answer：Alright, so I have this problem where I need to take a square piece of paper and fold it in such a way that I get the largest possible equilateral triangle. The catch is, I can't use any measurements or instruments, just folding. Hmm, interesting. I remember that an equilateral triangle has all sides equal and all angles equal to 60 degrees. So, I need to figure out how to fold the square to get a triangle with those properties.First, let me visualize the square. Let's say it's a standard square with sides of length 's'. If I try to fold it to make an equilateral triangle where all sides are equal to 's', that might not be the largest possible. The problem mentions that such a triangle won't be the largest, so there must be a way to get a bigger triangle by folding differently.Maybe if I fold the square in half first. If I fold it along the diagonal, I get a right-angled triangle, but that's not equilateral. If I fold it in half along the vertical or horizontal axis, I get a rectangle. Hmm, not helpful. Wait, what if I fold it diagonally? That gives me a triangle, but again, it's a right-angled triangle, not equilateral.I need to think about the angles. An equilateral triangle has 60-degree angles. How can I create a 60-degree angle by folding? Maybe by folding one corner to meet another point in such a way that it creates a 60-degree angle.Let me try this: take one corner of the square and fold it to meet the midpoint of the opposite side. If I do that, I might create a triangle with one 60-degree angle. But I'm not sure if that's the case. Maybe I need to fold it differently.Another idea: if I fold the square in such a way that one side becomes the base of the triangle and the other sides are folded to meet at a point, forming the apex of the triangle. But how do I ensure that all sides are equal?Wait, maybe I can use the property that in an equilateral triangle, the height is related to the side length by the formula ( h = frac{sqrt{3}}{2} s ). So, if I can somehow use the height of the square to determine the side length of the triangle, that might help.The square has a side length 's', so its height is also 's'. If I set the height of the triangle equal to 's', then the side length of the triangle would be ( frac{2h}{sqrt{3}} = frac{2s}{sqrt{3}} ). That's larger than 's', so maybe that's the way to go.But how do I fold the square to achieve that? Maybe by folding one corner up to meet a point that's ( frac{2s}{sqrt{3}} ) away from the base. But since I can't measure, I need to find a way to do this by folding.Perhaps I can fold the square in half vertically, then fold the top corner down to meet the midpoint of the bottom edge. That might create a 60-degree angle. Let me try to visualize that.If I fold the square in half vertically, I have a rectangle with width ( frac{s}{2} ) and height 's'. Then, folding the top corner down to meet the midpoint of the bottom edge would create a triangle. The distance from the top corner to the midpoint is ( frac{s}{2} ) horizontally and 's' vertically. So, the fold would create a hypotenuse that's the side of the triangle.Wait, but I need all sides to be equal. So, if I fold it this way, the sides might not be equal. Maybe I need to adjust the fold.Another approach: if I fold the square so that one corner touches the opposite side at a point that divides the side in a specific ratio. For an equilateral triangle, the ratio of the sides is related to the square's sides.I think the key is to find a fold that creates a 60-degree angle without measuring. Maybe by folding the corner to a specific point that ensures the angle is 60 degrees.Let me try this step-by-step:1. Start with the square.2. Fold it in half vertically to get a rectangle.3. Take the top right corner and fold it down to meet the midpoint of the bottom edge.4. This should create a triangle with a 60-degree angle at the bottom.But I'm not sure if this is accurate. Maybe I need to fold it differently.Alternatively, I can fold the square into thirds. If I fold one side to meet the one-third mark, that might create the necessary angle. But again, without measuring, it's tricky.Wait, maybe I can use the fact that in an equilateral triangle, the centroid, circumcenter, and incenter coincide. So, if I can find the centroid of the square and use that as a reference point for folding, it might help.The centroid of the square is at the intersection of its diagonals. If I fold the square so that one corner meets the centroid, that might create a triangle with the desired properties.Let me try that:1. Fold the square along one of its diagonals to find the centroid.2. Then, fold one corner to meet the centroid.3. This should create a triangle with one side equal to the square's side and the other sides adjusted accordingly.But I'm still not sure if this results in an equilateral triangle.Maybe I need to think about the proportions. If the side length of the triangle is ( frac{2s}{sqrt{3}} ), then the height of the triangle is 's'. So, if I can fold the square so that the height of the triangle is equal to the square's side, that should give me the largest possible equilateral triangle.To do this, I can fold the square in such a way that the height of the triangle aligns with the square's height. This would mean folding one side up to meet the top edge at a point that creates the necessary 60-degree angle.Perhaps by folding the bottom edge up to meet the top edge at a specific point, creating a triangle with the desired height and side length.I think the key is to create a fold that forms a 60-degree angle with the base of the square. This can be done by folding one corner to a point that is one-third of the way along the opposite side.Wait, if I fold the bottom left corner to a point one-third of the way along the top edge, that might create a 60-degree angle. Let me visualize that.Folding the corner to the one-third mark would create a triangle with sides in the ratio 1:2, which might not be equilateral. Hmm.Maybe I need to fold it so that the corner meets a point that is ( frac{sqrt{3}}{2} s ) away from the corner along the opposite side. But without measuring, how can I find that point?Perhaps by using the properties of the square and folding to create intersecting points that naturally lead to the 60-degree angle.I recall that in origami, certain folds can create specific angles by using the proportions of the paper. Maybe I can use a similar technique here.Let me try this method:1. Start with the square.2. Fold it in half vertically to get a rectangle.3. Fold the top right corner down to meet the midpoint of the left edge.4. This should create a triangle with a 60-degree angle.But I'm not sure if this is correct. Maybe I need to adjust the fold.Alternatively, I can fold the square into a hexagon, which has six equilateral triangles, and then isolate one of them. But that might be more complex.Wait, another idea: if I fold the square so that one corner touches the midpoint of the opposite side, and then fold again to create symmetry, that might result in an equilateral triangle.Let me try:1. Fold the square in half vertically.2. Fold the top right corner down to meet the midpoint of the bottom edge.3. This creates a triangle with a 60-degree angle at the bottom.But I need to verify if all sides are equal.Alternatively, I can fold the square into a smaller square and then fold that into an equilateral triangle. But that might not give the largest possible.I think the key is to find a fold that maximizes the side length of the triangle while maintaining the 60-degree angles. This likely involves folding the square in such a way that the triangle's height equals the square's side length.Given that the height of the equilateral triangle is ( frac{sqrt{3}}{2} s ), setting this equal to the square's side 's' gives the side length of the triangle as ( frac{2s}{sqrt{3}} ), which is larger than 's'. So, the triangle's side is longer than the square's side.To achieve this, I need to fold the square so that the triangle's height aligns with the square's height. This would involve folding one side up to meet the top edge at a specific point that creates the necessary angle.Perhaps by folding the bottom edge up to meet the top edge at a point that is ( frac{sqrt{3}}{2} s ) from the corner. But without measuring, I need to find a way to approximate this.Maybe by folding the corner to a point that is two-thirds of the way along the opposite side. This might create the necessary proportions for the equilateral triangle.Let me try:1. Start with the square.2. Fold the bottom left corner up to meet a point two-thirds of the way along the top edge.3. This should create a triangle with a 60-degree angle.But I'm not sure if this is accurate. I might need to adjust the fold.Alternatively, I can fold the square into a smaller equilateral triangle and then see how it fits within the square. But I need to ensure it's the largest possible.I think the best approach is to fold the square in such a way that the triangle's height equals the square's side, which would maximize the triangle's side length. This involves creating a fold that forms a 60-degree angle with the base.To do this without measuring, I can use the properties of the square and the equilateral triangle. Since the height of the triangle is ( frac{sqrt{3}}{2} s ), and the square's side is 's', I need to find a point along the top edge that is ( frac{sqrt{3}}{2} s ) from the corner.But without measuring, I can approximate this by folding the corner to a point that is roughly two-thirds of the way along the top edge. This should give me a triangle with a 60-degree angle.Let me try:1. Start with the square.2. Fold the bottom left corner up to meet a point two-thirds of the way along the top edge.3. Crease the fold and unfold.4. Repeat on the other side to create symmetry.5. Now, fold along the creases to form the equilateral triangle.This should give me the largest possible equilateral triangle by maximizing the side length while maintaining the 60-degree angles.I think this method works because it uses the proportions of the square to approximate the necessary fold for the equilateral triangle. By folding to two-thirds of the way, I'm effectively creating the height needed for the triangle, which in turn determines the side length.So, in summary, the steps are:1. Start with the square.2. Fold one corner to a point approximately two-thirds along the opposite side.3. Repeat on the other side to create symmetry.4. Fold along the creases to form the equilateral triangle.This should give me the largest possible equilateral triangle without using any measurements or instruments.

answer：Okay, so I have this problem where I need to find the maximum value of the function f(x) = |x - 1| - 2|x + 1|. Then, using that maximum value m, I have to find the maximum value of ab + 2bc given that a, b, c are positive real numbers and a² + 3b² + 2c² = m.Starting with part (I). I remember that absolute value functions can be tricky because they change their behavior depending on whether the expression inside is positive or negative. So, I should probably break down the function f(x) into different cases based on the critical points where the expressions inside the absolute values change sign. Looking at |x - 1|, the critical point is at x = 1, and for |x + 1|, the critical point is at x = -1. So, these points divide the real number line into three intervals: x < -1, -1 ≤ x < 1, and x ≥ 1. I'll need to analyze f(x) in each of these intervals.First interval: x < -1. In this case, both x - 1 and x + 1 are negative. So, |x - 1| becomes -(x - 1) = -x + 1, and |x + 1| becomes -(x + 1) = -x - 1. Plugging these into f(x):f(x) = (-x + 1) - 2*(-x - 1) = -x + 1 + 2x + 2 = x + 3.So, for x < -1, f(x) simplifies to x + 3. Since x is less than -1, the maximum value in this interval would be when x is as large as possible, approaching -1 from the left. Plugging x = -1 into x + 3 gives f(-1) = -1 + 3 = 2.Next interval: -1 ≤ x < 1. Here, x + 1 is non-negative, so |x + 1| = x + 1, but x - 1 is still negative, so |x - 1| = -(x - 1) = -x + 1. Plugging into f(x):f(x) = (-x + 1) - 2*(x + 1) = -x + 1 - 2x - 2 = -3x - 1.So, in this interval, f(x) is a linear function with a slope of -3, which is decreasing. Therefore, its maximum occurs at the left endpoint, which is x = -1. Plugging x = -1 into -3x - 1 gives -3*(-1) - 1 = 3 - 1 = 2. The minimum in this interval would be at x approaching 1 from the left: f(x) approaches -3*(1) - 1 = -4.Third interval: x ≥ 1. Both x - 1 and x + 1 are non-negative, so |x - 1| = x - 1 and |x + 1| = x + 1. Plugging into f(x):f(x) = (x - 1) - 2*(x + 1) = x - 1 - 2x - 2 = -x - 3.This is a linear function with a slope of -1, which is decreasing. So, its maximum occurs at the left endpoint, x = 1. Plugging x = 1 into -x - 3 gives -1 - 3 = -4. As x increases beyond 1, f(x) continues to decrease.Putting it all together, the function f(x) reaches its maximum value of 2 at x = -1. So, m = 2.Moving on to part (II). We have a constraint a² + 3b² + 2c² = 2, and we need to maximize ab + 2bc. All variables a, b, c are positive real numbers.This seems like an optimization problem with a quadratic constraint. I think I can use the method of Lagrange multipliers, but maybe there's a simpler way using inequalities like Cauchy-Schwarz or AM-GM.Let me think about how to express ab + 2bc in terms that can be bounded by the given quadratic form.First, notice that ab + 2bc can be factored as b(a + 2c). So, maybe I can write it as b(a + 2c). But I'm not sure if that helps directly.Alternatively, I can think of ab + 2bc as a combination of products. Perhaps I can use the Cauchy-Schwarz inequality or the AM-GM inequality to bound this expression.Let me try Cauchy-Schwarz. The Cauchy-Schwarz inequality states that for vectors u and v, (u ⋅ v)² ≤ (u ⋅ u)(v ⋅ v). Maybe I can set up vectors such that their dot product is ab + 2bc.Let me consider u = (a, √2 c) and v = (b, √2 b). Then, u ⋅ v = a*b + √2 c * √2 b = ab + 2bc, which is exactly what we have.Now, applying Cauchy-Schwarz:(ab + 2bc)² ≤ (a² + (√2 c)²)(b² + (√2 b)²) = (a² + 2c²)(b² + 2b²) = (a² + 2c²)(3b²)But from the constraint, we have a² + 3b² + 2c² = 2. Let me denote S = a² + 2c². Then, the constraint becomes S + 3b² = 2.So, substituting back into the inequality:(ab + 2bc)² ≤ S * 3b² = 3b² * SBut S = 2 - 3b², so:(ab + 2bc)² ≤ 3b²*(2 - 3b²) = 6b² - 9b⁴Now, we need to maximize 6b² - 9b⁴. Let me denote this as a function of b²:Let t = b². Then, the expression becomes 6t - 9t². To find its maximum, take derivative with respect to t:d/dt (6t - 9t²) = 6 - 18t. Setting equal to zero:6 - 18t = 0 ⇒ t = 6/18 = 1/3.So, maximum occurs at t = 1/3. Plugging back into 6t - 9t²:6*(1/3) - 9*(1/3)² = 2 - 9*(1/9) = 2 - 1 = 1.Therefore, (ab + 2bc)² ≤ 1 ⇒ ab + 2bc ≤ 1.So, the maximum value of ab + 2bc is 1.Wait, but I should check if equality can be achieved. For Cauchy-Schwarz, equality holds when u and v are proportional. That is, when (a, √2 c) = k*(b, √2 b) for some scalar k.So, a = k*b and √2 c = k*√2 b ⇒ c = k*b.So, a = k*b and c = k*b.From the constraint a² + 3b² + 2c² = 2, substituting a and c:(k*b)² + 3b² + 2*(k*b)² = 2 k² b² + 3b² + 2k² b² = 2 ( k² + 3 + 2k² ) b² = 2 (3k² + 3) b² = 2 3(k² + 1) b² = 2 b² = 2 / [3(k² + 1)]From earlier, t = b² = 1/3. So,1/3 = 2 / [3(k² + 1)] Multiply both sides by 3:1 = 2 / (k² + 1) So, k² + 1 = 2 k² = 1 k = 1 or k = -1But since a, b, c are positive, k must be positive. So, k = 1.Therefore, a = b and c = b.So, a = b = c.Wait, but from a = k*b and c = k*b with k =1, so a = b and c = b. So, a = b = c.But let's check if this satisfies the constraint:a² + 3b² + 2c² = b² + 3b² + 2b² = 6b² = 2 ⇒ b² = 1/3 ⇒ b = 1/√3.So, a = 1/√3, b = 1/√3, c = 1/√3.Then, ab + 2bc = (1/√3)(1/√3) + 2*(1/√3)(1/√3) = (1/3) + 2*(1/3) = 1.So, yes, equality is achieved when a = b = c = 1/√3. Therefore, the maximum value is indeed 1.Alternatively, I could have used the method of Lagrange multipliers, but this approach using Cauchy-Schwarz seems straightforward and effective.So, summarizing my findings:(I) The maximum value m of f(x) is 2.(II) The maximum value of ab + 2bc under the given constraint is 1.

answer：Alright, so I've got this problem here: Given that 0 < a < 1, and we have x = a^a, y = a, and z = log base a of a. I need to figure out the relationship between x, y, and z. The options are A: x > y > z, B: z > y > x, C: y > x > z, and D: z > x > y. Hmm, okay, let's break this down step by step.First off, let's recall what each of these expressions means. Starting with z, which is log base a of a. I remember that log base b of b is always 1, right? Because any number to the power of 1 is itself. So, z should be 1. That seems straightforward.Now, moving on to y, which is just a. Since 0 < a < 1, y is a number between 0 and 1. So, y is less than 1.Next, x is a^a. Hmm, exponentiation where both the base and the exponent are between 0 and 1. I think when you raise a number between 0 and 1 to a power that's also between 0 and 1, the result is actually larger than the original number. For example, if a is 1/2, then (1/2)^(1/2) is sqrt(1/2), which is approximately 0.707, which is larger than 1/2. So, in this case, x should be greater than y because x = a^a and y = a, and a^a > a when 0 < a < 1.So, putting that together, z is 1, which is greater than x, which is greater than y. So, z > x > y. That would correspond to option D.Wait, let me double-check to make sure I didn't make a mistake. Let's pick a specific value for a to test this out. Let's say a = 1/2.Then, z = log_{1/2}(1/2) = 1, as expected.y = a = 1/2.x = a^a = (1/2)^(1/2) = sqrt(1/2) ≈ 0.707.So, z = 1, x ≈ 0.707, y = 0.5. So, indeed, z > x > y, which is option D.Just to try another value, let's take a = 1/4.z = log_{1/4}(1/4) = 1.y = a = 1/4 = 0.25.x = a^a = (1/4)^(1/4). Let's compute that. (1/4)^(1/4) is the fourth root of 1/4, which is approximately 0.707 as well, since (0.707)^4 ≈ 0.25.Wait, actually, (1/4)^(1/4) is equal to (4)^(-1/4) = (2^2)^(-1/4) = 2^(-1/2) = 1/sqrt(2) ≈ 0.707. So, x ≈ 0.707, y = 0.25, z = 1. So again, z > x > y.Okay, that seems consistent. Let me think if there's another way to approach this without plugging in numbers. Maybe using properties of exponents and logarithms.We know that for 0 < a < 1, the function f(x) = a^x is a decreasing function. That means as x increases, f(x) decreases. So, since a < 1, a^a will be greater than a^1, which is a. So, x = a^a > a = y.Also, z = log_a(a) = 1, as we established earlier. So, z is 1, which is greater than both x and y because x and y are both less than 1.Therefore, the order is z > x > y, which is option D.I think that's solid. I don't see any mistakes in my reasoning. The key points are understanding that log base a of a is 1, recognizing that for 0 < a < 1, a^a is greater than a, and knowing that both x and y are less than 1, so z is the largest. That makes sense.Just to recap:- z = 1- x = a^a > a = y because 0 < a < 1- Therefore, z > x > yYep, that's it. So, the correct answer should be D.**Final Answer**boxed{D}

answer：First, I need to determine the principal amount using the compound interest formula.The formula for compound interest is:[ A = P left(1 + frac{r}{n}right)^{nt} ]Given:- ( A = 5324.000000000002 ) (the amount after interest)- ( r = 10% ) per annum, which is ( 0.10 ) in decimal- ( n = 1 ) (interest is compounded annually)- ( t = 3 ) yearsPlugging the values into the formula:[ 5324.000000000002 = P left(1 + 0.10right)^3 ][ 5324.000000000002 = P times 1.331 ]To find ( P ), divide both sides by 1.331:[ P = frac{5324.000000000002}{1.331} ][ P approx 4000 ]Therefore, the principal amount is approximately Rs. 4000.

answer：First, I need to determine the original price of the article before any discounts were applied. The final price after two successive discounts is 175.The first discount is 10%, which means the article is sold for 90% of its original price after this discount. The second discount is 20%, so the article is sold for 80% of the price after the first discount.To find the original price, I can set up the equation:Original Price × 0.90 × 0.80 = 175Multiplying 0.90 and 0.80 gives 0.72. So, the equation simplifies to:Original Price × 0.72 = 175To solve for the original price, I divide both sides of the equation by 0.72:Original Price = 175 / 0.72Calculating this gives:Original Price ≈ 243.06Therefore, the original price of the article before any discounts were applied is approximately 243.06.

answer：First, calculate the total runs scored by the batsman in 46 innings by multiplying the average runs per innings by the number of innings:63 runs/inning * 46 innings = 2898 runsNext, determine the lowest score by subtracting the given difference from the highest score:248 runs - 150 runs = 98 runsSubtract the highest and lowest scores from the total runs to find the total runs scored in the remaining 44 innings:2898 runs - 248 runs - 98 runs = 2552 runsFinally, calculate the new average by dividing the total runs excluding the highest and lowest scores by the number of remaining innings:2552 runs / 44 innings = 58 runs/inning