🧠:[<think>Okay, let me try to figure out this probability problem. So, the question is: a school needs to select 4 volunteers from 2 male and 4 female students. What is the probability that the selected volunteers include both male and female students? Hmm, right. I need to find the probability that when they pick 4 people, there's at least one male and at least one female. First, I should probably start by understanding the total number of students. There are 2 males and 4 females, so that's a total of 6 students. We need to select 4 volunteers out of these 6. The probability we're looking for is the number of favorable outcomes (selecting both males and females) divided by the total number of possible outcomes (all possible ways to select 4 students). So, first, let's compute the total number of possible ways to select 4 students from 6. That's a combination problem, right? The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number, and k is the number we want to choose. So here, n is 6 and k is 4. Let me calculate that. C(6, 4) = 6! / (4! * (6 - 4)!) = (6*5*4*3*2*1) / (4*3*2*1 * 2*1) = (720) / (24 * 2) = 720 / 48 = 15. So there are 15 possible ways to select 4 students out of 6. Now, for the favorable outcomes: selecting 4 students that include both male and female. Hmm. So this means the group can't be all male or all female. Since there are only 2 males, an all-male group isn't possible if we're selecting 4 volunteers. Wait, because there are only 2 males. So actually, it's impossible to have a group of 4 males. Therefore, the only way the group could be all one gender is if they are all female. So the unfavorable outcomes are the all-female groups. Therefore, the number of favorable outcomes would be total groups minus all-female groups. Let me check that logic. So total groups are 15. All-female groups: there are 4 female students, so we need to select 4 out of 4. The number of ways to do that is C(4, 4) which is 1. So only 1 all-female group. Therefore, the number of favorable groups is 15 - 1 = 14. Therefore, the probability is 14/15. Wait, but hold on. Let me verify again. So if there are 2 males and 4 females, and we need to pick 4 people. The possible gender distributions in the group could be:- 1 male and 3 females- 2 males and 2 femalesSince we can't have more than 2 males. So the groups that include both males and females would be the sum of these two cases. Alternatively, subtracting the all-female group from the total gives the same result. Let me compute it both ways to verify. First way: total groups = 15. Subtract all-female groups (1), so 14. So probability is 14/15. Second way: compute the number of groups with 1 male and 3 females plus groups with 2 males and 2 females. For 1 male and 3 females: number of ways is C(2,1) * C(4,3). C(2,1) is 2, and C(4,3) is 4. So 2*4=8. For 2 males and 2 females: C(2,2) * C(4,2). C(2,2)=1 and C(4,2)=6. So 1*6=6. Total favorable groups: 8 + 6 = 14. So 14/15, same as before. That checks out. So the probability is 14/15. Wait, but just to make sure, let's think again. Is there another possible mistake here? For example, when calculating the combinations. Let's check:C(6,4) is indeed 15. Because C(n,k) = C(n, n - k), so C(6,4)=C(6,2)=15. Alternatively, 6*5*4*3/(4*3*2*1) = 360/24 = 15. Correct. All-female groups: selecting 4 females out of 4. C(4,4)=1. Correct. So 15 total, 1 all-female, so 14 mixed. Alternatively, when calculating the number of groups with 1 male and 3 females: C(2,1) is 2, which is correct because there are 2 males. C(4,3) is 4, since there are 4 females. So 2*4=8. Then, groups with 2 males and 2 females: C(2,2)=1, C(4,2)=6. 1*6=6. 8+6=14. So yes, that's correct. Therefore, the probability is 14/15. But let me just confirm once more. Is there a way where the group could have 0 males (all female) or 1 male or 2 males. Since we need 4 volunteers. But since there are only 2 males, the maximum number of males in the group is 2, so possible male counts: 0,1,2. So total number of groups is sum over male=0 to 2 of C(2, male)*C(4, 4 - male). Compute each term:male=0: C(2,0)*C(4,4)=1*1=1male=1: C(2,1)*C(4,3)=2*4=8male=2: C(2,2)*C(4,2)=1*6=6Total:1+8+6=15. Which matches the total number of groups. So that's correct. Therefore, the groups with both male and female are male=1 and male=2. So 8 + 6 =14. Therefore, 14/15. Therefore, the probability is 14/15. I think that's solid. So unless I made a mistake in combination calculations, which I checked several times, this should be the answer. Wait, let me just verify that C(4,2)=6. Yes: 4*3/(2*1)=6. Correct. C(4,3)=4. Yes, because selecting 3 out of 4 is 4 ways. Correct. C(2,1)=2. Correct. So everything checks out. So the answer is 14/15. Final AnswerThe probability is boxed{dfrac{14}{15}}.</think>To determine the probability that the selected 4 volunteers include both male and female students, we start by calculating the total number of ways to select 4 students from 6. This is given by the combination formula ( C(n, k) ):[C(6, 4) = frac{6!}{4!(6 - 4)!} = frac{6 times 5 times 4 times 3}{4 times 3 times 2 times 1} = 15]Next, we consider the unfavorable outcome where all selected volunteers are female. The number of ways to select 4 females from 4 is:[C(4, 4) = 1]Thus, the number of favorable outcomes (selecting both male and female students) is the total number of ways minus the all-female selection:[15 - 1 = 14]To verify, we calculate the number of favorable outcomes directly by considering the possible gender distributions:1. 1 male and 3 females: ( C(2, 1) times C(4, 3) = 2 times 4 = 8 )2. 2 males and 2 females: ( C(2, 2) times C(4, 2) = 1 times 6 = 6 )Adding these gives:[8 + 6 = 14]The probability is then the ratio of favorable outcomes to the total outcomes:[frac{14}{15}]Thus, the probability that the selected volunteers include both male and female students is boxed{dfrac{14}{15}}.,<think>Okay, let's see. The problem is about selecting 4 volunteers from a group of 2 male and 4 female students. We need to find the probability that the selected volunteers include both male and female students. Hmm. Alright, let's break this down step by step.First, I remember that probability is usually the number of favorable outcomes divided by the total number of possible outcomes. So, I need to figure out two things: the total number of ways to select 4 volunteers, and then the number of those ways that include both male and female students.Let me start with the total number of possible outcomes. That would be the combination of all 6 students (2 males + 4 females) taken 4 at a time. The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number of items, and k is the number of items to choose. So here, n is 6 and k is 4.Calculating that, C(6, 4) = 6! / (4! * 2!) = (6*5*4!)/(4! * 2*1) = (6*5)/2 = 30/2 = 15. So there are 15 possible ways to select any 4 volunteers.Now, the favorable outcomes are those where there's at least one male and at least one female in the selected group. Hmm. But maybe it's easier to calculate the unfavorable outcomes and subtract them from the total. Unfavorable outcomes would be selecting all males or all females. But wait, there are only 2 males. Since we need to select 4 volunteers, we can't select all males because there aren't enough. So the only possible all-male selection would be selecting 2 males, but we need 4 volunteers, so that's not possible. Wait, wait, hold on. If we have 2 males and 4 females, and we need to select 4 volunteers, the only way to have an all-male team is to pick all 2 males and then 2 more males, which we don't have. Therefore, it's impossible to have an all-male selection. So the only possible single-gender selection is all females. So the unfavorable outcomes are the selections where all 4 volunteers are female.Therefore, the number of unfavorable outcomes is the number of ways to choose 4 females out of 4. That's C(4, 4) which is 1. So there's only 1 way to select all females. Therefore, the number of favorable outcomes is total outcomes (15) minus unfavorable (1), which is 14.Therefore, the probability is 14/15. Let me check that again to make sure.Wait, hold on. The total number of students is 6 (2 males, 4 females). Selecting 4 volunteers. The total combinations are indeed C(6, 4) = 15. The all-female selection is C(4, 4) = 1. So yes, the number of favorable is 15 - 1 = 14. So the probability is 14/15. Hmm. But let me confirm once more.Alternatively, maybe I can calculate the favorable outcomes directly. So the favorable cases are when we have both males and females. Since there are only 2 males, the possible number of males in the selection can be 1 or 2. Because if we select more than 2 males, we don't have them. So the number of ways to choose 1 male and 3 females plus the number of ways to choose 2 males and 2 females.Calculating each:1 male and 3 females: C(2, 1) * C(4, 3) = 2 * 4 = 8.2 males and 2 females: C(2, 2) * C(4, 2) = 1 * 6 = 6.Adding those together: 8 + 6 = 14. So that's 14 favorable outcomes. Total outcomes are 15, so probability is 14/15. That matches the previous result. So that's good.Therefore, the answer should be 14/15. Let me just make sure there's no other case. Could there be a case where you have 3 males? But there are only 2 males. So no, that's not possible. So the only possible male numbers are 0, 1, or 2. But 0 is all female, which is the only unfavorable case. So yes, 14/15 is correct.Alternatively, if I had made a mistake in calculating combinations, but C(2,1)=2, C(4,3)=4. 2*4=8. Then C(2,2)=1, C(4,2)=6. 1*6=6. 8+6=14. 14/15 is correct.So the probability is 14/15. So as a numerical value, that would be approximately 0.9333..., but since the question says to express as a numerical value, maybe they want the fraction. But the problem says to put the final answer in a box, so probably boxed{dfrac{14}{15}}.But let me check once again for any errors. Let's think about the total number of ways again. 6 students, choose 4. So 15. Correct. The all-female selection is 1. The rest 14 are mixed. Yes. Alternatively, let me list all possible combinations if it's manageable. Since 6 students, but maybe 15 is too many, but let's see.Wait, perhaps not necessary, but to be thorough. Let's denote the males as M1, M2 and females as F1, F2, F3, F4. The total combinations of 4 would include:All female: F1, F2, F3, F4. That's 1 combination.Then combinations with 1 male and 3 females. For each male, you choose 3 females. There are 2 males. For each male, the number of ways to choose 3 females from 4 is C(4,3)=4. So total for 1 male: 2*4=8. So those are 8 combinations.Combinations with 2 males and 2 females. For each pair of males (there's only 1 pair, since there's only 2 males), and then choose 2 females. C(4,2)=6. So 1*6=6 combinations.So total combinations: 1 + 8 + 6 = 15. That's correct. So the favorable ones are 8 + 6 = 14, which gives the probability of 14/15. So yes, that's correct.Therefore, the answer is 14/15.Final AnswerThe probability is boxed{dfrac{14}{15}}.</think>To find the probability that the selected volunteers include both male and female students, we start by calculating the total number of ways to select 4 volunteers from 6 students (2 males and 4 females). This is given by the combination formula ( C(n, k) ):[C(6, 4) = frac{6!}{4!(6 - 4)!} = frac{6 times 5}{2 times 1} = 15]Next, we consider the unfavorable outcomes, which are selections that include only female students. The number of ways to select 4 females from 4 is:[C(4, 4) = 1]Thus, the number of favorable outcomes (selecting both male and female students) is the total outcomes minus the unfavorable outcomes:[15 - 1 = 14]We can also verify this by calculating the favorable outcomes directly. These include selections with 1 male and 3 females, and 2 males and 2 females:- For 1 male and 3 females: ( C(2, 1) times C(4, 3) = 2 times 4 = 8 )- For 2 males and 2 females: ( C(2, 2) times C(4, 2) = 1 times 6 = 6 )Adding these together gives:[8 + 6 = 14]Therefore, the probability is:[frac{14}{15}]The final answer is boxed{dfrac{14}{15}}.]

🔑:To solve this problem, we consider the different ways the volunteers can be chosen. The total number of ways to select 4 students out of 2 males and 4 females is binom{6}{4} ways.The only case where the team doesn't include both males and females is when they are all female, which can happen in binom{4}{4} = 1 way. There's no case where all selected are male since only 2 males are available and we need to select 4 volunteers.So, the number of ways to select 4 volunteers that include at least one male and one female is the total number of ways minus the number of all-female ways, which is 15 - 1 = 14 ways.The probability is thus the number of favorable outcomes (both males and females included) over the total number of outcomes, which is:P(text{both males and females}) = frac{14}{15}Therefore, the probability that the selected volunteers include both male and female students is boxed{frac{14}{15}}.

🧠:[<think>Okay, let's see. I need to find the cube root of 253600, and I'm given the cube roots of 25.36 and 253.6 as 2.938 and 6.329 respectively. Hmm, how can I use these to find the cube root of 253600?First, let me write down the given information:∛25.36 = 2.938∛253.6 = 6.329And I need to find ∛253600.Hmm, the numbers 25.36, 253.6, and 253600 seem related by factors of 10. Let me check that.25.36 multiplied by 10 is 253.6, and 253.6 multiplied by 1000 is 253600. Wait, 253.6 * 1000 = 253600. So, 253600 is 253.6 times 1000. So, maybe I can express 253600 in terms of 253.6 and then take the cube root?Let me try that. So, 253600 = 253.6 * 1000. Therefore, the cube root of 253600 is the cube root of (253.6 * 1000). Since cube roots can be separated over multiplication, that would be ∛253.6 * ∛1000. I know that ∛1000 is 10, because 10^3 is 1000. And they already told me that ∛253.6 is 6.329. So then, ∛253600 = 6.329 * 10 = 63.29. Is that right?Wait, let me verify. Let's check the exponents. If I take 6.329 and cube it, I should get 253.6. Let me confirm that. 6.329^3. Let's calculate 6.329 * 6.329 first. 6 * 6 = 36. 6 * 0.329 = 1.974. 0.329 * 6 = 1.974. 0.329 * 0.329 ≈ 0.108. So adding up: 36 + 1.974 + 1.974 + 0.108 ≈ 36 + 3.948 + 0.108 ≈ 36 + 4.056 ≈ 40.056. So that's 6.329 squared is approximately 40.056. Then multiplying by 6.329 again: 40.056 * 6.329. Let's approximate.40 * 6 = 24040 * 0.329 = 13.160.056 * 6 = 0.3360.056 * 0.329 ≈ 0.0184Adding all these together: 240 + 13.16 = 253.16; 0.336 + 0.0184 ≈ 0.3544; total ≈ 253.16 + 0.3544 ≈ 253.5144. Which is approximately 253.5, close to 253.6. Considering rounding errors, that seems okay. So, ∛253.6 ≈ 6.329 is correct.Then, ∛253600 = ∛(253.6 * 1000) = ∛253.6 * ∛1000 = 6.329 * 10 = 63.29. That seems straightforward. Let me check if there's another way.Alternatively, let's look at the original numbers: 25.36, 253.6, 253600. Let's see how they relate. 25.36 to 253.6 is multiplying by 10, and 253.6 to 253600 is multiplying by 1000. So, 253600 is 25.36 * 10 * 1000 = 25.36 * 10^4. Therefore, 253600 = 25.36 * 10^4. Then, ∛253600 = ∛(25.36 * 10^4) = ∛25.36 * ∛10^4.We know ∛25.36 = 2.938, so that part is 2.938. Then, ∛10^4. 10^4 is 10,000. So, ∛10,000. Alternatively, 10^4 is (10^3)*10, so ∛(10^3 * 10) = ∛10^3 * ∛10 = 10 * ∛10. So, ∛10^4 = 10 * ∛10 ≈ 10 * 2.1544 ≈ 21.544. So, then ∛253600 ≈ 2.938 * 21.544. Let me compute that.2.938 * 21.544. Let's do 2 * 21.544 = 43.088. Then 0.938 * 21.544. Let's compute 0.9 * 21.544 = 19.3896, 0.038 * 21.544 ≈ 0.818. So total is 19.3896 + 0.818 ≈ 20.2076. Then total is 43.088 + 20.2076 ≈ 63.2956. Which is approximately 63.296, so rounding to 63.30. Hmm, but earlier calculation gave 63.29.Wait, that's a discrepancy. The two methods gave me slightly different results? Wait, first method: ∛253.6 * ∛1000 = 6.329 * 10 = 63.29. Second method: ∛25.36 * ∛10^4 ≈ 2.938 * 21.544 ≈ 63.2956 ≈ 63.30. The difference is due to approximation errors in ∛10.But which method is more accurate? Let me check. Since in the first method, we have exact value for ∛1000 = 10, and we use the given ∛253.6 = 6.329. So, 6.329 * 10 = 63.29, which is precise. In the second method, when I approximated ∛10 ≈ 2.1544, which is an approximate value, leading to a slightly different result. Therefore, the first method is more accurate because it uses exact values where possible.Therefore, the answer should be 63.29.Wait, but let me confirm once more. Let me compute 63.29^3 and see if it's approximately 253600.First, compute 63^3: 63 * 63 = 3969, 3969 * 63. Let's compute 3969 * 60 = 238,140, 3969 * 3 = 11,907. So total 238,140 + 11,907 = 250,047. So 63^3 = 250,047. Then, 63.29 is a bit more than 63, so 63.29^3 should be a bit more than 250,047. Let's compute the difference.First, let me write 63.29 as 63 + 0.29. Then, using the binomial expansion for (a + b)^3 where a = 63, b = 0.29.(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.We know a^3 = 250,047.Compute 3a^2b: 3*(63^2)*0.29. 63^2 = 3969. So 3*3969 = 11,907. 11,907*0.29. Let's calculate:11,907 * 0.2 = 2,381.411,907 * 0.09 = 1,071.63Total: 2,381.4 + 1,071.63 = 3,453.03Next, 3ab^2: 3*63*(0.29)^2. First, (0.29)^2 = 0.0841. Then, 3*63 = 189. 189*0.0841 ≈ 15.887.Then, b^3 = (0.29)^3 ≈ 0.024389.Adding all these together:250,047 + 3,453.03 = 253,500.03253,500.03 + 15.887 ≈ 253,515.917253,515.917 + 0.024389 ≈ 253,515.9414So, 63.29^3 ≈ 253,515.94But the actual value we're trying to find is ∛253,600. So, 253,600 - 253,515.94 ≈ 84.06. So, there's a difference of about 84. So, the cube of 63.29 is approximately 253,515.94, which is 84 less than 253,600. That suggests that 63.29 is slightly less than the actual cube root. Hmm, so maybe the answer is a little higher?Wait, but the original given values were approximations. Let me check ∛253.6 = 6.329. If I compute 6.329^3, as before, we had approximately 253.6, which is correct. Then, multiplying by 1000, the cube root becomes 63.29. So, according to that, 63.29^3 = (6.329 * 10)^3 = 6.329^3 * 10^3 = 253.6 * 1000 = 253,600. Wait, hold on! That seems contradictory to the previous calculation. Wait, but according to that, 6.329^3 is 253.6, so multiplying by 10^3 gives 253.6 * 1000 = 253,600. So, then (6.329 * 10)^3 = 253.6 * 1000 = 253,600. Therefore, 6.329 * 10 = 63.29 is indeed the cube root of 253,600. So why when I expanded (63 + 0.29)^3, I got 253,515.94 instead of 253,600?Ah, because my expansion was approximate. Let's check again. Wait, the exact value is (6.329 * 10)^3 = 6.329^3 * 10^3 = 253.6 * 1000 = 253,600. Therefore, 63.29 is indeed the exact cube root. The discrepancy in the binomial expansion is because I was using a linear approximation for 0.29, but actually, since 0.29 is 29/100, the exact computation would require precise multiplication. However, when I did the binomial expansion, I used decimal approximations which introduced some error, but the algebraic approach is exact.Therefore, the correct answer is 63.29. Therefore, the cube root of 253600 is 63.29. Therefore, the answer should be boxed{63.29}But let me just make sure there isn't a scaling factor I missed. For example, perhaps there's a different relationship between the numbers. Let me check the exponents again.Given that ∛25.36 = 2.938 and ∛253.6 = 6.329. So 253.6 is 25.36 * 10, so the cube root of 25.36 * 10 is 6.329. Then, since 253600 is 253.6 * 1000, then as before, the cube root is 6.329 * 10 = 63.29. So that's consistent.Alternatively, if I consider 253600 as 25.36 * 10,000, so ∛(25.36 * 10^4) = ∛25.36 * ∛10^4. Then, ∛25.36 = 2.938, and ∛10^4 = 10^(4/3) ≈ 10^1.333 ≈ 21.544. Then, 2.938 * 21.544 ≈ 63.295, which is close to 63.29. So, rounding to the same decimal places as the given values, which are three decimal places for 2.938 and three for 6.329. So, the answer is 63.29. But wait, let me check the exact decimal places.The problem gives ∛25.36 = 2.938 (three decimal places) and ∛253.6 = 6.329 (three decimal places). So, when we do 6.329 * 10, the result is 63.290, but written as 63.29 (removing the trailing zero). However, depending on the precision, maybe the answer is 63.29. Let me check if 6.329 is precise.Wait, 6.329^3 = 253.6. Let me compute 6.329 * 6.329 * 6.329 precisely.First, compute 6.329 * 6.329:6 * 6 = 366 * 0.329 = 1.9740.329 * 6 = 1.9740.329 * 0.329 = 0.108241Adding together: 36 + 1.974 + 1.974 + 0.108241 = 36 + 3.948 + 0.108241 = 39.948 + 0.108241 = 40.056241Then, 40.056241 * 6.329:First, multiply 40 * 6.329 = 253.16Then, 0.056241 * 6.329 ≈ 0.056241 * 6 = 0.337446; 0.056241 * 0.329 ≈ 0.0185So total ≈ 0.337446 + 0.0185 ≈ 0.355946Adding to 253.16: 253.16 + 0.355946 ≈ 253.515946Which is approximately 253.516, but the original problem says ∛253.6 = 6.329. So, 6.329^3 ≈ 253.516, which is about 0.084 less than 253.6. So, there is a small error here. So, 6.329 is an approximate value. Therefore, when we multiply by 10 to get 63.29, the error is scaled by 10 as well. But since the given values are approximate, we have to go with the provided approximations.Therefore, even though the precise calculation might show a slight discrepancy, according to the problem statement, we take ∛253.6 = 6.329 as given, so ∛253600 = 6.329 * 10 = 63.29.Therefore, the answer is 63.29.Final Answerboxed{63.29}</think>Given the problem of finding (sqrt[3]{253600}) and the provided values (sqrt[3]{25.36} = 2.938) and (sqrt[3]{253.6} = 6.329), we can proceed as follows:1. Recognize that (253600 = 253.6 times 1000).2. Use the property of cube roots that (sqrt[3]{a times b} = sqrt[3]{a} times sqrt[3]{b}).3. Therefore, (sqrt[3]{253600} = sqrt[3]{253.6 times 1000} = sqrt[3]{253.6} times sqrt[3]{1000}).4. We know (sqrt[3]{253.6} = 6.329) and (sqrt[3]{1000} = 10).5. Multiplying these results gives (6.329 times 10 = 63.29).Thus, the final answer is (boxed{63.29}).,<think>Okay, let's see. The problem is asking for the cube root of 253600, and they've given me two other cube roots: the cube root of 25.36 is 2.938, and the cube root of 253.6 is 6.329. Hmm, I need to figure out how these relate to each other so I can find the cube root of 253600 without a calculator. First, let me note down the given information:∛25.36 = 2.938∛253.6 = 6.329And we need ∛253600.I should probably look at the relationships between the numbers 25.36, 253.6, and 253600. Let me see how each number is scaled compared to the others.Starting with 25.36, if I multiply that by 10, I get 253.6. Then, if I multiply 253.6 by 1000, I get 253600. So 25.36 * 10 = 253.6, and 253.6 * 1000 = 253600. So the original number is scaled by 10 and then by 1000. But how does scaling affect cube roots?I remember that if a number is multiplied by a factor, its cube root is multiplied by the cube root of that factor. For example, ∛(a * b) = ∛a * ∛b. So maybe I can use that property here.Let me try to express 253600 in terms of 25.36 or 253.6. Let's see:253600 divided by 25.36 is 253600 / 25.36. Let me compute that. 25.36 * 10 = 253.6, 253.6 * 1000 = 253600. So 25.36 * 10 * 1000 = 25.36 * 10,000 = 253600. So 253600 is 25.36 multiplied by 10,000.Therefore, ∛253600 = ∛(25.36 * 10,000) = ∛25.36 * ∛10,000. I know ∛25.36 is given as 2.938. So now I need to compute ∛10,000.Wait, but 10,000 is 10^4. So ∛10^4 = 10^(4/3) = 10^(1 + 1/3) = 10 * 10^(1/3). But I don't have the value of ∛10, but maybe I can relate it to the given numbers?Alternatively, maybe another approach. Let's see, since 253.6 is 10 times 25.36, so the cube root of 253.6 is ∛(25.36 * 10) = ∛25.36 * ∛10. According to the given info, that equals 6.329. Given that ∛25.36 is 2.938, then 2.938 * ∛10 = 6.329. So perhaps I can solve for ∛10 here?Let me check:If 2.938 * ∛10 = 6.329, then ∛10 = 6.329 / 2.938.Let me compute that division. 6.329 divided by 2.938.Calculating this:Divide numerator and denominator by 1.469 to simplify? Wait, maybe just straight division.First, 2.938 * 2 = 5.8766.329 - 5.876 = 0.453So 2.938 goes into 6.329 two times with a remainder of 0.453.So 0.453 / 2.938 ≈ 0.453 / 2.938 ≈ 0.154So total is approximately 2.154. So ∛10 ≈ 2.154. Let me verify:2.154^3 = 2.154 * 2.154 * 2.154. First, 2.154 * 2.154: 2*2=4, 2*0.154=0.308, 0.154*2=0.308, 0.154*0.154≈0.0237. So adding up: 4 + 0.308 + 0.308 + 0.0237 ≈ 4.6397. Then multiply by 2.154: 4.6397 * 2 = 9.2794, 4.6397 * 0.154 ≈ 0.715. So total ≈9.2794 + 0.715 ≈9.9944. So approximately 10. So 2.154^3 ≈10. Close enough. So ∛10 ≈2.154.So going back, ∛10,000. Since 10,000 =10^4, so ∛10^4 =10^(4/3) =10^(1 +1/3)=10 *10^(1/3)=10*∛10≈10*2.154≈21.54. But wait, hold on. Alternatively, since 10,000 =1000 *10, and ∛1000=10, so ∛(1000*10)=10*∛10≈10*2.154≈21.54. Either way, same result.So ∛10,000 ≈21.54.Therefore, ∛253600= ∛25.36 * ∛10,000 ≈2.938 *21.54.Let me compute that. 2.938 *20=58.76, 2.938*1.54≈ (2.938*1)+(2.938*0.54)=2.938 + 1.586≈4.524. So total≈58.76 +4.524≈63.284.So approximately 63.284. Let me check if this makes sense.Alternatively, perhaps using the other given value, ∛253.6=6.329. Let's see if 253600 is 253.6 multiplied by 1000.Yes, 253.6 *1000=253600. So ∛253600=∛(253.6*1000)=∛253.6 *∛1000=6.329*10=63.29. Oh! Wait, that's simpler. Because 1000 is a perfect cube, its cube root is 10. So if I can express 253600 as 253.6 *1000, then the cube root is the cube root of 253.6 times the cube root of 1000, which is 6.329*10=63.29. Which is 63.29, which matches the previous approximate calculation of 63.284. So that's more precise and straightforward. So why did I even go through the first method? Maybe I didn't notice that 253600 is 253.6 *1000. That's a better approach. So the answer should be 6.329 *10=63.29. Since 253600 =253.6 *10^3, then ∛253600=∛253.6 *∛10^3=6.329*10=63.29.Therefore, the answer is 63.29. Let me check if that's consistent.Wait, but in the first approach, when I did 2.938*21.54≈63.28, which is very close to 63.29. So that's consistent. So both methods give approximately the same result, but the second method is exact because 1000 is a perfect cube, so we don't need to approximate ∛10.Therefore, the correct answer is 63.29. But let me verify once more. If ∛253.6 is 6.329, then multiplying that by ∛1000 which is 10, gives 63.29. Therefore, ∛253600=63.29. That seems correct.Alternatively, let's check by cubing 63.29. Let's compute 63.29^3 and see if it's approximately 253600.First, compute 63^3: 63*63=3969, 3969*63= (3969*60)+(3969*3)=238,140 +11,907=250,047.Then 0.29^3 is approximately 0.29*0.29*0.29. 0.29*0.29=0.0841, 0.0841*0.29≈0.024389. But actually, (63 + 0.29)^3 =63^3 +3*63^2*0.29 +3*63*(0.29)^2 + (0.29)^3. Compute each term:First term: 63^3 =250,047.Second term: 3*(63^2)*0.29. 63^2=3969. 3*3969=11,907. 11,907*0.29= let's compute 11,907*0.3=3,572.1, subtract 11,907*0.01=119.07, so 3,572.1 -119.07=3,453.03.Third term:3*63*(0.29)^2. First, 0.29^2=0.0841. So 3*63=189. 189*0.0841≈189*0.08 +189*0.0041=15.12 +0.7749≈15.8949.Fourth term:0.29^3≈0.024389.Now add all terms:250,047 +3,453.03=253,500.03 +15.8949≈253,515.9249 +0.024389≈253,515.9493.Wait, but 63.29^3≈253,515.95, but the original number is 253,600. So there's a difference of about 84.05. Hmm, that's a bit of a discrepancy. Wait, maybe my approximation is off because the given value ∛253.6=6.329 is approximate. Let me check with the given values.If ∛253.6=6.329, then 6.329^3=253.6. Let's verify that:6.329^3: 6*6*6=216, 6*6*0.329=6*6=36, 36*0.329≈11.844; 6*0.329*6 same thing. Wait, maybe better to compute step by step.First, 6.329*6.329: compute 6*6=36, 6*0.329=1.974, 0.329*6=1.974, 0.329*0.329≈0.108. So total: 36 +1.974 +1.974 +0.108≈36 +3.948 +0.108≈40.056. So (6.329)^2≈40.056. Then multiply by 6.329: 40.056*6=240.336, 40.056*0.329≈40*0.329=13.16, 0.056*0.329≈0.0184. So total≈13.16 +0.0184≈13.1784. So total≈240.336 +13.1784≈253.5144. Which is close to 253.6, but not exact. So 6.329^3≈253.5144, but the problem states that ∛253.6=6.329, so perhaps the given value is rounded. So 6.329 is an approximation. So when we cube it, we get approximately 253.5144, but the actual cube root of 253.6 is 6.329. So there's some rounding involved here.Therefore, when we multiply 6.329 by 10 to get 63.29, and cube that, we get approximately 253,515.95, but 253600 is a bit larger. However, since the given values are approximate, the answer is going to be approximate as well. Alternatively, perhaps I need to use the first given value to get a better approximation? Let's see.Given that ∛25.36=2.938 and ∛253.6=6.329. Then 253600 is 253.6*1000, so ∛253600=6.329*10=63.29. Since this is given, the answer is 63.29. The slight discrepancy when cubing is due to the rounding of the given cube roots.Therefore, the answer should be 63.29.Wait, but let's check another way. Let's see how much 253600 is compared to 253.6. It's 1000 times larger. So the cube root should be 10 times larger, because the cube root of 1000 is 10. Therefore, since ∛253.6=6.329, then ∛253600=6.329*10=63.29. That's straightforward. So the answer is 63.29.Therefore, the answer is boxed{63.29}Final Answerboxed{63.29}</think>Given the problem, we need to find the cube root of 253600 using the provided values (sqrt[3]{25.36} = 2.938) and (sqrt[3]{253.6} = 6.329).First, we observe the relationship between the numbers:- (253600) is (253.6 times 1000).Using the property of cube roots that (sqrt[3]{a times b} = sqrt[3]{a} times sqrt[3]{b}), we can express (sqrt[3]{253600}) as:[sqrt[3]{253600} = sqrt[3]{253.6 times 1000} = sqrt[3]{253.6} times sqrt[3]{1000}]We know:- (sqrt[3]{253.6} = 6.329)- (sqrt[3]{1000} = 10)Therefore:[sqrt[3]{253600} = 6.329 times 10 = 63.29]Thus, the final answer is (boxed{63.29}).]

🔑:To solve this problem, we need to understand how the cube root operation behaves with respect to scaling the argument by powers of 10. Given that sqrt[3]{25.36} = 2.938 and sqrt[3]{253.6} = 6.329, we observe that increasing the argument of the cube root by a factor of 10 (from 25.36 to 253.6) results in the cube root increasing by approximately a factor of 10^{1/3}, which changes the cube root from 2.938 to 6.329. Now, we are asked to find sqrt[3]{253600}. Notice that 253600 is 1000 times larger than 253.6. Since 1000 = 10^3, and we are dealing with cube roots, increasing the argument by a factor of 10^3 will increase the cube root by a factor of 10. Therefore, we can scale the cube root of 253.6 by a factor of 10 to find the cube root of 253600:[sqrt[3]{253600} = 10 times sqrt[3]{253.6} = 10 times 6.329 = 63.29]Thus, the cube root of 253600 is boxed{63.29}.

🧠:[<think>Okay, so the problem is asking about the angle between the bisector of an angle and the bisector of its adjacent supplementary angle. The options are 80°, 90°, 45°, or 180°. Hmm, let me try to visualize this.First, let me recall what supplementary angles are. Two angles are supplementary if their measures add up to 180°. So, if we have an angle, its adjacent supplementary angle would be the one right next to it, forming a straight line. Like, if angle A is, say, 60°, then its adjacent supplementary angle would be 120°, right? Because 60 + 120 = 180. But wait, actually, if they're adjacent and supplementary, they form a linear pair. So they are next to each other and their non-common sides form a straight line. So, each of those angles would be supplementary to each other.Now, the problem is talking about the bisector of an angle and the bisector of its adjacent supplementary angle. So, let's imagine angle ABC, and its adjacent supplementary angle would be angle CBD. So, points A, B, C, D are colinear in a way that angle ABC and angle CBD are supplementary. Then, we bisect angle ABC and angle CBD, and find the angle between those two bisectors.Let me try to draw this mentally. Let's say we have a straight line, and point B is somewhere in the middle. Let me take point B as the vertex. Then, one arm of angle ABC goes to point A, and the other arm goes to point C, which is on the straight line. Then, the supplementary angle adjacent to angle ABC would be angle CBD, where D is another point on the straight line extending from point C. So, angles ABC and CBD are supplementary, meaning they add up to 180°, and they are adjacent, forming a straight line.Now, if we bisect angle ABC and angle CBD, what does that look like? The bisector of angle ABC would split it into two equal parts. Similarly, the bisector of angle CBD would split that angle into two equal parts. The question is asking for the angle between these two bisectors.Let me assign some variables. Let's denote the measure of angle ABC as θ. Then, since angle ABC and angle CBD are supplementary, angle CBD would be 180° - θ. Now, if we bisect angle ABC, each of the resulting angles would be θ/2. Similarly, bisecting angle CBD would give two angles of (180° - θ)/2 each.Now, the bisectors of these angles would form some angle with each other. Let's try to find that angle. Let me think about the positions of these bisectors. If angle ABC is θ, then its bisector would be a line that splits θ into two angles of θ/2. Similarly, the bisector of angle CBD, which is 180° - θ, would split it into two angles of (180° - θ)/2.Since angles ABC and CBD are adjacent and supplementary, their sum is 180°, forming a straight line. So, if we have angle ABC with measure θ, then angle CBD is 180° - θ, and the two angles together form a straight angle at point B.Now, let's try to find the angle between the two bisectors. Let me consider the positions of the bisectors relative to the original angles. The bisector of angle ABC would be inside angle ABC, splitting it into two smaller angles. Similarly, the bisector of angle CBD would be inside angle CBD, splitting it into two smaller angles.Since angle ABC and angle CBD are adjacent, their bisectors are going to be on different sides of the original line. Wait, maybe not. Let me think again. If angle ABC is between BA and BC, then angle CBD is between BC and BD. So, the bisector of angle ABC would be between BA and BC, and the bisector of angle CBD would be between BC and BD. Therefore, the two bisectors are on either side of BC.Therefore, the angle between the two bisectors would be the sum of the two bisected angles from each original angle. Wait, is that correct? Let me try to visualize this.If angle ABC is θ, then the bisector splits it into θ/2 on each side. Similarly, angle CBD is 180° - θ, so the bisector splits it into (180° - θ)/2 on each side. Since the two original angles are adjacent and form a straight line, the angle between their bisectors would be θ/2 + (180° - θ)/2. Let's compute that.θ/2 + (180° - θ)/2 = [θ + 180° - θ]/2 = 180°/2 = 90°. So, that's 90°. Therefore, the angle between the two bisectors is 90°, regardless of the value of θ. So, the answer should be 90°, which is option B.Wait, let me verify this with an example. Suppose angle ABC is 60°, so angle CBD is 120°. Then, bisecting angle ABC gives 30°, and bisecting angle CBD gives 60°. The angle between the bisectors would be 30° + 60° = 90°, which is right. If angle ABC is 90°, then angle CBD is 90°, so each bisector splits them into 45°, and the angle between the bisectors is 45° + 45° = 90°. Another example, if angle ABC is 30°, then angle CBD is 150°, bisectors split into 15° and 75°, so angle between bisectors is 15° + 75° = 90°. Hmm, seems consistent.Therefore, regardless of the measure of the original angle, the angle between the bisectors is always 90°, so the answer should be B: 90°.Final Answerboxed{B}</think>Given the problem of finding the angle between the bisector of an angle and the bisector of its adjacent supplementary angle, we start by noting that supplementary angles add up to 180°. Let the measure of the given angle be (theta). Its adjacent supplementary angle will then be (180^circ - theta).When we bisect the given angle (theta), each resulting angle is (frac{theta}{2}). Similarly, bisecting the supplementary angle (180^circ - theta) results in angles of (frac{180^circ - theta}{2}).The angle between the two bisectors is the sum of these two bisected angles. Calculating this sum:[frac{theta}{2} + frac{180^circ - theta}{2} = frac{theta + 180^circ - theta}{2} = frac{180^circ}{2} = 90^circ]This result is consistent regardless of the value of (theta). Therefore, the angle between the bisector of the given angle and the bisector of its adjacent supplementary angle is always (90^circ).[boxed{B}],<think>Okay, so the problem is asking about the angle between the bisector of an angle and the bisector of its adjacent supplementary angle. The options are A: 80°, B: 90°, C: 45°, D: 180°. Hmm, let me think about how to approach this.First, let me recall that supplementary angles add up to 180 degrees. So if we have an angle, say θ, its supplementary angle would be 180° - θ. The problem mentions the bisector of the original angle and the bisector of its adjacent supplementary angle. I need to find the angle between these two bisectors.Let me visualize this. Imagine two adjacent angles forming a straight line, which is 180°. Let's denote the original angle as θ, so the adjacent supplementary angle would be 180° - θ. Each of these angles is bisected by their respective angle bisectors. The question is asking for the measure of the angle formed between these two bisectors.Maybe it would help to draw a diagram. Let me sketch this mentally. There's a point with two arms forming an angle θ. The supplementary angle adjacent to it would complete the straight line, so 180° - θ. The bisector of angle θ would split it into two angles of θ/2 each. Similarly, the bisector of the supplementary angle would split 180° - θ into two angles of (180° - θ)/2 each.Now, the angle between these two bisectors is what we need to find. Let me denote the original angle as θ, so the bisector of θ divides it into θ/2. The supplementary angle is 180° - θ, so its bisector divides it into (180° - θ)/2. The angle between the two bisectors would be the sum of these two bisected angles? Wait, not quite. Let me think.Wait, if we have two adjacent angles forming a straight line, θ and 180° - θ, their bisectors are each splitting their respective angles. The angle between the two bisectors would be the angle formed between the two bisectors. Let me consider the vertex where all these angles meet. Let's call this point O. The original angle is between two arms OA and OB, forming angle AOB = θ. The supplementary angle would then be between OB and OC, forming angle BOC = 180° - θ, such that OA, OB, and OC form a straight line.The bisector of angle AOB (which is θ) would be a line OD that splits angle AOB into two equal parts, each of θ/2. Similarly, the bisector of angle BOC (which is 180° - θ) would be a line OE that splits angle BOC into two equal parts, each of (180° - θ)/2.Now, the angle between the two bisectors OD and OE is the angle DOE. To find this angle, we can consider the positions of OD and OE relative to the original lines OA, OB, and OC.Since OA, OB, and OC form a straight line, OA to OC is 180°. The bisector OD is inside angle AOB, θ/2 away from OA and OB. Similarly, the bisector OE is inside angle BOC, (180° - θ)/2 away from OB and OC.But since both OD and OE are on the same side of OB (the common arm between angles AOB and BOC), the angle between OD and OE would be the sum of the angles between OD and OB, and between OB and OE. Wait, OD is θ/2 away from OB, and OE is (180° - θ)/2 away from OB in the opposite direction. Wait, no. Let me clarify.Actually, since angle AOB is θ, and OD is its bisector, the angle between OD and OB is θ/2. Similarly, angle BOC is 180° - θ, and OE is its bisector, so the angle between OE and OB is (180° - θ)/2. However, since OE is on the other side of OB compared to OD, the angle between OD and OE would actually be θ/2 + (180° - θ)/2.Let me compute that: θ/2 + (180° - θ)/2 = [θ + 180° - θ]/2 = 180°/2 = 90°. So, the angle between the two bisectors is 90 degrees. Therefore, the answer should be B: 90°. Hmm, that seems straightforward. Let me verify this with a concrete example.Suppose θ is 60°. Then the supplementary angle is 120°. The bisector of 60° would create two 30° angles, and the bisector of 120° would create two 60° angles. The angle between the two bisectors would be 30° + 60° = 90°. Yep, that works. Let's try another angle. Let θ be 90°, then the supplementary angle is 90°. The bisectors would each split 90° into 45°, so the angle between the bisectors would be 45° + 45° = 90°. Still 90°. What if θ is 120°, supplementary angle is 60°. The bisectors would split into 60° and 30°, so 60° + 30° = 90°. Same result.Therefore, regardless of the measure of θ, the angle between the bisectors is always 90°. So the correct answer is B: 90°.Final Answerboxed{B}</think>Given an angle θ and its adjacent supplementary angle 180° - θ, we need to find the angle between their respective bisectors.1. Supplementary Angles: The original angle θ and its adjacent supplementary angle add up to 180°.2. Angle Bisectors: - The bisector of θ divides it into two angles of θ/2 each. - The bisector of the supplementary angle (180° - θ) divides it into two angles of (180° - θ)/2 each.3. Position of Bisectors: - The bisector of θ is θ/2 away from one of its arms. - The bisector of the supplementary angle (180° - θ) is (180° - θ)/2 away from the common arm.4. Angle Between Bisectors: The angle between the two bisectors is the sum of θ/2 and (180° - θ)/2: [ frac{theta}{2} + frac{180° - theta}{2} = frac{theta + 180° - theta}{2} = frac{180°}{2} = 90° ]Thus, the angle between the bisector of an angle and the bisector of its adjacent supplementary angle is always 90°.[boxed{B}]]

🔑:Since the sum of an angle and its adjacent supplementary angle is 180°,[180° times frac{1}{2} = 90°.]Therefore, the angle formed by the bisector of an angle and the bisector of its adjacent supplementary angle is a right angle.Hence, the correct option is boxed{text{B}}.

🧠:[<think>Okay, so I need to figure out the minimum number of questions a student needs to answer correctly in the first round of this math competition to qualify for the second round. Let's see, the competition has 25 questions. For each correct answer, you get 4 points, and for each incorrect or unanswered question, they deduct 1 point. To qualify, you need a score of at least 60 points. Alright, let me start by understanding the problem. The total number of questions is fixed at 25. Each correct answer gives +4, each incorrect or unanswered gives -1. The total score needs to be ≥60. So, we need to find the minimum number of correct answers (let's call that 'c') such that the total score is at least 60. First, let's define variables. Let me let c be the number of correct answers. Then, the number of incorrect or unanswered questions would be 25 - c, right? Because there's no partial answering or anything; every question is either correct, incorrect, or unanswered. But since unanswered and incorrect are treated the same in terms of points, they can be grouped together. So, the total number of questions not answered correctly is 25 - c.Now, the total score is calculated by adding the points from correct answers and subtracting the points from incorrect/unanswered. So, that would be 4c - 1*(25 - c). This expression needs to be greater than or equal to 60. So, the inequality is:4c - (25 - c) ≥ 60Let me write that out again:4c - (25 - c) ≥ 60Let me simplify the left side. Distribute the negative sign:4c -25 + c ≥ 60Combine like terms:4c + c = 5c, so:5c -25 ≥ 60Now, add 25 to both sides:5c ≥ 60 +255c ≥ 85Then, divide both sides by 5:c ≥ 85 / 5c ≥ 17Wait, so c has to be at least 17? Hmm. Let me check that.If a student answers 17 correctly, then they get 17*4 = 68 points. Then, they have 25 -17 = 8 incorrect or unanswered questions, which deducts 8*1 = 8 points. So, total score is 68 -8 = 60. Exactly 60. So, 17 correct answers gives exactly 60, which is the minimum needed. So, the student needs at least 17 correct answers. Therefore, the answer should be 17.But let me test 16 just to be sure. If a student answers 16 correctly, then they get 16*4 = 64 points. Incorrect/unanswered is 25 -16 =9, so they lose 9 points. Total score is 64 -9 =55, which is less than 60. So, 16 correct answers would give 55, which is insufficient. Therefore, 17 is indeed the minimum.Wait, hold on. Wait, 16 correct answers: 16*4=64. But then 9 incorrect: 9*(-1)= -9. So total is 64-9=55. Correct. Then 17*4=68, 8 incorrect: 68-8=60. So exactly 60.Therefore, the student must have answered at least 17 questions correctly. So the answer is 17. But let me check the algebra again to make sure I didn't make a mistake.Starting with:Total score = 4c -1*(25 -c) ≥60So, 4c -25 +c ≥60Which is 5c -25 ≥605c ≥85c ≥17Yes, that's correct. So 17 is the minimal number. Therefore, the answer is 17.But wait, just to be thorough, let me consider if there are any constraints I might have missed. For example, can c be a non-integer? But since the number of questions answered correctly has to be an integer, c must be a whole number. So, even if the algebra had resulted in a fractional number, we would have to round up to the next whole number. But in this case, 85 divided by 5 is exactly 17, so no rounding needed. Therefore, 17 is indeed the correct answer.Alternatively, maybe another approach. Let's say the student answers c questions correctly and u questions incorrectly or leaves them unanswered. Then, c + u =25. The score is 4c -u. We need 4c -u ≥60. But since u=25 -c, substitute that in:4c - (25 -c) ≥60. Which is the same equation as before. So, same result.Alternatively, think of it as each correct answer gives +4 and each incorrect gives -1. The difference between a correct and incorrect is 5 points. But perhaps that's complicating it. But maybe another angle: If all questions were answered incorrectly, the score would be 25*(-1)= -25. Each correct answer you replace an incorrect with a correct, so each correct answer adds 4 - (-1) =5 points to the total. So, starting from -25, how many 5s do we need to add to reach 60?Let me see:Let S be the score. S = -25 +5c ≥60So, -25 +5c ≥605c ≥85c ≥17.Same result. So, same answer. So, that's another way to think about it, but arrives at the same conclusion. So, 17 is the minimal number.Therefore, the students must have answered at least 17 questions correctly.Final AnswerThe students who qualify for the second round must have answered at least boxed{17} questions correctly.</think>To determine the minimum number of correct answers needed to qualify for the second round, we start by defining ( c ) as the number of correct answers. The number of incorrect or unanswered questions is then ( 25 - c ). The total score is calculated by adding points for correct answers and subtracting points for incorrect or unanswered questions:[4c - (25 - c) geq 60]Simplifying this inequality:[4c - 25 + c geq 60][5c - 25 geq 60][5c geq 85][c geq 17]We verify this by checking the score for 17 correct answers: ( 17 times 4 = 68 ) points from correct answers and ( 25 - 17 = 8 ) points deducted for incorrect answers, resulting in a total score of ( 68 - 8 = 60 ). For 16 correct answers, the score is ( 16 times 4 = 64 ) points from correct answers and ( 25 - 16 = 9 ) points deducted, resulting in a total score of ( 64 - 9 = 55 ), which is insufficient. Thus, the minimum number of correct answers required is 17.[boxed{17}],<think>Okay, so I need to figure out how many questions a student needs to get right in the first round to qualify for the second round. The competition has 25 questions, right? For each correct answer, you get 4 points, and for each incorrect or unanswered question, they deduct 1 point. To qualify, you need a score of at least 60 points. Hmm, let me break this down step by step.First, let's define some variables to make this easier. Let me call the number of correct answers as 'x'. Then, since there are 25 questions in total, the number of incorrect or unanswered questions would be 25 minus x, right? So that's 25 - x.Now, the total score is calculated by the points from correct answers minus the points deducted for incorrect or unanswered ones. Each correct answer gives 4 points, so that's 4x. Each incorrect or unanswered question deducts 1 point, so that's -1*(25 - x). So the total score S would be 4x - (25 - x).They need this total score S to be at least 60 points. So we can set up the inequality:4x - (25 - x) ≥ 60.Let me solve this step by step. First, distribute the negative sign into the parentheses:4x - 25 + x ≥ 60.Combine like terms. 4x + x is 5x:5x - 25 ≥ 60.Now, add 25 to both sides to isolate the term with x:5x - 25 + 25 ≥ 60 + 25.Which simplifies to:5x ≥ 85.Then, divide both sides by 5 to solve for x:5x / 5 ≥ 85 / 5.So:x ≥ 17.Wait, so x has to be at least 17? Let me check that again to make sure I didn't make any mistakes.Starting with the total score:Total score = 4x - (25 - x).If x is 17, then the total score is 4*17 - (25 - 17) = 68 - 8 = 60. Okay, so 17 correct answers give exactly 60 points, which is the minimum needed to qualify. Therefore, students need to answer at least 17 questions correctly. Hmm, but wait, let me test with 16 to see what happens. If x is 16, then total score is 4*16 - (25 - 16) = 64 - 9 = 55. Which is less than 60. So 16 is not enough. If they answer 17, they get exactly 60, which is the cutoff. So the answer must be 17.But let me verify my equation again. The formula is correct? Each correct gives +4, each incorrect or blank gives -1, so yes. Total score is 4x - (25 - x). Then setting that equal to 60:4x -25 + x = 605x = 85x = 17. Yep, that seems right.Alternatively, maybe think in terms of total points. If all questions were answered correctly, you'd get 25*4 = 100 points. But if you get some wrong, each wrong answer not only doesn't give you the 4 points but also deducts 1. So the difference between a correct and an incorrect is 4 + 1 = 5 points. Because if you got a question right, you get 4, if you get it wrong, you lose 1, so the difference is 5 per question.Wait, so perhaps another way to think about it is: Starting from 100, each incorrect answer deducts 5 points. Because instead of gaining 4, you lose 1, so total difference is 5. So if you have y incorrect answers, then total score is 100 - 5y. We need 100 - 5y ≥ 60.Solving for y:100 - 5y ≥ 60Subtract 100:-5y ≥ -40Multiply both sides by -1 (which reverses the inequality):5y ≤ 40Divide by 5:y ≤ 8.So the number of incorrect answers must be at most 8. Since the total questions are 25, correct answers x = 25 - y. So x ≥ 25 - 8 = 17. So same answer, x is at least 17. That confirms the previous result.Therefore, the minimum number of correct answers needed is 17. So students must have answered at least 17 questions correctly.Just to make sure, let's do a quick check with x=17 and x=16.For x=17: Correct answers:17*4=68. Incorrect:8*-1=-8. Total:68-8=60. Exactly the qualifying score.For x=16: Correct:16*4=64. Incorrect:9*-1=-9. Total:64-9=55. Which is below 60. So indeed, 17 is the minimum.Another way to think about it: Each question contributes either +4 or -1. The maximum penalty for a wrong answer compared to leaving it blank is the same? Wait, no, the problem says "each incorrect or unanswered question deducts 1 point." So whether you leave it blank or answer incorrectly, you lose 1 point. So the strategy here is, it's better to leave a question blank if you don't know the answer because you might avoid a penalty? Wait, but in this case, both incorrect and unanswered deduct 1, so it's the same. So whether you guess or not, you lose 1 point. So in that case, maybe the optimal strategy is to answer all questions, but since incorrect and unanswered are treated the same, perhaps students should answer all the ones they know and leave the rest blank? But in any case, for the problem here, the calculation is the same. The number of correct is x, incorrect or blank is 25 - x, each of those deducts 1.So the equation holds as before. So, the conclusion is 17. So students need at least 17 correct answers.Final AnswerThe students who qualify for the second round must have answered at least boxed{17} questions correctly.</think>To determine the minimum number of correct answers needed to qualify for the second round, we define ( x ) as the number of correct answers. Each correct answer earns 4 points, and each incorrect or unanswered question deducts 1 point. The total score ( S ) is given by:[S = 4x - (25 - x)]We need this score to be at least 60 points:[4x - (25 - x) geq 60]Simplifying the inequality:[4x - 25 + x geq 60][5x - 25 geq 60][5x geq 85][x geq 17]To verify, if a student answers 17 questions correctly, the score is:[4 times 17 - (25 - 17) = 68 - 8 = 60]If a student answers 16 questions correctly, the score is:[4 times 16 - (25 - 16) = 64 - 9 = 55]Thus, 17 correct answers yield the minimum qualifying score of 60, while 16 correct answers do not. Therefore, the students must have answered at least (boxed{17}) questions correctly.]

🔑:【Analysis】17 questionsIf all answers are correct, the full score is 4 times 25=100 points, and for each wrong answer, 4+1=5 points will be deducted; since the maximum deduction is 100-60=40 points, which means a maximum of 40 div 5=8 questions can be answered incorrectly; therefore, at least 25-8=17 questions need to be answered correctly.

🧠:[<think>Okay, let's try to solve this problem step by step. So, we have three people: Ana, her mother, and her grandmother. All three celebrate their birthdays on June 23rd. Today is June 23rd, 2021, and the sum of their ages is 127. First, I need to figure out their current ages. Let me denote Ana's current age as A, her mother's current age as M, and her grandmother's current age as G. So, according to the problem, A + M + G = 127.Then, the problem says, "When Ana was 4 years old, her grandmother was twice as old as her mother." Hmm. Let me parse that. So, we need to look back in time to when Ana was 4. How many years ago was that? If Ana is currently A years old, then that was (A - 4) years ago. At that time, Ana's grandmother's age would have been G - (A - 4), and her mother's age would have been M - (A - 4). According to the problem, at that time, the grandmother was twice as old as her mother. So, the equation would be:G - (A - 4) = 2 * [M - (A - 4)]Let me write that down:G - A + 4 = 2M - 2A + 8Simplify that:G - A + 4 = 2M - 2A + 8Bring all terms to one side:G - A + 4 - 2M + 2A - 8 = 0Combine like terms:G + A - 2M - 4 = 0So, G + A - 2M = 4Let me note that as equation (1): G + A - 2M = 4Next part: "Two years ago, her mother was three times as old as Ana." Two years ago from today (2021), so in 2019. At that time, the mother's age was M - 2, and Ana's age was A - 2. According to the problem, the mother was three times as old as Ana then. So:M - 2 = 3*(A - 2)Let me write that as equation (2):M - 2 = 3A - 6Simplify:M = 3A - 6 + 2M = 3A - 4So, equation (2) is M = 3A - 4We also have the sum of their ages: A + M + G = 127. Let's denote that as equation (3).So, now we have three equations:1. G + A - 2M = 42. M = 3A - 43. A + M + G = 127Our variables are A, M, G. Let's try to substitute equation (2) into equations (1) and (3) to solve for A and G.Starting with equation (1):G + A - 2M = 4But since M = 3A - 4, substitute that into the equation:G + A - 2*(3A - 4) = 4Calculate:G + A - 6A + 8 = 4Combine like terms:G - 5A + 8 = 4So,G = 5A - 8 + 4G = 5A - 4So, G = 5A - 4. Let's note that as equation (4).Now, substitute equations (2) and (4) into equation (3):A + M + G = 127Replace M and G:A + (3A - 4) + (5A - 4) = 127Simplify:A + 3A - 4 + 5A - 4 = 127Combine terms:(1A + 3A + 5A) + (-4 -4) = 1279A - 8 = 127Add 8 to both sides:9A = 135Divide by 9:A = 15So, Ana is currently 15 years old. Now, let's find M (mother's age) using equation (2):M = 3A - 4 = 3*15 - 4 = 45 - 4 = 41So, the mother is 41 years old. Now, grandmother's age G is from equation (4):G = 5A - 4 = 5*15 - 4 = 75 - 4 = 71So, grandmother is 71 years old. Let's verify the sum: 15 + 41 + 71 = 127. Yes, that's correct.Now, the first question: In which year was Ana's mother born?If today is June 23rd, 2021, and the mother is 41 years old, then her birth year would be 2021 - 41 = 1980. But we need to check if her birthday has already occurred this year. However, since today is June 23rd, and their birthdays are on June 23rd, so if today is June 23rd, 2021, and the mother is 41, then she was born on June 23rd, 1980. So, the answer is 1980.Wait, but if today is June 23rd, and someone is 41 years old, then they were born on June 23rd, 2021 - 41 = 1980. So, yes, 1980 is the birth year.Second question: How old will Ana be when she and her mother together have the same number of years as her grandmother? In which year will that be?Hmm. So, we need to find a future year when Ana's age plus her mother's age equals her grandmother's age at that future time. Wait, but the way it's phrased: "when she and her mother together have the same number of years as her grandmother." So, Ana's age + mother's age = grandmother's age. But all three are aging at the same rate. So, let's denote x as the number of years from now (2021) until when Ana's age + mother's age = grandmother's age.Currently, Ana is 15, mother is 41, grandmother is 71.In x years:Ana's age: 15 + xMother's age: 41 + xGrandmother's age: 71 + xSo, the equation is:(15 + x) + (41 + x) = 71 + xSimplify:56 + 2x = 71 + xSubtract x from both sides:56 + x = 71Subtract 56:x = 15So, in 15 years from 2021, which is 2036, Ana will be 15 + 15 = 30 years old, her mother will be 41 + 15 = 56, and grandmother will be 71 + 15 = 86. Let's check: 30 + 56 = 86, which is correct. So, Ana will be 30 years old in the year 2036.Let me double-check the math. The equation:(15 + x) + (41 + x) = 71 + x56 + 2x = 71 + x2x - x = 71 - 56x = 15. Yep, correct.So, the answer to the second part is Ana will be 30 years old, and that will be in the year 2036.Let me recap all steps to ensure I didn't make a mistake.First, set up variables for current ages.Used the three conditions:1. Sum of ages is 127.2. When Ana was 4, grandmother was twice as old as mother.3. Two years ago, mother was three times as old as Ana.Solved the equations step by step, substituted, found Ana is 15, mother 41, grandmother 71.Checked the sum, correct.For the mother's birth year: 2021 - 41 = 1980. Since birthday is today, June 23rd, so yes, 1980.For the second part, set up the equation for future ages where Ana + mother = grandmother. Solved for x = 15, leading to Ana being 30 in 2036.Everything checks out. I think that's correct.Final AnswerAna's mother was born in the year boxed{1980}. Ana will be boxed{30} years old in the year boxed{2036}.</think>Let ( A ) be Ana's current age, ( M ) be her mother's current age, and ( G ) be her grandmother's current age. We are given the following information:1. The sum of their ages is 127: [ A + M + G = 127 ]2. When Ana was 4 years old, her grandmother was twice as old as her mother. This translates to the equation: [ G - (A - 4) = 2(M - (A - 4)) ] Simplifying this, we get: [ G + A - 2M = 4 ]3. Two years ago, her mother was three times as old as Ana: [ M - 2 = 3(A - 2) ] Simplifying this, we get: [ M = 3A - 4 ]We now solve these equations step by step:1. Substitute ( M = 3A - 4 ) into ( G + A - 2M = 4 ): [ G + A - 2(3A - 4) = 4 implies G + A - 6A + 8 = 4 implies G - 5A = -4 implies G = 5A - 4 ]2. Substitute ( M = 3A - 4 ) and ( G = 5A - 4 ) into ( A + M + G = 127 ): [ A + (3A - 4) + (5A - 4) = 127 implies 9A - 8 = 127 implies 9A = 135 implies A = 15 ]3. Calculate ( M ) and ( G ): [ M = 3(15) - 4 = 41 ] [ G = 5(15) - 4 = 71 ]Ana's mother is currently 41 years old, so she was born in ( 2021 - 41 = 1980 ).To find when Ana and her mother together have the same number of years as her grandmother, let ( x ) be the number of years from now. We need:[(15 + x) + (41 + x) = 71 + x]Solving this:[56 + 2x = 71 + x implies x = 15]In 15 years, Ana will be ( 15 + 15 = 30 ) years old, and the year will be ( 2021 + 15 = 2036 ). Final AnswerAna's mother was born in the year boxed{1980}. Ana will be boxed{30} years old in the year boxed{2036}.,<think>Alright, let's tackle this problem step by step. It's a bit complex with multiple questions, so I need to break it down carefully. Let me start by listing all the information given and then try to translate them into mathematical equations.First, the problem states:1. Today, June 23rd, 2021, the sum of Ana, her mother, and her grandmother's ages is 127.2. When Ana was 4 years old, her grandmother was twice as old as her mother.3. Two years ago, Ana's mother was three times as old as Ana.4. We need to find: a. The year Ana's mother was born. b. How old Ana will be when she and her mother together have the same number of years as her grandmother. c. The year when that will happen.Let me assign variables to their current ages. Let's denote:- Let A = Ana's current age (in 2021)- Let M = Mother's current age- Let G = Grandmother's current ageFrom the first piece of information, the sum of their ages is 127. So, equation 1 is:A + M + G = 127Now, the second piece of information: When Ana was 4 years old, her grandmother was twice as old as her mother.Wait, "When Ana was 4 years old" – that was (A - 4) years ago. Because Ana is currently A years old, so subtracting 4 gives the number of years ago that was. So, at that time:- Ana's age was 4- Mother's age was M - (A - 4)- Grandmother's age was G - (A - 4)And at that time, the grandmother's age was twice the mother's age. So:G - (A - 4) = 2 * [M - (A - 4)]That's equation 2.Third piece of information: Two years ago, the mother was three times as old as Ana. So two years ago:- Mother's age was M - 2- Ana's age was A - 2So:M - 2 = 3 * (A - 2)That's equation 3.So, now we have three equations:1. A + M + G = 1272. G - (A - 4) = 2[M - (A - 4)]3. M - 2 = 3(A - 2)Let me work through these equations step by step. Maybe start with equation 3 since it relates only A and M, so we can find a relationship between A and M first.Equation 3: M - 2 = 3(A - 2)Let me solve for M:M = 3(A - 2) + 2M = 3A - 6 + 2M = 3A - 4So, the mother's current age is 3A - 4. So we can express M in terms of A. That's helpful.Now, let's go to equation 2 and try to express G in terms of A as well, perhaps.Equation 2: G - (A - 4) = 2[M - (A - 4)]First, let's simplify both sides.Left side: G - (A - 4) = G - A + 4Right side: 2[M - (A - 4)] = 2[M - A + 4] = 2M - 2A + 8So, putting it together:G - A + 4 = 2M - 2A + 8Let's rearrange terms to solve for G:G = 2M - 2A + 8 + A - 4Simplify:G = 2M - A + 4Now, we can substitute M from equation 3 into this equation. Since M = 3A - 4, substitute:G = 2*(3A - 4) - A + 4Calculate:G = 6A - 8 - A + 4G = 5A - 4So now, we have expressions for both M and G in terms of A. That's great. Now, let's plug these into equation 1 to solve for A.Equation 1: A + M + G = 127Substituting M and G:A + (3A - 4) + (5A - 4) = 127Combine like terms:A + 3A + 5A - 4 - 4 = 1279A - 8 = 127Add 8 to both sides:9A = 135Divide by 9:A = 15So, Ana is currently 15 years old. Then, let's find M and G.M = 3A - 4 = 3*15 - 4 = 45 - 4 = 41G = 5A - 4 = 5*15 - 4 = 75 - 4 = 71So, currently (in 2021):- Ana is 15- Mother is 41- Grandmother is 71Let's check if these numbers add up to 127: 15 + 41 + 71 = 127. Yes, that's correct.Now, check the other conditions to make sure.First, when Ana was 4, that was 15 - 4 = 11 years ago. So, 2021 - 11 = 2010.In 2010:- Ana was 4- Mother was 41 - 11 = 30- Grandmother was 71 - 11 = 60Was the grandmother twice as old as the mother then? 60 vs. 30. Yes, 60 is twice 30. So that's good.Next, two years ago (2019):- Ana was 15 - 2 = 13- Mother was 41 - 2 = 39Was the mother three times as old as Ana? 39 vs. 3*13 = 39. Yes, that's correct. So all conditions are satisfied.Now, the first question: In which year was Ana's mother born?If the mother is currently 41 years old in 2021, then her birth year is 2021 - 41 = 1980. But we need to be careful about the exact date. The problem says their birthday is June 23rd, and today is June 23rd, 2021. So, if today is her birthday, then she turned 41 today, so her birth year is indeed 2021 - 41 = 1980. So the answer is 1980.Second question: How old will Ana be when she and her mother together have the same number of years as her grandmother? In which year will that be?So, we need to find a future year when Ana's age plus her mother's age equals her grandmother's age at that time.Let me denote the number of years from now until that future time as x. So, in the year 2021 + x:- Ana's age will be 15 + x- Mother's age will be 41 + x- Grandmother's age will be 71 + xWe need to find x such that:(15 + x) + (41 + x) = 71 + xSimplify the left side:15 + x + 41 + x = 56 + 2xSo:56 + 2x = 71 + xSubtract x from both sides:56 + x = 71Subtract 56:x = 15So, in 15 years, which would be 2021 + 15 = 2036, Ana will be 15 + 15 = 30 years old, her mother will be 41 + 15 = 56, and the grandmother will be 71 + 15 = 86.Check: 30 (Ana) + 56 (Mother) = 86, which equals the grandmother's age. Correct.Therefore, Ana will be 30 years old in the year 2036.So summarizing:a. Ana's mother was born in 1980.b. Ana will be 30 years old.c. That will be in the year 2036.Just to double-check, let's verify the calculations again.Mother's birth year: 2021 - 41 = 1980. Correct.For the second part, when Ana and her mother's combined age equals grandmother's age.Equation: (15 + x) + (41 + x) = 71 + xWhich gives x=15. So 2021 +15=2036, Ana's age=15+15=30. Correct.Everything checks out. Therefore, the answers are as above.Final AnswerAna's mother was born in the year boxed{1980}. Ana will be boxed{30} years old in the year boxed{2036}.</think>Let ( A ) be Ana's current age, ( M ) be her mother's current age, and ( G ) be her grandmother's current age. We are given the following information:1. The sum of their ages is 127: [ A + M + G = 127 ]2. When Ana was 4 years old, her grandmother was twice as old as her mother: [ G - (A - 4) = 2[M - (A - 4)] ]3. Two years ago, her mother was three times as old as Ana: [ M - 2 = 3(A - 2) ]First, solve equation 3 for ( M ):[M - 2 = 3(A - 2) implies M = 3A - 4]Next, substitute ( M = 3A - 4 ) into equation 2 and solve for ( G ):[G - (A - 4) = 2[(3A - 4) - (A - 4)] implies G - A + 4 = 2(2A) implies G - A + 4 = 4A implies G = 5A - 4]Substitute ( M = 3A - 4 ) and ( G = 5A - 4 ) into equation 1:[A + (3A - 4) + (5A - 4) = 127 implies 9A - 8 = 127 implies 9A = 135 implies A = 15]Thus, Ana's current age ( A ) is 15. Substitute back to find ( M ) and ( G ):[M = 3(15) - 4 = 41 quad text{and} quad G = 5(15) - 4 = 71]Ana's mother is currently 41 years old in 2021, so she was born in:[2021 - 41 = 1980]To find when Ana and her mother's combined age equals her grandmother's age, let ( x ) be the number of years from now:[(15 + x) + (41 + x) = 71 + x implies 56 + 2x = 71 + x implies x = 15]In 15 years, Ana will be:[15 + 15 = 30]and the year will be:[2021 + 15 = 2036]Final answers:- Ana's mother was born in the year boxed{1980}.- Ana will be boxed{30} years old in the year boxed{2036}.]

🔑:## Solution.This year, Ana is A years old, her mother is M years old, and her grandmother is B years old. It is given that A+M+B=127.Ana was 4 years old x years ago. Therefore, x=A-4.Furthermore, x years ago, her mother was M-x years old, and her grandmother was B-x years old. According to the condition of the problem,begin{aligned}B-x=2 cdot(M-x) B-(A-4)=2 cdot(M-(A-4)) B-A+4=2 M-2 A+8 A+B-2 M=4end{aligned}From the first equation, we have A+B=127-M, sobegin{gathered}127-M-2 M=4 3 M=123 M=41end{gathered}Ana's mother is 41 years old, which means she was born in 1980.Two years ago, her mother was 39 years old and three times as old as Ana. This means that Ana was 39 div 3 = 13 years old two years ago, and she is now 15 years old (A=15).The grandmother is 127-41-15=71 years old (B=71).In y years, Ana and her mother together will be as old as the grandmother. It holds thatbegin{gathered}15+y+41+y=71+y 56+y=71 y=15end{gathered}Ana will be 30 years old when she and her mother together will be as old as the grandmother.This will be in the year 2036.

🧠:[<think>Okay, let's see. I need to solve this geometry problem about triangle ABC. There are two parts: part I asks for the measure of angle A, and part II asks for the range of the perimeter l when a=1. Alright, let's start with part I.First, the problem mentions vectors m and n. Vector m is (a, 1/2) and vector n is (cos C, c - 2b). They are perpendicular, which means their dot product is zero. So, maybe I should start by setting up the dot product equation.The dot product of m and n is a*cos C + (1/2)*(c - 2b) = 0. Let me write that down:a*cos C + (1/2)(c - 2b) = 0.Hmm, simplifying this equation might help. Let's distribute the 1/2:a*cos C + (c/2 - b) = 0.So, a*cos C = b - c/2. That's equation (1).Now, I need to relate this to the triangle's angles and sides. Maybe using the Law of Cosines? The Law of Cosines says that c^2 = a^2 + b^2 - 2ab*cos C. Wait, but here we have a*cos C. Let's see if we can express cos C in terms of sides.From the Law of Cosines:cos C = (a^2 + b^2 - c^2)/(2ab).Plugging this into equation (1):a*(a^2 + b^2 - c^2)/(2ab) = b - c/2.Simplify the left side:(a*(a^2 + b^2 - c^2))/(2ab) = (a^2 + b^2 - c^2)/(2b).So, we have:(a^2 + b^2 - c^2)/(2b) = b - c/2.Multiply both sides by 2b to eliminate the denominator:a^2 + b^2 - c^2 = 2b^2 - bc.Simplify:a^2 + b^2 - c^2 - 2b^2 + bc = 0.Combine like terms:a^2 - b^2 - c^2 + bc = 0.Hmm, this seems a bit complicated. Let's rearrange terms:a^2 = b^2 + c^2 - bc.Wait a minute, the right-hand side is similar to the Law of Cosines. Let's recall that the Law of Cosines is:a^2 = b^2 + c^2 - 2bc*cos A.Comparing this to our equation:b^2 + c^2 - bc = b^2 + c^2 - 2bc*cos A.Subtract b^2 + c^2 from both sides:-bc = -2bc*cos A.Divide both sides by -bc (assuming bc ≠ 0, which they aren't in a triangle):1 = 2cos A.So, cos A = 1/2.Therefore, angle A is arccos(1/2), which is 60 degrees or π/3 radians.So, part I's answer is 60 degrees. Let me check if this makes sense.Wait, let's retrace the steps to ensure I didn't make a mistake. Starting from the dot product, set up the equation, substituted cos C using Law of Cosines, simplified, and ended up with cos A = 1/2. Seems correct. So angle A is 60 degrees. Good.Now, part II: If a=1, find the range of the perimeter l of triangle ABC.We know that a=1, which is the side opposite angle A, which we found to be 60 degrees. So, triangle ABC has side a=1 opposite angle A=60°, and sides b and c opposite angles B and C, respectively. The perimeter l = 1 + b + c. So we need to find the possible values of b and c such that the triangle is valid, given angle A=60° and a=1.First, using the Law of Sines might be helpful here. The Law of Sines states that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.Given that a=1 and angle A=60°, so:1/sin 60° = 2R => 2R = 1/(√3/2) = 2/√3. So R = 1/√3.Therefore, b = 2R sin B = (2/√3) sin B,c = 2R sin C = (2/√3) sin C.Since in a triangle, the sum of angles is 180°, so B + C = 120°. Therefore, C = 120° - B.So, c = (2/√3) sin(120° - B).Let me express c in terms of B. Using the sine subtraction formula:sin(120° - B) = sin 120° cos B - cos 120° sin B.We know sin 120° = √3/2, cos 120° = -1/2.So, sin(120° - B) = (√3/2)cos B - (-1/2)sin B = (√3/2)cos B + (1/2)sin B.Therefore, c = (2/√3)[ (√3/2)cos B + (1/2)sin B ] = (2/√3)(√3/2 cos B + 1/2 sin B ) = (2/√3)( (√3 cos B + sin B)/2 ) = (√3 cos B + sin B)/√3.Simplify:c = (√3 cos B + sin B)/√3 = cos B + (sin B)/√3.Similarly, b = (2/√3) sin B.So, perimeter l = 1 + b + c = 1 + (2/√3) sin B + cos B + (sin B)/√3.Let me combine the terms with sin B:(2/√3 + 1/√3) sin B + cos B + 1 = (3/√3) sin B + cos B + 1 = √3 sin B + cos B + 1.Therefore, perimeter l = √3 sin B + cos B + 1.Now, to find the range of l, we need to find the maximum and minimum values of the expression √3 sin B + cos B + 1, as angle B varies. Since angles B and C must be positive and add up to 120°, angle B must be between 0° and 120°. So B ∈ (0°, 120°).But in reality, in a triangle, each angle must be greater than 0° and less than 180°, but since A=60°, B and C must each be less than 120°, but greater than 0°, so B ∈ (0°, 120°). However, we also need to ensure that sides b and c are positive, which they will be as long as angles B and C are positive, so the domain is B ∈ (0°, 120°).Therefore, we need to find the range of the function f(B) = √3 sin B + cos B + 1 for B ∈ (0°, 120°). Let's work in radians for easier calculus. 0° = 0 radians, 120° = 2π/3 radians.First, let's analyze the function f(B) = √3 sin B + cos B + 1.This is a sinusoidal function. The expression √3 sin B + cos B can be written as a single sine (or cosine) function using the amplitude-phase form. Let's compute its amplitude.For the expression K sin B + M cos B, the amplitude is √(K² + M²).Here, K = √3 and M = 1, so amplitude is √( (√3)^2 + 1^2 ) = √(3 + 1) = √4 = 2.Therefore, √3 sin B + cos B = 2 sin(B + α), where α is the phase shift. To find α:tan α = M/K = 1/√3 => α = 30° or π/6 radians.Therefore, √3 sin B + cos B = 2 sin(B + π/6).Therefore, f(B) = 2 sin(B + π/6) + 1.So, the perimeter l = 2 sin(B + π/6) + 1.Now, we need to find the range of this function when B ∈ (0, 2π/3).So, B ranges from 0 to 2π/3 (120°). Then, B + π/6 ranges from π/6 to 2π/3 + π/6 = 5π/6.So, the argument of the sine function, θ = B + π/6, ranges from π/6 to 5π/6.The sine function sin θ reaches its maximum at θ = π/2 (which is 90°) with sin(π/2) = 1, and its minimum at the endpoints of the interval [π/6, 5π/6]. Let's check the sine values at θ = π/6 and θ = 5π/6:sin(π/6) = 1/2,sin(5π/6) = 1/2.So, sin θ ranges from 1/2 to 1 in this interval. Therefore, 2 sin θ ranges from 1 to 2, so 2 sin θ + 1 ranges from 2 to 3.Therefore, the perimeter l ranges from 2 to 3.But wait, hold on. Let me confirm:Wait, when θ ranges from π/6 to 5π/6, sin θ is maximum at π/2 (sin π/2 = 1), and minimum at π/6 and 5π/6 (sin π/6 = 1/2). Therefore, 2 sin θ ranges from 1 to 2, hence 2 sin θ + 1 ranges from 2 to 3. Therefore, l ∈ [2, 3].But wait, this seems too straightforward. Let me check with specific angles.For example, when B approaches 0°, then angle C approaches 120°, but sides b approaches 0 and c approaches... Let's see.If B approaches 0°, then sin B approaches 0, so b = (2/√3) sin B approaches 0. Then, c = cos B + (sin B)/√3. If B approaches 0°, cos B approaches 1, sin B approaches 0, so c approaches 1 + 0 = 1. Therefore, perimeter l approaches 1 + 0 + 1 = 2.Similarly, when B approaches 120°, angle C approaches 0°, so c approaches 0. Let's check:If B approaches 120°, then angle C approaches 0°, so sin C approaches 0, so c = (2/√3) sin C approaches 0. Also, sin B approaches sin 120° = √3/2, so b = (2/√3) sin B = (2/√3)(√3/2) = 1. Then, c = cos B + (sin B)/√3. cos 120° = -1/2, sin 120° = √3/2. So c = -1/2 + (√3/2)/√3 = -1/2 + 1/2 = 0. So perimeter l approaches 1 + 1 + 0 = 2. So in this case, the perimeter approaches 2 again.Wait, but according to our previous analysis, the perimeter ranges from 2 to 3. But when B is 30°, let's see:If B = 30°, then angle C = 120° - 30° = 90°. Then, b = (2/√3) sin 30° = (2/√3)(1/2) = 1/√3 ≈ 0.577. c = (2/√3) sin 90° = (2/√3)(1) = 2/√3 ≈ 1.154. Perimeter l = 1 + 1/√3 + 2/√3 = 1 + 3/√3 = 1 + √3 ≈ 2.732, which is less than 3.Wait, but according to our previous result, the maximum should be 3. Hmm, that's a problem. There must be a mistake here.Wait, where is the error? Let's check again.We had f(B) = √3 sin B + cos B + 1, which we converted to 2 sin(B + π/6) + 1. Then, as B ranges from 0 to 2π/3, B + π/6 ranges from π/6 to 5π/6. Then, sin(B + π/6) ranges from 1/2 (at π/6 and 5π/6) to 1 (at π/2). Therefore, 2 sin(B + π/6) + 1 ranges from 2*(1/2) + 1 = 2 to 2*1 + 1 = 3. So the perimeter l should be between 2 and 3. However, when I checked with B=30°, which is within the range, the perimeter was about 2.732, which is between 2 and 3, so that's okay.Wait, but when I considered B approaching 120°, I thought that c approaches 0, but let me check again. If B is 120°, then angle C = 0°, which is impossible because angles must be greater than 0. So B cannot actually reach 120°, it can only approach it. Similarly, B cannot be 0°, but can approach it. Therefore, the perimeter approaches 2 but never actually reaches 2. However, the maximum occurs when B + π/6 = π/2, so when B = π/2 - π/6 = π/3, which is 60°. Let's check B=60°:If B=60°, then angle C=60°, so it's an equilateral triangle? Wait, but a=1, and angle A=60°. If all angles are 60°, then all sides are equal. Wait, but a=1, so all sides should be 1. Then, perimeter would be 3. Let's check with the Law of Sines.If angle B=60°, then b = (2/√3) sin 60° = (2/√3)(√3/2) = 1.Similarly, angle C=60°, so c=1. Therefore, perimeter l=1+1+1=3. That matches the upper bound.Therefore, when B=60°, the triangle is equilateral with all sides 1, perimeter 3. So the maximum perimeter is 3, achieved when the triangle is equilateral.However, when I checked B=30°, the perimeter was about 2.732, which is between 2 and 3, as expected.Wait, but when B approaches 0°, the perimeter approaches 2, and when B approaches 120°, perimeter also approaches 2. So the perimeter can get as close to 2 as desired but never actually reaches 2 because B can't be 0° or 120°. Therefore, the range of l is (2, 3]. Wait, but in the problem statement, it says "the range of the perimeter l". Depending on the strictness of the inequalities, when B approaches 0° or 120°, l approaches 2, but never reaches 2. However, when B=60°, l=3. So the range is (2, 3]. But need to confirm if 3 is included. Yes, when B=60°, the triangle is valid (equilateral), so 3 is included. So the range is (2, 3].But wait, in the previous analysis using the function f(B) = 2 sin(B + π/6) + 1, when B is in (0, 2π/3), then θ = B + π/6 is in (π/6, 5π/6). The function sin θ in this interval reaches a maximum of 1 at θ=π/2, and the minimum approaches 1/2 as θ approaches π/6 or 5π/6. Therefore, sin θ is in (1/2, 1], so 2 sin θ is in (1, 2], so 2 sin θ + 1 is in (2, 3]. Therefore, the range of l is (2, 3].But in the problem statement, part II says "find the range of the perimeter l". Depending on the problem's requirements, sometimes they might accept [2, 3] if considering the limits. But strictly mathematically, since B cannot be 0° or 120°, the perimeter can't be 2. However, in some contexts, especially in optimization problems involving triangles, the interval might be considered closed if you can approach the limit as closely as desired. But in reality, since the sides must be positive, l can't actually be 2. Therefore, the range is 2 < l ≤ 3.But let me check with specific triangles. Suppose B is very small, say B=1°, then angle C=119°, compute sides b and c:b = (2/√3) sin(1°) ≈ (2/1.732)*0.01745 ≈ 0.0201,c = (2/√3) sin(119°) ≈ (2/1.732)*sin(119°). Wait, sin(119°) = sin(180° - 61°) = sin(61°) ≈ 0.8746,so c ≈ (2/1.732)*0.8746 ≈ 1.007.Therefore, perimeter l ≈ 1 + 0.0201 + 1.007 ≈ 2.027, which is just above 2.Similarly, if B approaches 120°, say B=119°, then angle C=1°, so c ≈ (2/√3) sin(1°) ≈ 0.0201,and b = (2/√3) sin(119°) ≈ 1.007,perimeter l ≈ 1 + 1.007 + 0.0201 ≈ 2.027.So the perimeter approaches 2 from above but never reaches it. The maximum perimeter is 3 when the triangle is equilateral.Therefore, the range of l is (2, 3]. So in boxed notation, we can write this as (2, 3].But sometimes, in exams, they might expect the interval to be written as 2 < l ≤ 3.However, depending on the problem's expectation, maybe they consider 2 as attainable, but strictly speaking, it's not possible. So perhaps the answer is (2, 3].But let me check once more. Wait, maybe there's an error in assuming that when B approaches 0°, the perimeter approaches 2. Wait, when B approaches 0°, angle C approaches 120°, so let's use the Law of Cosines to see the relationship between sides.Given angle A=60°, a=1.Law of Cosines: a² = b² + c² - 2bc cos A.So, 1 = b² + c² - 2bc*(1/2) => 1 = b² + c² - bc.If B approaches 0°, then angle C approaches 120°, and side b approaches 0. Let's set b approaching 0, then 1 ≈ 0 + c² - 0 => c² ≈ 1 => c ≈ 1. So perimeter l ≈ 1 + 0 + 1 = 2.But if b is exactly 0, it's not a triangle. So as b approaches 0, c approaches 1, and the perimeter approaches 2. Similarly, if c approaches 0, b approaches 1, perimeter approaches 2. So indeed, the perimeter can get arbitrarily close to 2 but never actually reach it. So the range is (2, 3].Therefore, the answer to part II is that the perimeter l is in the interval (2, 3].But let's check if there's another approach. Maybe using triangle inequalities. Since a=1, and the other sides must satisfy triangle inequalities.We have sides 1, b, c. So:1 + b > c,1 + c > b,b + c > 1.But since angle A=60°, perhaps we can use the Law of Cosines relation we had earlier: 1 = b² + c² - bc.So, maybe the perimeter l = 1 + b + c, with the constraint 1 = b² + c² - bc.We can try to express l in terms of one variable. Let me set variables to parameterize.Let’s let’s consider b and c such that 1 = b² + c² - bc. Maybe express c in terms of b or vice versa.Alternatively, use substitution. Let’s set x = b + c and y = bc. Then, the equation becomes 1 = (b² + c²) - y.But b² + c² = (b + c)^2 - 2bc = x² - 2y.Therefore, 1 = x² - 2y - y => x² - 3y = 1 => y = (x² - 1)/3.But perimeter l = 1 + x. So we need to relate x and y with the constraints.But maybe this approach is complicating things. Alternatively, since we have l = 1 + b + c, and 1 = b² + c² - bc, we can try to express this as an optimization problem with constraints.Let’s consider l = 1 + b + c, subject to the constraint b² + c² - bc = 1.We can use Lagrange multipliers, but maybe there's a simpler way.Let’s consider variables b and c. Let’s set t = b + c, s = bc. Then, from the constraint:b² + c² - bc = (b + c)^2 - 3bc = t² - 3s = 1.We need to express l = 1 + t. So, t = l - 1.So, substituting into the constraint:(l - 1)^2 - 3s = 1 => (l - 1)^2 - 1 = 3s => s = [ (l - 1)^2 - 1 ] / 3.But also, since b and c are real positive numbers, the discriminant of the quadratic equation must be non-negative. Let's consider that b and c are roots of the equation x² - tx + s = 0.For real roots, discriminant D = t² - 4s ≥ 0.Substituting s from above:D = t² - 4*[ ( (l - 1)^2 - 1 ) / 3 ] ≥ 0.But t = l - 1, so:(l - 1)^2 - (4/3)[ ( (l - 1)^2 - 1 ) ] ≥ 0.Simplify:Multiply through by 3 to eliminate denominator:3(l - 1)^2 - 4[ (l - 1)^2 - 1 ] ≥ 0Expand:3(l - 1)^2 - 4(l - 1)^2 + 4 ≥ 0Combine like terms:- (l - 1)^2 + 4 ≥ 0Multiply both sides by -1 (which reverses inequality):(l - 1)^2 - 4 ≤ 0Factor:[(l - 1) - 2][(l - 1) + 2] ≤ 0 => (l - 3)(l + 1) ≤ 0.So the product is ≤ 0 when one factor is non-positive and the other is non-negative. Since l + 1 > 0 (perimeter is positive), then (l - 3) ≤ 0. Therefore, l - 3 ≤ 0 => l ≤ 3.So combining this with the previous result, the discriminant is non-negative only when l ≤ 3. But we also know from the earlier analysis that l > 2. So combining both results, the range of l is 2 < l ≤ 3.Therefore, this confirms our previous conclusion. So part II's answer is (2, 3].Hence, summarizing:(I) Angle A is 60°, so the answer is 60 degrees or π/3 radians. Since the problem doesn't specify, but in Chinese exams, degrees are common unless specified. Let me check the original problem statement. It says "measure of angle A", which is commonly in degrees unless specified otherwise. However, in higher mathematics, radians are also possible, but since it's a triangle problem, degrees are more likely. But since π/3 is also 60°, both are correct. But in the context of the problem, since vectors are involved, but the question is about angle measure, probably degrees. Wait, no, vectors here have components with cos C and c - 2b, but angle measure can be in degrees or radians. The problem didn't specify. But given that it's a triangle problem from a competition, likely degrees. However, to be safe, both 60° and π/3 are correct. The problem says "measure", which could be either, but in many Chinese problems, degrees are standard. However, let's see the answer format. The user instructions say to put the final answer within boxed{}.For part I, angle A is 60°, so boxed as boxed{60^circ} or boxed{dfrac{pi}{3}}. Given that the problem uses "measure of angle A", which is often in degrees unless stated otherwise. So I think 60^circ is the expected answer.For part II, the range is (2, 3], which in boxed notation can be written as boxed{(2, 3]}.But sometimes, in some contexts, intervals are written with spaces, but in LaTeX, it's (2, 3]. Alternatively, if inclusive/exclusive is denoted with brackets, but since 2 is not attainable, and 3 is attainable, (2, 3] is correct.Final Answer(I) boxed{60^circ}(II) boxed{(2, 3]}</think>Given vectors (overrightarrow{m} = (a, frac{1}{2})) and (overrightarrow{n} = (cos C, c - 2b)) such that (overrightarrow{m} perp overrightarrow{n}), we start by noting that their dot product must be zero:[a cos C + frac{1}{2}(c - 2b) = 0]Simplifying this equation, we get:[a cos C + frac{c}{2} - b = 0 implies a cos C = b - frac{c}{2}]Using the Law of Cosines, (cos C = frac{a^2 + b^2 - c^2}{2ab}), we substitute this into the equation:[a left( frac{a^2 + b^2 - c^2}{2ab} right) = b - frac{c}{2}]Simplifying further:[frac{a^2 + b^2 - c^2}{2b} = b - frac{c}{2}]Multiplying both sides by (2b):[a^2 + b^2 - c^2 = 2b^2 - bc]Rearranging terms:[a^2 = b^2 + c^2 - bc]Comparing this with the Law of Cosines, (a^2 = b^2 + c^2 - 2bc cos A), we find:[2 cos A = 1 implies cos A = frac{1}{2} implies A = 60^circ]Thus, the measure of angle (A) is (boxed{60^circ}).For part II, given (a = 1), we use the Law of Sines. The perimeter (l = 1 + b + c). Using the Law of Sines, we express (b) and (c) in terms of angles (B) and (C):[b = frac{2}{sqrt{3}} sin B, quad c = frac{2}{sqrt{3}} sin C]Since (B + C = 120^circ), we convert the perimeter expression into a sinusoidal function:[l = 1 + sqrt{3} sin B + cos B]Rewriting this using the amplitude-phase form:[sqrt{3} sin B + cos B = 2 sin left( B + frac{pi}{6} right)]Thus, the perimeter becomes:[l = 2 sin left( B + frac{pi}{6} right) + 1]The range of (sin left( B + frac{pi}{6} right)) for (B in (0^circ, 120^circ)) is ((frac{1}{2}, 1]), leading to the perimeter range:[2 < l leq 3]Therefore, the range of the perimeter (l) is (boxed{(2, 3]}).,<think>Okay, so I have this problem about triangle ABC. The problem has two parts: part I asks for the measure of angle A, and part II asks for the range of the perimeter l if side a is 1. The given information involves vectors m and n that are perpendicular to each other. Let me try to break this down step by step.First, let's parse the problem again. We have triangle ABC with sides opposite angles A, B, and C being a, b, c respectively. The vectors given are m = (a, 1/2) and n = (cos C, c - 2b). These vectors are perpendicular, which means their dot product is zero. So, the first thing I need to do is set up the equation for the dot product being zero.So, the dot product of vectors m and n is a * cos C + (1/2) * (c - 2b) = 0. That's the equation we get from m ⊥ n. Let me write that down:a * cos C + (1/2)(c - 2b) = 0.Simplifying this equation, we can multiply through by 2 to eliminate the fraction:2a cos C + c - 2b = 0.So, 2a cos C = 2b - c.Therefore, cos C = (2b - c)/(2a). Hmm, that's an equation relating the sides of the triangle and the cosine of angle C. Maybe we can use the Law of Cosines here? Because the Law of Cosines relates the sides of a triangle with the cosine of one of its angles. Let's recall that:In any triangle, c² = a² + b² - 2ab cos C. Wait, no, actually, the Law of Cosines is:For angle C, which is opposite side c, the formula is:c² = a² + b² - 2ab cos C.Wait, but in our case, we have cos C expressed in terms of sides a, b, c. Let me note that. So from the given vectors, we have:cos C = (2b - c)/(2a).But from the Law of Cosines, we have:cos C = (a² + b² - c²)/(2ab).Therefore, setting these two expressions for cos C equal to each other:(a² + b² - c²)/(2ab) = (2b - c)/(2a).Multiplying both sides by 2ab to eliminate denominators:a² + b² - c² = (2b - c) * b.Expanding the right-hand side:a² + b² - c² = 2b² - bc.Let's bring all terms to the left side:a² + b² - c² - 2b² + bc = 0.Simplify:a² - b² - c² + bc = 0.Hmm, this seems a bit messy. Let me rearrange the terms:a² = b² + c² - bc.Wait a minute, this looks similar to the Law of Cosines again. The Law of Cosines states that a² = b² + c² - 2bc cos A. Comparing this with what we have:From the Law of Cosines: a² = b² + c² - 2bc cos A.But from our equation: a² = b² + c² - bc.Therefore, setting these equal:b² + c² - 2bc cos A = b² + c² - bc.Subtract b² + c² from both sides:-2bc cos A = -bc.Divide both sides by -bc (assuming bc ≠ 0, which they aren't in a triangle):2 cos A = 1.Therefore, cos A = 1/2.Which implies that angle A is 60 degrees or π/3 radians. So, angle A is 60 degrees. That answers part I. Let me verify that again to make sure I didn't make a mistake.Starting from the perpendicular vectors: dot product zero, which gives 2a cos C + c - 2b = 0. Then using Law of Cosines to express cos C and equating the two expressions. Then substituting into the equation, expanding, simplifying, and eventually getting cos A = 1/2, which gives A = 60 degrees. That seems correct.So part I's answer is 60 degrees. Now, moving on to part II: If a = 1, find the range of the perimeter l of triangle ABC.Given that a = 1, which is the side opposite angle A, which we now know is 60 degrees. So we have a triangle with side a = 1, angle A = 60 degrees, and we need to find the possible perimeters l = 1 + b + c.We need to determine the possible values of b and c such that the triangle with sides 1, b, c has angle A = 60 degrees, and also satisfies the earlier condition from the vectors being perpendicular, which led to a relationship between a, b, c. Wait, but we already used that to find angle A. So maybe the triangle is determined by angle A = 60 degrees and a = 1, but there might be additional constraints from the original vector condition? Wait, let's check.Wait, the condition from the vectors being perpendicular was already used in part I to find angle A. So once we have angle A fixed at 60 degrees, and a = 1, then the triangle is determined by the Law of Cosines and the Law of Sines, but perhaps there's still a relationship between sides b and c. Let me check.From part I, we had the equation a² = b² + c² - bc. Since a = 1, substituting that in:1 = b² + c² - bc.So, that's a constraint on b and c. Additionally, they must satisfy triangle inequalities:b + c > a = 1,a + c > b ⇒ 1 + c > b,a + b > c ⇒ 1 + b > c.So, we need to find all positive real numbers b and c such that:1. 1 = b² + c² - bc,2. b + c > 1,3. 1 + c > b,4. 1 + b > c.But maybe the equation 1 = b² + c² - bc already enforces some of the triangle inequalities? Let's see.First, let's work with the equation 1 = b² + c² - bc. Let's try to express this in terms of variables. Maybe express one variable in terms of the other.Alternatively, note that this equation resembles the Law of Cosines for angle A = 60 degrees. Indeed, since angle A is 60 degrees, the Law of Cosines gives:a² = b² + c² - 2bc cos A.But since angle A is 60 degrees, cos A = 1/2, so:a² = b² + c² - bc.But since a = 1, this is exactly the equation we have: 1 = b² + c² - bc.Therefore, that equation is equivalent to the Law of Cosines for angle A = 60 degrees. So, the condition from the vectors being perpendicular in part I led us to angle A being 60 degrees, and now in part II, given a = 1, the equation 1 = b² + c² - bc must hold. Therefore, the perimeter l = 1 + b + c. We need to find the range of possible l given that 1 = b² + c² - bc and the triangle inequalities.So, the key is to express the perimeter in terms of b and c, given that 1 = b² + c² - bc, and find the possible values of b + c.But since l = 1 + b + c, the range of l is 1 + the range of (b + c). So, we need to find the possible values of b + c given that 1 = b² + c² - bc.To find the range of b + c, let's set s = b + c and p = bc. Then, we know from algebra that b² + c² = s² - 2p. So, substituting into the equation:1 = s² - 2p - p = s² - 3p.Therefore, 1 = s² - 3p ⇒ 3p = s² - 1 ⇒ p = (s² - 1)/3.But also, since p = bc, and by the AM-GM inequality, bc ≤ (b + c)^2 / 4 = s² / 4. Therefore:p ≤ s² / 4.Substituting p = (s² - 1)/3 into this inequality:(s² - 1)/3 ≤ s² / 4.Multiply both sides by 12 to eliminate denominators:4(s² - 1) ≤ 3s²4s² - 4 ≤ 3s²4s² - 3s² - 4 ≤ 0s² - 4 ≤ 0s² ≤ 4s ≤ 2 (since s is positive, as the sum of sides)So, s ≤ 2. So the maximum possible value of s is 2. But we also need a lower bound for s.But in addition, since b and c are sides of a triangle with a = 1, we have triangle inequalities:b + c > 1,1 + b > c,1 + c > b.But since we already have a = 1, and angle A = 60 degrees, perhaps we can find more constraints.Alternatively, since we have the equation s² - 3p = 1, and p = bc. Also, since in a triangle, the sum of any two sides must be greater than the third. So:b + c > 1,1 + b > c ⇒ c < 1 + b,1 + c > b ⇒ b < 1 + c.But since b and c are positive, these inequalities are automatically satisfied if b + c > 1 and |b - c| < 1. Wait, because for example, if b >= c, then 1 + c > b implies that c > b - 1. But maybe we need to handle this carefully.Alternatively, since we have 1 = b² + c² - bc, maybe we can parametrize b and c. Let me consider setting variables to solve for possible b and c.Alternatively, perhaps express one variable in terms of the other.Let me solve for c in terms of b from the equation 1 = b² + c² - bc.Rewriting:c² - b c + (b² - 1) = 0.This is a quadratic in c:c² - b c + (b² - 1) = 0.Using the quadratic formula, c = [b ± sqrt(b² - 4*(b² - 1))]/2.Compute discriminant:sqrt(b² - 4*(b² - 1)) = sqrt(b² - 4b² + 4) = sqrt(-3b² + 4).For real solutions, discriminant must be non-negative:-3b² + 4 ≥ 0 ⇒ 3b² ≤ 4 ⇒ b² ≤ 4/3 ⇒ b ≤ 2/sqrt(3) ≈ 1.1547.Similarly, since b must be positive, we have 0 < b ≤ 2/sqrt(3).But we also need to satisfy the triangle inequalities.Wait, but c must be positive as well. So, let's check the solutions.c = [b ± sqrt(-3b² + 4)] / 2.Since c must be positive, we need to consider both roots:First, c = [b + sqrt(-3b² + 4)] / 2,and c = [b - sqrt(-3b² + 4)] / 2.The second solution requires that [b - sqrt(-3b² + 4)] / 2 > 0,which implies that b > sqrt(-3b² + 4).Square both sides (since both sides are positive):b² > -3b² + 44b² > 4b² > 1b > 1.But from earlier, b ≤ 2/sqrt(3) ≈ 1.1547. So, if b > 1, then c is positive.Therefore, c = [b - sqrt(-3b² + 4)] / 2 is positive when 1 < b ≤ 2/sqrt(3).Similarly, c = [b + sqrt(-3b² + 4)] / 2 is always positive because both terms in the numerator are positive.Therefore, for each b in (0, 2/sqrt(3)], there are two possible values of c, except when b = 2/sqrt(3), where the discriminant is zero, so only one value of c.But we need to also satisfy the triangle inequalities. So, for each b and c, we need to make sure that:1. b + c > 1,2. 1 + b > c,3. 1 + c > b.But since a = 1, these are equivalent to the usual triangle inequalities. Let's see.First, the equation 1 = b² + c² - bc comes from the Law of Cosines with angle A = 60 degrees, which already defines a valid triangle as long as the sides are positive and satisfy the triangle inequalities. But perhaps we need to check if these inequalities are automatically satisfied given the equation.Alternatively, maybe the equation and the constraints on b and c from the quadratic solution already ensure that the triangle inequalities are satisfied. Let me check.Suppose b is in (0, 2/sqrt(3)]. Let's take the case when c = [b + sqrt(-3b² + 4)] / 2.We need to check if 1 + b > c and 1 + c > b.First, check 1 + b > c:c = [b + sqrt(-3b² + 4)] / 2,So 1 + b > [b + sqrt(-3b² + 4)] / 2,Multiply both sides by 2:2 + 2b > b + sqrt(-3b² + 4),Simplify:2 + b > sqrt(-3b² + 4).Square both sides (since both sides are positive):(2 + b)^2 > -3b² + 4,Expand left side:4 + 4b + b² > -3b² + 4,Bring all terms to left:4 + 4b + b² + 3b² - 4 > 0,Simplify:4b + 4b² > 0,Factor:4b(1 + b) > 0.Since b > 0, this inequality is always true. Therefore, 1 + b > c is always satisfied for c = [b + sqrt(...)] / 2.Similarly, check 1 + c > b:1 + c > b,c > b - 1.But since c is positive, if b - 1 < 0, which is when b < 1, then c > b - 1 is automatically true. If b >= 1, then we need c > b - 1.Given c = [b + sqrt(-3b² + 4)] / 2,Let's check if [b + sqrt(-3b² + 4)] / 2 > b - 1.Multiply both sides by 2:b + sqrt(-3b² + 4) > 2b - 2,Simplify:sqrt(-3b² + 4) > b - 2.But since sqrt(-3b² + 4) is non-negative, and if b - 2 is negative (which it is when b < 2), then the inequality holds. Since b <= 2/sqrt(3) ≈1.1547 < 2, then b - 2 is always negative, so sqrt(...) >=0 > negative number. Therefore, 1 + c > b is always true.Similarly, the other inequality b + c > 1: we need to check that.Given that c = [b ± sqrt(-3b² + 4)] / 2,So b + c = b + [b ± sqrt(-3b² + 4)] / 2 = (2b + b ± sqrt(-3b² + 4)) / 2 = (3b ± sqrt(-3b² + 4)) / 2.We need this to be greater than 1.But this is for both solutions of c. Let's check for both cases.First, take the positive sqrt: c = [b + sqrt(-3b² + 4)] / 2,Then b + c = (3b + sqrt(-3b² + 4)) / 2.We need this > 1.So (3b + sqrt(-3b² + 4)) / 2 > 1,Multiply both sides by 2:3b + sqrt(-3b² + 4) > 2.Similarly, for the other solution c = [b - sqrt(-3b² + 4)] / 2,Then b + c = (3b - sqrt(-3b² + 4)) / 2.We need this to be > 1,So (3b - sqrt(-3b² + 4)) / 2 > 1,Multiply by 2:3b - sqrt(-3b² + 4) > 2.But let's first check the first case.Case 1: c = [b + sqrt(-3b² + 4)] / 2.So, 3b + sqrt(-3b² + 4) > 2.Let me analyze this inequality. Let’s denote D = sqrt(-3b² + 4). Then the inequality is 3b + D > 2.But D = sqrt(4 - 3b²). Let’s square both sides if necessary, but maybe first check for values of b.Since b is in (0, 2/sqrt(3)) ≈ (0, 1.1547). Let's check when b approaches 0:When b approaches 0, D approaches sqrt(4) = 2. So 3*0 + 2 = 2, which is equal to 2. So near b = 0, the expression is close to 2, but since b is positive, slightly above 0, 3b + D would be slightly above 2. So equality holds at b = 0, but b > 0, so maybe 3b + D > 2 for all b in (0, 2/sqrt(3)].Wait, let's check at b = 2/sqrt(3):D = sqrt(4 - 3*(4/3)) = sqrt(4 - 4) = 0. So 3*(2/sqrt(3)) + 0 = 2*sqrt(3) ≈3.464 > 2. So yes, at b = 2/sqrt(3), it's about 3.464, which is greater than 2. So perhaps 3b + D is always greater than 2 in the interval (0, 2/sqrt(3)]. Let's check at some intermediate value, say b = 1:D = sqrt(4 - 3*1) = sqrt(1) = 1. Then 3*1 +1 =4>2. So yes. So for all b in (0, 2/sqrt(3)], 3b + D >=2, with equality only when b=0, which is not allowed. Therefore, for the case of c = [b + sqrt(...)] / 2, the sum b + c >1 holds.Now check the other case: c = [b - sqrt(-3b² +4)] /2.Then, b + c = (3b - sqrt(-3b² +4))/2 >1.So,3b - sqrt(-3b² +4) > 2.Let’s analyze this. Let’s denote D = sqrt(-3b² +4).Then, 3b - D > 2.So, 3b - 2 > D.Since both sides are positive (for b > 2/3, since 3b -2 >0 requires b > 2/3), we can square both sides:(3b - 2)^2 > -3b² +4,Expand left side:9b² -12b +4 > -3b² +4,Bring all terms to left:9b² -12b +4 +3b² -4 >0,Simplify:12b² -12b >0,Factor:12b(b -1) >0.Since b >0, the inequality 12b(b -1) >0 holds when b >1.Therefore, 3b - D >2 holds when b >1.But remember that in this case, c = [b - D]/2 must be positive, which as we saw earlier requires that b >1. So, when b >1, we have this case, and the inequality b + c >1 reduces to 3b - D >2, which is true when b >1. Therefore, for b in (1, 2/sqrt(3)], c is positive, and b + c >1 is satisfied.Therefore, all solutions for c result in valid triangles, given that b is in (0, 2/sqrt(3))] for the positive sqrt case, and b in (1, 2/sqrt(3))] for the negative sqrt case.So, now, our task is to find the range of s = b + c, given that 1 = s² - 3p and p = bc. But we also found that s <=2. However, we need to find the minimum and maximum possible values of s.Earlier, we found that s <=2, but is 2 achievable? Let's check when s=2. Then p = (4 -1)/3 =1. So bc=1. Also, since s = b + c =2 and bc=1, the solutions are b and c being roots of x² -2x +1=0, which is (x-1)^2=0, so b=c=1. So, if b=c=1, then check if a=1, then it's an equilateral triangle. Indeed, if a=1, angle A=60 degrees, and sides b=c=1, then yes, it is an equilateral triangle with all sides 1, all angles 60 degrees. But wait, in that case, the perimeter would be 3. Wait, but s=2, so perimeter l=1 +2=3. Wait, but if a=1, b=c=1, perimeter is 1+1+1=3. But according to our earlier substitution, s = b + c =2, so perimeter l=1 +2=3, which matches.So, s can reach up to 2, which gives perimeter 3. But is that the maximum? Wait, but when s=2, b and c are both 1, which is allowed. So that's a valid triangle.What about the minimum value of s? Since s = b + c, and we have the equation 1 = s² - 3p, and p = bc. The minimum value of s would occur when p is maximized, since s² =1 +3p. To minimize s, we need to minimize s², which is 1 +3p. Therefore, to minimize s, we need to minimize p, but p=bc. Wait, actually, no: s² =1 + 3p. Therefore, if we minimize s, we need to minimize s², which requires minimizing p. But p=bc. However, given s =b +c, the product bc is maximized when b=c, and minimized when one variable approaches 0. Wait, but with s fixed, bc is maximized when b=c. But here, s is variable. So, if we want to minimize s, we need to minimize s² -3p =1. Hmm, this is getting confusing.Wait, let's think of it this way: s = b + c, and 1 = s² -3 bc. We need to find the range of s.Express bc in terms of s: bc = (s² -1)/3.But also, from the AM-GM inequality, bc ≤ (s²)/4. Therefore,(s² -1)/3 ≤ s²/4,Multiply both sides by 12:4(s² -1) ≤3s²,4s² -4 ≤3s²,s² -4 ≤0,s² ≤4,s ≤2.Which we already knew. So equality occurs when bc =s²/4, which is when b=c. Therefore, maximum s is 2 when b=c=1.To find the minimum s, we need to consider the minimum possible s such that there exist positive real numbers b and c with b + c = s and bc = (s² -1)/3.Additionally, from the triangle inequality, we have b + c >1. But s =b +c >1. So the minimum s is greater than1. But how much greater?We need to find the minimum possible s such that bc=(s² -1)/3 and b, c >0, with b +c =s.In other words, given s, the equation x² -s x + (s² -1)/3 =0 must have real positive roots. The discriminant must be non-negative:s² -4*(s² -1)/3 ≥0.Compute discriminant:s² - (4/3)(s² -1) ≥0,Multiply through by 3:3s² -4s² +4 ≥0,-s² +4 ≥0,s² ≤4,Which again gives s ≤2. But we already have s >1. But how does this help?Wait, but the discriminant must be non-negative for real roots to exist. However, the discriminant here is:sqrt(s² -4*(s² -1)/3) = sqrt[(3s² -4s² +4)/3] = sqrt[(-s² +4)/3].Therefore, the discriminant is real only when (-s² +4)/3 ≥0 ⇒ s² ≤4 ⇒ s ≤2. Which is already known.But we need to ensure that the roots are positive. The roots are:[b, c] = [s ± sqrt((-s² +4)/3)]/2.Wait, no, the quadratic equation is x² -s x + (s² -1)/3 =0,So roots are [s ± sqrt(s² -4*(s² -1)/3)] /2.Wait, compute the discriminant:D = s² -4*(s² -1)/3 = (3s² -4s² +4)/3 = (-s² +4)/3.Therefore, sqrt(D) = sqrt((4 -s²)/3).So roots are [s ± sqrt((4 -s²)/3)] /2.For the roots to be positive, both roots must be positive. Since s >0, and sqrt((4 -s²)/3) is real and positive (since s <2), then:The smaller root is [s - sqrt((4 -s²)/3)] /2.We need this to be positive:s - sqrt((4 -s²)/3) >0,s > sqrt((4 -s²)/3),Square both sides (since both sides positive):s² > (4 -s²)/3,Multiply both sides by3:3s² >4 -s²,4s² >4,s² >1,s>1.Which is consistent with our triangle inequality that s =b +c >1.Therefore, for s in (1, 2], the equation x² -s x + (s² -1)/3 =0 has two positive roots. Therefore, for each s in (1,2], there exist positive real numbers b and c such that b +c =s and bc=(s² -1)/3. Therefore, s can take any value in (1,2]. Therefore, the perimeter l=1 +s can range from (1 +1, 1 +2], i.e., (2,3]. But wait, when s approaches1 from above, the perimeter approaches 2. But is s=1 possible? No, because s must be greater than1. So the perimeter can be as close to2 as desired but greater than2, and up to3.Wait, but wait. Let me check. If s approaches1, then bc=(s² -1)/3 approaches(1 -1)/3=0. So one of the sides approaches0, which is not allowed in a triangle. But according to our earlier analysis, when s approaches1, then one of the sides approaches0, but the other approaches1. However, in that case, the triangle inequality 1 +c >b would require that if c approaches0 and b approaches1, then 1 +0 >1, which is 1>1, which is not true. Therefore, such a triangle would not be valid as s approaches1. Therefore, maybe there is a lower bound greater than1 for s.Wait, this is conflicting with our previous conclusion that s can be in (1,2]. But we need to check if for s approaching1, the triangle inequalities are satisfied. For example, take s=1.1. Then bc=(1.21 -1)/3=0.21/3=0.07.So bc=0.07, with b +c=1.1. Let's solve for b and c:b and c are roots of x² -1.1x +0.07=0.Discriminant: (1.1)^2 -4*1*0.07=1.21 -0.28=0.93>0.Therefore, roots are [1.1 ± sqrt(0.93)]/2≈ [1.1 ±0.964]/2.So,b≈(1.1 +0.964)/2≈2.064/2≈1.032,c≈(1.1 -0.964)/2≈0.136/2≈0.068.So, sides are approximately1.032,0.068, and a=1.Check triangle inequalities:1. a +b >c:1 +1.032>0.068: yes.2. a +c >b:1 +0.068>1.032≈1.068>1.032: yes.3. b +c >a:1.032 +0.068≈1.1>1: yes.So, even with s=1.1, the triangle inequalities are satisfied. Therefore, s can approach arbitrarily close to1, as long as s>1, and the triangle remains valid. Therefore, the perimeter can approach2 but cannot be equal to2. So the range of l=1 +s is (2,3].Wait, but when s approaches1 from above, l approaches2. But in reality, when s approaches1, the side c approaches0, but in a triangle, all sides must be positive, so c must be greater than0. So technically, s can be made as close to1 as desired, making l as close to2 as desired. Therefore, the range of l is (2,3].But we need to confirm if there's a lower bound greater than2. Let me consider when s=1. Then, bc=(1 -1)/3=0. So either b=0 or c=0, which is impossible. So s must be greater than1, but how much greater?But in our previous example, with s=1.1, we have valid sides. Therefore, s can be any real number greater than1 and up to2, so l ranges from (2,3]. Therefore, the perimeter can be made arbitrarily close to2 but not equal to2, and up to3.However, let's check another example where s is approaching1. Let’s take s=1 +ε, where ε is very small, say ε=0.01. Then s=1.01, bc=(1.0201 -1)/3≈0.0201/3≈0.0067.Solving x² -1.01x +0.0067=0.Discriminant: (1.01)^2 -4*0.0067≈1.0201 -0.0268≈0.9933>0.Roots: [1.01 ±sqrt(0.9933)]/2≈[1.01 ±0.9966]/2.So,b≈(1.01 +0.9966)/2≈2.0066/2≈1.0033,c≈(1.01 -0.9966)/2≈0.0134/2≈0.0067.Check triangle inequalities:1. a +b >c:1 +1.0033≈2.0033>0.0067: yes.2. a +c >b:1 +0.0067≈1.0067>1.0033: yes.3. b +c >a:1.0033 +0.0067≈1.01>1: yes.So even when s is very close to1, the triangle inequalities hold. Therefore, s can approach1 from above, making the perimeter approach2 from above, but never actually reaching2. Similarly, the maximum perimeter is3 when s=2.Therefore, the range of the perimeter l is (2,3]. But we need to express this in interval notation. However, the problem says "the range of the perimeter l of triangle ABC". Since the perimeter can get as close to2 as desired (but not equal to2) and up to3, including3.But in mathematical terms, since when s approaches1+, l approaches2+, and when s=2, l=3. So the range is (2,3].But we need to verify if there are any other constraints. Wait, but in the problem statement, is there any restriction on the triangle other than a=1 and angle A=60 degrees? The original condition from the vectors being perpendicular has already been used to determine that angle A=60 degrees, so once we've used that, the rest follows from the Law of Cosines and triangle inequalities.Therefore, the range of the perimeter l is all real numbers greater than2 and less than or equal to3. So, l ∈ (2,3].But the problem says "the range of the perimeter l", which can be expressed as 2 < l ≤3. So that's the answer for part II.Wait, but let me double-check with another approach. Let's parametrize the triangle. Since angle A=60 degrees and a=1, we can use the Law of Sines.Law of Sines: a/sin A = b/sin B = c/sin C.Given A=60°, so sin A=√3/2. Therefore,1/(√3/2) = 2/√3 = b/sin B = c/sin C.Therefore, b = (2/√3) sin B,c = (2/√3) sin C.Since in a triangle, B + C = 120°, so C = 120° - B.Thus, c = (2/√3) sin(120° - B).The perimeter l =1 + b + c =1 + (2/√3)(sin B + sin(120° - B)).We can express sin(120° - B) using sine subtraction formula:sin(120° - B) = sin 120° cos B - cos 120° sin B = (√3/2)cos B - (-1/2)sin B = (√3/2)cos B + (1/2)sin B.Therefore,sin B + sin(120° - B) = sin B + (√3/2 cos B + 1/2 sin B) = (3/2) sin B + (√3/2) cos B.Therefore, perimeter l =1 + (2/√3)[ (3/2 sin B + √3/2 cos B ) ] =1 + (2/√3)(3/2 sin B + √3/2 cos B).Simplify:=1 + (2/√3)*(3/2 sin B) + (2/√3)*(√3/2 cos B)=1 + (3/√3) sin B + ( (2/√3)*(√3/2) ) cos B=1 + √3 sin B + (1) cos B.Therefore, perimeter l =1 + √3 sin B + cos B.Now, we need to find the range of l as B varies over the possible angles in the triangle. Since angles B and C must be positive and add up to120°, B must be in (0°,120°). So B ∈ (0, 120°), which is equivalent to B ∈ (0, 2π/3) radians.Therefore, l =1 + √3 sin B + cos B.We can analyze the function f(B) = √3 sin B + cos B.This is a function of B, which can be written in the form R sin(B + φ), where R is the amplitude and φ is the phase shift.Compute R: R = √( (√3)^2 +1^2 )=√(3+1)=2.Compute φ: tan φ = 1/√3 ⇒ φ=30°=π/6.Therefore, f(B) = 2 sin(B + 30°).Therefore, l=1 +2 sin(B +30°).The range of sin(B +30°) is dependent on B ∈ (0°,120°). Therefore, B +30° ∈ (30°,150°).The sine function sinθ reaches its maximum at θ=90°, sin90°=1, and its minimum at the endpoints θ=30° and θ=150°, sin30°=1/2, sin150°=1/2.Therefore, sin(B +30°) ∈ [1/2,1].Thus, f(B) ∈ [2*(1/2),2*1] = [1,2].Therefore, l =1 + f(B) ∈ [1 +1,1 +2] = [2,3].Wait, but this contradicts our previous conclusion that l is in (2,3]. However, according to this analysis, l can be as low as2 and as high as3. But we need to check if the endpoints are attainable.When does sin(B +30°)=1/2? That occurs when B +30°=30° or150°, so B=0° or B=120°, but B must be in (0°,120°). Therefore, B approaching0°, sin(B +30°) approaches sin30°=1/2, so l approaches2. Similarly, B approaching120°, B+30°=150°, sin150°=1/2, so l approaches2 again. However, when B=60°, sin(90°)=1, so l=1 +2*1=3. Therefore, l=3 is attainable when B=60°, which corresponds to an equilateral triangle. So according to this, the range of l is [2,3]. But this conflicts with the previous analysis where s could not be1, leading to perimeter l approaching2 but not reaching it.But where is the mistake here?Wait, in the Law of Sines parametrization, angles B and C can approach0°, but in reality, in a triangle, all angles must be greater than0°, so B cannot be0° or120°, but can approach them. Therefore, l can approach2 from above but cannot be exactly2. But according to the Law of Sines parametrization, when B approaches0°, the side b approaches0 and c approaches (2/√3) sin120°= (2/√3)(√3/2)=1. So sides would be a=1, b approaching0, c approaching1. But in that case, the triangle inequality a +b >c would become1 +0 >1, which is1>1, which is false. Therefore, such a triangle is not valid. Therefore, the minimum perimeter is not actually achievable, but approaches2 as B approaches0° or120°, but those cases do not form valid triangles.Therefore, the correct range should be (2,3]. However, using the Law of Sines approach, we get l=1 +2 sin(B +30°), and since B ∈(0°,120°), then B +30° ∈(30°,150°), and sin(B +30°) ∈(1/2,1). Therefore, sin(B +30°) ∈(1/2,1), so l∈(1 +2*(1/2),1 +2*1)=(2,3]. Therefore, the perimeter is in the open interval (2,3]. So the range is2 < l ≤3.This aligns with our earlier conclusion. Therefore, the perimeter can get arbitrarily close to2 but cannot equal2, and reaches3 when the triangle is equilateral.Hence, part II's answer is that the perimeter l is in the interval (2,3].Final Answer(I) The measure of angle ( A ) is (boxed{60^circ}).(II) The range of the perimeter ( l ) of ( triangle ABC ) is (boxed{(2, 3]}).</think>Given vectors (overrightarrow{m} = (a, frac{1}{2})) and (overrightarrow{n} = (cos C, c - 2b)) are perpendicular, their dot product is zero:[a cos C + frac{1}{2}(c - 2b) = 0]Simplifying this equation:[2a cos C + c - 2b = 0 implies 2a cos C = 2b - c implies cos C = frac{2b - c}{2a}]Using the Law of Cosines for angle (C):[cos C = frac{a^2 + b^2 - c^2}{2ab}]Equating the two expressions for (cos C):[frac{a^2 + b^2 - c^2}{2ab} = frac{2b - c}{2a}]Multiplying both sides by (2ab):[a^2 + b^2 - c^2 = 2b^2 - bc implies a^2 = b^2 + c^2 - bc]Using the Law of Cosines for angle (A):[a^2 = b^2 + c^2 - 2bc cos A]Equating the two expressions for (a^2):[b^2 + c^2 - 2bc cos A = b^2 + c^2 - bc implies 2 cos A = 1 implies cos A = frac{1}{2} implies A = 60^circ]For part (II), given (a = 1), we use the equation derived from the Law of Cosines:[1 = b^2 + c^2 - bc]We need to find the range of the perimeter (l = 1 + b + c). Let (s = b + c). Then:[1 = s^2 - 3bc]Expressing (bc) in terms of (s):[bc = frac{s^2 - 1}{3}]Using the AM-GM inequality:[bc leq left(frac{s}{2}right)^2 implies frac{s^2 - 1}{3} leq frac{s^2}{4} implies s^2 leq 4 implies s leq 2]Considering the triangle inequalities and the quadratic nature of the solutions for (b) and (c), the minimum value of (s) approaches 1 (but cannot be 1), and the maximum value is 2. Thus, the perimeter (l = 1 + s) ranges from just above 2 to 3.Final answers:(I) The measure of angle (A) is (boxed{60^circ}).(II) The range of the perimeter (l) is (boxed{(2, 3]}).]

🔑:(I) From the given condition, overrightarrow{m} perp overrightarrow{n}, we know that overrightarrow{m} cdot overrightarrow{n} = 0,which implies acos C + frac{1}{2}c = b. Rearranging, we get sin A cos C + frac{1}{2} sin C = sin B.But, sin B = sin(A + C) = sin A cos C + cos A sin C.Thus, frac{1}{2} sin C = cos A sin C. Since sin C neq 0, we have cos A = frac{1}{2}.Given that 0 < A < pi, we conclude A = frac{pi}{3}.(II) By the sine law, we have b = frac{a sin B}{sin A} = frac{2}{sqrt{3}} sin B and c = frac{2}{sqrt{3}} sin C.Therefore, l = a + b + c = 1 + frac{2}{sqrt{3}} (sin B + sin C) = 1 + frac{2}{sqrt{3}}(sin B + sin(A + B)).Simplifying, we get l = 1 + 2sin(B + frac{pi}{6}).Since A = frac{pi}{3}, we have B in (0, frac{2pi}{3}), which implies B + frac{pi}{6} in (frac{pi}{6}, frac{5pi}{6}).As a result, sin(B + frac{pi}{6}) in (frac{1}{2}, 1].Thus, the range of the perimeter l of triangle ABC is boxed{(2, 3]}.