💡Okay, so I have this problem where a rectangle has a perimeter of 40 inches, and I need to find the maximum possible area in square inches. Hmm, let's see. I remember that the perimeter of a rectangle is calculated by adding up all the sides, which is 2 times the length plus 2 times the width. So, if I let the length be 'l' and the width be 'w', the perimeter formula would be 2l + 2w = 40 inches.Wait, maybe I can simplify that equation. If I divide both sides by 2, I get l + w = 20. That seems easier to work with. So, l + w = 20. Now, the area of a rectangle is length times width, so A = l * w. I need to maximize this area given that l + w = 20.Hmm, how do I maximize the area? I think this is a problem where I can use some algebra. Maybe I can express one variable in terms of the other using the perimeter equation and then substitute it into the area formula. Let's try that.From l + w = 20, I can solve for w: w = 20 - l. Now, substitute this into the area formula: A = l * (20 - l). So, A = 20l - l². That's a quadratic equation in terms of l. It looks like a downward-opening parabola because the coefficient of l² is negative.I remember that the maximum value of a quadratic function ax² + bx + c occurs at x = -b/(2a). In this case, a = -1 and b = 20. So, plugging those into the formula, the value of l that maximizes the area is l = -20/(2*(-1)) = -20/(-2) = 10. So, l = 10 inches.If l = 10 inches, then from w = 20 - l, w = 20 - 10 = 10 inches. So, both the length and the width are 10 inches. That means the rectangle is actually a square.Now, let's calculate the area: A = l * w = 10 * 10 = 100 square inches. So, the maximum area is 100 square inches.Wait, let me double-check to make sure I didn't make any mistakes. If the rectangle is a square with sides of 10 inches, the perimeter would be 4 * 10 = 40 inches, which matches the given perimeter. And the area is indeed 10 * 10 = 100. That seems correct.I think another way to look at this is by using calculus. If I take the area function A = 20l - l² and find its derivative with respect to l, I get dA/dl = 20 - 2l. Setting the derivative equal to zero to find critical points: 20 - 2l = 0, so 2l = 20, which gives l = 10. Then, w = 20 - 10 = 10. So, again, the maximum area is achieved when both sides are equal, making it a square with area 100 square inches.Alternatively, I could use the concept that for a given perimeter, the shape with the maximum area is a square. Since a rectangle can vary in length and width, but a square has all sides equal, it makes sense that it would enclose the largest area. So, without even doing the calculations, I could have reasoned that the maximum area occurs when the rectangle is a square.But just to be thorough, let's consider some other dimensions. Suppose the length is 12 inches, then the width would be 8 inches (since 12 + 8 = 20). The area would be 12 * 8 = 96 square inches, which is less than 100. If the length is 15 inches, the width would be 5 inches, and the area would be 15 * 5 = 75 square inches, which is even smaller. On the other hand, if the length is 9 inches, the width would be 11 inches, and the area would be 9 * 11 = 99 square inches, still less than 100. So, it seems that 10 inches for both length and width gives the maximum area.I guess another way to think about it is symmetry. Since the perimeter is fixed, making the sides equal distributes the perimeter equally among all sides, which intuitively should maximize the area. It's like when you have a fixed amount of fencing, the square enclosure will give you the largest area.Wait, let me also consider using the arithmetic mean-geometric mean inequality (AM-GM inequality). For two positive numbers l and w, the AM is (l + w)/2 and the GM is √(lw). The AM-GM inequality states that AM ≥ GM, with equality when all numbers are equal. In this case, l + w = 20, so AM = 10. Therefore, GM ≤ 10, which means √(lw) ≤ 10, so lw ≤ 100. Thus, the maximum area is 100 square inches when l = w = 10. That's another mathematical way to confirm it.So, after considering different methods—algebraic, calculus-based, intuitive reasoning, and even the AM-GM inequality—it all points to the same conclusion: the maximum area is 100 square inches when the rectangle is a square with sides of 10 inches each.I think I've covered all the angles here. I don't see any mistakes in my reasoning, and all the methods lead to the same result. So, I'm confident that the maximum possible area is indeed 100 square inches.

💡First, I need to determine how many large boats are required to accommodate all 104 students. Each large boat can hold 12 people, so I'll start by dividing the total number of students by the capacity of a large boat.104 divided by 12 equals approximately 8.666. Since I can't rent a fraction of a boat, I'll consider the whole numbers around this value, which are 8 and 9.Next, I'll check if 8 large boats can accommodate all the students. Multiplying 8 by 12 gives 96 seats. However, 96 is less than 104, meaning there are still 8 students left without seats. To accommodate these remaining students, I'll need additional small boats. Each small boat can hold 5 people, so I'll divide the remaining 8 students by 5, which gives approximately 1.6. Since I can't rent a fraction of a boat, I'll need to round up to 2 small boats. This means a total of 8 large boats and 2 small boats would be needed.Now, I'll check if 9 large boats can accommodate all the students. Multiplying 9 by 12 gives 108 seats, which is more than the total number of students. This means there are 4 extra seats available. However, since the problem states that all boats must be filled, having extra seats doesn't meet the requirement. Therefore, 9 large boats are not suitable.Based on this analysis, the only viable solution is to rent 8 large boats and 2 small boats to ensure all 104 students are accommodated without any empty seats.

💡Okay, so I have this function ( g(x) = frac{px + q}{rx + s} ), where ( p, q, r, s ) are all nonzero real numbers. The problem says that ( g(11) = 11 ), ( g(41) = 41 ), and ( g(g(x)) = x ) for all ( x ) except ( frac{-s}{r} ). I need to find the unique number that is not in the range of ( g ).First, let me understand what ( g(g(x)) = x ) means. This implies that ( g ) is its own inverse function. So, applying ( g ) twice brings me back to the original input. That means ( g ) is an involution.For a function to be its own inverse, it must satisfy ( g(g(x)) = x ). Let me write that out:[ g(g(x)) = x ]Substituting ( g(x) ) into itself:[ gleft( frac{px + q}{rx + s} right) = x ]Let me compute this:[ gleft( frac{px + q}{rx + s} right) = frac{p left( frac{px + q}{rx + s} right) + q}{r left( frac{px + q}{rx + s} right) + s} ]Simplify numerator and denominator:Numerator:[ p cdot frac{px + q}{rx + s} + q = frac{p(px + q) + q(rx + s)}{rx + s} = frac{p^2x + pq + qrx + qs}{rx + s} ]Denominator:[ r cdot frac{px + q}{rx + s} + s = frac{r(px + q) + s(rx + s)}{rx + s} = frac{rpx + rq + sr x + s^2}{rx + s} ]So, putting it together:[ frac{frac{p^2x + pq + qrx + qs}{rx + s}}{frac{rpx + rq + sr x + s^2}{rx + s}} = frac{p^2x + pq + qrx + qs}{rpx + rq + sr x + s^2} ]This must equal ( x ), so:[ frac{p^2x + pq + qrx + qs}{rpx + rq + sr x + s^2} = x ]Cross-multiplying:[ p^2x + pq + qrx + qs = x(rpx + rq + sr x + s^2) ]Let me expand the right side:[ x cdot rpx + x cdot rq + x cdot sr x + x cdot s^2 = r p x^2 + r q x + s r x^2 + s^2 x ]Combine like terms on the right:[ (r p + s r) x^2 + (r q + s^2) x ]So, the equation becomes:[ p^2x + pq + q r x + q s = (r p + s r) x^2 + (r q + s^2) x ]Now, let's bring all terms to one side:[ (r p + s r) x^2 + (r q + s^2) x - p^2x - pq - q r x - q s = 0 ]Simplify term by term:First, the ( x^2 ) term:[ (r p + s r) x^2 ]Then, the ( x ) terms:[ (r q + s^2 - p^2 - q r) x ]And the constant terms:[ -pq - q s ]So, the equation is:[ (r p + s r) x^2 + (r q + s^2 - p^2 - q r) x - pq - q s = 0 ]Since this must hold for all ( x ) except ( frac{-s}{r} ), the coefficients of each power of ( x ) must be zero. Therefore, we have the following system of equations:1. Coefficient of ( x^2 ):[ r p + s r = 0 ]2. Coefficient of ( x ):[ r q + s^2 - p^2 - q r = 0 ]3. Constant term:[ -pq - q s = 0 ]Let me write these equations:1. ( r p + s r = 0 ) → ( r(p + s) = 0 )2. ( r q + s^2 - p^2 - q r = 0 ) → Simplify: ( s^2 - p^2 = 0 ) → ( s^2 = p^2 )3. ( -pq - q s = 0 ) → ( -q(p + s) = 0 )Given that ( p, q, r, s ) are all nonzero, let's analyze these equations.From equation 1: ( r(p + s) = 0 ). Since ( r neq 0 ), this implies ( p + s = 0 ) → ( s = -p )From equation 3: ( -q(p + s) = 0 ). Again, since ( q neq 0 ), this also implies ( p + s = 0 ), which is consistent with equation 1.From equation 2: ( s^2 = p^2 ). Since ( s = -p ), this is satisfied because ( (-p)^2 = p^2 ).So, we have ( s = -p ). Let me substitute ( s = -p ) into the function ( g(x) ):[ g(x) = frac{px + q}{rx + s} = frac{px + q}{rx - p} ]Now, we also have the conditions ( g(11) = 11 ) and ( g(41) = 41 ). Let's use these to find relations between ( p, q, r ).First, ( g(11) = 11 ):[ frac{p cdot 11 + q}{r cdot 11 - p} = 11 ]Multiply both sides by denominator:[ 11p + q = 11(r cdot 11 - p) ]Simplify RHS:[ 11p + q = 121 r - 11p ]Bring all terms to left:[ 11p + q - 121 r + 11p = 0 ]Combine like terms:[ 22p + q - 121 r = 0 ] → ( 22p + q = 121 r ) → Equation ASimilarly, ( g(41) = 41 ):[ frac{p cdot 41 + q}{r cdot 41 - p} = 41 ]Multiply both sides by denominator:[ 41p + q = 41(r cdot 41 - p) ]Simplify RHS:[ 41p + q = 1681 r - 41p ]Bring all terms to left:[ 41p + q - 1681 r + 41p = 0 ]Combine like terms:[ 82p + q - 1681 r = 0 ] → ( 82p + q = 1681 r ) → Equation BNow, we have two equations:Equation A: ( 22p + q = 121 r )Equation B: ( 82p + q = 1681 r )Subtract Equation A from Equation B:[ (82p + q) - (22p + q) = 1681 r - 121 r ]Simplify:[ 60p = 1560 r ]Divide both sides by 60:[ p = 26 r ]So, ( p = 26 r ). Let me substitute this back into Equation A:[ 22(26 r) + q = 121 r ]Calculate 22*26:22*26 = 572So:[ 572 r + q = 121 r ]Subtract 572 r:[ q = 121 r - 572 r = -451 r ]So, ( q = -451 r )Now, we have ( p = 26 r ), ( q = -451 r ), ( s = -p = -26 r )So, the function ( g(x) ) can be written as:[ g(x) = frac{26 r x - 451 r}{r x - 26 r} ]Factor out ( r ) from numerator and denominator:[ g(x) = frac{r(26 x - 451)}{r(x - 26)} ]Since ( r neq 0 ), we can cancel ( r ):[ g(x) = frac{26 x - 451}{x - 26} ]So, ( g(x) = frac{26x - 451}{x - 26} )Now, the question is to find the unique number not in the range of ( g ). For a rational function like this, the range is all real numbers except the value that makes the equation ( y = frac{26x - 451}{x - 26} ) unsolvable for ( x ).Let me set ( y = frac{26x - 451}{x - 26} ) and solve for ( x ):Multiply both sides by ( x - 26 ):[ y(x - 26) = 26x - 451 ]Expand left side:[ yx - 26y = 26x - 451 ]Bring all terms to left:[ yx - 26y - 26x + 451 = 0 ]Factor terms with ( x ):[ x(y - 26) - 26y + 451 = 0 ]Solve for ( x ):[ x(y - 26) = 26y - 451 ]So,[ x = frac{26y - 451}{y - 26} ]This equation is solvable for all ( y ) except when the denominator is zero, i.e., when ( y - 26 = 0 ) → ( y = 26 ). Therefore, ( y = 26 ) is not in the range of ( g ).Hence, the unique number not in the range of ( g ) is 26.Wait, but let me double-check. Maybe I made a mistake in the calculation.Wait, in the function ( g(x) = frac{26x - 451}{x - 26} ), the vertical asymptote is at ( x = 26 ), which affects the domain, but the range is affected by the horizontal asymptote.The horizontal asymptote is found by the ratio of leading coefficients, which is 26/1 = 26. So, as ( x ) approaches infinity, ( g(x) ) approaches 26. Therefore, 26 is not in the range of ( g ).Yes, that seems correct. So, the unique number not in the range is 26.But wait, in the initial problem, the user had a different answer, 30. Did I make a mistake?Wait, let me check my steps again.I had ( p = 26 r ), ( q = -451 r ), ( s = -26 r ). So, ( g(x) = frac{26x - 451}{x - 26} ). Then, solving ( y = frac{26x - 451}{x - 26} ), I found ( y = 26 ) is excluded.But in the initial problem, the user's assistant had 30 as the answer. Maybe I made a miscalculation earlier.Wait, let me go back to where I found ( p = 26 r ) and ( q = -451 r ). Let me verify that.From Equation A: ( 22p + q = 121 r )From Equation B: ( 82p + q = 1681 r )Subtracting A from B: ( 60p = 1560 r ) → ( p = 26 r ). Correct.Then, substituting into Equation A: ( 22*(26 r) + q = 121 r )22*26: 20*26=520, 2*26=52, total 572. So, 572 r + q = 121 r → q = 121 r - 572 r = -451 r. Correct.So, ( g(x) = frac{26x - 451}{x - 26} ). Then, solving for ( y ), we get ( y = 26 ) is excluded.But the user's initial thought process had 30 as the answer. Maybe I made a mistake in the initial steps.Wait, in the initial problem, the user had:"Setting these equal gives:[ 11^2p + 12p = 41^2p + 42p rightarrow 40p = (41^2 - 11^2) p rightarrow p = frac{41^2 - 11^2}{40 + 12} ]This leads to [ frac{p}{r} = -frac{p}{p} = -1, ] however, this is outside the requirement. Re-examining coefficients and computation yields the correct ( frac{p}{-r} ).Thus, simplifying (41^2 - 11^2) / (30 + 1) = 30, hence[ boxed{30} ] is the number that is not in the range of g."Wait, so in the initial problem, the user had a different approach, leading to 30. But in my approach, I got 26.I need to see where the discrepancy is.Wait, perhaps the user made a mistake in their initial calculation, and I did it correctly.Alternatively, maybe I made a mistake in my own calculation.Let me check my function again.Given ( g(x) = frac{26x - 451}{x - 26} ). Let me compute ( g(11) ):[ g(11) = frac{26*11 - 451}{11 - 26} = frac{286 - 451}{-15} = frac{-165}{-15} = 11 ]. Correct.Similarly, ( g(41) = frac{26*41 - 451}{41 - 26} = frac{1066 - 451}{15} = frac{615}{15} = 41 ). Correct.So, my function satisfies the given conditions.Then, solving for ( y ), I get ( y = 26 ) is excluded.Alternatively, maybe the user's initial approach was flawed, leading to 30, but my approach is correct.Alternatively, perhaps I made a mistake in the horizontal asymptote.Wait, the horizontal asymptote is the ratio of leading coefficients, which is 26/1 = 26. So, as ( x ) approaches infinity, ( g(x) ) approaches 26, so 26 is not in the range.Alternatively, maybe the user's initial approach was considering the value that makes the denominator zero in the inverse function, but that would be 26, as above.Alternatively, perhaps the user made a mistake in their initial calculation.Wait, in the initial problem, the user had:"Setting these equal gives:[ 11^2p + 12p = 41^2p + 42p rightarrow 40p = (41^2 - 11^2) p rightarrow p = frac{41^2 - 11^2}{40 + 12} ]This leads to [ frac{p}{r} = -frac{p}{p} = -1, ] however, this is outside the requirement. Re-examining coefficients and computation yields the correct ( frac{p}{-r} ).Thus, simplifying (41^2 - 11^2) / (30 + 1) = 30, hence[ boxed{30} ] is the number that is not in the range of g."Wait, in this, the user is setting ( 11^2p + 12p = 41^2p + 42p ), which simplifies to ( (121 + 12)p = (1681 + 42)p ), which is ( 133p = 1723p ), which would imply ( 133 = 1723 ), which is not correct. So, the user made a mistake here.Wait, no, actually, the user wrote:"Setting these equal gives:[ 11^2p + 12p = 41^2p + 42p rightarrow 40p = (41^2 - 11^2) p rightarrow p = frac{41^2 - 11^2}{40 + 12} ]"Wait, let's compute ( 11^2 + 12 = 121 + 12 = 133 ), and ( 41^2 + 42 = 1681 + 42 = 1723 ). So, 133p = 1723p → 133 = 1723, which is not correct. So, the user made a mistake in their algebra here.Alternatively, perhaps the user meant to subtract instead of add.Wait, in my approach, I had:From ( g(11) = 11 ), I got ( 22p + q = 121 r )From ( g(41) = 41 ), I got ( 82p + q = 1681 r )Subtracting, I got ( 60p = 1560 r ) → ( p = 26 r )Then, substituting back, I got ( q = -451 r )So, my function is correct, leading to 26 being excluded.Therefore, the correct answer should be 26, not 30.But in the initial problem, the user had 30 as the answer, which seems incorrect.Alternatively, perhaps I made a mistake in my own calculation.Wait, let me check the function again.Given ( g(x) = frac{26x - 451}{x - 26} ), let me compute ( g(g(x)) ) to ensure it equals ( x ).Compute ( g(g(x)) ):First, ( g(x) = frac{26x - 451}{x - 26} )Then, ( g(g(x)) = frac{26 cdot frac{26x - 451}{x - 26} - 451}{frac{26x - 451}{x - 26} - 26} )Simplify numerator:[ 26 cdot frac{26x - 451}{x - 26} - 451 = frac{26(26x - 451) - 451(x - 26)}{x - 26} ]Compute numerator:[ 26*26x = 676x ][ 26*(-451) = -11726 ][ -451x + 451*26 = -451x + 11726 ]So, total numerator:[ 676x - 11726 - 451x + 11726 = (676x - 451x) + (-11726 + 11726) = 225x + 0 = 225x ]Denominator:[ frac{26x - 451}{x - 26} - 26 = frac{26x - 451 - 26(x - 26)}{x - 26} ]Compute numerator:[ 26x - 451 -26x + 676 = (26x -26x) + (-451 + 676) = 0 + 225 = 225 ]So, denominator is ( frac{225}{x - 26} )Thus, ( g(g(x)) = frac{225x / (x - 26)}{225 / (x - 26)} = frac{225x}{225} = x ). Correct.So, ( g(g(x)) = x ) as required.Therefore, my function is correct, and the excluded value is 26.Hence, the unique number not in the range of ( g ) is 26.But wait, in the initial problem, the user had 30 as the answer. Maybe the user made a mistake, or perhaps I misread the problem.Wait, let me check the initial problem again.The function ( g ) is defined by ( g(x)= frac{px+q}{rx+s} ), where ( p ), ( q ), ( r ), and ( s ) are nonzero real numbers. It has the properties ( g(11)=11 ), ( g(41)=41 ), and ( g(g(x))=x ) for all values except ( frac{-s}{r} ). Find the unique number that is not in the range of ( g ).So, the problem is correct as I interpreted it.Therefore, my conclusion is that the unique number not in the range is 26.But in the initial problem, the user had 30 as the answer. Maybe the user made a mistake in their calculation.Alternatively, perhaps I made a mistake in my own calculation.Wait, let me check the horizontal asymptote again.For ( g(x) = frac{26x - 451}{x - 26} ), as ( x ) approaches infinity, the function behaves like ( frac{26x}{x} = 26 ). So, the horizontal asymptote is at ( y = 26 ), meaning 26 is not in the range.Therefore, the unique number not in the range is 26.Hence, I think the correct answer is 26, not 30.But in the initial problem, the user had 30 as the answer. Maybe the user made a mistake in their calculation.Alternatively, perhaps I made a mistake in my own calculation.Wait, let me check the function again.Given ( g(x) = frac{26x - 451}{x - 26} ), let me compute ( g(11) ):[ g(11) = frac{26*11 - 451}{11 - 26} = frac{286 - 451}{-15} = frac{-165}{-15} = 11 ]. Correct.Similarly, ( g(41) = frac{26*41 - 451}{41 - 26} = frac{1066 - 451}{15} = frac{615}{15} = 41 ). Correct.So, the function satisfies the given conditions.Therefore, the unique number not in the range is 26.Hence, I think the correct answer is 26.But in the initial problem, the user had 30 as the answer. Maybe the user made a mistake in their calculation.Alternatively, perhaps the problem was different, but the user presented it correctly.Wait, perhaps the user's initial approach was incorrect, leading to 30, but my approach is correct, leading to 26.Therefore, I think the correct answer is 26.

💡Alright, so I have this problem here: If ( x ) and ( y ) are positive integers such that ( x^2 + y^2 - x ) is divisible by ( 2xy ), then ( x ) is a perfect square. Hmm, okay, let's try to unpack this step by step.First, let's understand what it means for ( x^2 + y^2 - x ) to be divisible by ( 2xy ). That means when we divide ( x^2 + y^2 - x ) by ( 2xy ), we get an integer. So, we can write this as:[ x^2 + y^2 - x = 2kxy ]where ( k ) is some positive integer. Okay, so that's our starting equation.Now, I need to show that ( x ) must be a perfect square. A perfect square is a number that can be expressed as ( n^2 ) where ( n ) is an integer. So, I need to somehow manipulate this equation to show that ( x ) has to be such a number.Let me rearrange the equation to see if I can get some insight:[ x^2 - x + y^2 = 2kxy ]Hmm, maybe I can rearrange terms to isolate ( y ) or ( x ). Let's try to rearrange terms involving ( y ):[ y^2 - 2kxy + (x^2 - x) = 0 ]This looks like a quadratic equation in terms of ( y ). So, if I consider this as a quadratic in ( y ), it would be:[ y^2 - (2kx)y + (x^2 - x) = 0 ]For ( y ) to be an integer, the discriminant of this quadratic must be a perfect square. The discriminant ( D ) of a quadratic equation ( ay^2 + by + c = 0 ) is given by ( D = b^2 - 4ac ). So, let's compute the discriminant here:[ D = (2kx)^2 - 4 times 1 times (x^2 - x) ][ D = 4k^2x^2 - 4x^2 + 4x ][ D = 4x^2(k^2 - 1) + 4x ]Hmm, so the discriminant is ( 4x^2(k^2 - 1) + 4x ). For ( y ) to be an integer, this discriminant must be a perfect square. Let's factor out a 4x:[ D = 4x(x(k^2 - 1) + 1) ]So, ( D = 4x times [x(k^2 - 1) + 1] ). Since ( D ) must be a perfect square, and 4 is already a perfect square, the product ( x times [x(k^2 - 1) + 1] ) must also be a perfect square.Let me denote ( A = x ) and ( B = x(k^2 - 1) + 1 ). So, ( A times B ) is a perfect square. Now, for the product of two numbers to be a perfect square, each of them must be such that their prime factors can pair up. But ( A = x ) and ( B = x(k^2 - 1) + 1 ). I need to analyze the relationship between ( A ) and ( B ). Let's see if ( A ) and ( B ) are coprime. If they are, then both ( A ) and ( B ) must individually be perfect squares. If they are not coprime, then their greatest common divisor (GCD) must be a square, and the remaining parts must also be squares.So, let's compute ( gcd(A, B) ). Since ( A = x ) and ( B = x(k^2 - 1) + 1 ), any common divisor of ( A ) and ( B ) must divide ( B - (k^2 - 1)A ), which is:[ B - (k^2 - 1)A = [x(k^2 - 1) + 1] - (k^2 - 1)x = 1 ]So, the GCD of ( A ) and ( B ) is 1. That means ( A ) and ( B ) are coprime. Therefore, since their product is a perfect square and they are coprime, both ( A ) and ( B ) must themselves be perfect squares.So, ( x ) is a perfect square, and ( x(k^2 - 1) + 1 ) is also a perfect square. That's a crucial point. So, we've established that ( x ) must be a perfect square.But wait, let me make sure I didn't skip any steps here. I concluded that since ( A times B ) is a perfect square and ( A ) and ( B ) are coprime, both must be perfect squares. Is that always true? Let me think.Yes, if two coprime integers multiply to a perfect square, then each must be a perfect square. Because in their prime factorizations, there are no common primes, so each prime must appear an even number of times in each factor. So, yes, that seems correct.Therefore, ( x ) must be a perfect square. So, that's the conclusion.But just to be thorough, let me see if I can find any counterexamples or if there are any special cases where this might not hold. Suppose ( x ) is not a perfect square. Then, ( A = x ) is not a perfect square, but ( A times B ) is a perfect square. Since ( A ) and ( B ) are coprime, ( A ) must be a perfect square, which contradicts our assumption. So, no, there are no such counterexamples.Therefore, the conclusion holds: ( x ) must be a perfect square.**Final Answer**boxed{x text{ is a perfect square}}

💡Okay, so I have this problem where someone is saving up to buy a car. They earn £4000 every month and they save £500 each month specifically for the car. They need £45,000 to buy their dream car. The question is asking how much money they will earn before they save enough to buy the car.Alright, let me break this down. First, I need to figure out how many months it will take for them to save up £45,000 if they're saving £500 each month. That seems straightforward. I can divide the total amount needed by the amount they save each month. So, £45,000 divided by £500 per month. Let me do that calculation: 45,000 divided by 500 equals 90. So, it will take them 90 months to save enough money.Now, the next part is figuring out how much money they will earn in those 90 months. They earn £4000 each month, so I need to multiply their monthly earnings by the number of months. That would be £4000 multiplied by 90. Let me calculate that: 4000 times 90 equals 360,000. So, they will earn £360,000 in total over those 90 months.Wait a second, let me double-check that. If they save £500 each month for 90 months, that's 500 times 90, which is indeed 45,000. And earning £4000 each month for 90 months is 4000 times 90, which is 360,000. That seems correct.I think I've got it. It will take them 90 months to save enough money, and during that time, they will earn a total of £360,000.

💡First, I'll define John's current age as ( J ).Five years ago, John's age was ( J - 5 ).In 8 years, John's age will be ( J + 8 ).According to the problem, five years ago, John's age was half of what it will be in 8 years. This gives the equation:[J - 5 = frac{1}{2}(J + 8)]To eliminate the fraction, I'll multiply both sides of the equation by 2:[2(J - 5) = J + 8]Expanding the left side:[2J - 10 = J + 8]Subtracting ( J ) from both sides to isolate ( J ):[J - 10 = 8]Finally, adding 10 to both sides to solve for ( J ):[J = 18]Therefore, John is currently 18 years old.