B) 4√2 -1 ≈ 4.656(C) 6√2 -1 ≈ 7.485(D) √61 -1 ≈ 6.81Hmm, none of these are 5√2 - 2 ≈ 5.07. So, maybe my approach is still wrong.Wait, perhaps I should consider that P is not just on the inner circle, but can be anywhere in the annulus. Maybe the closest point P is not along the line connecting the origin to (-5, -5), but somewhere else.Alternatively, perhaps I should parametrize P as (r cosθ, r sinθ) where 1 ≤ r ≤ 3, and then find the minimum distance from (r cosθ, r sinθ) to (-5, -5), and then subtract 1.Wait, that might work. Let me try that.So, the distance from P(r cosθ, r sinθ) to (-5, -5) is sqrt[(r cosθ + 5)^2 + (r sinθ + 5)^2].We need to minimize this expression over r ∈ [1, 3] and θ ∈ [0, 2π).Let me square the distance to make it easier:D = (r cosθ + 5)^2 + (r sinθ + 5)^2.Expanding this:= r² cos²θ + 10r cosθ + 25 + r² sin²θ + 10r sinθ + 25= r² (cos²θ + sin²θ) + 10r (cosθ + sinθ) + 50= r² + 10r (cosθ + sinθ) + 50.So, D = r² + 10r (cosθ + sinθ) + 50.We need to minimize D over r ∈ [1, 3] and θ ∈ [0, 2π).Let me consider the term 10r (cosθ + sinθ). The maximum and minimum of cosθ + sinθ are sqrt(2) and -sqrt(2), respectively.So, the term 10r (cosθ + sinθ) can vary between -10r sqrt(2) and 10r sqrt(2).Therefore, to minimize D, we need to minimize 10r (cosθ + sinθ). Since we're adding it to r² + 50, to minimize D, we need to take the minimum of 10r (cosθ + sinθ), which is -10r sqrt(2).Therefore, the minimum D is r² - 10r sqrt(2) + 50.So, now we have D_min(r) = r² - 10r sqrt(2) + 50.We need to find the minimum of D_min(r) over r ∈ [1, 3].Let me compute D_min(r):D_min(r) = r² - 10√2 r + 50.This is a quadratic in r, opening upwards (since the coefficient of r² is positive). The minimum occurs at the vertex.The vertex of a quadratic ax² + bx + c is at x = -b/(2a). Here, a = 1, b = -10√2, so the vertex is at r = (10√2)/(2*1) = 5√2 ≈ 7.07.But our r is restricted to [1, 3], so the minimum occurs at the endpoint closest to 5√2. Since 5√2 ≈ 7.07 > 3, the minimum occurs at r = 3.Wait, but that would mean that the minimum D_min(r) is at r = 3.Wait, but let me compute D_min(3):D_min(3) = 9 - 10√2 * 3 + 50 = 59 - 30√2 ≈ 59 - 42.426 ≈ 16.574.Similarly, D_min(1) = 1 - 10√2 * 1 + 50 = 51 - 10√2 ≈ 51 - 14.142 ≈ 36.858.So, D_min(r) is minimized at r = 3, giving D ≈ 16.574.Therefore, the minimum distance squared is approximately 16.574, so the minimum distance is sqrt(16.574) ≈ 4.07.But wait, that's just the distance from P to (-5, -5). Then, since M is on the circle centered at (-5, -5) with radius 1, the minimum distance from P to M would be this distance minus 1, so approximately 4.07 - 1 ≈ 3.07.But 3.07 is approximately 4√2 -1 ≈ 5.656 -1 ≈ 4.656, which is option B. Wait, but 3.07 is less than 4.656. Hmm, maybe I made a mistake.Wait, no. Let me clarify. The distance from P to (-5, -5) is sqrt(D) ≈ 4.07. Then, since M is on the circle centered at (-5, -5) with radius 1, the minimum distance from P to M is sqrt(D) - 1 ≈ 4.07 - 1 ≈ 3.07.But 3.07 is approximately 2√2 -1 ≈ 2.828 -1 ≈ 1.828, which is option A. Wait, no, 3.07 is more than 2.828.Wait, perhaps I should compute sqrt(16.574) more accurately.sqrt(16.574) ≈ 4.07, as I said. So, 4.07 - 1 ≈ 3.07.But 3.07 is approximately 4√2 -1 ≈ 5.656 -1 ≈ 4.656? No, that's not right. Wait, 4√2 is about 5.656, so 4√2 -1 is about 4.656, which is larger than 3.07.Wait, perhaps I made a mistake in interpreting the distance.Wait, actually, the distance from P to M is |PM| = |P - M|. Since M is on the circle centered at (-5, -5) with radius 1, the closest M to P is along the line connecting P to (-5, -5), at a distance of |P - (-5, -5)| - 1.So, if |P - (-5, -5)| is approximately 4.07, then |PM| is 4.07 - 1 ≈ 3.07.But 3.07 is approximately equal to sqrt(61) -1 ≈ 7.81 -1 ≈ 6.81? No, that's not right.Wait, maybe I should compute sqrt(16.574) more accurately.Wait, 4.07 squared is 16.5649, which is close to 16.574, so sqrt(16.574) ≈ 4.07.So, |PM| ≈ 4.07 -1 ≈ 3.07.But none of the answer choices are close to 3.07. The closest is option A: 2√2 -1 ≈ 2.828 -1 ≈ 1.828, which is too low, and option B: 4√2 -1 ≈ 5.656 -1 ≈ 4.656, which is higher than 3.07.Wait, maybe I made a mistake in the calculation of D_min(r).Let me go back.We had D = r² + 10r (cosθ + sinθ) + 50.To minimize D, we set cosθ + sinθ to its minimum, which is -sqrt(2). So, D_min(r) = r² - 10r sqrt(2) + 50.Then, we found that the minimum of D_min(r) occurs at r = 5√2, which is outside our range [1, 3]. Therefore, the minimum occurs at r = 3.So, D_min(3) = 9 - 10√2 *3 + 50 = 59 - 30√2.Compute 30√2 ≈ 30*1.414 ≈ 42.42.So, 59 - 42.42 ≈ 16.58.Therefore, sqrt(16.58) ≈ 4.07.So, |PM| = 4.07 -1 ≈ 3.07.But 3.07 is approximately sqrt(61) -1 ≈ 7.81 -1 ≈ 6.81? No, that's not right.Wait, maybe I should compute |PM| directly.Wait, if P is at (3 cosθ, 3 sinθ), and M is at (-5 - cosφ, -5 - sinφ), since M is on the circle (x +5)^2 + (y +5)^2 =1, so M can be parametrized as (-5 + cosφ, -5 + sinφ). Wait, no, because the circle is centered at (-5, -5), so M is (-5 + cosφ, -5 + sinφ). Wait, no, actually, the circle is (x +5)^2 + (y +5)^2 =1, so M is (-5 + cosφ, -5 + sinφ). Wait, no, that's not correct. The standard parametrization is center plus radius times (cosφ, sinφ). So, M is (-5 + cosφ, -5 + sinφ).Wait, no, actually, it's (-5 + cosφ, -5 + sinφ). So, the coordinates are (-5 + cosφ, -5 + sinφ).Wait, but that would make M be at (-5 + cosφ, -5 + sinφ). So, to find the distance between P and M, we have:|PM| = sqrt[(x - (-5 + cosφ))² + (y - (-5 + sinφ))²].But this seems complicated. Maybe instead of parametrizing both P and M, I should think differently.Wait, perhaps I can consider the distance from P to M as the distance from P to (-5, -5) minus the radius of the circle, which is 1. So, |PM| = |P - (-5, -5)| -1.Therefore, to minimize |PM|, I need to minimize |P - (-5, -5)|, which is the distance from P to (-5, -5), and then subtract 1.So, the minimum |PM| is (minimum distance from P to (-5, -5)) -1.So, what's the minimum distance from P to (-5, -5) where P is in the annulus 1 ≤ x² + y² ≤9.The minimum distance from P to (-5, -5) is the distance from (-5, -5) to the origin minus the maximum distance from the origin to P.Wait, no. The distance from (-5, -5) to the origin is 5√2. The maximum distance from the origin to P is 3. So, the minimum distance from P to (-5, -5) is 5√2 - 3.Wait, that makes sense because the closest point P to (-5, -5) would be in the direction towards (-5, -5), at the farthest point of the annulus, which is radius 3.Wait, no, actually, the closest point would be in the direction towards (-5, -5), but at the closest point of the annulus, which is radius 1.Wait, no, the annulus is between 1 and 3. So, the closest point to (-5, -5) would be in the direction towards (-5, -5), but at the closest possible point, which is radius 1.Wait, but if I go in the direction of (-5, -5) from the origin, the point at radius 1 would be closer to (-5, -5) than the point at radius 3.Wait, let me compute the distance from (-5, -5) to the origin, which is 5√2 ≈7.07.If I have a point P at radius 1 in the direction of (-5, -5), then the distance from P to (-5, -5) would be 5√2 -1 ≈7.07 -1≈6.07.Similarly, if P is at radius 3 in the direction of (-5, -5), the distance would be 5√2 -3≈7.07 -3≈4.07.Wait, so the closer P is to (-5, -5), the smaller the distance. So, the minimum distance from P to (-5, -5) is 5√2 -3≈4.07.Therefore, the minimum |PM| is 4.07 -1≈3.07.But again, 3.07 is not one of the answer choices. Wait, perhaps I should compute 5√2 -3 -1=5√2 -4≈7.07 -4≈3.07.But 5√2 -4 is approximately 3.07, which is not one of the options.Wait, let me check the answer choices again:(A) 2√2 -1 ≈1.828(B)4√2 -1≈5.656 -1≈4.656(C)6√2 -1≈8.485 -1≈7.485(D)√61 -1≈7.81 -1≈6.81Hmm, none of these are 5√2 -4≈3.07.Wait, maybe I made a mistake in assuming that the closest point P is along the line connecting the origin to (-5, -5). Maybe the closest point is not in that direction.Wait, perhaps the closest point P is not along that line. Maybe it's somewhere else.Let me consider the point P that minimizes |P - (-5, -5)|. That would be the point P in the annulus closest to (-5, -5). The closest point in the annulus to (-5, -5) would be the intersection of the line connecting (-5, -5) to the origin with the annulus.So, the line from (-5, -5) to the origin is the same as the line from the origin to (-5, -5). The point on this line that is closest to (-5, -5) and lies in the annulus would be the point where the line intersects the annulus.The parametric equation of the line from the origin to (-5, -5) is t*(-5, -5), where t ≥0.We need to find t such that the point ( -5t, -5t ) lies in the annulus 1 ≤ x² + y² ≤9.So, x² + y² = (25t²) + (25t²) =50t².We need 1 ≤50t² ≤9.So, t² ≥1/50≈0.02, so t≥1/√50≈0.1414.And t² ≤9/50≈0.18, so t≤sqrt(9/50)=3/√50≈0.424.So, the point on the line from the origin to (-5, -5) that lies in the annulus is t=3/√50≈0.424.So, the coordinates are (-5*(3/√50), -5*(3/√50)) = (-15/√50, -15/√50) = (-3√2/2, -3√2/2).Wait, because 15/√50 =15/(5√2)=3/√2= (3√2)/2.So, the point is (-3√2/2, -3√2/2).Now, the distance from this point to (-5, -5) is sqrt[(-5 + 3√2/2)^2 + (-5 + 3√2/2)^2].Let me compute that:First, (-5 + 3√2/2)^2 = (5 - 3√2/2)^2 =25 - 15√2 + (9*2)/4=25 -15√2 + 4.5=29.5 -15√2.Similarly for the y-coordinate.So, total distance squared is 2*(29.5 -15√2)=59 -30√2.Therefore, the distance is sqrt(59 -30√2).Compute sqrt(59 -30√2):First, compute 30√2≈42.426.So, 59 -42.426≈16.574.So, sqrt(16.574)≈4.07.Therefore, the distance from P to (-5, -5) is approximately4.07, and then subtracting the radius 1, we get |PM|≈3.07.But again, 3.07 is not one of the answer choices.Wait, maybe I should compute sqrt(59 -30√2) exactly.Wait, 59 -30√2 is approximately16.574, as before.Wait, but perhaps sqrt(59 -30√2) can be simplified.Let me see:Suppose sqrt(a - b√2) = sqrt(c) - sqrt(d), then squaring both sides:a - b√2 = c + d - 2sqrt(c d).Comparing, we have:c + d = a,2sqrt(c d) = b√2 => sqrt(c d) = (b/2)√2 => c d = (b²/4)*2= b²/2.So, for a=59, b=30,We have:c + d =59,c d= (30)^2 /2=900/2=450.So, we need two numbers c and d such that c + d=59 and c d=450.Let me solve for c and d.Let me set up the quadratic equation:x² -59x +450=0.Compute discriminant:59² -4*1*450=3481 -1800=1681=41².So, x=(59 ±41)/2.Thus, x=(59+41)/2=100/2=50,x=(59-41)/2=18/2=9.So, c=50, d=9.Therefore, sqrt(59 -30√2)=sqrt(50) - sqrt(9)=5√2 -3.Therefore, the distance from P to (-5, -5) is5√2 -3.Therefore, the minimum |PM| is (5√2 -3) -1=5√2 -4.But 5√2 -4≈7.07 -4≈3.07, which is still not one of the answer choices.Wait, but looking back at the answer choices, option D is sqrt(61) -1≈7.81 -1≈6.81, which is larger than 5√2 -4≈3.07.Wait, maybe I made a mistake in the direction of the point P.Wait, perhaps the closest point P is not in the direction towards (-5, -5), but in the opposite direction.Wait, if I take P in the direction away from (-5, -5), then the distance from P to (-5, -5) would be larger, so that's not helpful.Alternatively, maybe the closest point P is not along the line connecting the origin to (-5, -5), but somewhere else.Wait, perhaps I should use calculus to find the minimum.Let me consider P=(x, y) in the annulus 1 ≤x² + y² ≤9, and M=(-5 + cosφ, -5 + sinφ).We need to minimize |PM|=sqrt[(x +5 -cosφ)^2 + (y +5 -sinφ)^2].This seems complicated, but maybe I can fix φ and find the minimum over P, then minimize over φ.Alternatively, perhaps I can use the fact that the minimum distance from P to M is the distance from P to the circle, which is |P - (-5, -5)| -1.So, to minimize |PM|, we need to minimize |P - (-5, -5)| -1.Therefore, the minimum |PM| is (minimum |P - (-5, -5)|) -1.We already found that the minimum |P - (-5, -5)| is5√2 -3, so the minimum |PM| is5√2 -4.But again, that's not one of the answer choices.Wait, perhaps I made a mistake in the direction of P.Wait, if P is in the direction of (-5, -5), then P is (-3√2/2, -3√2/2). But maybe the closest point is actually in the opposite direction.Wait, let me compute the distance from P=(3√2/2, 3√2/2) to (-5, -5).So, distance squared is (3√2/2 +5)^2 + (3√2/2 +5)^2.=2*(3√2/2 +5)^2.Compute 3√2/2≈2.121, so 2.121 +5≈7.121.So, squared is≈50.7, so distance≈7.12.Then, |PM|≈7.12 -1≈6.12, which is close to option D: sqrt(61)-1≈6.81.Wait, but 7.12 -1=6.12, which is less than sqrt(61)-1≈6.81.Wait, but this is the distance when P is in the opposite direction.Wait, I'm getting confused.Wait, perhaps the minimum |PM| is achieved when P is as close as possible to M, which is on the circle.Alternatively, perhaps I should consider the point M on the circle closest to the origin, and then find the closest P to that M.Wait, the closest point on the circle to the origin is (-5 + cosφ, -5 + sinφ) where φ is such that the vector from (-5, -5) to M points towards the origin.Wait, the closest point on the circle to the origin is along the line connecting the origin to (-5, -5). So, the point M is (-5 + cosφ, -5 + sinφ) where φ is such that the vector from (-5, -5) to M is in the direction of the origin.Wait, the direction from (-5, -5) to the origin is (5,5), so the unit vector is (5/5√2, 5/5√2)=(1/√2, 1/√2).Therefore, the point M closest to the origin is (-5, -5) + (1/√2, 1/√2)*1= (-5 +1/√2, -5 +1/√2).So, M=(-5 +1/√2, -5 +1/√2).Now, the distance from M to the origin is sqrt[(-5 +1/√2)^2 + (-5 +1/√2)^2]=sqrt[2*(25 -10/√2 +1/2)]=sqrt[50 -20/√2 +1]=sqrt[51 -20/√2].But 20/√2=10√2, so sqrt[51 -10√2].Wait, earlier we saw that sqrt(51 -10√2)=5√2 -3≈4.07.Wait, so the distance from M to the origin is≈4.07.But P is in the annulus, so the closest P to M would be along the line connecting the origin to M, at radius 1.Wait, no, the closest P to M would be in the direction from M towards the origin, but within the annulus.Wait, perhaps the closest P to M is along the line connecting M to the origin, at radius 1.So, the point P would be (M normalized)*1.Wait, M is (-5 +1/√2, -5 +1/√2). The vector from origin to M is (-5 +1/√2, -5 +1/√2). The length of this vector is sqrt[(-5 +1/√2)^2 + (-5 +1/√2)^2]=sqrt[2*(25 -10/√2 +1/2)]=sqrt[50 -20/√2 +1]=sqrt[51 -20/√2]=sqrt(51 -10√2)≈4.07.So, the unit vector in the direction of M is [(-5 +1/√2)/sqrt(51 -10√2), (-5 +1/√2)/sqrt(51 -10√2)].Therefore, the point P on the inner circle (radius 1) in that direction is [(-5 +1/√2)/sqrt(51 -10√2), (-5 +1/√2)/sqrt(51 -10√2)].The distance from P to M is |M - P|=sqrt[( (-5 +1/√2 - (-5 +1/√2)/sqrt(51 -10√2) )^2 + same for y].This seems complicated, but perhaps we can compute it.Let me denote s = sqrt(51 -10√2)≈4.07.Then, P = [(-5 +1/√2)/s, (-5 +1/√2)/s].M = (-5 +1/√2, -5 +1/√2).So, M - P = [ (-5 +1/√2) - (-5 +1/√2)/s , same for y ].Factor out (-5 +1/√2):= [ (-5 +1/√2)(1 - 1/s), same for y ].Therefore, the distance squared is 2*[ (-5 +1/√2)^2 (1 -1/s)^2 ].Compute (-5 +1/√2)^2=25 -10/√2 +1/2=25.5 -10/√2≈25.5 -14.142≈11.358.(1 -1/s)^2≈(1 -1/4.07)^2≈(0.754)^2≈0.568.So, distance squared≈2*11.358*0.568≈2*6.46≈12.92.Therefore, distance≈sqrt(12.92)≈3.595.So, |PM|≈3.595.But again, this is not one of the answer choices.Wait, perhaps I should consider that the minimum |PM| is achieved when P is on the outer circle (radius 3) and M is on the circle centered at (-5, -5) with radius 1.Wait, let me compute the distance between the outer circle (radius 3) and the circle centered at (-5, -5) with radius 1.The distance between the centers is5√2≈7.07.The outer circle has radius 3, and the other circle has radius 1. So, the minimum distance between the two circles is5√2 -3 -1=5√2 -4≈3.07.But again, that's not one of the answer choices.Wait, maybe I should consider the point P=(3,0) and M=(-5, -5). Then, |PM|=sqrt[(3+5)^2 + (0+5)^2]=sqrt[64 +25]=sqrt(89)≈9.43.But that's larger than the previous distances.Wait, perhaps I should consider the point P=(0,3) and M=(-5, -5). Then, |PM|=sqrt[(0+5)^2 + (3+5)^2]=sqrt[25 +64]=sqrt(89)≈9.43.Alternatively, if P is at (3,3), then |PM|=sqrt[(3+5)^2 + (3+5)^2]=sqrt[64 +64]=sqrt(128)=8√2≈11.31.Wait, none of these are helpful.Wait, perhaps I should consider that the minimum |PM| is achieved when P is at (1,0) and M is at (-5, -5). Then, |PM|=sqrt[(1+5)^2 + (0+5)^2]=sqrt[36 +25]=sqrt(61)≈7.81.So, |PM|=sqrt(61)≈7.81, and since M is on the circle, the minimum |PM| would be sqrt(61)-1≈6.81, which is option D.Wait, that makes sense. Because if P is at (1,0), which is on the inner circle, and M is at (-5, -5), then the distance is sqrt(61). But since M is on the circle centered at (-5, -5) with radius 1, the closest M to P would be along the line connecting P to (-5, -5), at a distance of sqrt(61) -1.Therefore, the minimum |PM| is sqrt(61) -1, which is option D.I think that's the correct answer.

💡Alright, so I'm trying to figure out if there's a power of 2 where all the digits are non-zero, and you can rearrange those digits to form another power of 2. Hmm, sounds interesting. Let me break this down step by step.First, I know that powers of 2 grow exponentially, so their decimal representations get longer as the exponent increases. For example, 2^1 is 2, 2^2 is 4, 2^3 is 8, and so on. As we go higher, like 2^10 is 1024, which has a zero in it. But the problem specifies that all digits must be different from zero, so 1024 wouldn't qualify because of the zero.Okay, so I need to find two different powers of 2, say 2^a and 2^b, where a ≠ b, such that both have all non-zero digits and one can be rearranged to form the other. That means they must have the same number of digits, right? Because if they have different numbers of digits, you can't rearrange one to get the other.Let me think about the properties of numbers and their digit sums. I remember that numbers with the same digits have the same digit sum, and the digit sum modulo 9 is the same as the number itself modulo 9. So, if 2^a and 2^b are rearrangements of each other, then 2^a ≡ 2^b mod 9.But wait, powers of 2 modulo 9 cycle in a pattern. Let me list out some powers of 2 modulo 9 to see if there's a cycle:- 2^1 ≡ 2 mod 9- 2^2 ≡ 4 mod 9- 2^3 ≡ 8 mod 9- 2^4 ≡ 16 ≡ 7 mod 9- 2^5 ≡ 32 ≡ 5 mod 9- 2^6 ≡ 64 ≡ 1 mod 9- 2^7 ≡ 128 ≡ 2 mod 9- 2^8 ≡ 256 ≡ 4 mod 9- 2^9 ≡ 512 ≡ 8 mod 9- 2^10 ≡ 1024 ≡ 7 mod 9Okay, so the pattern repeats every 6 exponents: 2, 4, 8, 7, 5, 1, and then back to 2. So, if 2^a ≡ 2^b mod 9, then a ≡ b mod 6. That means the exponents a and b must differ by a multiple of 6.But wait, if a and b differ by 6, then 2^a and 2^b would have the same number of digits only if 2^6 is a factor that doesn't change the digit count. But 2^6 is 64, which is a two-digit number, so multiplying by 64 could change the number of digits. Hmm, this might complicate things.Let me think about specific examples. Let's list some powers of 2 with all non-zero digits:- 2^1 = 2- 2^2 = 4- 2^3 = 8- 2^4 = 16- 2^5 = 32- 2^6 = 64- 2^7 = 128- 2^8 = 256- 2^9 = 512- 2^10 = 1024 (has a zero, so exclude)- 2^11 = 2048 (has a zero, exclude)- 2^12 = 4096 (has a zero, exclude)- 2^13 = 8192 (has a zero, exclude)- 2^14 = 16384 (has a zero, exclude)- 2^15 = 32768 (has a zero, exclude)- 2^16 = 65536 (has a zero, exclude)- 2^17 = 131072 (has a zero, exclude)- 2^18 = 262144 (has a zero, exclude)- 2^19 = 524288 (has a zero, exclude)- 2^20 = 1048576 (has zeros, exclude)Okay, up to 2^20, the only powers of 2 without zeros are up to 2^9. Let's see if any of these can be rearranged to form another power of 2.Looking at 2^4 = 16. Can I rearrange 1 and 6 to get another power of 2? 61 is not a power of 2.2^5 = 32. Rearranging 3 and 2 gives 23, which is not a power of 2.2^6 = 64. Rearranging 6 and 4 gives 46, which is not a power of 2.2^7 = 128. Rearranging 1, 2, and 8. Let's see: 182, 218, 281, 812, 821. None of these are powers of 2.2^8 = 256. Rearranging 2, 5, and 6. Possible numbers: 265, 526, 562, 625, 652. None of these are powers of 2.2^9 = 512. Rearranging 5, 1, and 2. Possible numbers: 152, 125, 215, 251, 512, 521. None of these are powers of 2 except 512 itself.So, from the small exponents, it seems like there's no such pair. But maybe for larger exponents? However, as we saw earlier, larger exponents tend to have zeros in their decimal representations, which disqualifies them.Wait, but what if there's a larger power of 2 without zeros that can be rearranged? Let's check a few more:2^21 = 2097152 (has zeros)2^22 = 4194304 (has zeros)2^23 = 8388608 (has zeros)2^24 = 16777216 (has zeros)2^25 = 33554432 (has zeros)2^26 = 67108864 (has zeros)2^27 = 134217728 (has zeros)2^28 = 268435456 (has zeros)2^29 = 536870912 (has zeros)2^30 = 1073741824 (has zeros)It seems like beyond 2^9, all powers of 2 have zeros in their decimal representations. So, the only candidates are up to 2^9, and none of those can be rearranged to form another power of 2.Therefore, based on this exploration, it seems that no such power of 2 exists where all digits are non-zero and can be rearranged to form another power of 2.