answer：To determine by how many meters A beats B, we can analyze the relationships between the runners' performances.1. **Let’s denote**: - ( X ) as the distance by which A beats B. - When A finishes the race (1000 meters), B is ( X ) meters behind A, and C is 163 meters behind A.2. **Given that B beats C by 100 meters**, when B has run ( 1000 - X ) meters, C has run ( 1000 - X - 100 ) meters.3. **Since A beats C by 163 meters**, when A finishes the race, C has run ( 1000 - 163 = 837 ) meters.4. **Setting up the equation**: [ 1000 - X = 837 + 100 ] [ 1000 - X = 937 ] [ X = 1000 - 937 ] [ X = 63 ]**Therefore, A beats B by 63 meters.**

answer：Okay, so I have this equation to solve: [left(3^{x} - 27right)^{2} + left(5^{x} - 625right)^{2} = left(3^{x} + 5^{x} - 652right)^{2}]Hmm, that looks a bit complicated, but maybe I can simplify it. Let me think about how to approach this.First, I notice that both sides of the equation involve squares. Maybe if I expand both sides, I can find a way to simplify or factor it. Let me try expanding the right-hand side first.So, the right-hand side is:[left(3^{x} + 5^{x} - 652right)^{2}]If I expand this, it should be:[(3^{x})^2 + (5^{x})^2 + (652)^2 + 2 cdot 3^{x} cdot 5^{x} - 2 cdot 3^{x} cdot 652 - 2 cdot 5^{x} cdot 652]Wait, that seems a bit messy. Maybe there's a better way. Let me consider the left-hand side:[left(3^{x} - 27right)^{2} + left(5^{x} - 625right)^{2}]Expanding each term:[(3^{x})^2 - 2 cdot 3^{x} cdot 27 + 27^2 + (5^{x})^2 - 2 cdot 5^{x} cdot 625 + 625^2]So, combining these, the left-hand side becomes:[(3^{x})^2 + (5^{x})^2 - 2 cdot 3^{x} cdot 27 - 2 cdot 5^{x} cdot 625 + 27^2 + 625^2]Now, if I set the left-hand side equal to the right-hand side, I can write:[(3^{x})^2 + (5^{x})^2 - 2 cdot 3^{x} cdot 27 - 2 cdot 5^{x} cdot 625 + 27^2 + 625^2 = (3^{x})^2 + (5^{x})^2 + (652)^2 + 2 cdot 3^{x} cdot 5^{x} - 2 cdot 3^{x} cdot 652 - 2 cdot 5^{x} cdot 652]Hmm, okay, let's subtract the left-hand side from both sides to bring everything to one side:[0 = (652)^2 + 2 cdot 3^{x} cdot 5^{x} - 2 cdot 3^{x} cdot 652 - 2 cdot 5^{x} cdot 652 - (-2 cdot 3^{x} cdot 27 - 2 cdot 5^{x} cdot 625 + 27^2 + 625^2)]Wait, this seems too complicated. Maybe I made a mistake in expanding. Let me try a different approach.I remember that sometimes, equations like this can be simplified by letting variables represent the exponentials. Let me set:Let ( a = 3^{x} ) and ( b = 5^{x} ).Then, the equation becomes:[(a - 27)^2 + (b - 625)^2 = (a + b - 652)^2]That looks a bit simpler. Let me expand both sides now.Left-hand side:[(a - 27)^2 + (b - 625)^2 = a^2 - 54a + 27^2 + b^2 - 1250b + 625^2]Right-hand side:[(a + b - 652)^2 = a^2 + 2ab + b^2 - 1304a - 1304b + 652^2]Now, set them equal:[a^2 - 54a + 27^2 + b^2 - 1250b + 625^2 = a^2 + 2ab + b^2 - 1304a - 1304b + 652^2]Let me subtract ( a^2 + b^2 ) from both sides to simplify:[-54a + 27^2 - 1250b + 625^2 = 2ab - 1304a - 1304b + 652^2]Hmm, still a bit messy, but maybe I can collect like terms.Bring all terms to the left-hand side:[-54a + 27^2 - 1250b + 625^2 - 2ab + 1304a + 1304b - 652^2 = 0]Combine like terms:For ( a ): ( (-54a + 1304a) = 1250a )For ( b ): ( (-1250b + 1304b) = 54b )Constants: ( 27^2 + 625^2 - 652^2 )So, the equation becomes:[1250a + 54b - 2ab + (27^2 + 625^2 - 652^2) = 0]Let me compute the constants:First, ( 27^2 = 729 ), ( 625^2 = 390625 ), and ( 652^2 ). Let me compute ( 652^2 ):( 650^2 = 422500 ), and ( 652^2 = (650 + 2)^2 = 650^2 + 4 cdot 650 + 4 = 422500 + 2600 + 4 = 425104 ).So, ( 27^2 + 625^2 - 652^2 = 729 + 390625 - 425104 ).Calculate that:729 + 390625 = 391354391354 - 425104 = -33750So, the equation is:[1250a + 54b - 2ab - 33750 = 0]Hmm, this still looks complicated. Maybe I can factor something out.Let me rearrange the terms:[-2ab + 1250a + 54b - 33750 = 0]Factor out terms:Let me factor out -2a from the first two terms:[-2a(b - 625) + 54b - 33750 = 0]Wait, because 1250a = -2a*(-625). Let me check:-2a(b - 625) = -2ab + 1250a, which matches the first two terms.So, the equation becomes:[-2a(b - 625) + 54b - 33750 = 0]Hmm, maybe I can factor further or find a substitution.Wait, let's recall that ( a = 3^x ) and ( b = 5^x ). Maybe there's a relationship between a and b that can help.Alternatively, perhaps I can write this equation as:[-2ab + 1250a + 54b = 33750]Let me factor terms:Looking at -2ab + 1250a + 54b, maybe factor out a from the first two terms:a(-2b + 1250) + 54b = 33750Hmm, not sure if that helps. Alternatively, maybe factor out b from the last two terms:-2ab + 1250a + 54b = a(-2b + 1250) + b(54) = 33750Not sure. Maybe I can write it as:a(-2b + 1250) + 54b = 33750Let me see if I can express this in terms of a or b.Alternatively, perhaps I can write it as:a(1250 - 2b) + 54b = 33750Hmm, maybe I can solve for a in terms of b:a(1250 - 2b) = 33750 - 54bSo,a = (33750 - 54b)/(1250 - 2b)Simplify numerator and denominator:Factor numerator: 33750 - 54b = 54*(625) - 54b = 54*(625 - b)Denominator: 1250 - 2b = 2*(625 - b)So,a = [54*(625 - b)] / [2*(625 - b)] = 54/2 = 27Wait, that's interesting. So, a = 27, provided that 625 - b ≠ 0, which would make the denominator zero.So, if 625 - b ≠ 0, then a = 27.But if 625 - b = 0, then b = 625, which would make the denominator zero, but let's check if that's a valid solution.So, case 1: 625 - b ≠ 0, then a = 27.Case 2: 625 - b = 0, so b = 625.Let me analyze both cases.Case 1: a = 27.Since a = 3^x, so 3^x = 27.27 is 3^3, so x = 3.Case 2: b = 625.Since b = 5^x, so 5^x = 625.625 is 5^4, so x = 4.So, the solutions are x = 3 and x = 4.Therefore, the sum of all real solutions is 3 + 4 = 7.Wait, that seems straightforward. Let me verify if these solutions satisfy the original equation.First, x = 3:Compute each term:3^3 = 27, so 3^x - 27 = 0.5^3 = 125, so 5^x - 625 = 125 - 625 = -500.Left-hand side: (0)^2 + (-500)^2 = 0 + 250000 = 250000.Right-hand side: (27 + 125 - 652)^2 = (152 - 652)^2 = (-500)^2 = 250000.So, it checks out.Now, x = 4:Compute each term:3^4 = 81, so 3^x - 27 = 81 - 27 = 54.5^4 = 625, so 5^x - 625 = 0.Left-hand side: (54)^2 + (0)^2 = 2916 + 0 = 2916.Right-hand side: (81 + 625 - 652)^2 = (706 - 652)^2 = (54)^2 = 2916.So, that also checks out.Therefore, the solutions are indeed x = 3 and x = 4, and their sum is 7.

answer：Okay, so I have this problem here: Given that a is less than b, and both are less than zero, which of the following must be true? The options are A, B, C, and D. Let me try to figure this out step by step.First, let's understand what the given information tells us. It says a < b < 0. So, both a and b are negative numbers, and a is to the left of b on the number line, meaning a is more negative than b. For example, if a is -3 and b is -2, that fits the condition because -3 is less than -2, and both are less than zero.Now, let's look at each option one by one.**Option A: a - b > 0**Hmm, so if I subtract b from a, will that be positive? Let's test it with numbers. If a is -3 and b is -2, then a - b would be -3 - (-2) = -3 + 2 = -1. That's negative, so a - b is not greater than zero in this case. So, option A doesn't seem to hold true. Maybe I should try another set of numbers to confirm. Let's say a is -5 and b is -1. Then a - b is -5 - (-1) = -5 + 1 = -4, which is still negative. So, it looks like option A is not necessarily true.**Option B: ac < bc**This one is a bit trickier because it involves another variable, c. The problem doesn't tell us anything about c, so we have to consider different possibilities. If c is positive, then multiplying both sides of the inequality a < b by c would preserve the inequality direction, meaning ac < bc. But wait, if c is positive, then ac < bc would actually be true because a is less than b. However, if c is negative, multiplying both sides by c would reverse the inequality, making ac > bc. So, depending on the value of c, this statement could be either true or false. Since we don't know anything about c, we can't say for sure that ac < bc is always true. Therefore, option B isn't necessarily true.**Option C: 1/a < 1/b**Alright, let's think about reciprocals of negative numbers. If a and b are both negative, their reciprocals will also be negative. Let's take our earlier example where a is -3 and b is -2. Then 1/a is -1/3 and 1/b is -1/2. Now, comparing these two, -1/3 is greater than -1/2 because it's closer to zero. So, 1/a > 1/b in this case, which means 1/a < 1/b is false. Let's try another example to be sure. If a is -4 and b is -1, then 1/a is -1/4 and 1/b is -1. Again, -1/4 is greater than -1, so 1/a < 1/b is still false. It seems like option C is not true either.**Option D: a² > b²**Okay, let's check this one. Squaring negative numbers makes them positive, and since a is more negative than b, its square should be larger. Let's test it with our examples. If a is -3 and b is -2, then a² is 9 and b² is 4. So, 9 > 4, which is true. Another example: a is -5 and b is -1. Then a² is 25 and b² is 1, so 25 > 1, which is also true. It looks like squaring both a and b reverses the inequality because the number with the larger absolute value becomes larger when squared. Therefore, option D seems to hold true in all cases.So, after evaluating all the options, only option D must be true given that a < b < 0.

answer：To determine the number of digits in the base-7 representation of 956, I need to find the largest power of 7 that is less than or equal to 956.First, I calculate the powers of 7:- (7^1 = 7)- (7^2 = 49)- (7^3 = 343)- (7^4 = 2401)Since (7^4 = 2401) is greater than 956, the largest power of 7 that is less than 956 is (7^3 = 343).The number of digits in the base-7 representation is determined by the highest power of 7 needed to represent the number. Since (7^3) is the highest power used, the number of digits is (3 + 1 = 4).

answer：Okay, so I have this problem here: Tyler had some dogs. Each dog had 5 puppies. Tyler now has 75 puppies. How many dogs did Tyler have initially?Alright, let's break this down. First, Tyler has some dogs, and each of these dogs had 5 puppies. So, if I think about it, the number of puppies depends on how many dogs Tyler had initially. The more dogs he had, the more puppies there would be, right?So, if each dog has 5 puppies, and Tyler ends up with 75 puppies, I need to figure out how many dogs he started with. It seems like a multiplication problem because each dog contributes 5 puppies. So, if I let the number of dogs be 'd', then the total number of puppies would be 5 times 'd', which is 5d.Given that the total number of puppies is 75, I can set up the equation: 5d = 75.Now, to find out the value of 'd', I need to solve for it. To isolate 'd', I should divide both sides of the equation by 5. So, dividing both sides by 5 gives me d = 75 / 5.Calculating that, 75 divided by 5 is 15. So, d equals 15.Wait, does that make sense? If Tyler had 15 dogs and each dog had 5 puppies, then 15 times 5 is indeed 75 puppies. That checks out.But let me think again to make sure I didn't make any mistakes. Sometimes, when dealing with word problems, it's easy to misinterpret the relationships between the numbers.So, Tyler had some dogs. Each dog had 5 puppies. So, each dog contributes 5 puppies. The total number of puppies is 75. So, the number of dogs times 5 equals 75.Yes, that seems right. So, the number of dogs is 75 divided by 5, which is 15.I think that's correct. There's no indication that any puppies were lost or gained in another way, so it's a straightforward multiplication problem.Another way to think about it is to consider how many puppies come from each dog. If one dog has 5 puppies, then two dogs would have 10 puppies, three dogs would have 15 puppies, and so on. So, continuing this pattern, 15 dogs would have 75 puppies.Yes, that makes sense. So, Tyler initially had 15 dogs.I don't see any other factors mentioned in the problem, like some dogs not having puppies or puppies being adopted out. It's a simple relationship between the number of dogs and the number of puppies they had.Therefore, I'm confident that the initial number of dogs Tyler had is 15.

answer：Alright, so I have this integral problem here. It says that the integral of f(x) from 0 to 5 is 8 square units. Then, I need to find the area between the graph of y = 2f(2x - 1) and the x-axis from x = 0 to x = 2.5. Hmm, okay, let's break this down step by step.First, I know that the integral of f(x) from 0 to 5 is 8. That means the area under the curve f(x) between x = 0 and x = 5 is 8 square units. Now, I need to figure out how the function y = 2f(2x - 1) affects this area when we change the limits of integration from x = 0 to x = 2.5.Let me think about transformations of functions. The function f(2x - 1) involves two transformations: a horizontal compression by a factor of 2 and a horizontal shift to the right by 1 unit. Then, multiplying by 2 vertically scales the function by a factor of 2.So, to find the integral of 2f(2x - 1) from 0 to 2.5, I need to adjust for these transformations. I remember that when you have a function like f(kx + c), the integral over an interval can be found by substitution. Let me try that.Let’s set u = 2x - 1. Then, du/dx = 2, so du = 2dx, which means dx = du/2. Now, I need to change the limits of integration accordingly. When x = 0, u = 2(0) - 1 = -1. When x = 2.5, u = 2(2.5) - 1 = 5 - 1 = 4. So, the integral from x = 0 to x = 2.5 becomes the integral from u = -1 to u = 4.But wait, the original integral of f(x) is from 0 to 5. So, integrating f(u) from u = -1 to u = 4 includes some area outside the original interval. Specifically, from u = -1 to u = 0, which wasn't part of the original integral. However, since we don't have any information about f(u) for u < 0, I might need to make an assumption here. Maybe f(u) is zero outside the interval [0, 5]. If that's the case, then the integral from -1 to 0 would be zero.So, the integral from u = -1 to u = 4 would be the same as the integral from u = 0 to u = 4. But the original integral is from 0 to 5, which is 8. So, the integral from 0 to 4 would be less than 8. But without knowing the exact behavior of f(u) between 4 and 5, I can't determine the exact value. Hmm, this is a bit tricky.Wait, maybe I can express the integral from 0 to 4 in terms of the original integral. Let's denote the integral from 0 to 4 as I. Then, the integral from 4 to 5 would be 8 - I. But since I don't have information about f(u) between 4 and 5, I can't find I directly. Maybe I need to make another assumption or find another way.Alternatively, perhaps I can consider that the function f(2x - 1) is a transformation of f(x), and the scaling factors will affect the area. The horizontal compression by a factor of 2 would halve the area, and the vertical scaling by 2 would double the area. So, overall, the area might remain the same. But I need to verify this.Let me think about substitution again. The integral becomes:∫₀²·⁵ 2f(2x - 1) dxLet u = 2x - 1, so du = 2dx, dx = du/2.When x = 0, u = -1; when x = 2.5, u = 4.So, the integral becomes:∫₋₁⁴ 2f(u) * (du/2) = ∫₋₁⁴ f(u) duNow, if f(u) is zero for u < 0, then ∫₋₁⁴ f(u) du = ∫₀⁴ f(u) duBut we know ∫₀⁵ f(u) du = 8, so ∫₀⁴ f(u) du = 8 - ∫₄⁵ f(u) duWithout knowing ∫₄⁵ f(u) du, I can't find the exact value. Maybe the problem assumes that f(u) is zero beyond u = 5, but even then, I still have the integral from 4 to 5.Wait, maybe I'm overcomplicating this. Since the transformation is horizontal compression and shift, and vertical scaling, the area should scale accordingly. The horizontal compression by 2 would halve the width, and the vertical scaling by 2 would double the height, so the area remains the same.But let me check this with substitution. If I have ∫₀²·⁵ 2f(2x - 1) dx, and after substitution, it's ∫₋₁⁴ f(u) du. If f(u) is zero outside [0,5], then ∫₋₁⁴ f(u) du = ∫₀⁴ f(u) du. But ∫₀⁴ f(u) du is less than 8. So, unless f(u) is constant or something, I can't say it's exactly 8.Wait, maybe the problem is designed so that the area remains 8 despite the transformations. Let me think about the scaling factors. The horizontal compression by 2 would make the area half, and the vertical scaling by 2 would double it, so overall, the area remains the same. So, maybe the area is still 8.But I'm not entirely sure. Let me try with a simple function to test. Suppose f(x) = 1 for x in [0,5]. Then ∫₀⁵ f(x) dx = 5, but in the problem, it's 8. So, maybe f(x) is not constant. But if I take f(x) = 8/5, then ∫₀⁵ f(x) dx = 8. Then, y = 2f(2x -1) = 2*(8/5) = 16/5. Then, the area from x=0 to x=2.5 would be ∫₀²·⁵ 16/5 dx = (16/5)*2.5 = 8. So, in this case, the area is 8.Hmm, interesting. So, maybe regardless of the function, the area remains 8 because of the scaling factors. The horizontal compression by 2 and vertical scaling by 2 cancel each other out in terms of area.So, perhaps the answer is 8 square units.