💡Okay, so I have this problem about an ellipse, and I need to find its equation and then the area of a triangle formed by the origin and two intersection points of a line with the ellipse. Let me try to break this down step by step.First, part (1) asks for the equation of ellipse C. The standard form of an ellipse is given as (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a > b > 0). They also mention that the right focus is at (F(sqrt{3}, 0)) and the eccentricity (e = frac{sqrt{3}}{2}).I remember that for an ellipse, the distance from the center to each focus is (c), and the relationship between (a), (b), and (c) is (c^2 = a^2 - b^2). Also, the eccentricity (e) is defined as (e = frac{c}{a}).Given that the right focus is at ((sqrt{3}, 0)), this tells me that (c = sqrt{3}). The eccentricity is given as (frac{sqrt{3}}{2}), so I can set up the equation (e = frac{c}{a}) as:[frac{sqrt{3}}{2} = frac{sqrt{3}}{a}]Solving for (a), I can multiply both sides by (a) and then divide both sides by (frac{sqrt{3}}{2}):[a = sqrt{3} times frac{2}{sqrt{3}} = 2]So, (a = 2). Now, using the relationship (c^2 = a^2 - b^2), I can solve for (b):[(sqrt{3})^2 = 2^2 - b^2 3 = 4 - b^2 b^2 = 4 - 3 = 1 b = 1]Therefore, the equation of the ellipse is:[frac{x^2}{4} + y^2 = 1]Alright, that seems straightforward. Now moving on to part (2). A line (l) passes through the focus (F(sqrt{3}, 0)) and has an inclination angle of 45°, which means the slope of the line is 1. So, the equation of line (l) should be:[y = x - sqrt{3}]Wait, let me think. The line passes through (F(sqrt{3}, 0)) and has a slope of 1. Using the point-slope form:[y - 0 = 1 times (x - sqrt{3}) y = x - sqrt{3}]Yes, that's correct. Now, this line intersects the ellipse at two points M and N. I need to find these points of intersection.So, I'll substitute (y = x - sqrt{3}) into the ellipse equation:[frac{x^2}{4} + (x - sqrt{3})^2 = 1]Let me expand and simplify this equation:First, expand ((x - sqrt{3})^2):[(x - sqrt{3})^2 = x^2 - 2sqrt{3}x + 3]Now substitute back into the ellipse equation:[frac{x^2}{4} + x^2 - 2sqrt{3}x + 3 = 1]Combine like terms:[frac{x^2}{4} + x^2 = frac{5x^2}{4}]So, the equation becomes:[frac{5x^2}{4} - 2sqrt{3}x + 3 = 1]Subtract 1 from both sides:[frac{5x^2}{4} - 2sqrt{3}x + 2 = 0]To eliminate the fraction, multiply the entire equation by 4:[5x^2 - 8sqrt{3}x + 8 = 0]Now, this is a quadratic equation in terms of (x). Let me denote it as:[5x^2 - 8sqrt{3}x + 8 = 0]I can solve for (x) using the quadratic formula:[x = frac{8sqrt{3} pm sqrt{(8sqrt{3})^2 - 4 times 5 times 8}}{2 times 5}]First, compute the discriminant:[(8sqrt{3})^2 = 64 times 3 = 192 4 times 5 times 8 = 160 text{Discriminant} = 192 - 160 = 32]So,[x = frac{8sqrt{3} pm sqrt{32}}{10} sqrt{32} = 4sqrt{2} x = frac{8sqrt{3} pm 4sqrt{2}}{10} = frac{4sqrt{3} pm 2sqrt{2}}{5}]Therefore, the x-coordinates of points M and N are:[x_1 = frac{4sqrt{3} + 2sqrt{2}}{5}, quad x_2 = frac{4sqrt{3} - 2sqrt{2}}{5}]Now, let's find the corresponding y-coordinates using the equation (y = x - sqrt{3}):For (x_1):[y_1 = frac{4sqrt{3} + 2sqrt{2}}{5} - sqrt{3} = frac{4sqrt{3} + 2sqrt{2} - 5sqrt{3}}{5} = frac{-sqrt{3} + 2sqrt{2}}{5}]For (x_2):[y_2 = frac{4sqrt{3} - 2sqrt{2}}{5} - sqrt{3} = frac{4sqrt{3} - 2sqrt{2} - 5sqrt{3}}{5} = frac{-sqrt{3} - 2sqrt{2}}{5}]So, the coordinates of points M and N are:[Mleft( frac{4sqrt{3} + 2sqrt{2}}{5}, frac{-sqrt{3} + 2sqrt{2}}{5} right) Nleft( frac{4sqrt{3} - 2sqrt{2}}{5}, frac{-sqrt{3} - 2sqrt{2}}{5} right)]Now, I need to find the area of triangle OMN, where O is the origin (0,0). To find the area of a triangle given three points, I can use the determinant formula:[text{Area} = frac{1}{2} |x_1y_2 - x_2y_1|]Let me compute (x_1y_2 - x_2y_1):First, compute (x_1y_2):[x_1y_2 = left( frac{4sqrt{3} + 2sqrt{2}}{5} right) left( frac{-sqrt{3} - 2sqrt{2}}{5} right)]Multiply the numerators:[(4sqrt{3} + 2sqrt{2})(- sqrt{3} - 2sqrt{2}) = 4sqrt{3} times (-sqrt{3}) + 4sqrt{3} times (-2sqrt{2}) + 2sqrt{2} times (-sqrt{3}) + 2sqrt{2} times (-2sqrt{2}) = -4 times 3 - 8sqrt{6} - 2sqrt{6} - 4 times 2 = -12 - 8sqrt{6} - 2sqrt{6} - 8 = -20 - 10sqrt{6}]So,[x_1y_2 = frac{-20 - 10sqrt{6}}{25}]Now, compute (x_2y_1):[x_2y_1 = left( frac{4sqrt{3} - 2sqrt{2}}{5} right) left( frac{-sqrt{3} + 2sqrt{2}}{5} right)]Multiply the numerators:[(4sqrt{3} - 2sqrt{2})(- sqrt{3} + 2sqrt{2}) = 4sqrt{3} times (-sqrt{3}) + 4sqrt{3} times 2sqrt{2} - 2sqrt{2} times (-sqrt{3}) + (-2sqrt{2}) times 2sqrt{2} = -4 times 3 + 8sqrt{6} + 2sqrt{6} - 4 times 2 = -12 + 8sqrt{6} + 2sqrt{6} - 8 = -20 + 10sqrt{6}]So,[x_2y_1 = frac{-20 + 10sqrt{6}}{25}]Now, compute (x_1y_2 - x_2y_1):[frac{-20 - 10sqrt{6}}{25} - frac{-20 + 10sqrt{6}}{25} = frac{-20 - 10sqrt{6} + 20 - 10sqrt{6}}{25} = frac{-20sqrt{6}}{25}]Wait, that doesn't seem right. Let me double-check the subtraction:[frac{-20 - 10sqrt{6}}{25} - frac{-20 + 10sqrt{6}}{25} = frac{(-20 -10sqrt{6}) - (-20 +10sqrt{6})}{25} = frac{-20 -10sqrt{6} +20 -10sqrt{6}}{25} = frac{(-20 +20) + (-10sqrt{6} -10sqrt{6})}{25} = frac{0 -20sqrt{6}}{25} = frac{-20sqrt{6}}{25}]Yes, that's correct. So, the absolute value is:[|x_1y_2 - x_2y_1| = left| frac{-20sqrt{6}}{25} right| = frac{20sqrt{6}}{25} = frac{4sqrt{6}}{5}]Therefore, the area is:[text{Area} = frac{1}{2} times frac{4sqrt{6}}{5} = frac{2sqrt{6}}{5}]Hmm, that seems a bit small. Let me think if there's another way to compute the area, maybe using vectors or coordinates.Alternatively, I can use the formula for the area of a triangle with vertices at ((0,0)), ((x_1, y_1)), and ((x_2, y_2)):[text{Area} = frac{1}{2} |x_1y_2 - x_2y_1|]Which is exactly what I did. So, unless I made a mistake in calculating (x_1y_2) and (x_2y_1), the area should be correct.Wait, let me check the multiplication steps again for (x_1y_2) and (x_2y_1). It's easy to make a mistake with the signs.For (x_1y_2):[(4sqrt{3} + 2sqrt{2})(- sqrt{3} - 2sqrt{2}) = 4sqrt{3} times (-sqrt{3}) + 4sqrt{3} times (-2sqrt{2}) + 2sqrt{2} times (-sqrt{3}) + 2sqrt{2} times (-2sqrt{2}) = -12 - 8sqrt{6} - 2sqrt{6} - 8 = -20 - 10sqrt{6}]Yes, that's correct.For (x_2y_1):[(4sqrt{3} - 2sqrt{2})(- sqrt{3} + 2sqrt{2}) = 4sqrt{3} times (-sqrt{3}) + 4sqrt{3} times 2sqrt{2} - 2sqrt{2} times (-sqrt{3}) + (-2sqrt{2}) times 2sqrt{2} = -12 + 8sqrt{6} + 2sqrt{6} - 8 = -20 + 10sqrt{6}]Yes, that's also correct.So, subtracting these gives:[(-20 -10sqrt{6}) - (-20 +10sqrt{6}) = -20 -10sqrt{6} +20 -10sqrt{6} = -20sqrt{6}]Wait, no, that's not correct. Wait, no, the subtraction is:[frac{-20 -10sqrt{6}}{25} - frac{-20 +10sqrt{6}}{25} = frac{(-20 -10sqrt{6}) - (-20 +10sqrt{6})}{25} = frac{-20 -10sqrt{6} +20 -10sqrt{6}}{25} = frac{0 -20sqrt{6}}{25} = frac{-20sqrt{6}}{25}]Yes, so the absolute value is (frac{20sqrt{6}}{25}), which simplifies to (frac{4sqrt{6}}{5}). Then, half of that is (frac{2sqrt{6}}{5}).So, the area is indeed (frac{2sqrt{6}}{5}).Alternatively, another way to compute the area is by using vectors. The vectors OM and ON can be represented as (vec{OM} = (x_1, y_1)) and (vec{ON} = (x_2, y_2)). The area is half the magnitude of the cross product of these vectors, which is:[text{Area} = frac{1}{2} |x_1y_2 - x_2y_1|]Which is the same as before. So, I think my calculation is correct.Just to be thorough, maybe I can compute the distance between M and N and then compute the area using base and height. Let me try that.First, compute the distance between M and N. The coordinates are:[Mleft( frac{4sqrt{3} + 2sqrt{2}}{5}, frac{-sqrt{3} + 2sqrt{2}}{5} right) Nleft( frac{4sqrt{3} - 2sqrt{2}}{5}, frac{-sqrt{3} - 2sqrt{2}}{5} right)]Compute the differences in x and y coordinates:[Delta x = frac{4sqrt{3} + 2sqrt{2}}{5} - frac{4sqrt{3} - 2sqrt{2}}{5} = frac{4sqrt{2}}{5} Delta y = frac{-sqrt{3} + 2sqrt{2}}{5} - frac{-sqrt{3} - 2sqrt{2}}{5} = frac{4sqrt{2}}{5}]So, the distance |MN| is:[sqrt{(Delta x)^2 + (Delta y)^2} = sqrt{left( frac{4sqrt{2}}{5} right)^2 + left( frac{4sqrt{2}}{5} right)^2} = sqrt{2 times left( frac{32}{25} right)} = sqrt{frac{64}{25}} = frac{8}{5}]So, |MN| is (frac{8}{5}).Now, to find the area of triangle OMN, I can use the formula:[text{Area} = frac{1}{2} times text{base} times text{height}]Here, the base is |MN|, which is (frac{8}{5}). The height is the distance from the origin O to the line MN.First, let's find the equation of line MN. We already have it as (y = x - sqrt{3}).The distance from a point ((x_0, y_0)) to the line (Ax + By + C = 0) is:[d = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}]Rewriting the equation of line MN in standard form:[y = x - sqrt{3} implies x - y - sqrt{3} = 0]So, A = 1, B = -1, C = -sqrt{3}.The distance from O(0,0) to this line is:[d = frac{|1 times 0 + (-1) times 0 - sqrt{3}|}{sqrt{1^2 + (-1)^2}} = frac{|,-sqrt{3}|}{sqrt{2}} = frac{sqrt{3}}{sqrt{2}} = frac{sqrt{6}}{2}]Therefore, the area is:[text{Area} = frac{1}{2} times frac{8}{5} times frac{sqrt{6}}{2} = frac{1}{2} times frac{8}{5} times frac{sqrt{6}}{2} = frac{8sqrt{6}}{20} = frac{2sqrt{6}}{5}]So, this method also gives the same result, confirming that the area is indeed (frac{2sqrt{6}}{5}).I think that's thorough enough. I used two different methods to compute the area, both leading to the same answer, so I feel confident that it's correct.**Final Answer**(1) The equation of ellipse C is (boxed{dfrac{x^2}{4} + y^2 = 1}).(2) The area of triangle OMN is (boxed{dfrac{2sqrt{6}}{5}}).

💡Okay, so I have this math problem here: "Forty-two decreased by three times a number is 12. What is the number?" Hmm, let me try to understand what this is asking. Alright, first, "forty-two decreased by three times a number." So, that means I'm starting with 42, and then I'm subtracting three times some number. And the result of that is 12. So, I need to find out what that number is.Let me break it down. Let's call the unknown number "x." So, three times that number would be 3x. Now, forty-two decreased by that would be 42 minus 3x. And according to the problem, that equals 12. So, I can write that as an equation: 42 - 3x = 12.Okay, now I need to solve for x. Let me think about how to do that. I guess I need to isolate x on one side of the equation. So, first, maybe I should get rid of that 42 on the left side. To do that, I can subtract 42 from both sides of the equation. Let's see:42 - 3x = 12Subtracting 42 from both sides:42 - 3x - 42 = 12 - 42Simplifying that:-3x = -30Okay, so now I have -3x equals -30. I need to solve for x, so I should get rid of that -3 that's multiplied by x. To do that, I can divide both sides of the equation by -3. Let's try that:-3x / -3 = -30 / -3Simplifying:x = 10Wait, so x equals 10? Let me check if that makes sense. If I plug x back into the original equation:42 - 3(10) = 42 - 30 = 12Yes, that works out. So, the number is 10.But just to make sure I didn't make any mistakes, let me go through it again. Starting with the equation:42 - 3x = 12Subtract 42 from both sides:-3x = -30Divide both sides by -3:x = 10Yep, that seems correct. So, the number we're looking for is 10.

💡好，我现在要解决的问题是：找到最大的实数λ，使得对于所有根都是非负实数的三次多项式f(x) = x³ + a x² + b x + c，当x ≥ 0时，都有f(x) ≥ λ(x - a)³。同时，还要确定什么时候等号成立。首先，我需要理解题目中的条件。f(x)是一个三次多项式，且所有根都是非负实数。这意味着f(x)可以表示为(x - α)(x - β)(x - γ)，其中α、β、γ都是非负实数。然后，题目要求找到最大的λ，使得f(x) ≥ λ(x - a)³对于所有x ≥ 0成立。接下来，我需要明确a的值。根据多项式的展开，f(x) = x³ + a x² + b x + c，其中a = -(α + β + γ)，因为三次多项式展开后，x²项的系数是根的和的负数。所以，a = -(α + β + γ)。因此，x - a = x + (α + β + γ)。现在，我要比较f(x)和λ(x - a)³的大小。也就是比较(x - α)(x - β)(x - γ)和λ(x + α + β + γ)³的大小关系，对于所有x ≥ 0。为了找到最大的λ，使得f(x) ≥ λ(x - a)³对所有x ≥ 0成立，我需要分析f(x)和(x - a)³之间的关系。可能需要使用不等式，比如均值不等式，或者其他代数技巧。首先，我可以考虑当x = 0时的情况。代入x = 0，f(0) = (-α)(-β)(-γ) = -αβγ。而(x - a)³在x=0时为(-a)³ = (- (α + β + γ))³ = - (α + β + γ)³。因此，不等式变为-αβγ ≥ λ * (- (α + β + γ)³)。因为左边是负数，右边也是负数，所以两边同时乘以-1，不等式方向改变，得到αβγ ≤ λ (α + β + γ)³。因此，λ ≥ αβγ / (α + β + γ)³。为了找到最大的λ，使得这个不等式对所有非负α、β、γ成立，我需要找到αβγ / (α + β + γ)³的最大值。根据AM ≥ GM不等式，我们知道(α + β + γ)/3 ≥ (αβγ)^(1/3)，所以(α + β + γ)³ ≥ 27 αβγ，因此αβγ / (α + β + γ)³ ≤ 1/27。因此，λ ≥ 1/27。但是，这里需要注意的是，当x=0时，f(0) = -αβγ，而(x - a)³ = - (α + β + γ)³，所以不等式是-αβγ ≥ λ * (- (α + β + γ)³)，即αβγ ≤ λ (α + β + γ)³，所以λ ≥ αβγ / (α + β + γ)³。由于αβγ / (α + β + γ)³ ≤ 1/27，所以λ必须至少为1/27才能满足这个条件。但是，我需要确定的是，是否存在更大的λ，使得对于所有x ≥ 0，f(x) ≥ λ(x - a)³成立。接下来，我需要考虑x > 0的情况。可能需要分析f(x)和(x - a)³在x ≥ 0时的相对大小。也许可以考虑将f(x)和(x - a)³进行比较，找到它们的比值的最小值，从而确定λ的最大值。另外，我还需要考虑当x趋近于无穷大时的情况。当x趋近于无穷大时，f(x) ≈ x³，而(x - a)³ ≈ x³，所以它们的比值趋近于1。因此，如果λ太大，比如超过1，那么当x趋近于无穷大时，f(x) ≈ x³，而λ(x - a)³ ≈ λ x³，所以如果λ > 1，那么当x足够大时，f(x) < λ(x - a)³，这与题目要求的f(x) ≥ λ(x - a)³矛盾。因此，λ必须小于或等于1。但是，从x=0的情况来看，λ至少需要为1/27，而从x趋近于无穷大的情况，λ必须≤1。所以，我需要找到一个λ，使得在x=0时，f(x) ≥ λ(x - a)³，同时在x趋近于无穷大时，f(x) ≥ λ(x - a)³，而且在中间区域x ≥ 0时，f(x) ≥ λ(x - a)³。可能需要更深入地分析f(x)和(x - a)³之间的关系。也许可以考虑将f(x)和(x - a)³进行比较，找到它们的比值的最小值，从而确定λ的最大值。另外，我还需要考虑当f(x)和(x - a)³相等时的情况，即等号成立的条件。这可能发生在某些特定的x值和根的情况下。也许可以考虑将f(x)和(x - a)³进行比较，找到它们的比值的最小值，从而确定λ的最大值。或者，可以考虑使用导数来分析f(x)和(x - a)³的相对增长情况。另外，我还需要考虑当f(x)和(x - a)³相等时的情况，即等号成立的条件。这可能发生在某些特定的x值和根的情况下。也许可以考虑将f(x)和(x - a)³进行比较，找到它们的比值的最小值，从而确定λ的最大值。或者，可以考虑使用导数来分析f(x)和(x - a)³的相对增长情况。现在，我需要更系统地分析这个问题。首先，设f(x) = (x - α)(x - β)(x - γ)，其中α、β、γ ≥ 0。然后，a = -(α + β + γ)，所以x - a = x + α + β + γ。现在，我要比较f(x)和λ(x - a)³，即比较(x - α)(x - β)(x - γ)和λ(x + α + β + γ)³。为了找到最大的λ，使得对于所有x ≥ 0，有(x - α)(x - β)(x - γ) ≥ λ(x + α + β + γ)³。首先，考虑x=0的情况，如前所述，得到λ ≥ αβγ / (α + β + γ)³，而根据AM-GM不等式，这个比值的最大值为1/27，当且仅当α=β=γ时取得。接下来，考虑x趋近于无穷大时的情况，f(x) ≈ x³，而(x - a)³ ≈ x³，所以比值趋近于1。因此，λ必须≤1，否则当x足够大时，f(x) < λ(x - a)³，这与题目要求矛盾。但是，从x=0的情况，λ至少为1/27，而从x趋近于无穷大的情况，λ必须≤1。因此，λ的可能范围是1/27 ≤ λ ≤1。现在，我需要确定是否存在λ=1/27，使得对于所有x ≥ 0，f(x) ≥ (1/27)(x - a)³。或者，可能λ的最大值是1/27，因为当α=β=γ时，f(x) = (x - α)^3，而a = -3α，所以x - a = x + 3α。因此，f(x) = (x - α)^3，而(x - a)^3 = (x + 3α)^3。因此，比较(x - α)^3和λ(x + 3α)^3，当x=0时，左边为(-α)^3 = -α³，右边为λ*(3α)^3 = 27λ α³。因此，不等式为-α³ ≥ 27λ α³，即-1 ≥ 27λ，所以λ ≤ -1/27。这与之前的结论相反，可能哪里出错了。哦，这里可能有错误，因为当α=β=γ时，f(x) = (x - α)^3，而a = -3α，所以x - a = x + 3α。因此，f(x) = (x - α)^3，而(x - a)^3 = (x + 3α)^3。因此，比较f(x)和λ(x - a)^3，即比较(x - α)^3和λ(x + 3α)^3。当x=0时，f(0) = (-α)^3 = -α³，而(x - a)^3 = (3α)^3 = 27α³。因此，不等式为-α³ ≥ λ*27α³，即-1 ≥ 27λ，所以λ ≤ -1/27。但是，这与之前的结论相反，因为之前认为λ ≥1/27，但现在得到λ ≤ -1/27。这可能意味着我之前的分析有误，或者问题中的λ应该是负数。这可能是因为当x=0时，f(x) = -αβγ，而(x - a)^3 = (α + β + γ)^3，所以不等式为-αβγ ≥ λ*(α + β + γ)^3，即λ ≤ -αβγ / (α + β + γ)^3。由于αβγ / (α + β + γ)^3 ≤1/27，所以λ ≤ -1/27。因此，最大的λ是-1/27，因为当λ=-1/27时，对于所有x ≥0，f(x) ≥ -1/27*(x - a)^3成立，而当λ > -1/27时，可能存在某些x使得f(x) < λ*(x - a)^3。接下来，我需要验证这个结论是否正确，并确定等号成立的条件。当λ=-1/27时，不等式变为f(x) ≥ -1/27*(x - a)^3。对于所有x ≥0，这是否成立？考虑当x=0时，f(0) = -αβγ，而-1/27*(x - a)^3 = -1/27*(α + β + γ)^3。因此，不等式为-αβγ ≥ -1/27*(α + β + γ)^3，即αβγ ≤1/27*(α + β + γ)^3，这正是AM-GM不等式的结果，成立当且仅当α=β=γ。接下来，考虑x趋近于无穷大时，f(x) ≈x³，而-1/27*(x - a)^3 ≈ -1/27*x³，因此f(x) ≈x³ ≥ -1/27*x³，这对于x ≥0成立，因为x³ ≥0 ≥ -1/27*x³。现在，我需要检查在中间区域，比如x在0到某个值之间时，f(x)是否确实大于等于-1/27*(x - a)^3。考虑当x=α时，f(x)=0，而-1/27*(x - a)^3 = -1/27*(α + β + γ + α)^3 = -1/27*(2α + β + γ)^3。由于α、β、γ都是非负的，所以右边是负数，而f(x)=0 ≥负数，成立。同样地，当x=β或x=γ时，f(x)=0，而右边是负数，所以不等式成立。现在，考虑当x在0到α之间时，f(x) = (x - α)(x - β)(x - γ)。因为x ≤α ≤β ≤γ，所以x - α ≤0，x - β ≤0，x - γ ≤0，因此f(x) = (负数)(负数)(负数) = 负数。而(x - a)^3 = (x + α + β + γ)^3，因为x ≥0，所以x + α + β + γ ≥0，因此(x - a)^3 ≥0。因此，f(x) ≥ -1/27*(x - a)^3，即负数 ≥负数，需要比较它们的绝对值。因为f(x) = - (α - x)(β - x)(γ - x)，而(x - a)^3 = (x + α + β + γ)^3。因此，不等式变为- (α - x)(β - x)(γ - x) ≥ -1/27*(x + α + β + γ)^3，即(α - x)(β - x)(γ - x) ≤1/27*(x + α + β + γ)^3。根据AM-GM不等式，对于非负数，(α - x) + (β - x) + (γ - x) = α + β + γ - 3x，而(x + α + β + γ) = (α + β + γ) + x。因此，(α - x)(β - x)(γ - x) ≤ [(α + β + γ - 3x)/3]^3。而(x + α + β + γ) = (α + β + γ) + x，所以[(α + β + γ - 3x)/3]^3 ≤ [(α + β + γ + x)/3]^3，因为当x ≤ (α + β + γ)/3时，α + β + γ - 3x ≥0，而当x > (α + β + γ)/3时，左边可能为负数，但此时右边也为负数，可能需要更仔细的分析。不过，根据之前的分析，当x=0时，(α - x)(β - x)(γ - x) = αβγ ≤1/27*(α + β + γ)^3，成立。而当x增加时，左边(α - x)(β - x)(γ - x)会减小，而右边[(x + α + β + γ)/3]^3会增加，因此不等式可能成立。综上所述，最大的λ是-1/27，当且仅当α=β=γ时，等号在x=0时成立；或者当α=β=0，γ=2x时，等号在某些x值成立。不过，可能需要更准确地确定等号成立的条件。当α=β=γ时，f(x)=(x - α)^3，而a=-3α，所以x - a =x +3α。因此，f(x)=(x - α)^3，而-1/27*(x - a)^3 = -1/27*(x +3α)^3。当x=0时，f(0)=(-α)^3=-α³，而-1/27*(3α)^3=-1/27*27α³=-α³，所以等号成立。另外，当α=β=0，γ=2x时，f(x)=x(x)(x - 2x)=x²(-x)=-x³，而a=-(0+0+2x)=-2x，所以x - a =x +2x=3x，因此-1/27*(3x)^3=-1/27*27x³=-x³，所以f(x)=-x³= -1/27*(3x)^3，因此等号成立。因此，最大的λ是-1/27，当且仅当f(x)的所有根相等，即α=β=γ，或者当两个根为0，第三个根为2x时，等号成立。

💡To determine the probability that the product of Sophie's and Max's rolls is a multiple of 4, we'll consider the different scenarios where this condition is met.First, let's identify the numbers on each die that are multiples of 4:- Sophie's twelve-sided die has multiples of 4 at 4, 8, and 12.- Max's eight-sided die has multiples of 4 at 4 and 8.Next, we'll calculate the probabilities for each scenario:1. **Sophie rolls a multiple of 4:** There are 3 favorable outcomes out of 12 possible outcomes, so the probability is (frac{3}{12} = frac{1}{4}).2. **Sophie does not roll a multiple of 4 but rolls an even number (2, 6, 10):** There are 3 favorable outcomes out of 12 possible outcomes, so the probability is (frac{3}{12} = frac{1}{4}). In this case, Max must roll a multiple of 4, which has a probability of (frac{2}{8} = frac{1}{4}).Finally, we'll combine these probabilities:- The probability that both Sophie and Max roll multiples of 4 is (frac{1}{4} times frac{1}{4} = frac{1}{16}).- The probability that Sophie does not roll a multiple of 4 but Max does is (frac{1}{4} times frac{1}{4} = frac{1}{16}).Adding these probabilities together gives the total probability:[frac{1}{16} + frac{1}{16} = frac{2}{16} = frac{1}{8}]

💡First, I need to determine the increase in the annual interest rate from last year to this year. The interest rate increased by 10%, and last year's rate was 9.90990990990991%.To find 10% of last year's interest rate, I multiply 9.90990990990991% by 0.10. This gives me the amount of the increase.Next, I add this increase to last year's interest rate to find this year's interest rate. By adding 0.990990990990991% to 9.90990990990991%, I obtain the new annual interest rate for this year.Finally, I present the calculated interest rate as the solution.

💡First, I need to understand the problem. There are two objects: a flagstaff and a building. Both cast shadows under similar conditions. The building's height and shadow length are given, and I need to find the height of the flagstaff based on its shadow length.I recall that when two objects cast shadows under the same light source, the triangles formed by the objects and their shadows are similar. This means the ratios of their corresponding sides are equal.Let’s denote:- ( h ) as the height of the flagstaff- ( s_1 = 40.25 ) meters as the length of the flagstaff's shadow- ( h_2 = 12.5 ) meters as the height of the building- ( s_2 = 28.75 ) meters as the length of the building's shadowUsing the similarity of triangles, the ratio of the height of the flagstaff to the length of its shadow should be equal to the ratio of the height of the building to the length of its shadow:[frac{h}{s_1} = frac{h_2}{s_2}]Substituting the known values:[frac{h}{40.25} = frac{12.5}{28.75}]To find ( h ), I can rearrange the equation:[h = frac{12.5 times 40.25}{28.75}]Calculating the numerator:[12.5 times 40.25 = 503.125]Now, divide by the denominator:[h = frac{503.125}{28.75} = 17.5]Therefore, the height of the flagstaff is 17.5 meters.