💡Okay, so I have this problem where there are 49 natural numbers, let's call them (a_1, a_2, ldots, a_{49}), and their sum is 999. I need to find the greatest common divisor (GCD) of all these numbers, and specifically, I need to find the maximum possible value of this GCD, which is denoted as (d).First, I remember that the GCD of a set of numbers is the largest number that divides each of them without leaving a remainder. So, if (d) is the GCD of all these (a_i), then each (a_i) must be a multiple of (d). That means I can express each (a_i) as (d) multiplied by some integer. Let's say (a_i = d times k_i), where (k_i) is a natural number.Since all the (a_i) are multiples of (d), their sum will also be a multiple of (d). The sum of all (a_i) is given as 999, so (d) must be a divisor of 999. That makes sense because if (d) divides each (a_i), it should divide their total sum as well.Now, I need to find the maximum possible value of (d). To do that, I should look at the divisors of 999 and see which one is the largest possible that can still satisfy the condition that the sum of 49 multiples of (d) equals 999.Let me factorize 999 to find its divisors. Breaking it down:999 divided by 3 is 333.333 divided by 3 is 111.111 divided by 3 is 37.37 is a prime number.So, the prime factorization of 999 is (3^3 times 37). Therefore, the divisors of 999 are all the products of these prime factors. Listing them out:1, 3, 9, 27, 37, 111, 333, 999.These are all the possible values for (d). Now, I need to find the largest (d) such that when I divide 999 by (d), the result is at least 49 because I have 49 numbers, each of which is at least (d). Wait, actually, each (a_i) is a multiple of (d), so the smallest each (a_i) can be is (d). Therefore, the sum of the smallest possible (a_i)s would be (49 times d). This sum must be less than or equal to 999.So, the condition is:[49 times d leq 999]Solving for (d):[d leq frac{999}{49}]Calculating that:[frac{999}{49} approx 20.387]Since (d) has to be an integer, the maximum possible (d) can be is 20. But wait, 20 isn't a divisor of 999. From the list of divisors I made earlier, the largest divisor less than or equal to 20 is 9.Hold on, is 9 the maximum possible? Let me check.If (d = 9), then each (a_i) is a multiple of 9. The sum of all (a_i) is 999, so the sum of the (k_i)s (where (a_i = 9 times k_i)) would be:[frac{999}{9} = 111]So, we need 49 natural numbers (k_1, k_2, ldots, k_{49}) that add up to 111. Is that possible?Yes, because 49 is less than 111, so we can have some (k_i)s as 1 and others as higher numbers. For example, if 48 of the (k_i)s are 1, then the 49th (k_i) would be (111 - 48 = 63). That works because 63 is a natural number.So, in this case, each (a_i) would be either (9 times 1 = 9) or (9 times 63 = 567). The sum would be (48 times 9 + 567 = 432 + 567 = 999), which matches the given total.What if I tried a higher divisor, like 27? Let's see.If (d = 27), then the sum of the (k_i)s would be:[frac{999}{27} = 37]But we have 49 numbers to add up to 37. That's not possible because 49 is greater than 37, and each (k_i) has to be at least 1. So, the minimum sum would be 49, which is greater than 37. Therefore, (d = 27) is too large.Similarly, checking (d = 37):[frac{999}{37} = 27]Again, 27 is less than 49, so it's impossible to have 49 numbers adding up to 27 when each is at least 1. So, (d = 37) is also too large.Therefore, the next possible divisor is 9, which works as I showed earlier.Wait, just to make sure, what about (d = 3)?If (d = 3), then the sum of the (k_i)s is:[frac{999}{3} = 333]We need 49 numbers adding up to 333. That's definitely possible because 49 is much less than 333. For example, 48 numbers could be 1 and the last one could be (333 - 48 = 285). But since we're looking for the maximum (d), 3 is smaller than 9, so 9 is still better.So, after checking all possible divisors, the maximum value of (d) that satisfies the condition is 9.**Final Answer**The maximum value of (d) is boxed{9}.

💡First, identify the number of cookie packs sold by Lucy, which is 19.Next, identify the number of cookie packs sold by Robyn, which is 16.To find the total number of packs sold on their last day, add the packs sold by Lucy and Robyn together.So, 19 packs (Lucy) + 16 packs (Robyn) equals 35 packs.Therefore, the total number of packs they sold on their last day is 35.

💡Okay, so I have this problem where I need to find the minimum value of the expression ((a - x)^2 + (a + 2 - y)^2) given that (y = -x^2 + 3 ln x). Hmm, let me try to understand what this means.First, I know that ((a - x)^2 + (a + 2 - y)^2) looks like the square of the distance between two points: one is ((x, y)) and the other is ((a, a + 2)). So, essentially, I'm trying to find the minimum distance squared between a point on the curve (y = -x^2 + 3 ln x) and the point ((a, a + 2)). Wait, but (a) is a real number, so maybe I need to consider this for any (a)? Or is (a) fixed? The problem says "find the minimum value," so perhaps I need to find the minimum over all possible (a), (x), and (y) such that (y = -x^2 + 3 ln x). Hmm, that might be more complicated. Maybe I should think of it as minimizing the expression with respect to (x) and (y) given the constraint (y = -x^2 + 3 ln x), and then see how it relates to (a).Alternatively, maybe I can interpret this geometrically. The expression ((a - x)^2 + (a + 2 - y)^2) is the squared distance between the point ((x, y)) on the curve and the point ((a, a + 2)). So, if I can find the point on the curve that is closest to the point ((a, a + 2)), then the minimum value of this expression would be the square of that distance.But wait, ((a, a + 2)) is a point on the line (y = x + 2). So, actually, I'm looking for the minimum distance from the curve (y = -x^2 + 3 ln x) to the line (y = x + 2). Because if I can find the minimum distance between the curve and the line, then squaring it would give me the minimum value of the expression.So, perhaps I should rephrase the problem: find the minimum distance between the curve (y = -x^2 + 3 ln x) and the line (y = x + 2). Then, the minimum value of the expression would be the square of that minimum distance.To find the minimum distance between a curve and a line, I can use calculus. The distance from a point ((x, y)) on the curve to the line (y = x + 2) is given by the formula:[d = frac{|x - y + 2|}{sqrt{1^2 + (-1)^2}} = frac{|x - y + 2|}{sqrt{2}}]But since (y = -x^2 + 3 ln x), I can substitute that into the distance formula:[d = frac{|x - (-x^2 + 3 ln x) + 2|}{sqrt{2}} = frac{|x + x^2 - 3 ln x + 2|}{sqrt{2}}]To minimize (d), I can minimize the numerator, since the denominator is a constant. So, let me define a function:[f(x) = |x + x^2 - 3 ln x + 2|]Since the absolute value complicates things, maybe I can consider the square of the distance instead, which would be:[d^2 = frac{(x + x^2 - 3 ln x + 2)^2}{2}]But actually, since I'm looking for the minimum, I can just minimize the expression inside the absolute value squared, which is:[(x + x^2 - 3 ln x + 2)^2]But maybe it's simpler to consider the function without the absolute value, because the square will handle the positivity. So, let me define:[f(x) = (x + x^2 - 3 ln x + 2)^2]Now, to find the minimum of (f(x)), I can take its derivative and set it equal to zero.First, let me compute the derivative (f'(x)). Let me denote (g(x) = x + x^2 - 3 ln x + 2), so (f(x) = [g(x)]^2). Then, by the chain rule:[f'(x) = 2 g(x) cdot g'(x)]So, I need to compute (g'(x)):[g'(x) = 1 + 2x - frac{3}{x}]Therefore, the derivative of (f(x)) is:[f'(x) = 2 (x + x^2 - 3 ln x + 2) left(1 + 2x - frac{3}{x}right)]To find the critical points, set (f'(x) = 0). So, either (g(x) = 0) or (g'(x) = 0).First, let's consider (g'(x) = 0):[1 + 2x - frac{3}{x} = 0]Multiply both sides by (x) to eliminate the denominator:[x + 2x^2 - 3 = 0]Which simplifies to:[2x^2 + x - 3 = 0]This is a quadratic equation. Let's solve for (x):[x = frac{-1 pm sqrt{1 + 24}}{4} = frac{-1 pm 5}{4}]So, the solutions are:[x = frac{-1 + 5}{4} = 1 quad text{and} quad x = frac{-1 - 5}{4} = -frac{3}{2}]But since (x > 0) (because of the (ln x) term in the original equation), we discard (x = -frac{3}{2}). So, the critical point is at (x = 1).Now, let's check if (g(x) = 0) at (x = 1):[g(1) = 1 + 1 - 3 ln 1 + 2 = 1 + 1 - 0 + 2 = 4 neq 0]So, (g(x) neq 0) at (x = 1). Therefore, the critical point is only from (g'(x) = 0).Now, let's compute (f(x)) at (x = 1):[g(1) = 4 quad text{so} quad f(1) = 4^2 = 16]But wait, this is the square of the distance times (frac{1}{2}), so the actual distance squared would be (frac{16}{2} = 8). So, the minimum value of the expression ((a - x)^2 + (a + 2 - y)^2) is 8.But let me double-check my steps to make sure I didn't make a mistake.1. I interpreted the expression as the squared distance between ((x, y)) and ((a, a + 2)), which lies on the line (y = x + 2).2. I used the distance formula from a point to a line, substituted (y) from the curve, and set up the function to minimize.3. I considered the square of the distance to simplify differentiation.4. I found the derivative and set it to zero, leading to solving a quadratic equation.5. I found (x = 1) as the critical point.6. I computed (g(1)) and found it to be 4, leading to (f(1) = 16), which corresponds to a distance squared of 8.Seems correct. So, the minimum value is 8.

💡First, I need to determine how many books Mike has after selling some at a garage sale. He initially has 51.5 books and sells 45.75 books.Next, I'll calculate the number of books Mike has after selling:51.5 - 45.75 = 5.75 books remaining.Then, Mike receives 12.25 books as gifts. I'll add these to his current collection:5.75 + 12.25 = 18 books.Finally, Mike buys an additional 3.5 books from a store. I'll add these to his total:18 + 3.5 = 21.5 books.Therefore, Mike now has 21.5 books.

💡Okay, so I need to figure out how many more pins Richard knocked down than Patrick in this bowling game over four rounds. Let's see, the problem gives me the number of pins each person knocked down in each round, and there's also some bonus points for double strikes, but I think the main focus is on the total pins knocked down, right?Alright, let's start by breaking down each round and calculating the pins for each player.**First Round:**- Patrick knocked down 70 pins.- Richard knocked down 15 more pins than Patrick, so that's 70 + 15 = 85 pins.- Samantha knocked down 10 pins less than Richard, so that's 85 - 10 = 75 pins.Okay, so after the first round:- Patrick: 70 pins- Richard: 85 pins- Samantha: 75 pins**Second Round:**- Patrick knocked down twice as many pins as Richard did in the first round. Richard had 85 pins in the first round, so Patrick knocked down 2 * 85 = 170 pins.- Richard knocked down 3 less than Patrick in the same round. So, Richard knocked down 170 - 3 = 167 pins.- Samantha knocked down the same number of pins as Richard, so that's also 167 pins.After the second round:- Patrick: 170 pins- Richard: 167 pins- Samantha: 167 pins**Third Round:**- Patrick knocked down 1.5 times as many pins as he did in the first round. So, 1.5 * 70 = 105 pins.- Richard knocked down 25 more pins than he did in the second round. He had 167 pins in the second round, so 167 + 25 = 192 pins.- Samantha knocked down 1.5 times as many pins as she did in the second round. She had 167 pins in the second round, so 1.5 * 167 = 250.5 pins. Hmm, you can't knock down half a pin in bowling, so I guess we'll round that to 251 pins.After the third round:- Patrick: 105 pins- Richard: 192 pins- Samantha: 251 pins**Fourth Round:**- Patrick knocked down 1.2 times as many pins as he did in the third round. So, 1.2 * 105 = 126 pins.- Richard knocked down 1.3 times as many pins as he did in the third round. So, 1.3 * 192 = 249.6 pins. Again, we can't have a fraction of a pin, so we'll round that to 250 pins.- Samantha knocked down 1.1 times as many pins as she did in the third round. So, 1.1 * 251 = 276.1 pins. Rounding that to 276 pins.After the fourth round:- Patrick: 126 pins- Richard: 250 pins- Samantha: 276 pinsNow, let's sum up the total pins for each player over the four rounds.**Total Pins:**- Patrick: 70 (Round 1) + 170 (Round 2) + 105 (Round 3) + 126 (Round 4) = 70 + 170 = 240; 240 + 105 = 345; 345 + 126 = 471 pins.- Richard: 85 (Round 1) + 167 (Round 2) + 192 (Round 3) + 250 (Round 4) = 85 + 167 = 252; 252 + 192 = 444; 444 + 250 = 694 pins.- Samantha: 75 (Round 1) + 167 (Round 2) + 251 (Round 3) + 276 (Round 4) = 75 + 167 = 242; 242 + 251 = 493; 493 + 276 = 769 pins.So, Patrick knocked down a total of 471 pins, Richard knocked down 694 pins, and Samantha knocked down 769 pins.The question asks how many more pins Richard knocked down than Patrick. So, we subtract Patrick's total from Richard's total:694 (Richard) - 471 (Patrick) = 223 pins.Wait, but the problem also mentions bonus points for double strikes. Each double strike gives 20 points. I need to consider if this affects the total pins or just adds to the score. Since the question is about the total pins knocked down, and bonus points are additional, I think we don't need to add them to the pin count. They might affect the overall score, but the question specifically asks about the total pins knocked down, so I'll stick with the 223 pins difference.Just to be thorough, if there were double strikes, each would add 20 points, but since we don't have information about how many double strikes each player achieved, we can't calculate the exact bonus points. So, it's safe to ignore the bonus points for this calculation.Therefore, Richard knocked down 223 more pins than Patrick by the end of the four rounds.

💡Okay, so I have this problem about a square ABCD. Points M and N are midpoints of sides AD and BC respectively. I need to find the ratio of the area of triangle AMN to the area of the square ABCD. Hmm, let me visualize the square first.Let me label the square ABCD with A at the bottom-left corner, B at the bottom-right, C at the top-right, and D at the top-left. So, sides AB, BC, CD, and DA. Now, point M is the midpoint of AD, which is the left side, and point N is the midpoint of BC, which is the right side.If I consider the square on a coordinate plane, maybe it will help. Let's assume the square has side length 2 units for simplicity. So, placing point A at (0,0), then B would be at (2,0), C at (2,2), and D at (0,2). Therefore, the midpoint M of AD would be halfway between A(0,0) and D(0,2), so M is at (0,1). Similarly, the midpoint N of BC would be halfway between B(2,0) and C(2,2), so N is at (2,1).Now, triangle AMN has vertices at A(0,0), M(0,1), and N(2,1). I need to find the area of this triangle. One way to find the area of a triangle given its coordinates is by using the shoelace formula. The formula is:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|Plugging in the coordinates of A, M, and N:x1 = 0, y1 = 0 (Point A)x2 = 0, y2 = 1 (Point M)x3 = 2, y3 = 1 (Point N)So, substituting into the formula:Area = |(0*(1 - 1) + 0*(1 - 0) + 2*(0 - 1)) / 2|= |(0 + 0 + 2*(-1)) / 2|= |(-2) / 2|= | -1 | = 1So, the area of triangle AMN is 1 square unit.But wait, the square has side length 2, so its area is 2*2=4. Therefore, the ratio of the area of triangle AMN to the area of square ABCD is 1/4.Wait, that doesn't seem right. Because when I calculated the area using the shoelace formula, I got 1, but the square's area is 4, so 1/4 is the ratio. But I thought the answer was 1/8. Did I make a mistake somewhere?Let me double-check. Maybe I should have considered the side length as 1 instead of 2 to see if that changes the ratio.If the square has side length 1, then the coordinates would be A(0,0), B(1,0), C(1,1), D(0,1). Then, M is the midpoint of AD, so M is at (0,0.5), and N is the midpoint of BC, so N is at (1,0.5). Then, triangle AMN has vertices at A(0,0), M(0,0.5), and N(1,0.5).Using the shoelace formula again:Area = |(0*(0.5 - 0.5) + 0*(0.5 - 0) + 1*(0 - 0.5)) / 2|= |(0 + 0 + 1*(-0.5)) / 2|= |(-0.5) / 2|= | -0.25 | = 0.25So, the area of triangle AMN is 0.25, and the area of the square is 1, so the ratio is 0.25/1 = 0.25, which is 1/4.Hmm, so regardless of the side length, the ratio is 1/4. But I thought the answer was 1/8. Maybe I'm misunderstanding the problem.Wait, maybe I misassigned the points. Let me check the problem again: "In square ABCD, point M is the midpoint of side AD and point N is the midpoint of side BC." So, yes, M is on AD and N is on BC.Alternatively, maybe I should use vectors or another method to calculate the area.Another approach is to observe that triangle AMN is a right triangle. Let me see: from A(0,0) to M(0,1), that's vertical, and from M(0,1) to N(2,1), that's horizontal, and from N(2,1) back to A(0,0), which is a diagonal. Wait, no, that's not a right triangle because the sides AM and MN are perpendicular, but the triangle isn't right-angled at M because the third side is diagonal.Wait, actually, if I plot the points, A is at (0,0), M is at (0,1), and N is at (2,1). So, the triangle has a vertical side from A to M, a horizontal side from M to N, and a diagonal side from N back to A. So, actually, the triangle is a right triangle with legs of length 1 and 2, but that doesn't make sense because the legs would be along the axes, but the triangle is not in the corner.Wait, no, the legs are from A to M (length 1) and from M to N (length 2), but the angle at M is a right angle. So, the area would be (1*2)/2 = 1, which matches the shoelace formula result. So, the area is indeed 1 when the square has side length 2, making the ratio 1/4.But I'm confused because I thought the answer was 1/8. Maybe I need to reconsider the coordinates.Alternatively, perhaps I should use vectors or base and height.Let me try using base and height. The base can be AM, which is 1 unit (if side length is 2). The height would be the horizontal distance from N to the line AM. Since AM is vertical along the y-axis, the horizontal distance from N(2,1) to AM is 2 units. So, area = (base * height)/2 = (1 * 2)/2 = 1, which again gives the same result.So, the area is 1, square area is 4, ratio is 1/4.Wait, but maybe the problem is that I'm taking the side length as 2, but if I take it as 1, the ratio is 1/4 as well. So, regardless, the ratio is 1/4.But I'm still unsure because I thought the answer was 1/8. Maybe I need to visualize it differently.Alternatively, perhaps the triangle is not a right triangle. Let me plot the points again: A(0,0), M(0,1), N(2,1). So, connecting A to M is vertical, M to N is horizontal, and N to A is a diagonal. So, the triangle is a right triangle with legs 1 and 2, area 1, as before.Wait, but if I consider the triangle AMN, maybe it's not the triangle I'm thinking of. Maybe it's a different triangle. Let me make sure.Point A is at (0,0), M is midpoint of AD at (0,1), and N is midpoint of BC at (2,1). So, triangle AMN is formed by connecting these three points. Yes, that's correct.Alternatively, maybe the problem is referring to a different configuration. Let me check the problem statement again: "In square ABCD, point M is the midpoint of side AD and point N is the midpoint of side BC. What is the ratio of the area of triangle AMN to the area of square ABCD?"So, yes, that's correct. So, with side length 2, area of triangle is 1, area of square is 4, ratio 1/4.But I'm still confused because I thought the answer was 1/8. Maybe I made a mistake in the initial assumption.Wait, perhaps I should consider the square with side length 1. Then, the area of the square is 1, and the area of the triangle would be 1/4, making the ratio 1/4. So, regardless of the side length, the ratio is 1/4.But I'm still unsure because I thought the answer was 1/8. Maybe I need to approach it differently.Alternatively, perhaps I should use coordinate geometry with variables. Let the side length be 's'. Then, coordinates are A(0,0), B(s,0), C(s,s), D(0,s). Midpoints M(0, s/2) and N(s, s/2). So, triangle AMN has vertices at (0,0), (0, s/2), and (s, s/2).Using the shoelace formula:Area = |(0*(s/2 - s/2) + 0*(s/2 - 0) + s*(0 - s/2)) / 2|= |(0 + 0 + s*(-s/2)) / 2|= |(-s²/2) / 2|= | -s²/4 | = s²/4So, area of triangle is s²/4, area of square is s², ratio is (s²/4)/s² = 1/4.So, the ratio is 1/4.Wait, so why did I initially think it was 1/8? Maybe I confused it with another problem. But according to the calculations, it's 1/4.Alternatively, maybe I should use vectors or another method to confirm.Let me try using vectors. The vectors from A to M is (0, s/2) and from A to N is (s, s/2). The area of the triangle is half the magnitude of the cross product of these vectors.So, vector AM = (0, s/2), vector AN = (s, s/2). Cross product is (0)(s/2) - (s/2)(s) = 0 - s²/2 = -s²/2. The magnitude is s²/2, so area is half that, which is s²/4. So, same result.Therefore, the ratio is 1/4.Wait, so I think I was correct initially, the ratio is 1/4. Maybe I was confusing it with another problem where the ratio was 1/8. So, in this case, the answer is 1/4.But wait, let me think again. If I draw the square and the triangle, maybe I'm missing something. From A(0,0) to M(0,1) to N(2,1) back to A. So, the triangle is a right triangle with legs 1 and 2, area 1. Square area 4, ratio 1/4.Yes, that seems correct. So, I think the answer is 1/4.