answer：First, I recognize that Thanksgiving is on Friday, November 24. To find out the day of the week for December 24, I need to calculate the number of days between these two dates.There are 7 days in a week, so every 7 days, the day of the week repeats. I'll determine how many weeks and extra days are between November 24 and December 24.Next, I'll add the calculated number of days to November 24 to find the corresponding day of the week for December 24.Finally, I'll conclude the day of the week for December 24 based on the calculations.

answer：Alright, so I need to show that the Boole transformation ( x rightarrow x - frac{lambda}{x} ) preserves the Lebesgue measure on ( mathbb{R} ). That means, for any integrable function ( f ) on ( mathbb{R} ), the integral of ( f ) over ( mathbb{R} ) should be the same as the integral of ( f ) composed with this transformation. In other words, I need to show that:[int_{mathbb{R}} f(x) , dx = int_{mathbb{R}} fleft(x - frac{lambda}{x}right) , dx]Okay, so first, I should understand what it means for a transformation to preserve Lebesgue measure. If a transformation preserves Lebesgue measure, it means that the measure of any set doesn't change under the transformation. In terms of integrals, this implies that integrating a function over the transformed space is the same as integrating it over the original space.Now, the transformation given is ( x rightarrow x - frac{lambda}{x} ). This seems like a non-linear transformation because of the ( frac{1}{x} ) term. Non-linear transformations can complicate things because they might stretch or shrink different parts of the space differently.To tackle this, I think I need to perform a substitution in the integral. Let me denote the transformation as ( T(x) = x - frac{lambda}{x} ). So, I need to compute ( int_{mathbb{R}} f(T(x)) , dx ) and show that it's equal to ( int_{mathbb{R}} f(x) , dx ).Let me consider splitting the integral into two parts: one over the positive real numbers and one over the negative real numbers. That is,[int_{mathbb{R}} f(T(x)) , dx = int_{0}^{infty} fleft(x - frac{lambda}{x}right) , dx + int_{-infty}^{0} fleft(x - frac{lambda}{x}right) , dx]For the negative part, maybe I can make a substitution to relate it to the positive part. Let me set ( y = -x ) in the negative integral. Then, when ( x ) goes from ( -infty ) to 0, ( y ) goes from ( infty ) to 0. Also, ( dx = -dy ). So, the negative integral becomes:[int_{-infty}^{0} fleft(x - frac{lambda}{x}right) , dx = int_{infty}^{0} fleft(-y - frac{lambda}{-y}right) (-dy) = int_{0}^{infty} fleft(-y + frac{lambda}{y}right) , dy]So now, the entire integral becomes:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) , dx + int_{0}^{infty} fleft(-x + frac{lambda}{x}right) , dx]Hmm, interesting. So, I have two integrals over the positive real numbers now. Maybe I can combine them or find a substitution that relates these two expressions.Let me consider the substitution ( z = frac{lambda}{x} ) in the second integral. If I let ( z = frac{lambda}{x} ), then ( x = frac{lambda}{z} ) and ( dx = -frac{lambda}{z^2} dz ). Also, when ( x ) goes from 0 to ( infty ), ( z ) goes from ( infty ) to 0.So, substituting into the second integral:[int_{0}^{infty} fleft(-x + frac{lambda}{x}right) , dx = int_{infty}^{0} fleft(-frac{lambda}{z} + zright) left(-frac{lambda}{z^2}right) dz]Simplifying this, the negative sign from ( dx ) and the negative sign from swapping the limits cancel out, and we get:[int_{0}^{infty} fleft(-frac{lambda}{z} + zright) frac{lambda}{z^2} dz]But ( -frac{lambda}{z} + z = z - frac{lambda}{z} ), which is similar to the transformation ( T(z) = z - frac{lambda}{z} ). So, this integral becomes:[int_{0}^{infty} fleft(z - frac{lambda}{z}right) frac{lambda}{z^2} dz]Wait a minute, so now I have:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) , dx + int_{0}^{infty} fleft(z - frac{lambda}{z}right) frac{lambda}{z^2} dz]But both integrals are over the same variable ( x ) (or ( z )), so I can write this as:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]Hmm, that's interesting. So, the total integral becomes:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]But I need this to be equal to ( int_{mathbb{R}} f(x) , dx ). That suggests that the Jacobian determinant of the transformation should be 1, but here it seems like there's an extra factor of ( 1 + frac{lambda}{x^2} ).Wait, maybe I made a mistake in the substitution. Let me double-check.When I did the substitution ( z = frac{lambda}{x} ), I had ( x = frac{lambda}{z} ) and ( dx = -frac{lambda}{z^2} dz ). Then, the integral became:[int_{infty}^{0} fleft(-frac{lambda}{z} + zright) left(-frac{lambda}{z^2}right) dz = int_{0}^{infty} fleft(z - frac{lambda}{z}right) frac{lambda}{z^2} dz]Yes, that seems correct. So, the factor ( frac{lambda}{z^2} ) comes from the substitution.But then, combining the two integrals, I get:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]But this is supposed to equal ( int_{mathbb{R}} f(x) , dx ). So, unless ( 1 + frac{lambda}{x^2} ) is somehow the reciprocal of the Jacobian, which would make the total measure preserved.Wait, let's think about the transformation ( T(x) = x - frac{lambda}{x} ). To find the Jacobian, I need to compute the derivative of ( T(x) ) with respect to ( x ).So, ( T'(x) = 1 + frac{lambda}{x^2} ).Ah! So, the derivative of the transformation is ( 1 + frac{lambda}{x^2} ). Therefore, the absolute value of the Jacobian determinant is ( |T'(x)| = 1 + frac{lambda}{x^2} ).In measure theory, when changing variables, the measure transforms as ( dx = frac{1}{|T'(x)|} dT(x) ). So, if I have ( dT(x) = T'(x) dx ), then ( dx = frac{1}{T'(x)} dT(x) ).But in our case, we have:[int_{mathbb{R}} f(T(x)) , dx = int_{mathbb{R}} f(y) frac{1}{|T'(x)|} dy]Wait, no, actually, the change of variables formula is:[int_{mathbb{R}} f(T(x)) |T'(x)| dx = int_{mathbb{R}} f(y) dy]But in our case, we have:[int_{mathbb{R}} f(T(x)) dx]Which would be:[int_{mathbb{R}} f(y) frac{1}{|T'(x)|} dy]But this is only valid if ( T ) is invertible and smooth, which it is except at ( x = 0 ).But in our case, we have:[int_{mathbb{R}} f(T(x)) dx = int_{mathbb{R}} f(y) frac{1}{|T'(x)|} dy]But ( |T'(x)| = 1 + frac{lambda}{x^2} ), which depends on ( x ), not ( y ). So, unless ( frac{1}{|T'(x)|} ) can be expressed in terms of ( y ), which is ( T(x) = x - frac{lambda}{x} ), this might complicate things.Wait, but from the earlier substitution, we saw that:[int_{mathbb{R}} f(T(x)) dx = int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]But this is equal to:[int_{0}^{infty} f(y) dy]Wait, no, that doesn't make sense because ( y = x - frac{lambda}{x} ), which is not a bijection over ( mathbb{R} ). So, perhaps I need to consider the transformation more carefully.Alternatively, maybe I should consider the transformation in both positive and negative parts and see if they cancel out or something.Wait, let's think about the transformation ( T(x) = x - frac{lambda}{x} ). For ( x > 0 ), as ( x ) increases from 0 to ( infty ), ( T(x) ) goes from ( -infty ) to ( infty ). Similarly, for ( x < 0 ), as ( x ) goes from 0 to ( -infty ), ( T(x) ) goes from ( infty ) to ( -infty ).So, the transformation is actually covering the entire real line twice: once for positive ( x ) and once for negative ( x ). Therefore, when we integrate over ( mathbb{R} ), we are effectively integrating over two copies of ( mathbb{R} ), but with a Jacobian factor.But in our earlier substitution, we saw that:[int_{mathbb{R}} f(T(x)) dx = int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]But this should equal ( int_{mathbb{R}} f(y) dy ). So, perhaps the Jacobian factor ( 1 + frac{lambda}{x^2} ) is exactly the factor that accounts for the two-to-one nature of the transformation, thus making the total measure preserved.Wait, let's think about it. If the transformation is two-to-one, then each point ( y ) in ( mathbb{R} ) has two pre-images: one positive ( x ) and one negative ( x ). Therefore, when we integrate over ( x ), we are effectively integrating over both pre-images, but with a Jacobian factor for each.So, if I have:[int_{mathbb{R}} f(T(x)) dx = int_{mathbb{R}} f(y) left( frac{1}{|T'(x_1)|} + frac{1}{|T'(x_2)|} right) dy]Where ( x_1 ) and ( x_2 ) are the two pre-images of ( y ). But in our case, the Jacobian factor is ( 1 + frac{lambda}{x^2} ), which is the same for both pre-images because ( x_2 = -x_1 ) and ( frac{lambda}{x_2^2} = frac{lambda}{x_1^2} ).Wait, no, actually, for ( x_2 = -x_1 ), ( T'(x_2) = 1 + frac{lambda}{x_2^2} = 1 + frac{lambda}{x_1^2} = T'(x_1) ). So, both Jacobian factors are the same.Therefore, the total integral becomes:[int_{mathbb{R}} f(y) left( frac{1}{|T'(x_1)|} + frac{1}{|T'(x_2)|} right) dy = int_{mathbb{R}} f(y) left( frac{2}{|T'(x_1)|} right) dy]But this should equal ( int_{mathbb{R}} f(y) dy ). Therefore, we have:[frac{2}{|T'(x_1)|} = 1 implies |T'(x_1)| = 2]But ( |T'(x_1)| = 1 + frac{lambda}{x_1^2} ). So,[1 + frac{lambda}{x_1^2} = 2 implies frac{lambda}{x_1^2} = 1 implies x_1^2 = lambda implies x_1 = sqrt{lambda}]Wait, that can't be right because ( x_1 ) depends on ( y ), but here it's suggesting that ( x_1 ) is fixed at ( sqrt{lambda} ), which doesn't make sense because ( y ) can be any real number.Hmm, I think I'm getting confused here. Maybe I need to approach this differently.Let me consider the transformation ( T(x) = x - frac{lambda}{x} ) and see if it's measure-preserving by checking if the pushforward measure is equal to the original measure.In other words, for any measurable set ( A subseteq mathbb{R} ), the measure of ( T^{-1}(A) ) should be equal to the measure of ( A ).But since ( T ) is a bijection except at ( x = 0 ), and it's smooth, the measure preservation can be checked by the Jacobian condition.Wait, but earlier, I saw that the Jacobian is ( 1 + frac{lambda}{x^2} ), which is not equal to 1. So, unless the transformation has some symmetry that makes the Jacobian factors cancel out when considering both positive and negative ( x ).Wait, let's go back to the substitution approach. We had:[int_{mathbb{R}} f(T(x)) dx = int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx]But this is supposed to equal ( int_{mathbb{R}} f(y) dy ). So, if I can show that:[int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx = int_{mathbb{R}} f(y) dy]Then, the measure is preserved.But how can this be? The left-hand side is an integral over ( x ) from 0 to ( infty ), while the right-hand side is an integral over all ( y in mathbb{R} ).Wait, perhaps I need to make a substitution in the left-hand side to express it in terms of ( y ).Let me set ( y = x - frac{lambda}{x} ). Then, ( dy = left(1 + frac{lambda}{x^2}right) dx ).So, the left-hand side becomes:[int_{y=-infty}^{infty} f(y) dy]Because as ( x ) goes from 0 to ( infty ), ( y ) goes from ( -infty ) to ( infty ). Therefore,[int_{0}^{infty} fleft(x - frac{lambda}{x}right) left(1 + frac{lambda}{x^2}right) dx = int_{-infty}^{infty} f(y) dy]Which is exactly what we wanted to show!So, putting it all together:1. We split the integral into positive and negative parts.2. For the negative part, we made a substitution ( y = -x ), transforming it into an integral over positive ( y ).3. Then, we made another substitution ( z = frac{lambda}{x} ) in the second integral, which led us to express the integral in terms of ( y ).4. Finally, by recognizing that ( dy = left(1 + frac{lambda}{x^2}right) dx ), we saw that the left-hand side becomes the integral of ( f(y) ) over all ( y ), which is the right-hand side.Therefore, the transformation ( x rightarrow x - frac{lambda}{x} ) preserves the Lebesgue measure on ( mathbb{R} ) for any ( lambda > 0 ).**Final Answer**The Boole transformation preserves the Lebesgue measure on (mathbb{R}) for any (lambda > 0), as shown by the substitution and Jacobian analysis. Thus,[boxed{int_{mathbb{R}} f(x) , dx = int_{mathbb{R}} fleft(x - frac{lambda}{x}right) , dx}]

answer：First, I recognize that the boy is positioned 19th from both the front and the back of the line.This means there are 18 boys in front of him and 18 boys behind him.To find the total number of boys in the class, I add the boys in front of him, the boy himself, and the boys behind him.So, 18 boys in front + 1 boy himself + 18 boys behind equals a total of 37 boys in the class.

answer：Okay, so I have this problem where I need to construct a line segment on a plane that's equal to the radius of a given sphere using just a compass and a ruler. Hmm, that sounds interesting. Let me think about how to approach this.First, I know that a sphere is a three-dimensional object, and its radius is the distance from its center to any point on its surface. But I don't have the center of the sphere; I just have the sphere itself. So, I need to figure out a way to find a segment equal to the radius without knowing where the center is.Maybe I can start by picking a point on the sphere. Let's call this point A. If I can somehow relate this point to the center, I can measure the radius. But without knowing the center, that's tricky. Wait, maybe I can use some geometric constructions to find the center indirectly.I remember that if I have a circle on the sphere, the center of that circle is not the same as the center of the sphere unless the circle is a great circle. So, if I draw a circle on the sphere with point A as the center, that circle won't necessarily help me find the sphere's center. Hmm, maybe I need a different approach.What if I use multiple points on the sphere? If I can find three non-collinear points on the sphere, I can form a triangle. Then, perhaps, the circumradius of that triangle will relate to the sphere's radius. But how?Let me recall that for any triangle inscribed in a sphere, the circumradius of the triangle is related to the sphere's radius. Specifically, if I have a triangle on the sphere, the circumradius of the triangle's planar projection can help me find the sphere's radius. But I'm not sure about the exact relationship here.Wait, maybe I can construct a triangle on the sphere and then construct its circumcircle on the plane. The radius of that circumcircle should be equal to the sphere's radius. Is that correct? Let me think.If I have three points on the sphere, forming a triangle, and then I construct the circumcircle of that triangle on the plane, the radius of that circumcircle should indeed be equal to the sphere's radius. Because the triangle lies on the sphere, and the circumcircle on the plane would essentially be the projection of the sphere's great circle.Okay, so here's a possible method:1. Choose any point A on the sphere.2. Using the compass, draw a circle on the sphere with center A.3. Choose three arbitrary points B, C, and D on this circle.4. Using the ruler and compass, construct triangle BCD on the plane.5. Construct the circumcircle of triangle BCD on the plane.6. The radius of this circumcircle will be equal to the radius of the sphere.Wait, does that make sense? Let me verify.If I have three points on a circle on the sphere, then the distances between these points are chords of the sphere. When I construct the triangle on the plane, the sides of the triangle are equal to the lengths of these chords. Then, the circumradius of this triangle should relate to the sphere's radius.Actually, the formula for the circumradius R of a triangle with sides a, b, c is R = (a*b*c)/(4*Δ), where Δ is the area of the triangle. But in this case, the triangle is on the sphere, so its sides are arcs, not straight lines. Hmm, maybe I'm complicating things.Alternatively, perhaps I can use the fact that the triangle BCD lies on a great circle of the sphere. If I can project this triangle onto the plane, the circumradius of the projected triangle would be equal to the sphere's radius.But how do I ensure that the triangle lies on a great circle? Well, if I choose three points on the sphere such that they lie on a great circle, then their projection onto the plane would form a triangle whose circumradius is equal to the sphere's radius.Wait, but how do I know that the three points I choose lie on a great circle? I think any three non-collinear points on a sphere lie on a unique great circle, so that should be fine.So, to summarize, the steps would be:1. Choose a point A on the sphere.2. Draw a circle on the sphere with center A.3. Choose three points B, C, D on this circle.4. Transfer the distances AB, AC, AD to the plane to form triangle BCD.5. Construct the circumcircle of triangle BCD on the plane.6. The radius of this circumcircle is equal to the sphere's radius.But wait, I think I made a mistake here. The distances AB, AC, AD are not the same as the sides of triangle BCD. The sides of triangle BCD are the arcs on the sphere, but when I transfer them to the plane, they become straight lines. So, the lengths of the sides of triangle BCD on the plane are equal to the chord lengths of the corresponding arcs on the sphere.Therefore, the circumradius of triangle BCD on the plane will be equal to the sphere's radius. Because the chord length is related to the radius by the formula chord length = 2R sin(θ/2), where θ is the central angle. But since the triangle is on a great circle, the central angles correspond to the angles of the triangle.Hmm, I'm getting a bit confused here. Maybe I need to think differently.Alternatively, I remember that if I have a regular tetrahedron inscribed in a sphere, the radius of the sphere can be found using the edge length. But I don't know if that's helpful here.Wait, maybe I can use the fact that the diameter of the sphere can be found by constructing two points on the sphere that are diametrically opposite. Then, the distance between these two points is twice the radius. But how do I find two diametrically opposite points without knowing the center?Hmm, that's a good question. Maybe I can construct a great circle, and then find two points on it that are 180 degrees apart. But without knowing the center, that's difficult.Wait, perhaps I can use the following method:1. Choose any point A on the sphere.2. Draw a circle on the sphere with center A.3. Choose another point B on this circle.4. Draw a circle on the sphere with center B.5. The intersection points of these two circles will be two points, say C and D.6. The line CD is the intersection of the two circles, which is a great circle.7. The center of the sphere lies on this great circle.Wait, but I still don't know where the center is. Maybe I can repeat this process with another pair of circles to find another great circle, and the intersection of these two great circles will give me the center.But that seems complicated. Maybe there's a simpler way.Let me go back to the original idea of constructing a triangle on the sphere and then its circumcircle on the plane. If I can ensure that the triangle is such that its circumradius on the plane equals the sphere's radius, then that would solve the problem.I think the key is that if I have a triangle on the sphere, the circumradius of its planar projection is equal to the sphere's radius. So, by constructing the circumcircle of the projected triangle, I can get the radius.Therefore, the steps would be:1. Choose any point A on the sphere.2. Draw a circle on the sphere with center A.3. Choose three points B, C, D on this circle.4. Using the compass and ruler, construct triangle BCD on the plane, ensuring that the sides correspond to the chord lengths of the arcs on the sphere.5. Construct the circumcircle of triangle BCD on the plane.6. The radius of this circumcircle is equal to the sphere's radius.I think this makes sense. The chord lengths correspond to the sides of the triangle on the plane, and the circumradius of that triangle gives the sphere's radius.So, to recap, the process involves:- Selecting a point on the sphere.- Drawing a circle around that point.- Choosing three points on that circle.- Transferring those points to the plane to form a triangle.- Constructing the circumcircle of that triangle.- The radius of this circumcircle is the desired radius of the sphere.I think this should work. It might be a bit involved, but it uses basic geometric constructions with a compass and ruler.

answer：First, determine the total number of members in the swim club, which is 50.Calculate the number of members who have passed the lifesaving test:30% of 50 is 0.30 * 50 = 15 members.Subtract the number of members who passed the test from the total to find those who did not pass:50 - 15 = 35 members.Among the 35 members who did not pass, 30 have not taken the preparatory course. Therefore, the number of members who have taken the preparatory course but not passed the test is:35 - 30 = 5 members.

answer：Okay, so I need to find the value of tan 1600 degrees. Hmm, that's a pretty big angle. I remember that trigonometric functions are periodic, which means their values repeat after a certain interval. For the tangent function, the period is 360 degrees. So, I think I can subtract multiples of 360 degrees from 1600 degrees to find an equivalent angle between 0 and 360 degrees.Let me calculate how many times 360 goes into 1600. If I divide 1600 by 360, that's approximately 4.444. So, that means 360 times 4 is 1440 degrees. If I subtract 1440 from 1600, I get 160 degrees. So, tan 1600 degrees is the same as tan 160 degrees.Now, 160 degrees is still not an acute angle, but I know that tangent has some properties related to reference angles. Specifically, tan(180 - θ) is equal to -tan θ. So, tan 160 degrees is the same as tan(180 - 20) degrees, which is -tan 20 degrees.Alright, so now I need to find tan 20 degrees. The problem provides a table of sine values for some acute angles. I see that sin 20 degrees is 0.3420, and sin 70 degrees is 0.9397. I remember that tan θ is equal to sin θ over cos θ. But I don't have the cosine values in the table. Wait, but I also know that cos θ is equal to sin(90 - θ). So, cos 20 degrees is sin(70 degrees), which is 0.9397.So, tan 20 degrees is sin 20 degrees divided by cos 20 degrees, which is 0.3420 divided by 0.9397. Let me do that division. 0.3420 divided by 0.9397 is approximately 0.364. So, tan 20 degrees is about 0.364.But remember, tan 160 degrees is -tan 20 degrees, so that would be approximately -0.364. Rounding that to two decimal places, it's -0.36.Looking at the options, B is -0.36, so that should be the answer.