💡Alright, so I'm trying to solve these two sequence problems. Let me take them one at a time.**Problem 1:**We have a sequence defined by:[ a_1 = 1 ][ a_{n+1} = frac{10 + 4a_n}{1 + a_n} ]I need to find the general term formula for ( a_n ). Hmm, okay. This seems like a recursive sequence where each term is defined based on the previous one. The formula for ( a_{n+1} ) looks like a linear transformation of ( a_n ). Maybe I can find a pattern or express it in a closed form.First, let me compute the first few terms to see if I can spot a pattern.- ( a_1 = 1 )- ( a_2 = frac{10 + 4(1)}{1 + 1} = frac{14}{2} = 7 )- ( a_3 = frac{10 + 4(7)}{1 + 7} = frac{10 + 28}{8} = frac{38}{8} = 4.75 )- ( a_4 = frac{10 + 4(4.75)}{1 + 4.75} = frac{10 + 19}{5.75} = frac{29}{5.75} approx 5 )- ( a_5 = frac{10 + 4(5)}{1 + 5} = frac{10 + 20}{6} = frac{30}{6} = 5 )Wait, so starting from ( a_4 ), it seems like the sequence is approaching 5. Maybe 5 is a fixed point? Let me check.If ( a_{n+1} = a_n = x ), then:[ x = frac{10 + 4x}{1 + x} ]Multiply both sides by ( 1 + x ):[ x(1 + x) = 10 + 4x ][ x + x^2 = 10 + 4x ][ x^2 - 3x - 10 = 0 ]Solving this quadratic equation:[ x = frac{3 pm sqrt{9 + 40}}{2} = frac{3 pm 7}{2} ]So, ( x = 5 ) or ( x = -2 ). Interesting, so 5 and -2 are fixed points.But our sequence starts at 1 and seems to approach 5. Maybe the general term can be expressed in terms of these fixed points. I remember sometimes for such recursive sequences, especially linear fractional transformations, we can use substitution to make it linear.Let me try to manipulate the recursive formula. Let me set ( b_n = frac{a_n - 5}{a_n + 2} ). I'm choosing this because 5 and -2 are the fixed points, so maybe this substitution will simplify things.Compute ( b_{n+1} ):[ b_{n+1} = frac{a_{n+1} - 5}{a_{n+1} + 2} ]Substitute ( a_{n+1} = frac{10 + 4a_n}{1 + a_n} ):[ b_{n+1} = frac{frac{10 + 4a_n}{1 + a_n} - 5}{frac{10 + 4a_n}{1 + a_n} + 2} ]Simplify numerator and denominator:Numerator:[ frac{10 + 4a_n - 5(1 + a_n)}{1 + a_n} = frac{10 + 4a_n - 5 - 5a_n}{1 + a_n} = frac{5 - a_n}{1 + a_n} ]Denominator:[ frac{10 + 4a_n + 2(1 + a_n)}{1 + a_n} = frac{10 + 4a_n + 2 + 2a_n}{1 + a_n} = frac{12 + 6a_n}{1 + a_n} ]So, ( b_{n+1} = frac{5 - a_n}{12 + 6a_n} times frac{1 + a_n}{1 + a_n} ) Wait, actually, since both numerator and denominator have the same denominator ( 1 + a_n ), they cancel out:[ b_{n+1} = frac{5 - a_n}{12 + 6a_n} ]Factor numerator and denominator:Numerator: ( 5 - a_n = -(a_n - 5) )Denominator: ( 12 + 6a_n = 6(a_n + 2) )So,[ b_{n+1} = frac{-(a_n - 5)}{6(a_n + 2)} = -frac{1}{6} cdot frac{a_n - 5}{a_n + 2} = -frac{1}{6} b_n ]Ah! So, ( b_{n+1} = -frac{1}{6} b_n ). That's a simple linear recurrence relation. So, ( b_n ) is a geometric sequence with common ratio ( -frac{1}{6} ).Now, let's find ( b_1 ):[ b_1 = frac{a_1 - 5}{a_1 + 2} = frac{1 - 5}{1 + 2} = frac{-4}{3} ]Therefore, the general term for ( b_n ) is:[ b_n = b_1 left( -frac{1}{6} right)^{n - 1} = -frac{4}{3} left( -frac{1}{6} right)^{n - 1} ]Now, we can express ( a_n ) in terms of ( b_n ). From the substitution:[ b_n = frac{a_n - 5}{a_n + 2} ]Let me solve for ( a_n ):[ b_n (a_n + 2) = a_n - 5 ][ b_n a_n + 2b_n = a_n - 5 ][ a_n (b_n - 1) = -5 - 2b_n ][ a_n = frac{-5 - 2b_n}{b_n - 1} ]Simplify numerator and denominator:Factor out -1 in numerator:[ a_n = frac{- (5 + 2b_n)}{b_n - 1} = frac{5 + 2b_n}{1 - b_n} ]Now, substitute ( b_n = -frac{4}{3} left( -frac{1}{6} right)^{n - 1} ):[ a_n = frac{5 + 2 left( -frac{4}{3} left( -frac{1}{6} right)^{n - 1} right)}{1 - left( -frac{4}{3} left( -frac{1}{6} right)^{n - 1} right)} ]Simplify numerator and denominator:Numerator:[ 5 - frac{8}{3} left( -frac{1}{6} right)^{n - 1} ]Denominator:[ 1 + frac{4}{3} left( -frac{1}{6} right)^{n - 1} ]To make it cleaner, let me factor out the constants:Numerator:[ 5 - frac{8}{3} left( -frac{1}{6} right)^{n - 1} = 5 + frac{8}{3} left( frac{1}{6} right)^{n - 1} (-1)^{n} ]Wait, actually, ( left( -frac{1}{6} right)^{n - 1} = (-1)^{n - 1} left( frac{1}{6} right)^{n - 1} ). So, the numerator becomes:[ 5 + frac{8}{3} (-1)^n left( frac{1}{6} right)^{n - 1} ]Similarly, the denominator:[ 1 + frac{4}{3} left( -frac{1}{6} right)^{n - 1} = 1 + frac{4}{3} (-1)^{n - 1} left( frac{1}{6} right)^{n - 1} ]Hmm, this is getting a bit messy. Maybe there's a better way to express it.Alternatively, let's write ( b_n ) as:[ b_n = -frac{4}{3} left( -frac{1}{6} right)^{n - 1} = frac{4}{3} left( frac{1}{6} right)^{n - 1} (-1)^n ]So, substituting back:[ a_n = frac{5 + 2b_n}{1 - b_n} = frac{5 + 2 cdot frac{4}{3} left( frac{1}{6} right)^{n - 1} (-1)^n}{1 - frac{4}{3} left( frac{1}{6} right)^{n - 1} (-1)^n} ]Simplify:[ a_n = frac{5 + frac{8}{3} left( frac{1}{6} right)^{n - 1} (-1)^n}{1 - frac{4}{3} left( frac{1}{6} right)^{n - 1} (-1)^n} ]To make it look nicer, multiply numerator and denominator by 3 to eliminate fractions:[ a_n = frac{15 + 8 left( frac{1}{6} right)^{n - 1} (-1)^n}{3 - 4 left( frac{1}{6} right)^{n - 1} (-1)^n} ]We can write ( left( frac{1}{6} right)^{n - 1} = 6^{-(n - 1)} = 6^{1 - n} ). So:[ a_n = frac{15 + 8 cdot 6^{1 - n} (-1)^n}{3 - 4 cdot 6^{1 - n} (-1)^n} ]Alternatively, factor out constants:[ a_n = frac{15 + (-1)^n cdot 8 cdot 6^{1 - n}}{3 + (-1)^{n + 1} cdot 4 cdot 6^{1 - n}} ]This seems as simplified as it can get. Alternatively, we can write it as:[ a_n = 5 - frac{7}{1 + frac{3}{4} (-6)^{n - 1}} ]Wait, how did they get that? Let me check.Starting from:[ a_n = frac{15 + 8 cdot 6^{1 - n} (-1)^n}{3 - 4 cdot 6^{1 - n} (-1)^n} ]Let me factor out 3 from numerator and denominator:Numerator: ( 15 + 8 cdot 6^{1 - n} (-1)^n = 3 cdot 5 + 8 cdot 6^{1 - n} (-1)^n )Denominator: ( 3 - 4 cdot 6^{1 - n} (-1)^n )So, factor out 3:[ a_n = frac{3 left( 5 + frac{8}{3} cdot 6^{1 - n} (-1)^n right)}{3 left( 1 - frac{4}{3} cdot 6^{1 - n} (-1)^n right)} = frac{5 + frac{8}{3} cdot 6^{1 - n} (-1)^n}{1 - frac{4}{3} cdot 6^{1 - n} (-1)^n} ]Now, notice that ( 6^{1 - n} = frac{6}{6^n} = frac{6}{6^n} ), but I'm not sure if that helps. Alternatively, let's express ( 6^{1 - n} = 6 cdot 6^{-n} = 6 cdot (1/6)^n ).Wait, another approach: Let me write ( 6^{1 - n} (-1)^n = (-1)^n 6^{1 - n} = (-1)^n 6 cdot 6^{-n} = 6 (-1/6)^n ).So, substituting back:[ a_n = frac{5 + frac{8}{3} cdot 6 (-1/6)^n}{1 - frac{4}{3} cdot 6 (-1/6)^n} = frac{5 + 16 (-1/6)^n}{1 - 8 (-1/6)^n} ]Hmm, not sure if that helps. Maybe another substitution.Alternatively, let's consider the expression:[ a_n = frac{15 + 8 cdot 6^{1 - n} (-1)^n}{3 - 4 cdot 6^{1 - n} (-1)^n} ]Let me factor out 3 from numerator and denominator:[ a_n = frac{3 left(5 + frac{8}{3} cdot 6^{1 - n} (-1)^n right)}{3 left(1 - frac{4}{3} cdot 6^{1 - n} (-1)^n right)} = frac{5 + frac{8}{3} cdot 6^{1 - n} (-1)^n}{1 - frac{4}{3} cdot 6^{1 - n} (-1)^n} ]Now, let me write ( frac{8}{3} = frac{8}{3} ) and ( frac{4}{3} = frac{4}{3} ). Maybe factor out something else.Alternatively, let me set ( c_n = 6^{n - 1} ). Then ( 6^{1 - n} = 1/c_n ). So:[ a_n = frac{15 + 8 (-1)^n / c_n}{3 - 4 (-1)^n / c_n} ]Multiply numerator and denominator by ( c_n ):[ a_n = frac{15 c_n + 8 (-1)^n}{3 c_n - 4 (-1)^n} ]But ( c_n = 6^{n - 1} ), so:[ a_n = frac{15 cdot 6^{n - 1} + 8 (-1)^n}{3 cdot 6^{n - 1} - 4 (-1)^n} ]Factor out 3 from numerator and denominator:Numerator: ( 15 cdot 6^{n - 1} = 3 cdot 5 cdot 6^{n - 1} )Denominator: ( 3 cdot 6^{n - 1} )So,[ a_n = frac{3 cdot 5 cdot 6^{n - 1} + 8 (-1)^n}{3 cdot 6^{n - 1} - 4 (-1)^n} = frac{5 cdot 6^{n - 1} + frac{8}{3} (-1)^n}{6^{n - 1} - frac{4}{3} (-1)^n} ]Hmm, still not matching the suggested formula. Maybe another approach.Let me go back to the substitution ( b_n = frac{a_n - 5}{a_n + 2} ). We found that ( b_n = -frac{4}{3} left( -frac{1}{6} right)^{n - 1} ). So, ( b_n = frac{4}{3} left( frac{1}{6} right)^{n - 1} (-1)^n ).So, ( b_n = frac{4}{3} (-1)^n 6^{-(n - 1)} = frac{4}{3} (-1)^n 6^{1 - n} ).Now, from ( b_n = frac{a_n - 5}{a_n + 2} ), we have:[ a_n = frac{5 + 2b_n}{1 - b_n} ]Substitute ( b_n ):[ a_n = frac{5 + 2 cdot frac{4}{3} (-1)^n 6^{1 - n}}{1 - frac{4}{3} (-1)^n 6^{1 - n}} ]Simplify:[ a_n = frac{5 + frac{8}{3} (-1)^n 6^{1 - n}}{1 - frac{4}{3} (-1)^n 6^{1 - n}} ]Multiply numerator and denominator by 3:[ a_n = frac{15 + 8 (-1)^n 6^{1 - n}}{3 - 4 (-1)^n 6^{1 - n}} ]Now, notice that ( 6^{1 - n} = frac{6}{6^n} = frac{6}{6^n} ). So:[ a_n = frac{15 + 8 (-1)^n frac{6}{6^n}}{3 - 4 (-1)^n frac{6}{6^n}} = frac{15 + frac{48 (-1)^n}{6^n}}{3 - frac{24 (-1)^n}{6^n}} ]Factor out 3 from numerator and denominator:[ a_n = frac{3 cdot 5 + frac{48 (-1)^n}{6^n}}{3 - frac{24 (-1)^n}{6^n}} = frac{3 left(5 + frac{16 (-1)^n}{6^n}right)}{3 left(1 - frac{8 (-1)^n}{6^n}right)} = frac{5 + frac{16 (-1)^n}{6^n}}{1 - frac{8 (-1)^n}{6^n}} ]Hmm, still not matching. Maybe another way.Wait, let's consider the expression:[ a_n = 5 - frac{7}{1 + frac{3}{4} (-6)^{n - 1}} ]Let me check if this matches the terms I computed earlier.For ( n = 1 ):[ a_1 = 5 - frac{7}{1 + frac{3}{4} (-6)^{0}} = 5 - frac{7}{1 + frac{3}{4}} = 5 - frac{7}{frac{7}{4}} = 5 - 4 = 1 ] Correct.For ( n = 2 ):[ a_2 = 5 - frac{7}{1 + frac{3}{4} (-6)^{1}} = 5 - frac{7}{1 - frac{18}{4}} = 5 - frac{7}{1 - 4.5} = 5 - frac{7}{-3.5} = 5 + 2 = 7 ] Correct.For ( n = 3 ):[ a_3 = 5 - frac{7}{1 + frac{3}{4} (-6)^{2}} = 5 - frac{7}{1 + frac{3}{4} cdot 36} = 5 - frac{7}{1 + 27} = 5 - frac{7}{28} = 5 - 0.25 = 4.75 ] Correct.For ( n = 4 ):[ a_4 = 5 - frac{7}{1 + frac{3}{4} (-6)^{3}} = 5 - frac{7}{1 + frac{3}{4} (-216)} = 5 - frac{7}{1 - 162} = 5 - frac{7}{-161} = 5 + frac{7}{161} approx 5.043 ] Wait, but earlier I got ( a_4 approx 5 ). Hmm, discrepancy here.Wait, actually, when I computed ( a_4 ), I got approximately 5, but the exact value is ( frac{29}{5.75} = frac{29 times 4}{23} = frac{116}{23} approx 5.043 ). So, it's correct. So, the formula seems to hold.Therefore, the general term is:[ a_n = 5 - frac{7}{1 + frac{3}{4} (-6)^{n - 1}} ]**Problem 2:**Given the sequence:[ a_1 = 2 ][ a_{n+1} = frac{a_n - 1}{a_n + 1} ]Find ( a_{2019} ).Okay, so again, a recursive sequence. Let me compute the first few terms to see if there's a pattern.- ( a_1 = 2 )- ( a_2 = frac{2 - 1}{2 + 1} = frac{1}{3} )- ( a_3 = frac{frac{1}{3} - 1}{frac{1}{3} + 1} = frac{-frac{2}{3}}{frac{4}{3}} = -frac{1}{2} )- ( a_4 = frac{-frac{1}{2} - 1}{-frac{1}{2} + 1} = frac{-frac{3}{2}}{frac{1}{2}} = -3 )- ( a_5 = frac{-3 - 1}{-3 + 1} = frac{-4}{-2} = 2 )- ( a_6 = frac{2 - 1}{2 + 1} = frac{1}{3} )Oh, so it's periodic with period 4: 2, 1/3, -1/2, -3, 2, 1/3, -1/2, -3, ...Therefore, the sequence repeats every 4 terms. So, to find ( a_{2019} ), we can find the remainder when 2019 is divided by 4.Compute ( 2019 div 4 ):4 × 504 = 2016, remainder 3.So, ( 2019 = 4 × 504 + 3 ). Therefore, ( a_{2019} = a_3 = -frac{1}{2} ).**Final Answer**(1) The general term is (boxed{5 - dfrac{7}{1 + dfrac{3}{4}(-6)^{n-1}}}).(2) The value of (a_{2019}) is (boxed{-dfrac{1}{2}}).

💡Okay, so I have this problem here: If Diamond5 base 9 equals Diamond2 base 10, and Diamond represents a digit, I need to solve for Diamond. Hmm, let me try to figure this out step by step.First, I know that when a number is written in a certain base, each digit represents a power of that base. So, for example, in base 9, the number Diamond5 would have Diamond in the nines place and 5 in the ones place. Similarly, in base 10, Diamond2 would have Diamond in the tens place and 2 in the ones place.Let me write this out mathematically. If I convert both numbers to base 10, I can set up an equation and solve for Diamond. So, for Diamond5 in base 9, that should be equal to Diamond times 9 to the power of 1 plus 5 times 9 to the power of 0. Similarly, Diamond2 in base 10 would be Diamond times 10 to the power of 1 plus 2 times 10 to the power of 0.So, writing that out, I get:Diamond5_9 = Diamond * 9 + 5Diamond2_10 = Diamond * 10 + 2Since these two expressions are equal, I can set them equal to each other:Diamond * 9 + 5 = Diamond * 10 + 2Now, I need to solve for Diamond. Let me rearrange the equation to get all the Diamond terms on one side and the constants on the other. If I subtract Diamond * 9 from both sides, I get:5 = Diamond * 10 - Diamond * 9 + 2Simplifying the right side, Diamond * 10 - Diamond * 9 is just Diamond * (10 - 9) which is Diamond * 1, so:5 = Diamond + 2Now, subtracting 2 from both sides gives:Diamond = 5 - 2Diamond = 3Wait, let me check that again to make sure I didn't make a mistake. Starting from the equation:Diamond * 9 + 5 = Diamond * 10 + 2Subtract Diamond * 9 from both sides:5 = Diamond + 2Subtract 2:Diamond = 3Yes, that seems correct. Let me verify by plugging Diamond = 3 back into the original numbers.In base 9, Diamond5 would be 35_9. Converting that to base 10: 3*9 + 5 = 27 + 5 = 32.In base 10, Diamond2 would be 32_10, which is just 32. So both sides equal 32 in base 10, which confirms that Diamond = 3 is the correct solution.I think that makes sense. I didn't encounter any contradictions or errors in my steps, so I feel confident that 3 is the right answer.

💡First, I need to determine the total duration of the movie, which is 3 hours. Converting this to minutes gives me 180 minutes.Next, I'll calculate how long Hannah watched the movie before her laptop turned off. She started watching at 3:20 pm and the laptop turned off at 5:44 pm. From 3:20 pm to 4:20 pm is 1 hour, and from 4:20 pm to 5:20 pm is another hour. This accounts for 2 hours. From 5:20 pm to 5:44 pm is an additional 24 minutes. Adding these together, Hannah watched the movie for a total of 2 hours and 24 minutes, which is 144 minutes.Finally, to find out how much of the movie is remaining, I'll subtract the time Hannah has already watched from the total movie duration: 180 minutes minus 144 minutes equals 36 minutes.

💡Okay, so I have this problem about a right triangular prism, ABC-A₁B₁C₁, where angle BAC is 90 degrees. The area of the lateral face BCC₁B₁ is 16, and I need to find the minimum radius of the circumscribed sphere around this prism. Hmm, okay, let me try to break this down step by step.First, let me visualize the prism. It's a right triangular prism, so the bases are right triangles, and the sides are rectangles. Since angle BAC is 90 degrees, triangle ABC is a right triangle with the right angle at A. So, vertices A, B, and C form a right triangle, and then the prism extends to A₁, B₁, and C₁ respectively.Now, the lateral face BCC₁B₁ is a rectangle. The area of this face is given as 16. Let me recall that the area of a rectangle is length times width. In this case, the length would be BC, and the width would be BB₁ (which is the same as AA₁ and CC₁ since it's a prism). So, if I denote BC as some length and BB₁ as another, their product is 16.Let me denote BC as 2x and BB₁ as 2y. So, the area is (2x)(2y) = 4xy = 16. Therefore, xy = 4. That seems like a useful equation.Now, I need to find the radius of the circumscribed sphere around the prism. A circumscribed sphere (circum sphere) around a prism would pass through all its vertices. For a right prism, especially a right triangular prism, the center of the circumscribed sphere should be at the midpoint of the segment connecting the centroids of the two triangular bases. Wait, is that correct?Let me think. In a right prism, the two bases are congruent and parallel, and the lateral edges are perpendicular to the bases. So, the centroid of each triangular base is the point where the medians intersect. For a right triangle, the centroid is located at a distance of one-third the length of each median from each side.But if I consider the circumscribed sphere, it should pass through all eight vertices of the prism. So, the center of the sphere must be equidistant from all these vertices. Hmm, so maybe it's not just the midpoint between the centroids, but perhaps another point?Wait, actually, in a right prism, especially a right triangular prism, the circumscribed sphere's center is indeed the midpoint between the centroids of the two bases. Because the prism is symmetric with respect to this midpoint. So, if I can find the distance from this midpoint to any vertex, that should be the radius.Let me denote the centroid of triangle ABC as G and the centroid of triangle A₁B₁C₁ as G₁. Then, the center of the circumscribed sphere, let's call it O, is the midpoint of GG₁. So, O is located halfway between G and G₁.Now, to find the radius, I need to compute the distance from O to any vertex, say A. Let me figure out the coordinates of these points to make it easier.Let me set up a coordinate system. Let me place point A at the origin (0, 0, 0). Since angle BAC is 90 degrees, I can place point B along the x-axis and point C along the y-axis. Let me denote AB as length a and AC as length b. Then, the coordinates are:- A: (0, 0, 0)- B: (a, 0, 0)- C: (0, b, 0)- A₁: (0, 0, h)- B₁: (a, 0, h)- C₁: (0, b, h)Where h is the height of the prism, which is the length of BB₁, AA₁, etc.Now, the centroid G of triangle ABC is the average of the coordinates of A, B, and C. So,G = [(0 + a + 0)/3, (0 + 0 + b)/3, (0 + 0 + 0)/3] = (a/3, b/3, 0)Similarly, the centroid G₁ of triangle A₁B₁C₁ is:G₁ = [(0 + a + 0)/3, (0 + 0 + b)/3, (h + h + h)/3] = (a/3, b/3, h)So, the midpoint O between G and G₁ is:O = [(a/3 + a/3)/2, (b/3 + b/3)/2, (0 + h)/2] = (a/3, b/3, h/2)Now, to find the radius R of the circumscribed sphere, I can compute the distance from O to any vertex, say A.Point A is at (0, 0, 0), so the distance OA is:OA = sqrt[(a/3 - 0)^2 + (b/3 - 0)^2 + (h/2 - 0)^2] = sqrt[(a²/9) + (b²/9) + (h²/4)]Similarly, the distance from O to B is:OB = sqrt[(a/3 - a)^2 + (b/3 - 0)^2 + (h/2 - 0)^2] = sqrt[( - 2a/3)^2 + (b/3)^2 + (h/2)^2] = sqrt[(4a²/9) + (b²/9) + (h²/4)]Wait, but since O is the center of the sphere, OA should equal OB, right? Let me check if that's the case.OA² = (a²/9) + (b²/9) + (h²/4)OB² = (4a²/9) + (b²/9) + (h²/4)These are not equal unless a = 0, which isn't the case. Hmm, that seems contradictory. Maybe my assumption about the center being the midpoint of GG₁ is incorrect?Wait, perhaps I made a mistake in choosing the center. Let me think again. In a right prism, especially a right triangular prism, the circumscribed sphere's center is actually the midpoint of the segment connecting the midpoints of the two bases' hypotenuses. Hmm, maybe that's different.Wait, in a right triangle, the circumradius is half the hypotenuse. So, in triangle ABC, the circumradius is (BC)/2. Similarly, in triangle A₁B₁C₁, the circumradius is (B₁C₁)/2.But since it's a prism, the sphere needs to encompass both triangles and the lateral faces. Maybe the center is somewhere along the line connecting the circumcenters of the two triangular bases.Wait, in triangle ABC, the circumcenter is at the midpoint of BC because it's a right triangle. Similarly, in triangle A₁B₁C₁, the circumcenter is at the midpoint of B₁C₁.So, the line connecting these two midpoints is the line along which the center of the circumscribed sphere lies. So, the center O is somewhere along this line.Let me denote the midpoint of BC as M and the midpoint of B₁C₁ as M₁. Then, the line MM₁ is the line along which the center O lies.So, coordinates of M: midpoint of BC is [(a + 0)/2, (0 + b)/2, 0] = (a/2, b/2, 0)Coordinates of M₁: midpoint of B₁C₁ is [(a + 0)/2, (0 + b)/2, h] = (a/2, b/2, h)So, the line MM₁ goes from (a/2, b/2, 0) to (a/2, b/2, h). So, any point on this line is (a/2, b/2, t), where t ranges from 0 to h.So, the center O is somewhere on this line, so O = (a/2, b/2, t) for some t between 0 and h.Now, the radius R is the distance from O to any vertex, say A.Point A is (0, 0, 0), so OA² = (a/2 - 0)^2 + (b/2 - 0)^2 + (t - 0)^2 = (a²/4) + (b²/4) + t²Similarly, the distance from O to B is:OB² = (a/2 - a)^2 + (b/2 - 0)^2 + (t - 0)^2 = ( - a/2)^2 + (b/2)^2 + t² = (a²/4) + (b²/4) + t²Same as OA², which is good.Similarly, distance from O to C is:OC² = (a/2 - 0)^2 + (b/2 - b)^2 + t² = (a²/4) + ( - b/2)^2 + t² = (a²/4) + (b²/4) + t²Same as OA².Now, distance from O to A₁ is:OA₁² = (a/2 - 0)^2 + (b/2 - 0)^2 + (t - h)^2 = (a²/4) + (b²/4) + (t - h)^2Similarly, distance from O to B₁ is:OB₁² = (a/2 - a)^2 + (b/2 - 0)^2 + (t - h)^2 = ( - a/2)^2 + (b/2)^2 + (t - h)^2 = (a²/4) + (b²/4) + (t - h)^2Same as OA₁².Similarly, distance from O to C₁ is:OC₁² = (a/2 - 0)^2 + (b/2 - b)^2 + (t - h)^2 = (a²/4) + ( - b/2)^2 + (t - h)^2 = (a²/4) + (b²/4) + (t - h)^2Same as OA₁².So, for the sphere to be circumscribed, all these distances must be equal. So, OA² = OA₁².So, (a²/4 + b²/4 + t²) = (a²/4 + b²/4 + (t - h)^2)Simplify:t² = (t - h)^2Expanding the right side:t² = t² - 2th + h²Subtract t² from both sides:0 = -2th + h²So, 2th = h²Assuming h ≠ 0, we can divide both sides by h:2t = h => t = h/2So, the center O is at (a/2, b/2, h/2). So, the midpoint of MM₁ is indeed the center of the circumscribed sphere.Therefore, the radius R is the distance from O to A, which is sqrt[(a²/4) + (b²/4) + (h²/4)] = (1/2)sqrt(a² + b² + h²)Alternatively, since OA² = (a² + b² + h²)/4, so R = (sqrt(a² + b² + h²))/2So, R = (sqrt(a² + b² + h²))/2Now, I need to express this in terms of the given information. The area of the lateral face BCC₁B₁ is 16.As I mentioned earlier, the area of this face is BC * BB₁. BC is the length of side BC in triangle ABC, which is a right triangle with legs AB = a and AC = b. So, BC = sqrt(a² + b²). And BB₁ is the height of the prism, which is h.So, area of BCC₁B₁ is BC * BB₁ = sqrt(a² + b²) * h = 16So, sqrt(a² + b²) * h = 16Let me denote sqrt(a² + b²) as c, the length of the hypotenuse of triangle ABC. So, c * h = 16So, c = 16 / hBut c is sqrt(a² + b²), so sqrt(a² + b²) = 16 / hTherefore, a² + b² = (256) / (h²)So, going back to R:R = (sqrt(a² + b² + h²))/2 = (sqrt(256/h² + h²))/2So, R = (sqrt(256/h² + h²))/2Now, I need to find the minimum value of R. So, I need to minimize R with respect to h.Let me denote f(h) = sqrt(256/h² + h²)/2To find the minimum of f(h), I can first find the minimum of the expression inside the square root, which is g(h) = 256/h² + h²So, let's minimize g(h) = 256/h² + h²To find the minimum, take the derivative of g with respect to h and set it to zero.g'(h) = d/dh [256/h² + h²] = -512/h³ + 2hSet g'(h) = 0:-512/h³ + 2h = 0Multiply both sides by h³ to eliminate the denominator:-512 + 2h⁴ = 0So, 2h⁴ = 512Divide both sides by 2:h⁴ = 256Take the fourth root:h = sqrt(sqrt(256)) = sqrt(16) = 4Since h is a length, it must be positive, so h = 4So, the minimum occurs when h = 4Now, let's compute g(4):g(4) = 256/(4²) + (4)² = 256/16 + 16 = 16 + 16 = 32So, the minimum value of g(h) is 32, which occurs at h = 4Therefore, the minimum value of f(h) is sqrt(32)/2 = (4 * sqrt(2))/2 = 2 * sqrt(2)So, the minimum radius R is 2√2Therefore, the minimum radius of the circumscribed sphere is 2√2**Final Answer**The minimum radius of the circumscribed sphere is boxed{2sqrt{2}}.

💡First, I need to determine the probability that exactly one of the two pilots hits the target. This means either Pilot A hits and Pilot B does not, or Pilot B hits and Pilot A does not.I'll start by calculating the probability that Pilot A hits the target while Pilot B does not. Pilot A's probability of hitting is 0.4, and Pilot B's probability of missing is 1 - 0.5 = 0.5. Multiplying these probabilities gives 0.4 * 0.5 = 0.2.Next, I'll calculate the probability that Pilot B hits the target while Pilot A does not. Pilot B's probability of hitting is 0.5, and Pilot A's probability of missing is 1 - 0.4 = 0.6. Multiplying these probabilities gives 0.5 * 0.6 = 0.3.Finally, I'll add these two probabilities together to find the total probability that exactly one pilot hits the target: 0.2 + 0.3 = 0.5.

💡First, I need to determine the sum of Henry's and Jill's present ages, which are given as 27 and 16 years respectively.To find the sum, I will add Henry's age to Jill's age.Calculating the sum: 27 + 16 = 43.Therefore, the sum of their present ages is 43 years.