💡Okay, so I have this problem about a point P with coordinates (8 - 2m, m - 1). There are two parts to the problem. Let me try to understand each part step by step.**Part 1: If point P lies on the x-axis, find the value of m.**Hmm, I remember that any point on the x-axis has its y-coordinate equal to zero. So, for point P to be on the x-axis, the y-coordinate must be zero. The y-coordinate of P is given as (m - 1). So, I can set up the equation:m - 1 = 0Solving for m, I just add 1 to both sides:m = 1Okay, that seems straightforward. So, if m is 1, then the y-coordinate becomes zero, and the point P lies on the x-axis. Let me double-check by plugging m = 1 back into the coordinates:x-coordinate: 8 - 2(1) = 8 - 2 = 6y-coordinate: 1 - 1 = 0So, point P would be (6, 0), which is indeed on the x-axis. That makes sense.**Part 2: If the distance from point P to both coordinate axes is equal, find the coordinates of point P.**Alright, this part is a bit trickier. The distance from a point to the x-axis is the absolute value of its y-coordinate, and the distance to the y-axis is the absolute value of its x-coordinate. So, if these distances are equal, then:|x-coordinate| = |y-coordinate|Given the coordinates of P are (8 - 2m, m - 1), this translates to:|8 - 2m| = |m - 1|Okay, so I have an equation involving absolute values. I know that absolute value equations can have two cases each: one where the expression inside is positive and one where it's negative. So, I need to consider all possible cases here.Let me write down the equation again:|8 - 2m| = |m - 1|To solve this, I can square both sides to eliminate the absolute values, but maybe it's simpler to consider the different cases.Case 1: Both expressions inside the absolute values are non-negative.So, 8 - 2m ≥ 0 and m - 1 ≥ 0Which implies:8 - 2m ≥ 0 ⇒ -2m ≥ -8 ⇒ m ≤ 4andm - 1 ≥ 0 ⇒ m ≥ 1So, in this case, m is between 1 and 4, inclusive.So, the equation becomes:8 - 2m = m - 1Let me solve for m:8 - 2m = m - 1Bring variables to one side and constants to the other:8 + 1 = m + 2m9 = 3mm = 3Okay, so m = 3 is a solution in this case. Let me check if m = 3 satisfies the conditions for this case:m = 3 is between 1 and 4, so yes, it's valid.Now, let's find the coordinates of P when m = 3:x-coordinate: 8 - 2(3) = 8 - 6 = 2y-coordinate: 3 - 1 = 2So, point P is (2, 2). The distance to both axes is 2, which is equal. That works.Case 2: The expression inside the first absolute value is non-negative, and the second is negative.So, 8 - 2m ≥ 0 and m - 1 < 0Which implies:m ≤ 4 and m < 1So, m < 1The equation becomes:8 - 2m = -(m - 1)Simplify the right side:8 - 2m = -m + 1Now, solve for m:8 - 2m = -m + 1Bring variables to one side:8 - 1 = -m + 2m7 = mSo, m = 7Wait, but in this case, m < 1, but m = 7 is not less than 1. So, this solution doesn't satisfy the condition for this case. Therefore, m = 7 is not a valid solution here.Case 3: The expression inside the first absolute value is negative, and the second is non-negative.So, 8 - 2m < 0 and m - 1 ≥ 0Which implies:m > 4 and m ≥ 1So, m > 4The equation becomes:-(8 - 2m) = m - 1Simplify the left side:-8 + 2m = m - 1Now, solve for m:-8 + 2m = m - 1Bring variables to one side:2m - m = -1 + 8m = 7Okay, m = 7. Let's check if this satisfies the condition for this case:m = 7 > 4, so yes, it's valid.Now, let's find the coordinates of P when m = 7:x-coordinate: 8 - 2(7) = 8 - 14 = -6y-coordinate: 7 - 1 = 6So, point P is (-6, 6). The distance to the x-axis is |6| = 6, and the distance to the y-axis is |-6| = 6. They are equal, so this works.Case 4: Both expressions inside the absolute values are negative.So, 8 - 2m < 0 and m - 1 < 0Which implies:m > 4 and m < 1But m cannot be both greater than 4 and less than 1 at the same time. So, this case is impossible.Therefore, the only valid solutions are m = 3 and m = 7, giving points P(2, 2) and P(-6, 6).Wait, let me just verify if there are any other cases or if I missed something. I considered all four cases for the absolute values, but actually, since the equation is |A| = |B|, it's equivalent to A = B or A = -B. So, perhaps I should have considered those two cases instead of four.Let me try that approach to confirm.So, |8 - 2m| = |m - 1| implies:Either 8 - 2m = m - 1 or 8 - 2m = -(m - 1)First equation:8 - 2m = m - 18 + 1 = m + 2m9 = 3mm = 3Second equation:8 - 2m = -m + 18 - 1 = -m + 2m7 = mSo, m = 3 and m = 7, same as before.Therefore, the solutions are m = 3 and m = 7, leading to points (2, 2) and (-6, 6). That seems consistent.I think I covered all possibilities. So, the coordinates of point P are either (2, 2) or (-6, 6) when the distances to both axes are equal.Just to make sure, let me plug m = 3 and m = 7 back into the original coordinates and check the distances.For m = 3:x = 8 - 2(3) = 2y = 3 - 1 = 2Distance to x-axis: |2| = 2Distance to y-axis: |2| = 2Equal distances, good.For m = 7:x = 8 - 2(7) = -6y = 7 - 1 = 6Distance to x-axis: |6| = 6Distance to y-axis: |-6| = 6Equal distances, good.Yep, everything checks out. So, I think I've solved both parts correctly.

💡Okay, so I have this problem where I need to find the magnitude of a complex number z, which is given as z equals (1 minus i) divided by (2 plus i). Hmm, complex numbers can be tricky, but I think I remember some rules about how to handle them.First, I recall that the magnitude of a complex number is like its distance from the origin on the complex plane. So, if I have a complex number in the form a plus bi, its magnitude is the square root of (a squared plus b squared). But in this case, z is given as a fraction, so I probably need to simplify it first before I can find its magnitude.To simplify z, which is (1 - i)/(2 + i), I think I need to get rid of the imaginary unit i in the denominator. I remember that to do this, I can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi, right? So, the conjugate of (2 + i) should be (2 - i).Let me write that down:z = (1 - i)/(2 + i) * (2 - i)/(2 - i)Multiplying the numerators and denominators:Numerator: (1 - i)(2 - i)Denominator: (2 + i)(2 - i)Okay, let's expand the numerator first. Using the distributive property:(1)(2) + (1)(-i) + (-i)(2) + (-i)(-i) = 2 - i - 2i + i²Wait, i squared is equal to -1, so that becomes:2 - i - 2i + (-1) = 2 - 3i - 1 = (2 - 1) - 3i = 1 - 3iNow, the denominator:(2 + i)(2 - i) = (2)(2) + (2)(-i) + (i)(2) + (i)(-i) = 4 - 2i + 2i - i²Again, i squared is -1, so:4 - 2i + 2i - (-1) = 4 + 0i + 1 = 5So, putting it all together, z simplifies to:(1 - 3i)/5 = (1/5) - (3/5)iNow that I have z in the form a + bi, where a is 1/5 and b is -3/5, I can find its magnitude.The formula for magnitude is |z| = sqrt(a² + b²). Plugging in the values:|z| = sqrt((1/5)² + (-3/5)²) = sqrt(1/25 + 9/25) = sqrt(10/25) = sqrt(10)/5So, the magnitude of z is sqrt(10) divided by 5.Wait, let me double-check my steps to make sure I didn't make any mistakes. I multiplied numerator and denominator by the conjugate, expanded both correctly, simplified the terms, and then applied the magnitude formula. It all seems to add up. I think I did it right.

💡First, I need to determine the original price of the sundress before any discounts or price increases. The dress was on sale for 15% off, and the sale price was 68. To find the original price, I can set up the equation:0.85P = 68Solving for P gives:P = 68 / 0.85P = 80So, the original price of the dress was 80.Next, I know that the difference between the original price and the final price is 5. This means the final price after the store raised the sale price is:Final Price = Original Price - DifferenceFinal Price = 80 - 5Final Price = 75Now, I need to calculate the percentage increase from the sale price of 68 to the final price of 75. The increase in price is:Increase = Final Price - Sale PriceIncrease = 75 - 68Increase = 7To find the percentage increase, I divide the increase by the sale price and multiply by 100:Percentage Increase = (Increase / Sale Price) × 100Percentage Increase = (7 / 68) × 100Percentage Increase ≈ 10.29%Therefore, the store raised the new price by approximately 10.29%.

💡Okay, so I have this problem here: Given that (x) is real and (x^4 + frac{1}{x^4} = 2398), I need to find (x^2 + frac{1}{x^2}). Hmm, that seems a bit tricky at first glance, but maybe I can break it down step by step.First, I remember that expressions like (x^2 + frac{1}{x^2}) often come up when dealing with higher powers of (x) and (frac{1}{x}). Maybe there's a relationship between (x^4 + frac{1}{x^4}) and (x^2 + frac{1}{x^2}) that I can use. Let me think about how these expressions are connected.I recall that squaring (x^2 + frac{1}{x^2}) might give me something involving (x^4 + frac{1}{x^4}). Let me try that. So, if I let (y = x^2 + frac{1}{x^2}), then squaring both sides would give:[y^2 = left(x^2 + frac{1}{x^2}right)^2]Expanding the right side, I get:[y^2 = x^4 + 2 cdot x^2 cdot frac{1}{x^2} + frac{1}{x^4}]Simplifying the middle term, (x^2 cdot frac{1}{x^2}) is just 1, so:[y^2 = x^4 + 2 + frac{1}{x^4}]Wait, that's almost the expression I have in the problem! The problem gives me (x^4 + frac{1}{x^4} = 2398). So, if I substitute that into the equation above, I can solve for (y).Let me write that out:[y^2 = 2398 + 2][y^2 = 2400]So, to find (y), I take the square root of both sides:[y = sqrt{2400}]Hmm, (sqrt{2400}) seems a bit complicated. Let me simplify that. I know that 2400 can be broken down into 100 times 24, so:[sqrt{2400} = sqrt{100 times 24} = sqrt{100} times sqrt{24} = 10 times sqrt{24}]Now, (sqrt{24}) can be simplified further because 24 is 4 times 6, and 4 is a perfect square:[sqrt{24} = sqrt{4 times 6} = sqrt{4} times sqrt{6} = 2 times sqrt{6}]So, putting it all together:[sqrt{2400} = 10 times 2 times sqrt{6} = 20sqrt{6}]Therefore, (y = 20sqrt{6}). But wait, since (y = x^2 + frac{1}{x^2}), and (x) is real, (x^2) is positive, so (y) must also be positive. That makes sense because the square root of 2400 is positive.Let me just double-check my steps to make sure I didn't make a mistake. I started by letting (y = x^2 + frac{1}{x^2}), squared both sides, expanded, substituted the given value, solved for (y), and simplified the square root. Everything seems to add up.So, I think I've got it. The value of (x^2 + frac{1}{x^2}) is (20sqrt{6}).**Final Answer**The value of (x^2 + frac{1}{x^2}) is boxed{20sqrt{6}}.

💡Okay, so I have this arithmetic sequence problem here. Let me try to understand what it's asking. There's an arithmetic sequence {a_n}, and I know two things: the sum of the 16th, 17th, and 18th terms is equal to the 9th term, and both of these are equal to -36. Also, the sum of the first n terms is S_n. The first part asks me to find the general term formula for the sequence and the sum of the first n terms. The second part wants the minimum value of S_n and the value of n when this minimum occurs. Hmm, okay, let's start with part 1.I remember that in an arithmetic sequence, each term is the previous term plus a common difference, d. So, the general term is a_n = a_1 + (n-1)d, where a_1 is the first term. Given that a_16 + a_17 + a_18 = -36, and also a_9 = -36. Let me write that down:a_16 + a_17 + a_18 = -36 a_9 = -36Since a_9 is given, maybe I can express a_16, a_17, a_18 in terms of a_9. Let me see:a_16 = a_9 + 7d a_17 = a_9 + 8d a_18 = a_9 + 9dSo, adding these up:a_16 + a_17 + a_18 = (a_9 + 7d) + (a_9 + 8d) + (a_9 + 9d) = 3a_9 + (7d + 8d + 9d) = 3a_9 + 24dBut we know this sum is equal to -36, and a_9 is also -36. So:3*(-36) + 24d = -36 -108 + 24d = -36 24d = -36 + 108 24d = 72 d = 72 / 24 d = 3Okay, so the common difference d is 3. Now, since a_9 = -36, let's find a_1. a_9 = a_1 + 8d = -36 a_1 + 8*3 = -36 a_1 + 24 = -36 a_1 = -36 - 24 a_1 = -60So, the first term is -60, and the common difference is 3. Therefore, the general term formula is:a_n = a_1 + (n-1)d a_n = -60 + (n-1)*3 Let me simplify that:a_n = -60 + 3n - 3 a_n = 3n - 63Okay, so that's the general term. Now, for the sum of the first n terms, S_n. I remember the formula for the sum of an arithmetic sequence is:S_n = n/2 * (2a_1 + (n-1)d) Alternatively, it can also be written as S_n = n*(a_1 + a_n)/2Let me use the first formula:S_n = n/2 * [2*(-60) + (n-1)*3] Simplify inside the brackets:2*(-60) = -120 (n-1)*3 = 3n - 3 So, total inside is -120 + 3n - 3 = 3n - 123Thus, S_n = n/2 * (3n - 123) Simplify:S_n = (3n^2 - 123n)/2 I can factor out a 3:S_n = (3/2)(n^2 - 41n)Hmm, maybe I can write this in vertex form to find the minimum later. Vertex form is a quadratic in the form a(n - h)^2 + k, where (h, k) is the vertex. Let me complete the square.Starting with S_n = (3/2)(n^2 - 41n). Let me factor out the coefficient of n^2 first:S_n = (3/2)[n^2 - 41n]To complete the square, take half of 41, which is 20.5, square it: (20.5)^2 = 420.25So, add and subtract 420.25 inside the brackets:S_n = (3/2)[(n^2 - 41n + 420.25) - 420.25] = (3/2)[(n - 20.5)^2 - 420.25]Distribute the (3/2):S_n = (3/2)(n - 20.5)^2 - (3/2)*420.25 Calculate (3/2)*420.25:420.25 * 3 = 1260.75 1260.75 / 2 = 630.375So, S_n = (3/2)(n - 20.5)^2 - 630.375Since the coefficient of the squared term is positive, the parabola opens upwards, meaning the vertex is the minimum point. The vertex occurs at n = 20.5. But n has to be an integer because it's the number of terms. So, the minimum occurs at n = 20 or n = 21.Let me compute S_20 and S_21 to see which is smaller.First, S_20:S_20 = (3/2)(20^2 - 41*20) = (3/2)(400 - 820) = (3/2)(-420) = (3/2)*(-420) = -630Now, S_21:S_21 = (3/2)(21^2 - 41*21) = (3/2)(441 - 861) = (3/2)(-420) = -630So, both S_20 and S_21 are equal to -630. Therefore, the minimum value of S_n is -630, achieved at n = 20 and n = 21.Wait, but in the problem statement, it says "the value of n when S_n reaches its minimum." Since both n=20 and n=21 give the same minimum, maybe I should mention both? Or is it sufficient to say n=20.5, but since n must be integer, the minimum occurs at both 20 and 21.I think in the context of the problem, since n must be an integer, both 20 and 21 are valid answers. But maybe the problem expects just one value? Hmm, let me check my calculations again.Wait, when I calculated S_20 and S_21, both gave me -630. So, the minimum is achieved at both n=20 and n=21. So, perhaps I should state both.But in the initial problem, part 2 says "the value of n when S_n reaches its minimum." So, maybe it's expecting both values? Or maybe just the smaller one? Hmm, not sure. But in any case, both give the same minimum.So, to recap:1. The general term is a_n = 3n - 63, and the sum S_n = (3/2)n^2 - (123/2)n.2. The minimum value of S_n is -630, achieved at n=20 and n=21.I think that's it. Let me just double-check my calculations to make sure I didn't make any mistakes.First, for the general term:a_n = a_1 + (n-1)d = -60 + 3(n-1) = -60 + 3n - 3 = 3n - 63. That looks correct.For the sum:S_n = n/2 [2a_1 + (n-1)d] = n/2 [2*(-60) + 3(n-1)] = n/2 [-120 + 3n - 3] = n/2 [3n - 123] = (3n^2 - 123n)/2. Correct.Completing the square:S_n = (3/2)(n^2 - 41n) = (3/2)[(n - 20.5)^2 - (20.5)^2] = (3/2)(n - 20.5)^2 - (3/2)(420.25) = (3/2)(n - 20.5)^2 - 630.375Since 630.375 is approximately 630.375, but since we're dealing with exact values, 420.25 is 1681/4, so (3/2)*(1681/4) = (3*1681)/(8) = 5043/8 = 630.375. So, that's correct.Therefore, the minimum occurs at n=20.5, but since n must be integer, n=20 and n=21 both give the same minimum sum of -630. So, that's correct.I think I'm confident with this solution.

💡Okay, so I have this problem about a triangular pyramid, which is also called a tetrahedron, right? It's labeled ABCD, and there's a plane P that intersects it, forming a quadrilateral EFGH. The points E and F are on edges AB and AC, respectively. The plane P is parallel to edges AD and BC, and it's given that AD and BC are equal in length. Also, the ratio of EA to EB is 2, which means EA is twice as long as EB. I need to find the ratio EF to EH.Hmm, let me visualize this. So, we have a tetrahedron ABCD. Let me sketch it mentally: A is the apex, and B, C, D are the base vertices. Plane P cuts through the tetrahedron, intersecting edges AB and AC at E and F, and presumably edges AD and BC at G and H, forming a quadrilateral EFGH.Since plane P is parallel to edges AD and BC, that should mean that the lines of intersection of P with the faces of the tetrahedron are parallel to AD and BC. So, for example, the intersection of P with face ABD should be a line parallel to AD, and similarly, the intersection with face ABC should be a line parallel to BC.Wait, but EFGH is a quadrilateral. So, E is on AB, F is on AC, G is on CD, and H is on BD? Or maybe not. Let me think. Since plane P is parallel to AD and BC, which are edges, the intersection lines should be parallel to these edges.Let me try to break it down step by step.1. **Understanding the Plane Intersection:** - Plane P intersects the tetrahedron ABCD. - The intersection is a quadrilateral EFGH. - E is on AB, F is on AC. - Plane P is parallel to edges AD and BC.2. **Properties of Parallel Planes:** - If a plane is parallel to an edge, the intersection line with a face containing that edge will be parallel to the edge. - So, since P is parallel to AD, the intersection of P with face ABD (which contains AD) must be a line parallel to AD. Similarly, the intersection with face ACD (which also contains AD) will also be parallel to AD. - Similarly, since P is parallel to BC, the intersection with face ABC (which contains BC) will be a line parallel to BC, and the intersection with face BCD (which also contains BC) will be parallel to BC.3. **Identifying the Intersection Points:** - On face ABD, the intersection line is EH, which is parallel to AD. - On face ABC, the intersection line is EF, which is parallel to BC. - On face ACD, the intersection line is FG, which is parallel to AD. - On face BCD, the intersection line is GH, which is parallel to BC.4. **Conclusion on Quadrilateral EFGH:** - Since EH is parallel to AD and FG is parallel to AD, EH is parallel to FG. - Similarly, EF is parallel to BC and GH is parallel to BC, so EF is parallel to GH. - Therefore, EFGH is a parallelogram because both pairs of opposite sides are parallel.5. **Given Ratios and Similar Triangles:** - The ratio EA:EB is 2:1. So, EA = 2 EB. - Since E is on AB, we can say that E divides AB in the ratio 2:1. So, if AB is divided into three equal parts, E is two parts away from A and one part away from B.6. **Applying Similar Triangles:** - Let's consider triangle ABD. Since EH is parallel to AD, triangle BEH is similar to triangle BAD. - The ratio of similarity is the same as the ratio of BE to BA. Since EA:EB = 2:1, BA = EA + EB = 2 + 1 = 3 parts. So, BE is 1 part, BA is 3 parts. Therefore, the ratio is 1:3. - Therefore, EH = (1/3) AD.7. **Similarly for EF:** - Consider triangle ABC. Since EF is parallel to BC, triangle AEF is similar to triangle ABC. - The ratio of similarity is EA:AB = 2:3. - Therefore, EF = (2/3) BC.8. **But AD = BC:** - It's given that AD and BC are equal. So, AD = BC. - Therefore, EH = (1/3) AD = (1/3) BC. - And EF = (2/3) BC.9. **Calculating the Ratio EF:EH:** - EF = (2/3) BC - EH = (1/3) BC - Therefore, EF:EH = (2/3 BC) : (1/3 BC) = 2:1.Wait, that seems straightforward, but let me double-check.10. **Verification:** - Since EFGH is a parallelogram, opposite sides are equal and parallel. - We found that EF is parallel and proportional to BC, and EH is parallel and proportional to AD. - Given AD = BC, the lengths of EF and EH are directly proportional to the ratios we found. - So, EF is twice as long as EH, hence the ratio 2:1.Hmm, seems consistent. I don't see any flaws in this reasoning. The key was recognizing that EFGH is a parallelogram due to the plane being parallel to two edges, and then using similar triangles to find the proportions.**Final Answer**The ratio (EF:EH) is boxed{2}.