💡Okay, so Piravena needs to travel from A to B, then B to C, and finally C back to A. Each leg of the trip can be done entirely by bus or entirely by airplane. The cities form a right-angled triangle, with C being 3000 km from A, and B being 3250 km from A. I need to figure out the cheapest way for her to make this trip.First, I should probably figure out the distances between all the cities. I know AC is 3000 km and AB is 3250 km. Since it's a right-angled triangle, I can use the Pythagorean theorem to find the distance from B to C. Let me write that down.So, in a right-angled triangle, the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). Here, AB is the hypotenuse because it's the longest distance given. So, AB² = AC² + BC².Plugging in the numbers: 3250² = 3000² + BC².Let me calculate 3250 squared. Hmm, 3250 times 3250. That's a big number. Maybe I can break it down. 3250 is 3000 + 250, so (3000 + 250)² = 3000² + 2*3000*250 + 250². That would be 9,000,000 + 1,500,000 + 62,500 = 10,562,500.Similarly, 3000 squared is 9,000,000.So, BC² = AB² - AC² = 10,562,500 - 9,000,000 = 1,562,500.Taking the square root of 1,562,500 gives BC. Let me see, 1250 squared is 1,562,500 because 1000² is 1,000,000 and 250² is 62,500, and 2*1000*250 is 500,000, so adding them up gives 1,562,500. So BC is 1250 km.Alright, now I know all the distances: AB is 3250 km, BC is 1250 km, and AC is 3000 km.Next, I need to figure out the cost for each leg of the trip by both bus and airplane and then choose the cheaper option for each.The cost by bus is 0.15 per kilometer. The cost by airplane is a 100 booking fee plus 0.10 per kilometer.Let me calculate the cost for each leg.Starting with A to B, which is 3250 km.By bus: 3250 km * 0.15/km = 487.50.By airplane: 100 + (3250 km * 0.10/km) = 100 + 325 = 425.So, airplane is cheaper for A to B.Next, B to C, which is 1250 km.By bus: 1250 km * 0.15/km = 187.50.By airplane: 100 + (1250 km * 0.10/km) = 100 + 125 = 225.Here, bus is cheaper for B to C.Finally, C to A, which is 3000 km.By bus: 3000 km * 0.15/km = 450.By airplane: 100 + (3000 km * 0.10/km) = 100 + 300 = 400.Again, airplane is cheaper for C to A.So, summarizing:- A to B: Airplane at 425- B to C: Bus at 187.50- C to A: Airplane at 400Now, adding these up: 425 + 187.50 + 400.Let me calculate that step by step.First, 425 + 187.50 = 612.50.Then, 612.50 + 400 = 1012.50.So, the total cost would be 1012.50.Wait, let me double-check my calculations to make sure I didn't make a mistake.For A to B:- Bus: 3250 * 0.15 = 487.50- Airplane: 100 + (3250 * 0.10) = 100 + 325 = 425- Correct, airplane is cheaper.For B to C:- Bus: 1250 * 0.15 = 187.50- Airplane: 100 + (1250 * 0.10) = 100 + 125 = 225- Correct, bus is cheaper.For C to A:- Bus: 3000 * 0.15 = 450- Airplane: 100 + (3000 * 0.10) = 100 + 300 = 400- Correct, airplane is cheaper.Adding the cheapest options: 425 + 187.50 + 400.425 + 187.50 = 612.50612.50 + 400 = 1012.50Yes, that seems correct.I think I've considered all the legs and chosen the cheapest option for each. So, the total cost should be 1012.50.

💡Alright, so I've got this problem here where I need to insert words that can be preceded by certain letters on the left. The first three words should have four letters each, the next two should have five letters, and the last two should have six letters. The letters provided are B, M, Γ, ЯГ, ШК, and БОР. First, I need to understand what it means for a word to be "preceded" by these letters. I think it means that these letters are the starting letters of the words I need to find. So, for example, if I have the letter B, I need to find a four-letter word that starts with B. Similarly, for Γ, I need to find a four-letter word that starts with Γ, and so on.Okay, let's start with the first letter, B. I need a four-letter word that starts with B. In English, that could be "ball," "book," "back," etc. But since the example uses Russian words, maybe I should stick to Russian. In Russian, B is "В," so I need a four-letter Russian word starting with В. "Врач" means "doctor," which is a four-letter word starting with В. That seems good.Next is M. In English, M could be "moon," "make," etc., but again, in Russian, M is "М." So I need a four-letter Russian word starting with М. "Мама" means "mother," which fits perfectly.Now, Γ. Γ is the Greek letter Gamma. In Russian, Γ is pronounced like "G." So I need a four-letter Russian word starting with G. "Гора" means "mountain," which is a four-letter word starting with Г. That works.Next up is ЯГ. This is a bit tricky because it's two letters combined. In Russian, Я is "ya" and Г is "g." So I need a five-letter Russian word starting with ЯГ. "Ягода" means "berry," which is a five-letter word starting with ЯГ. Perfect.Then we have ШК. Ш is "sh" and К is "k" in Russian. So I need a five-letter Russian word starting with ШК. "Школа" means "school," which is a five-letter word starting with ШК. That fits.Finally, we have БОР. Б is "b," О is "o," and Р is "r" in Russian. So I need a six-letter Russian word starting with БОР. "Борода" means "beard," which is a six-letter word starting with БОР. That works as well.So, putting it all together, the words I've found are:- B: Врач- M: Мама- Γ: Гора- ЯГ: Ягода- ШК: Школа- БОР: БородаI think that covers all the requirements. Each word starts with the given letters and has the correct number of letters. I hope I didn't make any mistakes with the translations or the letter counts. It seems to fit the problem's instructions.

💡First, I need to determine the total number of possible words with up to 5 letters using the standard English alphabet. Since each position in the word can be any of the 26 letters, the total number of words for each length is calculated as:- One-letter words: (26)- Two-letter words: (26^2 = 676)- Three-letter words: (26^3 = 17,576)- Four-letter words: (26^4 = 456,976)- Five-letter words: (26^5 = 11,881,376)Next, I calculate the number of words that do not contain the letter 'A'. For each word length, this is the total number of words minus the number of words that include 'A' in at least one position. The number of words without 'A' for each length is:- One-letter words without 'A': (25)- Two-letter words without 'A': (25^2 = 625)- Three-letter words without 'A': (25^3 = 15,625)- Four-letter words without 'A': (25^4 = 390,625)- Five-letter words without 'A': (25^5 = 9,765,625)Finally, to find the number of words that contain the letter 'A' at least once, I subtract the number of words without 'A' from the total number of words for each length and then sum these results:- One-letter words with 'A': (26 - 25 = 1)- Two-letter words with 'A': (676 - 625 = 51)- Three-letter words with 'A': (17,576 - 15,625 = 1,951)- Four-letter words with 'A': (456,976 - 390,625 = 66,351)- Five-letter words with 'A': (11,881,376 - 9,765,625 = 2,115,751)Adding these together gives the total number of words that contain the letter 'A' at least once:(1 + 51 + 1,951 + 66,351 + 2,115,751 = 2,184,105)

💡First, I need to determine the total distance the train travels while passing the platform. This is the sum of the train's length and the platform's length.Next, I'll calculate the speed of the train in meters per second by dividing the total distance by the time it takes to pass the platform.Finally, I'll convert the speed from meters per second to kilometers per hour using the appropriate conversion factor.

💡Okay, so I have this problem where z is a complex number, and it's given that z plus 1 over z equals 2 times the cosine of 5 degrees. I need to find z to the power of 1000 plus 1 over z to the power of 1000. Hmm, that seems a bit intimidating at first because of the high exponent, but maybe I can break it down step by step.First, I remember that for complex numbers, expressions like z + 1/z often relate to Euler's formula or something with roots of unity. Since it's given as 2 cos 5 degrees, that makes me think of Euler's formula, which states that e^(iθ) = cos θ + i sin θ. So, if I let z be e^(iθ), then 1/z would be e^(-iθ), and adding them together would give 2 cos θ. That seems to fit perfectly here because the given equation is z + 1/z = 2 cos 5 degrees. So, maybe z is e^(i5°) or e^(-i5°). That would make sense because both would satisfy the equation.So, if z is e^(i5°), then z^1000 would be e^(i5°*1000) = e^(i5000°). Similarly, 1/z^1000 would be e^(-i5000°). So, adding them together would give 2 cos 5000°, right? But 5000 degrees is a huge angle. I need to simplify that angle to something within 0 to 360 degrees because trigonometric functions are periodic with a period of 360 degrees.To simplify 5000 degrees, I can divide it by 360 and find the remainder. Let me do that. 5000 divided by 360. Let's see, 360 times 13 is 4680, and 5000 minus 4680 is 320. So, 5000 degrees is equivalent to 320 degrees in standard position. Therefore, cos 5000° is the same as cos 320°, and sin 5000° is the same as sin 320°. So, z^1000 + 1/z^1000 would be 2 cos 320°.Wait, but 320 degrees is in the fourth quadrant, where cosine is positive and sine is negative. But I'm just dealing with cosine here, so 2 cos 320° is a positive value. However, I remember that cos(360° - θ) = cos θ, so cos 320° is the same as cos(40°), because 360° - 320° = 40°. So, cos 320° = cos 40°, which means 2 cos 320° = 2 cos 40°. Therefore, z^1000 + 1/z^1000 equals 2 cos 40°.But wait a second, is that correct? Let me double-check. If z is e^(i5°), then z^1000 is e^(i5000°), which is e^(i(360°*13 + 320°)), which simplifies to e^(i320°). Similarly, 1/z^1000 is e^(-i320°). Adding them together gives e^(i320°) + e^(-i320°) = 2 cos 320°, which is indeed 2 cos 40°, since cos 320° = cos(360° - 40°) = cos 40°. So, that seems right.But hold on, what if z is e^(-i5°)? Then z^1000 would be e^(-i5000°), which is e^(-i(360°*13 + 320°)) = e^(-i320°). Similarly, 1/z^1000 would be e^(i320°). Adding them together would still give e^(i320°) + e^(-i320°) = 2 cos 320°, which is 2 cos 40°. So, regardless of whether z is e^(i5°) or e^(-i5°), the result is the same.Wait, but in the original problem, it's given that z + 1/z = 2 cos 5°, which is a positive value. So, does that mean z is on the unit circle? Yes, because if |z| = 1, then 1/z is just the complex conjugate of z, and adding them together gives twice the real part, which is 2 cos θ. So, z must lie on the unit circle, which is why we can represent it as e^(iθ).Another way to think about this is using De Moivre's theorem, which states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). So, if z = cos 5° + i sin 5°, then z^1000 would be cos(5000°) + i sin(5000°), and 1/z^1000 would be cos(5000°) - i sin(5000°). Adding them together cancels out the imaginary parts, leaving 2 cos 5000°, which simplifies to 2 cos 320°, and then to 2 cos 40°, as before.But I want to make sure I didn't make a mistake in simplifying 5000 degrees. Let me do that division again. 5000 divided by 360. 360 times 10 is 3600, subtract that from 5000, we get 1400. 360 times 3 is 1080, subtract that from 1400, we get 320. So, 5000 = 360*13 + 320, so 5000 degrees is 13 full circles plus 320 degrees. So, yes, 5000 degrees is equivalent to 320 degrees.Alternatively, I could have used modulo operation: 5000 mod 360. Let me compute that. 5000 divided by 360 is approximately 13.888..., so the integer part is 13, and 13*360 is 4680. Subtracting that from 5000 gives 320. So, 5000 mod 360 is 320. So, that confirms it.Now, 320 degrees is in the fourth quadrant, as I thought earlier. The reference angle for 320 degrees is 360 - 320 = 40 degrees. So, cos 320° = cos 40°, and sin 320° = -sin 40°. But since we're only dealing with cosine here, it's positive.Wait, but in the original problem, z + 1/z = 2 cos 5°, which is a positive value. So, does that mean z is in the first quadrant? Because if z were in the fourth quadrant, say e^(-i5°), then z + 1/z would still be 2 cos 5°, which is positive. So, both e^(i5°) and e^(-i5°) satisfy the equation z + 1/z = 2 cos 5°, right?Yes, because e^(iθ) + e^(-iθ) = 2 cos θ, regardless of the sign of θ. So, whether z is e^(i5°) or e^(-i5°), the equation holds. Therefore, both possibilities for z lead to the same result when raised to the 1000th power and added to their reciprocals.So, putting it all together, z^1000 + 1/z^1000 = 2 cos 40°. But wait, is that the final answer? Let me check if there's a way to express this differently or if I missed a negative sign somewhere.Wait, when I simplified 5000 degrees, I got 320 degrees, which is in the fourth quadrant, and cos 320° is positive, so 2 cos 320° is positive. But the problem didn't specify whether the answer should be positive or negative. However, since 40 degrees is in the first quadrant, cos 40° is positive, so 2 cos 40° is positive. Therefore, the answer is positive.But just to be thorough, let me consider another approach. Suppose I let z = cos θ + i sin θ, then z + 1/z = 2 cos θ. So, given that z + 1/z = 2 cos 5°, θ must be 5°, so z = cos 5° + i sin 5°. Then, z^n + 1/z^n = 2 cos(nθ). So, for n = 1000, it's 2 cos(1000*5°) = 2 cos(5000°). As before, 5000° mod 360° is 320°, so 2 cos 320° = 2 cos 40°. So, same result.Alternatively, using Euler's formula, z = e^(i5°), so z^1000 = e^(i5000°) = e^(i320°), and 1/z^1000 = e^(-i320°). Adding them gives 2 cos 320°, which is 2 cos 40°. So, consistent.Wait, but sometimes when dealing with exponents, especially with complex numbers, there can be multiple roots or something, but in this case, since z is given to satisfy z + 1/z = 2 cos 5°, it uniquely determines z up to the choice of e^(i5°) or e^(-i5°), but as we saw, both lead to the same result when raised to the 1000th power and added to their reciprocals.Another thing to consider: is 1000*5° equal to 5000°, which is correct. 1000*5 is 5000. So, that's straightforward.Wait, but 5000 degrees is a large angle, but since trigonometric functions are periodic, we can reduce it modulo 360°, which we did, getting 320°, which is correct.Wait, but 320° is in the fourth quadrant, so cos 320° is positive, as I thought earlier. So, 2 cos 320° is positive, which is consistent with the original equation where z + 1/z was positive.But just to make sure, let me compute 5000 divided by 360 again. 360*13 is 4680, subtract that from 5000, we get 320. So, 5000° = 13*360° + 320°, so yes, 320° is the equivalent angle.Alternatively, I could have subtracted 360° repeatedly until I get an angle between 0° and 360°. So, 5000 - 360*13 = 5000 - 4680 = 320°, which is the same as before.Wait, but 320° is greater than 270°, so it's in the fourth quadrant, right? Yes, because 270° is the start of the fourth quadrant, and 320° is between 270° and 360°, so it's in the fourth quadrant.So, in the fourth quadrant, cosine is positive, sine is negative. So, cos 320° is positive, which matches our earlier conclusion.Wait, but I also remember that cos(θ) = cos(-θ), so cos 320° = cos(-320°) = cos(40°), because -320° is equivalent to 40° when considering the unit circle (since -320° + 360° = 40°). So, that's another way to see that cos 320° = cos 40°.Therefore, 2 cos 320° = 2 cos 40°, so z^1000 + 1/z^1000 = 2 cos 40°.But wait, the problem didn't specify whether to leave it in terms of cosine or to express it differently. I think 2 cos 40° is a valid answer, but sometimes problems might expect it in a different form, but in this case, I think 2 cos 40° is the simplest form.Alternatively, if I wanted to express it in terms of sine, since cos 40° = sin 50°, but I don't think that's necessary here. The problem just asks for z^1000 + 1/z^1000, so 2 cos 40° is the answer.Wait, but let me think again. If z is e^(i5°), then z^1000 is e^(i5000°), which is e^(i320°), and 1/z^1000 is e^(-i320°). So, adding them gives 2 cos 320°, which is 2 cos 40°, as we saw. So, that seems consistent.Alternatively, if I use De Moivre's theorem directly on z = cos 5° + i sin 5°, then z^1000 = cos(5000°) + i sin(5000°), which is cos(320°) + i sin(320°), and 1/z^1000 = cos(320°) - i sin(320°). Adding them together gives 2 cos 320°, which is 2 cos 40°.So, all approaches lead to the same result, which is 2 cos 40°. Therefore, I'm confident that the answer is 2 cos 40°, which can be written as 2 cos 40°, or if needed, in a box as boxed{2cos 40^circ}.Wait, but in the original problem, the angle was 5°, and after multiplying by 1000, we got 5000°, which reduced to 320°, which is 40° less than 360°, so cos 320° is cos 40°, which is correct.But just to make sure, let me compute cos 320° numerically and see if it's equal to cos 40°. Let's compute cos 320°. 320° is in the fourth quadrant, so cos 320° = cos(360° - 40°) = cos 40°, which is approximately 0.7660. So, 2 cos 40° is approximately 1.532, which is the same as 2 cos 320°, since cos 320° is also approximately 0.7660.Wait, but let me check with a calculator. Let me compute cos 320°. 320° is 360° - 40°, so cos 320° = cos 40°, which is approximately 0.7660. So, 2 cos 320° is approximately 1.532. Similarly, 2 cos 40° is also approximately 1.532. So, that confirms it numerically as well.Therefore, I'm confident that the answer is 2 cos 40°, which is approximately 1.532, but since the problem doesn't specify a numerical value, the exact form is 2 cos 40°, so I should present that as the answer.Wait, but in the initial problem, the angle was 5°, and after raising to the 1000th power, we got 5000°, which reduced to 320°, which is 40° less than 360°, so cos 320° = cos 40°, which is correct.Another way to think about it is that 5000° divided by 360° is approximately 13.888..., so the fractional part is 0.888..., which multiplied by 360° gives approximately 320°, which is consistent with our earlier calculation.Wait, but 5000 divided by 360 is exactly 13 and 320/360, which simplifies to 13 and 8/9, but that's not necessary here.In any case, all the methods I've tried lead to the same conclusion: z^1000 + 1/z^1000 equals 2 cos 40°, so that's the answer.Wait, but just to make sure, let me think about another approach. Suppose I use the identity that z^n + 1/z^n = 2 cos(nθ), where z = e^(iθ). So, given z + 1/z = 2 cos θ, then z^n + 1/z^n = 2 cos(nθ). So, in this case, θ is 5°, so nθ is 1000*5° = 5000°, which reduces to 320°, so 2 cos 320° = 2 cos 40°, as before.Therefore, using this identity directly gives the same result, so I think that's solid.Wait, but let me consider if there's any possibility that z could be a different root or something. For example, if z satisfies z + 1/z = 2 cos 5°, then z could be e^(i5°) or e^(-i5°), as we saw earlier. But in both cases, raising to the 1000th power would give e^(i5000°) or e^(-i5000°), which both reduce to e^(i320°) or e^(-i320°), and adding them to their reciprocals would still give 2 cos 320°, which is 2 cos 40°. So, regardless of which root we take, the result is the same.Therefore, I think I've covered all the bases here. I've used multiple methods: Euler's formula, De Moivre's theorem, and the identity for z^n + 1/z^n, and all lead to the same conclusion. I've also checked the angle reduction and verified it numerically. So, I'm pretty confident that the answer is 2 cos 40°.Just to recap:1. Given z + 1/z = 2 cos 5°, so z is e^(i5°) or e^(-i5°).2. z^1000 is e^(i5000°) or e^(-i5000°), which reduces to e^(i320°) or e^(-i320°).3. Adding z^1000 + 1/z^1000 gives 2 cos 320°.4. Since cos 320° = cos 40°, the result is 2 cos 40°.Therefore, the final answer is 2 cos 40°, which can be written as boxed{2cos 40^circ}.Wait, but in the initial problem, the user wrote "find z^{1000} + 1/z^{1000}", and in the previous response, the assistant wrote "-2 cos 40°", but in my thinking, I arrived at 2 cos 40°. So, is there a discrepancy here?Wait, let me check that. In the initial problem, the user wrote:"Given that z is a complex number such that z + 1/z = 2 cos 5°, find z^{1000} + 1/z^{1000}."And the previous response was:"Thus, we have boxed{-2cos 40^circ}"Wait, but in my thinking, I arrived at 2 cos 40°, not negative. So, why is there a discrepancy?Wait, perhaps I made a mistake in my thinking. Let me go back.Wait, in the previous response, the assistant wrote:"Recall that cos 140° = -cos 40°, so 2 cos 140° = -2 cos 40°"Wait, but in my thinking, I arrived at 2 cos 40°, but in the assistant's solution, they arrived at -2 cos 40°. So, which one is correct?Wait, let me see. In the assistant's solution, they started with z = cos 5° ± i sin 5°, then z^1000 = cos 5000° ± i sin 5000°, which reduces to cos 140° ± i sin 140°, because 5000 mod 360 is 140°, not 320°. Wait, that's different from what I thought earlier.Wait, hold on, 5000 divided by 360 is 13 with a remainder of 320°, as I calculated earlier. But in the assistant's solution, they said 5000° mod 360° is 140°, which is incorrect. Wait, that must be a mistake.Wait, 5000 divided by 360: 360*13=4680, 5000-4680=320. So, 5000 mod 360 is 320°, not 140°. So, the assistant made a mistake in their calculation, leading to an incorrect answer of -2 cos 40°, whereas the correct answer should be 2 cos 40°.Wait, that's concerning. So, in my thinking, I correctly reduced 5000° to 320°, leading to 2 cos 40°, but the assistant incorrectly reduced it to 140°, leading to -2 cos 40°. So, which one is correct?Wait, let me check 5000 divided by 360 again. 360*13=4680, 5000-4680=320. So, 5000°=360°*13 +320°, so 5000° mod 360°=320°, which is correct. So, the assistant's solution is wrong because they said 5000 mod 360 is 140°, which is incorrect.Therefore, the correct answer is 2 cos 40°, not -2 cos 40°. So, I think the assistant made a mistake in their calculation, leading to the wrong sign.Wait, but in the assistant's solution, they wrote:"Simplifying the angle 5000° modulo 360°: 5000° mod 360° = 140°"But that's incorrect. 5000 mod 360 is 320°, not 140°. So, that's a mistake.Therefore, the correct answer is 2 cos 40°, not -2 cos 40°. So, I think the assistant's solution is wrong, and my thinking is correct.Wait, but let me double-check the angle reduction. 5000 divided by 360: 360*13=4680, 5000-4680=320. So, 5000=360*13+320, so 5000 mod 360=320°, which is correct. So, the assistant's solution is wrong because they said 5000 mod 360=140°, which is incorrect.Therefore, the correct answer is 2 cos 40°, not -2 cos 40°. So, I think the assistant made a mistake in their calculation, leading to an incorrect answer.Wait, but in the assistant's solution, they wrote:"z^{1000} = (cos 5° ± i sin 5°)^{1000} = cos 5000° ± i sin 5000°"Then they said 5000 mod 360=140°, which is wrong. So, they concluded z^1000=cos 140° +i sin 140°, and 1/z^1000=cos 140° -i sin 140°, so adding them gives 2 cos 140°, which is -2 cos 40°, because cos 140°= -cos 40°.But since 5000 mod 360 is actually 320°, not 140°, then z^1000=cos 320° +i sin 320°, and 1/z^1000=cos 320° -i sin 320°, so adding them gives 2 cos 320°, which is 2 cos 40°, because cos 320°=cos 40°.Therefore, the correct answer is 2 cos 40°, not -2 cos 40°, so the assistant's solution is incorrect.Wait, but in my initial thinking, I arrived at 2 cos 40°, which is correct, whereas the assistant's solution was wrong because of an incorrect angle reduction.Therefore, I think the correct answer is 2 cos 40°, and the assistant's solution is incorrect.Wait, but to be thorough, let me compute 5000 mod 360 again. 360*13=4680, 5000-4680=320. So, 5000 mod 360=320°, which is correct. So, the assistant's solution is wrong because they said 5000 mod 360=140°, which is incorrect.Therefore, the correct answer is 2 cos 40°, not -2 cos 40°.Wait, but just to make sure, let me compute 5000 divided by 360. 360*13=4680, 5000-4680=320. So, 5000=360*13+320, so 5000 mod 360=320°, which is correct.Therefore, the correct answer is 2 cos 40°, and the assistant's solution is incorrect because they made a mistake in the angle reduction.So, in conclusion, the answer should be 2 cos 40°, which is boxed{2cos 40^circ}.But wait, in the initial problem, the user wrote "find z^{1000} + 1/z^{1000}", and in the previous response, the assistant wrote "-2 cos 40°", but in my thinking, I arrived at 2 cos 40°, which is correct. So, the assistant made a mistake in their calculation, leading to an incorrect answer.Therefore, the correct answer is 2 cos 40°, and the assistant's solution is wrong.Wait, but let me think again. If z is e^(i5°), then z^1000 is e^(i5000°)=e^(i320°), and 1/z^1000 is e^(-i320°). So, adding them gives 2 cos 320°, which is 2 cos 40°, as we saw.Alternatively, if z is e^(-i5°), then z^1000 is e^(-i5000°)=e^(-i320°), and 1/z^1000 is e^(i320°). Adding them gives 2 cos 320°, which is 2 cos 40°, same result.Therefore, regardless of whether z is e^(i5°) or e^(-i5°), the result is 2 cos 40°, which is positive.Wait, but in the assistant's solution, they said 5000 mod 360=140°, which is wrong, leading to 2 cos 140°, which is -2 cos 40°, which is incorrect.Therefore, the correct answer is 2 cos 40°, and the assistant's solution is wrong.So, in conclusion, the answer is 2 cos 40°, which is boxed{2cos 40^circ}.

💡First, I need to determine the total amount of time Sam spends studying all three subjects. Sam studies Science for 60 minutes, Math for 80 minutes, and Literature for an unknown amount of time. The total study time is given as 3 hours.I should convert the total study time from hours to minutes to make the units consistent. Since 1 hour equals 60 minutes, 3 hours is equal to 180 minutes.Next, I'll add the time spent on Science and Math:60 minutes (Science) + 80 minutes (Math) = 140 minutes.To find the time spent on Literature, I'll subtract the combined time spent on Science and Math from the total study time:180 minutes (total) - 140 minutes (Science and Math) = 40 minutes.Therefore, Sam spends 40 minutes studying Literature.