💡Okay, so I have this problem about an equilateral triangle ABC with side length 1. Points D and E are on sides AB and AC respectively, and when the triangle is folded along DE, vertex A falls exactly on side BC. I need to find the minimum length of AD.Hmm, let me try to visualize this. Since ABC is equilateral, all sides are 1, and all angles are 60 degrees. Points D and E are somewhere on AB and AC. When I fold along DE, A lands on BC. So, DE must be some kind of crease that reflects A onto BC.I think reflection is key here. When you fold a point over a line, the line is the perpendicular bisector of the segment joining the original point and its image. So, DE must be the perpendicular bisector of the segment joining A and its image after folding, which is a point on BC.Let me denote the image of A after folding as G. So, G lies on BC, and DE is the perpendicular bisector of AG. That means that D and E are equidistant from A and G.Since ABC is equilateral, BC is the base, and G is somewhere along BC. Let me assign coordinates to make this more concrete. Maybe coordinate geometry can help.Let's place the triangle ABC with point A at (0, h), and B at (-0.5, 0), C at (0.5, 0). Since it's equilateral with side length 1, the height h can be calculated as h = sqrt(1^2 - (0.5)^2) = sqrt(3)/2.So, coordinates:- A: (0, sqrt(3)/2)- B: (-0.5, 0)- C: (0.5, 0)Now, points D and E are on AB and AC. Let me parameterize D and E.Let me let AD = x, so since AB has length 1, then DB = 1 - x. Similarly, let AE = y, so EC = 1 - y.But since D is on AB and E is on AC, and DE is the fold line, which reflects A to G on BC.So, DE is the perpendicular bisector of AG. Therefore, the midpoint of AG lies on DE, and DE is perpendicular to AG.Let me denote G as a point on BC. Since BC is from (-0.5, 0) to (0.5, 0), let me parameterize G as (g, 0), where g is between -0.5 and 0.5.So, point G is (g, 0). Then, the midpoint of AG is ((0 + g)/2, (sqrt(3)/2 + 0)/2) = (g/2, sqrt(3)/4).Since DE is the perpendicular bisector, it must pass through this midpoint. Also, DE must be perpendicular to AG.The slope of AG is (0 - sqrt(3)/2)/(g - 0) = (-sqrt(3)/2)/g. Therefore, the slope of DE, being perpendicular, is the negative reciprocal, which is (2g)/sqrt(3).So, DE has slope (2g)/sqrt(3) and passes through (g/2, sqrt(3)/4). So, the equation of DE is:y - sqrt(3)/4 = (2g)/sqrt(3) (x - g/2)Now, points D and E lie on AB and AC respectively. Let me find the coordinates of D and E.First, equation of AB: from A(0, sqrt(3)/2) to B(-0.5, 0). The slope is (0 - sqrt(3)/2)/(-0.5 - 0) = (-sqrt(3)/2)/(-0.5) = sqrt(3). So, equation is y = sqrt(3)x + sqrt(3)/2.Wait, let me check that. When x = 0, y = sqrt(3)/2, which is correct. When x = -0.5, y = sqrt(3)*(-0.5) + sqrt(3)/2 = -sqrt(3)/2 + sqrt(3)/2 = 0, which is correct.Similarly, equation of AC: from A(0, sqrt(3)/2) to C(0.5, 0). Slope is (0 - sqrt(3)/2)/(0.5 - 0) = (-sqrt(3)/2)/0.5 = -sqrt(3). So, equation is y = -sqrt(3)x + sqrt(3)/2.Now, point D is on AB, so its coordinates can be written as (d_x, sqrt(3)d_x + sqrt(3)/2). Similarly, point E is on AC, so its coordinates are (e_x, -sqrt(3)e_x + sqrt(3)/2).But since D and E lie on DE, their coordinates must satisfy the equation of DE.So, let's plug D into DE's equation:sqrt(3)d_x + sqrt(3)/2 - sqrt(3)/4 = (2g)/sqrt(3) (d_x - g/2)Simplify left side: sqrt(3)d_x + sqrt(3)/4Right side: (2g)/sqrt(3) d_x - (2g)/sqrt(3) * g/2 = (2g)/sqrt(3) d_x - (g^2)/sqrt(3)So, equation becomes:sqrt(3)d_x + sqrt(3)/4 = (2g)/sqrt(3) d_x - (g^2)/sqrt(3)Multiply both sides by sqrt(3) to eliminate denominators:3d_x + 3/4 = 2g d_x - g^2Bring all terms to left:3d_x + 3/4 - 2g d_x + g^2 = 0Factor d_x:(3 - 2g)d_x + (3/4 + g^2) = 0Solve for d_x:d_x = -(3/4 + g^2)/(3 - 2g)Similarly, plug point E into DE's equation:-sqrt(3)e_x + sqrt(3)/2 - sqrt(3)/4 = (2g)/sqrt(3) (e_x - g/2)Simplify left side: -sqrt(3)e_x + sqrt(3)/4Right side: (2g)/sqrt(3) e_x - (g^2)/sqrt(3)Multiply both sides by sqrt(3):-3e_x + 3/4 = 2g e_x - g^2Bring all terms to left:-3e_x + 3/4 - 2g e_x + g^2 = 0Factor e_x:(-3 - 2g)e_x + (3/4 + g^2) = 0Solve for e_x:e_x = -(3/4 + g^2)/(-3 - 2g) = (3/4 + g^2)/(3 + 2g)So, now we have expressions for d_x and e_x in terms of g.But we also know that AD = x, which is the distance from A to D. Since A is at (0, sqrt(3)/2) and D is at (d_x, sqrt(3)d_x + sqrt(3)/2), the distance AD is:AD = sqrt[(d_x - 0)^2 + (sqrt(3)d_x + sqrt(3)/2 - sqrt(3)/2)^2]= sqrt[d_x^2 + (sqrt(3)d_x)^2]= sqrt[d_x^2 + 3d_x^2]= sqrt[4d_x^2]= 2|d_x|Since d_x is negative (because D is on AB, which goes from (0, sqrt(3)/2) to (-0.5, 0)), so d_x is negative, so AD = -2d_x.Similarly, AE = y = distance from A to E. Since E is on AC, which goes from (0, sqrt(3)/2) to (0.5, 0), e_x is positive, so AE = 2e_x.But since ABC is equilateral, and DE is the fold line, I think AD and AE might be related in some way. Maybe AD = AE? Wait, not necessarily, because the fold is along DE, which could be asymmetric.But perhaps there's a relationship between d_x and e_x.Wait, from the expressions above, d_x = -(3/4 + g^2)/(3 - 2g) and e_x = (3/4 + g^2)/(3 + 2g). So, if I let k = 3/4 + g^2, then d_x = -k/(3 - 2g) and e_x = k/(3 + 2g). So, d_x and e_x are related through g.Also, since D and E are points on AB and AC, their coordinates must satisfy the triangle's boundaries. So, d_x must be between -0.5 and 0, and e_x must be between 0 and 0.5.Moreover, since DE is a line segment inside the triangle, the coordinates of D and E must be such that DE doesn't go outside the triangle.But perhaps instead of working with coordinates, there's a more geometric approach.Since DE is the perpendicular bisector of AG, and G is on BC, maybe we can use some properties of reflections and triangles.Alternatively, maybe using the Law of Sines or Cosines in some triangles.Let me consider triangle ADG. Since DE is the perpendicular bisector, AD = GD and AE = GE. Wait, no, that's not necessarily true. Wait, DE is the perpendicular bisector of AG, so any point on DE is equidistant from A and G. So, D and E are equidistant from A and G.Therefore, AD = GD and AE = GE.So, in triangle ADG, AD = GD, so it's an isosceles triangle with AD = GD. Similarly, in triangle AEG, AE = GE.So, maybe I can use the Law of Cosines in triangle ADG.Wait, but I don't know where G is on BC. Maybe I can express AG in terms of g.Since G is (g, 0), AG is the distance from A(0, sqrt(3)/2) to G(g, 0):AG = sqrt[(g - 0)^2 + (0 - sqrt(3)/2)^2] = sqrt[g^2 + 3/4]Since AD = GD, and AD is x, then GD = x. So, in triangle ADG, sides AD = GD = x, and AG = sqrt(g^2 + 3/4). So, by the Law of Cosines:AG^2 = AD^2 + GD^2 - 2 AD * GD * cos(theta)Where theta is the angle at D. But since AD = GD, this simplifies to:AG^2 = 2x^2 - 2x^2 cos(theta)But I don't know theta. Alternatively, maybe using coordinates is better.Wait, since DE is the perpendicular bisector of AG, the midpoint M of AG is (g/2, sqrt(3)/4), and DE has slope (2g)/sqrt(3). So, DE is the line we derived earlier.But DE also connects points D and E on AB and AC. So, perhaps we can find the relationship between x and g.From earlier, we have:AD = -2d_x, and d_x = -(3/4 + g^2)/(3 - 2g). So,AD = -2 * [ -(3/4 + g^2)/(3 - 2g) ] = 2(3/4 + g^2)/(3 - 2g)Similarly, AE = 2e_x = 2 * (3/4 + g^2)/(3 + 2g)So, AD and AE are expressed in terms of g.But since ABC is equilateral, perhaps there's a symmetry or another condition that relates AD and AE.Alternatively, maybe we can express g in terms of x, and then find the minimum x.Wait, let's express g in terms of x.From AD = 2(3/4 + g^2)/(3 - 2g) = xSo,x = 2(3/4 + g^2)/(3 - 2g)Multiply both sides by (3 - 2g):x(3 - 2g) = 2(3/4 + g^2)Expand left side:3x - 2xg = 2*(3/4) + 2g^2Simplify:3x - 2xg = 3/2 + 2g^2Bring all terms to one side:2g^2 + 2xg + 3/2 - 3x = 0Multiply through by 2 to eliminate fractions:4g^2 + 4xg + 3 - 6x = 0So, quadratic in g:4g^2 + 4xg + (3 - 6x) = 0For real solutions, discriminant must be non-negative.Discriminant D = (4x)^2 - 4*4*(3 - 6x) = 16x^2 - 16*(3 - 6x) = 16x^2 - 48 + 96xSo,16x^2 + 96x - 48 ≥ 0Divide by 16:x^2 + 6x - 3 ≥ 0Solve x^2 + 6x - 3 = 0:x = [-6 ± sqrt(36 + 12)]/2 = [-6 ± sqrt(48)]/2 = [-6 ± 4sqrt(3)]/2 = -3 ± 2sqrt(3)Since x is a length between 0 and 1, we discard the negative solution:x = -3 + 2sqrt(3) ≈ -3 + 3.464 = 0.464So, the minimum x is -3 + 2sqrt(3), which is approximately 0.464.Wait, but is this the minimum? Because the quadratic in g must have real solutions, so x must satisfy x ≥ -3 + 2sqrt(3). Since x is positive, the minimum x is -3 + 2sqrt(3).So, the minimum length of AD is 2sqrt(3) - 3.Wait, let me check: 2sqrt(3) ≈ 3.464, so 2sqrt(3) - 3 ≈ 0.464, which matches the earlier approximation.So, yes, the minimum AD is 2sqrt(3) - 3.**Final Answer**The minimum length of ( AD ) is (boxed{2sqrt{3} - 3}).

💡Alright, so I've got this problem here about Bob and Johnny who have a leaf raking business. They charge a certain amount for each bag of leaves they rake. On Monday, they raked 5 bags, on Tuesday, 3 bags, and on Wednesday, 9 bags. Altogether, they made 68 over these three days. The question is asking how much they charge for each bag of leaves.Okay, let's break this down. First, I need to figure out the total number of bags they raked over the three days. So, on Monday, it's 5 bags, Tuesday is 3 bags, and Wednesday is 9 bags. If I add those up, that should give me the total number of bags.So, 5 bags on Monday plus 3 bags on Tuesday is 8 bags. Then, adding the 9 bags from Wednesday, that makes a total of 17 bags. Okay, so they raked 17 bags in total over the three days.Now, they made 68 in total from all these bags. To find out how much they charge per bag, I think I need to divide the total amount of money they made by the total number of bags they raked. That should give me the price per bag.So, if I take 68 and divide it by 17 bags, that should give me the charge per bag. Let me do that calculation: 68 divided by 17 equals 4. So, that means they charge 4 per bag of leaves.Wait, let me double-check to make sure I didn't make a mistake. If they charge 4 per bag, then on Monday, 5 bags would earn them 5 times 4, which is 20. On Tuesday, 3 bags would be 3 times 4, which is 12. On Wednesday, 9 bags would be 9 times 4, which is 36. Adding those up: 20 plus 12 is 32, and 32 plus 36 is 68. That matches the total amount they made, so it seems correct.I think that makes sense. They charge 4 for each bag of leaves they rake. It's a straightforward division problem once you know the total number of bags and the total amount earned.

💡Okay, so I have this problem here with four different equations labeled A, B, C, and D. I need to figure out which one is correct. Let me take it step by step.Starting with **Option A**: The equation is -5(x - y) = -5x + 5y. Hmm, okay, so I need to distribute the -5 across both terms inside the parentheses. If I multiply -5 by x, I get -5x, and if I multiply -5 by -y, that should be +5y. So, putting it together, it becomes -5x + 5y. That matches the right side of the equation, so A seems correct.Moving on to **Option B**: The equation is -2(-a + c) = -2a - 2c. Let me distribute the -2 here. Multiplying -2 by -a gives me +2a, and multiplying -2 by c gives me -2c. So, combining those, it should be 2a - 2c. But the right side of the equation is -2a - 2c, which is different. So, B is incorrect.Next is **Option C**: The equation is 3 - (x + y + z) = -x + y - z. Okay, so I need to distribute the negative sign inside the parentheses. That would make it 3 - x - y - z. But the right side is -x + y - z, which is not the same as 3 - x - y - z. So, C is also incorrect.Lastly, **Option D**: The equation is 3(a + 2b) = 3a + 2b. Distributing the 3, I get 3a + 6b. But the right side is 3a + 2b, which is different. So, D is incorrect as well.After checking all four options, only **Option A** is correct.

💡Alright, so I have this problem about three points on a parabola, and I need to figure out the relationship between their y-values. Let me try to break this down step by step.First, the points given are A(1/2, y₁), B(0, y₂), and C(-1, y₃). They all lie on the parabola defined by the equation y = ax² + bx + c. There's also this condition: 0 < 2a < b. Hmm, okay, so a is positive because 2a is greater than 0, and b is greater than 2a. That might be important later.Since all three points are on the parabola, I can plug their x-coordinates into the equation to find expressions for y₁, y₂, and y₃.Let's start with point A: x = 1/2. Plugging that into the equation:y₁ = a*(1/2)² + b*(1/2) + cy₁ = a*(1/4) + b*(1/2) + cy₁ = (a/4) + (b/2) + cOkay, that's y₁. Now, point B is at x = 0, so:y₂ = a*(0)² + b*(0) + cy₂ = 0 + 0 + cy₂ = cThat's straightforward. Now, point C is at x = -1:y₃ = a*(-1)² + b*(-1) + cy₃ = a*1 + (-b) + cy₃ = a - b + cAlright, so now I have expressions for y₁, y₂, and y₃ in terms of a, b, and c. The question is about the relationship between y₁, y₂, and y₃. So, I need to compare these three expressions.Let me write them down again:y₁ = (a/4) + (b/2) + cy₂ = cy₃ = a - b + cHmm. So, y₂ is just c, which is the constant term. y₁ has some positive terms added to c, and y₃ has a - b term added to c.Given that a is positive (since 2a > 0) and b is greater than 2a, which is also positive. So, b is positive as well.Let me see if I can express y₁ and y₃ in terms of y₂.Starting with y₁:y₁ = (a/4) + (b/2) + cBut y₂ = c, so:y₁ = (a/4) + (b/2) + y₂Similarly, for y₃:y₃ = a - b + cAgain, since y₂ = c:y₃ = a - b + y₂So, now I have:y₁ = (a/4) + (b/2) + y₂y₃ = a - b + y₂Now, I need to compare y₁, y₂, and y₃. Let's see.First, let's look at y₁ compared to y₂.y₁ = y₂ + (a/4) + (b/2)Since both a and b are positive, (a/4) and (b/2) are positive. Therefore, y₁ is greater than y₂.So, y₁ > y₂.Now, let's compare y₃ to y₂.y₃ = y₂ + a - bGiven that 0 < 2a < b, which implies that b > 2a.So, a - b = a - (something greater than 2a) = negative number.Therefore, y₃ = y₂ + (negative number), which means y₃ < y₂.So, y₃ < y₂.Putting it all together:y₁ > y₂ > y₃So, the relationship is y₁ is greater than y₂, which is greater than y₃.Wait, but let me double-check to make sure I didn't make any mistakes.Starting with y₁:y₁ = (a/4) + (b/2) + cSince a and b are positive, y₁ is definitely greater than c, which is y₂.For y₃:y₃ = a - b + cGiven that b > 2a, so a - b = a - (something greater than 2a) = negative.Therefore, y₃ = c + negative number, so y₃ < c = y₂.Yes, that seems correct.Just to be thorough, let's pick some numbers that satisfy 0 < 2a < b and see if the relationship holds.Let's choose a = 1 (so 2a = 2), and b = 3 (which is greater than 2a = 2). Let c be any constant, say c = 0 for simplicity.So, y₁ = (1/4) + (3/2) + 0 = 0.25 + 1.5 = 1.75y₂ = 0y₃ = 1 - 3 + 0 = -2So, y₁ = 1.75, y₂ = 0, y₃ = -2Indeed, 1.75 > 0 > -2, so y₁ > y₂ > y₃Let's try another set of numbers. Let a = 0.5, so 2a = 1, and b = 2 (which is greater than 1). Let c = 5.y₁ = (0.5/4) + (2/2) + 5 = 0.125 + 1 + 5 = 6.125y₂ = 5y₃ = 0.5 - 2 + 5 = 3.5So, y₁ = 6.125, y₂ = 5, y₃ = 3.5Again, 6.125 > 5 > 3.5, so y₁ > y₂ > y₃Wait, but in this case, y₃ is still less than y₂, but it's positive. Hmm, but in the first example, y₃ was negative. So, depending on the value of c, y₃ could be positive or negative, but it's always less than y₂.Wait, in the second example, y₃ = 3.5, which is less than y₂ = 5, but still positive. So, regardless of c, y₃ is less than y₂ because y₃ = y₂ + (a - b), and since a - b is negative, y₃ is less than y₂.Yes, that makes sense.Just to confirm, let's try with a different c. Let a = 1, b = 3, c = 10.y₁ = (1/4) + (3/2) + 10 = 0.25 + 1.5 + 10 = 11.75y₂ = 10y₃ = 1 - 3 + 10 = 8So, y₁ = 11.75, y₂ = 10, y₃ = 8Again, y₁ > y₂ > y₃Alright, I think that's consistent. So, the relationship is y₁ > y₂ > y₃.I don't see any flaws in this reasoning. I've checked it with different values, and it holds true. So, I'm confident that the relationship is y₁ > y₂ > y₃.

💡First, I need to determine the number of horses on the farm based on the given ratio of sheep to horses, which is 4:7. Since there are 32 sheep, I can set up a proportion to find the number of horses.Next, I'll solve the proportion to find the exact number of horses. Once I have the number of horses, I can calculate the amount of horse food each horse needs per day by dividing the total horse food required by the number of horses.Finally, I'll present the calculated amount of horse food per horse in a clear and concise manner.

💡Okay, so I have this problem about a company restructuring into a joint-stock company. Initially, everyone was supposed to invest an equal amount. But then, some people decided not to invest, which caused the remaining people to have to contribute more. It happened twice: first, 10 people opted out, and then another 15 people withdrew. Each time, the remaining investors had to pay more. I need to find out how many people were initially prepared to invest and how much each person was supposed to contribute before anyone opted out.Alright, let me try to break this down. Let's denote the initial number of people as ( x ) and the initial amount each person was supposed to contribute as ( y ). So, the total amount of money the company was expecting to raise is ( x times y ).First, 10 people opted out. That means the number of people left is ( x - 10 ). Because of this, each remaining person had to contribute an additional 10,000 yuan. So, the new contribution per person becomes ( y + 10,000 ). But wait, the problem mentions that the increase is 10,000 yuan each time, but in the second instance, it's 20,000 yuan. Hmm, maybe I should think in terms of increments.Wait, actually, the problem says that after 10 people opted out, the remaining had to contribute an additional 10,000 yuan each. Then, when another 15 people withdrew, the remaining had to contribute an additional 20,000 yuan each. So, the total increase after the first withdrawal is 10,000, and after the second withdrawal, it's another 20,000, making it a total increase of 30,000? Or is it that each withdrawal caused an increase, but the second increase is 20,000 on top of the previous 10,000?Let me read the problem again: "when the official count was taken, 10 people opted out, resulting in the remaining individuals having to contribute an additional 10,000 yuan each. By the time the actual payment was due, another 15 people decided to withdraw, leading to a further increase of 20,000 yuan per remaining person."So, the first withdrawal caused an increase of 10,000 yuan per person, and the second withdrawal caused an additional increase of 20,000 yuan per person. So, the total increase after both withdrawals is 30,000 yuan per person? Or is it that each withdrawal caused a separate increase, so the first increase is 10,000, and the second is 20,000, making the total increase 30,000? Hmm, that might be.But let's think in terms of equations. The total amount of money the company is raising should remain the same, right? Because the company needs a fixed amount. So, initially, the total amount is ( x times y ).After the first withdrawal, the number of people is ( x - 10 ), and each contributes ( y + 10,000 ). So, the total amount is ( (x - 10)(y + 10,000) ). This should be equal to the original total amount ( x times y ).Similarly, after the second withdrawal, the number of people is ( x - 10 - 15 = x - 25 ), and each contributes ( y + 10,000 + 20,000 = y + 30,000 ). So, the total amount is ( (x - 25)(y + 30,000) ), which should also equal the original total amount ( x times y ).So, we have two equations:1. ( (x - 10)(y + 10,000) = x y )2. ( (x - 25)(y + 30,000) = x y )These are two equations with two variables, so we can solve for ( x ) and ( y ).Let me write them out:First equation:( (x - 10)(y + 10,000) = x y )Second equation:( (x - 25)(y + 30,000) = x y )Let me expand the first equation:( x y + 10,000 x - 10 y - 100,000 = x y )Subtract ( x y ) from both sides:( 10,000 x - 10 y - 100,000 = 0 )Simplify:( 10,000 x - 10 y = 100,000 )Divide both sides by 10:( 1,000 x - y = 10,000 )So, equation (1) simplifies to:( 1,000 x - y = 10,000 ) --- (1)Now, let's expand the second equation:( (x - 25)(y + 30,000) = x y )Multiply out:( x y + 30,000 x - 25 y - 750,000 = x y )Subtract ( x y ) from both sides:( 30,000 x - 25 y - 750,000 = 0 )Simplify:( 30,000 x - 25 y = 750,000 )Divide both sides by 25:( 1,200 x - y = 30,000 ) --- (2)Now, we have two equations:1. ( 1,000 x - y = 10,000 ) --- (1)2. ( 1,200 x - y = 30,000 ) --- (2)Let me subtract equation (1) from equation (2):( (1,200 x - y) - (1,000 x - y) = 30,000 - 10,000 )Simplify:( 200 x = 20,000 )Divide both sides by 200:( x = 100 )So, the initial number of people is 100.Now, plug ( x = 100 ) into equation (1):( 1,000 times 100 - y = 10,000 )Calculate:( 100,000 - y = 10,000 )Subtract 100,000 from both sides:( - y = -90,000 )Multiply both sides by -1:( y = 90,000 )Wait, but in the problem, the increases are 10,000 and 20,000 yuan. So, if the initial contribution was 90,000 yuan, then after 10 people left, each had to pay 100,000 yuan, and after another 15 left, each had to pay 120,000 yuan. That seems correct because 10,000 and 20,000 increases.But let me check the calculations again.From equation (1):( 1,000 x - y = 10,000 )With ( x = 100 ):( 1,000 * 100 - y = 10,000 )( 100,000 - y = 10,000 )So, ( y = 90,000 ). That seems correct.But let me verify with the second equation:Equation (2):( 1,200 x - y = 30,000 )Plug in ( x = 100 ):( 1,200 * 100 - y = 30,000 )( 120,000 - y = 30,000 )So, ( y = 90,000 ). That matches.So, the initial number of people is 100, and each was supposed to contribute 90,000 yuan.But wait, in the problem, the increases are 10,000 and 20,000. So, 90,000 + 10,000 = 100,000, and then 100,000 + 20,000 = 120,000. Let's check if the total amount remains the same.Original total: 100 * 90,000 = 9,000,000 yuan.After first withdrawal: 90 people * 100,000 = 9,000,000 yuan. Correct.After second withdrawal: 75 people * 120,000 = 9,000,000 yuan. Correct.So, everything checks out.Therefore, the initial number of people was 100, and each was supposed to contribute 90,000 yuan.But wait, in the initial problem statement, the increases are 10,000 and 20,000, but in my equations, I used 10,000 and 30,000. Wait, no, actually, the second increase is 20,000 on top of the previous 10,000, so total increase is 30,000. But in the equations, I correctly accounted for that by adding 10,000 and then 20,000, so the total increase is 30,000. So, the equations are correct.Alternatively, if I had considered the increases separately, I might have set up the equations differently, but I think the way I did it is correct.So, final answer: 100 people initially, each supposed to contribute 90,000 yuan.**Final Answer**The company was initially prepared to have boxed{100} people invest, and each was supposed to contribute boxed{90000} yuan.