answer：Okay, so I have this problem here where I'm given that a and b are both positive numbers, and their sum is 1. I need to figure out which of the four inequalities (A, B, C, D) hold true under these conditions. Let me take it step by step.First, let me write down what's given:- a > 0- b > 0- a + b = 1And the inequalities to check are:A: ab ≤ 1/4B: 1/a + 1/b ≤ 4C: √a + √b ≤ √2D: a² + b² ≥ 1/2Alright, let's tackle each one by one.**Option A: ab ≤ 1/4**Hmm, I remember something about the arithmetic mean and geometric mean inequality (AM-GM). It says that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. So, for a and b, the arithmetic mean is (a + b)/2, and the geometric mean is √(ab). Given that a + b = 1, the arithmetic mean is 1/2. So, according to AM-GM:1/2 ≥ √(ab)If I square both sides to get rid of the square root:(1/2)² ≥ abWhich simplifies to:1/4 ≥ abSo, ab ≤ 1/4. That means option A is correct. Cool, that wasn't too bad.**Option B: 1/a + 1/b ≤ 4**Alright, let's see. I need to find if 1/a + 1/b is less than or equal to 4. Let me express this in terms of a and b. Since a + b = 1, maybe I can find a relationship between 1/a and 1/b.First, let's write 1/a + 1/b as (b + a)/(ab). Since a + b = 1, this becomes 1/(ab).So, 1/a + 1/b = 1/(ab)From option A, we know that ab ≤ 1/4. So, 1/(ab) ≥ 4 because if ab is small, its reciprocal is large.Wait, that means 1/(ab) ≥ 4, which implies that 1/a + 1/b ≥ 4. But the option says 1/a + 1/b ≤ 4. That seems contradictory.Let me test with specific values to confirm. Suppose a = b = 1/2. Then ab = 1/4, and 1/a + 1/b = 2 + 2 = 4. So, in this case, it's equal to 4.What if a is very small, say a approaches 0, then b approaches 1. Then 1/a approaches infinity, and 1/b approaches 1. So, 1/a + 1/b would be very large, definitely greater than 4. So, the inequality 1/a + 1/b ≤ 4 doesn't hold in general. It only holds when a and b are both 1/2.Therefore, option B is not always true. So, B is incorrect.**Option C: √a + √b ≤ √2**Hmm, let's think about this. I need to see if the sum of the square roots of a and b is less than or equal to √2.Maybe I can square both sides to make it easier. Let's square √a + √b:(√a + √b)² = a + 2√(ab) + bSince a + b = 1, this simplifies to:1 + 2√(ab)So, (√a + √b)² = 1 + 2√(ab)We need to see if this is less than or equal to (√2)², which is 2.So, 1 + 2√(ab) ≤ 2Subtract 1 from both sides:2√(ab) ≤ 1Divide both sides by 2:√(ab) ≤ 1/2Square both sides:ab ≤ 1/4Wait, that's exactly what we found in option A. So, since ab ≤ 1/4, then √(ab) ≤ 1/2, which implies that 2√(ab) ≤ 1, and thus (√a + √b)² ≤ 2. Taking square roots on both sides, we get √a + √b ≤ √2.Therefore, option C is correct.**Option D: a² + b² ≥ 1/2**Alright, let's see. I need to check if the sum of the squares of a and b is at least 1/2.I remember that (a + b)² = a² + 2ab + b². Since a + b = 1, this becomes:1 = a² + 2ab + b²So, a² + b² = 1 - 2abWe need to find if a² + b² ≥ 1/2. Substituting the expression we have:1 - 2ab ≥ 1/2Subtract 1 from both sides:-2ab ≥ -1/2Multiply both sides by -1 (remembering to reverse the inequality):2ab ≤ 1/2Divide both sides by 2:ab ≤ 1/4Again, this is exactly what we found in option A. So, since ab ≤ 1/4, then 2ab ≤ 1/2, which implies that 1 - 2ab ≥ 1/2. Therefore, a² + b² ≥ 1/2.So, option D is correct.**Summary:**- A: Correct- B: Incorrect- C: Correct- D: CorrectTherefore, the correct options are A, C, and D.

answer：Alright, so I've got this geometry problem here about a rhombus with an inscribed circle. The center of the circle is labeled as O. Points E, F, G, and H are on the sides AB, BC, CD, and DA respectively, and it's given that EF and GH are tangent to the inscribed circle. I need to show that EH and FG are parallel.Hmm, okay. Let me start by recalling some properties of a rhombus. A rhombus has all sides equal, and its diagonals bisect each other at right angles. Also, the center of the inscribed circle in a rhombus is the intersection point of its diagonals. So, O is where the diagonals meet, and it's equidistant from all sides.Since EF and GH are tangent to the inscribed circle, I know that the distance from O to each of these lines must be equal to the radius of the circle. That might be useful later.Let me try to visualize this. I imagine a rhombus ABCD with AB, BC, CD, and DA as sides. Points E on AB, F on BC, G on CD, and H on DA. EF and GH are tangent to the incircle. I need to show that EH and FG are parallel.Maybe I can use coordinate geometry to model this. Let me assign coordinates to the rhombus. Let's place the rhombus such that its center O is at the origin (0,0). Since the diagonals bisect each other at right angles, I can let the diagonals be along the x-axis and y-axis.Let’s denote the lengths of the diagonals as 2p and 2q, so the vertices of the rhombus can be at (p,0), (0,q), (-p,0), and (0,-q). So, the sides of the rhombus can be determined using these points.Wait, actually, in a rhombus, all sides are equal, so the distance between (p,0) and (0,q) should be equal to the distance between (0,q) and (-p,0), and so on. Let me calculate the side length.The distance between (p,0) and (0,q) is sqrt((p)^2 + (q)^2). So, all sides are of length sqrt(p² + q²). That makes sense.Now, the inscribed circle in a rhombus touches all four sides. The radius r of the inscribed circle can be calculated as the area divided by the perimeter. The area of the rhombus is (2p * 2q)/2 = 2pq. The perimeter is 4 * sqrt(p² + q²). So, the radius r = (2pq) / (4 * sqrt(p² + q²)) = (pq) / (2 * sqrt(p² + q²)).Okay, so the radius is known in terms of p and q.Now, points E, F, G, H are on AB, BC, CD, DA respectively, such that EF and GH are tangent to the incircle. I need to find the coordinates of E, F, G, H such that EF and GH are tangent to the circle centered at O with radius r.Let me parameterize the points. Let’s denote E on AB. Since AB goes from (p,0) to (0,q), I can write E as (p(1 - t), qt) for some t between 0 and 1. Similarly, F on BC can be parameterized. BC goes from (0,q) to (-p,0), so F can be written as (-pt, q(1 - t)) for the same t? Wait, maybe not. Maybe each point has its own parameter. Let me think.Alternatively, since EF is tangent to the circle, the distance from O to EF must be equal to r. Similarly for GH. So, if I can write the equations of lines EF and GH, set their distance from O equal to r, and solve for the coordinates of E, F, G, H.But this might get complicated. Maybe there's a better approach using symmetry or similar triangles.Let me consider the properties of tangents. If EF is tangent to the circle, then the line EF is at a distance r from O. Similarly, GH is also at a distance r from O.Since the rhombus is symmetric, perhaps the lines EF and GH are symmetric with respect to the center O. That might imply that EH and FG are also symmetric, leading to them being parallel.Wait, but I need to be more precise. Let me think about the slopes of these lines.Suppose I can find the slopes of EF and GH, and then find the slopes of EH and FG, and show that they are equal.Alternatively, maybe I can use vectors or coordinate geometry to express the lines and then compute their slopes.Let me try to assign coordinates as I initially thought. Let’s set O at (0,0), with the rhombus vertices at (a,0), (0,b), (-a,0), and (0,-b). So, sides AB is from (a,0) to (0,b), BC from (0,b) to (-a,0), CD from (-a,0) to (0,-b), and DA from (0,-b) to (a,0).Now, the inscribed circle has radius r = (ab)/sqrt(a² + b²). Wait, earlier I had p and q, but let's stick with a and b for simplicity.So, the equation of the incircle is x² + y² = r², where r = (ab)/sqrt(a² + b²).Now, let me parameterize point E on AB. AB goes from (a,0) to (0,b). So, any point on AB can be written as (a - ta, tb) where t is between 0 and 1. Similarly, point F on BC can be parameterized as (-ta, b - tb), point G on CD as (-a + ta, -tb), and point H on DA as (ta, -b + tb). Wait, is that correct?Wait, actually, for BC, which goes from (0,b) to (-a,0), a parameterization would be (0 - ta, b - tb) = (-ta, b - tb). Similarly, CD goes from (-a,0) to (0,-b), so a point on CD can be (-a + ta, 0 - tb) = (-a + ta, -tb). DA goes from (0,-b) to (a,0), so a point on DA can be (0 + ta, -b + tb) = (ta, -b + tb).So, points E, F, G, H can be written as:E = (a - ta, tb)F = (-ta, b - tb)G = (-a + ta, -tb)H = (ta, -b + tb)Now, EF is the line connecting E and F. Let me find the equation of line EF.First, find the coordinates:E = (a - ta, tb) = (a(1 - t), tb)F = (-ta, b - tb) = (-ta, b(1 - t))So, the line EF goes from (a(1 - t), tb) to (-ta, b(1 - t)).Let me compute the slope of EF.Slope m_EF = [b(1 - t) - tb] / [-ta - a(1 - t)] = [b - bt - tb] / [-ta - a + at] = [b - 2bt] / [-a + 0] = (b - 2bt)/(-a) = (2bt - b)/a = b(2t - 1)/aSo, the slope of EF is b(2t - 1)/a.Similarly, let me compute the equation of line EF.Using point E: y - tb = [b(2t - 1)/a](x - a(1 - t))Simplify:y = [b(2t - 1)/a]x - [b(2t - 1)/a] * a(1 - t) + tbSimplify the second term:= [b(2t - 1)/a]x - b(2t - 1)(1 - t) + tbNow, let me compute the distance from O(0,0) to line EF. The distance should be equal to r.The formula for the distance from a point (x0,y0) to the line Ax + By + C = 0 is |Ax0 + By0 + C| / sqrt(A² + B²).First, let me write the equation of EF in standard form.From above:y = [b(2t - 1)/a]x - b(2t - 1)(1 - t) + tbBring all terms to one side:y - [b(2t - 1)/a]x + b(2t - 1)(1 - t) - tb = 0So, the standard form is:-[b(2t - 1)/a]x + y + [b(2t - 1)(1 - t) - tb] = 0Let me denote A = -b(2t - 1)/a, B = 1, C = b(2t - 1)(1 - t) - tbThen, the distance from O(0,0) to EF is |A*0 + B*0 + C| / sqrt(A² + B²) = |C| / sqrt(A² + B²)This distance should be equal to r = ab / sqrt(a² + b²)So,|C| / sqrt(A² + B²) = ab / sqrt(a² + b²)Compute C:C = b(2t - 1)(1 - t) - tb = b[(2t - 1)(1 - t) - t]Expand (2t - 1)(1 - t):= 2t(1 - t) - 1(1 - t) = 2t - 2t² - 1 + t = 3t - 2t² - 1So,C = b[3t - 2t² - 1 - t] = b[2t - 2t² - 1] = b(-2t² + 2t - 1)So, |C| = |b(-2t² + 2t - 1)| = |b(2t² - 2t + 1)| since the quadratic inside is always positive? Let me check the discriminant: 4 - 8 = -4 < 0, so yes, it's always positive. So, |C| = b(2t² - 2t + 1)Now, compute sqrt(A² + B²):A = -b(2t - 1)/a, so A² = b²(2t - 1)² / a²B = 1, so B² = 1Thus,sqrt(A² + B²) = sqrt(b²(2t - 1)² / a² + 1) = sqrt[(b²(4t² - 4t + 1) + a²) / a²] = sqrt[(4b²t² - 4b²t + b² + a²) / a²] = sqrt(4b²t² - 4b²t + b² + a²) / aSo, putting it all together:|C| / sqrt(A² + B²) = [b(2t² - 2t + 1)] / [sqrt(4b²t² - 4b²t + b² + a²) / a] = [ab(2t² - 2t + 1)] / sqrt(4b²t² - 4b²t + b² + a²)Set this equal to r = ab / sqrt(a² + b²):[ab(2t² - 2t + 1)] / sqrt(4b²t² - 4b²t + b² + a²) = ab / sqrt(a² + b²)Cancel ab from both sides:(2t² - 2t + 1) / sqrt(4b²t² - 4b²t + b² + a²) = 1 / sqrt(a² + b²)Cross-multiplying:(2t² - 2t + 1) * sqrt(a² + b²) = sqrt(4b²t² - 4b²t + b² + a²)Square both sides to eliminate the square roots:(2t² - 2t + 1)² (a² + b²) = 4b²t² - 4b²t + b² + a²Let me expand the left side:(4t⁴ - 8t³ + 8t² - 4t + 1)(a² + b²) = 4b²t² - 4b²t + b² + a²This seems complicated. Maybe there's a better way.Alternatively, perhaps I can use the fact that EF and GH are both tangent to the circle, so they must be symmetric in some way. Maybe the parameters for E and F are related to G and H in a symmetric manner.Wait, in the parameterization I used earlier, E and F are determined by the same parameter t, but maybe that's not necessarily the case. Perhaps E and F are determined by different parameters, but EF is tangent to the circle, which imposes a condition on t.This is getting too algebraic and messy. Maybe I should try a different approach.Let me consider the properties of the rhombus and the incircle. Since the rhombus is tangential, all sides are tangent to the circle. The points of tangency divide the sides into segments whose lengths are equal. So, from each vertex, the two adjacent sides have segments of equal length.But in this problem, EF and GH are also tangent to the circle. So, EF and GH are additional tangent lines inside the rhombus.Maybe I can use the fact that the lengths from E to the point of tangency on AB and from F to the point of tangency on BC are equal, and similarly for G and H.Wait, in a tangential quadrilateral, the lengths from a vertex to the points of tangency are equal. But here, EF is a tangent, so the lengths from E to the point of tangency on EF and from F to the point of tangency on EF are equal.Hmm, not sure if that helps directly.Alternatively, maybe I can use homothety. Since EF and GH are both tangent to the incircle, perhaps there's a homothety that maps EF to GH, and this homothety would also map EH to FG, implying they are parallel.But I'm not too familiar with homothety in this context.Wait, another idea: in a rhombus, the sides are all equal and opposite sides are parallel. If I can show that the slopes of EH and FG are equal, then they must be parallel.So, maybe I can compute the slopes of EH and FG in terms of the coordinates of E, F, G, H and show that they are equal.But earlier, when I tried to parameterize E, F, G, H, it got too complicated. Maybe I need a different parameterization.Alternatively, perhaps I can use vectors. Let me denote vectors for the sides and points.Let me consider the rhombus as a parallelogram with vectors **a** and **b**. Then, the sides are vectors **a** and **b**, and the center O is at the midpoint.But I'm not sure if this approach will simplify things.Wait, another thought: since EF and GH are both tangent to the incircle, and the incircle is tangent to all four sides of the rhombus, maybe EF and GH are midlines or something similar.But midlines in a rhombus are the lines connecting midpoints of sides, which are parallel to the diagonals. But EF and GH are tangent to the incircle, so they might not be midlines.Alternatively, maybe EF and GH are at a constant distance from the center O, equal to the radius r.So, both EF and GH are lines at distance r from O, which is the center.In a rhombus, the sides are all at distance r from O, but EF and GH are additional lines inside the rhombus, also at distance r from O.So, EF and GH are two lines inside the rhombus, both tangent to the incircle, and intersecting the sides AB, BC, CD, DA at E, F, G, H.I need to show that EH and FG are parallel.Maybe I can use the fact that in a rhombus, the opposite sides are parallel, and the lines EF and GH are symmetric with respect to the center O.Wait, if EF is tangent to the circle, then GH must be another tangent such that the figure remains symmetric. So, perhaps EH and FG are symmetric with respect to O, leading them to be parallel.But I need a more concrete argument.Alternatively, maybe I can use the concept of harmonic division or projective geometry, but that might be overcomplicating.Wait, perhaps I can use similar triangles.Let me consider triangles formed by the center O and the points E, F, G, H.Since EF is tangent to the circle at some point, say P, then OP is perpendicular to EF. Similarly, GH is tangent at some point Q, so OQ is perpendicular to GH.Given that, OP and OQ are both radii perpendicular to EF and GH respectively.Since EF and GH are both tangent to the circle, and the rhombus is symmetric, maybe OP and OQ are related in a way that makes EH and FG parallel.Alternatively, since OP and OQ are both radii, and EF and GH are both tangent lines, the angles between OP and OE, and OQ and OG might be equal, leading to similar triangles.Wait, maybe I can consider the angles formed at O by the lines OE, OF, OG, OH.Since EF is tangent to the circle, the angle between OE and OP is equal to the angle between OF and OP, because OP is the radius perpendicular to EF.Similarly, for GH tangent at Q, the angle between OG and OQ is equal to the angle between OH and OQ.But I'm not sure how this helps with EH and FG.Wait, perhaps I can consider the cyclic quadrilateral formed by E, F, G, H. If I can show that EH and FG are sides of a parallelogram, then they must be parallel.But I don't know if E, F, G, H form a parallelogram.Alternatively, maybe I can use the fact that in a rhombus, the diagonals bisect the angles, and since EF and GH are tangent to the incircle, they create equal angles with the sides, leading to EH and FG being parallel.This is getting a bit abstract. Maybe I need to go back to coordinate geometry but simplify the parameterization.Let me try to parameterize E and F such that EF is tangent to the circle.Given the incircle equation x² + y² = r², and EF is a tangent line. The condition for a line to be tangent to a circle is that the distance from the center to the line equals the radius.So, if I can write the equation of EF in terms of E and F, then set its distance from O equal to r, I can find a relationship between E and F.But this seems similar to what I did earlier, which got too complicated.Wait, maybe I can use the fact that in a rhombus, the sides are symmetric with respect to the center. So, if EF is tangent to the circle, then the reflection of EF over the center O would be GH, since GH is also tangent. Therefore, GH is the reflection of EF over O.If that's the case, then the lines EH and FG would be reflections of each other over O, implying they are parallel.Wait, that sounds promising. Let me elaborate.If EF is tangent to the circle, then its reflection over O, which is the center, would also be tangent to the circle because the circle is symmetric about O. So, GH is the reflection of EF over O.Therefore, the points G and H are the reflections of E and F over O.So, if E is reflected over O to G, and F is reflected over O to H, then the lines EH and FG are reflections of each other over O.But reflections over a point preserve parallelism. So, if EH is reflected to FG, then EH must be parallel to FG.Therefore, EH and FG are parallel.Wait, that seems like a solid argument. Let me check if the reflection of EF over O is indeed GH.Since EF is tangent to the circle, reflecting it over O would map the circle to itself (since it's centered at O), so the image of EF under reflection would also be tangent to the circle. Therefore, GH must be the reflection of EF over O.Hence, G is the reflection of E, and H is the reflection of F.Therefore, the line EH is the reflection of the line FG over O, which implies that EH is parallel to FG.Yes, that makes sense. So, by symmetry and reflection properties, EH and FG must be parallel.I think this is a more straightforward approach than getting bogged down in coordinate algebra. Using the reflection symmetry of the rhombus and the incircle, we can deduce the parallelism of EH and FG.

answer：Alright, so I've got this sequence problem here. The sequence is defined by the equation ( a_{n+1} + (-1)^n a_n = 3n - 1 ) for ( n in mathbb{N}^* ). I need to find the sum of the first 40 terms of this sequence. The options given are A: 860, B: 1240, C: 1830, and D: 2420. Hmm, okay, let's break this down step by step.First, let me understand the recurrence relation. It says that each term ( a_{n+1} ) is related to the previous term ( a_n ) with an alternating sign because of the ( (-1)^n ) factor. So, depending on whether ( n ) is odd or even, the sign of ( a_n ) changes in the equation. That might mean the behavior of the sequence alternates in some way between odd and even terms.Let me write out the first few terms to see if I can spot a pattern.For ( n = 1 ):( a_2 + (-1)^1 a_1 = 3(1) - 1 )Simplify:( a_2 - a_1 = 2 )So, ( a_2 = a_1 + 2 )For ( n = 2 ):( a_3 + (-1)^2 a_2 = 3(2) - 1 )Simplify:( a_3 + a_2 = 5 )So, ( a_3 = 5 - a_2 )But since ( a_2 = a_1 + 2 ), substitute:( a_3 = 5 - (a_1 + 2) = 3 - a_1 )For ( n = 3 ):( a_4 + (-1)^3 a_3 = 3(3) - 1 )Simplify:( a_4 - a_3 = 8 )So, ( a_4 = a_3 + 8 )But ( a_3 = 3 - a_1 ), so:( a_4 = (3 - a_1) + 8 = 11 - a_1 )For ( n = 4 ):( a_5 + (-1)^4 a_4 = 3(4) - 1 )Simplify:( a_5 + a_4 = 11 )So, ( a_5 = 11 - a_4 )Substituting ( a_4 = 11 - a_1 ):( a_5 = 11 - (11 - a_1) = a_1 )Wait, that's interesting. ( a_5 = a_1 ). So, the fifth term is equal to the first term. Let's see what happens next.For ( n = 5 ):( a_6 + (-1)^5 a_5 = 3(5) - 1 )Simplify:( a_6 - a_5 = 14 )So, ( a_6 = a_5 + 14 )But ( a_5 = a_1 ), so:( a_6 = a_1 + 14 )For ( n = 6 ):( a_7 + (-1)^6 a_6 = 3(6) - 1 )Simplify:( a_7 + a_6 = 17 )So, ( a_7 = 17 - a_6 )Substituting ( a_6 = a_1 + 14 ):( a_7 = 17 - (a_1 + 14) = 3 - a_1 )Hmm, ( a_7 = 3 - a_1 ). Wait, that's the same as ( a_3 ). So, ( a_7 = a_3 ). Interesting, so the sequence seems to have a periodicity or some repeating pattern every four terms? Let me check the next few terms.For ( n = 7 ):( a_8 + (-1)^7 a_7 = 3(7) - 1 )Simplify:( a_8 - a_7 = 20 )So, ( a_8 = a_7 + 20 )But ( a_7 = 3 - a_1 ), so:( a_8 = (3 - a_1) + 20 = 23 - a_1 )For ( n = 8 ):( a_9 + (-1)^8 a_8 = 3(8) - 1 )Simplify:( a_9 + a_8 = 23 )So, ( a_9 = 23 - a_8 )Substituting ( a_8 = 23 - a_1 ):( a_9 = 23 - (23 - a_1) = a_1 )Wait, so ( a_9 = a_1 ). So, every four terms, the sequence repeats? Let me list out the terms I have so far:- ( a_1 = a_1 )- ( a_2 = a_1 + 2 )- ( a_3 = 3 - a_1 )- ( a_4 = 11 - a_1 )- ( a_5 = a_1 )- ( a_6 = a_1 + 14 )- ( a_7 = 3 - a_1 )- ( a_8 = 23 - a_1 )- ( a_9 = a_1 )So, it seems that every four terms, the sequence cycles back to ( a_1 ). So, the pattern is:( a_1, a_1 + 2, 3 - a_1, 11 - a_1, a_1, a_1 + 14, 3 - a_1, 23 - a_1, a_1, ldots )So, every four terms, the sequence repeats with an added constant. Let me see the differences between each cycle.From ( a_1 ) to ( a_5 ), the terms are:( a_1, a_1 + 2, 3 - a_1, 11 - a_1 )Then, ( a_5 = a_1 ), so the next four terms are:( a_1, a_1 + 14, 3 - a_1, 23 - a_1 )Wait, so the second term in the first cycle is ( a_1 + 2 ), and in the second cycle, it's ( a_1 + 14 ). So, the difference between the second terms is 12. Similarly, the fourth term in the first cycle is ( 11 - a_1 ), and in the second cycle, it's ( 23 - a_1 ). The difference is 12 as well.So, it seems that every four terms, the sequence increases by 12 in some way. Let me check the next cycle.For ( n = 9 ):( a_{10} + (-1)^9 a_9 = 3(9) - 1 )Simplify:( a_{10} - a_9 = 26 )So, ( a_{10} = a_9 + 26 )But ( a_9 = a_1 ), so:( a_{10} = a_1 + 26 )For ( n = 10 ):( a_{11} + (-1)^{10} a_{10} = 3(10) - 1 )Simplify:( a_{11} + a_{10} = 29 )So, ( a_{11} = 29 - a_{10} )Substituting ( a_{10} = a_1 + 26 ):( a_{11} = 29 - (a_1 + 26) = 3 - a_1 )So, ( a_{11} = 3 - a_1 ), same as ( a_3 ) and ( a_7 ). Then, for ( n = 11 ):( a_{12} + (-1)^{11} a_{11} = 3(11) - 1 )Simplify:( a_{12} - a_{11} = 32 )So, ( a_{12} = a_{11} + 32 )But ( a_{11} = 3 - a_1 ), so:( a_{12} = (3 - a_1) + 32 = 35 - a_1 )For ( n = 12 ):( a_{13} + (-1)^{12} a_{12} = 3(12) - 1 )Simplify:( a_{13} + a_{12} = 35 )So, ( a_{13} = 35 - a_{12} )Substituting ( a_{12} = 35 - a_1 ):( a_{13} = 35 - (35 - a_1) = a_1 )So, ( a_{13} = a_1 ). So, the pattern continues every four terms, with each cycle adding 12 to the second and fourth terms.So, summarizing the pattern:- Every four terms, the sequence cycles back to ( a_1 ).- The second term increases by 12 each cycle.- The fourth term increases by 12 each cycle.So, the first four terms are:1. ( a_1 )2. ( a_1 + 2 )3. ( 3 - a_1 )4. ( 11 - a_1 )The next four terms (5-8):5. ( a_1 )6. ( a_1 + 14 ) (which is 2 + 12)7. ( 3 - a_1 )8. ( 23 - a_1 ) (which is 11 + 12)Then terms 9-12:9. ( a_1 )10. ( a_1 + 26 ) (14 + 12)11. ( 3 - a_1 )12. ( 35 - a_1 ) (23 + 12)And so on.So, each cycle of four terms adds 12 to the second and fourth terms.Now, since we need the sum of the first 40 terms, which is 10 cycles of four terms each.Let me calculate the sum of one cycle (four terms) and then multiply by 10.Sum of first four terms:( a_1 + (a_1 + 2) + (3 - a_1) + (11 - a_1) )Simplify:( a_1 + a_1 + 2 + 3 - a_1 + 11 - a_1 )Combine like terms:( (a_1 + a_1 - a_1 - a_1) + (2 + 3 + 11) )Simplify:( 0 + 16 = 16 )Wait, the sum of the first four terms is 16? Let me check that again.Wait, ( a_1 + (a_1 + 2) + (3 - a_1) + (11 - a_1) )Let me compute term by term:- ( a_1 )- ( a_1 + 2 )- ( 3 - a_1 )- ( 11 - a_1 )Adding them up:( a_1 + (a_1 + 2) + (3 - a_1) + (11 - a_1) )= ( a_1 + a_1 + 2 + 3 - a_1 + 11 - a_1 )= ( (a_1 + a_1 - a_1 - a_1) + (2 + 3 + 11) )= ( 0 + 16 )= 16Yes, that's correct. So, each cycle of four terms sums to 16.But wait, let me check the next cycle to see if the sum is the same.Sum of terms 5-8:( a_1 + (a_1 + 14) + (3 - a_1) + (23 - a_1) )Simplify:( a_1 + a_1 + 14 + 3 - a_1 + 23 - a_1 )= ( (a_1 + a_1 - a_1 - a_1) + (14 + 3 + 23) )= ( 0 + 40 )= 40Wait, that's different. So, the sum of the first four terms is 16, the next four terms sum to 40, which is 16 + 24.Wait, 16, 40, so the difference is 24. Hmm, so each subsequent cycle's sum increases by 24?Wait, let me check the third cycle.Sum of terms 9-12:( a_1 + (a_1 + 26) + (3 - a_1) + (35 - a_1) )Simplify:( a_1 + a_1 + 26 + 3 - a_1 + 35 - a_1 )= ( (a_1 + a_1 - a_1 - a_1) + (26 + 3 + 35) )= ( 0 + 64 )= 64So, the sums of the cycles are 16, 40, 64, etc. So, each cycle's sum increases by 24.So, the first cycle (terms 1-4): 16Second cycle (terms 5-8): 40 (16 + 24)Third cycle (terms 9-12): 64 (40 + 24)Fourth cycle (terms 13-16): 88 (64 + 24)And so on.So, the sum of each cycle forms an arithmetic sequence with the first term 16 and common difference 24.Since we have 10 cycles (40 terms), the sum of all 40 terms is the sum of the first 10 terms of this arithmetic sequence.The formula for the sum of the first ( n ) terms of an arithmetic sequence is:( S_n = frac{n}{2} [2a_1 + (n - 1)d] )Where:- ( S_n ) is the sum of the first ( n ) terms- ( a_1 ) is the first term- ( d ) is the common difference- ( n ) is the number of termsIn our case:- ( a_1 = 16 )- ( d = 24 )- ( n = 10 )Plugging into the formula:( S_{10} = frac{10}{2} [2(16) + (10 - 1)(24)] )Simplify:( S_{10} = 5 [32 + 9(24)] )Calculate ( 9 times 24 ):( 9 times 24 = 216 )So,( S_{10} = 5 [32 + 216] = 5 [248] = 1240 )So, the sum of the first 40 terms is 1240, which corresponds to option B.Wait, let me just double-check to make sure I didn't make a mistake in the cycle sums.First cycle: 16Second cycle: 40 (difference of 24)Third cycle: 64 (difference of 24)Fourth cycle: 88Fifth cycle: 112Sixth cycle: 136Seventh cycle: 160Eighth cycle: 184Ninth cycle: 208Tenth cycle: 232Wait, hold on, if the first cycle is 16, then the second is 40, which is 16 + 24, third is 64, which is 40 + 24, and so on. So, the tenth cycle should be 16 + 9*24 = 16 + 216 = 232. So, the sum of the first 10 cycles is the sum from 16 to 232 with a common difference of 24.Wait, but earlier I used the formula for the sum of an arithmetic sequence with 10 terms, starting at 16, difference 24, which gave me 1240. Let me verify that.Alternatively, I can compute the sum as:Sum = 16 + 40 + 64 + 88 + 112 + 136 + 160 + 184 + 208 + 232Let me add them step by step:16 + 40 = 5656 + 64 = 120120 + 88 = 208208 + 112 = 320320 + 136 = 456456 + 160 = 616616 + 184 = 800800 + 208 = 10081008 + 232 = 1240Yes, that adds up to 1240. So, that's correct.Alternatively, using the formula:( S_{10} = frac{10}{2} [2(16) + (10 - 1)(24)] = 5 [32 + 216] = 5 [248] = 1240 )So, that's consistent.Therefore, the sum of the first 40 terms is 1240, which is option B.**Final Answer**boxed{B}

answer：Okay, so I need to factor the polynomial ( x^4 - 81 ) as far as possible, and the factors should be monic polynomials with real coefficients. Hmm, let me think about how to approach this.First, I remember that ( x^4 ) is a perfect square because it's ( (x^2)^2 ). Similarly, 81 is also a perfect square since it's ( 9^2 ). So, this looks like a difference of squares. The difference of squares formula is ( a^2 - b^2 = (a - b)(a + b) ). Applying that here, I can write:[ x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9) ]Alright, so now I have two factors: ( x^2 - 9 ) and ( x^2 + 9 ). Let me check if these can be factored further.Starting with ( x^2 - 9 ), this is also a difference of squares because ( x^2 = (x)^2 ) and 9 is ( 3^2 ). So, applying the difference of squares formula again:[ x^2 - 9 = (x - 3)(x + 3) ]Great, so now I have ( (x - 3)(x + 3) ) as factors. What about ( x^2 + 9 )? This is a sum of squares. I know that over the real numbers, a sum of squares doesn't factor into real polynomials because it doesn't have real roots. If I try to factor it, I would end up with complex numbers, which aren't allowed here since we need real coefficients.So, ( x^2 + 9 ) can't be factored further with real coefficients. That means the complete factorization of ( x^4 - 81 ) into monic polynomials with real coefficients is:[ x^4 - 81 = (x - 3)(x + 3)(x^2 + 9) ]Let me double-check to make sure I didn't miss anything. Starting from the original expression, I recognized it as a difference of squares, factored it into two quadratics. Then, I saw that one of those quadratics could be factored further into linear terms, while the other couldn't be factored any further without using complex numbers. Since the problem specifies real coefficients, I can't factor ( x^2 + 9 ) any further.So, I think I've factored it as far as possible under the given constraints. The factors are all monic, which they are because each polynomial has a leading coefficient of 1. The linear factors ( (x - 3) ) and ( (x + 3) ) are monic, and ( (x^2 + 9) ) is also monic.Just to be thorough, let me multiply them back together to see if I get the original polynomial:First, multiply ( (x - 3)(x + 3) ):[ (x - 3)(x + 3) = x^2 - 9 ]Then, multiply that result by ( (x^2 + 9) ):[ (x^2 - 9)(x^2 + 9) = x^4 + 9x^2 - 9x^2 - 81 = x^4 - 81 ]Yep, that checks out. So, the factorization is correct.I think I'm confident with this answer. It's fully factored into monic polynomials with real coefficients, and I can't factor it any further without going into complex numbers, which isn't allowed here.**Final Answer**The factorization of ( x^4 - 81 ) is boxed{(x-3)(x+3)(x^2+9)}.

answer：Okay, so I have this problem about a sign pyramid. It says that each cell in the pyramid gets a "+" if the two cells directly below it have the same sign, and a "-" if they have different signs. The pyramid has five levels, which means the bottom row has five cells. I need to figure out how many ways I can fill those five cells with "+" and "-" signs so that the very top of the pyramid is a "+".Hmm, let me try to visualize this pyramid. The bottom row has five cells, let's call them A, B, C, D, E. Then the row above that would have four cells, each determined by the two cells below it. So the first cell in the fourth row would depend on A and B, the next on B and C, then C and D, and finally D and E. The third row would then have three cells, each determined by the two cells in the fourth row. Continuing this way, the second row would have two cells, and finally, the top of the pyramid would be determined by those two cells.So, the top cell is determined by the entire bottom row. I need to find all possible combinations of "+" and "-" in the bottom row that result in a "+" at the top.Let me think about how the signs propagate up the pyramid. If two cells below are the same, the cell above is "+", and if they're different, it's "-". So, it's like a logical operation where "+" is like equality and "-" is like inequality.Maybe I can represent the signs as +1 and -1 instead of "+" and "-", so that I can use multiplication to represent the operations. If two cells are the same, their product is +1, and if they're different, their product is -1. That might make it easier to handle.So, if I denote the bottom row as A, B, C, D, E, each being either +1 or -1, then the fourth row would be:- A*B, B*C, C*D, D*EThe third row would then be:- (A*B)*(B*C), (B*C)*(C*D), (C*D)*(D*E)Simplifying each term:- (A*B)*(B*C) = A*B^2*C = A*C (since B^2 = 1)- Similarly, (B*C)*(C*D) = B*D- And (C*D)*(D*E) = C*ESo the third row is A*C, B*D, C*E.Then the second row would be:- (A*C)*(B*D), (B*D)*(C*E)Simplifying:- (A*C)*(B*D) = A*B*C*D- (B*D)*(C*E) = B*C*D*ESo the second row is A*B*C*D and B*C*D*E.Finally, the top cell is:- (A*B*C*D)*(B*C*D*E) = A*B^2*C^2*D^2*E = A*E (since B^2, C^2, D^2 are all 1)So, the top cell is A*E. We want this to be +1, which means A*E = 1. Therefore, A and E must be the same.So, the condition for the top cell to be "+" is that the first and last cells in the bottom row are the same.Now, how many ways can we fill the five cells in the bottom row such that A and E are the same?Each of the five cells can be either + or -, so there are 2^5 = 32 possible combinations. But we need to count only those where A and E are the same.Let's fix A and E first. If A is +, then E must be +. If A is -, then E must be -. So, for each choice of A (either + or -), E is determined. Then, the middle three cells B, C, D can be anything.So, there are 2 choices for A (and thus E), and 2^3 = 8 choices for B, C, D. So, total combinations are 2*8 = 16.Wait, but the answer choices don't include 16 as the correct answer. The options are 8, 10, 12, 14, 16. So, 16 is an option, but I need to check if my reasoning is correct.Wait, maybe I made a mistake in simplifying the expressions. Let me go back.I said that the top cell is A*E, but let me verify that step again.Starting from the second row: A*B*C*D and B*C*D*E.Multiplying these two gives (A*B*C*D)*(B*C*D*E) = A*B^2*C^2*D^2*E = A*E.Yes, that seems correct because B^2, C^2, D^2 are all 1.So, the top cell is indeed A*E. Therefore, A*E must be +1, so A and E must be the same.Therefore, the number of valid combinations is 2*2^3 = 16.But the answer choices include 16 as option E. So, is 16 the correct answer? But wait, the initial problem says "how many possible ways to fill the five cells in the bottom row to produce a '+' at the top of the pyramid." So, according to my reasoning, it's 16.But let me think again. Maybe I'm missing something. Because sometimes in these pyramids, the parity or some other constraints might affect the count.Wait, another way to think about it is that the top cell is determined by the product of A and E. So, if A and E are the same, the top is +, otherwise, it's -. So, the number of configurations where A and E are the same is indeed 16.But let me consider smaller cases to verify.Suppose the bottom row has only two cells. Then, the top cell is determined by those two. If they are the same, top is +; different, top is -. So, for two cells, the number of ways to get + at the top is 2 (both + or both -).Similarly, for three cells, the top is determined by the first and last cells. So, same logic: if first and last are same, top is +. So, number of ways is 2*2^(3-2) = 4.Wait, for three cells, the bottom row is A, B, C. The top is determined by A and C. So, if A and C are same, top is +. So, number of configurations is 2*2^(3-2) = 4.But actually, for three cells, the number of configurations where A and C are same is 2*2^(1) = 4, which is correct.Similarly, for four cells, the top is determined by A and D. So, number of configurations is 2*2^(4-2) = 8.Wait, but in the case of four cells, the top is determined by A and D, so same logic: 2*2^(4-2) = 8.So, for five cells, it should be 2*2^(5-2) = 16.But the answer choices include 16, so maybe that's correct. But I'm a bit confused because sometimes these pyramids have more constraints.Wait, maybe I'm oversimplifying. Let me think about the actual pyramid structure.Starting from the bottom row: A, B, C, D, E.Fourth row: A⊕B, B⊕C, C⊕D, D⊕E, where ⊕ represents the operation: same sign gives "+", different gives "-".Third row: (A⊕B)⊕(B⊕C), (B⊕C)⊕(C⊕D), (C⊕D)⊕(D⊕E).Second row: [(A⊕B)⊕(B⊕C)]⊕[(B⊕C)⊕(C⊕D)], [(B⊕C)⊕(C⊕D)]⊕[(C⊕D)⊕(D⊕E)].Top cell: [((A⊕B)⊕(B⊕C))⊕((B⊕C)⊕(C⊕D))]⊕[((B⊕C)⊕(C⊕D))⊕((C⊕D)⊕(D⊕E))].This seems complicated. Maybe I can represent the operations in terms of XOR, where "+" is 0 and "-" is 1, and XOR is the operation. But actually, in this case, the operation is not exactly XOR because "+" is same signs and "-" is different. So, it's similar to XOR but with different symbols.Wait, actually, if I map "+" to 0 and "-" to 1, then the operation is equivalent to XOR: if two below are same (0 and 0 or 1 and 1), the result is 0 ("+"); if different, result is 1 ("-"). So, yes, it's equivalent to XOR.Therefore, the top cell is the XOR of the entire bottom row in some way. But in the case of five cells, it's not just the XOR of all five, but a more complex combination.Wait, but earlier I simplified it to A*E, which seems too simple. Maybe I made a mistake in that simplification.Let me try to represent the operations algebraically. Let me assign each cell a value of +1 or -1, as I did before.So, bottom row: A, B, C, D, E.Fourth row: A*B, B*C, C*D, D*E.Third row: (A*B)*(B*C) = A*C, (B*C)*(C*D) = B*D, (C*D)*(D*E) = C*E.Second row: (A*C)*(B*D) = A*B*C*D, (B*D)*(C*E) = B*C*D*E.Top cell: (A*B*C*D)*(B*C*D*E) = A*E.Wait, so according to this, the top cell is A*E. So, if A and E are the same, top is +1; if different, top is -1.So, the number of configurations where A and E are the same is 2*2^3 = 16, as I thought earlier.But let me test this with a small example. Suppose A = +, E = +, and B, C, D arbitrary.Then, top cell should be +. Let's see:A = +, B = +, C = +, D = +, E = +.Fourth row: +, +, +, +.Third row: +, +, +.Second row: +, +.Top cell: +.Good.Another example: A = +, B = -, C = +, D = -, E = +.Fourth row: +*- = -, -*+ = -, +*- = -, -*+ = -.So, fourth row: -, -, -, -.Third row: -* - = +, -* - = +, -* - = +.Second row: +*+ = +, +*+ = +.Top cell: +*+ = +.Yes, works.Another example: A = -, B = +, C = -, D = +, E = -.Fourth row: -*+ = -, +*- = -, -*+ = -, +*- = -.Third row: -* - = +, -* - = +, -* - = +.Second row: +*+ = +, +*+ = +.Top cell: +*+ = +.Yes, works.Another example: A = +, B = +, C = -, D = -, E = +.Fourth row: +*+ = +, +*- = -, -* - = +, -*+ = -.Third row: +* - = -, -*+ = -, +*- = -.Second row: -* - = +, -* - = +.Top cell: +*+ = +.Wait, but A and E are both +, so top should be +. Correct.Another example: A = +, E = -, so top should be -.Let me try: A = +, B = +, C = +, D = +, E = -.Fourth row: +, +, +, -*+ = -.Third row: +, +, -*+ = -.Second row: +*+ = +, +*- = -.Top cell: +*- = -.Yes, correct.So, it seems that the top cell is indeed determined by A and E. Therefore, the number of configurations where A and E are the same is 2*2^3 = 16.But wait, the answer choices include 16 as option E. So, is 16 the correct answer?Wait, but in the initial problem, the options are A)8, B)10, C)12, D)14, E)16.So, according to my reasoning, it's 16, which is option E.But I'm a bit confused because sometimes these problems have more constraints, but in this case, the top cell is solely determined by A and E, so the count should be 16.Wait, but let me think again. Maybe I'm missing something about the parity or the number of "-" signs.Wait, no, because the top cell is determined by A and E, regardless of what's in between. So, as long as A and E are the same, the top is "+", regardless of B, C, D.Therefore, the number of configurations is indeed 2*2^3 = 16.So, the answer should be 16, which is option E.

answer：Okay, so I have this problem here where I need to find the set of all x such that f(x - 2) is greater than zero. The function f(x) is given as an even function, and for x greater than or equal to zero, it's defined as 2^x minus 4. Hmm, okay, let me try to unpack this step by step.First, I remember that an even function satisfies the condition f(x) = f(-x) for all x in its domain. So, if I know the function for x >= 0, I can figure it out for x < 0 by reflecting it over the y-axis. That makes sense. So, for x >= 0, f(x) = 2^x - 4, and for x < 0, f(x) should be equal to f(-x), which would be 2^{-x} - 4.Wait, is that right? Let me double-check. If x is negative, say x = -a where a is positive, then f(-a) = f(a) because it's even. So, f(-a) = 2^a - 4. But 2^a is the same as 2^{-(-a)}, so yeah, f(x) = 2^{|x|} - 4 for all x. That seems correct.So, f(x) = 2^{|x|} - 4. Got it. Now, the problem is asking for the set {x | f(x - 2) > 0}. So, I need to find all x such that when I plug (x - 2) into f, the result is positive.Let me write that down: f(x - 2) > 0. Since f is defined in terms of absolute value, let's substitute. So, f(x - 2) = 2^{|x - 2|} - 4. Therefore, the inequality becomes 2^{|x - 2|} - 4 > 0.Simplify that: 2^{|x - 2|} > 4. Hmm, 4 is 2 squared, so 2^{|x - 2|} > 2^2. Since the base 2 is greater than 1, the inequality holds when the exponent on the left is greater than the exponent on the right. So, |x - 2| > 2.Alright, now I have to solve |x - 2| > 2. Let me recall how to solve absolute value inequalities. The expression |A| > B (where B is positive) implies that A > B or A < -B. So, applying that here, |x - 2| > 2 implies that x - 2 > 2 or x - 2 < -2.Solving the first inequality: x - 2 > 2. Add 2 to both sides: x > 4.Solving the second inequality: x - 2 < -2. Add 2 to both sides: x < 0.So, putting it all together, the solution set is x < 0 or x > 4. Therefore, the set {x | f(x - 2) > 0} is all real numbers less than 0 or greater than 4.Wait, let me verify that. If I pick a number less than 0, say x = -1. Then x - 2 = -3. So, f(-3) = f(3) = 2^3 - 4 = 8 - 4 = 4, which is greater than 0. Okay, that works.If I pick x = 5, which is greater than 4, then x - 2 = 3. So, f(3) = 2^3 - 4 = 8 - 4 = 4, which is also greater than 0. Good.What about a number between 0 and 4, say x = 2. Then x - 2 = 0. f(0) = 2^0 - 4 = 1 - 4 = -3, which is less than 0. So, that's not in the solution set, which matches our result.Another test: x = 1. x - 2 = -1. f(-1) = f(1) = 2^1 - 4 = 2 - 4 = -2, which is less than 0. So, again, not in the solution set.And x = 4: x - 2 = 2. f(2) = 2^2 - 4 = 4 - 4 = 0, which is not greater than 0, so x = 4 is not included. Similarly, x = 0: x - 2 = -2. f(-2) = f(2) = 0, which is not greater than 0. So, the endpoints are excluded, which is consistent with our solution.I think that makes sense. So, the solution set is all real numbers less than 0 or greater than 4. Therefore, in interval notation, that would be (-∞, 0) union (4, ∞). But since the question asks for the set {x | f(x - 2) > 0}, I can express it as x < 0 or x > 4.Just to recap the steps:1. Recognize that f(x) is even, so f(x) = f(|x|).2. Express f(x - 2) in terms of absolute value: f(x - 2) = 2^{|x - 2|} - 4.3. Set up the inequality 2^{|x - 2|} - 4 > 0.4. Simplify to 2^{|x - 2|} > 4, which leads to |x - 2| > 2.5. Solve the absolute value inequality to find x < 0 or x > 4.6. Verify with test points to ensure the solution is correct.I think that covers everything. I don't see any mistakes in my reasoning, so I feel confident about the answer.**Final Answer**The set is boxed{{x | x < 0 text{ or } x > 4}}.