💡Alright, so I have this problem here where I need to find all possible values of ( a ) given some conditions about the sum of distances from twenty consecutive natural numbers to ( a ) and to ( a^2 ). Let me try to break this down step by step.First, let me make sure I understand the problem correctly. We have twenty consecutive natural numbers, say starting from some number ( k ), so the numbers are ( k, k+1, k+2, ldots, k+19 ). The distance between two numbers is defined as the absolute value of their difference. So, the sum of the distances from these twenty numbers to some number ( a ) is 4460, and the sum of the distances from these same numbers to ( a^2 ) is 2755. I need to find all possible values of ( a ).Hmm, okay. So, I need to find ( a ) such that:1. ( sum_{i=0}^{19} |a - (k + i)| = 4460 )2. ( sum_{i=0}^{19} |a^2 - (k + i)| = 2755 )I think the key here is to figure out where ( a ) and ( a^2 ) lie relative to the sequence of numbers ( k, k+1, ldots, k+19 ). Because the sum of absolute differences depends on whether the point is inside or outside the range of the numbers.Let me recall that for a set of consecutive numbers, the sum of distances to a point outside the range can be calculated using a formula. If the point is outside, say to the right of the largest number, the sum is ( 20 times text{distance} ) minus the sum of the numbers. Similarly, if it's to the left, it's ( 20 times text{distance} ) plus the sum of the numbers. Wait, actually, let me think more carefully.Suppose the numbers are ( x_1, x_2, ldots, x_{20} ) in increasing order. If ( a ) is less than ( x_1 ), then the sum of distances is ( sum_{i=1}^{20} (x_i - a) = sum x_i - 20a ). Similarly, if ( a ) is greater than ( x_{20} ), the sum is ( sum_{i=1}^{20} (a - x_i) = 20a - sum x_i ). If ( a ) is somewhere inside the range, the sum is a bit more complicated, but since the numbers are consecutive, maybe we can find a median or something.Wait, but the problem doesn't specify whether ( a ) is inside or outside the range. So, maybe I should consider both cases.But before that, let me compute the sum of the numbers ( k, k+1, ldots, k+19 ). That's an arithmetic series with first term ( k ), last term ( k+19 ), and 20 terms. The sum is ( frac{20}{2} times (k + (k + 19)) = 10 times (2k + 19) = 20k + 190 ).So, the sum of the numbers is ( 20k + 190 ).Now, if ( a ) is outside the range, say ( a > k + 19 ), then the sum of distances is ( 20a - (20k + 190) ). Similarly, if ( a < k ), the sum is ( (20k + 190) - 20a ).Similarly, for ( a^2 ), the sum of distances would be ( 20a^2 - (20k + 190) ) if ( a^2 > k + 19 ), or ( (20k + 190) - 20a^2 ) if ( a^2 < k ).But since the sums are given as 4460 and 2755, which are positive numbers, we can set up equations.Let me denote ( S = 20k + 190 ). Then, depending on where ( a ) is, we have:If ( a > k + 19 ), then ( 20a - S = 4460 ).If ( a < k ), then ( S - 20a = 4460 ).Similarly, for ( a^2 ):If ( a^2 > k + 19 ), then ( 20a^2 - S = 2755 ).If ( a^2 < k ), then ( S - 20a^2 = 2755 ).So, we have four possible cases:1. ( a > k + 19 ) and ( a^2 > k + 19 )2. ( a > k + 19 ) and ( a^2 < k )3. ( a < k ) and ( a^2 > k + 19 )4. ( a < k ) and ( a^2 < k )But let's think about these cases.Case 1: ( a > k + 19 ) and ( a^2 > k + 19 ). Then, both sums are ( 20a - S = 4460 ) and ( 20a^2 - S = 2755 ).Case 2: ( a > k + 19 ) and ( a^2 < k ). Then, sums are ( 20a - S = 4460 ) and ( S - 20a^2 = 2755 ).Case 3: ( a < k ) and ( a^2 > k + 19 ). Then, sums are ( S - 20a = 4460 ) and ( 20a^2 - S = 2755 ).Case 4: ( a < k ) and ( a^2 < k ). Then, sums are ( S - 20a = 4460 ) and ( S - 20a^2 = 2755 ).Now, let's analyze these cases.First, note that ( a^2 ) is either greater than ( k + 19 ) or less than ( k ). But if ( a ) is a natural number, ( a^2 ) is also a natural number, and if ( a ) is greater than ( k + 19 ), ( a^2 ) is likely to be much larger, so probably greater than ( k + 19 ). Similarly, if ( a ) is less than ( k ), ( a^2 ) could be either, depending on ( a ).But let's see.Let me first consider Case 1: ( a > k + 19 ) and ( a^2 > k + 19 ).Then, we have:( 20a - S = 4460 ) --> ( 20a = S + 4460 )( 20a^2 - S = 2755 ) --> ( 20a^2 = S + 2755 )So, subtracting the first equation from the second:( 20a^2 - 20a = 2755 - 4460 )( 20(a^2 - a) = -1705 )( a^2 - a = -1705 / 20 = -85.25 )So, ( a^2 - a + 85.25 = 0 )This is a quadratic equation. Let's compute the discriminant:( D = 1 - 4 * 1 * 85.25 = 1 - 341 = -340 )Negative discriminant, so no real solutions. Thus, Case 1 is impossible.Case 2: ( a > k + 19 ) and ( a^2 < k )So, we have:( 20a - S = 4460 ) --> ( 20a = S + 4460 )( S - 20a^2 = 2755 ) --> ( S = 20a^2 + 2755 )Substituting S from the second equation into the first:( 20a = (20a^2 + 2755) + 4460 )( 20a = 20a^2 + 7215 )( 20a^2 - 20a + 7215 = 0 )Divide both sides by 5:( 4a^2 - 4a + 1443 = 0 )Compute discriminant:( D = 16 - 4 * 4 * 1443 = 16 - 16 * 1443 = 16(1 - 1443) = 16*(-1442) = -23072 )Again, negative discriminant. So, no real solutions. Thus, Case 2 is impossible.Case 3: ( a < k ) and ( a^2 > k + 19 )So, we have:( S - 20a = 4460 ) --> ( S = 20a + 4460 )( 20a^2 - S = 2755 ) --> ( 20a^2 = S + 2755 )Substitute S from the first equation into the second:( 20a^2 = (20a + 4460) + 2755 )( 20a^2 = 20a + 7215 )( 20a^2 - 20a - 7215 = 0 )Divide by 5:( 4a^2 - 4a - 1443 = 0 )Compute discriminant:( D = 16 + 4 * 4 * 1443 = 16 + 16 * 1443 = 16(1 + 1443) = 16 * 1444 = 23104 )Square root of D is 152.So, solutions:( a = [4 ± 152] / (2 * 4) = (4 ± 152)/8 )So,( a = (4 + 152)/8 = 156/8 = 19.5 )( a = (4 - 152)/8 = (-148)/8 = -18.5 )Now, since ( a ) is a natural number? Wait, the problem says "some number ( a )", not necessarily a natural number. So, ( a ) could be a real number.But let's see. If ( a = 19.5 ), then ( a^2 = 380.25 ). Now, we need to check if ( a < k ) and ( a^2 > k + 19 ).From the first equation, ( S = 20a + 4460 ). Remember ( S = 20k + 190 ). So,( 20k + 190 = 20a + 4460 )Divide both sides by 20:( k + 9.5 = a + 223 )So,( k = a + 223 - 9.5 = a + 213.5 )So, ( k = a + 213.5 ). Since ( a = 19.5 ), then ( k = 19.5 + 213.5 = 233 ).So, ( k = 233 ). Now, check if ( a < k ): 19.5 < 233, which is true.Check if ( a^2 > k + 19 ): ( a^2 = 380.25 ), ( k + 19 = 233 + 19 = 252 ). So, 380.25 > 252, which is true.So, ( a = 19.5 ) is a possible solution.Similarly, check ( a = -18.5 ). Then, ( a^2 = 342.25 ).From ( S = 20a + 4460 ), ( S = 20*(-18.5) + 4460 = -370 + 4460 = 4090 ).But ( S = 20k + 190 = 4090 ), so ( 20k = 4090 - 190 = 3900 ), so ( k = 195 ).Now, check if ( a < k ): -18.5 < 195, which is true.Check if ( a^2 > k + 19 ): ( a^2 = 342.25 ), ( k + 19 = 195 + 19 = 214 ). So, 342.25 > 214, which is true.So, ( a = -18.5 ) is also a possible solution.Case 4: ( a < k ) and ( a^2 < k )So, we have:( S - 20a = 4460 ) --> ( S = 20a + 4460 )( S - 20a^2 = 2755 ) --> ( S = 20a^2 + 2755 )Set equal:( 20a + 4460 = 20a^2 + 2755 )( 20a^2 - 20a - 1705 = 0 )Divide by 5:( 4a^2 - 4a - 341 = 0 )Compute discriminant:( D = 16 + 4 * 4 * 341 = 16 + 16 * 341 = 16(1 + 341) = 16 * 342 = 5472 )Square root of 5472: Let's see, 74^2 = 5476, so sqrt(5472) ≈ 74 - 4/74 ≈ 73.945So, approximate solutions:( a = [4 ± 73.945]/8 )So,( a ≈ (4 + 73.945)/8 ≈ 77.945/8 ≈ 9.743 )( a ≈ (4 - 73.945)/8 ≈ (-69.945)/8 ≈ -8.743 )Now, check if these satisfy ( a < k ) and ( a^2 < k ).First, ( a ≈ 9.743 ). Then, ( a^2 ≈ 94.93 ).From ( S = 20a + 4460 ), ( S ≈ 20*9.743 + 4460 ≈ 194.86 + 4460 ≈ 4654.86 )But ( S = 20k + 190 ≈ 4654.86 ), so ( 20k ≈ 4654.86 - 190 ≈ 4464.86 ), so ( k ≈ 223.243 )Check if ( a < k ): 9.743 < 223.243, which is true.Check if ( a^2 < k ): 94.93 < 223.243, which is true.So, ( a ≈ 9.743 ) is a possible solution.Similarly, ( a ≈ -8.743 ). Then, ( a^2 ≈ 76.43 ).From ( S = 20a + 4460 ≈ 20*(-8.743) + 4460 ≈ -174.86 + 4460 ≈ 4285.14 )So, ( 20k + 190 ≈ 4285.14 ), so ( 20k ≈ 4285.14 - 190 ≈ 4095.14 ), so ( k ≈ 204.757 )Check if ( a < k ): -8.743 < 204.757, which is true.Check if ( a^2 < k ): 76.43 < 204.757, which is true.So, ( a ≈ -8.743 ) is also a possible solution.Wait, but these are approximate solutions. Let me try to find exact solutions.From Case 3, we had exact solutions: ( a = 19.5 ) and ( a = -18.5 ).From Case 4, we had approximate solutions, but let's see if we can get exact forms.In Case 4, the quadratic equation was:( 4a^2 - 4a - 341 = 0 )Solutions:( a = [4 ± sqrt(16 + 4*4*341)] / (2*4) = [4 ± sqrt(16 + 5456)] / 8 = [4 ± sqrt(5472)] / 8 )Simplify sqrt(5472):5472 = 16 * 342 = 16 * 9 * 38 = 16 * 9 * 2 * 19 = 16 * 9 * 2 * 19So, sqrt(5472) = 4 * 3 * sqrt(2*19) = 12 * sqrt(38)Thus,( a = [4 ± 12sqrt(38)] / 8 = [1 ± 3sqrt(38)] / 2 )So, exact solutions are ( a = frac{1 + 3sqrt{38}}{2} ) and ( a = frac{1 - 3sqrt{38}}{2} )Compute numerical values:sqrt(38) ≈ 6.1644So,( a ≈ (1 + 3*6.1644)/2 ≈ (1 + 18.4932)/2 ≈ 19.4932/2 ≈ 9.7466 )( a ≈ (1 - 18.4932)/2 ≈ (-17.4932)/2 ≈ -8.7466 )Which matches our earlier approximations.So, in total, we have four possible solutions:1. ( a = 19.5 )2. ( a = -18.5 )3. ( a ≈ 9.7466 )4. ( a ≈ -8.7466 )But wait, let's check if these solutions are valid in their respective cases.For ( a = 19.5 ):- ( a = 19.5 ) is less than ( k = 233 ), which is true.- ( a^2 = 380.25 ) is greater than ( k + 19 = 252 ), which is true.So, valid.For ( a = -18.5 ):- ( a = -18.5 ) is less than ( k = 195 ), which is true.- ( a^2 = 342.25 ) is greater than ( k + 19 = 214 ), which is true.So, valid.For ( a ≈ 9.7466 ):- ( a ≈ 9.7466 ) is less than ( k ≈ 223.243 ), which is true.- ( a^2 ≈ 94.93 ) is less than ( k ≈ 223.243 ), which is true.So, valid.For ( a ≈ -8.7466 ):- ( a ≈ -8.7466 ) is less than ( k ≈ 204.757 ), which is true.- ( a^2 ≈ 76.43 ) is less than ( k ≈ 204.757 ), which is true.So, valid.But wait, the problem says "natural numbers". So, the sequence ( k, k+1, ldots, k+19 ) must be natural numbers, so ( k ) must be a natural number.In Case 3, ( k = 233 ), which is natural.In Case 4, ( k ≈ 223.243 ) or ( k ≈ 204.757 ), which are not integers. So, these solutions would require ( k ) to be non-integer, which contradicts the fact that ( k ) is a natural number.Wait, so in Case 4, we have ( k = (S - 190)/20 ). From ( S = 20a + 4460 ), so ( k = (20a + 4460 - 190)/20 = (20a + 4270)/20 = a + 213.5 ). So, ( k = a + 213.5 ). Since ( k ) must be a natural number, ( a + 213.5 ) must be an integer. So, ( a ) must be of the form ( n - 213.5 ), where ( n ) is an integer. But in our solutions for Case 4, ( a ≈ 9.7466 ) and ( a ≈ -8.7466 ), which are not of the form ( n - 213.5 ). So, these solutions would require ( k ) to be non-integer, which is not allowed.Therefore, only the solutions from Case 3 are valid, where ( k ) is an integer.So, the valid solutions are ( a = 19.5 ) and ( a = -18.5 ).But let me double-check.For ( a = 19.5 ):- ( k = 233 ), which is natural.- The numbers are 233 to 252.- ( a = 19.5 ) is less than 233, so the sum of distances is ( S - 20a = 4460 ).- ( a^2 = 380.25 ) is greater than 252, so the sum of distances is ( 20a^2 - S = 2755 ).Let me compute ( S = 20*233 + 190 = 4660 + 190 = 4850 ).Compute ( S - 20a = 4850 - 20*19.5 = 4850 - 390 = 4460 ). Correct.Compute ( 20a^2 - S = 20*(380.25) - 4850 = 7605 - 4850 = 2755 ). Correct.Similarly, for ( a = -18.5 ):- ( k = 195 ), which is natural.- The numbers are 195 to 214.- ( a = -18.5 ) is less than 195, so the sum of distances is ( S - 20a = 4460 ).- ( a^2 = 342.25 ) is greater than 214, so the sum of distances is ( 20a^2 - S = 2755 ).Compute ( S = 20*195 + 190 = 3900 + 190 = 4090 ).Compute ( S - 20a = 4090 - 20*(-18.5) = 4090 + 370 = 4460 ). Correct.Compute ( 20a^2 - S = 20*(342.25) - 4090 = 6845 - 4090 = 2755 ). Correct.So, both solutions are valid.But wait, the problem says "natural numbers". So, ( a ) doesn't have to be natural, just the sequence is natural. So, ( a ) can be any real number.Therefore, the possible values of ( a ) are ( 19.5 ) and ( -18.5 ).But let me check if there are other cases where ( a ) is inside the range ( [k, k+19] ). Because in the earlier analysis, I considered only cases where ( a ) is outside the range. But if ( a ) is inside the range, the sum of distances is different.Wait, actually, the sum of distances when ( a ) is inside the range is more complicated because it's the sum of distances to each number, which would involve a different formula.But in the problem, we have two sums: one for ( a ) and one for ( a^2 ). So, if ( a ) is inside the range, then ( a^2 ) could be either inside or outside.But since we already found solutions where ( a ) is outside the range, and those satisfy the conditions, maybe those are the only solutions.But just to be thorough, let me consider the case where ( a ) is inside the range ( [k, k+19] ).If ( a ) is inside the range, the sum of distances is minimized when ( a ) is the median. For twenty numbers, the median is between the 10th and 11th terms, so ( k + 9.5 ).So, if ( a = k + 9.5 ), the sum of distances is minimized. But in our case, the sum is 4460, which is much larger than the minimal sum.Wait, let me compute the minimal sum.The minimal sum when ( a ) is the median is:For numbers ( k, k+1, ldots, k+19 ), the sum of distances to the median ( k + 9.5 ) is:Sum = ( sum_{i=0}^{9} (k + 9.5 - (k + i)) + sum_{i=10}^{19} ((k + i) - (k + 9.5)) )= ( sum_{i=0}^{9} (9.5 - i) + sum_{i=10}^{19} (i - 9.5) )= ( sum_{j=0}^{9} (9.5 - j) + sum_{j=1}^{10} (j + 0.5) )Wait, let me compute it more carefully.For ( i = 0 ) to ( 9 ):Distance = ( (k + 9.5) - (k + i) = 9.5 - i )So, sum is ( sum_{i=0}^{9} (9.5 - i) = sum_{i=0}^{9} 9.5 - sum_{i=0}^{9} i = 10*9.5 - 45 = 95 - 45 = 50 )For ( i = 10 ) to ( 19 ):Distance = ( (k + i) - (k + 9.5) = i - 9.5 )Sum is ( sum_{i=10}^{19} (i - 9.5) = sum_{j=0}^{9} (j + 0.5) = sum_{j=0}^{9} j + sum_{j=0}^{9} 0.5 = 45 + 4.5 = 49.5 )Total minimal sum = 50 + 49.5 = 99.5But in our problem, the sum is 4460, which is way larger. So, ( a ) cannot be inside the range because the sum would be much smaller. Therefore, our earlier consideration that ( a ) is outside the range is correct.Similarly, for ( a^2 ), if ( a^2 ) were inside the range, the sum would be minimal, but since the sum is 2755, which is also larger than the minimal sum, ( a^2 ) must also be outside the range.Therefore, only the solutions from Case 3 are valid, where ( a ) is outside the range and ( a^2 ) is also outside the range.So, the possible values of ( a ) are ( 19.5 ) and ( -18.5 ).But let me write them as fractions for exactness.19.5 = 39/2-18.5 = -37/2So, the possible values of ( a ) are ( frac{39}{2} ) and ( -frac{37}{2} ).I think that's it. So, the final answer is ( boxed{left{ -frac{37}{2}, frac{39}{2} right}} )

💡Okay, so I need to figure out the least positive integer that isn't a factor of some factorial number and isn't a prime number. The answer is given as 62, and I need to find the factorial number. Hmm, let me break this down step by step.First, I know that factorial numbers, like 1!, 2!, 3!, and so on, are products of all positive integers up to that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. So, factorial numbers get really big very quickly.Now, the question is about finding the smallest number that isn't a factor of any factorial and isn't prime. Since 62 is given as the answer, I need to understand why 62 isn't a factor of some factorial. Let's start by looking at 62 itself.62 is an even number, so it's divisible by 2. Dividing 62 by 2 gives me 31. So, 62 = 2 × 31. Both 2 and 31 are prime numbers. That means 62 is a composite number, specifically a product of two primes.Now, to find out if 62 is a factor of a factorial, I need to see if both 2 and 31 are present in the prime factors of that factorial. For example, 5! = 120, which factors into 2^3 × 3 × 5. Here, 2 is present, but 31 isn't. So, 62 isn't a factor of 5!.Let's check 6!. 6! = 720, which factors into 2^4 × 3^2 × 5. Again, 2 is present, but 31 isn't. So, 62 isn't a factor of 6!.Continuing this way, I realize that 31 is a prime number, and it won't appear in the prime factors of any factorial until we reach 31!. That's because 31! is the first factorial that includes 31 as one of its factors.So, before 31!, none of the factorials will have 31 as a factor, which means none of them will have 62 as a factor either. Therefore, 62 isn't a factor of any factorial less than 31!.But wait, is 62 the smallest such number? Let me think. The numbers smaller than 62 that aren't prime are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60.I need to check if any of these numbers aren't factors of some factorial. For example, 4 is a factor of 4! and beyond, 6 is a factor of 3! and beyond, 8 is a factor of 4! and beyond, and so on. Most of these numbers are factors of some factorial because their prime factors are included in the factorial's prime factors.However, 62 is different because it includes 31, which isn't present in factorials until 31!. So, 62 is the smallest composite number that isn't a factor of any factorial less than 31!.Therefore, the factorial number we're looking for is 31! because that's the first factorial that includes 31, making 62 a factor of it. But since the question is about the least positive integer that isn't a factor of some factorial and isn't prime, 62 fits that description, and the corresponding factorial is 31!.Wait, but the question says "the least positive integer that is not a factor of some factorial number and is not a prime number." So, 62 isn't a factor of any factorial less than 31!, but it is a factor of 31! and beyond. So, does that mean 62 is a factor of 31!?Yes, because 31! includes all numbers up to 31, including 2 and 31, so 62 is a factor of 31!. But the question is about the least integer that isn't a factor of some factorial and isn't prime. So, 62 isn't a factor of any factorial less than 31!, making it the smallest such number.Therefore, the factorial number in question is 31! because that's the first factorial where 62 becomes a factor.

💡First, I need to determine the total number of ways to select three jellybeans from the 12 available. This can be calculated using the combination formula:[{12 choose 3} = frac{12 times 11 times 10}{3 times 2 times 1} = 220]Next, I need to find the number of ways to pick exactly two blue jellybeans from the three available:[{3 choose 2} = 3]The third jellybean must be of a different color, which can be either red or white. There are five red and four white jellybeans, making a total of nine non-blue jellybeans:[{9 choose 1} = 9]Multiplying the number of ways to choose two blue jellybeans by the number of ways to choose one non-blue jellybean gives the total number of favorable outcomes:[3 times 9 = 27]Finally, the probability of picking exactly two blue jellybeans is the ratio of favorable outcomes to the total number of possible outcomes:[frac{27}{220}]This fraction is already in its simplest form, so the probability is:[boxed{frac{27}{220}}]

💡Alright, so I've been given this problem about renovating an underground pipe network in a city to prevent waterlogging during the rainy season. The problem has two parts, and I need to solve both. Let me try to understand what's being asked here.First, the city is renovating a 3600-meter section of the underground pipe network. The construction efficiency per day is 20% higher than originally planned, and because of this increased efficiency, the task can be completed 10 days ahead of schedule. Part (1) asks for the length of pipe network renovated per day during actual construction. So, I need to find the daily renovation length when the efficiency is 20% higher than planned.Part (2) is a bit more complex. After 20 days of construction, the construction unit decides to accelerate the progress again to ensure that the total construction period doesn't exceed 40 days. I need to find out how many more meters of pipe network should be renovated per day in the future to meet this requirement.Let me start with part (1). I think I need to set up some variables to represent the original planned daily renovation length and the actual daily renovation length. Let's denote the original planned daily renovation length as ( x ) meters. Since the efficiency is 20% higher, the actual daily renovation length would be ( x + 0.2x = 1.2x ) meters.Now, the total length to be renovated is 3600 meters. If the original plan was to renovate ( x ) meters per day, the total time required would be ( frac{3600}{x} ) days. With the increased efficiency, the time required becomes ( frac{3600}{1.2x} ) days. According to the problem, this increased efficiency allows the project to be completed 10 days earlier. So, the difference between the original time and the actual time is 10 days.Mathematically, this can be written as:[frac{3600}{x} - frac{3600}{1.2x} = 10]Let me simplify this equation step by step. First, let's compute ( frac{3600}{1.2x} ). Since ( 1.2 = frac{6}{5} ), ( frac{3600}{1.2x} = frac{3600 times 5}{6x} = frac{18000}{6x} = frac{3000}{x} ).So, the equation becomes:[frac{3600}{x} - frac{3000}{x} = 10][frac{600}{x} = 10]Now, solving for ( x ):[x = frac{600}{10} = 60]So, the original planned daily renovation length was 60 meters. But since the efficiency is 20% higher, the actual daily renovation length is:[1.2 times 60 = 72 text{ meters}]Okay, that seems straightforward. So, for part (1), the answer is 72 meters per day.Now, moving on to part (2). After 20 days of construction at the rate of 72 meters per day, the construction unit wants to accelerate the progress to ensure the total construction period doesn't exceed 40 days. I need to find out how many more meters should be renovated per day in the future.First, let's calculate how much of the pipe network has already been renovated in the first 20 days. If they've been renovating 72 meters per day for 20 days, the total length renovated so far is:[72 times 20 = 1440 text{ meters}]So, the remaining length to be renovated is:[3600 - 1440 = 2160 text{ meters}]Now, the total construction period should not exceed 40 days. Since 20 days have already passed, the remaining time available is:[40 - 20 = 20 text{ days}]So, they have 20 days left to renovate the remaining 2160 meters. Let ( m ) be the additional meters per day they need to renovate beyond the current rate of 72 meters per day. Therefore, the new daily renovation rate would be ( 72 + m ) meters per day.The total length renovated in the remaining 20 days should be at least 2160 meters. So, we can set up the inequality:[20 times (72 + m) geq 2160]Let's solve this inequality step by step. First, expand the left side:[20 times 72 + 20m geq 2160][1440 + 20m geq 2160]Subtract 1440 from both sides:[20m geq 2160 - 1440][20m geq 720]Now, divide both sides by 20:[m geq frac{720}{20}][m geq 36]So, they need to increase the daily renovation rate by at least 36 meters per day to ensure the total construction period does not exceed 40 days.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with part (1):- Original daily rate: ( x = 60 ) meters.- Actual daily rate: ( 1.2x = 72 ) meters.- Time saved: ( frac{3600}{60} - frac{3600}{72} = 60 - 50 = 10 ) days. That checks out.For part (2):- After 20 days: ( 72 times 20 = 1440 ) meters.- Remaining: ( 3600 - 1440 = 2160 ) meters.- Remaining time: ( 40 - 20 = 20 ) days.- Required rate: ( frac{2160}{20} = 108 ) meters per day.- Additional rate needed: ( 108 - 72 = 36 ) meters per day.Yes, that seems correct. So, they need to increase the daily renovation rate by 36 meters per day.I think I've got it right. The key was setting up the equations correctly and making sure the units and time frames matched up. It's important to carefully translate the problem into mathematical expressions and then solve step by step.

💡Okay, so I've got this problem here about functions defined as sums of absolute values. It has two parts: first, solving an inequality involving f₂(x), and second, finding the minimum value of f₅(x). Let me take this step by step.Starting with part (1): Solve the inequality f₂(x) < x + 1. First, I need to understand what f₂(x) is. The function is given as fₙ(x) = Σ from i=1 to n of |x - i|. So for n=2, that means f₂(x) = |x - 1| + |x - 2|. So the inequality becomes |x - 1| + |x - 2| < x + 1. Alright, so I need to solve |x - 1| + |x - 2| < x + 1. Absolute value functions can be tricky because they change their behavior based on whether the expression inside is positive or negative. So, I think the best way to approach this is to consider different cases based on the value of x.Let me recall that |a| is equal to a if a ≥ 0 and -a if a < 0. So, for |x - 1| and |x - 2|, the expressions inside the absolute values change sign at x=1 and x=2, respectively. So, I should consider the intervals divided by these points: x < 1, 1 ≤ x < 2, and x ≥ 2.Case 1: x < 1.In this case, both x - 1 and x - 2 are negative, so |x - 1| = -(x - 1) = 1 - x, and |x - 2| = -(x - 2) = 2 - x. So, f₂(x) becomes (1 - x) + (2 - x) = 3 - 2x. So the inequality becomes 3 - 2x < x + 1. Let me solve that:3 - 2x < x + 1 3 - 1 < x + 2x 2 < 3x 2/3 < x.But wait, in this case, we're assuming x < 1. So, the solution here is x > 2/3 and x < 1. So, the interval is (2/3, 1).Case 2: 1 ≤ x < 2.Here, x - 1 is non-negative, so |x - 1| = x - 1. But x - 2 is still negative, so |x - 2| = 2 - x. So, f₂(x) becomes (x - 1) + (2 - x) = 1.So the inequality is 1 < x + 1. Simplifying:1 < x + 1 0 < x.But in this case, x is between 1 and 2, so x is already greater than 0. So, the inequality holds for all x in [1, 2). So, the solution here is [1, 2).Case 3: x ≥ 2.In this interval, both x - 1 and x - 2 are non-negative, so |x - 1| = x - 1 and |x - 2| = x - 2. Therefore, f₂(x) becomes (x - 1) + (x - 2) = 2x - 3.So the inequality is 2x - 3 < x + 1. Let's solve that:2x - 3 < x + 1 2x - x < 1 + 3 x < 4.But in this case, x is ≥ 2, so combining with x < 4, the solution here is [2, 4).Now, combining all three cases:From Case 1: (2/3, 1)From Case 2: [1, 2)From Case 3: [2, 4)So overall, the solution set is (2/3, 4). Wait, let me check if x=2/3 is included. In Case 1, x > 2/3, so 2/3 is not included. Similarly, in Case 3, x < 4, so 4 is not included. So, the solution is all real numbers x such that 2/3 < x < 4.I think that's the solution for part (1). Now, moving on to part (2): Find the minimum value of f₅(x).So, f₅(x) = |x - 1| + |x - 2| + |x - 3| + |x - 4| + |x - 5|. I remember that the sum of absolute values like this has a minimum at the median of the points. So, since we have points at 1, 2, 3, 4, 5, which are five points, the median is the third one, which is 3. So, the minimum should occur at x=3.But let me verify that.Alternatively, I can think of the function f₅(x) as a piecewise linear function, and its minimum occurs where the slope changes from negative to positive. The slope of f₅(x) changes at each integer point from 1 to 5.Let me compute the slope in different intervals.For x < 1: All terms |x - i| will have negative derivatives, so the slope is -5.Between 1 and 2: The term |x - 1| changes slope to +1, so total slope is -5 + 2 = -3.Between 2 and 3: Similarly, |x - 2| changes slope, so total slope becomes -3 + 2 = -1.Between 3 and 4: |x - 3| changes slope, so total slope becomes -1 + 2 = +1.Between 4 and 5: |x - 4| changes slope, so total slope becomes +1 + 2 = +3.For x > 5: All terms have positive derivatives, so slope is +5.So, the slope changes from negative to positive at x=3. So, the minimum occurs at x=3.Therefore, the minimum value is f₅(3). Let's compute that.f₅(3) = |3 - 1| + |3 - 2| + |3 - 3| + |3 - 4| + |3 - 5| = 2 + 1 + 0 + 1 + 2 = 6.So, the minimum value is 6.Wait, let me make sure I didn't make a calculation error.|3-1|=2, |3-2|=1, |3-3|=0, |3-4|=1, |3-5|=2. Adding them up: 2+1=3, 3+0=3, 3+1=4, 4+2=6. Yes, that's correct.Alternatively, I could have thought about pairing terms symmetrically around 3.For example, |x - 1| + |x - 5| is minimized when x is between 1 and 5, and the minimum value is 4 (since it's the distance between 1 and 5). Similarly, |x - 2| + |x - 4| is minimized when x is between 2 and 4, with a minimum value of 2. And |x - 3| is minimized at x=3 with value 0. So, adding them up: 4 + 2 + 0 = 6. That's another way to see it.So, I think that's solid. The minimum value is indeed 6.**Final Answer**(1) The solution is boxed{left( dfrac{2}{3}, 4 right)}.(2) The minimum value is boxed{6}.

💡Okay, so I need to find the minimum value of the expression (frac{x^2 + 9}{sqrt{x^2 + 5}}) for all real numbers (x). Hmm, let me think about how to approach this. First, the expression looks a bit complicated, but maybe I can simplify it somehow. Let me see, the numerator is (x^2 + 9) and the denominator is (sqrt{x^2 + 5}). Maybe I can rewrite the numerator in terms of the denominator to make it easier to handle. I notice that (x^2 + 9) can be expressed as (x^2 + 5 + 4). So, substituting that in, the expression becomes:[frac{x^2 + 5 + 4}{sqrt{x^2 + 5}} = frac{x^2 + 5}{sqrt{x^2 + 5}} + frac{4}{sqrt{x^2 + 5}}]Simplifying the first term, (frac{x^2 + 5}{sqrt{x^2 + 5}}) is just (sqrt{x^2 + 5}). So now the expression is:[sqrt{x^2 + 5} + frac{4}{sqrt{x^2 + 5}}]Alright, so now the expression is the sum of two terms: (sqrt{x^2 + 5}) and (frac{4}{sqrt{x^2 + 5}}). This looks like something where I can apply the AM-GM inequality, which states that for non-negative real numbers (a) and (b), the arithmetic mean is greater than or equal to the geometric mean. That is:[frac{a + b}{2} geq sqrt{ab}]Multiplying both sides by 2 gives:[a + b geq 2sqrt{ab}]In our case, let (a = sqrt{x^2 + 5}) and (b = frac{4}{sqrt{x^2 + 5}}). Applying AM-GM:[sqrt{x^2 + 5} + frac{4}{sqrt{x^2 + 5}} geq 2sqrt{sqrt{x^2 + 5} cdot frac{4}{sqrt{x^2 + 5}}}]Simplifying the right side:[2sqrt{sqrt{x^2 + 5} cdot frac{4}{sqrt{x^2 + 5}}} = 2sqrt{4} = 2 times 2 = 4]So, the expression is greater than or equal to 4. That means the minimum value of the original expression is at least 4. But wait, I should check if this minimum is actually achievable. For the AM-GM inequality, equality holds when (a = b). So, setting (a = b):[sqrt{x^2 + 5} = frac{4}{sqrt{x^2 + 5}}]Multiplying both sides by (sqrt{x^2 + 5}):[(sqrt{x^2 + 5})^2 = 4]Which simplifies to:[x^2 + 5 = 4]Subtracting 5 from both sides:[x^2 = -1]Hmm, that's a problem because (x^2) can't be negative for real numbers (x). So, equality doesn't hold here, which means the minimum value of 4 isn't actually achieved by any real (x). That suggests that the expression might have a higher minimum value. Wait, maybe I made a mistake in setting up the AM-GM inequality. Let me double-check. I had (a = sqrt{x^2 + 5}) and (b = frac{4}{sqrt{x^2 + 5}}), so their product is indeed 4, and the geometric mean is 2. So, the AM-GM step seems correct. But since equality isn't achievable, perhaps I need another approach to find the minimum.Alternatively, maybe I can use calculus to find the minimum. Let me define the function:[f(x) = frac{x^2 + 9}{sqrt{x^2 + 5}}]To find its minimum, I can take the derivative and set it equal to zero. Let's compute (f'(x)). First, rewrite (f(x)) as:[f(x) = (x^2 + 9)(x^2 + 5)^{-1/2}]Using the product rule and chain rule, the derivative (f'(x)) is:[f'(x) = 2x(x^2 + 5)^{-1/2} + (x^2 + 9)(-1/2)(x^2 + 5)^{-3/2}(2x)]Simplifying each term:First term: (2x(x^2 + 5)^{-1/2})Second term: (- (x^2 + 9)x(x^2 + 5)^{-3/2})So, combining these:[f'(x) = 2x(x^2 + 5)^{-1/2} - x(x^2 + 9)(x^2 + 5)^{-3/2}]Factor out (x(x^2 + 5)^{-3/2}):[f'(x) = x(x^2 + 5)^{-3/2} [2(x^2 + 5) - (x^2 + 9)]]Simplify inside the brackets:[2(x^2 + 5) - (x^2 + 9) = 2x^2 + 10 - x^2 - 9 = x^2 + 1]So, the derivative becomes:[f'(x) = x(x^2 + 5)^{-3/2}(x^2 + 1)]To find critical points, set (f'(x) = 0):[x(x^2 + 5)^{-3/2}(x^2 + 1) = 0]Since ((x^2 + 5)^{-3/2}) is always positive for real (x), and (x^2 + 1) is always positive, the only solution is when (x = 0).So, the critical point is at (x = 0). Let's evaluate (f(x)) at this point:[f(0) = frac{0 + 9}{sqrt{0 + 5}} = frac{9}{sqrt{5}} approx 4.0249]Hmm, that's just a bit above 4. But earlier, using AM-GM, I found that the expression is at least 4, but it's not achievable. So, the minimum must be slightly above 4. But wait, maybe I can find another critical point or check the behavior as (x) approaches infinity. Let's see what happens as (x) becomes very large. As (x to infty), the expression behaves like:[frac{x^2}{sqrt{x^2}} = frac{x^2}{|x|} = |x|]Which goes to infinity. So, the function tends to infinity as (x) becomes large in magnitude. Therefore, the minimum must occur at some finite (x). We found a critical point at (x = 0), but that gives a value of approximately 4.0249. Is there another critical point? Let me check my derivative again.Wait, in the derivative, I had:[f'(x) = x(x^2 + 5)^{-3/2}(x^2 + 1)]Which is zero only when (x = 0). So, that's the only critical point. Therefore, the function has a minimum at (x = 0) with value (frac{9}{sqrt{5}}). But earlier, using AM-GM, I thought the minimum was 4, but that's not achievable. So, perhaps the actual minimum is (frac{9}{sqrt{5}}), which is approximately 4.0249. Wait, but let me check another approach. Maybe using substitution. Let me set (y = sqrt{x^2 + 5}). Then, since (x^2 = y^2 - 5), the original expression becomes:[frac{y^2 - 5 + 9}{y} = frac{y^2 + 4}{y} = y + frac{4}{y}]So, now the problem reduces to minimizing (y + frac{4}{y}) where (y geq sqrt{5}) because (y = sqrt{x^2 + 5} geq sqrt{5}).Now, to minimize (y + frac{4}{y}), we can take the derivative with respect to (y):[frac{d}{dy}left(y + frac{4}{y}right) = 1 - frac{4}{y^2}]Set this equal to zero:[1 - frac{4}{y^2} = 0 implies y^2 = 4 implies y = 2]But wait, (y geq sqrt{5} approx 2.236), so (y = 2) is less than (sqrt{5}), which is not in the domain. Therefore, the minimum occurs at the boundary of the domain, which is (y = sqrt{5}).So, substituting back, the minimum value is:[sqrt{5} + frac{4}{sqrt{5}} = frac{5 + 4}{sqrt{5}} = frac{9}{sqrt{5}} approx 4.0249]So, this confirms that the minimum value is (frac{9}{sqrt{5}}), which is approximately 4.0249. Wait, but earlier when I tried AM-GM, I thought the minimum was 4, but that's not achievable because it would require (x^2 = -1), which isn't possible. So, the actual minimum is higher, specifically (frac{9}{sqrt{5}}). But let me rationalize the denominator to make it look nicer:[frac{9}{sqrt{5}} = frac{9sqrt{5}}{5}]So, the minimum value is (frac{9sqrt{5}}{5}).But wait, earlier when I did the substitution, I set (y = sqrt{x^2 + 5}), and found that the minimum occurs at (y = sqrt{5}), which corresponds to (x = 0). So, that's consistent with the calculus approach where the critical point was at (x = 0).Therefore, the minimum value of the expression is (frac{9sqrt{5}}{5}).But hold on, let me double-check everything because initially, I thought the minimum was 4, but that's not achievable, so the actual minimum is higher. Alternatively, maybe I can use another substitution or method. Let me try completing the square or another inequality.Wait, another approach could be to let (t = x^2), so (t geq 0). Then, the expression becomes:[frac{t + 9}{sqrt{t + 5}}]Let me define (f(t) = frac{t + 9}{sqrt{t + 5}}). To find the minimum, take the derivative with respect to (t):[f'(t) = frac{sqrt{t + 5} - frac{1}{2}(t + 9)(t + 5)^{-1/2}}{t + 5}]Wait, that might be a bit messy. Alternatively, let me rewrite (f(t)) as:[f(t) = (t + 9)(t + 5)^{-1/2}]Then, the derivative is:[f'(t) = (1)(t + 5)^{-1/2} + (t + 9)(-1/2)(t + 5)^{-3/2}(1)]Simplifying:[f'(t) = (t + 5)^{-1/2} - frac{1}{2}(t + 9)(t + 5)^{-3/2}]Factor out ((t + 5)^{-3/2}):[f'(t) = (t + 5)^{-3/2} left[ (t + 5) - frac{1}{2}(t + 9) right]]Simplify inside the brackets:[(t + 5) - frac{1}{2}(t + 9) = t + 5 - frac{t}{2} - frac{9}{2} = frac{t}{2} + frac{10}{2} - frac{t}{2} - frac{9}{2} = frac{1}{2}]So, the derivative is:[f'(t) = frac{1}{2}(t + 5)^{-3/2}]Wait, that's always positive for (t geq 0). That suggests that (f(t)) is increasing for all (t geq 0). Therefore, the minimum occurs at the smallest (t), which is (t = 0).So, substituting (t = 0):[f(0) = frac{0 + 9}{sqrt{0 + 5}} = frac{9}{sqrt{5}} = frac{9sqrt{5}}{5}]This confirms again that the minimum value is (frac{9sqrt{5}}{5}).Wait, but earlier when I used substitution (y = sqrt{x^2 + 5}), I found that the minimum occurs at (y = sqrt{5}), which corresponds to (t = 0). So, everything is consistent.Therefore, the minimum value of the expression is (frac{9sqrt{5}}{5}).But just to be thorough, let me check another point. Suppose (x = 1), then:[f(1) = frac{1 + 9}{sqrt{1 + 5}} = frac{10}{sqrt{6}} approx 4.0825]Which is higher than (frac{9sqrt{5}}{5} approx 4.0249).Similarly, (x = sqrt{11}), as I thought earlier, gives:[f(sqrt{11}) = frac{11 + 9}{sqrt{11 + 5}} = frac{20}{4} = 5]Which is higher.So, it seems that the minimum is indeed at (x = 0), giving (frac{9sqrt{5}}{5}).Wait, but earlier when I tried AM-GM, I thought the minimum was 4, but that's not achievable. So, perhaps the correct minimum is (frac{9sqrt{5}}{5}).But let me check another angle. Maybe using substitution (u = x^2 + 5), so (u geq 5). Then, the expression becomes:[frac{u + 4}{sqrt{u}} = sqrt{u} + frac{4}{sqrt{u}}]Now, to minimize (sqrt{u} + frac{4}{sqrt{u}}) for (u geq 5). Let me set (v = sqrt{u}), so (v geq sqrt{5}). Then, the expression is (v + frac{4}{v}).Taking the derivative with respect to (v):[frac{d}{dv}left(v + frac{4}{v}right) = 1 - frac{4}{v^2}]Set to zero:[1 - frac{4}{v^2} = 0 implies v^2 = 4 implies v = 2]But (v geq sqrt{5} approx 2.236), so (v = 2) is not in the domain. Therefore, the minimum occurs at (v = sqrt{5}), which gives:[sqrt{5} + frac{4}{sqrt{5}} = frac{5 + 4}{sqrt{5}} = frac{9}{sqrt{5}} = frac{9sqrt{5}}{5}]So, again, the same result.Therefore, after multiple approaches—AM-GM, substitution, calculus—the minimum value is (frac{9sqrt{5}}{5}).But wait, let me just confirm if I didn't make a mistake in the substitution approach. When I set (y = sqrt{x^2 + 5}), then (y geq sqrt{5}), and the expression becomes (y + frac{4}{y}). The derivative of this with respect to (y) is (1 - frac{4}{y^2}), which is zero at (y = 2), but since (y geq sqrt{5}), the minimum is at (y = sqrt{5}), giving the value (frac{9}{sqrt{5}}).Yes, that seems correct.So, in conclusion, the minimum value of the expression is (frac{9sqrt{5}}{5}).