💡Okay, so I have this problem here about 1200 students taking a math test. It says that the 80th percentile of all students' scores is 103 points. And the question is asking for the minimum number of students who scored at least 103 points. The options are A: 220, B: 240, C: 250, D: 300.Alright, let me think about what percentile means. I remember that the nth percentile is the value below which n% of the data falls. So, the 80th percentile means that 80% of the students scored below 103 points. That would imply that 20% scored 103 or above, right?So, if 80% scored below 103, then 20% scored 103 or higher. Let me calculate 20% of 1200 students. 20% is 0.2, so 0.2 times 1200 is 240. So, does that mean at least 240 students scored 103 or higher? Hmm, that seems straightforward.But wait, let me make sure I'm not missing anything. The question says "at least" how many students scored at least 103. So, is 240 the minimum number? Or could it be more?Well, percentiles can sometimes be a bit tricky because they depend on how the data is distributed. For example, if multiple students scored exactly 103, that could affect the percentile. But in this case, since it's the 80th percentile, it means that 80% scored less than or equal to 103, and 20% scored more than or equal to 103.Wait, no, actually, the 80th percentile is the value where 80% of the data is less than or equal to that value. So, 80% of the students scored less than or equal to 103, which means 20% scored more than 103. Hmm, so does that mean exactly 20% scored more than 103, or at least 20%?I think it's at least 20%. Because if some students scored exactly 103, then the percentage above could be less. But in terms of the minimum number, we can assume that the 20% is the exact number above 103, so that would be 240 students.Let me double-check. If 80% is 960 students, then the remaining 20% is 240 students. So, 240 students scored 103 or higher. Since the question is asking for the number of students with a score of at least 103, that would be 240.But wait, could it be more? If some students scored exactly 103, then the number of students above 103 would be less than 240. But the question is asking for the minimum number of students with at least 103. So, the minimum would be when as many as possible scored exactly 103, right?Wait, no. If we want the minimum number of students with at least 103, that would occur when as few students as possible scored above 103. But since the 80th percentile is 103, that means that 80% scored less than or equal to 103, and 20% scored more than or equal to 103.Actually, I think I might have confused myself. Let me try to clarify. The 80th percentile is the value where 80% of the data is less than or equal to it. So, 80% of 1200 is 960 students scored less than or equal to 103. That leaves 240 students who scored more than 103. So, the number of students who scored at least 103 is at least 240, because some of those 240 could have scored exactly 103.Wait, no, if 80% scored less than or equal to 103, then the remaining 20% scored more than 103. So, the number of students who scored more than 103 is exactly 240. But the question is asking for the number of students with a score of at least 103, which includes those who scored exactly 103 and those who scored more than 103.But since the 80th percentile is 103, that means that 80% scored less than or equal to 103, so the number of students who scored exactly 103 could be more than zero. Therefore, the number of students who scored at least 103 is at least 240, because 240 is the number who scored more than 103, and there could be more who scored exactly 103.Wait, no, that doesn't make sense. If 80% scored less than or equal to 103, then the number of students who scored exactly 103 is included in that 80%. So, the number of students who scored more than 103 is 20%, which is 240. But the number of students who scored at least 103 is the number who scored exactly 103 plus those who scored more than 103.But since the 80th percentile is 103, it's possible that some students scored exactly 103, but we don't know how many. So, the minimum number of students who scored at least 103 would be when the number of students who scored exactly 103 is as small as possible, which is zero. But that can't be, because the 80th percentile is 103, which means that at least 80% scored less than or equal to 103, but it doesn't necessarily mean that 80% scored strictly less than 103.Wait, I'm getting confused. Let me think differently. The 80th percentile is the smallest value such that at least 80% of the data is less than or equal to it. So, in this case, 103 is the smallest value where at least 80% of the students scored less than or equal to 103. That means that the number of students who scored less than or equal to 103 is at least 80% of 1200, which is 960.Therefore, the number of students who scored more than 103 is at most 1200 - 960 = 240. But the question is asking for the number of students who scored at least 103, which includes those who scored exactly 103 and those who scored more than 103.So, the number of students who scored at least 103 is equal to the number who scored exactly 103 plus the number who scored more than 103. Since the number who scored more than 103 is at most 240, and the number who scored exactly 103 could be more than zero, the total number of students who scored at least 103 is at least 240.Wait, no, that's not right. If the number who scored more than 103 is at most 240, then the number who scored at least 103 is at least 240, because some of the 960 students who scored less than or equal to 103 could have scored exactly 103. So, the minimum number of students who scored at least 103 is 240, because if none of the 960 scored exactly 103, then all 240 scored more than 103. But if some of the 960 scored exactly 103, then the number who scored at least 103 would be more than 240.But the question is asking for the minimum number of students who scored at least 103. So, the minimum would occur when the number of students who scored exactly 103 is as small as possible, which is zero. But wait, can the number of students who scored exactly 103 be zero? If so, then all 240 students scored more than 103, making the number of students who scored at least 103 equal to 240.But is it possible for the 80th percentile to be 103 with zero students scoring exactly 103? Let me think. The 80th percentile is the value where 80% of the data is less than or equal to it. If no one scored exactly 103, then the 80th percentile would have to be the next higher score. So, if the 80th percentile is 103, that means that at least 80% of the students scored less than or equal to 103, which could include some students scoring exactly 103.Therefore, it's possible that some students scored exactly 103, but it's also possible that none did. So, the minimum number of students who scored at least 103 would be when the number of students who scored exactly 103 is zero, making the number of students who scored at least 103 equal to 240.Wait, but if no one scored exactly 103, then the 80th percentile would have to be the next higher score, right? Because the 80th percentile is defined as the smallest value where at least 80% of the data is less than or equal to it. So, if no one scored exactly 103, then the 80th percentile would have to be higher than 103.But the problem states that the 80th percentile is 103, so that means that at least 80% of the students scored less than or equal to 103. Therefore, it's possible that some students scored exactly 103, but it's also possible that none did. However, if none did, then the 80th percentile would have to be higher than 103, which contradicts the given information.Therefore, there must be at least some students who scored exactly 103. So, the number of students who scored exactly 103 is at least 1, which would mean that the number of students who scored at least 103 is at least 240 plus 1, which is 241. But 241 isn't one of the options, so maybe my reasoning is off.Wait, perhaps I'm overcomplicating this. Let's go back to the definition. The 80th percentile is the value where 80% of the data is less than or equal to it. So, 80% of 1200 is 960 students. That means that 960 students scored less than or equal to 103. Therefore, the number of students who scored more than 103 is 1200 - 960 = 240.But the question is asking for the number of students who scored at least 103, which includes those who scored exactly 103 and those who scored more than 103. Since 960 students scored less than or equal to 103, the number of students who scored exactly 103 is 960 minus the number of students who scored less than 103.But we don't know how many scored less than 103. It could be that all 960 scored less than 103, meaning that 240 scored exactly 103, but that would make the 80th percentile higher than 103, which contradicts the given information.Wait, no, if 960 students scored less than or equal to 103, and some of them scored exactly 103, then the number of students who scored more than 103 is 240. So, the number of students who scored at least 103 is the number who scored exactly 103 plus 240.But since we don't know how many scored exactly 103, the minimum number of students who scored at least 103 would be when the number who scored exactly 103 is as small as possible, which is zero. But as I thought earlier, if no one scored exactly 103, then the 80th percentile would have to be higher than 103, which contradicts the given information.Therefore, there must be at least one student who scored exactly 103, making the number of students who scored at least 103 at least 241. But since 241 isn't an option, maybe the question is considering that the 80th percentile is the smallest value where at least 80% scored less than or equal to it, so it's possible that exactly 80% scored less than or equal to 103, meaning that 20% scored more than 103, which is 240 students.Therefore, the number of students who scored at least 103 is at least 240, because 240 scored more than 103, and some could have scored exactly 103, but the minimum is 240.So, the answer should be 240, which is option B.

💡Okay, so I have this problem about a sequence where each term is defined by the recurrence relation ( a_{n+2} = a_{n+1} - a_n ). The initial terms are given as ( a_1 = 2 ) and ( a_2 = 3 ). I need to find the sum of the first 2016 terms, denoted as ( S_{2016} ). The options are A: 0, B: 2, C: 5, D: 6.Hmm, let me start by trying to understand the sequence. Since it's defined by a recurrence relation, maybe it's periodic or has some repeating pattern. That would make calculating the sum easier because I could find the period and then see how many full periods fit into 2016 terms.First, let me compute the first few terms to see if there's a pattern.Given:- ( a_1 = 2 )- ( a_2 = 3 )Now, using the recurrence relation ( a_{n+2} = a_{n+1} - a_n ):- ( a_3 = a_2 - a_1 = 3 - 2 = 1 )- ( a_4 = a_3 - a_2 = 1 - 3 = -2 )- ( a_5 = a_4 - a_3 = -2 - 1 = -3 )- ( a_6 = a_5 - a_4 = -3 - (-2) = -3 + 2 = -1 )- ( a_7 = a_6 - a_5 = -1 - (-3) = -1 + 3 = 2 )- ( a_8 = a_7 - a_6 = 2 - (-1) = 2 + 1 = 3 )Wait a minute, ( a_7 = 2 ) and ( a_8 = 3 ), which are the same as ( a_1 ) and ( a_2 ). So does this mean the sequence repeats every 6 terms? Let me check the next few terms to confirm.- ( a_9 = a_8 - a_7 = 3 - 2 = 1 ) (same as ( a_3 ))- ( a_{10} = a_9 - a_8 = 1 - 3 = -2 ) (same as ( a_4 ))- ( a_{11} = a_{10} - a_9 = -2 - 1 = -3 ) (same as ( a_5 ))- ( a_{12} = a_{11} - a_{10} = -3 - (-2) = -1 ) (same as ( a_6 ))- ( a_{13} = a_{12} - a_{11} = -1 - (-3) = 2 ) (same as ( a_7 ))Yes, it seems like the sequence repeats every 6 terms. So the period is 6.Now, since the sequence is periodic with period 6, the sum of each full period (6 terms) should be the same. Let me calculate the sum of the first 6 terms:( S_6 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 2 + 3 + 1 + (-2) + (-3) + (-1) )Calculating step by step:- 2 + 3 = 5- 5 + 1 = 6- 6 + (-2) = 4- 4 + (-3) = 1- 1 + (-1) = 0So, the sum of one full period (6 terms) is 0.Now, since 2016 is a multiple of 6, specifically ( 2016 = 6 times 336 ), the sum ( S_{2016} ) will be 336 times the sum of one period.Therefore, ( S_{2016} = 336 times 0 = 0 ).So, the answer should be A: 0.Wait, let me double-check my calculations to make sure I didn't make a mistake.First, the terms:- ( a_1 = 2 )- ( a_2 = 3 )- ( a_3 = 1 )- ( a_4 = -2 )- ( a_5 = -3 )- ( a_6 = -1 )- ( a_7 = 2 ) (same as ( a_1 ))- ( a_8 = 3 ) (same as ( a_2 ))Yes, the pattern repeats every 6 terms.Sum of 6 terms:2 + 3 + 1 - 2 - 3 - 1 = 0. Correct.Number of periods in 2016 terms: 2016 / 6 = 336. So, 336 periods.Sum is 336 * 0 = 0.Therefore, the answer is A: 0.I think that's solid. I don't see any mistakes in my reasoning.**Final Answer**boxed{A}

💡First, I need to determine the relative speed of the two trains since they are moving in opposite directions. The relative speed is the sum of their individual speeds.Next, I'll convert the relative speed from kilometers per hour to meters per second to match the units of the train lengths and the given time.Once I have the relative speed in meters per second, I can calculate the total distance covered by both trains during the time they take to completely pass each other.Finally, by subtracting the length of the first train from the total distance covered, I can find the length of the second train.

💡First, I need to solve the inequality (1.5n - 6 < 7.5) to find the range of positive integers (n) that satisfy this condition.I'll start by isolating the term with (n). To do this, I'll add 6 to both sides of the inequality:[1.5n - 6 + 6 < 7.5 + 6]Simplifying this, I get:[1.5n < 13.5]Next, I'll solve for (n) by dividing both sides of the inequality by 1.5:[n < frac{13.5}{1.5} = 9]This tells me that (n) must be less than 9. Since (n) is a positive integer, the possible values of (n) are 1, 2, 3, 4, 5, 6, 7, and 8.Finally, I'll sum these integers to find the total:[1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36]

💡First, I need to solve the quadratic equation ( x^{2} - 6x + 1 = 0 ) using the completing the square method. To start, I'll move the constant term to the other side of the equation:[ x^{2} - 6x = -1 ]Next, I'll complete the square by adding the square of half the coefficient of ( x ) to both sides. The coefficient of ( x ) is (-6), so half of that is (-3), and squaring it gives ( 9 ):[ x^{2} - 6x + 9 = -1 + 9 ]Simplifying the right side:[ x^{2} - 6x + 9 = 8 ]Now, I can express the left side as a perfect square:[ (x - 3)^{2} = 8 ]This matches option B.

💡First, I need to simplify the given complex number (frac{2 - ai}{1 + i}). To do this, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is (1 - i), to eliminate the imaginary unit from the denominator.After multiplying, I'll expand both the numerator and the denominator. In the numerator, I'll use the distributive property to multiply out the terms, and in the denominator, I'll apply the difference of squares formula.Once simplified, I'll separate the resulting expression into its real and imaginary parts. Since the problem states that the complex number is purely imaginary, the real part must be equal to zero. This will give me an equation to solve for the value of (a).Finally, I'll solve the equation to find the value of (a) that makes the real part zero, ensuring that the complex number is indeed purely imaginary.