🧠:Okay, so I have this tetrahedron ABCD, and I need to show that AD is longer than BC. The given information is that angles ABD and ACD are obtuse. Hmm, let me try to visualize this tetrahedron first. A tetrahedron has four triangular faces, and in this case, two of the angles at vertex D in triangles ABD and ACD are obtuse. First, maybe I should recall some properties of tetrahedrons and the relationships between their edges and angles. Since we're dealing with angles in triangles, perhaps the Law of Cosines could be helpful here. Let me think. In a triangle, if an angle is obtuse, then the side opposite that angle is longer than the other sides. Wait, no, actually, the Law of Cosines states that for a triangle with sides a, b, c opposite angles A, B, C respectively, c² = a² + b² - 2ab cos C. So if angle C is obtuse (greater than 90 degrees), then cos C is negative, making the term -2ab cos C positive. Therefore, c² > a² + b². So, in triangle ABD, since angle ABD is obtuse, the side opposite to angle ABD, which is AD, should satisfy AD² > AB² + BD². Similarly, in triangle ACD, angle ACD is obtuse, so AD² > AC² + CD². Wait, so from both triangles ABD and ACD, we get that AD squared is greater than AB² + BD² and also greater than AC² + CD². Hmm, but how does that help us relate AD and BC? Maybe we need to relate BC to AB, AC, BD, CD somehow. Let me think. BC is a edge of the tetrahedron, connecting vertices B and C. To relate BC with other edges, perhaps considering triangles involving B and C. For example, in triangle ABC, BC² = AB² + AC² - 2 AB AC cos(angle BAC). But I don't know if angle BAC is known. Alternatively, maybe consider the edges BD and CD, and how they relate to BC. Alternatively, maybe use the fact that in a tetrahedron, the edges can be related through the triangle inequalities in various faces. But since we have specific angles being obtuse, maybe there's a way to combine the inequalities from the two triangles ABD and ACD. Let me write down the inequalities:From triangle ABD with obtuse angle at B:AD² > AB² + BD². (1)From triangle ACD with obtuse angle at C:AD² > AC² + CD². (2)So, if I can combine these two inequalities, perhaps I can find something about BC. Let's think about BC. To relate BC with BD and CD, maybe consider triangle BCD. In triangle BCD, BC² < BD² + CD² if angle BDC is acute, but we don't know that. Alternatively, maybe use another approach. Alternatively, maybe use vectors. Let me assign coordinates to the tetrahedron. Let me place point D at the origin, since angles at D might be relevant, but wait, the angles ABD and ACD are at B and C, respectively. Hmm. Alternatively, maybe place point A somewhere, and express vectors in terms of coordinates. Let me try that.Let me set point D at the origin (0,0,0). Let me let point B be at coordinates (b, 0, 0) on the x-axis. Similarly, point C can be at coordinates (c, 0, 0), but wait, but then points B and C would be colinear with D, which might not be a good idea since in a tetrahedron, all four points shouldn't be coplanar. Maybe a better coordinate system.Alternatively, set point D at the origin, point B somewhere on the x-axis, point C somewhere on the y-axis, and point A somewhere in 3D space. Hmm. Let me try this:Let D = (0,0,0).Let B = (x, 0, 0), along the x-axis.Let C = (0, y, 0), along the y-axis.Then point A can be somewhere in 3D space, say (p, q, r), where r ≠ 0 to ensure it's not coplanar with D, B, C.Now, angles ABD and ACD are obtuse. Let's figure out what that means in terms of coordinates.First, angle ABD is the angle at point B in triangle ABD. The vectors BA and BD would form this angle. Similarly, angle ACD is the angle at point C in triangle ACD, formed by vectors CA and CD.Since angle at B is obtuse, the dot product of BA and BD should be negative because the angle between them is greater than 90 degrees. Similarly, at point C, the dot product of CA and CD should be negative.Let me compute BA and BD vectors.Point A is (p, q, r), point B is (x,0,0), point D is (0,0,0).Vector BA = A - B = (p - x, q - 0, r - 0) = (p - x, q, r).Vector BD = D - B = (0 - x, 0 - 0, 0 - 0) = (-x, 0, 0).The dot product BA · BD = (p - x)(-x) + q*0 + r*0 = -x(p - x) = -xp + x².Since angle ABD is obtuse, the dot product is negative. Therefore:-xp + x² < 0=> x² - xp < 0=> x(x - p) < 0So this inequality holds if x and (x - p) have opposite signs. So either x > 0 and x - p < 0 (i.e., 0 < x < p) or x < 0 and x - p > 0 (i.e., x < 0 and p < x). Since we placed point B at (x, 0, 0), x is non-zero. Let's assume x > 0 for simplicity. Then, x(x - p) < 0 implies x - p < 0, so p > x. So the x-coordinate of point A is greater than x. Similarly, if x were negative, but let's stick with x > 0 for simplicity.Similarly, for angle ACD being obtuse. Let's compute the vectors CA and CD.Point C is (0, y, 0), so vector CA = A - C = (p - 0, q - y, r - 0) = (p, q - y, r).Vector CD = D - C = (0 - 0, 0 - y, 0 - 0) = (0, -y, 0).Dot product CA · CD = p*0 + (q - y)(-y) + r*0 = -y(q - y) = -yq + y².Since angle ACD is obtuse, the dot product is negative:-yq + y² < 0=> y² - yq < 0=> y(y - q) < 0Again, assuming y > 0 (since we placed C on the positive y-axis), then y(y - q) < 0 implies y - q < 0 => q > y. So the y-coordinate of point A is greater than y.So from the obtuse angles at B and C, we have that in our coordinate system, the coordinates of point A satisfy p > x and q > y, where B is (x,0,0), C is (0,y,0), and D is the origin.Now, we need to compare AD and BC. Let's compute their lengths.AD is the distance from A to D, which is sqrt(p² + q² + r²).BC is the distance from B to C, which is sqrt((x - 0)^2 + (0 - y)^2 + (0 - 0)^2) = sqrt(x² + y²).So we need to show that sqrt(p² + q² + r²) > sqrt(x² + y²). Squaring both sides (since lengths are positive), this is equivalent to p² + q² + r² > x² + y².Given that p > x and q > y, from earlier. Also, r is non-zero because otherwise, point A would be coplanar with D, B, C, which would collapse the tetrahedron. So r² > 0.Therefore, p² > x² (since p > x > 0), q² > y² (since q > y > 0), and r² > 0. Adding them up:p² + q² + r² > x² + y² + 0 = x² + y².Therefore, AD² > BC², so AD > BC.Wait, that seems straightforward. But did I make any assumptions in the coordinate system? I placed D at the origin, B on the x-axis, C on the y-axis, and A somewhere in space. Is this a valid setup? Since we can always choose a coordinate system to simplify the problem without loss of generality. By choosing D as the origin, and aligning B and C along the axes, we are essentially fixing the coordinate system to make calculations easier. The relations we derived (p > x and q > y) are due to the obtuse angles at B and C. Then, since r² is positive, adding everything up gives AD longer than BC. But let me verify if there is another way to approach this without coordinates. Maybe using the properties of tetrahedrons and triangle inequalities. Let's think again.From triangle ABD: AD² > AB² + BD² (since angle at B is obtuse).From triangle ACD: AD² > AC² + CD² (since angle at C is obtuse).If we can relate BC to AB, AC, BD, CD, then maybe combine these inequalities. Let's see. For instance, in triangle ABC, by the triangle inequality, BC < AB + AC. But I don't see how that directly helps. Alternatively, in triangle BCD, BC² < BD² + CD² + 2 BD CD cos(angle BDC). But we don't know angle BDC. Alternatively, consider adding the two inequalities:AD² > AB² + BD²AD² > AC² + CD²So adding them: 2 AD² > AB² + AC² + BD² + CD²But how does this relate to BC? If we can find a relationship between BC and the sum AB² + AC² + BD² + CD².Alternatively, maybe consider the following approach: use vectors in 3D space. Let vector AD be the vector from A to D. Then BC is the vector from B to C. Wait, but perhaps if we express BC in terms of vectors from A. Hmm. Let me think.Wait, another idea: use the fact that in 3D space, the distance between two points can be related through other points. For example, BC can be expressed as the vector from B to C, which is C - B. If we can relate this to vectors involving A and D.But maybe this is complicating things. The coordinate approach seems to have worked. Let me check again.In the coordinate system, with D at (0,0,0), B at (x,0,0), C at (0,y,0), and A at (p,q,r). Then, from the obtuse angles, we found p > x and q > y, and r ≠ 0. Then, AD is sqrt(p² + q² + r²), BC is sqrt(x² + y²). Since p² > x², q² > y², and r² > 0, adding these gives AD² > x² + y² = BC², hence AD > BC. That seems correct.But wait, maybe there's a case where even if p > x and q > y, the sum p² + q² + r² isn't necessarily greater than x² + y². Wait, but if p > x, then p² > x². Similarly, q² > y². So p² + q² > x² + y². Then adding r² (which is positive) would make the left side even larger. So p² + q² + r² > x² + y². Therefore, AD² > BC², so AD > BC. That seems solid.But maybe the coordinate system imposes some restrictions? For example, if we had chosen different axes, would the conclusion still hold? But since we can always choose a coordinate system without loss of generality, the result should hold regardless. Alternatively, think of it this way: in the coordinate system, BC is the hypotenuse of a right triangle with legs x and y. AD is the distance from A to D, which is like the hypotenuse of a 3D right triangle with legs p, q, r. Since p > x and q > y and r > 0, then each corresponding leg of AD is longer than that of BC, making AD longer overall. Therefore, the conclusion AD > BC follows from the given obtuse angles and the coordinate analysis. Is there another way to see this without coordinates? Let me try using vectors and dot products.In triangle ABD, angle at B is obtuse, so vectors BA and BD satisfy BA · BD < 0.Similarly, in triangle ACD, angle at C is obtuse, so vectors CA and CD satisfy CA · CD < 0.If we can relate these dot products to the lengths AD and BC. Let me denote vectors:Let’s denote vector BD = D - B, vector CD = D - C.Wait, in triangle ABD, vectors BA = A - B, BD = D - B.The dot product BA · BD < 0.Similarly, in triangle ACD, vectors CA = A - C, CD = D - C, and their dot product < 0.Let me write these out:BA · BD = (A - B) · (D - B) < 0CA · CD = (A - C) · (D - C) < 0Expanding these:(A - B) · (D - B) = (A · D - A · B - B · D + B · B) < 0Similarly, (A - C) · (D - C) = (A · D - A · C - C · D + C · C) < 0Assuming D is the origin, then D = 0, so these simplify:(A - B) · (-B) < 0 => (A · (-B)) + B · B < 0 => -A · B + |B|² < 0 => |B|² < A · BSimilarly, for the second inequality:(A - C) · (-C) < 0 => -A · C + |C|² < 0 => |C|² < A · CSo in vector terms, we have |B|² < A · B and |C|² < A · C.Now, we need to compare |A| (which is AD, since D is the origin) and |B - C| (which is BC).We need to show |A| > |B - C|.Let’s square both sides: |A|² > |B - C|².Which is A · A > (B - C) · (B - C) = |B|² + |C|² - 2 B · C.So we need to show |A|² > |B|² + |C|² - 2 B · C.But from the inequalities we have:From |B|² < A · B => A · B > |B|²From |C|² < A · C => A · C > |C|²So if we add these two inequalities: A · B + A · C > |B|² + |C|².But how does that relate to |A|²?Hmm. Let’s consider using Cauchy-Schwarz inequality or something else. Let me think.We have A · B > |B|² and A · C > |C|².Let me denote vectors B and C. Let’s consider A as a vector in space.If I consider A · (B + C) = A · B + A · C > |B|² + |C|².But we can also write:|A|² |B + C|² ≥ (A · (B + C))² by Cauchy-Schwarz.But I don't know if that helps. Alternatively, perhaps use the fact that A · B > |B|² implies that the projection of A onto B is longer than |B|. Which would mean that the component of A in the direction of B is greater than |B|. Similarly for C.This suggests that vector A has components in both B and C directions that are longer than |B| and |C| respectively. Which would imply that |A| is larger than |B| and |C| individually. But BC is the distance between B and C, which is sqrt(|B|² + |C|² - 2 B · C) if B and C are orthogonal. Wait, in our coordinate system, B and C were orthogonal, so BC would be sqrt(|B|² + |C|²). But in the general case, if B and C are not orthogonal, BC would be sqrt(|B|² + |C|² - 2 B · C cos θ), where θ is the angle between B and C. But since we placed B and C along axes in the coordinate system, they were orthogonal, so BC is sqrt(|B|² + |C|²). But in the general case, they might not be orthogonal.Wait, but in our coordinate system, B and C are orthogonal, so BC = sqrt(x² + y²). Then AD = sqrt(p² + q² + r²), and since p > x, q > y, r ≠ 0, AD is longer. But if B and C are not orthogonal, then BC could be longer? Hmm, but in the problem statement, there's no assumption about the angle between B and C, so the result must hold regardless. Wait, maybe the key is that in our coordinate system, we can always align B and C along axes, making them orthogonal. Because the problem doesn't specify any particular orientation, so we can choose coordinates such that B and C are orthogonal. Therefore, BC is sqrt(|B|² + |C|²), and AD is sqrt(p² + q² + r²) with p > |B|, q > |C|, and r ≠ 0, hence AD > BC. Therefore, in this coordinate system, the result holds, and since we can always choose such coordinates, the result holds in general.Therefore, regardless of the original positions, by choosing a suitable coordinate system where B and C are orthogonal, and using the obtuse angle conditions to derive p > |B| and q > |C|, we can conclude AD > BC.Alternatively, without coordinates, maybe use the Pythagorean theorem in 3D. If we consider AD as the space diagonal of a rectangular box with edges p, q, r, then BC would be the diagonal of the face with edges x and y. Since p > x, q > y, and r > 0, the space diagonal AD must be longer than the face diagonal BC. But this is similar to the coordinate approach.But in reality, the tetrahedron isn't necessarily a rectangular box, but by choosing coordinates appropriately, we can model it in such a way. The critical point is the obtuse angles forcing the coordinates of A to exceed those of B and C in their respective axes, plus an additional non-zero component, making AD longer.Therefore, through both coordinate analysis and vector dot product properties, we can conclude that AD must be longer than BC in the given tetrahedron with the specified obtuse angles.Final Answerboxed{AD > BC}

✅:Given a tetrahedron (ABCD) where the angles (angle ABD) and (angle ACD) are obtuse, we need to show that (AD > BC).1. Using the Law of Cosines in Triangles ABD and ACD: - In (triangle ABD), since (angle ABD) is obtuse, we have: [ AD^2 > AB^2 + BD^2 ] - In (triangle ACD), since (angle ACD) is obtuse, we have: [ AD^2 > AC^2 + CD^2 ]2. Coordinate System Setup: - Place (D) at the origin ((0, 0, 0)). - Place (B) on the x-axis at ((x, 0, 0)) and (C) on the y-axis at ((0, y, 0)). - Let (A) be at ((p, q, r)) in 3D space.3. Dot Product Conditions for Obtuse Angles: - For (angle ABD) to be obtuse, the dot product of vectors (overrightarrow{BA}) and (overrightarrow{BD}) must be negative: [ -xp + x^2 < 0 implies p > x ] - For (angle ACD) to be obtuse, the dot product of vectors (overrightarrow{CA}) and (overrightarrow{CD}) must be negative: [ -yq + y^2 < 0 implies q > y ]4. Comparing Distances: - The length (AD) is: [ AD = sqrt{p^2 + q^2 + r^2} ] - The length (BC) is: [ BC = sqrt{x^2 + y^2} ]5. Conclusion: - Since (p > x), (q > y), and (r neq 0), we have: [ p^2 + q^2 + r^2 > x^2 + y^2 ] - Therefore, (AD^2 > BC^2) implies (AD > BC).[boxed{AD > BC}]

🔑:1. We are given that the angles ( angle ABD ) and ( angle ACD ) within the tetrahedron ( ABCD ) are obtuse.2. In the triangle ( ABD ), since ( angle ABD ) is obtuse, point ( B ) must lie inside the circumcircle with diameter ( AD ) (Thales' theorem states that a point on the circle subtended by a diameter forms a right angle with said diameter, so an obtuse angle indicates ( B ) is inside the circle).3. With point ( B ) lying inside the circumcircle of diameter ( AD ), it means ( B ) is also inside the sphere that has ( AD ) as its diameter.4. Similarly, since the angle ( angle ACD ) is obtuse in the triangle ( ACD ), point ( C ) must lie inside the circumcircle of ( AD ).5. Therefore, points ( B ) and ( C ) both lie inside the sphere having ( AD ) as the diameter, but they do not lie on the surface of the sphere.6. The maximum possible distance between any two points inside this sphere is the length of the diameter ( AD ), and this maximum distance can only be achieved if the points lie on the surface of the sphere (as the endpoints of the diameter).7. Since neither ( B ) nor ( C ) lies on the sphere's surface, the distance between points ( B ) and ( C ), which is the length of segment ( BC ), must be less than the diameter ( AD ).Conclusion:[boxed{AD > BC}]

🧠:Okay, so I need to find the equation of a circle that passes through the intersection points of two given circles, and its center lies on the line 2x - y - 4 = 0. Hmm, let me start by recalling some concepts about circles and their intersections.First, when two circles intersect, the set of all circles passing through their intersection points can be described by the family of circles given by the equation S1 + λS2 = 0, where S1 and S2 are the equations of the two circles, and λ is a parameter. But wait, I think it might actually be S1 + λ(S1 - S2) = 0? No, maybe I should double-check. Actually, the family of circles passing through the intersection points of S1 and S2 is S1 + λS2 = 0, but actually, no, that's not quite right. Let me think. The general equation is S1 + λ(S2 - S1) = 0, which varies as λ changes. Alternatively, maybe it's just S1 + λ(S2) = 0? Wait, maybe it's simpler. If we have two circles S1: x² + y² + D1x + E1y + F1 = 0 and S2: x² + y² + D2x + E2y + F2 = 0, then the equation of any circle passing through their intersection can be written as S1 + λ(S2 - S1) = 0. Alternatively, subtract the two equations to get the radical axis, which is the line equation. But since the problem is about a circle passing through both intersection points, the family can be represented by S1 + λ(S2) = k, but actually, I need to recall the correct method.Wait, let's take it step by step. The two given circles are:1. x² + y² + 4x - 3 = 02. x² + y² - 4y - 3 = 0Let me denote the first circle as S1: x² + y² + 4x - 3 = 0, and the second as S2: x² + y² - 4y - 3 = 0.If we subtract S2 from S1, we get:(S1) - (S2) = (x² + y² + 4x - 3) - (x² + y² - 4y - 3) = 4x + 4y = 0.Simplifying, 4x + 4y = 0 ⇒ x + y = 0. So, this is the radical axis of the two circles, which is the line containing their points of intersection. Therefore, the points of intersection lie on the line x + y = 0. However, any circle passing through these two intersection points must have its equation as a combination of S1 and S2. The general equation for such a circle would be S1 + λS2 = 0, where λ is a parameter. Wait, actually, if we take the linear combination S1 + λS2 = 0, this represents a family of circles passing through the intersection points of S1 and S2. But let's verify.If we write S1 + λS2 = 0:(x² + y² + 4x - 3) + λ(x² + y² - 4y - 3) = 0Combine like terms:(1 + λ)x² + (1 + λ)y² + 4x - 4λy - 3(1 + λ) = 0Divide both sides by (1 + λ) (assuming λ ≠ -1) to get the standard circle equation:x² + y² + [4/(1 + λ)]x - [4λ/(1 + λ)]y - 3 = 0So, this is the general equation of a circle passing through the intersection points of S1 and S2. The center of this circle is at (-2/(1 + λ), 2λ/(1 + λ)), since the general form of a circle x² + y² + Dx + Ey + F = 0 has center (-D/2, -E/2). Wait, in the equation above, the coefficients are D = 4/(1 + λ) and E = -4λ/(1 + λ), so the center would be (-D/2, -E/2) = (-2/(1 + λ), 2λ/(1 + λ)).Now, the problem states that the center of the desired circle lies on the line 2x - y - 4 = 0. So, we need the center coordinates (-2/(1 + λ), 2λ/(1 + λ)) to satisfy 2x - y - 4 = 0. Let's substitute x and y:2*(-2/(1 + λ)) - (2λ/(1 + λ)) - 4 = 0Simplify:(-4/(1 + λ)) - (2λ/(1 + λ)) - 4 = 0Combine the first two terms:[-4 - 2λ]/(1 + λ) - 4 = 0Multiply through by (1 + λ) to eliminate the denominator:-4 - 2λ - 4(1 + λ) = 0Wait, let's check that step again. If we have:[-4 - 2λ]/(1 + λ) - 4 = 0Then, moving the -4 to the other side:[-4 - 2λ]/(1 + λ) = 4Multiply both sides by (1 + λ):-4 - 2λ = 4(1 + λ)Expand the right side:-4 - 2λ = 4 + 4λBring all terms to one side:-4 - 2λ - 4 - 4λ = 0 ⇒ -8 - 6λ = 0Solving for λ:-6λ = 8 ⇒ λ = -8/6 = -4/3So, λ = -4/3.Therefore, substituting back into the general equation S1 + λS2 = 0:(x² + y² + 4x - 3) + (-4/3)(x² + y² - 4y - 3) = 0Let me compute this step by step.First, expand the terms with λ:= x² + y² + 4x - 3 - (4/3)x² - (4/3)y² + (16/3)y + 4Wait, let's compute each term:Multiply (-4/3) into S2:-4/3*(x² + y² - 4y - 3) = -4/3 x² -4/3 y² + 16/3 y + 4Therefore, the entire equation becomes:S1 + λS2 = (x² + y² + 4x - 3) + (-4/3 x² -4/3 y² + 16/3 y + 4) = 0Combine like terms:x² - 4/3 x² = (1 - 4/3)x² = (-1/3)x²Similarly for y²: y² -4/3 y² = (-1/3)y²Then, the linear terms: 4x + 16/3 yConstants: -3 + 4 = 1So the equation becomes:(-1/3)x² -1/3 y² + 4x + (16/3)y + 1 = 0Multiply both sides by -3 to eliminate the fractions:x² + y² - 12x - 16y - 3 = 0Wait, let's check:Multiplying each term:(-1/3)x² * (-3) = x²(-1/3)y² * (-3) = y²4x * (-3) = -12x(16/3)y * (-3) = -16y1 * (-3) = -3So indeed, the equation becomes x² + y² - 12x - 16y - 3 = 0.But let me verify if this is correct. Let's check if the center lies on the line 2x - y - 4 = 0.The center of the circle x² + y² - 12x - 16y - 3 = 0 is (D/2, E/2) where D = -12 and E = -16. Wait, no. Wait, the general equation is x² + y² + Dx + Ey + F = 0, so the center is (-D/2, -E/2). So here, D = -12, E = -16. Therefore, center is (12/2, 16/2) = (6, 8).Now, check if (6,8) lies on 2x - y -4 = 0:2*6 -8 -4 = 12 -8 -4 = 0. Yes, it does. So that's correct.But let me also check if this circle actually passes through the intersection points of the original two circles.Alternatively, perhaps another approach is to find the intersection points first and then find the circle passing through them with the center on the given line. Let me try that as a verification.First, find the intersection points of the two circles S1 and S2.S1: x² + y² +4x -3 =0S2: x² + y² -4y -3 =0Subtracting S2 - S1:(x² + y² -4y -3) - (x² + y² +4x -3) = -4y -4x =0 ⇒ -4x -4y =0 ⇒ x + y =0, which is the radical axis as before.So the intersection points lie on x + y =0. Let's solve for y = -x, substitute into S1:x² + (-x)^2 +4x -3 =0 ⇒ 2x² +4x -3=0Solving quadratic equation:x = [-4 ±√(16 +24)]/(2*2) = [-4 ±√40]/4 = [-4 ±2√10]/4 = [-2 ±√10]/2Therefore, the intersection points are:x = (-2 + √10)/2, y = -x = (2 - √10)/2andx = (-2 - √10)/2, y = -x = (2 + √10)/2So the two points are ((-2 + √10)/2, (2 - √10)/2) and ((-2 - √10)/2, (2 + √10)/2).Now, the desired circle passes through these two points and has its center on 2x - y -4 =0.Let me denote the desired circle as (x - h)^2 + (y - k)^2 = r^2, with center (h,k) on 2h -k -4=0 ⇒ k =2h -4.So, the circle's equation is (x - h)^2 + (y - (2h -4))^2 = r^2.Since it passes through both intersection points, substituting each into the equation should satisfy it.Let's take the first point ((-2 + √10)/2, (2 - √10)/2).Plug into the circle equation:[(-2 + √10)/2 - h]^2 + [(2 - √10)/2 - (2h -4)]^2 = r^2Similarly for the second point ((-2 - √10)/2, (2 + √10)/2):[(-2 - √10)/2 - h]^2 + [(2 + √10)/2 - (2h -4)]^2 = r^2Since both equal r^2, we can set them equal to each other:[(-2 + √10)/2 - h]^2 + [(2 - √10)/2 - (2h -4)]^2 = [(-2 - √10)/2 - h]^2 + [(2 + √10)/2 - (2h -4)]^2This looks complicated, but maybe simplifies. Let's compute each term step by step.First, simplify the expressions inside the squares.For the x-coordinate difference:For the first point: x1 = (-2 + √10)/2 - hFor the second point: x2 = (-2 - √10)/2 - hSimilarly for the y-coordinate difference:For the first point: y1 = (2 - √10)/2 - (2h -4) = (2 - √10)/2 - 2h +4 = (2 - √10 +8)/2 -2h = (10 - √10)/2 -2hWait, let's compute that again:(2 - √10)/2 - (2h -4) = (2 - √10)/2 -2h +4 = 4 + (2 - √10)/2 -2h = convert 4 to 8/2:= (8/2 + 2 - √10)/2 -2h = (10 - √10)/2 -2hSimilarly for the second point:y2 = (2 + √10)/2 - (2h -4) = (2 + √10)/2 -2h +4 = (2 + √10 +8)/2 -2h = (10 + √10)/2 -2hTherefore, the equation becomes:[x1]^2 + [y1]^2 = [x2]^2 + [y2]^2Substituting:[ (-2 + √10)/2 - h ]^2 + [ (10 - √10)/2 -2h ]^2 = [ (-2 - √10)/2 - h ]^2 + [ (10 + √10)/2 -2h ]^2This is going to be messy, but perhaps expanding both sides and simplifying.Let me denote A = (-2 ±√10)/2 - h and B = (10 ∓√10)/2 -2h. Let's compute the left-hand side (LHS) and right-hand side (RHS):LHS: [ (-2 + √10)/2 - h ]^2 + [ (10 - √10)/2 -2h ]^2RHS: [ (-2 - √10)/2 - h ]^2 + [ (10 + √10)/2 -2h ]^2Let me expand each term:First term LHS:Let me compute (-2 + √10)/2 - h = (-2 - 2h + √10)/2So squaring this:[ (-2 -2h + √10)/2 ]^2 = [ ( (-2 -2h) + √10 ) /2 ]^2 = [ (-2 -2h)^2 + 2*(-2 -2h)*(√10) + (√10)^2 ] /4Similarly, the second term LHS:(10 - √10)/2 -2h = (10 - √10 -4h)/2Squaring:[ (10 - √10 -4h)/2 ]^2 = [ (10 -4h) - √10 ) ]^2 /4 = [ (10 -4h)^2 - 2*(10 -4h)*√10 + (√10)^2 ] /4Similarly for RHS:First term RHS: [ (-2 - √10)/2 - h ]^2 = [ (-2 -2h - √10)/2 ]^2 = [ (-2 -2h)^2 + 2*(-2 -2h)*(-√10) + (√10)^2 ] /4Second term RHS: [ (10 + √10)/2 -2h ]^2 = [ (10 + √10 -4h)/2 ]^2 = [ (10 -4h) + √10 ) ]^2 /4 = [ (10 -4h)^2 + 2*(10 -4h)*√10 + (√10)^2 ] /4Therefore, combining LHS and RHS:LHS:[ (-2 -2h)^2 + 2*(-2 -2h)*√10 + 10 ] /4 + [ (10 -4h)^2 - 2*(10 -4h)*√10 +10 ] /4RHS:[ (-2 -2h)^2 + 2*(-2 -2h)*(-√10) +10 ] /4 + [ (10 -4h)^2 + 2*(10 -4h)*√10 +10 ] /4Simplify both sides:LHS numerator:[ (-2 -2h)^2 + 10 + (10 -4h)^2 +10 ] + [ 2*(-2 -2h)*√10 - 2*(10 -4h)*√10 ]Similarly, RHS numerator:[ (-2 -2h)^2 +10 + (10 -4h)^2 +10 ] + [ 2*(-2 -2h)*(-√10) + 2*(10 -4h)*√10 ]Notice that the non-radical terms are the same on both sides:[ (-2 -2h)^2 + (10 -4h)^2 +20 ] on both LHS and RHS.Therefore, subtract these from both sides, and equate the remaining terms involving √10:On LHS: [2*(-2 -2h)*√10 - 2*(10 -4h)*√10 ] = [ (-4 -4h -20 +8h ) ]√10 = ( -24 +4h )√10On RHS: [2*(-2 -2h)*(-√10) + 2*(10 -4h)*√10 ] = [ (4 +4h)√10 + (20 -8h)√10 ] = (24 -4h )√10Thus, setting LHS = RHS:(-24 +4h )√10 = (24 -4h )√10Move all terms to left side:(-24 +4h -24 +4h )√10 =0 ⇒ (-48 +8h )√10 =0Since √10 ≠0, we have:-48 +8h =0 ⇒ 8h=48 ⇒ h=6So, h=6. Then since k=2h -4=2*6 -4=12 -4=8. Therefore, the center is (6,8).Now, compute the radius squared by plugging one of the intersection points into the circle equation.Let's take the first intersection point: ((-2 + √10)/2, (2 - √10)/2)Compute (x -6)^2 + (y -8)^2:x -6 = [ (-2 + √10)/2 -6 ] = [ (-2 + √10 -12)/2 ] = [ (-14 +√10)/2 ]y -8 = [ (2 - √10)/2 -8 ] = [ (2 - √10 -16)/2 ] = [ (-14 -√10)/2 ]So squared terms:[ (-14 +√10)/2 ]^2 + [ (-14 -√10)/2 ]^2Compute each:First term: [ (-14 +√10)^2 ] /4 = (196 -28√10 +10)/4 = (206 -28√10)/4Second term: [ (-14 -√10)^2 ] /4 = (196 +28√10 +10)/4 = (206 +28√10)/4Add them together:(206 -28√10 +206 +28√10)/4 = (412)/4 = 103Therefore, radius squared is 103. Hence, the equation is (x -6)^2 + (y -8)^2 =103.Expanding this:x² -12x +36 + y² -16y +64 =103Combine constants: 36+64=100Thus: x² + y² -12x -16y +100 =103 ⇒ x² + y² -12x -16y -3=0, which matches the earlier result. Good, so both methods confirm the same equation.Therefore, the required circle is x² + y² -12x -16y -3=0, with center at (6,8) and radius √103.But the problem didn't specify the form, so likely the standard form is acceptable, but maybe they want the expanded form. Let me check if the question says "Find the circle...", so perhaps either form is okay, but since it's passing through intersections, maybe the general equation is expected.Alternatively, since we have the center (6,8) and radius √103, the standard form is (x -6)^2 + (y -8)^2 =103, which can also be written as x² + y² -12x -16y -3=0.Yes, either is correct. The problem might prefer one over the other, but since they mention "the circle that passes through the intersection points...", which is unique given the condition on the center. Therefore, the answer is x² + y² -12x -16y -3=0.Final AnswerThe equation of the circle is boxed{x^2 + y^2 - 12x - 16y - 3 = 0}.

✅:To find the circle that passes through the intersection points of the circles (x^2 + y^2 + 4x - 3 = 0) and (x^2 + y^2 - 4y - 3 = 0) with its center lying on the line (2x - y - 4 = 0), we follow these steps:1. Identify the family of circles passing through the intersection points: The general equation of a circle passing through the intersection points of the given circles can be written as: [ (x^2 + y^2 + 4x - 3) + lambda(x^2 + y^2 - 4y - 3) = 0 ] Simplifying, we get: [ (1 + lambda)x^2 + (1 + lambda)y^2 + 4x - 4lambda y - 3(1 + lambda) = 0 ] Dividing by (1 + lambda) (assuming (lambda neq -1)): [ x^2 + y^2 + frac{4}{1 + lambda}x - frac{4lambda}{1 + lambda}y - 3 = 0 ] The center of this circle is (left(-frac{2}{1 + lambda}, frac{2lambda}{1 + lambda}right)).2. Determine the parameter (lambda): The center must lie on the line (2x - y - 4 = 0). Substituting the coordinates of the center into this line equation: [ 2left(-frac{2}{1 + lambda}right) - frac{2lambda}{1 + lambda} - 4 = 0 ] Simplifying: [ -frac{4 + 2lambda}{1 + lambda} - 4 = 0 ] Solving for (lambda), we find (lambda = -frac{4}{3}).3. Substitute (lambda = -frac{4}{3}) back into the family equation: [ (x^2 + y^2 + 4x - 3) - frac{4}{3}(x^2 + y^2 - 4y - 3) = 0 ] Simplifying this equation: [ -frac{1}{3}x^2 - frac{1}{3}y^2 + 4x + frac{16}{3}y + 1 = 0 ] Multiplying by (-3) to clear the fractions: [ x^2 + y^2 - 12x - 16y - 3 = 0 ]4. Verification: The center of the resulting circle ((6, 8)) lies on the line (2x - y - 4 = 0) and passes through the intersection points of the original circles.Thus, the equation of the circle is (boxed{x^2 + y^2 - 12x - 16y - 3 = 0}).

🔑:First, consider the equations of the two given circles:1. (x^2 + y^2 + 4x - 3 = 0)2. (x^2 + y^2 - 4y - 3 = 0)We are looking for a circle that passes through the intersection points of these two circles and whose center lies on the line (2x - y - 4 = 0).First, form the equation of the required circle by adding the given equations with a linear combination:[(x^2 + y^2 + 4x - 3) + lambda (x^2 + y^2 - 4y - 3) = 0]This yields:[(1 + lambda)x^2 + (1 + lambda)y^2 + 4x - 4lambda y - 3 - 3lambda = 0]Now, reformat the equation:[(1 + lambda)x^2 + (1 + lambda)y^2 + 4x - 4lambda y - 3 - 3lambda = 0]To identify the center of this circle, rewrite it in a standard form by completing the square. The general form of a circle's center ((h, k)) and radius (r) is:[(x - h)^2 + (y - k)^2 = r^2]But we can identify the center ( left( -frac{B}{2A}, -frac{D}{2A} right) ) from the general form (Ax^2 + Ay^2 + Bx + Dy + F = 0).Thus, the center for our circle is:[left(-frac{2}{1 + lambda}, frac{2lambda}{1 + lambda}right)]Given that the center of the circle lies on the line (2x - y - 4 = 0), substitute the center coordinates into this line equation:[2left(-frac{2}{1 + lambda}right) - frac{2lambda}{1 + lambda} - 4 = 0]Simplify and solve for (lambda):[- frac{4}{1 + lambda} - frac{2lambda}{1 + lambda} - 4 = 0][-frac{4 + 2lambda}{1 + lambda} - 4 = 0][- frac{4 + 2lambda + 4 + 4lambda}{1 + lambda} = 0][- frac{8 + 6lambda}{1 + lambda} = 0][8 + 6lambda = 0][lambda = -frac{4}{3}]Now that we have (lambda), substitute it back into our linear combination equation:[(1 - frac{4}{3})x^2 + (1 - frac{4}{3})y^2 + 4x - 4left(-frac{4}{3}right)y - 3 - 3left(-frac{4}{3}right) = 0][-frac{1}{3}x^2 - frac{1}{3}y^2 + 4x + frac{16}{3}y - 3 + 4 = 0]Multiplying through by (-3) to clear the fractions:[x^2 + y^2 - 12x - 16y - 3 = 0]Thus, the equation of the circle that passes through the intersection points of the given circles and whose center lies on the specified line is:[boxed{x^2 + y^2 - 12x - 16y - 3 = 0}]

🧠:Okay, so I need to find all functions ( f(x) ) defined on the positive real numbers that take positive real values, satisfying two conditions: the functional equation ( f(x f(y)) = y f(x) ) and the limit condition ( lim_{x to infty} f(x) = 0 ). Let's start by breaking down the problem step by step.First, functional equations can sometimes be tricky, but maybe I can find some substitutions or properties that the function must have. Let's look at condition (1): ( f(x f(y)) = y f(x) ). This equation relates the value of the function at ( x f(y) ) to ( y f(x) ). Maybe I can plug in specific values for ( x ) or ( y ) to simplify this equation.Let me try setting ( x = 1 ). Then the equation becomes ( f(1 cdot f(y)) = y f(1) ). Let's denote ( f(1) ) as a constant, say ( c ). So, ( f(f(y)) = c y ). Hmm, that's interesting. This suggests that ( f ) composed with itself is a linear function. That might be a useful property. Also, if ( f(f(y)) = c y ), then ( f ) is invertible? Maybe, but I need to check if ( f ) is injective or surjective.Wait, since the function maps positive reals to positive reals, and if ( f ) is injective (which we might be able to prove), then it would have an inverse. Let's see if we can show injectivity. Suppose ( f(a) = f(b) ). Then, using the functional equation, maybe we can show ( a = b ). Let's see: If ( f(a) = f(b) ), then for any ( x ), ( f(x f(a)) = f(x f(b)) ). By condition (1), this implies ( a f(x) = b f(x) ). Since ( f(x) ) is positive, it's never zero, so we can divide both sides by ( f(x) ), getting ( a = b ). Therefore, ( f ) is injective.Great, so ( f ) is injective. That means it has an inverse function on its image. Now, since ( f(f(y)) = c y ), applying ( f^{-1} ) to both sides gives ( f(y) = f^{-1}(c y) ). Maybe this can help us find the form of ( f ).Alternatively, let's try to find a function that satisfies ( f(f(y)) = c y ). If ( f ) is linear, say ( f(y) = k y ), then ( f(f(y)) = k (k y) = k^2 y ). So, we would have ( k^2 y = c y ), implying ( k^2 = c ). Let's check if such a linear function satisfies the original functional equation.Suppose ( f(y) = k y ). Then the left-hand side of condition (1) is ( f(x f(y)) = f(x k y) = k (x k y) = k^2 x y ). The right-hand side is ( y f(x) = y (k x) = k x y ). For these to be equal for all ( x, y > 0 ), we need ( k^2 x y = k x y ), which simplifies to ( k^2 = k ). So, ( k = 1 ) or ( k = 0 ). But ( f ) takes positive values, so ( k ) can't be zero. Therefore, ( k = 1 ). But then ( f(y) = y ), but let's check if this satisfies the limit condition. If ( f(x) = x ), then ( lim_{x to infty} f(x) = infty ), which doesn't satisfy condition (2). So, this linear function doesn't work. Hmm.Wait, but earlier we thought ( f(f(y)) = c y ). If ( f ) is linear, ( f(y) = k y ), then ( c = k^2 ). But in the original functional equation, we saw that ( k^2 = k ), which only allows ( k = 1 ). Since that doesn't work, maybe ( f ) isn't linear. So, perhaps another functional form.Let me think of other standard functions. Maybe exponential functions? Suppose ( f(y) = y^k ) for some constant ( k ). Let's test this. Then ( f(x f(y)) = f(x y^k) = (x y^k)^k = x^k y^{k^2} ). On the other hand, the right-hand side is ( y f(x) = y x^k ). So, we need ( x^k y^{k^2} = y x^k ), which implies ( y^{k^2} = y ). For this to hold for all ( y > 0 ), we need ( k^2 = 1 ), so ( k = 1 ) or ( k = -1 ). But ( f(y) = y^{-1} = 1/y ). Let's check this.If ( f(y) = 1/y ), then the left-hand side of condition (1) is ( f(x f(y)) = f(x / y) = 1 / (x / y) = y / x ). The right-hand side is ( y f(x) = y (1 / x ) = y / x ). So, equality holds! So, ( f(y) = 1/y ) satisfies the functional equation. Now, does it satisfy the limit condition? As ( x to infty ), ( f(x) = 1/x to 0 ). Yes, so that works. So, ( f(x) = 1/x ) is a solution.But the problem says "find all functions", so maybe this is the only solution? Or are there others?Wait, earlier when I tried the linear function, the only possible linear solution was ( f(x) = x ), which didn't satisfy the limit, but ( f(x) = 1/x ) is a reciprocal function. Maybe there's a family of solutions, but perhaps only ( f(x) = 1/x ) satisfies both conditions.But let's check if there are other functions. Maybe we can derive the general solution.Starting again from the functional equation ( f(x f(y)) = y f(x) ). Let me try to find some substitutions or variable changes. Let's denote ( z = x f(y) ). Then, the equation becomes ( f(z) = y f(x) ). But ( z = x f(y) ), so maybe express ( x ) in terms of ( z ): ( x = z / f(y) ). Substituting back, we get ( f(z) = y f(z / f(y)) ). Hmm, not sure if that helps.Alternatively, let's look for multiplicative or additive properties. Suppose we set ( x = 1 ), which gives ( f(f(y)) = c y ), where ( c = f(1) ). Also, maybe set ( y = 1 ) in the original equation. Let's try that.Setting ( y = 1 ), equation becomes ( f(x f(1)) = 1 cdot f(x) ). So, ( f(c x) = f(x) ), since ( c = f(1) ). Therefore, ( f(c x) = f(x) ). Let's denote this as equation (3).So, equation (3) says that scaling the argument by ( c ) doesn't change the function's value. If ( c neq 1 ), this suggests some periodicity or scaling property. For example, if ( c > 1 ), then ( f(c x) = f(x) ), which implies ( f(x) = f(x / c) ), so ( f(x) ) is a periodic function in the logarithmic scale. Similarly, if ( c < 1 ), same idea. However, given that the limit as ( x to infty ) is 0, this might restrict possible values of ( c ).Wait, let's suppose ( c neq 1 ). If ( c > 1 ), then ( f(c x) = f(x) ). Applying this repeatedly, ( f(c^n x) = f(x) ) for any integer ( n ). If we take ( x = 1 ), then ( f(c^n) = f(1) = c ). But as ( n to infty ), ( c^n to infty ) (since ( c > 1 )), so ( f(c^n) = c ). But the limit condition says ( lim_{x to infty} f(x) = 0 ). However, ( c ) is a positive constant, so unless ( c = 0 ), which it can't be since ( f ) maps to positive reals, this would contradict the limit condition. Therefore, ( c > 1 ) is impossible.Similarly, if ( c < 1 ), then ( c^n to 0 ) as ( n to infty ). Then, ( f(c^n x) = f(x) ). If we fix ( x ) and let ( n to infty ), then ( c^n x to 0 ). But the function is defined for positive reals, and we don't have information about the limit as ( x to 0 ). However, the limit condition is as ( x to infty ), so maybe this isn't a contradiction. However, even if ( c < 1 ), let's see. If we set ( x = c^{-n} ), then ( f(c cdot c^{-n}) = f(c^{-(n-1)}) = f(c^{-n}) ). So, by induction, ( f(c^{-n}) = f(1) = c ). But as ( n to infty ), ( c^{-n} to infty ), so ( f(c^{-n}) = c ), but the limit condition requires that ( f(x) to 0 ) as ( x to infty ). Therefore, unless ( c = 0 ), which is impossible, this would also contradict the limit condition. Thus, ( c < 1 ) is also impossible.Therefore, the only possibility is ( c = 1 ). So, ( f(1) = 1 ). Then, equation (3) becomes ( f(1 cdot x) = f(x) ), which is trivial. So, this case doesn't give us new information. But earlier, when we set ( x = 1 ), we had ( f(f(y)) = c y = 1 cdot y ), so ( f(f(y)) = y ). Therefore, ( f ) is an involution, meaning ( f ) is its own inverse: ( f^{-1}(y) = f(y) ).So, ( f ) is bijective (since it's injective and surjective onto its codomain, which is positive reals because it's invertible). So, ( f ) is a bijective function satisfying ( f(f(y)) = y ). So, it's an involution.Now, combining this with the original functional equation: ( f(x f(y)) = y f(x) ). Since ( f ) is invertible and ( f^{-1}(y) = f(y) ), maybe we can apply ( f ) to both sides of the equation. Let's see:Applying ( f ) to both sides gives ( f(f(x f(y))) = f(y f(x)) ). But the left-hand side is ( f(f(x f(y))) = x f(y) ), since ( f(f(z)) = z ). The right-hand side is ( f(y f(x)) ). So, we have ( x f(y) = f(y f(x)) ).But the original functional equation is ( f(x f(y)) = y f(x) ). Comparing the two, we have:From the original: ( f(x f(y)) = y f(x) ).From applying ( f ) to both sides: ( x f(y) = f(y f(x)) ).So, equating these two expressions for ( f(y f(x)) ):From the original equation, swapping x and y: If we swap x and y in the original equation, we get ( f(y f(x)) = x f(y) ). But that's exactly the same as the equation we just got by applying ( f ) to both sides. So, no new information here.Alternatively, maybe we can relate these two equations. Let me write them again:1. ( f(x f(y)) = y f(x) ).2. ( x f(y) = f(y f(x)) ).So, equation 2 is derived from equation 1 by applying ( f ) to both sides and using the involution property. So, these are consistent.Perhaps we can find a relationship between ( f(x) ) and ( x ). Let's assume that ( f ) is a multiplicative function, or maybe something else. Let me test the function ( f(x) = k/x ), where ( k ) is a constant. Then, check if this satisfies the functional equation.Let ( f(x) = k/x ). Then, compute the left-hand side of equation (1): ( f(x f(y)) = f(x cdot k/y) = k / (x cdot k/y) = k / ( (x k)/y ) = (k y)/(x k) ) = y / x ).The right-hand side is ( y f(x) = y cdot (k / x) = (k y)/x ).Comparing the two sides: left-hand side is ( y / x ), right-hand side is ( k y / x ). Therefore, to have equality, we need ( k = 1 ). So, ( f(x) = 1/x ). Which matches the earlier solution. Therefore, ( f(x) = 1/x ) is indeed a solution. And since we had to set ( k = 1 ), this is the only function of the form ( k/x ) that works.But maybe there's a more general family. Let's suppose that ( f ) is a power function, ( f(x) = x^k ). Wait, we tried this earlier and found ( k = -1 ) works. Let me confirm again. If ( f(x) = x^k ), then ( f(x f(y)) = (x f(y))^k = (x y^k)^k = x^k y^{k^2} ). The right-hand side is ( y f(x) = y x^k ). So, equating these, ( x^k y^{k^2} = y x^k ). Therefore, ( y^{k^2} = y ), which implies ( k^2 = 1 ), so ( k = 1 ) or ( k = -1 ). As before, ( k = 1 ) doesn't satisfy the limit, so only ( k = -1 ). So, ( f(x) = 1/x ) is the only power function solution.But could there be non-power function solutions? Let's see. Suppose there exists another function that isn't a power function. For example, maybe a function combined with exponentials or logarithms. Let's think.Given that ( f ) is an involution (( f(f(y)) = y )), perhaps logarithmic or exponential functions could satisfy this. For instance, ( f(y) = ln(a/y) ) or something, but that might not preserve positive real numbers. Wait, if ( f(y) = ln(a/y) ), then for ( y ) positive, ( a/y ) must be greater than 1 to have a positive logarithm. But that complicates things. Maybe not.Alternatively, consider functions of the form ( f(y) = k/y ), but we already saw that only ( k = 1 ) works. So perhaps ( 1/x ) is the only solution. But how can we be sure?Let me try to find the general solution. Suppose ( f ) is a bijection (since it's an involution) and satisfies ( f(x f(y)) = y f(x) ). Let's make a substitution. Let me set ( x = f(z) ). Then, the left-hand side becomes ( f(f(z) f(y)) ). Let's compute this:( f(f(z) f(y)) = y f(f(z)) ) [by the original equation, since ( x = f(z) )]But ( f(f(z)) = z ), so this becomes ( y z ).Therefore, ( f(f(z) f(y)) = y z ).But since ( f ) is an involution, ( f(f(z) f(y)) = f( f(y) f(z) ) ). So, ( f( f(y) f(z) ) = y z ).Therefore, ( f( f(y) f(z) ) = y z ). Let me denote ( u = f(y) ) and ( v = f(z) ). Then, since ( f ) is a bijection, ( u ) and ( v ) can be any positive real numbers. Therefore, ( f(u v) = f^{-1}(u) f^{-1}(v) ). But since ( f ) is an involution, ( f^{-1}(u) = f(u) ). Therefore, ( f(u v) = f(u) f(v) ).Ah, so ( f(u v) = f(u) f(v) ) for all ( u, v > 0 ). That is, ( f ) is a multiplicative function. And combined with the fact that ( f ) is bijective and satisfies ( lim_{x to infty} f(x) = 0 ).So, now we have a multiplicative function: ( f(u v) = f(u) f(v) ). The multiplicative functions on the positive reals are known to be power functions, provided they are continuous, but the problem doesn't specify continuity. However, often in functional equations without continuity assumptions, we can still derive that the function must be a power function if we have other conditions. Let's see.But first, combining this with the involution property ( f(f(u)) = u ). Suppose ( f(u) = u^k ). Then, ( f(f(u)) = (u^k)^k = u^{k^2} ). For this to equal ( u ), we must have ( k^2 = 1 ), so ( k = 1 ) or ( k = -1 ). As before, ( k = 1 ) doesn't satisfy the limit, so ( k = -1 ), so ( f(u) = 1/u ).But if we don't assume continuity, are there other solutions? For multiplicative functions, even non-continuous ones, they can be constructed using Hamel bases, but such functions are not expressible in closed form and are generally considered pathological. However, the problem doesn't restrict to continuous functions, but the limit condition might constrain us.Suppose there is a non-continuous multiplicative function ( f ) satisfying ( f(f(u)) = u ) and ( lim_{x to infty} f(x) = 0 ). However, multiplicative functions can be written as ( f(x) = x^k ) for some exponent ( k ) if they are continuous, but without continuity, there could be more solutions. However, the condition ( f(f(u)) = u ) forces ( f ) to be an involution, which for multiplicative functions implies ( f(f(u)) = f(u)^k = u^{k^2} = u ). So, even for non-continuous multiplicative functions, this would require ( k^2 = 1 ), hence ( k = 1 ) or ( k = -1 ). But ( k = 1 ) doesn't satisfy the limit, so only ( k = -1 ). Therefore, even considering non-continuous functions, ( f(x) = 1/x ) is the only multiplicative solution.Therefore, combining all these steps, the only function that satisfies both conditions is ( f(x) = 1/x ).Wait, let's verify once more. If ( f(x) = 1/x ), then:1. Check condition (1): ( f(x f(y)) = f(x cdot 1/y) = 1 / (x/y) = y/x ). On the other hand, ( y f(x) = y cdot 1/x = y/x ). So, equality holds.2. Check condition (2): ( lim_{x to infty} f(x) = lim_{x to infty} 1/x = 0 ). So, that's satisfied.Therefore, ( f(x) = 1/x ) works. And based on the above reasoning, it's the only function that does so.Just to ensure there are no other possible functions, let's recap the steps:- By setting ( x = 1 ), we found ( f(f(y)) = c y ), and later deduced ( c = 1 ) due to the limit condition, leading to ( f(f(y)) = y ).- Setting ( y = 1 ), we found ( f(c x) = f(x) ), which forced ( c = 1 ).- This led us to the conclusion that ( f ) is an involution and multiplicative, ( f(u v) = f(u) f(v) ).- Multiplicative involution functions must satisfy ( f(u) = u^{-1} ), given the limit condition.Therefore, the only solution is ( f(x) = 1/x ).Final AnswerThe function is boxed{dfrac{1}{x}}.

✅:To find all functions ( f(x) ) defined on the set of positive real numbers and taking values in positive real numbers that satisfy the given conditions, we start by analyzing the functional equation ( f(x f(y)) = y f(x) ) and the limit condition ( lim_{x to infty} f(x) = 0 ).1. Setting ( x = 1 ) in the functional equation gives ( f(f(y)) = f(1) y ). Let ( c = f(1) ), then ( f(f(y)) = c y ).2. Setting ( y = 1 ) in the functional equation gives ( f(c x) = f(x) ). This implies that ( f ) is periodic with period ( c ). However, considering the limit condition ( lim_{x to infty} f(x) = 0 ), we deduce that ( c ) must be 1. Thus, ( f(f(y)) = y ), meaning ( f ) is an involution.3. Considering the functional equation ( f(x f(y)) = y f(x) ) and using the involution property ( f(f(y)) = y ), we substitute ( x = f(z) ) and find that ( f(f(z) f(y)) = y z ). This leads to the multiplicative property ( f(u v) = f(u) f(v) ).4. Multiplicative functions on the positive reals are power functions. Given the involution property ( f(f(u)) = u ), the only power function that satisfies this is ( f(u) = u^{-1} ). This function also satisfies the limit condition ( lim_{x to infty} f(x) = 0 ).5. Verification: Checking ( f(x) = frac{1}{x} ) in the functional equation confirms it satisfies both conditions. For ( f(x f(y)) = fleft(x cdot frac{1}{y}right) = frac{y}{x} ) and ( y f(x) = y cdot frac{1}{x} = frac{y}{x} ). The limit condition is satisfied as ( lim_{x to infty} frac{1}{x} = 0 ).Thus, the only function that satisfies both conditions is (boxed{dfrac{1}{x}}).

🔑:We need to find all functions ( f(x) ) defined on the set of positive real numbers, satisfying the following properties:1. ( f(x f(y)) = y f(x) )2. ( lim_{x to infty} f(x) = 0 )Let's break this problem down step by step.1. First, we will show that 1 is in the range of ( f ). - For any ( x_0 > 1 ), let ( y_0 = frac{1}{f(x_0)} ). - Substitute ( y = y_0 ) into property (1): [ f(x_0 f(y_0)) = y_0 f(x_0) ] Given ( y_0 = frac{1}{f(x_0)} ), it follows that: [ y_0 f(x_0) = 1 ] Thus, [ f(x_0 f(y_0)) = 1 ] Hence, 1 is in the range of ( f ).2. Next, let ( y_0 ) be such that ( f(y_0) = 1 ). - Substituting ( y = y_0 ) into property (1), we get: [ f(x f(y_0)) = y_0 f(x) ] Since ( f(y_0) = 1 ): [ f(x) = y_0 f(x) ] This implies ( y_0 = 1 ). Hence, ( f(1) = 1 ).3. For any positive real number ( x ), let ( x f(x) = a ). - Substituting ( y = x ) into property (1), we get: [ f(a) = x f(x) ] Since ( x f(x) = a ), it follows that ( f(a) = a ). Thus, ( f(a) = a ).4. If ( a > 1 ): - Then we have: [ f(a^2) = f(a cdot f(a)) = a f(a) = a^2 ] - Using induction and the property of ( f ), it is easy to see: [ f(a^{2^n}) = a^{2^n} rightarrow +infty quad text{as} quad n rightarrow infty ] - This contradicts property (2) that ( lim_{x to infty} f(x) = 0 ).5. If ( a < 1 ): - Then we have: [ f(1) = fleft( frac{1}{a} f(a) right) = a fleft( frac{1}{a} right) ] Since ( f(a) = a ), it follows that: [ fleft( frac{1}{a} right) = frac{1}{a} ] - Given ( frac{1}{a} > 1 ), using the earlier discussion (replacing ( frac{1}{a} ) with ( a )), we see that this leads to a similar contradiction.Therefore, the only remaining possibility is:[a = 1]Thus, ( f(x) = frac{1}{x} ) for all ( x ).It is easy to verify that ( f(x) = frac{1}{x} ) satisfies both properties (1) and (2). Hence, the solution is:[boxed{f(x) = frac{1}{x}}]

🧠:Okay, let's see. I need to find the maximum value of the modulus of the complex expression |z³ + z² -5z +3| where |z| = 1. Hmm, complex numbers on the unit circle. Since |z|=1, z can be represented as e^(iθ) for some angle θ. Maybe converting it to trigonometric form will help? Let me think.First, since |z| =1, z = cosθ + i sinθ. Then z³, z², etc., can be expanded using De Moivre's theorem. But expanding all of them might get complicated. Alternatively, maybe using the fact that for |z|=1, the expression can be simplified somehow? Let me try substituting z = e^(iθ) and compute the expression.So, let's write z = e^(iθ). Then:z³ + z² -5z +3 = e^(3iθ) + e^(2iθ) -5e^(iθ) +3.Now, we need the modulus of this expression. The modulus squared might be easier to compute. Remember that |a + b|² = |a|² + |b|² + 2Re(aoverline{b}). But this might get messy with four terms. Alternatively, maybe we can express the entire expression in terms of sine and cosine.Expressing each term:e^(inθ) = cos(nθ) + i sin(nθ). So,z³ = cos3θ + i sin3θ,z² = cos2θ + i sin2θ,-5z = -5cosθ - i5sinθ,and the constant term is 3. So adding all together:Real parts: cos3θ + cos2θ -5cosθ +3,Imaginary parts: sin3θ + sin2θ -5sinθ.So the entire expression is [cos3θ + cos2θ -5cosθ +3] + i[sin3θ + sin2θ -5sinθ].The modulus of this is sqrt[(real part)^2 + (imaginary part)^2]. To find the maximum value of this modulus, we need to maximize the expression sqrt[(cos3θ + cos2θ -5cosθ +3)^2 + (sin3θ + sin2θ -5sinθ)^2].Hmm, squaring it, we can just maximize the square to make it easier: (cos3θ + cos2θ -5cosθ +3)^2 + (sin3θ + sin2θ -5sinθ)^2.Let me compute this squared modulus. Let's denote A = cos3θ + cos2θ -5cosθ +3 and B = sin3θ + sin2θ -5sinθ, so the modulus squared is A² + B².Expanding A² + B²:= [cos3θ + cos2θ -5cosθ +3]^2 + [sin3θ + sin2θ -5sinθ]^2.This looks like the sum of squares of real and imaginary parts, which for a complex number expression is equal to the modulus squared. So perhaps we can consider this entire expression as the modulus squared of (z³ + z² -5z +3). Alternatively, maybe there's a smarter way to compute this modulus without expanding everything.Wait, since z has modulus 1, then overline{z} = 1/z. Maybe using that property to simplify the expression. Let's see.Let me consider the original expression: z³ + z² -5z +3. Let's see if we can factor or rewrite it somehow.Alternatively, perhaps considering that |z³ + z² -5z +3| can be rewritten by grouping terms. For example, z³ + z² = z²(z +1), so maybe:|z²(z +1) -5z +3|. Hmm, not sure if that helps. Or maybe factor out z from some terms? Let's see:z³ + z² -5z +3 = z²(z +1) -5z +3. Still not obvious.Alternatively, note that z³ + z² = z²(z +1), and then the rest is -5z +3. But maybe that's not helpful.Alternatively, perhaps note that for |z|=1, we can write the expression in terms of z and overline{z}. But since overline{z} =1/z, maybe substituting that in. Let's see.But perhaps another approach is to use the fact that |a + b| ≤ |a| + |b|, but since we need the maximum value, triangle inequality gives upper bounds, but maybe not tight. Alternatively, using the reverse triangle inequality? Not sure.Alternatively, perhaps parametrizing θ and then optimizing over θ. Since z is on the unit circle, θ is from 0 to 2π. Then we can write the modulus squared as a function of θ and find its maximum.So let's compute the modulus squared:|z³ + z² -5z +3|² = |e^(3iθ) + e^(2iθ) -5e^(iθ) +3|².Expanding this using |a + b + c + d|²:= |e^(3iθ) + e^(2iθ)|² + | -5e^(iθ) +3|² + 2Re[(e^(3iθ) + e^(2iθ))(overline{-5e^(iθ) +3})].Wait, that might be too complicated. Alternatively, compute the square directly:Let me denote each term:Term1: e^(3iθ),Term2: e^(2iθ),Term3: -5e^(iθ),Term4: 3.So, modulus squared is:|Term1 + Term2 + Term3 + Term4|²= |Term1 + Term2|² + |Term3 + Term4|² + 2Re[(Term1 + Term2)(overline{Term3 + Term4})]Wait, actually, no. The general formula is |A + B|² = |A|² + |B|² + 2Re(Aoverline{B}), where A and B are complex numbers.But here, A = Term1 + Term2, B = Term3 + Term4. So yes, it's |A + B|² = |A|² + |B|² + 2Re(Aoverline{B}).First compute |A|² where A = e^(3iθ) + e^(2iθ):|A|² = |e^(3iθ) + e^(2iθ)|² = |e^(2iθ)(e^(iθ) +1)|² = |e^(2iθ)|² |e^(iθ) +1|² = 1 * |e^(iθ) +1|² = |e^(iθ) +1|².But |e^(iθ) +1|² = (cosθ +1)^2 + (sinθ)^2 = cos²θ + 2cosθ +1 + sin²θ = 2 + 2cosθ. So |A|² = 2 + 2cosθ.Then |B|² where B = -5e^(iθ) +3:|B|² = |-5e^(iθ) +3|² = (-5cosθ +3)^2 + (-5sinθ)^2 = 25cos²θ -30cosθ +9 +25sin²θ = 25(cos²θ + sin²θ) -30cosθ +9 = 25 -30cosθ +9 = 34 -30cosθ.Now the cross term 2Re(Aoverline{B}).First compute Aoverline{B}:A = e^(3iθ) + e^(2iθ),overline{B} = overline{-5e^(iθ) +3} = -5e^(-iθ) +3.Therefore, Aoverline{B} = (e^(3iθ) + e^(2iθ))(-5e^(-iθ) +3)= -5e^(3iθ)e^(-iθ) + 3e^(3iθ) -5e^(2iθ)e^(-iθ) +3e^(2iθ)= -5e^(2iθ) +3e^(3iθ) -5e^(iθ) +3e^(2iθ)Combine like terms:-5e^(2iθ) +3e^(3iθ) -5e^(iθ) +3e^(2iθ) = (-5 +3)e^(2iθ) +3e^(3iθ) -5e^(iθ)= -2e^(2iθ) +3e^(3iθ) -5e^(iθ)Therefore, Re(Aoverline{B}) is the real part of this expression:Re(-2e^(2iθ) +3e^(3iθ) -5e^(iθ)) = -2cos2θ +3cos3θ -5cosθ.Hence, the cross term is 2*(-2cos2θ +3cos3θ -5cosθ) = -4cos2θ +6cos3θ -10cosθ.Therefore, putting it all together, the modulus squared is:|A|² + |B|² + 2Re(Aoverline{B}) = (2 + 2cosθ) + (34 -30cosθ) + (-4cos2θ +6cos3θ -10cosθ)Simplify term by term:Constants: 2 +34 = 36.cosθ terms: 2cosθ -30cosθ -10cosθ = (2 -30 -10)cosθ = -38cosθ.Then the other terms: -4cos2θ +6cos3θ.So modulus squared = 36 -38cosθ -4cos2θ +6cos3θ.So the problem reduces to maximizing the function f(θ) = 36 -38cosθ -4cos2θ +6cos3θ over θ ∈ [0, 2π).To find the maximum of this expression, we can take the derivative and set it to zero. But trigonometric functions might be a bit involved, but let's try.First, compute the derivative f’(θ):f’(θ) = 38sinθ +8sin2θ -18sin3θ.Set derivative to zero:38sinθ +8sin2θ -18sin3θ =0.Hmm, solving this equation might be complicated. Alternatively, we can use trigonometric identities to simplify the expression f(θ).Let me recall that cos3θ = 4cos³θ -3cosθ, cos2θ = 2cos²θ -1.Substituting these into f(θ):f(θ) = 36 -38cosθ -4*(2cos²θ -1) +6*(4cos³θ -3cosθ)= 36 -38cosθ -8cos²θ +4 +24cos³θ -18cosθCombine like terms:Constants: 36 +4 =40.cosθ terms: -38cosθ -18cosθ = -56cosθ.cos²θ terms: -8cos²θ.cos³θ terms: 24cos³θ.So, f(θ) = 24cos³θ -8cos²θ -56cosθ +40.Let me denote x = cosθ. Then since θ ∈ [0, 2π), x ∈ [-1, 1]. Therefore, f(θ) becomes:f(x) = 24x³ -8x² -56x +40.We need to find the maximum of f(x) over x ∈ [-1, 1].So now the problem is a calculus problem: find maximum of f(x) =24x³ -8x² -56x +40 on [-1,1].First, compute critical points by taking derivative:f’(x) = 72x² -16x -56.Set f’(x)=0:72x² -16x -56 =0.Divide all terms by 8:9x² -2x -7 =0.Use quadratic formula:x = [2 ± sqrt(4 + 252)] / (2*9) = [2 ± sqrt(256)] /18 = [2 ±16]/18.Thus,x = (2 +16)/18 =18/18=1,x=(2 -16)/18 = (-14)/18 = -7/9 ≈ -0.777...So critical points at x=1 and x=-7/9.But we need to check if these are in the interval [-1,1]. x=1 is at the endpoint, x=-7/9 ≈-0.777 is inside.So now evaluate f(x) at x=-1, x=1, and x=-7/9.First, at x=1:f(1)=24*(1)^3 -8*(1)^2 -56*(1)+40=24 -8 -56 +40= (24+40) - (8+56)=64 -64=0.At x=-1:f(-1)=24*(-1)^3 -8*(-1)^2 -56*(-1)+40= -24 -8 +56 +40= (-24-8)+(56+40)= -32 +96=64.At x=-7/9:Compute f(-7/9):First, compute each term:24x³: 24*(-7/9)^3 =24*(-343)/(729)= (-24*343)/729.Compute 24*343: 24*300=7200, 24*43=1032, total=7200+1032=8232. So 8232/729. Negative: -8232/729.-8x²: -8*(49/81)= -392/81.-56x: -56*(-7/9)=392/9.+40: 40.So f(-7/9)= (-8232/729) -392/81 +392/9 +40.Convert all terms to 729 denominator:-8232/729,-392/81 = (-392*9)/729= -3528/729,392/9 = (392*81)/729= 31752/729,40 = 40*729/729=29160/729.So total:(-8232 -3528 +31752 +29160)/729.Calculate numerator:First, -8232 -3528 = -11760.Then, -11760 +31752= 19992.19992 +29160= 49152.Thus, f(-7/9)=49152/729.Simplify:Divide numerator and denominator by 3:49152 ÷3=16384,729 ÷3=243.16384/243 ≈16384 ÷243≈67.42.Wait, 243*67=243*60=14580, 243*7=1701, so 14580+1701=16281. Then 16384 -16281=103. So 67 +103/243≈67.42.So approximately 67.42. Let me check exact value:49152 ÷729:Divide numerator and denominator by 3:49152 ÷3=16384; 729 ÷3=243.16384 ÷243: 243*67=16281, 16384-16281=103. So 16384/243=67 +103/243≈67.423.So f(-7/9)≈67.423.Comparing the values:At x=-1, f(-1)=64,At x=1, f(1)=0,At x=-7/9, f≈67.423.Therefore, the maximum on [-1,1] is approximately 67.423 at x=-7/9. So the maximum modulus squared is approximately 67.423, so the modulus is sqrt(67.423)≈8.21.But let's compute it exactly. Since f(-7/9)=49152/729.So sqrt(49152/729)=sqrt(49152)/sqrt(729)=sqrt(49152)/27.Compute sqrt(49152):Note that 49152= 16384*3, because 16384*3=49152. And sqrt(16384*3)=sqrt(16384)*sqrt(3)=128*sqrt(3). Therefore, sqrt(49152)=128√3. Hence, sqrt(49152)/27= (128√3)/27.Therefore, the maximum modulus is sqrt(49152/729)=128√3/27.Wait, let's check that again:49152=16384*3, yes.sqrt(16384*3)=sqrt(16384)*sqrt(3)=128√3. Then, sqrt(49152)/27=128√3/27.Yes, because sqrt(49152)=sqrt(16384*3)=128√3. Therefore, modulus is 128√3 /27.Compute 128/27≈4.7407, √3≈1.732, so 4.7407*1.732≈8.21, which matches the approximation.Therefore, the maximum modulus is 128√3/27.But let me verify the calculation step by step to be sure. Starting from f(-7/9)=24*(-7/9)^3 -8*(-7/9)^2 -56*(-7/9) +40.First term: 24*(-343)/(729)= -24*343/729.24*343: 24*300=7200, 24*43=1032, total=7200+1032=8232. So first term= -8232/729.Second term: -8*(49/81)= -392/81.Third term: -56*(-7/9)= 392/9.Fourth term: +40.Now, converting all to denominator 729:First term: -8232/729.Second term: -392/81= -392*9/729= -3528/729.Third term: 392/9=392*81/729=31752/729.Fourth term:40=40*729/729=29160/729.Adding them up:-8232 -3528 +31752 +29160= (-8232 -3528) + (31752 +29160)= (-11760) + (60912)=49152.So 49152/729.Then sqrt(49152/729)=sqrt(49152)/sqrt(729)=sqrt(16384*3)/27= (128√3)/27. Yes, correct.So the maximum modulus is 128√3/27.But let me check if there could be a mistake in the modulus squared expression.Earlier steps: computed modulus squared as 36 -38cosθ -4cos2θ +6cos3θ, then converted to polynomial in x=cosθ, leading to f(x)=24x³ -8x² -56x +40. Then found critical points at x=1 and x=-7/9. Then evaluated f at x=-1, x=1, x=-7/9.Wait, but when converting cos3θ and cos2θ in terms of x=cosθ, the substitution steps:Original expression after expansion:36 -38cosθ -4cos2θ +6cos3θ.Expressed as:36 -38x -4*(2x² -1) +6*(4x³ -3x)= 36 -38x -8x² +4 +24x³ -18x=24x³ -8x² -56x +40.Yes, that's correct.So, all steps are correct, so maximum modulus squared is 49152/729, which simplifies to (128√3)/27 when square rooted.So the maximum value of |z³ + z² -5z +3| is 128√3 /27.But let me check with another approach to verify.Alternatively, perhaps using Lagrange multipliers. Let me consider z as a complex number on the unit circle, so z = x + iy with x² + y²=1. Then the expression is |z³ + z² -5z +3|. But this might be complicated as we have to deal with real and imaginary parts.Alternatively, parametrize z as e^(iθ) and compute the modulus. Wait, that's what I did earlier.Alternatively, note that 128√3 /27 is approximately 128*1.732/27≈221.7/27≈8.21. Let me check with specific θ values.For example, when θ=π, z=-1. Compute |(-1)^3 + (-1)^2 -5*(-1) +3| = |-1 +1 +5 +3| = |8| =8. Which is less than 8.21, so that's consistent.At θ=0, z=1. |1 +1 -5 +3|=|0|=0. Which matches f(1)=0.At θ=π/3, z=0.5 +i√3/2. Compute the expression:z³: (0.5 +i√3/2)^3. Let's compute:First, (0.5 +i√3/2) is e^(iπ/3). So z³ = e^(iπ) = -1.z² = e^(i2π/3)= -0.5 +i√3/2.-5z = -5*(0.5 +i√3/2)= -2.5 -i(5√3)/2.3 remains 3.Adding together: z³ + z² -5z +3 = (-1) + (-0.5 +i√3/2) + (-2.5 -i5√3/2) +3.Real parts: -1 -0.5 -2.5 +3= (-4) +3= -1.Imaginary parts: √3/2 -5√3/2= (-4√3)/2= -2√3.Modulus: sqrt((-1)^2 + (-2√3)^2)=sqrt(1 +12)=sqrt(13)≈3.605. Less than 8.21, so not maximum.Another angle, maybe θ where cosθ=-7/9. Let's compute θ such that cosθ=-7/9. Then sinθ=±sqrt(1 -49/81)=±sqrt(32/81)=±4√2/9.Then let's compute the expression |z³ + z² -5z +3| when z is at θ where cosθ=-7/9.But computing this directly might be tedious, but since we've already done the calculus approach and found maximum at x=-7/9, which corresponds to θ= arccos(-7/9). So the maximum modulus is indeed 128√3 /27≈8.21, which is achieved when cosθ=-7/9.Therefore, after thorough calculations and verification, the maximum value is 128√3 divided by 27.Final AnswerThe maximum value is boxed{dfrac{128sqrt{3}}{27}}.

✅:Given the complex number ( z ) satisfying ( |z| = 1 ), we need to find the maximum value of ( left|z^3 + z^2 - 5z + 3right| ).1. Representation of ( z ): Since ( |z| = 1 ), we can write ( z = e^{itheta} ) where ( theta ) is a real number.2. Expression in terms of ( theta ): Substitute ( z = e^{itheta} ) into the expression: [ z^3 + z^2 - 5z + 3 = e^{3itheta} + e^{2itheta} - 5e^{itheta} + 3 ]3. Modulus squared: The modulus squared of the expression is: [ left| e^{3itheta} + e^{2itheta} - 5e^{itheta} + 3 right|^2 ] Expanding this, we separate real and imaginary parts: [ text{Real part: } cos 3theta + cos 2theta - 5cos theta + 3 ] [ text{Imaginary part: } sin 3theta + sin 2theta - 5sin theta ] The modulus squared is: [ (cos 3theta + cos 2theta - 5cos theta + 3)^2 + (sin 3theta + sin 2theta - 5sin theta)^2 ]4. Simplification: Using trigonometric identities and simplifying, we find the modulus squared expression: [ 36 - 38cos theta - 4cos 2theta + 6cos 3theta ]5. Conversion to polynomial: Substitute ( x = cos theta ) and use trigonometric identities to convert the expression into a polynomial: [ f(x) = 24x^3 - 8x^2 - 56x + 40 ]6. Critical points: Find the critical points by solving the derivative ( f'(x) = 72x^2 - 16x - 56 ): [ x = 1 quad text{and} quad x = -frac{7}{9} ]7. Evaluation: Evaluate ( f(x) ) at ( x = -1 ), ( x = 1 ), and ( x = -frac{7}{9} ): - ( f(1) = 0 ) - ( f(-1) = 64 ) - ( fleft(-frac{7}{9}right) = frac{49152}{729} )8. Maximum modulus: The maximum value of the modulus squared is ( frac{49152}{729} ). Taking the square root gives the maximum modulus: [ sqrt{frac{49152}{729}} = frac{128sqrt{3}}{27} ]Thus, the maximum value of ( left|z^3 + z^2 - 5z + 3right| ) is boxed{dfrac{128sqrt{3}}{27}}.

🔑:1. We start with the given condition that ( |z| = 1 ). This implies that ( z ) lies on the unit circle in the complex plane.2. The expression we need to maximize is ( left|z^3 + z^2 - 5z + 3right| ).3. Factor the expression ( z^3 + z^2 - 5z + 3 ). We get [z^3 + z^2 - 5z + 3 = (z-1)^2(z+3)]4. Next, let us consider ( z ) on the unit circle, and represent ( z ) in terms of its polar form:[z = cos theta + mathrm{i} sin theta]5. Using the modulus properties, we have:[left|z^3 + z^2 - 5z + 3 right| = |(z-1)^2 (z+3)|]6. Calculate ( |z-1| ):[|z-1| = left| cos theta + mathrm{i} sin theta - 1 right| = sqrt{(cos theta - 1)^2 + sin^2 theta} = sqrt{(cos theta - 1)^2 + (1 - cos^2 theta)} = sqrt{2 - 2cos theta} = sqrt{2(1 - cos theta)}]7. Calculate ( |z+3| ):[|z+3| = left| cos theta + 3 + mathrm{i} sin theta right| = sqrt{( cos theta + 3 )^2 + sin^2 theta} = sqrt{cos^2 theta + 6 cos theta + 9 + sin^2 theta} = sqrt{10 + 6 cos theta}]8. Therefore, the expression becomes:[left|z^3 + z^2 - 5z + 3 right| = |(z-1)^2| |(z+3)| = |z-1|^2 |z+3|]Substituting the above values, we get:[|z-1|^2 = ( sqrt{2(1 - cos theta)} )^2 = 2(1 - cos theta)]And[|z+3| = sqrt{10 + 6 cos theta}]9. Thus:[left|z^3 + z^2 - 5z + 3 right| = 2(1 - cos theta) sqrt{10 + 6 cos theta}]10. Next, we simplify the right-hand side using the identity:[= frac{2}{3} sqrt{(3 - 3 cos theta)^2 (10 + 6 cos theta)}]11. Now, we need to maximize this expression. Consequently, we solve:[= frac{2}{3} sqrt{left( frac{16}{3} right)^3} = frac{2}{3} cdot frac{64 cdot sqrt{3}}{3} = frac{128 sqrt{3}}{27}]12. Solving for (theta), we find (cos theta = -frac{7}{9}). Conclusion:The maximum value of ( left|z^3 + z^2 - 5z + 3 right| ) is thus:[boxed{frac{128 sqrt{3}}{27}}]

🧠:Okay, so I have this problem here where I need to prove that the average of a sequence of real numbers is at least -1/2. The sequence starts with a₁ = 0, and each subsequent term satisfies |aₖ| = |aₖ₋₁ + 1| for k from 2 to n. Hmm, let's see. I need to show that the average (1/n)(a₁ + a₂ + ... + aₙ) ≥ -1/2. First, let me try to understand the recurrence relation here. The absolute value of each term aₖ is equal to the absolute value of the previous term plus 1. That is, |aₖ| = |aₖ₋₁ + 1|. But since we're dealing with absolute values, this means that aₖ can be either aₖ₋₁ + 1 or -(aₖ₋₁ + 1). So each term has two possibilities. But starting from a₁ = 0, let's see how the sequence can progress. Let me compute the first few terms manually to get a sense of the pattern.For k = 2: |a₂| = |a₁ + 1| = |0 + 1| = 1. So a₂ can be 1 or -1.Case 1: a₂ = 1.Then for k = 3: |a₃| = |a₂ + 1| = |1 + 1| = 2. So a₃ = 2 or -2.Case 1a: a₃ = 2.k = 4: |a₄| = |2 + 1| = 3. So a₄ = 3 or -3.Case 1a continues growing by 1 each time. The sequence would be 0, 1, 2, 3, ..., n-1. The average here would be (0 + 1 + 2 + ... + (n-1))/n = [ (n-1)n / 2 ] / n = (n-1)/2, which is definitely greater than -1/2. So this case is fine.Case 1b: a₃ = -2.Then k = 4: |a₄| = | -2 + 1 | = | -1 | = 1. So a₄ = 1 or -1.If a₄ = 1, then k = 5: |a₅| = |1 + 1| = 2, similar to before. So this seems like a possible oscillation.Alternatively, a₄ = -1. Then k = 5: |a₅| = |-1 + 1| = 0. So a₅ = 0. Then k = 6: |a₆| = |0 + 1| = 1, so it starts over. This might form a cycle.Hmm, interesting. So depending on the choices made at each step (whether to take the positive or negative), the sequence can either keep increasing, decrease, or oscillate. But our goal is to find the minimal possible average, so we need to consider the case where the terms are as negative as possible to minimize the sum. Therefore, we need to analyze the path that would lead to the most negative sum, hence the minimal average.Therefore, perhaps we need to construct a sequence where as many terms as possible are negative, but under the constraint given by the absolute value recurrence. Let me try to model this.Starting with a₁ = 0.a₂ can be 1 or -1. To minimize the sum, we choose a₂ = -1.Then for k=3: |a₃| = |a₂ + 1| = |-1 + 1| = |0| = 0. Therefore, a₃ must be 0. Wait, that's interesting. If we choose a₂ = -1, then a₃ is forced to be 0. Then a₄ would then be |0 + 1| = 1 or -1. If we again choose the negative option, a₄ = -1, then a₅ is | -1 + 1 | = 0, and so on. So this creates a cycle: 0, -1, 0, -1, 0, -1, ..., alternating between 0 and -1.Let's see what the average would be in this case. If n is even, say n=2m, then the sequence is 0, -1, 0, -1, ..., 0, -1. The sum is m*(-1) = -m, so average is -m/(2m) = -1/2. If n is odd, n=2m+1, the sum is -m, average is -m/(2m+1) > -1/2 because m < (2m+1)/2. Therefore, the minimal average is exactly -1/2 when n is even, and slightly higher when n is odd. Therefore, this suggests that the minimal average is indeed -1/2, achieved when n is even with the alternating sequence. Hence, the inequality to prove is tight for even n.But to confirm this, let's check for small n.For n=2: sequence is 0, -1. Average is (-1)/2 = -0.5, which is exactly -1/2.For n=3: The sequence would be 0, -1, 0. Sum is -1, average is -1/3 ≈ -0.333, which is greater than -1/2.For n=4: 0, -1, 0, -1. Sum is -2, average is -0.5.For n=5: 0, -1, 0, -1, 0. Sum is -2, average is -2/5 = -0.4.So it seems like when n is even, the average is exactly -1/2, and when n is odd, it's higher. Therefore, the inequality holds, and the minimal average is -1/2. So the problem is to prove that regardless of the choices made at each step (whether to take positive or negative), the average cannot be less than -1/2.But how do we approach this for a general n? Maybe by induction, or by considering the properties of the sequence. Alternatively, since each step has a relation involving absolute values, perhaps we can model this as a walk on the real line where at each step, you either add 1 or subtract 1, but constrained such that the absolute value of the current position equals the absolute value of the previous position plus 1. Wait, that might not be exactly the case.Wait, the relation is |aₖ| = |aₖ₋₁ + 1|. So it's not that the step is ±1, but rather that the magnitude of aₖ is equal to the magnitude of (aₖ₋₁ + 1). Therefore, aₖ can be either aₖ₋₁ + 1 or -(aₖ₋₁ + 1). So each term is determined by either adding 1 to the previous term and keeping the sign, or adding 1 and flipping the sign. Hmm.Alternatively, the recursion can be written as aₖ = ±(aₖ₋₁ + 1). So each term is either (aₖ₋₁ + 1) or -(aₖ₋₁ + 1). So starting from 0, the next term is ±1, then the next term is ±(1 + 1) = ±2 or ±0, but wait, no. Wait, if a₂ is ±1, then a₃ is ±(a₂ + 1). If a₂ = 1, then a₃ is ±2. If a₂ = -1, then a₃ is ±0. But |a₃| must equal |a₂ + 1|. So if a₂ = -1, then |a₃| = | -1 + 1 | = 0, so a₃ must be 0.Ah, so actually, if aₖ₋₁ is such that when you add 1, the absolute value is taken, then aₖ is forced to be either (aₖ₋₁ + 1) or -(aₖ₋₁ + 1). However, in some cases, this might lead to aₖ being zero if aₖ₋₁ + 1 is zero. Wait, no. If aₖ₋₁ + 1 is zero, then |aₖ| = 0, so aₖ = 0. But in other cases, when aₖ₋₁ + 1 is not zero, aₖ can be either positive or negative of that.But in the case where aₖ₋₁ = -1, then aₖ must be 0. Then the next term aₖ₊₁ would be |0 + 1| = 1, so aₖ₊₁ can be 1 or -1. So this allows for cycles.Therefore, the sequence can alternate between -1 and 0 if we choose the negative option each time. Let me formalize this.Suppose starting from a₁ = 0.If we choose a₂ = -1.Then a₃ must be 0 (since |a₂ + 1| = |-1 + 1| = 0, so |a₃| = 0 => a₃ = 0.Then a₄ = |a₃ + 1| = |0 + 1| = 1, so a₄ can be 1 or -1. If we choose a₄ = -1 again.Then a₅ = 0, and so on. So the sequence alternates between -1 and 0: 0, -1, 0, -1, 0, ..., which gives the minimal average as we saw before.But what if, at some point, instead of choosing the negative option, we choose the positive one? Let's see.Suppose we have the sequence: 0, -1, 0, 1. Then a₄ = 1. Then a₅ would be |1 + 1| = 2, so a₅ can be 2 or -2. If we choose a₅ = -2, then a₆ = | -2 + 1 | = 1, so a₆ can be 1 or -1. Choosing a₆ = -1 leads to a₇ = 0, and so on. But in this case, the average might be different. Let's compute the sum up to n=5: 0 + (-1) + 0 + 1 + (-2) = -2. Average is -2/5 = -0.4, which is still better than -0.5. Wait, but that's actually better (i.e., higher) than the minimal case. So if we sometimes choose positive terms, the sum might actually be higher, hence the average is less negative. Therefore, the minimal average occurs when we choose the negative option as much as possible, leading to the alternating -1 and 0 sequence.Therefore, to formalize this, perhaps we can show that any deviation from the alternating -1, 0 sequence would result in a higher average. Hence, the minimal average is achieved by that alternating sequence, which gives exactly -1/2 when n is even, and -m/(2m+1) when n is odd (which is greater than -1/2).But how can we turn this into a formal proof? Maybe by induction or by considering the cumulative sum and showing that at each step, the minimal possible sum is achieved by choosing the negative option whenever possible.Alternatively, consider defining Sₖ = a₁ + a₂ + ... + aₖ. We need to show that Sₙ ≥ -n/2. Let's try to analyze Sₖ recursively.Given that a₁ = 0, S₁ = 0.For each k ≥ 2, we have aₖ = ±(aₖ₋₁ + 1). So Sₖ = Sₖ₋₁ + aₖ = Sₖ₋₁ ± (aₖ₋₁ + 1).But note that aₖ₋₁ is part of the previous sum Sₖ₋₁. Let's express Sₖ in terms of Sₖ₋₁ and the choice of sign.Wait, Sₖ = Sₖ₋₁ + aₖ.But aₖ can be either (aₖ₋₁ + 1) or -(aₖ₋₁ + 1).So Sₖ = Sₖ₋₁ + (aₖ₋₁ + 1) or Sₖ = Sₖ₋₁ - (aₖ₋₁ + 1).But Sₖ₋₁ includes aₖ₋₁, so let's write Sₖ₋₁ = Sₖ₋₂ + aₖ₋₁.Therefore, substituting:Case 1: aₖ = (aₖ₋₁ + 1)Then Sₖ = Sₖ₋₁ + aₖ₋₁ + 1 = Sₖ₋₂ + aₖ₋₁ + aₖ₋₁ + 1 = Sₖ₋₂ + 2aₖ₋₁ + 1.Case 2: aₖ = -(aₖ₋₁ + 1)Then Sₖ = Sₖ₋₁ - aₖ₋₁ - 1 = Sₖ₋₂ + aₖ₋₁ - aₖ₋₁ - 1 = Sₖ₋₂ - 1.Hmm, this is interesting. So depending on the choice of aₖ, the sum Sₖ can either be Sₖ₋₂ + 2aₖ₋₁ + 1 or Sₖ₋₂ - 1.Therefore, the evolution of the sum depends on the previous term aₖ₋₁. To minimize the total sum Sₙ, we would want to choose the option that gives the smaller Sₖ at each step.But since aₖ₋₁ can be positive or negative, we need to see when choosing case 2 (subtracting 1) is better (i.e., gives a smaller sum) than case 1.Suppose we have aₖ₋₁. If we choose case 2, Sₖ = Sₖ₋₂ - 1. If we choose case 1, Sₖ = Sₖ₋₂ + 2aₖ₋₁ + 1. So to minimize Sₖ, we should choose case 2 if Sₖ₋₂ -1 ≤ Sₖ₋₂ + 2aₖ₋₁ + 1, which simplifies to -1 ≤ 2aₖ₋₁ + 1 => -2 ≤ 2aₖ₋₁ => -1 ≤ aₖ₋₁.Therefore, if aₖ₋₁ ≥ -1, then choosing case 2 (subtracting 1) is better (since it gives a smaller Sₖ). If aₖ₋₁ < -1, then choosing case 1 (adding 2aₖ₋₁ + 1) might be better.But wait, in our problem, the terms are generated such that |aₖ| = |aₖ₋₁ + 1|. Let's see if aₖ₋₁ can ever be less than -1.Suppose we start with a₁ = 0.a₂ can be 1 or -1. If we choose a₂ = 1, then a₃ can be 2 or -2. If we choose a₃ = -2, then a₄ would be | -2 + 1 | = 1, so a₄ can be 1 or -1. Then a₅ = |1 + 1| = 2 or -2, and so on. In this case, the terms can become more negative than -1. For example, a₃ = -2, which is less than -1.But in such a case, can we get a term less than -1? Let's see:Suppose we have a sequence: 0, 1, -2, -1, 0, -1, 0, ... Here, a₁=0, a₂=1, a₃=-2, a₄= -1, a₅=0, a₆=-1, etc. In this case, a₃=-2, which is less than -1. Then, according to the previous reasoning, when aₖ₋₁ = -2, which is less than -1, then choosing case 1 would give Sₖ = Sₖ₋₂ + 2*(-2) + 1 = Sₖ₋₂ - 4 + 1 = Sₖ₋₂ -3. Choosing case 2 would give Sₖ = Sₖ₋₂ -1. So which is smaller?If aₖ₋₁ = -2, then case 1 leads to Sₖ = Sₖ₋₂ -3, which is smaller (more negative) than case 2's Sₖ = Sₖ₋₂ -1. Therefore, to minimize the sum, we should choose case 1 when aₖ₋₁ < -1. Wait, but in the problem statement, the sequence is fixed; that is, we are given that the sequence satisfies the recurrence, but we have to consider all possible such sequences and prove that the average is at least -1/2. Therefore, to find the minimal possible average, we need to consider the sequence constructed in such a way to minimize the sum, which involves choosing the sign at each step to make the sum as small as possible.Therefore, the problem reduces to analyzing the minimal possible sum Sₙ over all possible sequences generated by the recurrence |aₖ| = |aₖ₋₁ + 1|, starting from a₁=0, and then proving that Sₙ ≥ -n/2.To model this, perhaps we can define a function f(k, s) which represents the minimal sum achievable for the first k terms, given that the k-th term is s. But this might get complicated. Alternatively, maybe we can track the possible values of aₖ and the corresponding minimal sum up to step k.Alternatively, let's model this as a dynamic programming problem where at each step k, we have possible values of aₖ and the corresponding minimal sum Sₖ.But perhaps there's a smarter way. Let's think recursively. Let's denote by mₖ the minimal possible sum for the first k terms. We need to find mₖ and show that mₖ ≥ -k/2.Starting with k=1: m₁ = 0.For k=2: a₂ can be 1 or -1. To minimize the sum, choose a₂ = -1. So m₂ = 0 + (-1) = -1.For k=3: |a₃| = |a₂ + 1|. Since a₂ = -1, |a₃| = | -1 + 1 | = 0, so a₃ must be 0. Therefore, m₃ = m₂ + 0 = -1.For k=4: |a₄| = |a₃ + 1| = |0 + 1| = 1. Choose a₄ = -1 to minimize the sum. So m₄ = m₃ + (-1) = -2.For k=5: |a₅| = |a₄ + 1| = | -1 + 1 | = 0, so a₅ = 0. m₅ = m₄ + 0 = -2.k=6: |a₆| = |0 + 1| = 1, choose a₆ = -1. m₆ = -3.So the pattern here is that for even k, mₖ = -k/2, and for odd k, mₖ = -(k-1)/2. Indeed, for k=2: -1 = -2/2, k=4: -2 = -4/2, k=6: -3 = -6/2. For odd k: k=1: 0, k=3: -1, k=5: -2, which is -(k-1)/2.Therefore, for general k:If k is even, mₖ = -k/2.If k is odd, mₖ = -(k-1)/2.Therefore, the minimal sum Sₙ is -floor(n/2). But let's check for n=5: floor(5/2)=2, so -2, which matches. For n=6: -3, which is -6/2. Therefore, indeed, the minimal sum Sₙ is -floor(n/2). Therefore, the average would be -floor(n/2)/n.But floor(n/2)/n is equal to (n/2 - 1/2)/n = 1/2 - 1/(2n) when n is odd, and exactly 1/2 when n is even. Therefore, -floor(n/2)/n = -1/2 + 1/(2n) when n is odd, and -1/2 when n is even. Therefore, the minimal average is -1/2 + 1/(2n) for odd n, which is greater than -1/2, and exactly -1/2 for even n. Hence, the average is always ≥ -1/2.Therefore, this suggests that the minimal average is indeed bounded below by -1/2, achieved when n is even. Thus, the inequality holds.But to formalize this, we need to prove that for any sequence {aₖ} satisfying the given conditions, the sum Sₙ = a₁ + ... + aₙ satisfies Sₙ ≥ -n/2. Alternatively, we can use induction. Let's try mathematical induction.Base case: n=1. The average is 0, which is ≥ -1/2. True.n=2. The sum can be -1, average -1/2. True.Assume that for all k ≤ m, the average is ≥ -1/2. Now consider k = m + 1. But induction might not be straightforward here because the sequence depends on previous terms in a non-trivial way.Alternatively, consider the sequence in pairs. For even n, we can pair the terms as (a₁, a₂), (a₃, a₄), ..., (a_{n-1}, a_n}. For each pair, let's compute the sum.But wait, in the minimal case, the sequence alternates between 0 and -1. So each pair (0, -1) sums to -1, and there are n/2 pairs, so total sum is -n/2. For odd n, there's an extra 0 at the end, so sum is -(n-1)/2, average is -(n-1)/(2n) = -1/2 + 1/(2n) > -1/2.But how do we know that in any other sequence, the sum cannot be lower than this?Suppose there exists a sequence where the sum is lower than -n/2. Let's suppose for contradiction that there is such a sequence. Then, in that sequence, there must be some terms that are more negative than the alternating 0, -1 sequence. But is that possible?Wait, earlier example: if we have a sequence like 0, -1, 0, -1, ..., which gives the minimal sum. If we choose a different path where some terms are more negative, like 0, 1, -2, -1, 0, -1, ..., does that lead to a lower sum?Let's compute the sum for such a sequence.Take n=4: 0, 1, -2, -1. Sum: 0 + 1 + (-2) + (-1) = -2. Average: -2/4 = -0.5, same as the minimal case. But what about n=5: 0,1,-2,-1,0. Sum: 0 +1 -2 -1 +0 = -2. Average: -2/5 = -0.4, which is higher than -0.5.Alternatively, another sequence: 0, -1, -2, -1, 0. Wait, but can we have such a sequence?Starting with a₁=0.a₂ = -1.a₃: |a₂ +1| = |-1 +1| = 0, so a₃ = 0.a₄: |0 +1| = 1, so a₄ = -1.a₅: | -1 +1 | = 0. So the sequence is 0, -1, 0, -1, 0. Sum: -2, average -2/5.But if we tried to get a more negative term, say a₃ = something else. Wait, but given a₂ = -1, |a₃| = | -1 +1 | = 0, so a₃ must be 0. So you can't get a more negative term here. Therefore, in this case, you can't have a₃ being anything other than 0. So even if you wanted to make the sum more negative, the constraints of the problem prevent you from doing so.Alternatively, let's try another path. Starting with a₁=0, a₂=1.a₃ can be 2 or -2. Choose -2.a₄ = | -2 +1 | = 1, so a₄ can be 1 or -1. Choose -1.a₅ = | -1 +1 | = 0.a₆ = |0 +1| =1, choose -1.a₇ = 0, etc.For n=6, the sequence would be: 0,1,-2,-1,0,-1. Sum: 0 +1 -2 -1 +0 -1 = -3. Average: -3/6 = -0.5, same as the minimal case. So even though we have a term -2, the sum ends up being the same as the minimal case. Interesting. So having a more negative term doesn't necessarily make the sum more negative, because subsequent terms are constrained by the previous terms.So even if we try to make a term more negative, the following terms might compensate by being less negative or positive. Therefore, it seems that the minimal sum is achieved by the alternating 0, -1 sequence.Hence, perhaps the key is to show that any deviation from this minimal path does not lead to a lower sum. To formalize this, we can use induction with a careful analysis of the possible steps.Let me attempt an inductive proof.Base cases:n=1: Average is 0 ≥ -1/2. True.n=2: Possible sums are 1 or -1. The minimal average is -1/2 ≥ -1/2. True.Inductive step:Assume that for all sequences of length k ≤ m, the average is ≥ -1/2. Now consider a sequence of length m+1.But this seems tricky because the sequence of length m+1 is built from the sequence of length m, but the term a_{m+1} depends on a_m. However, the previous terms can vary depending on the choices made at each step.Alternatively, maybe we can use a different approach. Let's consider the cumulative sum Sₙ and analyze its lower bound.Given the recurrence |aₖ| = |aₖ₋₁ + 1|, we can square both sides to remove the absolute value:aₖ² = (aₖ₋₁ + 1)².Expanding this, we get:aₖ² = aₖ₋₁² + 2aₖ₋₁ + 1.Rearranging:aₖ² - aₖ₋₁² - 2aₖ₋₁ - 1 = 0.Summing this equation from k=2 to n:Σ_{k=2}^n (aₖ² - aₖ₋₁² - 2aₖ₋₁ - 1) = 0.This telescopes:(aₙ² - a₁²) - 2Σ_{k=1}^{n-1} aₖ - (n - 1) = 0.Given that a₁ = 0, this simplifies to:aₙ² - 2Σ_{k=1}^{n-1} aₖ - (n - 1) = 0.Rearranging:2Σ_{k=1}^{n-1} aₖ = aₙ² - (n - 1).But we need to relate this to the total sum Sₙ = Σ_{k=1}^n aₖ = Σ_{k=1}^{n-1} aₖ + aₙ.Let me denote S = Σ_{k=1}^n aₖ, so Σ_{k=1}^{n-1} aₖ = S - aₙ.Substituting into the equation:2(S - aₙ) = aₙ² - (n - 1).Rearranging:2S - 2aₙ = aₙ² - n + 1.Bring all terms to one side:aₙ² + 2aₙ - 2S + (n - 1) = 0.Solve for S:2S = aₙ² + 2aₙ + (n - 1).Therefore,S = (aₙ² + 2aₙ + n - 1)/2.Our goal is to show that S ≥ -n/2.So, substituting S:(aₙ² + 2aₙ + n - 1)/2 ≥ -n/2.Multiply both sides by 2:aₙ² + 2aₙ + n - 1 ≥ -n.Rearranging:aₙ² + 2aₙ + 2n - 1 ≥ 0.Hmm, so we need to show that for any real number aₙ (which is generated by the recurrence), the inequality aₙ² + 2aₙ + 2n - 1 ≥ 0 holds.But is this always true? Let's analyze the quadratic in aₙ:aₙ² + 2aₙ + (2n - 1) ≥ 0.The discriminant of this quadratic is D = 4 - 4*(2n - 1) = 4 - 8n + 4 = 8 - 8n = 8(1 - n).For n ≥ 1, D = 8(1 - n) ≤ 0. Therefore, the quadratic is always non-negative if the leading coefficient is positive and the discriminant is ≤ 0. Since the leading coefficient is 1 (positive), this quadratic is always non-negative for all real aₙ when n ≥ 1.Therefore, aₙ² + 2aₙ + 2n - 1 ≥ 0 holds for all real aₙ, which implies that S ≥ -n/2. Hence, the average S/n ≥ -1/2.Wow, that's a neat approach! By squaring the recurrence relation and telescoping, we derived an expression for the sum S in terms of aₙ, and then showed that the resulting inequality must hold due to the quadratic being non-negative. This effectively proves the required inequality without needing to analyze specific sequences or use induction.Therefore, the key steps were:1. Square both sides of the recurrence relation to eliminate absolute values.2. Telescope the summed equations to relate the sum S to the last term aₙ.3. Express S in terms of aₙ and manipulate the inequality to form a quadratic in aₙ.4. Show that this quadratic is always non-negative, hence proving the inequality.This method elegantly circumvents the need to consider individual sequences or minimal cases by leveraging algebraic manipulation and properties of quadratic functions. It demonstrates the power of telescoping sums and quadratic analysis in solving such recurrence-related inequalities.

✅:To prove that the average of the sequence (a_1, a_2, ldots, a_n) satisfies (frac{1}{n}(a_1 + a_2 + cdots + a_n) geq -frac{1}{2}), we proceed as follows:Step 1: Square the Recurrence RelationGiven (|a_k| = |a_{k-1} + 1|) for (k geq 2), squaring both sides yields:[a_k^2 = (a_{k-1} + 1)^2.]Expanding and rearranging:[a_k^2 - a_{k-1}^2 - 2a_{k-1} - 1 = 0.]Step 2: Sum Over All TermsSumming from (k = 2) to (n):[sum_{k=2}^n left(a_k^2 - a_{k-1}^2 - 2a_{k-1} - 1right) = 0.]Telescoping the squared terms and simplifying:[a_n^2 - a_1^2 - 2sum_{k=1}^{n-1} a_k - (n-1) = 0.]Since (a_1 = 0), this reduces to:[a_n^2 - 2sum_{k=1}^{n-1} a_k - (n-1) = 0.]Step 3: Express Total Sum (S_n)Let (S_n = sum_{k=1}^n a_k). Then:[sum_{k=1}^{n-1} a_k = S_n - a_n.]Substitute into the equation:[a_n^2 - 2(S_n - a_n) - (n-1) = 0.]Rearranging gives:[2S_n = a_n^2 + 2a_n + (n - 1).]Thus:[S_n = frac{a_n^2 + 2a_n + (n - 1)}{2}.]Step 4: Derive the InequalityWe need (S_n geq -frac{n}{2}). Substitute (S_n):[frac{a_n^2 + 2a_n + (n - 1)}{2} geq -frac{n}{2}.]Multiply both sides by 2:[a_n^2 + 2a_n + n - 1 geq -n.]Simplify:[a_n^2 + 2a_n + 2n - 1 geq 0.]Step 5: Analyze the Quadratic InequalityThe quadratic in (a_n):[a_n^2 + 2a_n + (2n - 1) geq 0,]has a discriminant:[Delta = 4 - 4(2n - 1) = 8(1 - n).]For (n geq 1), (Delta leq 0). Since the coefficient of (a_n^2) is positive, the quadratic is always non-negative. Hence, the inequality holds for all real (a_n).ConclusionTherefore, (frac{1}{n}(a_1 + a_2 + cdots + a_n) geq -frac{1}{2}) is always satisfied, completing the proof.[boxed{frac{1}{n}left(a_{1}+a_{2}+cdots+a_{n}right) geqslant -frac{1}{2}}]

🔑:1. We start with the given conditions: [ a_{1} = 0, quad left| a_{k} right| = left| a_{k-1} + 1 right| quad text{for } k=2,3, ldots, n. ]2. To explore the properties of the sequence, we consider the square of (a_k): [ a_{k}^{2} = (a_{k-1} + 1)^{2} = a_{k-1}^{2} + 2a_{k-1} + 1 quad text{for } k=2,3, ldots, n. ]3. Summing both sides of the equation from (k=2) to (k=n): [ sum_{k=2}^{n} a_{k}^{2} = sum_{k=2}^{n} left(a_{k-1}^{2} + 2a_{k-1} + 1right). ]4. This can be split into three sums: [ sum_{k=2}^{n} a_{k}^{2} = sum_{k=2}^{n} a_{k-1}^{2} + 2sum_{k=2}^{n} a_{k-1} + sum_{k=2}^{n} 1. ]5. Simplify the sums: [ sum_{k=2}^{n} a_{k-1}^{2} = sum_{k=1}^{n-1} a_{k}^{2}, ] [ 2sum_{k=2}^{n} a_{k-1} = 2sum_{k=1}^{n-1} a_{k}, ] and [ sum_{k=2}^{n} 1 = n-1. ]6. Thus, we can rewrite the equation as: [ sum_{k=2}^{n} a_{k}^{2} = sum_{k=1}^{n-1} a_{k}^{2} + 2sum_{k=1}^{n-1} a_{k} + (n-1). ]7. Rearranging the terms: [ sum_{k=2}^{n} a_{k}^{2} - sum_{k=1}^{n-1} a_{k}^{2} = 2 sum_{k=1}^{n-1} a_{k} + (n-1). ]8. Observe the telescoping nature on the left side: [ a_{n}^{2} - a_{1}^{2} = 2sum_{k=1}^{n-1} a_{k} + (n-1). ]9. Given (a_{1} = 0), [ a_{n}^{2} = 2sum_{k=1}^{n-1} a_{k} + (n-1). ]10. Rearrange to find the sum: [ 2 sum_{k=1}^{n-1} a_{k} = a_{n}^{2} - (n-1). ]11. The average of the sequence: [ frac{1}{n} left( a_{1} + a_{2} + cdots + a_{n} right) = frac{1}{n} left(a_{1} + sum_{k=2}^{n-1} a_{k} + a_{n} right). ]12. Since (a_{1} = 0), focus on: [ frac{1}{n} sum_{k=1}^{n} a_{k} = frac{1}{n} left( sum_{k=1}^{n-1} a_{k} + a_{n} right). ]13. Substitute the obtained sum: [ frac{1}{n} left( frac{a_{n}^{2} - (n-1)}{2} + a_{n} right). ]14. Simplifying further: [ frac{1}{n}left( frac{a_{n}^{2} + 2a_{n} - (n-1)}{2} right), ] which simplifies to: [ frac{1}{2n} left( a_{n}^{2} + 2a_{n} + 1 - n right), ] [ frac{1}{2n} left( (a_{n} + 1)^{2} - n right). ]15. Notice, ((a_{n} + 1)^{2} geq 0), thus: [ frac{1}{2n} left( (a_{n} + 1)^{2} - n right) geq frac{1}{2n}(-n) = -frac{1}{2}. ] Conclusion:[boxed{frac{1}{n} left(a_{1} + a_{2} + cdots + a_{n} right) geq -frac{1}{2}}.

🧠:Okay, let's tackle this problem step by step. So, we have an alloy that's 12 kg in total, and it's 45% copper. The rest must be tin, right? So, first, I need to find out how much copper and how much tin are in the original alloy. Let me write that down.Original alloy:- Total mass = 12 kg- Copper percentage = 45%- Tin percentage = 100% - 45% = 55%So, mass of copper = 45% of 12 kg. Let me calculate that. 45% is 0.45 in decimal, so 0.45 * 12 kg = 5.4 kg of copper. Then, the tin would be 12 kg - 5.4 kg = 6.6 kg. Alternatively, 55% of 12 kg is 0.55 * 12 = 6.6 kg. Yep, that checks out.Now, the problem is asking how much pure tin needs to be added to this alloy so that the new alloy contains 40% copper. Hmm. So, when we add pure tin, we're increasing the total mass of the alloy, but the amount of copper stays the same, right? Because we're only adding tin. So, the copper remains at 5.4 kg, but the total mass becomes 12 kg plus whatever amount of tin we add. Let's denote the amount of pure tin to add as 'x' kg.So, after adding x kg of tin:- Total mass of new alloy = 12 kg + x kg = (12 + x) kg- Mass of copper remains 5.4 kg- Percentage of copper in new alloy = 40%So, the percentage of copper is (mass of copper) / (total mass of new alloy) * 100% = 40%. So, setting up the equation:5.4 kg / (12 + x) kg = 40% (or 0.4)Let me write that equation:5.4 / (12 + x) = 0.4Now, solve for x.Multiply both sides by (12 + x):5.4 = 0.4*(12 + x)Compute 0.4*(12 + x):0.4*12 = 4.80.4*x = 0.4xSo, 5.4 = 4.8 + 0.4xSubtract 4.8 from both sides:5.4 - 4.8 = 0.4x0.6 = 0.4xThen, divide both sides by 0.4:x = 0.6 / 0.4 = 1.5So, x = 1.5 kg. Therefore, 1.5 kg of pure tin needs to be added.Wait, let me check this again to be sure. Let's verify.Original copper: 5.4 kg. After adding 1.5 kg of tin, total mass is 12 + 1.5 = 13.5 kg. Percentage of copper is 5.4 / 13.5 = 0.4, which is 40%. Yep, that's correct.Alternatively, thinking in terms of ratios. The percentage of copper is being reduced from 45% to 40%. Since the amount of copper doesn't change, the ratio of the original total mass to the new total mass should be inversely proportional to the ratio of the copper percentages. Wait, maybe that's overcomplicating, but let's see.Original percentage / New percentage = (45%) / (40%) = 9/8. So, the new total mass should be (9/8) times the original total mass? Wait, original mass is 12 kg. (9/8)*12 = 13.5 kg. So, the new mass is 13.5 kg, which is 1.5 kg more. Therefore, 1.5 kg of tin is added. That also matches the previous result.Another way to think about it: the amount of copper is constant. If we want the concentration to go down, we have to increase the total mass. The required total mass can be found by dividing the copper mass by the desired percentage.So, desired total mass = copper mass / desired percentage = 5.4 kg / 0.4 = 13.5 kg. Therefore, the added tin is 13.5 - 12 = 1.5 kg. Yep, same answer.So, all methods lead to 1.5 kg. Therefore, the answer is 1.5 kg of pure tin needs to be added.But just to ensure I haven't missed anything, let me think again. The problem is about alloy of copper and tin. Adding pure tin. So, the initial tin is 6.6 kg. After adding x kg of tin, total tin becomes 6.6 + x kg. Total alloy becomes 12 + x kg. The copper percentage is 40%, which is 5.4 kg copper. So, maybe another way to set up the equation: (mass of copper)/(mass of copper + mass of tin + added tin) = 40%.Wait, but copper is 5.4 kg, tin is 6.6 + x. So total is 5.4 + 6.6 + x = 12 + x, which is the same as before. So, 5.4 / (12 + x) = 0.4. So, same equation. So, same result. So, 1.5 kg.Another way: suppose I don't use algebra. Let's think in terms of how much tin is needed to dilute the copper from 45% to 40%. The ratio of tin to copper will change. Initially, tin is 55%, so the ratio of tin to copper is 55/45 = 11/9. After adding tin, the copper is 40%, so tin is 60%, so the ratio becomes 60/40 = 3/2. So, we need the ratio to go from 11/9 to 3/2. Let's compute the difference.3/2 - 11/9 = (27 - 22)/18 = 5/18. So, the ratio needs to increase by 5/18 per unit of copper. But maybe this approach is more complicated.Alternatively, the amount of tin needed can be calculated by the difference in the ratios. But I think algebraically it's straightforward. Since the copper remains the same, the total mass must be such that 5.4 kg is 40%, so total mass is 13.5, so adding 1.5 kg tin. Yep.Alternatively, maybe using alligation. If we have the original alloy which is 45% copper, and we add pure tin which is 0% copper. The desired concentration is 40% copper.So, the alligation method would have:Percentage of copper in original alloy: 45%Percentage of copper in added tin: 0%Desired percentage: 40%The ratio of the amounts is (0% - 40%) : (40% - 45%) = (-40):(-5) = 8:1. Wait, alligation typically is (high - desired) : (desired - low). So, (45 - 40) : (40 - 0) = 5:40 = 1:8. So, the ratio of the amounts of the added tin to the original alloy is 1:8. Therefore, the amount of tin to add is (1/8)*12 kg = 1.5 kg. Yep, same answer. That's another way to confirm.So, all methods converge to 1.5 kg. Therefore, the answer is 1.5 kg.I think that's thorough. Checked multiple methods: direct algebra, ratio approach, alligation. All confirm the same result. So, confident that 1.5 kg is correct.

✅:To solve the problem, we start by determining the amount of copper and tin in the original alloy:- Original alloy mass: 12 kg - Copper content: 45% of 12 kg = (0.45 times 12 = 5.4 , text{kg}) - Tin content: (12 - 5.4 = 6.6 , text{kg}) When pure tin (x kg) is added, the total mass becomes (12 + x , text{kg}), while the copper mass remains unchanged at 5.4 kg. The new alloy must contain 40% copper: [frac{5.4}{12 + x} = 0.4]Solving for (x): [5.4 = 0.4(12 + x) implies 5.4 = 4.8 + 0.4x implies 0.6 = 0.4x implies x = frac{0.6}{0.4} = 1.5]Verification: After adding 1.5 kg of tin, the total mass is (12 + 1.5 = 13.5 , text{kg}). The copper percentage is: [frac{5.4}{13.5} = 0.4 , (40%)]Final Answer: (boxed{1.5}) kg of pure tin must be added.

🔑:1. Determine the mass of copper in the original alloy: We know the total mass of the original alloy is 12 kg and it contains 45% copper. Therefore, the mass of copper in the original alloy can be calculated as: [ text{Mass of copper} = 12 , text{kg} times 0.45 = 5.4 , text{kg} ]2. Set up the equation to find the new total mass of the alloy: Let ( x ) be the mass of pure tin that needs to be added to the alloy. The new alloy should contain 40% copper. Therefore, the mass of copper in the new alloy, which remains 5.4 kg, should be 40% of the total mass of the new alloy. Denote the total mass of the new alloy as ( 12 + x ). Hence, we have: [ 5.4 = 0.4 times (12 + x) ]3. Solve for ( x ): To isolate ( x ), first divide both sides of the equation by 0.4: [ frac{5.4}{0.4} = 12 + x ] Simplify the left side: [ 13.5 = 12 + x ] Now, solve for ( x ): [ x = 13.5 - 12 = 1.5 ]4. Conclusion: Therefore, the mass of pure tin that needs to be added to the original alloy is: [ boxed{1.5 , text{kg}} ]