answer：Okay, so I need to find the minimum value of the function ( f(x) = x + frac{4}{x - 2} ) where ( x > 2 ). Hmm, let's see. I remember that to find the minimum or maximum of a function, calculus is useful. Specifically, taking the derivative and setting it equal to zero to find critical points. Then, I can check if those points are minima or maxima.First, let me write down the function again to make sure I have it right: ( f(x) = x + frac{4}{x - 2} ). Yeah, that looks correct. So, since ( x > 2 ), the denominator ( x - 2 ) is positive, which is good because it means the function is defined and we don't have to worry about division by zero or negative values here.Alright, let's find the derivative of ( f(x) ). The derivative of ( x ) with respect to ( x ) is 1. Now, for the second term, ( frac{4}{x - 2} ), I can rewrite it as ( 4(x - 2)^{-1} ) to make differentiation easier. Using the power rule, the derivative of ( (x - 2)^{-1} ) is ( -1(x - 2)^{-2} ), so multiplying by the constant 4, we get ( -4(x - 2)^{-2} ). Therefore, the derivative of the second term is ( -frac{4}{(x - 2)^2} ).Putting it all together, the derivative ( f'(x) ) is:[f'(x) = 1 - frac{4}{(x - 2)^2}]Okay, that seems right. Now, to find the critical points, I need to set this derivative equal to zero and solve for ( x ):[1 - frac{4}{(x - 2)^2} = 0]Let me solve this equation step by step. First, move the fraction to the other side:[1 = frac{4}{(x - 2)^2}]Now, multiply both sides by ( (x - 2)^2 ) to eliminate the denominator:[(x - 2)^2 = 4]Taking the square root of both sides, we get:[x - 2 = pm 2]So, solving for ( x ), we have two possibilities:1. ( x - 2 = 2 ) which gives ( x = 4 )2. ( x - 2 = -2 ) which gives ( x = 0 )But wait, the original condition is ( x > 2 ), so ( x = 0 ) is not in the domain we're considering. Therefore, the only critical point is at ( x = 4 ).Now, I need to determine whether this critical point is a minimum or a maximum. One way to do this is by using the second derivative test. Let's compute the second derivative ( f''(x) ).First, recall that the first derivative is:[f'(x) = 1 - frac{4}{(x - 2)^2}]So, the second derivative is the derivative of ( f'(x) ). The derivative of 1 is 0. For the second term, ( -4(x - 2)^{-2} ), using the power rule, the derivative is ( (-4)(-2)(x - 2)^{-3} ), which simplifies to ( 8(x - 2)^{-3} ) or ( frac{8}{(x - 2)^3} ).Therefore, the second derivative is:[f''(x) = frac{8}{(x - 2)^3}]Now, let's evaluate the second derivative at the critical point ( x = 4 ):[f''(4) = frac{8}{(4 - 2)^3} = frac{8}{2^3} = frac{8}{8} = 1]Since ( f''(4) = 1 ) which is positive, this means that the function is concave upwards at ( x = 4 ), so this critical point is a local minimum.Therefore, the function ( f(x) ) has a minimum at ( x = 4 ). To find the minimum value, plug ( x = 4 ) back into the original function:[f(4) = 4 + frac{4}{4 - 2} = 4 + frac{4}{2} = 4 + 2 = 6]So, the minimum value of the function is 6.Wait, just to make sure I didn't make a mistake, let me verify my steps again.1. I found the derivative correctly: ( f'(x) = 1 - frac{4}{(x - 2)^2} ). That seems right.2. Setting the derivative equal to zero: ( 1 - frac{4}{(x - 2)^2} = 0 ) leads to ( (x - 2)^2 = 4 ), so ( x - 2 = pm 2 ). Since ( x > 2 ), only ( x = 4 ) is valid. That makes sense.3. Calculating the second derivative: ( f''(x) = frac{8}{(x - 2)^3} ). At ( x = 4 ), that's ( 8 / (2)^3 = 1 ), which is positive, so it's a minimum. Correct.4. Plugging ( x = 4 ) back into ( f(x) ): ( 4 + 4/(4 - 2) = 4 + 2 = 6 ). Yep, that's correct.Alternatively, I remember that sometimes for functions of the form ( x + frac{a}{x - b} ), you can use the AM-GM inequality to find the minimum. Let me try that approach to double-check.The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. So, if I can express the function in a way that allows me to apply AM-GM, that might work.Looking at ( f(x) = x + frac{4}{x - 2} ), let me set ( y = x - 2 ). Since ( x > 2 ), ( y > 0 ). Then, ( x = y + 2 ), so substituting into the function:[f(x) = (y + 2) + frac{4}{y} = y + 2 + frac{4}{y}]So, now the function is ( f(y) = y + frac{4}{y} + 2 ). Now, the terms ( y ) and ( frac{4}{y} ) are both positive since ( y > 0 ). So, applying AM-GM on ( y ) and ( frac{4}{y} ):The arithmetic mean is ( frac{y + frac{4}{y}}{2} ) and the geometric mean is ( sqrt{y cdot frac{4}{y}} = sqrt{4} = 2 ). According to AM-GM:[frac{y + frac{4}{y}}{2} geq 2]Multiplying both sides by 2:[y + frac{4}{y} geq 4]Therefore, ( f(y) = y + frac{4}{y} + 2 geq 4 + 2 = 6 ). So, the minimum value of ( f(y) ) is 6, achieved when ( y = frac{4}{y} ), which implies ( y^2 = 4 ), so ( y = 2 ) (since ( y > 0 )). Then, ( x = y + 2 = 2 + 2 = 4 ). So, this confirms that the minimum value is indeed 6 at ( x = 4 ).That's reassuring. Both calculus and AM-GM give the same result, so I feel confident that the minimum value is 6.Just to be thorough, let me also check the behavior of the function as ( x ) approaches 2 from the right and as ( x ) approaches infinity.As ( x ) approaches 2 from the right, ( x - 2 ) approaches 0, so ( frac{4}{x - 2} ) approaches positive infinity. Therefore, ( f(x) ) approaches infinity.As ( x ) approaches infinity, the term ( x ) dominates, so ( f(x) ) also approaches infinity.Therefore, the function has a minimum somewhere in between, which we found at ( x = 4 ), and the value is 6. So, there are no other minima or maxima in the domain ( x > 2 ), so this must be the global minimum.I think that covers all the bases. I used calculus to find the critical point and confirmed it's a minimum with the second derivative. Then, I used the AM-GM inequality as an alternative method to verify the result. I also checked the behavior of the function at the boundaries of the domain, which showed that the function tends to infinity on both ends, so the critical point must indeed be the global minimum.Therefore, I can confidently say that the minimum value of ( x + frac{4}{x - 2} ) for ( x > 2 ) is 6.**Final Answer**The minimum value is boxed{6}.

answer：Okay, so I need to figure out how many positive divisors of 18,800 are divisible by 235. Hmm, let me start by understanding the problem step by step.First, I know that to find the number of divisors of a number that are divisible by another number, I can use prime factorization. That makes sense because prime factors help break down the number into its building blocks, which can then be used to determine all possible divisors.So, let me begin by finding the prime factorization of both 18,800 and 235. That should give me a solid foundation to work with.Starting with 18,800. I need to break this down into its prime factors. Let me recall that 18,800 is 188 multiplied by 100. I know that 100 is 10 squared, which is 2^2 * 5^2. So, 18,800 is 188 * 2^2 * 5^2.Now, let's factor 188. 188 divided by 2 is 94, and 94 divided by 2 is 47. Wait, 47 is a prime number, right? So, 188 is 2^2 * 47^1. Therefore, putting it all together, 18,800 is 2^2 * 47^1 * 2^2 * 5^2. Wait, hold on, that doesn't seem right. Let me recast that.Wait, 18,800 is 188 * 100, which is (2^2 * 47) * (2^2 * 5^2). So when multiplying these together, I can combine like terms. So, 2^2 * 2^2 is 2^(2+2) = 2^4. Then, 5^2 remains, and 47^1 remains. So, the prime factorization of 18,800 is 2^4 * 5^2 * 47^1. Let me double-check that: 2^4 is 16, 5^2 is 25, 47 is 47. Multiplying 16 * 25 gives 400, and 400 * 47 is 18,800. Yep, that's correct.Now, moving on to 235. Let me factor that as well. 235 divided by 5 is 47, right? Because 5 times 47 is 235. And 47 is a prime number. So, 235 is 5^1 * 47^1. Got it.So, now I have both numbers broken down into their prime factors:- 18,800 = 2^4 * 5^2 * 47^1- 235 = 5^1 * 47^1The next step is to find the number of positive divisors of 18,800 that are divisible by 235. From what I remember, if a number D is a divisor of N and D is divisible by another number M, then D must be a multiple of M and also a divisor of N. So, essentially, we're looking for divisors of 18,800 that are multiples of 235.To find such divisors, I can express D as 235 multiplied by some integer k, where k must be such that D divides 18,800. So, D = 235 * k, and D | 18,800. Therefore, 235 * k | 18,800, which implies that k | (18,800 / 235). So, if I compute 18,800 divided by 235, I can find the number of divisors of that result, which will give me the number of such k's, and thus the number of D's.Let me compute 18,800 divided by 235. Let's see, 235 times 80 is 18,800 because 235 * 80 = (200 * 80) + (35 * 80) = 16,000 + 2,800 = 18,800. So, 18,800 / 235 = 80.Therefore, k must be a divisor of 80. So, the number of positive divisors of 18,800 that are divisible by 235 is equal to the number of positive divisors of 80.Now, I need to find how many positive divisors 80 has. To do this, I can perform the prime factorization of 80 and then use the formula for the number of divisors.Let's factor 80. 80 divided by 2 is 40, divided by 2 is 20, divided by 2 is 10, divided by 2 is 5, and 5 is prime. So, 80 is 2^4 * 5^1.The formula for the number of divisors is to take the exponents of the prime factors, add 1 to each, and multiply them together. So, for 80, which is 2^4 * 5^1, the number of divisors is (4 + 1) * (1 + 1) = 5 * 2 = 10.Therefore, 80 has 10 positive divisors, which means there are 10 positive divisors of 18,800 that are divisible by 235.Wait, let me just recap to make sure I didn't skip any steps or make a mistake. So, we have 18,800, which factors into 2^4 * 5^2 * 47^1. 235 factors into 5^1 * 47^1. So, any divisor of 18,800 that is divisible by 235 must include at least 5^1 and 47^1. So, when we factor out 235 from 18,800, we get 80, which is 2^4 * 5^1. So, the remaining part is 2^4 * 5^1, and the number of divisors of this remaining part is (4 + 1)*(1 + 1) = 5*2=10. That seems correct.Alternatively, another way to think about it is that if D is a divisor of 18,800 and divisible by 235, then D can be written as 235 * k, where k is a divisor of 18,800 / 235 = 80. So, the number of such D's is equal to the number of divisors of 80, which is 10. So, that confirms it.Just to make sure, let me list the divisors of 80 and see if that makes sense. The divisors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. That's 10 divisors. So, if I multiply each of these by 235, I should get the divisors of 18,800 that are divisible by 235.Let me test a couple of them. For example, 1 * 235 = 235. Is 235 a divisor of 18,800? Yes, because 18,800 divided by 235 is 80, which is an integer. Similarly, 2 * 235 = 470. Is 470 a divisor of 18,800? Let's check: 18,800 / 470 = 40, which is an integer. Another one: 5 * 235 = 1,175. 18,800 / 1,175 = 16, which is an integer. And 80 * 235 = 18,800, which is obviously a divisor. So, all these seem to check out.Therefore, I can be confident that the number of positive divisors of 18,800 that are divisible by 235 is 10.**Final Answer**The number of positive divisors of 18800 that are divisible by 235 is boxed{10}.

answer：Okay, so I have this problem where I'm given that (tan x = 3 tan y) and both (x) and (y) are between 0 and (frac{pi}{2}), with (y) less than or equal to (x). I need to find the maximum value of the function (u = x - y). Hmm, okay, let's break this down step by step.First, I know that both (x) and (y) are in the first quadrant since they're between 0 and (frac{pi}{2}). That means all the trigonometric functions for these angles will be positive, which is helpful. The equation given is (tan x = 3 tan y). So, I can write this as:[tan x = 3 tan y]I need to express (u = x - y) in terms of a single variable so that I can find its maximum. Maybe I can express (x) in terms of (y) or vice versa. Let me try expressing (x) in terms of (y). Since (tan x = 3 tan y), I can take the arctangent of both sides:[x = arctan(3 tan y)]So, substituting this into (u), we get:[u = arctan(3 tan y) - y]Now, I need to find the maximum value of (u) with respect to (y). To do this, I can take the derivative of (u) with respect to (y) and set it equal to zero to find critical points. Let's compute the derivative (du/dy).First, let me recall the derivative of (arctan(z)) with respect to (z) is (frac{1}{1 + z^2}). So, using the chain rule, the derivative of (arctan(3 tan y)) with respect to (y) is:[frac{d}{dy} arctan(3 tan y) = frac{1}{1 + (3 tan y)^2} cdot 3 sec^2 y]Simplifying that:[frac{3 sec^2 y}{1 + 9 tan^2 y}]Now, the derivative of (-y) with respect to (y) is just (-1). So, putting it all together, the derivative (du/dy) is:[frac{3 sec^2 y}{1 + 9 tan^2 y} - 1]I need to set this equal to zero to find critical points:[frac{3 sec^2 y}{1 + 9 tan^2 y} - 1 = 0]Let me rewrite (sec^2 y) in terms of (tan y). I remember that (sec^2 y = 1 + tan^2 y). So substituting that in:[frac{3(1 + tan^2 y)}{1 + 9 tan^2 y} - 1 = 0]Let me denote (t = tan y) to make the equation simpler. Then, the equation becomes:[frac{3(1 + t^2)}{1 + 9 t^2} - 1 = 0]Simplify this:[frac{3 + 3 t^2}{1 + 9 t^2} - 1 = 0]Combine the terms:[frac{3 + 3 t^2 - (1 + 9 t^2)}{1 + 9 t^2} = 0]Simplify the numerator:[3 + 3 t^2 - 1 - 9 t^2 = (3 - 1) + (3 t^2 - 9 t^2) = 2 - 6 t^2]So, the equation becomes:[frac{2 - 6 t^2}{1 + 9 t^2} = 0]A fraction is zero when the numerator is zero (provided the denominator isn't zero, which it isn't here since (1 + 9 t^2) is always positive). So:[2 - 6 t^2 = 0]Solving for (t^2):[6 t^2 = 2 t^2 = frac{2}{6} = frac{1}{3} t = pm frac{sqrt{3}}{3}]But since (y) is between 0 and (frac{pi}{2}), (tan y) is positive, so (t = frac{sqrt{3}}{3}).So, (t = frac{sqrt{3}}{3}), which means:[tan y = frac{sqrt{3}}{3}]I know that (tan frac{pi}{6} = frac{sqrt{3}}{3}), so (y = frac{pi}{6}).Now, let's find (x). From the original equation:[tan x = 3 tan y = 3 cdot frac{sqrt{3}}{3} = sqrt{3}]So, (tan x = sqrt{3}), which means (x = frac{pi}{3}), since (tan frac{pi}{3} = sqrt{3}).Therefore, the critical point occurs at (y = frac{pi}{6}) and (x = frac{pi}{3}). Now, let's compute (u = x - y):[u = frac{pi}{3} - frac{pi}{6} = frac{pi}{6}]But before concluding that this is the maximum, I should check the endpoints of the interval to ensure that this critical point indeed gives the maximum value.The interval for (y) is from 0 to (x), but since (x) is also dependent on (y), we need to consider the behavior as (y) approaches 0 and as (y) approaches (x). However, since (x) is determined by (y), we can think of (y) approaching 0 and (y) approaching (frac{pi}{2}), but since (x < frac{pi}{2}), (y) can't approach (frac{pi}{2}) because (x) would have to be greater than (y), but (x) is less than (frac{pi}{2}).Wait, actually, let's think about the domain. Since (0 leq y leq x < frac{pi}{2}), as (y) approaches (frac{pi}{2}), (x) would have to approach (frac{pi}{2}) as well, but let's see what happens to (u = x - y) in that case.When (y) approaches 0, (x) is determined by (tan x = 3 tan y). As (y) approaches 0, (tan y approx y), so (tan x approx 3 y), which would make (x approx 3 y). Therefore, (u = x - y approx 3 y - y = 2 y), which approaches 0 as (y) approaches 0.When (y) approaches (frac{pi}{2}), (tan y) approaches infinity, so (tan x = 3 tan y) would also approach infinity, meaning (x) approaches (frac{pi}{2}). Therefore, (u = x - y) approaches (frac{pi}{2} - frac{pi}{2} = 0). So, at both endpoints, (u) approaches 0.Therefore, the maximum must occur at the critical point we found earlier, which is (u = frac{pi}{6}).But just to be thorough, let's check another point. Suppose (y = frac{pi}{4}). Then, (tan y = 1), so (tan x = 3), which means (x = arctan 3). Calculating (u = arctan 3 - frac{pi}{4}). Let me approximate this:(arctan 3) is approximately 1.249 radians, and (frac{pi}{4}) is approximately 0.785 radians. So, (u approx 1.249 - 0.785 = 0.464) radians.Compare this to (frac{pi}{6}), which is approximately 0.523 radians. So, (u) is indeed larger at the critical point.Another test: let me take (y = frac{pi}{12}), which is 15 degrees. Then, (tan y = tan frac{pi}{12} approx 0.2679). So, (tan x = 3 times 0.2679 approx 0.8037), so (x approx arctan(0.8037) approx 0.674 radians). Then, (u = 0.674 - frac{pi}{12} approx 0.674 - 0.2618 approx 0.412 radians), which is still less than (frac{pi}{6}).So, it seems consistent that the maximum occurs at (u = frac{pi}{6}).Therefore, the maximum value of (u = x - y) is (frac{pi}{6}).**Final Answer**The maximum value of (u = x - y) is boxed{dfrac{pi}{6}}.

answer：Okay, so I have this geometry problem here. It says that in triangle PQR, angle PQR is equal to angle PRQ. Hmm, okay, so that means two angles in the triangle are equal. Let me try to visualize the triangle. There's an Asymptote diagram provided, but since I can't see images, I'll just imagine it based on the description.Alright, so triangle PQR has points Q, R, and P. The sides QR and PR are given as 5 and 7 units, respectively. The angles at Q and R are equal. Wait, angle PQR is at point Q, and angle PRQ is at point R. So, if those two angles are equal, that means triangle PQR is isosceles with two equal sides opposite those equal angles.In an isosceles triangle, the sides opposite the equal angles are equal in length. So, if angles at Q and R are equal, then the sides opposite them should be equal. Let me recall: in triangle PQR, side opposite angle Q is PR, and the side opposite angle R is PQ. So, if angles Q and R are equal, then sides PR and PQ must be equal.Wait, hold on. PR is given as 7. So, if PR equals PQ, then PQ is also 7. So, that would make two sides of the triangle equal: PQ = PR = 7. Then, the third side is QR, which is given as 5.But wait, let me double-check that. If angle Q equals angle R, then sides opposite them, which are PR and PQ, should be equal. So, yes, PQ = PR = 7. Then, QR is 5. So, the sides of the triangle are 7, 7, and 5.Therefore, the perimeter would be the sum of all sides: 7 + 7 + 5. Let me compute that: 7 + 7 is 14, plus 5 is 19. So, the perimeter should be 19.Wait, but let me make sure I didn't make a mistake here. Maybe I got the sides mixed up. Let me label the triangle properly. Point Q is connected to R and P. So, side QR is 5, and side PR is 7. So, side PQ is opposite angle R, and side PR is opposite angle Q. Since angles Q and R are equal, their opposite sides should be equal. So, side PQ equals side PR. But side PR is given as 7, so PQ is also 7.Therefore, sides PQ = 7, PR = 7, and QR = 5. So, perimeter is 7 + 7 + 5 = 19. That seems straightforward.But wait, let me think again. Sometimes, in these problems, it's easy to mix up which sides are opposite which angles. Let me draw a rough sketch on paper to visualize.So, triangle PQR with points Q, R, and P. QR is the base, which is 5 units. PR is 7 units, which connects point P to R. PQ connects point P to Q, which is also 7 units because angles at Q and R are equal. So, yeah, that makes sense.Alternatively, maybe I can use the Law of Sines to confirm this. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. So, if angles at Q and R are equal, then their opposite sides should be equal.So, let's denote angle at Q as angle Q and angle at R as angle R. Given that angle Q = angle R, then side opposite angle Q is PR, which is 7, and side opposite angle R is PQ. Therefore, PR = PQ, so PQ = 7.Thus, sides PQ = 7, PR = 7, QR = 5. Perimeter is 7 + 7 + 5 = 19.Alternatively, maybe I can use the Law of Cosines to find another side, but since we already have two sides equal, it's straightforward.Wait, but hold on. Let me think if there's another way this could be interpreted. Maybe the triangle is labeled differently? For example, sometimes diagrams can be misleading or labels can be confusing.Looking back at the Asymptote code, it draws the triangle from (0,0) to (2.5,7.43) to (5,0) and back. So, point Q is at (0,0), R is at (5,0), and P is at (2.5,7.43). So, QR is the base from (0,0) to (5,0), which is 5 units. PR is from (5,0) to (2.5,7.43), which is given as 7 units. PQ is from (0,0) to (2.5,7.43), which should also be 7 units because angles at Q and R are equal.So, in this coordinate system, PQ is from (0,0) to (2.5,7.43). Let me compute the distance between (0,0) and (2.5,7.43) to verify if it's 7.Using the distance formula: sqrt[(2.5 - 0)^2 + (7.43 - 0)^2] = sqrt[(6.25) + (55.2049)] = sqrt[61.4549] ≈ 7.84. Hmm, that's approximately 7.84, which is not 7. So, that's confusing.Wait, maybe I made a mistake here. The Asymptote code labels "5" at (2.5,0), which is the midpoint of QR, so that makes sense because QR is 5 units. Then, it labels "7" at (4.2,3.7), which is somewhere along PR. So, maybe the length PR is 7, but PQ is not necessarily 7.Wait, hold on, this contradicts my earlier conclusion. So, if in the coordinate system, PQ is approximately 7.84, which is not equal to PR, which is 7, then my earlier assumption that PQ = PR might be wrong.Hmm, so perhaps I need to re-examine the problem. The problem states that angle PQR = angle PRQ. So, angle at Q is equal to angle at R. So, in triangle PQR, angle Q = angle R. Therefore, sides opposite these angles should be equal.In triangle PQR, side opposite angle Q is PR, and side opposite angle R is PQ. So, if angle Q = angle R, then PR = PQ. But in the coordinate system, PR is 7, but PQ is approximately 7.84. So, that seems contradictory.Wait, maybe the Asymptote code is just a rough drawing, not to scale. So, perhaps in reality, PQ should be equal to PR, both being 7, making the triangle isosceles with sides 7,7,5. But in the drawing, it's not precise. So, maybe I shouldn't rely on the coordinates but just on the given information.So, the problem says angle PQR = angle PRQ. So, angle at Q equals angle at R. Therefore, sides opposite those angles, which are PR and PQ, must be equal. So, PR = PQ. Given that PR is 7, so PQ is also 7. Then, QR is 5. So, the sides are 7,7,5, perimeter is 19.But then why does the Asymptote code show PQ as approximately 7.84? Maybe it's just an illustrative diagram, not to scale. So, perhaps I shouldn't worry about that.Alternatively, maybe I can compute the length PQ using coordinates to see if it's actually 7. Let me compute the distance between (0,0) and (2.5,7.43).So, distance formula: sqrt[(2.5)^2 + (7.43)^2] = sqrt[6.25 + 55.2049] = sqrt[61.4549] ≈ 7.84. So, that's approximately 7.84, not 7. So, that suggests that in the diagram, PQ is longer than PR, which is 7.But according to the problem, angle Q = angle R, so sides PR and PQ should be equal. So, this seems contradictory.Wait, perhaps the Asymptote code is incorrect? Or maybe I misread the problem. Let me check again.The problem says: "In the diagram, angle PQR = angle PRQ. If QR=5 and PR=7, what is the perimeter of triangle PQR?"So, angle PQR is at point Q, between points P, Q, R. Angle PRQ is at point R, between points P, R, Q. So, those two angles are equal. Therefore, sides opposite them, which are PR and PQ, should be equal. So, PR = PQ.Given that PR is 7, so PQ is 7. QR is 5. So, perimeter is 7 + 7 + 5 = 19.But in the Asymptote code, the coordinates are given as (0,0), (2.5,7.43), (5,0). So, if I compute the lengths:QR: distance from (0,0) to (5,0) is 5, which matches.PR: distance from (5,0) to (2.5,7.43): sqrt[(5 - 2.5)^2 + (0 - 7.43)^2] = sqrt[(2.5)^2 + (7.43)^2] = sqrt[6.25 + 55.2049] ≈ sqrt[61.4549] ≈ 7.84, which is approximately 7.84, not 7.Wait, so in the Asymptote code, PR is approximately 7.84, but the problem states PR is 7. So, that suggests that the Asymptote diagram is not to scale, or perhaps there's a mistake in the coordinates.Alternatively, maybe I need to calculate the correct coordinates based on the given side lengths.Wait, perhaps I can use coordinates to solve the problem. Let me try that.Let me place point Q at (0,0) and point R at (5,0), since QR is 5 units. Then, point P is somewhere above the x-axis. Let me denote point P as (x,y). Then, the distance from P to R is 7, so:sqrt[(x - 5)^2 + (y - 0)^2] = 7So, (x - 5)^2 + y^2 = 49.Also, since angle PQR = angle PRQ, which are angles at Q and R. So, in triangle PQR, angles at Q and R are equal, so sides opposite them are equal. So, side PR = side PQ.Wait, but side PR is given as 7, so side PQ should also be 7. So, distance from P to Q is 7:sqrt[(x - 0)^2 + (y - 0)^2] = 7So, x^2 + y^2 = 49.So, now I have two equations:1. (x - 5)^2 + y^2 = 492. x^2 + y^2 = 49Subtracting equation 2 from equation 1:(x - 5)^2 + y^2 - (x^2 + y^2) = 49 - 49Expanding (x - 5)^2: x^2 -10x +25 + y^2 - x^2 - y^2 = 0Simplify: -10x +25 = 0So, -10x = -25 => x = 2.5So, x is 2.5. Then, plug back into equation 2: (2.5)^2 + y^2 = 49So, 6.25 + y^2 = 49 => y^2 = 42.75 => y = sqrt(42.75) ≈ 6.545Wait, but in the Asymptote code, point P is at (2.5,7.43). So, according to my calculation, y should be approximately 6.545, but in the diagram, it's 7.43. So, that's a discrepancy.Hmm, so perhaps the Asymptote code is just an approximate drawing, not to scale. So, in reality, if we have triangle PQR with QR=5, PR=7, and angles at Q and R equal, then point P should be at (2.5, approximately 6.545), not (2.5,7.43). So, the diagram might be slightly off.But regardless, the problem gives us QR=5 and PR=7, and tells us angles at Q and R are equal. So, from that, we can conclude that sides PQ and PR are equal, so PQ=7, making the perimeter 7+7+5=19.Alternatively, maybe I can use coordinates to find the lengths and confirm.Wait, if I use the coordinates from the Asymptote code, point P is at (2.5,7.43). So, distance from P to Q is sqrt[(2.5)^2 + (7.43)^2] ≈ sqrt[6.25 + 55.2049] ≈ sqrt[61.4549] ≈ 7.84, as before. Distance from P to R is sqrt[(2.5)^2 + (7.43)^2] as well? Wait, no.Wait, distance from P to R is sqrt[(5 - 2.5)^2 + (0 - 7.43)^2] = sqrt[(2.5)^2 + (7.43)^2] ≈ sqrt[6.25 + 55.2049] ≈ sqrt[61.4549] ≈ 7.84 as well. Wait, so in the Asymptote code, both PQ and PR are approximately 7.84, but the problem states PR=7. So, that suggests that the Asymptote code is incorrect, or the labels are wrong.Alternatively, maybe I misread the labels. Let me check the Asymptote code again:draw((0,0)--(2.5,7.43)--(5,0)--cycle);label("5",(2.5,0),S);label("Q",(0,0),SW);label("R",(5,0),SE);label("P",(2.5,7.43),N);label("7",(4.2,3.7));So, the label "5" is at (2.5,0), which is the midpoint of QR, so that makes sense for the length QR=5. The label "7" is at (4.2,3.7), which is somewhere along PR. So, that suggests that PR is 7 units. So, in the Asymptote code, PR is from (5,0) to (2.5,7.43), which we calculated as approximately 7.84, but the label says 7. So, that's inconsistent.Therefore, the Asymptote code is not accurate. So, perhaps I should ignore the coordinates and just go by the given information.Given that, in triangle PQR, angle Q = angle R, so sides opposite are equal: PR = PQ. Given PR=7, so PQ=7. QR=5. Therefore, perimeter is 7+7+5=19.Alternatively, maybe I can use the Law of Cosines to find the other sides.Wait, if I consider triangle PQR with QR=5, PR=7, and angle Q = angle R.Let me denote angle Q = angle R = θ. Then, angle P can be found since the sum of angles in a triangle is 180 degrees. So, angle P = 180 - 2θ.Using the Law of Sines:PR / sin(angle Q) = QR / sin(angle P) = PQ / sin(angle R)But since angle Q = angle R = θ, and PR = PQ = 7, QR =5.So, PR / sin(θ) = QR / sin(180 - 2θ)But sin(180 - 2θ) = sin(2θ). So,7 / sin(θ) = 5 / sin(2θ)But sin(2θ) = 2 sinθ cosθ. So,7 / sinθ = 5 / (2 sinθ cosθ)Multiply both sides by sinθ:7 = 5 / (2 cosθ)So, 7 = 5 / (2 cosθ)Then, 2 cosθ = 5 /7So, cosθ = 5 /14Therefore, θ = arccos(5/14)So, angle Q = angle R = arccos(5/14). Then, angle P = 180 - 2θ.But I don't think I need to compute the angles; I just need to confirm that sides PQ and PR are equal, which they are, both being 7, so perimeter is 19.Alternatively, maybe I can use the Law of Cosines to find side PQ.Wait, if I consider triangle PQR, with sides QR=5, PR=7, and PQ=7. So, sides are 5,7,7. So, perimeter is 19.Alternatively, if I didn't know that, I could use the Law of Cosines to find the third side.Wait, but in this case, since two sides are equal, it's straightforward.Wait, maybe I can compute the length of PQ using the coordinates, but since the Asymptote code is inconsistent with the given lengths, it's better to ignore it.So, in conclusion, since angles at Q and R are equal, sides PR and PQ are equal, both being 7. QR is 5. So, perimeter is 7+7+5=19.Therefore, the perimeter of triangle PQR is 19.**Final Answer**The perimeter of triangle PQR is boxed{19}.

answer：To determine the 50th number in the series, I start by identifying the first term and the common difference. The first term is -48, and each subsequent term increases by 2.Next, I use the formula for the nth term of an arithmetic sequence: aₙ = a₁ + (n - 1)d. Plugging in the values, aₙ = -48 + (50 - 1) * 2.Calculating this, I find that the 50th term is 52.

answer：First, I recognize that converting a number from base 9 to base 3 can be simplified by converting each base 9 digit to its equivalent in base 3, since 9 is a power of 3.Next, I break down the number (813_9) into its individual digits: 8, 1, and 3.I then convert each of these digits to base 3:- 8 in base 9 is equal to 22 in base 3.- 1 in base 9 is equal to 01 in base 3.- 3 in base 9 is equal to 10 in base 3.Finally, I concatenate these base 3 equivalents to get the final result.