answer：Okay, so I have this problem about folding a rectangular piece of paper. The paper is 8 inches wide and 10 inches long. When folded, one corner touches the midpoint of the opposite longer side, creating a crease. I need to find the length of this crease, L, in terms of the folding angle φ from the horizontal. The options are given, so I need to figure out which one is correct.First, let me visualize the problem. I have a rectangle, let's say with width 8 inches and length 10 inches. Let me label the corners as A, B, C, D, where A is the top-left corner, B is the top-right, C is the bottom-right, and D is the bottom-left. So, the rectangle is 8 inches wide (from A to B) and 10 inches long (from A to D).Now, the paper is folded such that one corner touches the midpoint of the opposite longer side. Let's assume we're folding corner D (the bottom-left corner) to touch the midpoint of the opposite longer side, which would be the side BC. The midpoint of BC would be at (8, 5) if we consider the coordinate system with A at (0,10), B at (8,10), C at (8,0), and D at (0,0). So, the midpoint is (8,5).When we fold corner D to this midpoint, a crease is formed. I need to find the length of this crease in terms of the angle φ, which is the angle from the horizontal to the crease.Let me think about how folding works. When you fold a point to another point, the crease is the perpendicular bisector of the segment joining those two points. So, in this case, the crease should be the perpendicular bisector of the segment joining D(0,0) and E(8,5).Wait, is that correct? Or is it the line along which the paper is folded, which maps D to E. So, the crease is the set of points equidistant from D and E. Hmm, that might be the case.Alternatively, maybe I can model this using triangles. When folding, the crease will form a triangle with the sides of the rectangle. Let me try to sketch this mentally.If I fold D to E, the crease will be a line that connects somewhere on the top side AB to somewhere on the right side BC. Let me denote the crease as line EF, where F is on AB and E is on BC.Wait, no, E is the midpoint, so E is fixed at (8,5). So, the crease will be the line that connects E(8,5) to some point F on AD or AB? Hmm, maybe.Alternatively, perhaps the crease is the line that is the fold, so it's the line along which we fold D to E. So, the crease is the perpendicular bisector of DE.Let me calculate the coordinates. D is (0,0) and E is (8,5). The midpoint of DE is ((0+8)/2, (0+5)/2) = (4, 2.5). The slope of DE is (5-0)/(8-0) = 5/8. Therefore, the slope of the perpendicular bisector is the negative reciprocal, which is -8/5.So, the crease is the line with slope -8/5 passing through (4, 2.5). But how does this help me find the length of the crease?Wait, the crease is a line segment on the paper, so its length depends on where it intersects the sides of the rectangle. Let me find the intersection points of this crease with the sides of the rectangle.The crease has equation y - 2.5 = (-8/5)(x - 4). Let me write this in slope-intercept form: y = (-8/5)x + (32/5) + 2.5. Converting 2.5 to fifths, that's 12.5/5, so total y-intercept is (32 + 12.5)/5 = 44.5/5 = 8.9. Wait, that can't be right because 2.5 is 5/2, which is 12.5/5. So, 32/5 + 12.5/5 = 44.5/5 = 8.9. Hmm, but the rectangle only goes up to y=10, so maybe that's okay.Wait, let me double-check the calculation. Starting from point-slope form: y - 2.5 = (-8/5)(x - 4). So, y = (-8/5)x + (32/5) + 2.5. 32/5 is 6.4, and 2.5 is 2.5, so total y-intercept is 6.4 + 2.5 = 8.9. So, the equation is y = (-8/5)x + 8.9.Now, I need to find where this line intersects the sides of the rectangle. The rectangle has sides at x=0, x=8, y=0, and y=10.First, let's find intersection with the top side y=10. Plugging y=10 into the equation: 10 = (-8/5)x + 8.9. Solving for x: (-8/5)x = 10 - 8.9 = 1.1 => x = (1.1)*(-5/8) = -5.5/8 = -0.6875. But x cannot be negative since the rectangle is from x=0 to x=8. So, the crease does not intersect the top side.Next, intersection with the right side x=8. Plugging x=8 into the equation: y = (-8/5)*8 + 8.9 = (-64/5) + 8.9 = (-12.8) + 8.9 = -3.9. But y cannot be negative, so the crease does not intersect the right side.Next, intersection with the bottom side y=0. Plugging y=0: 0 = (-8/5)x + 8.9 => (8/5)x = 8.9 => x = (8.9)*(5/8) = (44.5)/8 ≈ 5.5625. So, the crease intersects the bottom side at approximately (5.5625, 0).Finally, intersection with the left side x=0: y = (-8/5)*0 + 8.9 = 8.9. So, the crease intersects the left side at (0, 8.9).Wait, but the crease is supposed to be within the rectangle, so it goes from (0, 8.9) to (5.5625, 0). But when we fold the paper, the crease should be such that point D(0,0) maps to E(8,5). So, perhaps the crease is not the entire line from (0,8.9) to (5.5625,0), but only a segment within the rectangle.Wait, maybe I made a mistake. The crease is the fold line, so it's the set of points equidistant from D and E. Therefore, the crease is the perpendicular bisector of DE, which we found as y = (-8/5)x + 8.9. But since the paper is folded along this line, the crease within the paper would be the segment of this line that lies within the rectangle.So, the crease starts at (0,8.9) and goes to (5.5625,0). But let me check if these points are indeed on the rectangle. (0,8.9) is on the left side, and (5.5625,0) is on the bottom side. So, the crease is this segment.Now, I need to find the length of this crease. The length can be calculated using the distance formula between (0,8.9) and (5.5625,0). Let me compute that.First, let's write 8.9 as 89/10 and 5.5625 as 89/16 (since 5.5625 = 5 + 9/16 = 89/16). Wait, 5.5625 is actually 5 + 9/16? Let me check: 9/16 is 0.5625, so yes, 5.5625 = 89/16.So, the two points are (0,89/10) and (89/16,0). The distance between them is sqrt[(89/16 - 0)^2 + (0 - 89/10)^2] = sqrt[(89/16)^2 + (89/10)^2].Let me factor out (89)^2: sqrt[(89)^2*(1/16^2 + 1/10^2)] = 89*sqrt(1/256 + 1/100).Compute 1/256 + 1/100: find a common denominator, which is 256*100=25600.So, 1/256 = 100/25600 and 1/100 = 256/25600. Adding them: 100 + 256 = 356, so 356/25600.Simplify 356/25600: divide numerator and denominator by 4: 89/6400.So, sqrt(89/6400) = sqrt(89)/80.Therefore, the distance is 89*(sqrt(89)/80) = (89*sqrt(89))/80.Wait, that seems complicated. Maybe I made a mistake in calculations.Alternatively, let's compute numerically:89/16 ≈ 5.562589/10 = 8.9So, the distance is sqrt[(5.5625)^2 + (8.9)^2] ≈ sqrt[30.9414 + 79.21] ≈ sqrt[110.1514] ≈ 10.495 inches.But looking at the answer choices, they are in terms of trigonometric functions, not numerical values. So, maybe I need a different approach.Let me think about the triangle formed by the fold. When we fold D to E, the crease EF is the hypotenuse of a right triangle where one leg is the vertical distance from D to the crease, and the other leg is the horizontal distance.Wait, perhaps using trigonometry with angle φ. The angle φ is from the horizontal to the crease. So, if I consider the crease as the hypotenuse, then the adjacent side is along the horizontal, and the opposite side is vertical.In this case, the length of the crease L can be expressed in terms of φ. Let me denote the horizontal component as L*cosφ and the vertical component as L*sinφ.But how does this relate to the dimensions of the paper?Wait, when we fold D to E, the distance from D to the crease is equal to the distance from E to the crease. Since the crease is the perpendicular bisector, the distances from D and E to the crease are equal.Alternatively, maybe I can consider the triangle formed by D, E, and the midpoint of DE, which is (4,2.5). The crease is the perpendicular bisector, so it passes through (4,2.5) and has a slope of -8/5.But perhaps using similar triangles or trigonometric identities would be better.Let me consider the angle φ. Since φ is the angle from the horizontal to the crease, the slope of the crease is tanφ. Wait, but earlier I found the slope to be -8/5, which is negative. So, tanφ = |slope| = 8/5.Wait, but in the coordinate system, the crease has a negative slope, so the angle φ is measured from the horizontal to the crease in the downward direction. So, tanφ = 8/5.Therefore, φ = arctan(8/5). But I need to express L in terms of φ, not the other way around.Wait, maybe I can express L using the relationship between the sides of the triangle and the angle φ.In the right triangle formed by the crease, the horizontal distance from the crease to D, and the vertical distance from the crease to D.But I'm getting confused. Maybe I should use the distance from D to the crease.The distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2).The crease is y = (-8/5)x + 8.9, which can be rewritten as (8/5)x + y - 8.9 = 0.So, the distance from D(0,0) to the crease is |(8/5)(0) + (1)(0) - 8.9| / sqrt((8/5)^2 + 1^2) = | -8.9 | / sqrt(64/25 + 1) = 8.9 / sqrt(89/25) = 8.9 / (sqrt(89)/5) = (8.9 * 5)/sqrt(89).But 8.9 is 89/10, so (89/10 * 5)/sqrt(89) = (89/2)/sqrt(89) = (89)/(2*sqrt(89)) = sqrt(89)/2.So, the distance from D to the crease is sqrt(89)/2.But since the crease is the perpendicular bisector, the distance from E to the crease is also sqrt(89)/2.Now, considering the right triangle formed by D, the foot of the perpendicular from D to the crease, and the crease itself. The distance from D to the crease is sqrt(89)/2, and the angle between the crease and the horizontal is φ.In this right triangle, the distance sqrt(89)/2 is the opposite side to angle φ, and the length of the crease L is the hypotenuse.So, sinφ = opposite/hypotenuse = (sqrt(89)/2)/L => L = (sqrt(89)/2)/sinφ = sqrt(89)/(2 sinφ).But sqrt(89) is approximately 9.433, which doesn't match the answer choices. Wait, maybe I made a mistake.Alternatively, perhaps the distance from D to the crease is related to the length of the crease through the angle φ.Wait, let's think differently. The crease length L can be found using the relationship in the triangle where φ is the angle from the horizontal.If I consider the crease as the hypotenuse, then the horizontal component is L*cosφ and the vertical component is L*sinφ.But how does this relate to the dimensions of the paper?When folding D to E, the horizontal distance covered is from x=0 to x=8, but the crease only covers part of that. Similarly, the vertical distance is from y=0 to y=5, but again, the crease covers part of that.Wait, maybe I can set up equations based on similar triangles.Let me denote the point where the crease intersects the left side as F(0, y1) and where it intersects the bottom side as G(x1, 0). Then, the crease is FG with length L.From earlier, we found F is (0,8.9) and G is approximately (5.5625,0). But let's use exact values.From the equation y = (-8/5)x + 8.9, when x=0, y=8.9, and when y=0, x= (8.9)*(5/8) = (89/10)*(5/8) = 445/80 = 89/16 ≈5.5625.So, F is (0,89/10) and G is (89/16,0).Now, the length L is the distance between F and G, which is sqrt[(89/16)^2 + (89/10)^2].Let me compute this:(89/16)^2 = (7921)/(256)(89/10)^2 = (7921)/(100)So, L = sqrt[(7921/256) + (7921/100)] = sqrt[7921*(1/256 + 1/100)].Factor out 7921: sqrt[7921*(100 + 256)/(256*100)] = sqrt[7921*(356)/(25600)].Simplify 356/25600: divide numerator and denominator by 4: 89/6400.So, L = sqrt[7921*(89/6400)] = sqrt[(7921*89)/6400].But 7921 is 89^2, so this becomes sqrt[(89^2 *89)/6400] = sqrt[(89^3)/6400] = (89^(3/2))/80.But 89^(3/2) is 89*sqrt(89), so L = (89*sqrt(89))/80.This seems complicated, and none of the answer choices have sqrt(89). So, perhaps I'm overcomplicating it.Let me try a different approach. Let's consider the triangle formed by D, E, and the midpoint M of DE.Point M is (4,2.5). The crease is the perpendicular bisector of DE, passing through M with slope -8/5.The length of DE is sqrt[(8-0)^2 + (5-0)^2] = sqrt(64 +25) = sqrt(89).The crease is the perpendicular bisector, so the distance from M to the crease is zero, but that doesn't help directly.Wait, maybe using trigonometry with angle φ.If φ is the angle from the horizontal to the crease, then tanφ = (slope of crease). But the slope is -8/5, so tanφ = 8/5 (since angle is measured from horizontal to crease, regardless of direction).So, tanφ = 8/5 => φ = arctan(8/5).But I need L in terms of φ, not φ in terms of L.From the triangle, we have tanφ = 8/5, so we can express 8 = 5 tanφ.But how does this relate to L?Wait, in the right triangle formed by the crease, the horizontal component is L*cosφ, and the vertical component is L*sinφ.But the horizontal component should relate to the width of the paper, and the vertical component to the height.Wait, when folding, the horizontal distance from D to the crease is L*cosφ, and the vertical distance is L*sinφ.But the total horizontal distance from D to E is 8 inches, and the vertical distance is 5 inches.Wait, maybe the horizontal component L*cosφ should equal 8, and the vertical component L*sinφ should equal 5.But that would mean L*cosφ =8 and L*sinφ=5. Then, dividing these, tanφ=5/8, which contradicts earlier where tanφ=8/5.Wait, perhaps I have the angle in the wrong orientation. If φ is from the horizontal to the crease, then the adjacent side is along the horizontal, and the opposite side is vertical.So, tanφ = opposite/adjacent = (vertical component)/(horizontal component) = 5/8.Wait, that makes sense because the vertical distance from D to E is 5, and horizontal is 8.So, tanφ = 5/8.Therefore, φ = arctan(5/8).Now, from the right triangle, we have:sinφ = 5/sqrt(89)cosφ = 8/sqrt(89)But we need L in terms of φ.From the triangle, L is the hypotenuse, so L = 5/sinφ.Because in the triangle, the vertical side is 5, which is opposite to φ, so sinφ = 5/L => L = 5/sinφ = 5 cscφ.Yes, that makes sense. So, the length of the crease L is 5 cscφ.Looking at the answer choices, option B is 5 cscφ, so that should be the correct answer.

answer：Okay, so I have this geometry problem about a tetrahedron where the radii of the circumcircles of all four faces are equal. I need to prove that the opposite edges are equal, meaning AB equals CD, AC equals BD, and AD equals BC. Hmm, tetrahedrons can be a bit tricky because they have four triangular faces, and all these faces are triangles, so maybe I can use some properties of triangles and their circumcircles.First, let me recall that the radius of a circumcircle of a triangle can be calculated using the formula ( R = frac{abc}{4K} ), where ( a, b, c ) are the lengths of the sides and ( K ) is the area of the triangle. So, if all four faces have equal circumradii, then for each face, ( frac{abc}{4K} ) must be the same.Let me denote the edges of the tetrahedron as follows: AB = c, AC = b, AD = a, BC = d, BD = e, and CD = f. So, the four faces are:1. Face ABC with sides AB = c, AC = b, BC = d.2. Face ABD with sides AB = c, AD = a, BD = e.3. Face ACD with sides AC = b, AD = a, CD = f.4. Face BCD with sides BC = d, BD = e, CD = f.Since all four circumradii are equal, I can write equations for each face:For face ABC: ( R = frac{bcd}{4K_{ABC}} )For face ABD: ( R = frac{ace}{4K_{ABD}} )For face ACD: ( R = frac{abf}{4K_{ACD}} )For face BCD: ( R = frac{def}{4K_{BCD}} )So, all these expressions equal the same R. Maybe I can set them equal to each other.Let me set the first equal to the second: ( frac{bcd}{4K_{ABC}} = frac{ace}{4K_{ABD}} )Simplify: ( frac{bcd}{K_{ABC}} = frac{ace}{K_{ABD}} )Cancel out c: ( frac{bd}{K_{ABC}} = frac{ae}{K_{ABD}} )Hmm, not sure if that helps directly. Maybe I need another approach.Wait, maybe I can use the fact that in a tetrahedron, if all the circumradii of the faces are equal, then the tetrahedron is isohedral, meaning it's face-transitive. But I'm not sure if that's the case here. Maybe it's a more specific type of tetrahedron.Alternatively, perhaps I can use vector geometry. Let me assign coordinates to the vertices. Let me place vertex A at (0,0,0), vertex B at (c,0,0), vertex C at (p,q,0), and vertex D at (r,s,t). Then, I can compute the circumradii for each face and set them equal.But that might get too complicated with too many variables. Maybe there's a simpler way.Wait, another thought: in a tetrahedron, if the opposite edges are equal, then it's called an isosceles tetrahedron. Maybe the condition given (equal circumradii on all faces) implies that the tetrahedron is isosceles.I should check if in an isosceles tetrahedron, the circumradii of all faces are equal. If that's the case, then the converse might also hold.Let me recall that in an isosceles tetrahedron, all four faces are congruent triangles. If all faces are congruent, then their circumradii would naturally be equal. But in our problem, the tetrahedron might not necessarily have congruent faces, but just equal circumradii. So, maybe it's a weaker condition.Hmm, perhaps I need to use some properties of the circumradius in terms of the edges and the volume or something else.Wait, another formula for the circumradius of a triangle is ( R = frac{a}{2sin A} ), where a is a side and A is the opposite angle. Maybe I can use this for each face.But in a tetrahedron, the angles are not independent because the dihedral angles affect each other. So, maybe that's not straightforward.Alternatively, perhaps I can use the formula for the circumradius in terms of the sides and the volume of the tetrahedron. Wait, for a tetrahedron, the radius of the circumscribed sphere is given by ( R = frac{|mathbf{AB} cdot (mathbf{AC} times mathbf{AD})|}{6V} ), where V is the volume. But that's for the circumscribed sphere of the entire tetrahedron, not the faces.Wait, but in our problem, we're dealing with the circumradii of the faces, which are triangles. So, maybe I need to relate the circumradii of the faces to the edges of the tetrahedron.Let me consider face ABC. Its circumradius is ( R = frac{AB cdot BC cdot CA}{4K_{ABC}} ). Similarly for the other faces.Since all four R's are equal, I can set up equations:( frac{AB cdot BC cdot CA}{4K_{ABC}} = frac{AB cdot BD cdot DA}{4K_{ABD}} )Simplify: ( frac{BC cdot CA}{K_{ABC}} = frac{BD cdot DA}{K_{ABD}} )Hmm, not sure. Maybe I need to express the areas ( K_{ABC} ) and ( K_{ABD} ) in terms of the edges.Alternatively, perhaps I can use Heron's formula for the areas. For face ABC, the area is ( sqrt{s(s - AB)(s - BC)(s - CA)} ), where s is the semi-perimeter.But that might complicate things further.Wait, maybe I can consider the fact that if all four circumradii are equal, then the tetrahedron is equifacial, meaning all faces are congruent. But I'm not sure if equal circumradii imply congruent faces.Wait, no, equal circumradii don't necessarily mean congruent faces, because different triangles can have the same circumradius. For example, a right-angled triangle with legs 1 and 1 has a circumradius of ( frac{sqrt{2}}{2} ), and a triangle with sides 2, 2, 2 also has a circumradius of 1, which is different. Wait, no, actually, in that case, the circumradius would be different. Wait, maybe another example.Wait, actually, maybe it's not possible for two different triangles to have the same circumradius unless they are similar or something. Hmm, not necessarily. For example, a triangle with sides 3,4,5 has a circumradius of 2.5, and a triangle with sides 5,5,6 has a circumradius of ( frac{5 cdot 5 cdot 6}{4 cdot sqrt{5^2 cdot 5^2 - (6/2)^2}} ) which is more complicated, but maybe it's different.Wait, maybe I'm overcomplicating. Let me think differently.If all four faces have equal circumradii, then for each face, the product of the sides divided by the area is constant. So, for each face, ( frac{a b c}{K} = 4R ), which is constant.So, for each face, ( a b c = 4 R K ). Since R is the same for all faces, the product of the sides times the area is the same for each face.But I'm not sure how to relate this to the edges of the tetrahedron.Wait, maybe I can use the fact that in a tetrahedron, the areas of the faces are related to the volume. The volume can be expressed in terms of the areas and the dihedral angles, but that might not help directly.Alternatively, perhaps I can use the formula for the area of a triangle in terms of two sides and the included angle. For face ABC, the area is ( frac{1}{2} AB cdot BC cdot sin theta ), where ( theta ) is the angle between AB and BC.Similarly, for face ABD, the area is ( frac{1}{2} AB cdot BD cdot sin phi ), where ( phi ) is the angle between AB and BD.But since the circumradii are equal, we have:For face ABC: ( R = frac{AB cdot BC cdot CA}{4 cdot frac{1}{2} AB cdot BC cdot sin theta} = frac{CA}{2 sin theta} )Similarly, for face ABD: ( R = frac{AB cdot BD cdot DA}{4 cdot frac{1}{2} AB cdot BD cdot sin phi} = frac{DA}{2 sin phi} )Since both equal R, we have ( frac{CA}{2 sin theta} = frac{DA}{2 sin phi} ), so ( frac{CA}{sin theta} = frac{DA}{sin phi} )Hmm, interesting. So, the ratio of the length of CA to the sine of the angle between AB and BC is equal to the ratio of DA to the sine of the angle between AB and BD.But I'm not sure how to proceed from here.Wait, maybe I can consider the law of sines for each face. For face ABC, the law of sines says ( frac{AB}{sin alpha} = frac{BC}{sin beta} = frac{CA}{sin gamma} = 2R ), where ( alpha, beta, gamma ) are the angles opposite to AB, BC, CA respectively.Similarly, for face ABD, ( frac{AB}{sin delta} = frac{BD}{sin epsilon} = frac{DA}{sin zeta} = 2R ), where ( delta, epsilon, zeta ) are the angles opposite to AB, BD, DA respectively.Since all faces have the same R, the ratios for each face are equal.So, for face ABC: ( frac{AB}{sin alpha} = 2R )For face ABD: ( frac{AB}{sin delta} = 2R )Therefore, ( frac{AB}{sin alpha} = frac{AB}{sin delta} ), which implies ( sin alpha = sin delta ). So, either ( alpha = delta ) or ( alpha = pi - delta ). But since angles in a tetrahedron are less than ( pi ), both cases are possible.Hmm, but I'm not sure how this helps me relate the edges.Wait, maybe I can consider that if the opposite edges are equal, then the angles opposite to them in the faces would be equal, leading to equal circumradii. But I need to show the converse.Alternatively, perhaps I can use the fact that in a tetrahedron, if the opposite edges are equal, then the tetrahedron is isohedral, and thus all faces are congruent, leading to equal circumradii. But I need to show that equal circumradii imply equal opposite edges.Wait, maybe I can use the formula for the circumradius in terms of the edges and the volume. For a triangle, ( R = frac{abc}{4K} ), and for a tetrahedron, the volume V can be related to the areas of the faces and the dihedral angles.But I'm not sure.Wait, another idea: maybe I can use the fact that in a tetrahedron, the circumradius of a face is related to the circumradius of the tetrahedron and the distance from the center of the tetrahedron's circumsphere to the face.Specifically, if O is the center of the circumscribed sphere of the tetrahedron, and O_a is the center of the circumcircle of face BCD, then the distance from O to O_a is related to R and R_a.Wait, actually, in 3D geometry, the circumradius of a face can be related to the circumradius of the tetrahedron and the distance from the center to the face. The formula is ( R_a^2 = R^2 - d_a^2 ), where ( d_a ) is the distance from O to face BCD.Similarly, for all faces, ( R_b^2 = R^2 - d_b^2 ), ( R_c^2 = R^2 - d_c^2 ), ( R_d^2 = R^2 - d_d^2 ).Since all ( R_a = R_b = R_c = R_d = R ), then ( R^2 - d_a^2 = R^2 - d_b^2 = R^2 - d_c^2 = R^2 - d_d^2 ). Therefore, all ( d_a = d_b = d_c = d_d ). So, the distances from the center O to all four faces are equal.But in a tetrahedron, if the distances from the center of the circumscribed sphere to all four faces are equal, then the tetrahedron is isohedral, meaning it's regular. Wait, no, isohedral means that the symmetry group acts transitively on the faces, which is a stronger condition than just equal distances.Wait, but if all the distances from O to the faces are equal, then the tetrahedron is equidistant from all faces, which might imply that it's isohedral, but I'm not sure.Wait, actually, in a tetrahedron, if the distances from the circumcenter to all four faces are equal, then the tetrahedron is called "equidistant" or "isohedral". But I think that such tetrahedrons are exactly the isohedral ones, which are the regular tetrahedrons. But wait, no, isohedral doesn't necessarily mean regular; it just means that the faces are congruent and can be mapped onto each other by symmetries.But in our case, the tetrahedron might not be regular, but just have equal circumradii on all faces, leading to equal distances from the circumcenter to all faces. So, maybe it's a more general class.Wait, but if the distances from O to all faces are equal, then the tetrahedron is isohedral. So, maybe it's a regular tetrahedron, but I'm not sure.Wait, no, a regular tetrahedron is a special case of an isohedral tetrahedron where all faces are congruent equilateral triangles. But in our case, the faces might not be congruent, but just have equal circumradii.Wait, maybe I'm overcomplicating. Let me think again.If the distances from O to all four faces are equal, then the tetrahedron is isohedral, meaning that all faces are congruent. Therefore, all faces are congruent triangles, so their edges must be equal in pairs. Therefore, AB = CD, AC = BD, AD = BC.Wait, but I'm not sure if equal distances from O to all faces imply that the faces are congruent. Maybe it's a theorem that if the distances from the circumcenter to all faces are equal, then the tetrahedron is isohedral.Alternatively, maybe I can use the fact that in such a tetrahedron, the opposite edges are equal. Let me try to show that AB = CD.Consider the face ABC and face ADC. Both have circumradius R. Let me write the formula for R for both faces.For face ABC: ( R = frac{AB cdot BC cdot CA}{4K_{ABC}} )For face ADC: ( R = frac{AD cdot DC cdot CA}{4K_{ADC}} )Since R is the same, we have ( frac{AB cdot BC cdot CA}{4K_{ABC}} = frac{AD cdot DC cdot CA}{4K_{ADC}} )Simplify: ( frac{AB cdot BC}{K_{ABC}} = frac{AD cdot DC}{K_{ADC}} )Hmm, not sure. Maybe I can relate the areas K_{ABC} and K_{ADC}.Wait, the areas can be expressed in terms of the volume V of the tetrahedron and the distances from the opposite vertices to the faces.Specifically, ( K_{ABC} = frac{3V}{h_D} ), where ( h_D ) is the height from D to face ABC.Similarly, ( K_{ADC} = frac{3V}{h_B} ), where ( h_B ) is the height from B to face ADC.But I don't know if that helps.Wait, but if the distances from O to all faces are equal, then the heights from O to all faces are equal. Let me denote this distance as d.Then, the volume can also be expressed as ( V = frac{1}{3} K_{ABC} cdot h_D ), and similarly for other faces.But since the distances from O to all faces are equal, maybe the heights from O are equal, but I'm not sure how that relates to the heights from the vertices.Wait, maybe I can use the fact that in a tetrahedron, the distance from the circumcenter to a face is related to the circumradius of that face.Wait, earlier I thought that ( R_a^2 = R^2 - d_a^2 ), where ( R ) is the circumradius of the tetrahedron and ( d_a ) is the distance from O to face a.If all ( R_a ) are equal, then ( R^2 - d_a^2 ) is constant, so all ( d_a ) are equal. Therefore, the distances from O to all faces are equal.So, in such a tetrahedron, the center O is equidistant from all four faces. Now, in a tetrahedron, if the circumcenter is equidistant from all four faces, then the tetrahedron is isohedral, meaning all faces are congruent. Therefore, all faces are congruent triangles, so their corresponding edges must be equal.Therefore, AB = CD, AC = BD, AD = BC.Wait, but I'm not sure if being equidistant from all faces implies that the tetrahedron is isohedral. Maybe I need to look up a theorem or something.Alternatively, maybe I can use the fact that if all four faces have equal circumradii, then the tetrahedron is isohedral, and thus the opposite edges are equal.But I'm not entirely sure about the exact theorem here. Maybe I can try to find a relationship between the edges.Wait, another approach: let's consider the face ABC and face ADC. Both have edge AC in common. Let me denote AC = b.For face ABC: ( R = frac{AB cdot BC cdot b}{4K_{ABC}} )For face ADC: ( R = frac{AD cdot DC cdot b}{4K_{ADC}} )Since R is the same, we have ( frac{AB cdot BC}{K_{ABC}} = frac{AD cdot DC}{K_{ADC}} )Similarly, for faces ABD and BCD, which share edge BD.For face ABD: ( R = frac{AB cdot BD cdot AD}{4K_{ABD}} )For face BCD: ( R = frac{BC cdot CD cdot BD}{4K_{BCD}} )So, ( frac{AB cdot AD}{K_{ABD}} = frac{BC cdot CD}{K_{BCD}} )Hmm, not sure.Wait, maybe I can use the fact that the areas of the faces can be related to the volume. For example, ( K_{ABC} = frac{3V}{h_D} ), where ( h_D ) is the height from D to face ABC.Similarly, ( K_{ADC} = frac{3V}{h_B} ), where ( h_B ) is the height from B to face ADC.But I don't know if that helps directly.Wait, but if the distances from O to all faces are equal, then the heights from O to all faces are equal. Let me denote this distance as d.Then, the volume can be expressed as ( V = frac{1}{3} K_{ABC} cdot d ), and similarly for other faces.But since all ( K_{ABC}, K_{ABD}, K_{ACD}, K_{BCD} ) are multiplied by the same d to get the same volume, that would imply that all areas are equal. Wait, no, because the volume is the same, so ( K_{ABC} cdot d = K_{ABD} cdot d = K_{ACD} cdot d = K_{BCD} cdot d ), which would imply that all areas are equal.Wait, but that can't be right because in a general tetrahedron, the areas of the faces can be different even if the circumradii are equal. Or can they?Wait, if the distances from O to all faces are equal, then ( K_{ABC} = K_{ABD} = K_{ACD} = K_{BCD} ), because ( V = frac{1}{3} K cdot d ), and V is the same for all. So, if d is the same, then K must be the same for all faces.Therefore, all four faces have equal areas. So, not only do they have equal circumradii, but their areas are also equal.Now, if all four faces have equal areas and equal circumradii, maybe that's enough to conclude that the opposite edges are equal.Wait, let's think about two triangles with equal areas and equal circumradii. Does that imply that they are congruent? Not necessarily. For example, two different triangles can have the same area and circumradius but different side lengths.Wait, let me check. Suppose triangle 1 has sides 3,4,5, area 6, circumradius 2.5. Triangle 2 has sides 5,5,6, area 12, circumradius 2.5. Wait, no, the area is different. Wait, can I find two triangles with the same area and same circumradius but different side lengths?Let me try. Let me take triangle 1: sides 5,5,6. Its area is ( sqrt{8(8-5)(8-5)(8-6)} = sqrt{8 cdot 3 cdot 3 cdot 2} = sqrt{144} = 12 ). Its circumradius is ( frac{5 cdot 5 cdot 6}{4 cdot 12} = frac{150}{48} = frac{25}{8} = 3.125 ).Now, triangle 2: sides 4,5,6. Its area is ( sqrt{7.5(7.5-4)(7.5-5)(7.5-6)} = sqrt{7.5 cdot 3.5 cdot 2.5 cdot 1.5} ). Let me compute that: 7.5*3.5=26.25, 2.5*1.5=3.75, so 26.25*3.75=98.4375. So, area is ( sqrt{98.4375} approx 9.921567 ). Its circumradius is ( frac{4 cdot 5 cdot 6}{4 cdot 9.921567} = frac{120}{39.686268} approx 3.025 ).Hmm, not equal. Maybe another example.Wait, maybe it's hard to find such triangles. Maybe equal area and equal circumradius imply congruence? I'm not sure.Wait, let me think about the formula ( R = frac{abc}{4K} ). If two triangles have the same R and same K, then ( abc ) must be the same. So, if two triangles have the same area and same circumradius, then the product of their sides is the same.But that doesn't necessarily mean the triangles are congruent. For example, triangle 1: sides 3,4,5, product 60, area 6, R=2.5. Triangle 2: sides 2,5,6, product 60, area ( sqrt{6.5(6.5-2)(6.5-5)(6.5-6)} = sqrt{6.5 cdot 4.5 cdot 1.5 cdot 0.5} ). Let's compute: 6.5*4.5=29.25, 1.5*0.5=0.75, so 29.25*0.75=21.9375. Area is ( sqrt{21.9375} approx 4.683 ). Its circumradius is ( frac{2 cdot 5 cdot 6}{4 cdot 4.683} = frac{60}{18.732} approx 3.203 ). So, different R.Wait, so maybe if two triangles have the same area and same circumradius, then they must be congruent? Because ( R = frac{abc}{4K} ), so if K and R are the same, then abc is the same. But abc being the same doesn't necessarily mean the triangles are congruent, as shown in my previous example, but in that case, the areas were different.Wait, in my first example, triangle 1: 3,4,5, area 6, R=2.5. Triangle 2: sides 5,5,6, area 12, R=3.125. So, different areas and different R.Wait, maybe if two triangles have the same area and same R, then they must be congruent. Let me see.Suppose triangle 1 has sides a,b,c, area K, R.Triangle 2 has sides d,e,f, area K, R.Then, ( a b c = 4 R K ), and ( d e f = 4 R K ). So, ( a b c = d e f ).But does that imply that the triangles are congruent? Not necessarily. For example, triangle 1: 3,4,5, product 60, area 6, R=2.5.Triangle 2: 2,5,6, product 60, area≈4.683, R≈3.203. Wait, but in this case, R is different. So, if K and R are the same, then the product abc must be the same, but also, the area K is the same, so maybe the triangles must be congruent.Wait, let me think. If two triangles have the same area and same circumradius, then they must be congruent. Because from ( R = frac{abc}{4K} ), if K and R are the same, then abc is the same. Also, the area K is the same. So, maybe the triangles must be congruent.Wait, let me test with actual numbers. Let me find two triangles with the same area and same R.Let me take triangle 1: sides 5,5,6, area 12, R=25/8=3.125.Now, let me find another triangle with area 12 and R=3.125.Using ( R = frac{abc}{4K} ), so ( abc = 4 R K = 4 * 3.125 * 12 = 150 ).So, I need a triangle with sides a,b,c such that a*b*c=150 and area=12.Let me try sides 5,5,6: product 150, area 12. That's the same triangle.Another example: sides 3,4,5: product 60, area 6, R=2.5.If I want another triangle with area 6 and R=2.5, then abc=4*2.5*6=60.So, sides 3,4,5 is one, another could be 2,5,6: product 60, but area≈4.683, which is different.Wait, so maybe it's not possible to have two different triangles with the same area and same R. Therefore, if two triangles have the same area and same circumradius, they must be congruent.Therefore, in our tetrahedron, since all four faces have equal areas and equal circumradii, all four faces must be congruent triangles. Therefore, the tetrahedron is isohedral, meaning all its faces are congruent, and thus the opposite edges must be equal.Therefore, AB=CD, AC=BD, AD=BC.Wait, but I'm not entirely sure if my reasoning about the triangles is correct. Maybe I need to look for a theorem or a more rigorous proof.Alternatively, maybe I can use the fact that in a tetrahedron, if all four faces have equal circumradii, then the tetrahedron is isohedral, and thus the opposite edges are equal.But I think the key idea is that equal circumradii and equal areas of the faces imply congruent faces, leading to equal opposite edges.

answer：First, I need to determine the total sum of all six numbers. Given that their average is 3.95, I can calculate the total sum by multiplying the average by the number of numbers:6 × 3.95 = 23.7Next, I'll find the sum of the first two numbers with an average of 3.8:2 × 3.8 = 7.6Then, I'll calculate the sum of the next two numbers with an average of 3.85:2 × 3.85 = 7.7Now, I'll add the sums of these four numbers to find their combined total:7.6 + 7.7 = 15.3To find the sum of the remaining two numbers, I'll subtract the combined total of the first four numbers from the overall total sum:23.7 - 15.3 = 8.4Finally, to determine the average of the remaining two numbers, I'll divide their sum by 2:8.4 ÷ 2 = 4.2Therefore, the average of the remaining two numbers is 4.2.

answer：Okay, so I have this function g(x) which is equal to x cubed plus x squared minus one. And I need to find the value of g at 2, which means I need to plug in 2 into this function and calculate the result. Alright, let's break this down step by step.First, I should recall what it means to evaluate a function at a specific value. When we say g(2), we're essentially replacing every x in the function with the number 2 and then simplifying the expression to find the result. So, starting with the function:g(x) = x³ + x² - 1Now, substituting x with 2:g(2) = (2)³ + (2)² - 1Okay, let's compute each term separately to avoid mistakes. I'll start with the first term, which is 2 cubed. Cubing a number means multiplying it by itself three times. So, 2 times 2 times 2. Let me do that:2 × 2 = 44 × 2 = 8So, 2 cubed is 8. Got that down.Next, the second term is 2 squared. Squaring a number means multiplying it by itself twice. So, 2 times 2:2 × 2 = 4Alright, 2 squared is 4.Now, the last term is just -1, so that stays as it is.Putting it all together:g(2) = 8 + 4 - 1Now, let's add these numbers step by step. First, add 8 and 4:8 + 4 = 12Then, subtract 1 from that result:12 - 1 = 11So, after evaluating each term and performing the arithmetic operations, I end up with 11.Wait a second, let me double-check to make sure I didn't make any mistakes. It's easy to mix up the exponents or misplace a negative sign. So, going back:g(2) = 2³ + 2² - 1= 8 + 4 - 1= 12 - 1= 11Yep, that looks correct. Each step checks out. I calculated the exponents properly and performed the addition and subtraction in the right order. So, I'm confident that g(2) equals 11.I guess another way to think about it is to visualize the function. If I were to graph g(x), it's a cubic function because the highest exponent is 3. Cubic functions can have various shapes, but in this case, since the coefficient of x³ is positive, it will rise to the right and fall to the left. But that's probably beyond what I need for this problem. I just need the specific value at x=2.Just to be thorough, maybe I can plug in x=2 into the function again to confirm:g(2) = (2)^3 + (2)^2 - 1= 8 + 4 - 1= 12 - 1= 11Same result. Okay, that's reassuring.I wonder if there's a quicker way to do this without computing each term separately. Maybe by factoring or something? Let's see:g(x) = x³ + x² - 1Is there a way to factor this? Hmm, factoring cubics can be tricky. Maybe I can try to factor by grouping, but looking at the terms, it doesn't seem straightforward. The first two terms have x squared in common:x³ + x² = x²(x + 1)So, g(x) = x²(x + 1) - 1But that doesn't really help me evaluate at x=2 any faster. It's still going to require me to compute x squared times (x + 1) and then subtract 1.Alternatively, I could think about the function in terms of its components. The function is a combination of a cubic term, a quadratic term, and a constant. Each of these contributes differently to the overall value. At x=2, the cubic term (8) is larger than the quadratic term (4), so the function is growing rapidly here.But again, that's more about understanding the behavior of the function rather than computing g(2). For this problem, straightforward substitution seems to be the most efficient method.Another thing I could consider is using the order of operations to ensure I'm calculating everything correctly. Remembering PEMDAS—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In this case, there are no parentheses other than the ones around the 2, which don't affect the order. The exponents are calculated first, so 2³ and 2² come before the addition and subtraction.So, following PEMDAS:1. Exponents: - 2³ = 8 - 2² = 42. Addition and Subtraction from left to right: - 8 + 4 = 12 - 12 - 1 = 11Everything lines up perfectly. No steps skipped, no errors in calculation.I suppose if I were teaching someone else how to do this, I'd emphasize the importance of substituting carefully and performing each operation step by step. It's easy to rush and maybe add instead of subtract or miscalculate an exponent, especially with negative numbers or larger exponents.But in this case, with positive numbers and small exponents, it's pretty straightforward. Still, attention to detail is key. Even simple calculations can lead to wrong answers if not approached methodically.Just to explore a bit more, what if I wanted to find g(2) without calculating each term? Maybe by recognizing a pattern or using some algebraic identity? Let's see:g(x) = x³ + x² - 1Is there a way to write this differently? Perhaps factor something out or complete the square? For example, if I factor x² from the first two terms:g(x) = x²(x + 1) - 1But as I thought earlier, this doesn't really help in evaluating at x=2. It's still two separate terms that need to be calculated.Alternatively, could I write this as (x³ + x²) - 1? Well, that's essentially what it is. So, unless there's a way to simplify x³ + x² further, which I don't think there is without additional context or constraints, this approach doesn't offer any advantage.Another angle might be to consider the function's behavior around x=2. For instance, if I wanted to approximate g(2) using derivatives or something, but that's definitely overcomplicating things for this problem. We're just asked for the exact value, so direct substitution is the way to go.I could also think about this in terms of polynomial evaluation. Polynomials are some of the simplest functions to evaluate because they only involve addition, subtraction, multiplication, and non-negative integer exponents. So, for any given x, you can compute the value by performing these basic operations step by step.In this case, since we're dealing with a cubic polynomial, the process is a bit longer than, say, a linear function, but still manageable. It's all about breaking it down into smaller, more digestible parts.One thing I might consider is using the Horner's method for evaluating polynomials, which can sometimes be more efficient, especially for higher-degree polynomials. Let's see how that would work here.Horner's method involves rewriting the polynomial in a nested form to minimize the number of multiplications. For g(x) = x³ + x² - 1, we can rewrite it as:g(x) = x(x + 1) - 1Wait, that's similar to what I did earlier when factoring out x². But perhaps I can take it a step further.Let me try to express it as:g(x) = ((x) * x) * x + x² - 1Hmm, that doesn't seem particularly helpful. Maybe I'm not applying Horner's method correctly here. Let me recall: Horner's method for a polynomial like a x³ + b x² + c x + d would be written as ((a x + b) x + c) x + d.Applying that to our function:g(x) = 1 x³ + 1 x² + 0 x - 1So, using Horner's method:Start with the coefficients: 1 (for x³), 1 (for x²), 0 (for x), and -1 (constant term).Now, evaluate at x=2:Start with the leading coefficient: 1Multiply by x (which is 2) and add the next coefficient:1 * 2 + 1 = 2 + 1 = 3Multiply by x again:3 * 2 = 6Add the next coefficient (which is 0):6 + 0 = 6Multiply by x:6 * 2 = 12Add the last coefficient (-1):12 + (-1) = 11So, using Horner's method, we also get g(2) = 11.Interesting! I didn't think of using Horner's method initially, but it's a good reminder that there are different techniques to evaluate polynomials efficiently, especially useful for higher-degree polynomials where it can save computation time.But for this problem, since it's a cubic, both methods—direct substitution and Horner's—are manageable and lead to the same result.I think I've covered all the bases here. I've evaluated the function directly, double-checked my work, considered alternative methods like factoring and Horner's method, and even reflected on why these steps are important. It's a good practice to verify your answers, especially when first learning these concepts, to build confidence and catch any potential mistakes.In summary, by substituting x=2 into the function g(x) = x³ + x² - 1 and carefully computing each term, I arrived at the conclusion that g(2) equals 11. This process involved understanding function evaluation, applying the order of operations, and considering alternative methods to confirm the result.

answer：Okay, so I have this problem here: I need to find a positive number such that its decimal part, integer part, and the number itself form three consecutive terms of a geometric sequence. Hmm, let me try to break this down step by step.First, let me recall what a geometric sequence is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if I have three terms a, b, c in a geometric sequence, then b/a = c/b, which means b² = a*c.In this problem, the three terms are the decimal part, the integer part, and the number itself. Let me denote the number as x. So, if x is a positive number, it can be written as x = [x] + {x}, where [x] is the integer part and {x} is the decimal part. For example, if x is 3.27, then [x] is 3 and {x} is 0.27.So, according to the problem, {x}, [x], and x should form a geometric sequence. That means the ratio between [x] and {x} should be the same as the ratio between x and [x]. So, mathematically, that can be written as:[x] / {x} = x / [x]Let me write that down:[x] / {x} = x / [x]Since x = [x] + {x}, I can substitute that into the equation:[x] / {x} = ([x] + {x}) / [x]Simplify the right side:[x] / {x} = 1 + {x} / [x]Hmm, so now I have:[x] / {x} = 1 + {x} / [x]Let me denote [x] as n and {x} as r for simplicity. So, n is an integer greater than or equal to 1, and r is a decimal between 0 and 1.So substituting, we have:n / r = 1 + r / nLet me write that:n / r = 1 + (r / n)Multiply both sides by r to eliminate the denominator:n = r + (r² / n)Multiply both sides by n to get rid of the denominator on the right side:n² = nr + r²So, n² = r(n + r)But since x = n + r, we can write:n² = r*xSo, n² = r*xBut x = n + r, so substituting back:n² = r*(n + r)Let me write that:n² = nr + r²Bring all terms to one side:n² - nr - r² = 0Hmm, this is a quadratic equation in terms of n or r. Let me see if I can solve for r in terms of n.Let me rearrange the equation:r² + nr - n² = 0Yes, that's a quadratic in r. So, using the quadratic formula:r = [-n ± sqrt(n² + 4n²)] / 2Simplify under the square root:sqrt(n² + 4n²) = sqrt(5n²) = n*sqrt(5)So, r = [-n ± n*sqrt(5)] / 2Since r is a decimal part, it must be positive and less than 1. So, we discard the negative solution:r = [-n + n*sqrt(5)] / 2Factor out n:r = n(-1 + sqrt(5)) / 2So, r = n*(sqrt(5) - 1)/2But remember that r must be less than 1 because it's the decimal part. So:n*(sqrt(5) - 1)/2 < 1Let me compute (sqrt(5) - 1)/2 approximately. sqrt(5) is about 2.236, so 2.236 - 1 = 1.236, divided by 2 is approximately 0.618.So, 0.618 * n < 1Which means n < 1 / 0.618 ≈ 1.618Since n is an integer greater than or equal to 1, the only possible value is n = 1.So, n = 1.Then, plugging back into r:r = 1*(sqrt(5) - 1)/2 ≈ (2.236 - 1)/2 ≈ 1.236/2 ≈ 0.618So, r ≈ 0.618, which is less than 1, which is good.Therefore, the number x is n + r = 1 + (sqrt(5) - 1)/2Simplify that:x = 1 + (sqrt(5) - 1)/2 = (2 + sqrt(5) - 1)/2 = (1 + sqrt(5))/2So, x = (1 + sqrt(5))/2Let me verify if this satisfies the original condition.Compute the decimal part {x} = x - [x] = (1 + sqrt(5))/2 - 1 = (sqrt(5) - 1)/2 ≈ 0.618So, the three terms are:1. {x} ≈ 0.6182. [x] = 13. x ≈ 1.618Check if they form a geometric sequence.Compute the ratio between the second and first term: 1 / 0.618 ≈ 1.618Compute the ratio between the third and second term: 1.618 / 1 = 1.618Yes, both ratios are equal, so it is indeed a geometric sequence with common ratio (1 + sqrt(5))/2.Therefore, the solution is x = (1 + sqrt(5))/2.**Final Answer**The positive number is boxed{dfrac{1 + sqrt{5}}{2}}.

answer：Okay, so I have this problem about sets, and I need to solve two parts. Let me first make sure I understand what each part is asking.We have three sets: A, B, and U. Set A is {1,2,3,4,5}, set B is {-1,1,2,3}, and set U is defined as all integers x such that -1 ≤ x < 6. So, U should be {-1,0,1,2,3,4,5}. I think that's right because it includes all integers from -1 up to but not including 6.Part (1) asks me to find A ∩ B and A ∪ B. Okay, so intersection and union of sets A and B. Let me recall: the intersection of two sets is the set of elements that are in both sets, and the union is the set of elements that are in either of the sets.So, for A ∩ B, I need to find elements that are in both A and B. Let's see: A has 1,2,3,4,5 and B has -1,1,2,3. So, the common elements are 1,2,3. Therefore, A ∩ B should be {1,2,3}.For A ∪ B, I need all elements that are in A or in B. So, combining both sets, we have -1,1,2,3,4,5. So, A ∪ B is {-1,1,2,3,4,5}.Wait, let me double-check. A is {1,2,3,4,5}, B is {-1,1,2,3}. So, when taking the union, we include all elements from both sets without duplication. So yes, -1 is only in B, and 4,5 are only in A, so the union should indeed be {-1,1,2,3,4,5}. That seems correct.Okay, moving on to part (2): Find (C_U B) ∩ A. Hmm, I need to figure out what C_U B means. I think this notation refers to the complement of B with respect to U. So, the complement of B in U would be all elements in U that are not in B. Then, we take the intersection of that complement with set A.Let me write down U again: U is {-1,0,1,2,3,4,5}. Set B is {-1,1,2,3}. So, the complement of B in U would be all elements in U that are not in B. Let's list them: U has -1,0,1,2,3,4,5. B has -1,1,2,3. So, the elements not in B are 0,4,5. Therefore, C_U B is {0,4,5}.Now, we need to find the intersection of this complement with set A. Set A is {1,2,3,4,5}. So, {0,4,5} ∩ {1,2,3,4,5} would be the elements common to both sets. The common elements are 4 and 5. So, (C_U B) ∩ A is {4,5}.Wait, let me make sure I didn't make a mistake. The complement of B in U is U minus B, which is {-1,0,1,2,3,4,5} minus {-1,1,2,3}, so that leaves 0,4,5. Then, intersecting that with A, which is {1,2,3,4,5}, so 0 is not in A, but 4 and 5 are. So, yes, {4,5} is correct.Just to recap:1. A ∩ B: Common elements between A and B are 1,2,3.2. A ∪ B: All elements from both sets, which are -1,1,2,3,4,5.3. Complement of B in U: Elements in U not in B, which are 0,4,5.4. Intersection of that complement with A: 4 and 5.I think that's all correct. I don't see any mistakes in my reasoning.**Final Answer**(1) Acap B = boxed{{1, 2, 3}} and Acup B = boxed{{-1, 1, 2, 3, 4, 5}};(2) (C_{U}B) cap A = boxed{{4, 5}}.