💡Alright, so I have this problem here about a 5000-member association that includes both men and women. The problem states that exactly 15% of the men and 25% of the women are homeowners. I need to find the least number of members who are homeowners. Hmm, okay, let's break this down.First, I need to understand what exactly is being asked. We have a total of 5000 members, some are men, and some are women. Out of these, 15% of the men own homes, and 25% of the women own homes. I need to find the minimum number of homeowners possible in this association.So, to minimize the number of homeowners, I should aim to have as few men and women as possible who are homeowners. But wait, the percentages are fixed—15% for men and 25% for women. So, I can't just arbitrarily reduce the number of homeowners; I have to work within these percentages.Let me denote the number of men as M and the number of women as W. So, M + W = 5000. That's our total membership.Now, the number of male homeowners would be 15% of M, which is 0.15M, and the number of female homeowners would be 25% of W, which is 0.25W. Therefore, the total number of homeowners is 0.15M + 0.25W.Since I need to find the minimum number of homeowners, I need to minimize 0.15M + 0.25W, given that M + W = 5000.To minimize this expression, I should try to maximize the number of people in the group with the lower percentage, which is men in this case because 15% is less than 25%. So, if I have more men, the total number of homeowners will be lower, and fewer women will lead to fewer homeowners as well.Therefore, to minimize the total number of homeowners, I should maximize M and minimize W. But I need to ensure that both M and W are whole numbers because you can't have a fraction of a person.So, let's express W in terms of M: W = 5000 - M.Substituting this into the total number of homeowners:Total homeowners = 0.15M + 0.25(5000 - M)= 0.15M + 1250 - 0.25M= -0.10M + 1250So, the total number of homeowners is -0.10M + 1250. To minimize this, I need to maximize M because the coefficient of M is negative. Therefore, the larger M is, the smaller the total number of homeowners.But M can't exceed 5000 because the total membership is 5000. So, theoretically, if M = 5000, then W = 0, and total homeowners would be 0.15*5000 + 0.25*0 = 750.However, the problem is that M and W need to be such that 15% of M and 25% of W are whole numbers because you can't have a fraction of a person being a homeowner.So, 0.15M must be an integer, and 0.25W must also be an integer.Let's express these conditions mathematically:0.15M = 3M/20 must be an integer. Therefore, M must be a multiple of 20.Similarly, 0.25W = W/4 must be an integer. Therefore, W must be a multiple of 4.Given that M + W = 5000, and M is a multiple of 20, and W is a multiple of 4, we need to find the maximum possible M that is a multiple of 20, and the corresponding W = 5000 - M must be a multiple of 4.So, let's find the largest M that is a multiple of 20 such that W = 5000 - M is a multiple of 4.Let's denote M = 20k, where k is an integer.Then W = 5000 - 20k.We need W to be a multiple of 4, so 5000 - 20k must be divisible by 4.Since 5000 is divisible by 4 (5000 / 4 = 1250), and 20k is also divisible by 4 (20k / 4 = 5k), their difference will also be divisible by 4. Therefore, W will always be a multiple of 4 as long as M is a multiple of 20.Therefore, the maximum M is 5000, but we need to check if W = 0 is acceptable. If W = 0, then the number of female homeowners is 0, which is fine.But wait, if M = 5000, then W = 0, and total homeowners = 0.15*5000 + 0.25*0 = 750.But is this the minimum? Let's see if we can have a slightly smaller M that still satisfies the conditions and results in a smaller total number of homeowners.Wait, but since the total number of homeowners is -0.10M + 1250, as M increases, the total decreases. Therefore, the maximum M gives the minimum total homeowners.But let's double-check. Suppose M = 5000, W = 0, total homeowners = 750.If M = 4980, which is still a multiple of 20, then W = 20.Total homeowners = 0.15*4980 + 0.25*20 = 747 + 5 = 752.Wait, that's actually more than 750. Hmm, that's contradictory.Wait, no, because when M decreases, the total number of homeowners increases. So, the maximum M gives the minimum total homeowners.But in this case, when M = 5000, W = 0, total homeowners = 750.But if M = 4980, W = 20, total homeowners = 747 + 5 = 752, which is higher than 750.So, indeed, the maximum M gives the minimum total homeowners.But wait, is M = 5000 acceptable? Because W = 0, which means there are no women in the association. Is that allowed? The problem says it's an association consisting of men and women, so I think W must be at least 1.Therefore, M cannot be 5000; it must be less than 5000, and W must be at least 1.So, the maximum M is 4996, which is a multiple of 20? Wait, 4996 divided by 20 is 249.8, which is not an integer. So, the largest multiple of 20 less than 5000 is 4980.Wait, 5000 divided by 20 is 250, so 250*20 = 5000. Therefore, the largest multiple of 20 less than 5000 is 4980.So, M = 4980, W = 20.Total homeowners = 0.15*4980 + 0.25*20 = 747 + 5 = 752.But earlier, when M = 5000, W = 0, total homeowners = 750, which is less than 752.But since W must be at least 1, we can't have M = 5000. Therefore, the next best is M = 4980, W = 20, total homeowners = 752.Wait, but is there a way to have W = 4, which is the smallest multiple of 4 greater than 0?Let's see. If W = 4, then M = 5000 - 4 = 4996.Is 4996 a multiple of 20? 4996 divided by 20 is 249.8, which is not an integer. Therefore, M = 4996 is not a multiple of 20, so 0.15*4996 = 749.4, which is not an integer. Therefore, W cannot be 4 because M would not be a multiple of 20.So, the next possible W is 8.If W = 8, then M = 5000 - 8 = 4992.Is 4992 a multiple of 20? 4992 / 20 = 249.6, which is not an integer. So, M = 4992 is not a multiple of 20.Next, W = 12, M = 5000 - 12 = 4988.4988 / 20 = 249.4, not an integer.W = 16, M = 4984.4984 / 20 = 249.2, not an integer.W = 20, M = 4980.4980 / 20 = 249, which is an integer.Therefore, the smallest possible W that makes M a multiple of 20 is W = 20, M = 4980.Therefore, total homeowners = 0.15*4980 + 0.25*20 = 747 + 5 = 752.But wait, earlier, when M = 5000, W = 0, total homeowners = 750, which is less than 752. But since W must be at least 1, we can't have W = 0.Therefore, the minimum number of homeowners is 752.But let's double-check if there's a way to have W = 4 and M = 4996, but M must be a multiple of 20. Since 4996 is not a multiple of 20, we can't have W = 4.Similarly, W = 8, M = 4992, which is not a multiple of 20.W = 12, M = 4988, not a multiple of 20.W = 16, M = 4984, not a multiple of 20.W = 20, M = 4980, which is a multiple of 20.Therefore, the minimum number of homeowners is 752.But wait, is there a way to have W = 25, M = 4975, but M must be a multiple of 20. 4975 / 20 = 248.75, not an integer.Alternatively, W = 24, M = 4976.4976 / 20 = 248.8, not an integer.W = 28, M = 4972.4972 / 20 = 248.6, not an integer.W = 32, M = 4968.4968 / 20 = 248.4, not an integer.W = 36, M = 4964.4964 / 20 = 248.2, not an integer.W = 40, M = 4960.4960 / 20 = 248, which is an integer.So, W = 40, M = 4960.Total homeowners = 0.15*4960 + 0.25*40 = 744 + 10 = 754.Wait, that's more than 752.So, 752 is still the minimum.Wait, but if I go further, W = 20, M = 4980, total homeowners = 752.If I increase W to 40, M decreases to 4960, but total homeowners increases to 754.Therefore, 752 is indeed the minimum.But let's see if there's a way to have W = 20, M = 4980, total homeowners = 752.Is there a way to have W = 20, M = 4980, and both 0.15M and 0.25W are integers.Yes, because 0.15*4980 = 747, which is an integer, and 0.25*20 = 5, which is also an integer.Therefore, the minimum number of homeowners is 752.But wait, earlier, I thought that if M = 5000, W = 0, total homeowners = 750, which is less than 752. But since W must be at least 1, we can't have W = 0.Therefore, the minimum number of homeowners is 752.But let's see if there's a way to have W = 4, M = 4996, but M must be a multiple of 20. Since 4996 is not a multiple of 20, we can't have W = 4.Similarly, W = 8, M = 4992, not a multiple of 20.W = 12, M = 4988, not a multiple of 20.W = 16, M = 4984, not a multiple of 20.W = 20, M = 4980, which is a multiple of 20.Therefore, the minimum number of homeowners is 752.But wait, let's try to see if there's a way to have W = 25, M = 4975, but M must be a multiple of 20. 4975 / 20 = 248.75, not an integer.Alternatively, W = 24, M = 4976.4976 / 20 = 248.8, not an integer.W = 28, M = 4972.4972 / 20 = 248.6, not an integer.W = 32, M = 4968.4968 / 20 = 248.4, not an integer.W = 36, M = 4964.4964 / 20 = 248.2, not an integer.W = 40, M = 4960.4960 / 20 = 248, which is an integer.So, W = 40, M = 4960.Total homeowners = 0.15*4960 + 0.25*40 = 744 + 10 = 754.Wait, that's more than 752.So, 752 is still the minimum.Therefore, the least number of members who are homeowners is 752.

💡Okay, so I have this problem where I need to figure out which cards can be obtained from the card (5,19). I'm not entirely sure what the rules are for obtaining other cards from this one, but I think it has something to do with the numbers 5 and 19. Maybe it's about arithmetic operations or some kind of transformation.First, I should probably understand what the card (5,19) represents. It could be a pair of numbers, maybe like coordinates or something else. If I think of it as coordinates, maybe I can move in some way from (5,19) to other points. But I'm not sure. Maybe it's about the difference between the two numbers? Let me calculate that: 19 minus 5 is 14. Hmm, 14 is a multiple of 7. Maybe that's important.If the difference is 14, which is 2 times 7, maybe I can add or subtract multiples of 7 to one of the numbers to get new cards. Let me try that. If I add 7 to 5, I get 12, so one card could be (12,19). If I subtract 7 from 5, I get -2, which doesn't make much sense in this context. Maybe I should only consider positive numbers. Alternatively, I could add or subtract 7 to 19.If I add 7 to 19, I get 26, so another card could be (5,26). If I subtract 7 from 19, I get 12, so that would give me (5,12). Wait, I already had (12,19) and (5,12). Are these the same? No, they're different pairs. So maybe both are valid.But I'm not sure if adding or subtracting 7 is the only way. Maybe there's another rule. Let me think about the original problem statement again. It says "determine which cards can be obtained from the card (5,19), and which cannot." So, it's about transformations from (5,19) to other cards.Perhaps it's similar to some kind of game where you can move a certain number of steps or something. Maybe like in a board game where you can move forward or backward by a specific number. If the difference is 14, which is 2 times 7, maybe I can move in steps of 7. So, from 5, I can go to 12, 19, 26, etc., or from 19, I can go to 12, 5, -2, etc. But negative numbers might not be allowed.Alternatively, maybe it's about modular arithmetic. If the difference is 14, which is 0 modulo 7, then maybe any card where the difference is a multiple of 7 can be obtained. So, if I have a card (a,b), and b - a is a multiple of 7, then it can be obtained from (5,19). Otherwise, it cannot.Let me test this idea. If I have (5,12), the difference is 7, which is a multiple of 7. So, that should be obtainable. Similarly, (5,26) has a difference of 21, which is also a multiple of 7. So, that should be obtainable too. What about (5,19)? The difference is 14, which is 2 times 7, so that's also a multiple of 7. Makes sense.What if I have a card like (5,10)? The difference is 5, which is not a multiple of 7. So, according to my idea, it shouldn't be obtainable from (5,19). Similarly, (5,20) has a difference of 15, which is not a multiple of 7, so that shouldn't be obtainable either.But wait, is this the only rule? What if I have a card where both numbers are changed? For example, (12,19). The difference is 7, which is a multiple of 7. So, that should be obtainable. Similarly, (12,26) has a difference of 14, which is also a multiple of 7. So, that should be obtainable too.What about (12,20)? The difference is 8, which is not a multiple of 7. So, that shouldn't be obtainable. Okay, this seems consistent.But I'm still not entirely sure if this is the correct approach. Maybe there's another way to transform the card. Perhaps I can add or subtract 7 to both numbers? Let's try that. If I add 7 to both 5 and 19, I get (12,26). The difference is still 14, which is a multiple of 7. So, that works. If I subtract 7 from both, I get (-2,12). Negative numbers might not be allowed, so maybe that's not valid.Alternatively, maybe I can only add or subtract 7 to one of the numbers. If I add 7 to 5, I get 12, so (12,19). If I subtract 7 from 19, I get 12, so (5,12). Both of these have differences that are multiples of 7.What if I add 14 to 5? That would give me 19, so (19,19). But that's not a valid card because the two numbers are the same, and I think the first number should be less than the second. So, maybe that's not allowed.Alternatively, if I subtract 14 from 19, I get 5, so (5,5). Again, same issue. So, maybe the only valid transformations are adding or subtracting 7 to one of the numbers, as long as the resulting numbers are positive and the first is less than the second.So, summarizing my thoughts:1. The original card is (5,19), with a difference of 14, which is a multiple of 7.2. To obtain other cards, I can add or subtract multiples of 7 to one of the numbers, ensuring the first number is less than the second and both are positive.3. Therefore, any card (a,b) where b - a is a multiple of 7 can be obtained from (5,19).4. Cards where the difference is not a multiple of 7 cannot be obtained.I think this makes sense. Let me test it with a few examples:- (5,12): Difference is 7, which is a multiple of 7. Obtainable.- (5,26): Difference is 21, which is a multiple of 7. Obtainable.- (12,19): Difference is 7, which is a multiple of 7. Obtainable.- (5,10): Difference is 5, not a multiple of 7. Not obtainable.- (5,20): Difference is 15, not a multiple of 7. Not obtainable.- (12,20): Difference is 8, not a multiple of 7. Not obtainable.This seems consistent with my earlier reasoning. I think I've got it. So, the answer is that any card where the difference between the two numbers is a multiple of 7 can be obtained from (5,19), and those that don't have such a difference cannot be obtained.

💡Okay, so I have this problem here: In the expansion of the binomial ((x - dfrac{a}{x})^6), the constant term is 20. I need to find the value of (a). Hmm, let me think about how to approach this.First, I remember that when expanding a binomial like ((A + B)^n), each term can be found using the binomial theorem, which states that the expansion is the sum from (r = 0) to (n) of (dbinom{n}{r} A^{n - r} B^r). So, in this case, (A) is (x) and (B) is (-dfrac{a}{x}), right?So, applying the binomial theorem, the general term (T_{r+1}) in the expansion of ((x - dfrac{a}{x})^6) would be:[T_{r+1} = dbinom{6}{r} x^{6 - r} left(-dfrac{a}{x}right)^r]Let me simplify this term. First, I can write (left(-dfrac{a}{x}right)^r) as ((-a)^r cdot x^{-r}). So, substituting that back into the term:[T_{r+1} = dbinom{6}{r} x^{6 - r} cdot (-a)^r cdot x^{-r}]Combining the exponents of (x), since (x^{6 - r} cdot x^{-r} = x^{6 - 2r}):[T_{r+1} = (-a)^r cdot dbinom{6}{r} cdot x^{6 - 2r}]Now, the problem is asking for the constant term. A constant term in the expansion is the term where the exponent of (x) is zero because (x^0 = 1), which is a constant. So, I need to find the value of (r) such that the exponent (6 - 2r = 0).Let me solve for (r):[6 - 2r = 0 2r = 6 r = 3]Okay, so when (r = 3), the term is the constant term. Let me write that term out:[T_{4} = (-a)^3 cdot dbinom{6}{3} cdot x^{0}]Simplifying this, since (x^0 = 1), we have:[T_{4} = (-a)^3 cdot dbinom{6}{3}]I know that (dbinom{6}{3}) is the binomial coefficient, which is calculated as:[dbinom{6}{3} = dfrac{6!}{3! cdot (6 - 3)!} = dfrac{720}{6 cdot 6} = dfrac{720}{36} = 20]Wait, hold on, that doesn't seem right. Let me recalculate that:[6! = 720 3! = 6 So, dbinom{6}{3} = dfrac{720}{6 cdot 6} = dfrac{720}{36} = 20]Hmm, actually, that is correct. So, (dbinom{6}{3} = 20). Therefore, the constant term is:[T_{4} = (-a)^3 cdot 20]According to the problem, this constant term is equal to 20. So, I can set up the equation:[(-a)^3 cdot 20 = 20]Let me solve for (a). First, divide both sides of the equation by 20:[(-a)^3 = 1]So, ((-a)^3 = 1). To solve for (a), I can take the cube root of both sides:[-a = sqrt[3]{1}]Since the cube root of 1 is 1, we have:[-a = 1 a = -1]Wait, let me double-check that. If ((-a)^3 = 1), then (-a) must be the real cube root of 1, which is indeed 1. So, (-a = 1) implies (a = -1). That seems correct.Just to make sure I didn't make any mistakes, let me recap:1. I used the binomial theorem to find the general term in the expansion.2. I set the exponent of (x) to zero to find the constant term, which gave me (r = 3).3. I calculated the binomial coefficient (dbinom{6}{3}) correctly as 20.4. I set up the equation with the constant term equal to 20 and solved for (a), resulting in (a = -1).Everything seems to check out. I don't think I made any calculation errors. So, I'm confident that (a = -1) is the correct answer.**Final Answer**The value of (a) is boxed{-1}.

💡First, I need to determine the capacities of the pails used by Timmy, Tommy, and Tina. I know that Tina's pail holds 4 gallons of water. Tommy's pail holds 2 gallons more than Tina's, so Tommy's pail capacity is 4 + 2 = 6 gallons.Next, I need to find out how much water Timmy's pail can hold. The total amount of water used to fill the pool is 66 gallons, and each person makes 3 trips. This means that the combined capacity of their pails for one trip is 66 gallons divided by 3 trips, which equals 22 gallons per trip.Now, I can set up an equation to find Timmy's pail capacity. The sum of the capacities of all three pails for one trip is Timmy's pail capacity plus Tommy's pail capacity plus Tina's pail capacity. So, T + 6 + 4 = 22. Solving for T, I subtract 10 from both sides to get T = 12 gallons.Finally, to find the ratio of Timmy's pail capacity to Tommy's pail capacity, I compare their capacities: 12 gallons to 6 gallons. Simplifying this ratio by dividing both numbers by 6 gives me 2:1.

💡Okay, so I have this problem about a parabola and a line intersecting it. Let me try to understand what it's asking. The parabola is given by the equation ( y^2 = 3x ). I know that the standard form of a parabola that opens to the right is ( y^2 = 4px ), where ( p ) is the distance from the vertex to the focus. So in this case, comparing ( y^2 = 3x ) with ( y^2 = 4px ), I can see that ( 4p = 3 ), which means ( p = frac{3}{4} ). So the focus ( F ) of this parabola should be at ( left( frac{3}{4}, 0 right) ). That makes sense because the vertex is at the origin, and it opens to the right.Now, the problem says there's a line passing through this focus ( F ) with an inclination angle of ( 30^circ ). I need to find the points where this line intersects the parabola, which are points ( A ) and ( B ), and then find the length of the segment ( |AB| ).First, let me figure out the equation of this line. An inclination angle of ( 30^circ ) means the slope ( m ) of the line is ( tan(30^circ) ). I remember that ( tan(30^circ) = frac{sqrt{3}}{3} ). So the slope ( m ) is ( frac{sqrt{3}}{3} ).Since the line passes through the focus ( F ) at ( left( frac{3}{4}, 0 right) ), I can write the equation of the line in point-slope form. The point-slope formula is ( y - y_1 = m(x - x_1) ). Plugging in the values, I get:[y - 0 = frac{sqrt{3}}{3}left( x - frac{3}{4} right)]Simplifying this, the equation of the line is:[y = frac{sqrt{3}}{3}x - frac{sqrt{3}}{4}]Okay, so now I have the equation of the parabola ( y^2 = 3x ) and the equation of the line ( y = frac{sqrt{3}}{3}x - frac{sqrt{3}}{4} ). To find the points of intersection ( A ) and ( B ), I need to solve these two equations simultaneously.Let me substitute the expression for ( y ) from the line equation into the parabola equation. So replacing ( y ) in ( y^2 = 3x ) with ( frac{sqrt{3}}{3}x - frac{sqrt{3}}{4} ), I get:[left( frac{sqrt{3}}{3}x - frac{sqrt{3}}{4} right)^2 = 3x]Now, I need to expand the left side of this equation. Let me compute that step by step.First, square the binomial:[left( frac{sqrt{3}}{3}x - frac{sqrt{3}}{4} right)^2 = left( frac{sqrt{3}}{3}x right)^2 - 2 cdot frac{sqrt{3}}{3}x cdot frac{sqrt{3}}{4} + left( frac{sqrt{3}}{4} right)^2]Calculating each term:1. ( left( frac{sqrt{3}}{3}x right)^2 = frac{3}{9}x^2 = frac{1}{3}x^2 )2. ( -2 cdot frac{sqrt{3}}{3}x cdot frac{sqrt{3}}{4} = -2 cdot frac{3}{12}x = -2 cdot frac{1}{4}x = -frac{1}{2}x )3. ( left( frac{sqrt{3}}{4} right)^2 = frac{3}{16} )Putting it all together:[frac{1}{3}x^2 - frac{1}{2}x + frac{3}{16} = 3x]Now, let's bring all terms to one side to form a quadratic equation:[frac{1}{3}x^2 - frac{1}{2}x + frac{3}{16} - 3x = 0]Combine like terms:[frac{1}{3}x^2 - left( frac{1}{2} + 3 right)x + frac{3}{16} = 0]Convert 3 into halves to combine with ( frac{1}{2} ):[3 = frac{6}{2}, text{ so } frac{1}{2} + frac{6}{2} = frac{7}{2}]So the equation becomes:[frac{1}{3}x^2 - frac{7}{2}x + frac{3}{16} = 0]To make this easier to solve, let's eliminate the fractions by finding a common denominator. The denominators are 3, 2, and 16. The least common multiple of 3, 2, and 16 is 48. So, multiply each term by 48:[48 cdot frac{1}{3}x^2 - 48 cdot frac{7}{2}x + 48 cdot frac{3}{16} = 0]Calculating each term:1. ( 48 cdot frac{1}{3}x^2 = 16x^2 )2. ( 48 cdot frac{7}{2}x = 24 cdot 7x = 168x )3. ( 48 cdot frac{3}{16} = 3 cdot 3 = 9 )So the equation becomes:[16x^2 - 168x + 9 = 0]Now, I have a quadratic equation in the form ( ax^2 + bx + c = 0 ), where ( a = 16 ), ( b = -168 ), and ( c = 9 ). To find the solutions for ( x ), I can use the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Plugging in the values:[x = frac{-(-168) pm sqrt{(-168)^2 - 4 cdot 16 cdot 9}}{2 cdot 16}]Simplify step by step:1. ( -(-168) = 168 )2. ( (-168)^2 = 28224 )3. ( 4 cdot 16 cdot 9 = 576 )4. So, the discriminant is ( 28224 - 576 = 27648 )Now, compute the square root of 27648. Let me see:First, factor 27648 to simplify the square root:27648 ÷ 16 = 17281728 is a known cube: ( 12^3 = 1728 ), but we need square factors. Let's factor 1728:1728 = 144 × 12 = 12 × 12 × 12 = ( 12^3 )But 144 is ( 12^2 ), so 1728 = ( 12^3 = (12^{1.5})^2 ) Hmm, maybe another approach.Alternatively, 27648 = 64 × 432432 = 16 × 27So, 27648 = 64 × 16 × 27 = ( 8^2 × 4^2 × 3^3 )Therefore, ( sqrt{27648} = 8 × 4 × 3^{1.5} = 32 × 3 sqrt{3} = 96sqrt{3} )Wait, let me check:( 8^2 = 64 ), ( 4^2 = 16 ), ( 3^3 = 27 )So, ( 64 × 16 = 1024 ), ( 1024 × 27 = 27648 ). So, ( sqrt{27648} = sqrt{64 × 16 × 27} = 8 × 4 × 3sqrt{3} = 32 × 3sqrt{3} = 96sqrt{3} ). Yes, that's correct.So, the square root of the discriminant is ( 96sqrt{3} ).Now, plug back into the quadratic formula:[x = frac{168 pm 96sqrt{3}}{32}]Simplify numerator and denominator:Divide numerator and denominator by 16:[x = frac{168 ÷ 16 pm 96sqrt{3} ÷ 16}{32 ÷ 16} = frac{10.5 pm 6sqrt{3}}{2}]Wait, 168 ÷ 16 is 10.5, which is ( frac{21}{2} ), and 96 ÷ 16 is 6, so:[x = frac{frac{21}{2} pm 6sqrt{3}}{2} = frac{21}{4} pm 3sqrt{3}]So, the two solutions for ( x ) are:1. ( x = frac{21}{4} + 3sqrt{3} )2. ( x = frac{21}{4} - 3sqrt{3} )Now, I need to find the corresponding ( y ) values for each ( x ) using the equation of the line ( y = frac{sqrt{3}}{3}x - frac{sqrt{3}}{4} ).Let me compute ( y ) for each ( x ):First, for ( x = frac{21}{4} + 3sqrt{3} ):[y = frac{sqrt{3}}{3} left( frac{21}{4} + 3sqrt{3} right) - frac{sqrt{3}}{4}]Let me compute each term:1. ( frac{sqrt{3}}{3} times frac{21}{4} = frac{21sqrt{3}}{12} = frac{7sqrt{3}}{4} )2. ( frac{sqrt{3}}{3} times 3sqrt{3} = frac{3 times 3}{3} = 3 )3. Subtract ( frac{sqrt{3}}{4} ): ( frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} = frac{6sqrt{3}}{4} = frac{3sqrt{3}}{2} )So, combining these:[y = frac{7sqrt{3}}{4} + 3 - frac{sqrt{3}}{4} = 3 + frac{6sqrt{3}}{4} = 3 + frac{3sqrt{3}}{2}]Wait, actually, let me correct that. The computation should be:First, compute ( frac{sqrt{3}}{3} times left( frac{21}{4} + 3sqrt{3} right) ):[frac{sqrt{3}}{3} times frac{21}{4} = frac{21sqrt{3}}{12} = frac{7sqrt{3}}{4}][frac{sqrt{3}}{3} times 3sqrt{3} = frac{3 times 3}{3} = 3]So, adding these together:[frac{7sqrt{3}}{4} + 3]Then subtract ( frac{sqrt{3}}{4} ):[frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} = frac{6sqrt{3}}{4} = frac{3sqrt{3}}{2}]So, total ( y ):[3 + frac{3sqrt{3}}{2}]Wait, no, that's not correct. Because the expression is:[y = frac{sqrt{3}}{3}x - frac{sqrt{3}}{4}]So, substituting ( x ):[y = frac{sqrt{3}}{3} left( frac{21}{4} + 3sqrt{3} right) - frac{sqrt{3}}{4}]Which is:[frac{sqrt{3}}{3} times frac{21}{4} + frac{sqrt{3}}{3} times 3sqrt{3} - frac{sqrt{3}}{4}]Calculating each term:1. ( frac{sqrt{3}}{3} times frac{21}{4} = frac{21sqrt{3}}{12} = frac{7sqrt{3}}{4} )2. ( frac{sqrt{3}}{3} times 3sqrt{3} = frac{3 times 3}{3} = 3 )3. ( - frac{sqrt{3}}{4} )So, combining all terms:[frac{7sqrt{3}}{4} + 3 - frac{sqrt{3}}{4} = 3 + left( frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} right) = 3 + frac{6sqrt{3}}{4} = 3 + frac{3sqrt{3}}{2}]So, ( y = 3 + frac{3sqrt{3}}{2} )Similarly, for ( x = frac{21}{4} - 3sqrt{3} ):[y = frac{sqrt{3}}{3} left( frac{21}{4} - 3sqrt{3} right) - frac{sqrt{3}}{4}]Compute each term:1. ( frac{sqrt{3}}{3} times frac{21}{4} = frac{7sqrt{3}}{4} )2. ( frac{sqrt{3}}{3} times (-3sqrt{3}) = -frac{3 times 3}{3} = -3 )3. Subtract ( frac{sqrt{3}}{4} ): ( frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} = frac{6sqrt{3}}{4} = frac{3sqrt{3}}{2} )So, combining these:[frac{7sqrt{3}}{4} - 3 - frac{sqrt{3}}{4} = -3 + left( frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} right) = -3 + frac{6sqrt{3}}{4} = -3 + frac{3sqrt{3}}{2}]So, ( y = -3 + frac{3sqrt{3}}{2} )Wait, that doesn't seem right. Let me double-check:For ( x = frac{21}{4} - 3sqrt{3} ):[y = frac{sqrt{3}}{3} times left( frac{21}{4} - 3sqrt{3} right) - frac{sqrt{3}}{4}]Which is:[frac{sqrt{3}}{3} times frac{21}{4} - frac{sqrt{3}}{3} times 3sqrt{3} - frac{sqrt{3}}{4}]Calculating each term:1. ( frac{sqrt{3}}{3} times frac{21}{4} = frac{7sqrt{3}}{4} )2. ( - frac{sqrt{3}}{3} times 3sqrt{3} = - frac{3 times 3}{3} = -3 )3. ( - frac{sqrt{3}}{4} )So, combining all terms:[frac{7sqrt{3}}{4} - 3 - frac{sqrt{3}}{4} = -3 + left( frac{7sqrt{3}}{4} - frac{sqrt{3}}{4} right) = -3 + frac{6sqrt{3}}{4} = -3 + frac{3sqrt{3}}{2}]Yes, that's correct. So, the two points of intersection are:1. ( A left( frac{21}{4} + 3sqrt{3}, 3 + frac{3sqrt{3}}{2} right) )2. ( B left( frac{21}{4} - 3sqrt{3}, -3 + frac{3sqrt{3}}{2} right) )Now, I need to find the distance between points ( A ) and ( B ). The distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is:[|AB| = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]Let me compute ( x_2 - x_1 ) and ( y_2 - y_1 ):First, ( x_1 = frac{21}{4} + 3sqrt{3} ) and ( x_2 = frac{21}{4} - 3sqrt{3} ). So,[x_2 - x_1 = left( frac{21}{4} - 3sqrt{3} right) - left( frac{21}{4} + 3sqrt{3} right) = -6sqrt{3}]Similarly, ( y_1 = 3 + frac{3sqrt{3}}{2} ) and ( y_2 = -3 + frac{3sqrt{3}}{2} ). So,[y_2 - y_1 = left( -3 + frac{3sqrt{3}}{2} right) - left( 3 + frac{3sqrt{3}}{2} right) = -6]So, the differences are ( Delta x = -6sqrt{3} ) and ( Delta y = -6 ).Now, plug these into the distance formula:[|AB| = sqrt{(-6sqrt{3})^2 + (-6)^2} = sqrt{(36 times 3) + 36} = sqrt{108 + 36} = sqrt{144} = 12]So, the length of segment ( AB ) is 12 units.Wait, that seems straightforward, but let me verify if there's another way to compute this without going through all the quadratic solutions. Maybe using properties of parabolas?I recall that for a parabola ( y^2 = 4px ), the length of the focal chord (a chord passing through the focus) making an angle ( theta ) with the x-axis is given by ( frac{4p}{sin^2 theta} ). Let me check if this formula applies here.Given our parabola is ( y^2 = 3x ), so ( 4p = 3 ) which means ( p = frac{3}{4} ). The angle of inclination is ( 30^circ ), so ( theta = 30^circ ).Plugging into the formula:[text{Length} = frac{4p}{sin^2 theta} = frac{4 times frac{3}{4}}{sin^2 30^circ} = frac{3}{left( frac{1}{2} right)^2} = frac{3}{frac{1}{4}} = 12]So, using this formula, I also get the length as 12. That confirms my previous result.Therefore, the length of ( |AB| ) is 12.**Final Answer**The length of ( |AB| ) is boxed{12}.

💡Okay, so I have this problem where there are 1000 integers, a₁, a₂, ..., a₁₀₀₀. Their squares are written in a circle, and it's given that the sum of any 41 consecutive numbers on this circle is a multiple of 41². The question is whether every integer a₁, a₂, ..., a₁₀₀₀ must necessarily be a multiple of 41.Hmm, let me try to break this down. First, the squares of these integers are arranged in a circle, which means that the sequence is cyclic. So, the first number comes after the last one. The key condition here is that any 41 consecutive numbers in this circle sum up to a multiple of 41².I think I should start by considering the properties of these sums. Since the sum of any 41 consecutive squares is divisible by 41², that means each of these sums is congruent to 0 modulo 41². Let me write that down:For any k, the sum from i = k to i = k+40 of a_i² ≡ 0 mod 41².Since the sequence is circular, this condition holds for all k from 1 to 1000, with the understanding that the indices wrap around.Now, I wonder if this condition implies something about each individual a_i. Specifically, does it force each a_i to be divisible by 41?Let me think about what it means for a number to be divisible by 41². If a number is divisible by 41², then it's also divisible by 41. But the converse isn't necessarily true. So, if a_i² is divisible by 41², then a_i must be divisible by 41 because 41 is prime. If a_i were not divisible by 41, then a_i² wouldn't be divisible by 41². So, maybe this condition does imply that each a_i is divisible by 41.But wait, the condition is about the sum of 41 consecutive squares, not about individual squares. So, even if each a_i² is not necessarily divisible by 41², their sum could still be. Is that possible?Let me explore that. Suppose that some a_i² is not divisible by 41². Then, a_i² mod 41² would be some non-zero value. But since the sum of 41 such terms is 0 mod 41², maybe these non-zero values can cancel each other out in some way.But considering that 41 is a prime number, and the modulus is 41², which is a prime power, the behavior might be different. Let me think about the properties of squares modulo 41².In modular arithmetic, if we have a sum of numbers that is 0 mod 41², it doesn't necessarily mean each term is 0 mod 41². However, in this case, the condition is that every possible sum of 41 consecutive terms is 0 mod 41². That seems like a very strong condition.Maybe I can use the fact that the sequence is periodic. Since the sum of any 41 consecutive terms is 0 mod 41², shifting the window by one should also give 0 mod 41². So, if I subtract the sum from k to k+40 from the sum from k+1 to k+41, I should get 0 mod 41².Mathematically, that would be:(a_{k+1}² + a_{k+2}² + ... + a_{k+41}²) - (a_k² + a_{k+1}² + ... + a_{k+40}²) ≡ 0 mod 41².Simplifying this, most terms cancel out, and we're left with:a_{k+41}² - a_k² ≡ 0 mod 41².Which implies that a_{k+41}² ≡ a_k² mod 41².Since the sequence is circular, and 41 and 1000 are coprime (because 41 is prime and doesn't divide 1000), this suggests that the sequence is periodic with period 41. But since 41 and 1000 are coprime, the period would have to be 1, meaning that all a_i² are congruent modulo 41².Wait, that might not be right. Let me think again. If a_{k+41}² ≡ a_k² mod 41² for all k, and since 41 and 1000 are coprime, this would imply that all a_i² are equal modulo 41². So, each a_i² ≡ c mod 41² for some constant c.But then, the sum of 41 such terms would be 41c mod 41². Since the sum is 0 mod 41², we have 41c ≡ 0 mod 41², which implies that c ≡ 0 mod 41. Therefore, each a_i² ≡ 0 mod 41, meaning that each a_i is divisible by 41.Wait, does that mean each a_i is divisible by 41? Because if a_i² is divisible by 41, then a_i must be divisible by 41 since 41 is prime.Yes, that seems to be the case. So, each a_i must be divisible by 41.But let me double-check. Suppose that a_i is not divisible by 41. Then, a_i² mod 41² would not be 0. But we just concluded that each a_i² ≡ 0 mod 41, which is a weaker condition. However, since the sum of 41 such terms is 0 mod 41², and each term is 0 mod 41, their sum is 0 mod 41² only if each term is 0 mod 41². Because 41 times 0 mod 41² is 0 mod 41², but if each term is only 0 mod 41, their sum would be 0 mod 41, but not necessarily 0 mod 41².Wait, that contradicts my earlier conclusion. So, maybe I made a mistake there.Let me clarify. If each a_i² ≡ c mod 41², and the sum of 41 terms is 0 mod 41², then 41c ≡ 0 mod 41², which implies c ≡ 0 mod 41. So, each a_i² ≡ 0 mod 41, meaning each a_i is divisible by 41. But does that necessarily mean that a_i² is divisible by 41²?No, because if a_i is divisible by 41, say a_i = 41k, then a_i² = 41²k², which is divisible by 41². So, actually, if each a_i is divisible by 41, then each a_i² is divisible by 41², and hence the sum of any 41 consecutive terms would be divisible by 41².But wait, the condition is that the sum is divisible by 41², not necessarily that each term is. So, could it be that the sum is divisible by 41² even if individual terms are not?I think not, because if each a_i² is only divisible by 41, then a_i² = 41b_i, where b_i is an integer. Then, the sum of 41 such terms would be 41(b₁ + b₂ + ... + b₄₁). For this sum to be divisible by 41², the sum of the b_i's must be divisible by 41. But since this must hold for any 41 consecutive terms, it would imply that each b_i is the same modulo 41, leading to all b_i being equal, and hence all a_i² being equal modulo 41², which would again force each a_i² to be 0 mod 41².Wait, I'm getting confused. Let me try a different approach.Suppose that not all a_i are divisible by 41. Then, there exists some a_j such that a_j is not divisible by 41. Then, a_j² mod 41² is not 0. Let's denote a_j² = 41²m + r, where r is not 0 mod 41².Now, consider the sum of 41 consecutive terms including a_j². Since the sum is 0 mod 41², the sum of the other 40 terms must be -r mod 41². But this would mean that the sum of those 40 terms is not 0 mod 41², which contradicts the condition that any 41 consecutive terms sum to 0 mod 41².Wait, no, because the sum of 41 terms is 0 mod 41², but the sum of 40 terms could be anything. Hmm, maybe that's not the right way to think about it.Alternatively, let's consider the difference between two consecutive sums. As I did earlier, subtracting the sum from k to k+40 from the sum from k+1 to k+41 gives a_{k+41}² - a_k² ≡ 0 mod 41². This implies that a_{k+41}² ≡ a_k² mod 41².Since the sequence is circular and 41 and 1000 are coprime, this recurrence would imply that all a_i² are congruent modulo 41². Let's denote this common value as c. So, each a_i² ≡ c mod 41².Then, the sum of 41 such terms would be 41c mod 41². For this to be 0 mod 41², we must have 41c ≡ 0 mod 41², which simplifies to c ≡ 0 mod 41. Therefore, each a_i² ≡ 0 mod 41, meaning each a_i is divisible by 41.But wait, if each a_i is divisible by 41, then a_i² is divisible by 41², so each a_i² ≡ 0 mod 41². Therefore, the sum of any 41 consecutive terms would be 0 mod 41², which satisfies the given condition.So, putting it all together, the condition that any 41 consecutive squares sum to 0 mod 41² implies that each a_i must be divisible by 41.I think that makes sense. So, the answer should be yes, every integer a₁, a₂, ..., a₁₀₀₀ must be a multiple of 41.