🧠:[<think>Okay, so I need to figure out the number of rectangles that are not squares in a square grid that's divided into 5 equal parts on each side. Let me start by visualizing the problem. The square has side length 5, and each side is divided into 5 equal segments. Then, lines are drawn parallel to the sides through these division points. This should create a grid of smaller squares inside the larger square. But the question isn't asking for the number of squares; instead, it wants the number of rectangles that are not squares. First, maybe I should recall how to calculate the total number of rectangles (including squares) in a grid. I remember that for a grid with m vertical lines and n horizontal lines, the number of rectangles is given by the combination formula C(m, 2) * C(n, 2). But in this case, the square is divided into 5 equal parts on each side. If each side is divided into 5 equal segments, then there are 6 division points on each side (including the corners). So, the number of vertical lines and horizontal lines would each be 5 (since the lines are drawn through the division points, which are 5 per side). Wait, hold on. If the side is divided into 5 equal parts, there are 5+1=6 points on each side, right? So, the number of vertical lines would be 5-1=4? Wait, no. Let me think carefully.If each side is divided into 5 equal segments, each of length 1 (since the total length is 5). So, the division points are at 1, 2, 3, 4, 5 units from the corner. Wait, but 5 is the total length, so actually, the divisions would be at 1, 2, 3, 4 units from each corner, right? Because if you divide a side of length 5 into 5 equal parts, each part is 1 unit long. So, the first division is at 1 unit from the corner, the next at 2 units, and so on until 4 units. Then, the number of vertical lines drawn parallel to the sides would be 4 on each side? Wait, no, the lines are drawn through the division points. Each side has 5 division points (including the corners). So, for each side, excluding the corners, there are 3 division points? Wait, no. Let me clarify.If we have a square with side length 5, and each side is divided into 5 equal segments, that means between the two corners, there are 5 segments. Therefore, there must be 5+1=6 points on each side (including the corners). So, each side has 6 points: the two corners and 4 internal division points. Therefore, if we draw lines parallel to the sides through these division points, we need to consider both vertical and horizontal lines. For each side, the vertical lines would be drawn through the division points on the top and bottom sides, and the horizontal lines through the division points on the left and right sides.But actually, all four sides are divided into 5 equal parts, so each side has 6 division points (including the corners). Then, lines parallel to the sides are drawn through these division points. So, for the vertical lines (parallel to the left and right sides), we draw them through the division points on the top and bottom sides. Similarly, for the horizontal lines (parallel to the top and bottom sides), we draw them through the division points on the left and right sides.Since each side has 6 division points (including corners), but the vertical lines are drawn through the division points on the top and bottom. Wait, but the division points on the top and bottom sides are the same as the ones on the left and right? Hmm, no. Let me think again.Wait, in a square, all sides are equal. If each side is divided into 5 equal parts, each of length 1. The division points on the top side would be at (1,5), (2,5), (3,5), (4,5). Similarly, the division points on the bottom side would be at (1,0), (2,0), (3,0), (4,0). Similarly, the left side division points are at (0,1), (0,2), (0,3), (0,4), and the right side at (5,1), (5,2), (5,3), (5,4). So, when we draw lines parallel to the sides through these division points, we have vertical lines through the division points on the top and bottom, and horizontal lines through the division points on the left and right.But since the vertical lines are drawn through the horizontal division points (on top and bottom), each vertical line corresponds to a horizontal coordinate (x-coordinate). Similarly, horizontal lines are drawn through vertical division points (left and right), corresponding to y-coordinates. Since each side (top and bottom) has 5 division points (including the corners). Wait, no. Wait, each side is divided into 5 equal parts, so there are 6 division points per side (including the corners). For example, on the top side from (0,5) to (5,5), divided into 5 parts of length 1 each, so division points at x=0,1,2,3,4,5. Similarly for the bottom side. Therefore, the vertical lines are drawn at x=1,2,3,4 (since the corners at x=0 and x=5 are already the sides of the square). Similarly, horizontal lines are drawn at y=1,2,3,4.So, in total, there are 4 vertical lines (at x=1,2,3,4) and 4 horizontal lines (at y=1,2,3,4) inside the square. Therefore, the grid formed has 5 columns and 5 rows (since 4 vertical lines divide the square into 5 vertical columns, and 4 horizontal lines divide it into 5 horizontal rows). Each small cell is a 1x1 square.Now, to compute the total number of rectangles (including squares) in this grid, the formula is C(m+1, 2) * C(n+1, 2), where m is the number of vertical divisions and n is the number of horizontal divisions. Wait, actually, the number of vertical lines is 4, so the number of vertical segments (columns) is 5. Similarly, horizontal lines are 4, so horizontal segments (rows) are 5. Therefore, the number of rectangles is C(5+1, 2) * C(5+1, 2)? Wait, no. Wait, the formula is C(number of vertical lines +1, 2) * C(number of horizontal lines +1, 2). Wait, perhaps I confused something here.Wait, let's recall: To find the number of rectangles in a grid formed by m vertical lines and n horizontal lines, the number of rectangles is C(m,2)*C(n,2). But in this case, the vertical lines include the sides of the square. Wait, in our case, the square has side length 5, and we have drawn 4 vertical lines inside, so total vertical lines are 4 internal + 2 sides = 6 vertical lines. Similarly, horizontal lines are 4 internal + 2 sides = 6 horizontal lines.Therefore, the total number of rectangles is C(6,2) * C(6,2). Let me check that. Because each rectangle is defined by choosing two vertical lines and two horizontal lines. So, if there are 6 vertical lines, the number of ways to choose two is C(6,2)=15. Similarly, C(6,2)=15 for horizontal lines. Therefore, total rectangles are 15*15=225. But wait, the original square is also a rectangle, but in this count, we include all possible rectangles, including the entire square itself.But in our problem, the square is divided into 5 equal parts on each side, leading to 4 internal lines each in vertical and horizontal directions, so total vertical lines: 0,1,2,3,4,5 (the sides at 0 and 5, and internal lines at 1,2,3,4). So yes, 6 vertical lines and 6 horizontal lines. Therefore, total rectangles: 15*15=225. Then, the total number of squares can be calculated by summing over all possible square sizes.How many squares are there? In a 5x5 grid, the number of squares is the sum from k=1 to 5 of (5 - k + 1)^2. Wait, no. Wait, in an m x n grid, the number of squares is sum_{k=1 to min(m,n)} (m - k +1)(n -k +1). Since in our case, the grid is 5x5 (5 rows and 5 columns of 1x1 squares), the number of squares would be sum_{k=1 to 5} (5 - k +1)^2 = sum_{k=1 to5} (6 -k)^2 = 5^2 +4^2 +3^2 +2^2 +1^2=25+16+9+4+1=55.Therefore, total number of squares is 55. Therefore, the total number of rectangles not squares is total rectangles minus total squares: 225 - 55=170. So, the answer should be 170. But let me verify this again step by step to make sure.First, confirming total number of rectangles. If there are 6 vertical lines and 6 horizontal lines, then indeed C(6,2)=15 for each, so 15*15=225 rectangles. Then, the number of squares in a 5x5 grid. For 1x1 squares: there are 5 rows and 5 columns, so 25. For 2x2 squares: 4x4=16. For 3x3: 3x3=9. For 4x4: 2x2=4. For 5x5: 1x1=1. So total squares:25+16+9+4+1=55. So, 225-55=170. But wait, is there a possibility that the lines drawn through the division points do not form a 5x5 grid? Wait, if each side is divided into 5 equal parts, each of length 1, then drawing lines through those division points would create a grid of 1x1 squares. So, yes, it's a 5x5 grid. Therefore, the calculation above is correct. Therefore, the answer should be 170.But wait, let me think again. The problem says "lines parallel to the sides through the 5 equal division points on each side". Wait, each side has 5 equal division points. If the side is divided into 5 equal parts, there are 6 division points (including the endpoints). But the problem says "the 5 equal division points on each side". Hmm, this is confusing. Maybe there's a misinterpretation here. If each side is divided into 5 equal parts, the number of division points is 6 (including the corners). But the problem mentions "the 5 equal division points on each side", which might exclude the corners. So, perhaps each side has 5 internal division points, making the total division points 5 (excluding the corners). Wait, but if you divide a line segment into 5 equal parts, you need 4 division points between the 5 parts. So, if the side length is 5, dividing it into 5 parts each of length 1, the division points would be at 1,2,3,4 units from the corner. So, 4 points per side, excluding the corners. Therefore, if the problem says "the 5 equal division points", that might be incorrect. Wait, perhaps there's a translation issue. Maybe the original problem in Chinese specifies something slightly different.But given the problem as stated: "draw lines parallel to the sides through the 5 equal division points on each side". If each side has 5 equal division points, then those would be points dividing the side into 6 equal parts. Wait, no. Wait, if you have a side of length 5, and you want 5 equal division points, that would divide the side into 6 equal segments. But that's conflicting. Wait, the number of division points (excluding the endpoints) needed to divide a segment into n equal parts is n-1. So, to divide a side into 5 equal parts, you need 4 division points. Therefore, if the problem says "5 equal division points", that would mean dividing the side into 6 equal parts, resulting in division points at 5/6, 10/6, 15/6, 20/6, 25/6. But the problem says "equal division points on each side", so maybe the problem actually means dividing each side into 5 equal parts, resulting in 4 division points, but the problem states there are 5. This is confusing.Wait, the problem says "through the 5 equal division points on each side". So, perhaps each side has 5 points that divide it into equal parts. If the side is divided into 5 equal parts, then those 5 division points would divide the side into 6 equal parts? Wait, no. If you have 5 division points on a side, including the endpoints, then dividing into 4 parts. Wait, this is very confusing.Hold on, perhaps the problem is translated from Chinese, and in Chinese, "5 equal division points" might mean dividing the side into 5 equal parts, resulting in 5 points, but in reality, dividing into 5 equal parts requires 4 points. Hmm. Maybe there is a mistranslation here. Alternatively, maybe the problem considers the corners as division points as well. So, if each side is divided into 5 equal segments, each of length 1, then the division points are at positions 0,1,2,3,4,5 along each side. So, 6 points in total. If the problem refers to the 5 equal division points excluding the corners, then that's 4 points. But the problem says 5. This is unclear.Wait, maybe the problem is using "equal division points" to mean the points that divide the side into equal parts, not counting the endpoints. For example, if you divide a side into 5 equal parts, each part is 1 unit, so the division points are at 1, 2, 3, 4. So, 4 points. But the problem says 5. Hmm. Alternatively, maybe the problem is considering the side divided into 5 equal parts, resulting in 5 points including the endpoints. But that would be 6 points. Wait, no. If you have a side from 0 to 5, and divide it into 5 equal parts, each part is length 1. Then, the division points are at 1,2,3,4, so 4 points. Therefore, the problem statement is conflicting.Given that the problem states "5 equal division points on each side", maybe we need to take it as given, even if mathematically it's a bit confusing. If there are 5 equal division points on each side, then those points would divide each side into 6 equal segments. Because, for a line segment divided into n equal parts, you need n-1 division points. So, if there are 5 division points, that would divide the side into 6 equal parts. Therefore, each segment is 5/6 units long. But then, the lines drawn through these division points would form a grid where each small rectangle has sides of length 5/6. But in the problem statement, the square has side length 5. If each side is divided into 6 equal parts, each of length 5/6, then we have a 6x6 grid of small squares. But the problem mentions "5 equal division points on each side", leading to 6 divisions. This seems possible.Wait, but the problem says "5 equal division points", not 6. Maybe the problem is using "equal division points" to refer to the points that split the side into equal parts, including the endpoints. So, if a side is divided into 5 equal parts, each part is 1 unit, then the division points (including endpoints) are 0,1,2,3,4,5. So, 6 points. Then, perhaps the problem is considering 5 internal division points (excluding the endpoints). But that would be 4 points. Wait, this is very confusing.Alternatively, maybe in Chinese terminology, "5 equal division points" refers to 5 intervals, hence 6 points. So, if you divide a side into 5 equal parts, you get 5 intervals, which requires 6 points. But the problem says "5 equal division points", which might mean 5 points. Given that this is a math competition problem, and the answer is likely an integer, maybe the intended interpretation is that each side is divided into 5 equal parts, each of length 1, leading to division points at 1,2,3,4, so 4 points per side (excluding the corners). Therefore, lines drawn through these 4 points on each side. Then, the grid formed is 5x5 (since 4 lines divide the side into 5 parts). Therefore, total vertical lines: 4 internal + 2 sides = 6. Similarly for horizontal. Then, total rectangles C(6,2)^2=225. Total squares sum_{k=1}^5 (6 -k)^2=55. So, 225-55=170. Hence, the answer is 170.But since the problem statement says "5 equal division points on each side", if each side has 5 points (including the corners?), then dividing into 4 equal parts. Wait, but 4 equal parts would mean division points at 1.25, 2.5, 3.75, which is probably not integers. But the problem says the square has side length 5. If the division points are at 1,2,3,4, which divides the side into 5 equal parts of length 1, then those are 4 internal division points. But the problem mentions 5 equal division points. So maybe the problem is using "equal division points" to include the endpoints. Then, each side has 6 division points (including endpoints), so 5 equal parts. Then, we have 5 vertical lines and 5 horizontal lines drawn through these division points (excluding the sides?), leading to a 5x5 grid. Wait, no. Wait, if each side has 6 division points (0,1,2,3,4,5), then drawing lines through these division points would create a grid with lines at x=0,1,2,3,4,5 and y=0,1,2,3,4,5. Therefore, the grid is 5x5 squares. But that's the same as the original square. Wait, no. If you draw lines through all division points, including the sides, then you just have the grid. But in that case, the number of vertical lines is 6 (from x=0 to x=5), same for horizontal. Therefore, total rectangles C(6,2)^2=225. Total squares sum_{k=1}^5 (6 -k)^2=55. Therefore, 225-55=170.But the problem says "draw lines parallel to the sides through the 5 equal division points on each side". If the "equal division points" include the corners (0 and 5), then each side has 5 equal division points. Wait, but if each side is divided into 5 equal parts, you need 6 division points (including the corners). Therefore, perhaps the problem is mistranslated or misstated, and the correct interpretation is 4 division points per side (excluding corners), leading to a 5x5 grid. Then, the answer is 170.Alternatively, maybe the problem is correct as stated, and "5 equal division points" on each side, which would divide each side into 6 equal parts. Then, each small segment is 5/6 units. Then, the grid would have 5 vertical lines and 5 horizontal lines inside the square, leading to 6 vertical lines in total (including sides) and 6 horizontal lines. Then, total rectangles would still be C(6,2)^2=225. Then, total squares would be sum_{k=1}^5 (6 -k)^2 * (5/6)^2? Wait, no. Wait, the squares in this grid would have sides that are multiples of 5/6. But since the problem is about counting the number of rectangles that are not squares, regardless of their size. However, in this case, the grid is not made up of integer lengths, so calculating squares would be different. But the problem states the square has side length 5. If the grid is divided into 5 equal parts on each side, each part is length 1, leading to a 5x5 grid of 1x1 squares. Therefore, the total number of squares is 55, as calculated earlier, and rectangles 225.Therefore, the answer is 170. I think this is the correct approach. Therefore, the total number of rectangles formed that are not squares is 170.Final Answerboxed{170}</think>To solve the problem of finding the number of rectangles formed that are not squares in a square with side length 5, divided into 5 equal parts on each side, we follow these steps:1. Understanding the Grid: - Each side of the square is divided into 5 equal parts, resulting in 6 division points (including the corners). - Drawing lines parallel to the sides through these division points creates a grid of 5x5 smaller squares inside the larger square.2. Total Number of Rectangles: - The total number of rectangles (including squares) in a grid is calculated using the combination formula ( binom{m}{2} times binom{n}{2} ), where ( m ) and ( n ) are the number of vertical and horizontal lines respectively. - Here, there are 6 vertical lines and 6 horizontal lines (including the sides of the square). - Total number of rectangles: ( binom{6}{2} times binom{6}{2} = 15 times 15 = 225 ).3. Total Number of Squares: - The number of squares in a 5x5 grid is calculated by summing the squares of all possible sizes. - For each size ( k times k ) (where ( k ) ranges from 1 to 5), the number of squares is ( (6 - k)^2 ). - Total number of squares: ( 5^2 + 4^2 + 3^2 + 2^2 + 1^2 = 25 + 16 + 9 + 4 + 1 = 55 ).4. Number of Rectangles Not Squares: - Subtract the total number of squares from the total number of rectangles. - Number of rectangles not squares: ( 225 - 55 = 170 ).Thus, the total number of rectangles formed that are not squares is (boxed{170}).,<think>Okay, so I need to figure out how many rectangles that are not squares are formed when you divide each side of a square with side length 5 into 5 equal parts and draw lines parallel to the sides through those division points. Hmm, let me start by visualizing the problem. First, the square has a side length of 5 units. If each side is divided into 5 equal parts, then each division is 1 unit apart. So, drawing lines parallel to the sides through these division points would create a grid of smaller squares inside the original square. Wait, but if each division is 1 unit, then the grid lines would be at positions 1, 2, 3, and 4 units from each side. That would divide the original square into a 5x5 grid of 1x1 squares. But the problem isn't asking for the number of squares; it's asking for the number of rectangles that are not squares. So, first, I might need to find the total number of rectangles (including squares) and then subtract the number of squares to get the number of non-square rectangles. Let me recall that in a grid with m vertical lines and n horizontal lines, the number of rectangles formed is given by the combination formula C(m, 2) * C(n, 2). In this case, the original square is divided into 5 equal parts on each side, which means there are 6 lines per side (including the sides of the original square). Wait, yes, because dividing a side into 5 equal parts requires 5+1=6 points, but the lines are drawn through the 5 division points. Hmm, maybe I need to clarify this. Wait, the problem says "through the 5 equal division points on each side". So each side is divided into 5 equal segments, each of length 1, so the division points are at positions 1, 2, 3, 4, and 5. But the original square has side length 5, so the division points are at 1, 2, 3, 4 from the starting corner. Wait, maybe I'm overcomplicating. Let me think again.If the side length is 5, dividing each side into 5 equal parts means each part is 1 unit long. Therefore, the division points are at 1, 2, 3, 4 units from each corner. So, drawing lines parallel to the sides through these points would create a grid. For each side, there are 5 division points, but how many lines does that correspond to?Each side is divided into 5 equal segments, so there are 5-1=4 division points between the segments? Wait, no, if you divide a line into 5 equal parts, you need 4 division points. For example, if you have a line from 0 to 5, dividing it into 5 segments of 1 each would require marks at 1, 2, 3, 4. So, 4 division points. But the problem says "5 equal division points". Hmm, that's confusing. Maybe the problem is written such that each side is divided into 5 equal parts, resulting in 5 segments, each 1 unit long, with 4 division points. But the problem states "through the 5 equal division points on each side". Wait, maybe it's 5 division points, which would mean 6 segments? No, that doesn't make sense.Wait, perhaps the problem is not in English originally, so there might be a translation issue. Let me read again: "draw lines parallel to the sides through the 5 equal division points on each side". So, if each side has 5 equal division points, those points divide the side into 6 equal parts? Because if you have n division points, you divide the side into n+1 parts. Wait, but that's only if the division points are between the parts. If you have 5 division points on a side, that would split the side into 6 segments. But the problem says "5 equal division points", which might mean dividing the side into 5 equal parts, resulting in 4 division points. Hmm. This is a critical point because it affects the entire grid.Let me try to resolve this ambiguity. The problem says "5 equal division points on each side". If the side is divided into 5 equal parts, then the number of division points would be 4 (since each division is between two segments). But the problem mentions 5 division points. Maybe in Chinese, the term "equal division points" refers to the points that divide the side into equal parts, including the endpoints. For example, if you divide a side into 5 equal parts, each of length 1, then the division points would be at 0, 1, 2, 3, 4, 5. But 0 and 5 are the original corners. So, if we are to draw lines through the division points, excluding the corners, then the lines would be at positions 1, 2, 3, 4. But if including the corners, then 0,1,2,3,4,5. But since the original square already has sides, drawing lines through the corners would just be the original sides. So, perhaps the lines are drawn through the internal division points (1,2,3,4). Hence, 4 lines per side, resulting in a 5x5 grid of 1x1 squares.But the problem says "5 equal division points on each side". If each side has 5 equal division points, then perhaps each side is divided into 5 intervals with 5 points? Wait, that still doesn't make sense. If you divide a side into 5 equal parts, you need 4 division points. So maybe the problem is translated incorrectly, and it should be 4 division points, resulting in 5 equal parts. Then, drawing lines through those 4 points per side, which would create a 5x5 grid.Alternatively, if the problem says 5 division points, then each side is divided into 6 equal parts of length 5/6 each. But the problem states the square has side length 5, so each division would be 5/6. However, the problem mentions "5 equal division points", which might mean 5 intervals. Wait, this is getting confusing. Let me check the standard terminology.In combinatorics, when you divide a line segment into n equal parts, you need n-1 division points. For example, dividing into 2 parts needs 1 division point. So, if the problem says 5 equal division points, that would imply n-1 = 5, so n=6 parts. Therefore, each side is divided into 6 equal parts, each of length 5/6. Then, the lines are drawn through these 5 division points, which are at positions 5/6, 10/6=5/3, 15/6=2.5, 20/6=10/3, and 25/6≈4.166... But this seems messy, and the problem mentions the square has side length 5. If the divisions are 1 unit each, then 5 divisions would make the side length 5. So, if each side is divided into 5 equal parts of length 1, then there are 4 division points at 1,2,3,4. Therefore, drawing lines through these 4 points per side.But the problem says "5 equal division points on each side". So, if each side has 5 equal division points, then each side is divided into 5 parts, each of length 1, with division points at 1,2,3,4,5. Wait, but 5 is the endpoint. So perhaps they consider the corners as division points as well. If that's the case, then drawing lines through all 5 division points (including the corners) would mean drawing lines at positions 0,1,2,3,4,5. But the original square already has sides at 0 and 5, so drawing lines through these would just be the original sides. Therefore, the internal division points would be at 1,2,3,4, and drawing lines through these would create 4 lines per direction, leading to a 5x5 grid.But the problem states "5 equal division points on each side", which perhaps includes the corners. So, each side is divided into 5 equal segments, each of length 1, with division points at positions 1,2,3,4 (excluding the corners). Therefore, 4 division points, leading to 5 segments. But the problem says 5 division points. Hmm. Maybe the problem is translated as "5 equal parts", which would have 4 division points. The wording is confusing.Alternatively, maybe the problem is from a source where "equal division points" includes both the starting and ending points. For example, if you divide a side into 5 equal parts, the division points are at 0,1,2,3,4,5, making 6 points. But if they are referring to the internal points only, it's 4 points. But the problem says "5 equal division points on each side". Maybe in Chinese terminology, when they say "5 equal division points", it refers to the internal points that divide the side into 5 equal parts, which would be 4 points. However, 4 points divide the side into 5 equal segments, each of length 1. Then, drawing lines through those 4 points per side, leading to a grid of 5x5 squares. Therefore, total lines: vertical lines at x=1,2,3,4 and horizontal lines at y=1,2,3,4. Thus, the grid has 5 rows and 5 columns of 1x1 squares.Given that confusion, but trying to proceed with the assumption that each side is divided into 5 equal parts (each 1 unit), with 4 division points (at 1,2,3,4), leading to a 5x5 grid of 1x1 squares. Then, the total number of rectangles in such a grid is calculated by choosing two distinct vertical lines and two distinct horizontal lines. The number of vertical lines is 6 (including the original sides), so C(6,2) ways to choose vertical lines, and similarly C(6,2) for horizontal lines. Therefore, total rectangles would be C(6,2) * C(6,2) = 15 * 15 = 225.But wait, in a 5x5 grid (i.e., 6 vertical and 6 horizontal lines), the number of rectangles is indeed C(6,2)*C(6,2)=15*15=225. Then, the number of squares can be calculated by summing up all possible squares of different sizes. For a square of size 1x1, there are 5*5=25. For 2x2, there are 4*4=16. For 3x3, 3*3=9. For 4x4, 2*2=4. For 5x5, 1*1=1. So total squares are 25+16+9+4+1=55. Therefore, the number of rectangles that are not squares would be total rectangles minus squares, i.e., 225 - 55 = 170. So the answer is 170.But wait, let me confirm the calculations again. Total rectangles: C(6,2) is 15, so 15*15=225. Squares: sum from k=1 to 5 of (6 - k)^2. Wait, no. Wait, in a grid with n rows and n columns of squares, the number of squares is sum_{k=1}^{n} (n - k + 1)^2. In this case, n=5. So sum_{k=1}^5 (5 - k + 1)^2 = sum_{k=1}^5 (6 - k)^2 = 5^2 + 4^2 + 3^2 + 2^2 +1^2=25+16+9+4+1=55. So that's correct. Therefore, 225 total rectangles -55 squares=170 non-square rectangles. But wait, the problem says "lines parallel to the sides through the 5 equal division points on each side". If the division points are 5, does that mean 5 internal points? If so, the grid would be different. Wait, perhaps my initial assumption was wrong. Let me check again.If each side has 5 equal division points, and we draw lines through those points, then how many lines do we have? For each side, if there are 5 division points, then along each side, you have 5 points where the lines are drawn. Since the square has four sides, but the lines are drawn parallel to the sides. So, for example, on the top and bottom sides, the division points are marked, and lines are drawn vertically through them. Similarly, on the left and right sides, division points are marked, and lines are drawn horizontally through them. Wait, perhaps that's another way to interpret it.Alternatively, maybe the problem is that on each side of the square, there are 5 points that divide the side into 5 equal segments. So each side is divided into 5 equal parts, meaning each part is 1 unit in length (since the side is 5 units). Therefore, on each side, the division points are at 1, 2, 3, 4 units from the corner. Therefore, on each of the four sides, there are 4 division points (excluding the corners). But the problem says "5 equal division points on each side". Hmm, this is conflicting.Wait, perhaps the problem is as follows: each side is divided into 5 equal parts, resulting in 5 points per side (including the corners). So, each side has 5 segments, each of length 1, with division points at 0,1,2,3,4,5 (including the corners). Therefore, each side has 6 points, but 5 intervals. Then, the lines are drawn through the division points, which include the corners. But the original square's sides are already there, so drawing lines through the corners would not add any new lines. Therefore, the only new lines are those through the internal division points at 1,2,3,4. Therefore, 4 vertical lines and 4 horizontal lines, creating a grid of 5x5 squares.But in this case, the number of vertical lines would be original two sides plus 4 internal lines, totaling 6 vertical lines. Similarly, 6 horizontal lines. Therefore, total rectangles would be C(6,2)*C(6,2)=15*15=225. Number of squares is 55, as before, so non-squares are 225-55=170.But given the problem statement says "through the 5 equal division points on each side", if each side has 5 division points, and we draw lines through them, then those lines would be 5 per side. Wait, but each side is a side of the square. If we take, say, the top side, which is a horizontal side, and we have 5 division points on it. Then, drawing lines parallel to the sides through these points would mean drawing vertical lines through those points, right? Similarly, on the left side, which is vertical, the 5 division points would have horizontal lines drawn through them.But each side has 5 division points. So, for the top and bottom sides (horizontal sides), each has 5 division points, through which vertical lines are drawn. Similarly, left and right sides (vertical sides) each have 5 division points, through which horizontal lines are drawn. Therefore, in total, we would have 5 vertical lines and 5 horizontal lines drawn inside the square. But wait, the original square has sides, so including them, there would be 5+2=7 vertical lines and 7 horizontal lines? Wait, no.Wait, each vertical line is drawn through a division point on a horizontal side (top or bottom). If each horizontal side has 5 division points, then there are 5 vertical lines. Similarly, each vertical side has 5 division points, leading to 5 horizontal lines. But the original square's sides are separate. Wait, this is a different approach.Let me try to parse the problem again: "In a square with side length 5, draw lines parallel to the sides through the 5 equal division points on each side, then the total number of rectangles formed that are not squares is..." So, the square has side length 5. On each side, there are 5 equal division points. Through each of these points, we draw lines parallel to the sides.So, each side is divided into 5 equal parts, so each part is length 1. Therefore, the division points on, say, the top side would be at 1,2,3,4,5 units from the left corner. Wait, but the side is length 5, so 5 equal parts would mean the points are at 1,2,3,4 from the corner (since the fifth division point would be at the end, which is the corner itself). Therefore, each side has 4 division points (excluding the corners). So, if we draw lines through these 4 points per side, but the problem says "5 equal division points on each side". So, maybe including the corners?If we include the corners as division points, then each side has 6 points (including both ends) dividing it into 5 equal parts. Then, drawing lines through all 5 division points (excluding the corners?), but the problem states "through the 5 equal division points on each side". If we take that literally, each side has 5 division points, which would divide the side into 4 equal parts? Wait, that would mean each part is length 5/4=1.25. But the problem mentions "5 equal division points", which could be confusing.Alternatively, perhaps "5 equal division points" refers to dividing each side into 5 equal parts, resulting in 5 intervals, each of length 1. In that case, the division points would be at 1,2,3,4 (excluding the corners). Therefore, 4 points per side. Then, drawing lines through these 4 points on each side. But since the sides are part of the square, each side is a horizontal or vertical side.For example, on the top side (a horizontal side), the division points are at x=1,2,3,4 (assuming the square is from (0,0) to (5,5)). Drawing vertical lines through these points. Similarly, on the bottom side, division points at x=1,2,3,4, drawing vertical lines through them. But these are the same vertical lines. Similarly, on the left and right sides (vertical sides), division points at y=1,2,3,4, drawing horizontal lines through them. Therefore, in total, we have 4 vertical lines (at x=1,2,3,4) and 4 horizontal lines (at y=1,2,3,4). Combined with the original square's sides (x=0, x=5, y=0, y=5), the total number of vertical lines is 6 (0,1,2,3,4,5) and horizontal lines is also 6. Therefore, the grid is 5x5 squares, each of size 1x1. Therefore, total number of rectangles is C(6,2)*C(6,2)=15*15=225, as before.Number of squares is sum_{k=1}^5 (6 - k)^2 = 25 + 16 + 9 + 4 +1=55. Therefore, non-square rectangles=225-55=170. But let me confirm once more. Suppose the square is divided into a 5x5 grid. Then, the number of rectangles of all sizes is indeed C(6,2)^2=225. The number of squares is as calculated 55, so the answer should be 170. However, I want to check if there's another interpretation where the division points include more lines, leading to a different count.Alternatively, if each side is divided into 5 equal parts, but considering both horizontal and vertical sides. Wait, perhaps each side (each of the four sides) is divided into 5 equal parts, and lines are drawn through all division points. For example, each vertical side (left and right) has 5 division points, so drawing horizontal lines through them, and each horizontal side (top and bottom) has 5 division points, drawing vertical lines through them. Wait, but how does that work?If each vertical side (left and right) is divided into 5 equal parts, then each part is 1 unit in length, so division points at y=1,2,3,4 on both left and right sides. Drawing horizontal lines through these points would create horizontal lines at y=1,2,3,4. Similarly, each horizontal side (top and bottom) is divided into 5 equal parts, division points at x=1,2,3,4 on both top and bottom sides. Drawing vertical lines through these points would create vertical lines at x=1,2,3,4. Therefore, again, creating a 5x5 grid. So same as before.Therefore, regardless of how I approach it, as long as the division is into 5 equal parts on each side, leading to a 5x5 grid, the total number of rectangles is 225, squares are 55, so non-squares are 170. But I need to make sure that the initial division is indeed into 5 equal parts. If the side length is 5, each part is 1, division points at 1,2,3,4. Drawing lines through those points, creating a grid. Then, yes, the rest follows.Alternatively, if the problem had said that each side is divided into 5 equal parts with 5 division points, meaning 6 intervals, each of length 5/6, then the calculation would be different. Then, the number of vertical lines would be 5 (from division points) plus the original sides, totaling 7 vertical lines. Similarly, 7 horizontal lines. Then, total rectangles would be C(7,2)*C(7,2)=21*21=441. Squares would be sum_{k=1}^6 (6 - k +1)^2= sum_{k=1}^6 (7 - k)^2=36+25+16+9+4+1=91. Then non-squares would be 441-91=350. But this is under the assumption that each side is divided into 6 equal parts, which would require 5 division points. However, the problem says "5 equal division points on each side". So, if "equal division points" refers to the number of points that divide the side into equal parts, then 5 points would divide the side into 6 equal parts, each of length 5/6. But in that case, the problem's answer would be 350, but since the problem mentions side length 5, it's more intuitive that the divisions are integer lengths. Hence, likely 1 unit each, leading to 5x5 grid.Given that, I think the answer is 170. Therefore, the total number of rectangles formed that are not squares is 170. So, boxed{170} is the answer.Final Answerboxed{170}</think>To solve the problem of finding the number of rectangles formed that are not squares in a square with side length 5, we start by dividing each side into 5 equal parts. This creates division points at 1, 2, 3, and 4 units from each corner, resulting in a 5x5 grid of 1x1 squares. First, we calculate the total number of rectangles (including squares) in this grid. The number of vertical lines and horizontal lines is 6 each (including the original sides of the square). The number of ways to choose two vertical lines and two horizontal lines to form a rectangle is given by the combination formula:[binom{6}{2} times binom{6}{2} = 15 times 15 = 225]Next, we calculate the number of squares. The number of 1x1 squares is 25, 2x2 squares is 16, 3x3 squares is 9, 4x4 squares is 4, and 5x5 squares is 1. Summing these gives:[25 + 16 + 9 + 4 + 1 = 55]Subtracting the number of squares from the total number of rectangles gives the number of rectangles that are not squares:[225 - 55 = 170]Thus, the total number of rectangles formed that are not squares is (boxed{170}).]

🔑:21. In 6 parallel lines, the number of ways to choose 2 lines each time is C_{6}^{2}, so the total number of different rectangles (including squares) is C_{6}^{2} cdot C_{6}^{2}. Also, the number of squares with side length 1 is 5^{2}, the number of squares with side length 2 is 4^{2}, cdots, so the total number of squares is 5^{2}+4^{2}+3^{2}+2^{2}+ 1^{2}=55. Therefore, the total number of non-square rectangles is left(C_{6}^{2}right)^{2}-55=170.

🧠:[<think>Okay, so I need to find the volume of a cone where the slant height is given as 5, and the lateral surface area is 15π. Hmm, let me recall the formulas related to cones. The lateral surface area (which is also called the curved surface area) of a cone is πrl, where r is the radius and l is the slant height. The volume of a cone is (1/3)πr²h, where h is the height. Alright, the problem says the slant height (l) is 5, and the lateral surface area is 15π. So first, maybe I can use the lateral surface area formula to find the radius. Let me write that down:Lateral Surface Area = πrl = 15π.Given that l = 5, substitute that into the equation:πr*5 = 15π.Hmm, okay. Let's solve for r. Divide both sides by π:5r = 15So then r = 15/5 = 3. Okay, so the radius is 3. Got that.Now, I need to find the volume. For the volume, I need the radius (which I now know is 3) and the height h. But I don't have h yet. However, I remember that in a cone, the slant height (l), radius (r), and height (h) form a right triangle, so by the Pythagorean theorem:l² = r² + h².We know l is 5 and r is 3, so let's plug those in:5² = 3² + h²25 = 9 + h²Subtract 9 from both sides:h² = 25 - 9 = 16So h = √16 = 4. Great, the height is 4.Now, plug the radius and height into the volume formula:Volume = (1/3)πr²h = (1/3)π*(3)²*4Let me compute that step by step. 3 squared is 9, so:(1/3)*π*9*4First, multiply 1/3 and 9: 9*(1/3) = 3.So then we have 3*π*4 = 12π.So the volume is 12π. Let me just verify the steps again to make sure I didn't make a mistake.1. Lateral surface area is πrl = 15π. Slant height l=5, so π*r*5 =15π. Divide both sides by π: 5r=15 → r=3. That seems right.2. Then, using Pythagorean theorem: l²=r² + h². 25=9 + h² → h²=16 → h=4. Correct.3. Volume is (1/3)πr²h. Plug in r=3, h=4: (1/3)π*9*4 = (1/3)*36π =12π. Yes, that's correct.Wait, let me check if I confused lateral surface area with total surface area. The problem specifically says lateral surface area, which is just the curved part, not including the base. So lateral surface area is indeed πrl, not πr(r + l). So that part is okay. The calculation steps seem all correct. Therefore, the volume is 12π. Final AnswerThe volume of the cone is boxed{12pi}.</think>Given that the slant height ( l ) of a cone is 5 and its lateral surface area is ( 15pi ), we need to find the volume of the cone.First, we use the formula for the lateral surface area of a cone, ( pi r l ), where ( r ) is the radius and ( l ) is the slant height:[pi r l = 15pi]Substituting ( l = 5 ):[pi r cdot 5 = 15pi]Dividing both sides by ( pi ):[5r = 15]Solving for ( r ):[r = frac{15}{5} = 3]Next, we use the Pythagorean theorem to find the height ( h ) of the cone. The slant height ( l ), radius ( r ), and height ( h ) form a right triangle:[l^2 = r^2 + h^2]Substituting ( l = 5 ) and ( r = 3 ):[5^2 = 3^2 + h^2][25 = 9 + h^2]Solving for ( h^2 ):[h^2 = 25 - 9 = 16]Taking the square root:[h = sqrt{16} = 4]Finally, we calculate the volume of the cone using the formula ( frac{1}{3} pi r^2 h ):[text{Volume} = frac{1}{3} pi (3)^2 (4)][= frac{1}{3} pi cdot 9 cdot 4][= frac{1}{3} cdot 36 pi][= 12 pi]The volume of the cone is (boxed{12pi}).,<think>Okay, let's see. The problem says that the slant height of a cone is 5, and its lateral surface area is 15π. We need to find the volume of the cone and keep the result in terms of π. Hmm, alright, let me remember the formulas related to cones. First, the lateral surface area of a cone is also called the curved surface area, right? The formula for that is πrl, where r is the radius and l is the slant height. Wait, the problem gives the slant height as 5. So, l = 5. The lateral surface area is given as 15π. So, substituting into the formula, we have π * r * 5 = 15π. Let me write that down:π * r * 5 = 15πOkay, so if I divide both sides by π, that cancels out. Then we have 5r = 15. Dividing both sides by 5, we get r = 3. So, the radius is 3 units. That seems straightforward.Now, we need to find the volume of the cone. The formula for the volume of a cone is (1/3)πr²h, where h is the height. We already have r, which is 3, but we need to find h. To find h, we can use the Pythagorean theorem because in a right triangle formed by the radius, height, and slant height of the cone, the slant height is the hypotenuse. So, according to the Pythagorean theorem, r² + h² = l². We know r = 3 and l = 5. Plugging those values in:3² + h² = 5² 9 + h² = 25 h² = 25 - 9 h² = 16 Taking the square root of both sides, h = 4. Wait, so the height is 4 units. Then, plugging r = 3 and h = 4 into the volume formula:Volume = (1/3)π*(3)²*4 Calculating that step by step: 3 squared is 9, multiplied by 4 gives 36, and then divided by 3 is 12. So, the volume is 12π.Let me double-check the steps to make sure I didn't make a mistake. 1. Lateral surface area: πrl = 15π. Given l = 5, solved for r: 5r = 15 => r=3. Correct.2. Then, using Pythagorean theorem: 3² + h² = 5². 9 + h² =25 => h²=16 => h=4. Correct.3. Volume: (1/3)πr²h = (1/3)π*9*4 = (36/3)π = 12π. Yep, that adds up.Wait, is there another way to approach this problem? Maybe check the lateral surface area formula again. Let me confirm the formula. Yes, lateral surface area of a cone is indeed πrl. So, that step is correct.Another thing to check: units. The problem doesn't mention units, so the answer will just be numerical with π. So, 12π is the correct volume. I think that's it. All steps seem to check out. So, the volume of the cone is 12π.Final AnswerThe volume of the cone is boxed{12pi}.</think>Given that the slant height ( l ) of a cone is 5 and its lateral surface area is ( 15pi ), we need to find the volume of the cone.1. Finding the radius ( r ): The lateral surface area of a cone is given by ( pi r l ). Substituting the given values: [ pi r cdot 5 = 15pi ] Dividing both sides by ( pi ): [ 5r = 15 ] Solving for ( r ): [ r = 3 ]2. Finding the height ( h ): Using the Pythagorean theorem in the right triangle formed by ( r ), ( h ), and ( l ): [ r^2 + h^2 = l^2 ] Substituting ( r = 3 ) and ( l = 5 ): [ 3^2 + h^2 = 5^2 ] [ 9 + h^2 = 25 ] Solving for ( h ): [ h^2 = 16 ] [ h = 4 ]3. Calculating the volume: The volume ( V ) of a cone is given by ( frac{1}{3} pi r^2 h ). Substituting ( r = 3 ) and ( h = 4 ): [ V = frac{1}{3} pi (3)^2 cdot 4 ] [ V = frac{1}{3} pi cdot 9 cdot 4 ] [ V = frac{36}{3} pi ] [ V = 12 pi ]The volume of the cone is (boxed{12pi}).]

🔑:Let the radius of the base of the cone be r, the slant height be l, and the height be h.Since the slant height of the cone is l=5, and the lateral surface area is 15pi,we have frac{1}{2} times l times r = 15pi. Solving this equation, we find the base radius r=3.Therefore, the height of the cone h= sqrt{l^2 - r^2} = 4.Thus, the volume of the cone is: V= frac{1}{3}pi r^2 h = frac{1}{3} times pi times 9 times 4 = 12pi.Hence, the answer is boxed{12pi}.To solve this problem, we calculate the base radius r=3 using the formula for the lateral surface area, and then use the Pythagorean theorem to find the height h= sqrt{l^2 - r^2} = 4. Substituting these values into the formula for the volume of a cone gives us the volume of this cone.This problem, by providing the slant height and lateral surface area of a cone and asking for its volume, mainly tests knowledge of the formulas for the lateral surface area and volume of a cone. It is considered a basic question.

🧠:[<think>Okay, so I need to prove these two inequalities given that a, b, c are positive real numbers and a² + b² + c² = 9. Let's start with the first one: abc ≤ 3√3. Hmm, I remember that for inequalities involving products and sums of variables with constraints, methods like AM-GM or Lagrange multipliers might be useful. Since all variables are positive, AM-GM is probably a good starting point.The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. But here we have a constraint involving squares. Let me think. If I can relate a², b², c² to their arithmetic mean, maybe I can use that to bound abc.Wait, the given condition is a² + b² + c² = 9. If I apply AM-GM to a², b², c², then (a² + b² + c²)/3 ≥ (a²b²c²)^(1/3). Plugging in the sum, 9/3 = 3 ≥ (a²b²c²)^(1/3). So, 3 ≥ (a²b²c²)^(1/3). Cubing both sides gives 27 ≥ a²b²c², so a²b²c² ≤ 27. Taking square roots on both sides, abc ≤ √27 = 3√3. Oh! That works out perfectly. So that proves the first inequality using AM-GM on the squares. That was straightforward. Maybe I need to check if equality holds. For AM-GM equality, all the terms must be equal, so a² = b² = c². Then each would be 3, so a = b = c = √3. Then abc would be (√3)^3 = 3√3, which matches the upper bound. So equality is achieved when a = b = c = √3. Okay, part (1) is done.Now part (2): (a²)/(b + c) + (b²)/(c + a) + (c²)/(a + b) > (a + b + c)/3. Hmm, this seems trickier. Let's see. The left side is similar to the form of Nesbitt's inequality, which is (a/(b + c) + b/(a + c) + c/(a + b)) ≥ 3/2. But here we have a²/(b + c) instead. Maybe we can use Cauchy-Schwarz inequality here. Let me recall Cauchy-Schwarz: (Σ (x_i y_i))² ≤ (Σ x_i²)(Σ y_i²). Alternatively, in the form of Σ (a_i² / b_i) ≥ (Σ a_i)² / Σ b_i. That's the Titu's lemma, which is a specific case of Cauchy-Schwarz.Let me try applying Titu's lemma to the left side. Let's set each term as a²/(b + c). Then according to Titu's lemma, the sum would be ≥ (a + b + c)² / [2(a + b + c)] = (a + b + c)/2. Wait, that gives (a²)/(b + c) + (b²)/(c + a) + (c²)/(a + b) ≥ (a + b + c)/2. But the problem asks to prove that the left side is greater than (a + b + c)/3. But (a + b + c)/2 is certainly greater than (a + b + c)/3, since all variables are positive. So if the left side is at least (a + b + c)/2, then it's definitely greater than (a + b + c)/3. But wait, the problem says "prove that the left side is greater than (a + b + c)/3". But given that we can show it's at least (a + b + c)/2, which is a stronger statement, why does the problem ask to prove the weaker inequality? Maybe there's a mistake here.Wait, let me check Titu's lemma again. If I have Σ (a²/(b + c)), then indeed by Cauchy-Schwarz:Σ (a²/(b + c)) ≥ (a + b + c)² / [2(a + b + c)] = (a + b + c)/2.Yes, that's correct. So the left-hand side is at least half the sum of a + b + c, which is stronger than what needs to be proven. But then the problem states to prove it's greater than (a + b + c)/3. So perhaps this is a misstep? Wait, but the problem says "prove that ... > ...", but according to Titu's lemma, we have it's ≥ (a + b + c)/2, which is certainly ≥ (a + b + c)/3 since a, b, c are positive. But why is the problem asking for a weaker inequality? Maybe the original problem had a typo, or perhaps there's a condition I'm missing.Wait, the given condition is a² + b² + c² = 9. Maybe the variables can take values such that a + b + c is large, but given that a² + b² + c² = 9, the maximum possible value of a + b + c is when all variables are equal, so by Cauchy-Schwarz, (a + b + c)² ≤ 3(a² + b² + c²) = 27, so a + b + c ≤ 3√3. So (a + b + c)/3 ≤ √3. However, the left-hand side, by Titu's lemma, is at least (a + b + c)/2, which would be ≥ (something up to 3√3)/2. Wait, but since a, b, c are positive, maybe even when a + b + c is minimal? But a + b + c can be as small as... Let's see, by Cauchy-Schwarz, (a + b + c)^2 ≤ 3(a² + b² + c²) = 27, but also, the minimum of a + b + c given a² + b² + c² = 9. The minimum occurs when one variable approaches √9 and the others approach 0, so a + b + c approaches 3. So (a + b + c)/3 approaches 1, and the left-hand side, by Titu's lemma, approaches (3)/2 = 1.5, which is greater than 1. So in all cases, the left-hand side is at least (a + b + c)/2, which is more than (a + b + c)/3. Therefore, the inequality (2) holds. But why does the problem state it as "greater than" rather than "greater than or equal"? Because when do we have equality in Titu's lemma? If all denominators are equal, so when (b + c) = (c + a) = (a + b), which would imply a = b = c. If a = b = c, then a² + b² + c² = 3a² = 9, so a = √3. Then left-hand side would be 3*( (√3)^2 )/(2√3) ) = 3*(3)/(2√3) = 9/(2√3) = (9√3)/6 = (3√3)/2. The right-hand side would be (3√3)/3 = √3. Comparing (3√3)/2 and √3, we have (3√3)/2 ≈ 2.598 and √3 ≈ 1.732, so indeed the left-hand side is greater. Therefore, equality in Titu's lemma would require a = b = c, but even then, the left-hand side is larger than the right-hand side. Therefore, the inequality is strict. Therefore, the original problem's statement of ">" is correct because even in the case of equality in Titu's lemma, the left-hand side is still larger. Wait, but if we have a case where variables are not equal, then the left-hand side would be even larger. Wait, but when a = b = c, the left-hand side is (3√3)/2, and the right-hand side is √3, so (3√3)/2 ≈ 2.598 > √3 ≈ 1.732. Therefore, the inequality is strict. Hence, the problem's second part is proven by Titu's lemma, as the left-hand side is at least (a + b + c)/2, which is always greater than (a + b + c)/3. Therefore, the inequality holds. So problem (2) is proven as well.Wait, but maybe I need to write a more detailed step-by-step for the proof. Let me verify again.Given that a, b, c are positive real numbers with a² + b² + c² = 9.First inequality: Using AM-GM on a², b², c²:(a² + b² + c²)/3 ≥ (a²b²c²)^(1/3)=> 3 ≥ (a²b²c²)^(1/3)=> 27 ≥ a²b²c²=> abc ≤ 3√3. Equality when a = b = c = √3.Second inequality: Apply Cauchy-Schwarz (Titu's lemma) on the terms a²/(b + c):Σ [a²/(b + c)] ≥ (a + b + c)² / [2(a + b + c)] = (a + b + c)/2.Since a, b, c > 0, (a + b + c)/2 > (a + b + c)/3.Therefore, Σ [a²/(b + c)] > (a + b + c)/3.Hence, the inequality is proven.Wait, but the problem uses ">" rather than "≥", but according to the steps, since (a + b + c)/2 is strictly greater than (a + b + c)/3, because (a + b + c) is positive, so dividing by 2 vs. 3 gives a larger number. Therefore, the inequality is strict. Therefore, the second part is proven.Therefore, both inequalities are proven with these methods. Let me check if there's any possible mistake here. For the first part, AM-GM is applied correctly. For the second part, Titu's lemma is correctly applied, leading to a lower bound which is already higher than the required upper bound. Therefore, the conclusion holds. So I think that's solid.Final Answer(1) boxed{abc leq 3sqrt{3}} (2) boxed{frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} > frac{a+b+c}{3}}</think>Given (a, b, c in mathbb{R}^{+}) such that (a^2 + b^2 + c^2 = 9), we need to prove the following inequalities:1. (abc leq 3sqrt{3})2. (frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} > frac{a+b+c}{3}) Proof of (1)Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality on (a^2, b^2, c^2):[frac{a^2 + b^2 + c^2}{3} geq sqrt[3]{a^2 b^2 c^2}]Given (a^2 + b^2 + c^2 = 9), we substitute:[frac{9}{3} geq sqrt[3]{a^2 b^2 c^2} implies 3 geq sqrt[3]{a^2 b^2 c^2}]Cubing both sides:[27 geq a^2 b^2 c^2]Taking the square root of both sides:[abc leq sqrt{27} = 3sqrt{3}]Equality holds when (a = b = c = sqrt{3}). Proof of (2)Applying Titu's lemma (a specific case of the Cauchy-Schwarz inequality) to the left-hand side:[frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} geq frac{(a + b + c)^2}{2(a + b + c)} = frac{a + b + c}{2}]Since (a, b, c > 0), we have:[frac{a + b + c}{2} > frac{a + b + c}{3}]Thus,[frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} > frac{a + b + c}{3}] Final Answers1. (boxed{abc leq 3sqrt{3}})2. (boxed{frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} > frac{a+b+c}{3}}),<think>Okay, so I have this problem here where I need to prove two inequalities given that a, b, c are positive real numbers and a² + b² + c² = 9. Let me start with the first one: proving that abc ≤ 3√3. Hmm, I remember that inequalities often involve techniques like AM-GM or Cauchy-Schwarz. Let me think about AM-GM here. The AM-GM inequality states that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. Since a², b², c² are all positive, maybe I can apply AM-GM to them. The arithmetic mean of a², b², c² is (a² + b² + c²)/3 = 9/3 = 3. The geometric mean would be (a²b²c²)^(1/3). So according to AM-GM, 3 ≥ (a²b²c²)^(1/3). If I cube both sides, I get 27 ≥ a²b²c². Taking the square root of both sides, we get √27 ≥ abc. Since √27 is 3√3, that gives abc ≤ 3√3. Wait, that seems to work out! So the first part is proved using AM-GM on the squares of a, b, c. Let me check if equality holds. For AM-GM equality, we need all terms equal, so a² = b² = c². Since a, b, c are positive, this implies a = b = c. Then a² + b² + c² = 3a² = 9 ⇒ a² = 3 ⇒ a = √3. Then abc = (√3)^3 = 3√3, which matches the upper bound. So equality holds when a = b = c = √3. Alright, part (1) seems sorted.Now moving on to part (2): proving that (a²)/(b + c) + (b²)/(c + a) + (c²)/(a + b) > (a + b + c)/3. Hmm, this looks a bit trickier. Let me recall some inequalities that might help here. Maybe Cauchy-Schwarz or Nesbitt's inequality? Wait, Nesbitt's inequality states that for positive a, b, c, (a/(b + c) + b/(a + c) + c/(a + b)) ≥ 3/2. But here we have a²/(b + c) instead of a/(b + c). So it's similar but not the same. Let me see. Alternatively, perhaps using Cauchy-Schwarz in the form of (sum x_i^2)(sum y_i^2) ≥ (sum x_i y_i)^2. If I set up the terms appropriately. Let me consider the numerator a², b², c² and denominators b + c, c + a, a + b. Maybe applying Cauchy-Schwarz to the numerators and denominators.Alternatively, maybe use the Titu's lemma, which is a particular case of Cauchy-Schwarz: sum(a_i^2 / b_i) ≥ (sum a_i)^2 / sum b_i. Let's try that. If I apply Titu's lemma to the left-hand side, we have:Sum (a² / (b + c)) ≥ (a + b + c)^2 / [2(a + b + c)] = (a + b + c)/2. So that gives us that the left-hand side is at least (a + b + c)/2. But the problem states that the left-hand side is greater than (a + b + c)/3. Well, since (a + b + c)/2 is definitely greater than (a + b + c)/3, as 1/2 > 1/3, this would imply that the left-hand side is indeed greater than (a + b + c)/3. But wait, the problem says to prove it's strictly greater. So maybe Titu's lemma gives a stronger result here. However, the problem specifies to prove it's greater than (a + b + c)/3, which is weaker. So in that case, this approach works. But wait, the problem says "prove that [...] > [...]", so if Titu's lemma gives a lower bound of (a + b + c)/2, which is greater than (a + b + c)/3, then that would suffice. But wait, hold on. Let me double-check.Wait, Titu's lemma says that sum(a_i^2 / b_i) ≥ (sum a_i)^2 / sum b_i. Here, the sum b_i would be sum (b + c) = 2(a + b + c). So sum(a²/(b + c)) ≥ (a + b + c)^2 / [2(a + b + c)] = (a + b + c)/2. Therefore, indeed, the left-hand side is at least (a + b + c)/2. Since (a + b + c)/2 is greater than (a + b + c)/3 (because 1/2 > 1/3 and a + b + c is positive), then we have (a²/(b + c) + ... ) ≥ (a + b + c)/2 > (a + b + c)/3. Therefore, the inequality holds. Wait, but the problem states to prove that the left-hand side is greater than (a + b + c)/3. But according to this, it's actually at least (a + b + c)/2, which is a stronger statement. So maybe the problem is a bit easier than expected. But wait, let me confirm if Titu's lemma is applicable here. Yes, since all denominators are positive (a, b, c are positive real numbers), so b + c, c + a, a + b are all positive. Therefore, Titu's lemma applies. Therefore, the conclusion holds. Therefore, the left-hand side is indeed greater than or equal to (a + b + c)/2, which is greater than (a + b + c)/3. Therefore, the inequality (2) holds. But the problem states ">", not "≥". So is it possible that equality could occur? Let's check when equality holds in Titu's lemma. For Titu's lemma, equality holds when a_i / b_i is constant for all i. So in this case, a/(b + c) = b/(a + c) = c/(a + b). Let's see if such a case is possible. Suppose a = b = c. Then a/(b + c) = a/(2a) = 1/2, so all terms are equal, so equality holds in Titu's lemma. So if a = b = c, then the left-hand side would be 3*(a²)/(2a) = 3*(a/2) = (3a)/2. The right-hand side would be (3a)/3 = a. So (3a)/2 > a? Since a is positive, 3/2 > 1, so yes, (3a)/2 > a. Therefore, even when a = b = c, the left-hand side is (3a)/2 and the right-hand side is a, so (3a)/2 > a. Therefore, even in the equality case of Titu's lemma, the inequality (2) is strict. Therefore, the original inequality (2) is strict. Wait, but in Titu's lemma, we have sum(a²/(b + c)) ≥ (a + b + c)/2. And (a + b + c)/2 > (a + b + c)/3, so the inequality (2) is actually a consequence of Titu's lemma. Therefore, the problem's second part is proved by Titu's lemma. However, the problem says ">", so I need to make sure that in all cases, it's strictly greater. But in Titu's lemma, equality holds when a/(b + c) = b/(a + c) = c/(a + b). Let me see if this is possible. Let's suppose a = b = c. Then each ratio is 1/2, so equal. But as we saw, even in that case, the left-hand side is (3a)/2, and the right-hand side is a, so (3a)/2 is greater than a. So in the case where equality holds in Titu's lemma, the inequality (2) is still strict. Therefore, there is no case where sum(a²/(b + c)) equals (a + b + c)/3. Therefore, the inequality is strict. Hence, part (2) is proved.But let me think again. Wait, suppose we have a case where two variables approach zero. Let's say a approaches sqrt(9 - 0 - 0) = 3, and b, c approach 0. Then the left-hand side becomes approximately (9)/(0 + 0) + 0 + 0. Wait, but b and c are approaching zero, so the denominators b + c would approach zero as well, making the first term approach infinity. But the right-hand side would be (a + b + c)/3 ≈ 3/3 = 1. So in this case, the left-hand side tends to infinity, which is definitely greater than 1. Similarly, if two variables are very small, the left-hand side becomes large. If all variables are equal, it's still greater. If one variable is larger and the others are smaller but positive, it seems the left-hand side is still greater. Therefore, the inequality should hold in all cases. Alternatively, maybe another approach. Let's consider the Cauchy-Schwarz inequality for each term. For example, (a²)/(b + c) + (b + c)/4 ≥ a, by the inequality (x²)/y + y ≥ 2x, which is AM-GM. Wait, but that might not be directly applicable here. Let me try:For each term, (a²)/(b + c) + (b + c)/4 ≥ a. This is because of AM-GM: [(a²)/(b + c) + (b + c)/4] ≥ 2*sqrt[(a²)/(b + c) * (b + c)/4] = 2*(a/2) = a. So summing over all three terms:Sum [(a²)/(b + c) + (b + c)/4] ≥ Sum a. So sum (a²/(b + c)) + (1/4)(2(a + b + c)) ≥ a + b + c. Therefore, sum (a²/(b + c)) + (a + b + c)/2 ≥ a + b + c. Therefore, sum (a²/(b + c)) ≥ (a + b + c) - (a + b + c)/2 = (a + b + c)/2. So again, we get the same result as Titu's lemma, that the sum is at least (a + b + c)/2. Therefore, this approach also gives the same bound. Hence, since (a + b + c)/2 > (a + b + c)/3, the inequality is proved. So both methods lead us to the conclusion. Therefore, part (2) is proved. Let me just check if there are any edge cases where the inequality might not hold. For example, when a, b, c are all equal, as before, sum (a²/(b + c)) = 3*(a²)/(2a) = (3a)/2. The right-hand side is (3a)/3 = a. Since (3a)/2 > a, holds. If one variable is much larger, say a approaches 3, and b, c approach 0, sum becomes approximately 9/(0 + 0) + 0 + 0, which is infinity, which is greater than (3)/3 = 1. If two variables are equal, and the third is different, say a = b, c different. Let me pick a = b = 2, then a² + b² + c² = 4 + 4 + c² = 8 + c² = 9 ⇒ c² = 1 ⇒ c = 1. Then sum (a²/(b + c)) = (4)/(2 + 1) + (4)/(1 + 2) + (1)/(2 + 2) = 4/3 + 4/3 + 1/4 = 8/3 + 1/4 = 32/12 + 3/12 = 35/12 ≈ 2.9167. The right-hand side is (2 + 2 + 1)/3 = 5/3 ≈ 1.6667. So indeed, 2.9167 > 1.6667. Another example: a = √8, b = 1, c = 0. But wait, c has to be positive. Let's take c approaching 0. Then sum becomes (8)/(1 + 0) + (1)/(√8 + 0) + 0/(√8 + 1) ≈ 8 + 1/(2.828) + 0 ≈ 8 + 0.353 ≈ 8.353. The right-hand side is (√8 + 1 + 0)/3 ≈ (2.828 + 1)/3 ≈ 3.828/3 ≈ 1.276. So 8.353 > 1.276. Therefore, in all test cases, the inequality holds. Therefore, I think part (2) is proved by Titu's lemma or Cauchy-Schwarz, giving a lower bound of (a + b + c)/2, which is greater than (a + b + c)/3. Hence, the inequality is strict.So both parts seem to be proved with these methods. Let me just recap:1. For abc ≤ 3√3: Applied AM-GM on a², b², c², which gave a bound on (abc)^(2/3) ≤ 3, leading to abc ≤ 3√3. Equality at a = b = c = √3.2. For the sum of a²/(b + c) > (a + b + c)/3: Applied Titu's lemma (or Cauchy-Schwarz) to show that the sum is at least (a + b + c)/2, which is strictly greater than (a + b + c)/3. Since even in the equality case of Titu's lemma, the inequality remains strict, the result holds.Therefore, both statements are proved.Final Answer(1) boxed{abc leq 3sqrt{3}} (2) boxed{frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} > frac{a+b+c}{3}}</think>Given (a, b, c in mathbb{R}^{+}) with (a^2 + b^2 + c^2 = 9), we need to prove the following inequalities:1. (abc leq 3sqrt{3})2. (frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} > frac{a+b+c}{3})Proof for (1):Using the AM-GM inequality on (a^2, b^2, c^2):[frac{a^2 + b^2 + c^2}{3} geq sqrt[3]{a^2 b^2 c^2}]Given (a^2 + b^2 + c^2 = 9), we have:[frac{9}{3} = 3 geq sqrt[3]{a^2 b^2 c^2}]Cubing both sides:[27 geq a^2 b^2 c^2]Taking the square root:[sqrt{27} geq abc implies 3sqrt{3} geq abc]Equality holds when (a = b = c = sqrt{3}). Thus, (abc leq 3sqrt{3}).Proof for (2):Using Titu's lemma (a consequence of the Cauchy-Schwarz inequality):[frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} geq frac{(a + b + c)^2}{2(a + b + c)} = frac{a + b + c}{2}]Since (frac{a + b + c}{2} > frac{a + b + c}{3}) (as (1/2 > 1/3)), we have:[frac{a^2}{b+c} + frac{b^2}{c+a} + frac{c^2}{a+b} geq frac{a + b + c}{2} > frac{a + b + c}{3}]Even in the case where equality holds in Titu's lemma (when (a = b = c)), the left-hand side is (frac{3a}{2}) and the right-hand side is (a), confirming the strict inequality. Thus, the inequality is proved. Final Answer(1) boxed{abc leq 3sqrt{3}} (2) boxed{frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} > frac{a+b+c}{3}}]

🔑:Proof of (1):Given a, b, c in mathbb{R}^{+} and a^{2}+b^{2}+c^{2}=9, we apply the AM-GM inequality to the left side of the equation:[a^{2}+b^{2}+c^{2} geq 3sqrt[3]{a^{2}b^{2}c^{2}}]Substituting the given condition into the inequality gives:[9 geq 3sqrt[3]{a^{2}b^{2}c^{2}}]Dividing both sides by 3:[3 geq sqrt[3]{a^{2}b^{2}c^{2}}]Raising both sides to the power of 3:[27 geq a^{2}b^{2}c^{2}]Taking the cube root of both sides:[3sqrt{3} geq abc]Therefore, we have proven that:[boxed{abc leq 3sqrt{3}}]Proof of (2):Given a, b, c in mathbb{R}^{+}, we apply the AM-GM inequality to each fraction and its corresponding sum:[frac{a^2}{b+c}+frac{b+c}{4} geq 2sqrt{frac{a^2}{b+c}cdotfrac{b+c}{4}} = a][frac{b^2}{c+a}+frac{c+a}{4} geq 2sqrt{frac{b^2}{c+a}cdotfrac{c+a}{4}} = b][frac{c^2}{a+b}+frac{a+b}{4} geq 2sqrt{frac{c^2}{a+b}cdotfrac{a+b}{4}} = c]Adding these inequalities together:[frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b}+frac{a+b+c}{2} geq a+b+c]Rearranging gives:[frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} geq frac{a+b+c}{2}]Since a+b+c > 0, it follows that:[frac{a+b+c}{2} > frac{a+b+c}{3}]Therefore, combining the above results:[boxed{frac{a^2}{b+c}+frac{b^2}{c+a}+frac{c^2}{a+b} > frac{a+b+c}{3}}]

🧠:[<think>Okay, let's see. The teacher and the student are talking about their ages. The problem says that when the teacher was the student's current age, the student was just 3 years old. Then, when the student becomes the teacher's current age, the teacher will be 39. Hmm, I need to figure out how old the teacher is now.Alright, let me break this down. Let's denote the teacher's current age as T and the student's current age as S. We need to find T. The first part of the problem says, "When I was your age, you were just 3 years old." So, the teacher is referring to a time in the past when they were the same age as the student is now. Let's figure out how many years ago that was. The difference between the teacher's current age and the student's current age is T - S years. So, that many years ago, the teacher was the student's current age. At that time, the student's age would have been S - (T - S). Because if the teacher is T now and was S age T - S years ago, then the student, who is S now, was S - (T - S) years old at that time. According to the problem, that age was 3. So:S - (T - S) = 3Let me simplify that:S - T + S = 3So, 2S - T = 3. Let's keep that as equation (1).Now the second part says, "When you are as old as I am now, I will be 39 years old." This is referring to the future when the student reaches the teacher's current age, which is T. Since the student is currently S, the number of years into the future that will take is T - S. At that future time, the teacher's age will be T + (T - S). Because the teacher is currently T, so adding T - S years to that gives T + (T - S). The problem states that at that time, the teacher will be 39. So:T + (T - S) = 39Simplify that:2T - S = 39Let's call this equation (2).So now, we have two equations:1) 2S - T = 32) 2T - S = 39We can solve these two equations simultaneously. Let's use substitution or elimination. Let me try elimination. From equation (1): T = 2S - 3Let's substitute this into equation (2):2(2S - 3) - S = 39Simplify:4S - 6 - S = 393S - 6 = 39Add 6 to both sides:3S = 45Divide by 3:S = 15So the student is 15 years old now. Then substitute S = 15 into equation (1):2(15) - T = 330 - T = 3So, T = 30 - 3 = 27Wait, the teacher is 27? Let me check that again. If the teacher is 27 and the student is 15, then the age difference is 12 years. So when the teacher was 15 (which is 27 - 15 = 12 years ago), the student was 15 - 12 = 3. That checks out. Now, when the student becomes 27 (which is 27 - 15 = 12 years from now), the teacher will be 27 + 12 = 39. That also checks out. So, even though 27 seems young for a teacher, mathematically it works. Maybe it's a tutor or something. But according to the equations, that's the answer.Final AnswerThe teacher is boxed{27} years old.</think>Let ( T ) be the teacher's current age and ( S ) be the student's current age. We need to find ( T ).From the first part of the problem:- When the teacher was the student's current age (( S )), the student was 3 years old. The age difference between the teacher and the student is ( T - S ). Therefore, ( S - (T - S) = 3 ).Simplifying this equation:[S - T + S = 3 implies 2S - T = 3 quad text{(1)}]From the second part of the problem:- When the student becomes the teacher's current age (( T )), the teacher will be 39 years old. The number of years until the student reaches age ( T ) is ( T - S ). Therefore, ( T + (T - S) = 39 ).Simplifying this equation:[2T - S = 39 quad text{(2)}]We solve these equations simultaneously. From equation (1), we express ( T ) in terms of ( S ):[T = 2S - 3]Substituting this into equation (2):[2(2S - 3) - S = 39 implies 4S - 6 - S = 39 implies 3S - 6 = 39 implies 3S = 45 implies S = 15]Substituting ( S = 15 ) back into equation (1):[2(15) - T = 3 implies 30 - T = 3 implies T = 27]Thus, the teacher is (boxed{27}) years old.,<think>Okay, let's try to figure out the teacher's age. Hmm, the problem says that when the teacher was the student's age, the student was just 3 years old. And when the student becomes as old as the teacher is now, the teacher will be 39. Alright, let's break this down step by step.First, let's assign variables to their current ages. Let me call the teacher's current age T and the student's current age S. So right now, the teacher is T years old, and the student is S years old. The teacher mentions a time in the past when the teacher was the student's current age. So, how many years ago was that? Well, if the teacher is now T and was S years old at that time, the difference in years would be T - S. So that many years ago, the teacher was S years old. At that time, the student's age would have been their current age minus those years. The student's age then was S - (T - S). Let me check that. If the teacher is T now and was S years old T - S years ago, then the student, who is S now, was S - (T - S) back then. Simplifying that: S - T + S = 2S - T. But the teacher says that at that time, the student was 3 years old. So, 2S - T = 3. That's our first equation: 2S - T = 3.Now, the second part of the problem says when the student is as old as the teacher is now, the teacher will be 39. So, in the future, when the student's age is T (the teacher's current age), how many years will have passed? The student is currently S, so it will take T - S years for the student to reach T. At that future time, the teacher's age will be their current age plus those T - S years. So, teacher's age then will be T + (T - S) = 2T - S. According to the problem, the teacher will be 39 then. So, 2T - S = 39. That's our second equation: 2T - S = 39.Now we have two equations:1) 2S - T = 32) 2T - S = 39We need to solve this system of equations to find T (teacher's age). Let's solve one of the equations for one variable and substitute into the other. Let's take the first equation and solve for T:From equation 1: 2S - T = 3 => T = 2S - 3Now plug this expression for T into equation 2:2T - S = 39Substitute T:2*(2S - 3) - S = 39Multiply out:4S - 6 - S = 39Combine like terms:3S - 6 = 39Add 6 to both sides:3S = 45Divide by 3:S = 15So the student is currently 15 years old. Now substitute S = 15 back into the expression for T:T = 2S - 3 = 2*15 - 3 = 30 - 3 = 27Wait, that would make the teacher 27 years old. Let me check if that works.First scenario: When the teacher was the student's current age (15), how many years ago was that? The teacher is now 27, so that was 27 - 15 = 12 years ago. At that time, the student was 15 - 12 = 3 years old. That checks out with the first statement.Second scenario: When the student is as old as the teacher is now (27), how many years from now will that be? The student is currently 15, so that's 27 - 15 = 12 years in the future. The teacher's age then will be 27 + 12 = 39, which matches the second statement. So both conditions are satisfied. Therefore, the teacher is 27 years old.Final AnswerThe teacher is boxed{27} years old.</think>Let ( T ) be the teacher's current age and ( S ) be the student's current age.1. When the teacher was the student's current age (i.e., ( S ) years old), the student was 3 years old. The difference in years between now and then is ( T - S ). At that time, the student's age was ( S - (T - S) ). This gives the equation: [ 2S - T = 3 ]2. When the student is as old as the teacher is now (i.e., ( T ) years old), the teacher will be 39 years old. The number of years from now until then is ( T - S ). The teacher's age at that time will be ( T + (T - S) ). This gives the equation: [ 2T - S = 39 ]We solve these equations step by step:1. From the first equation, solve for ( T ): [ T = 2S - 3 ]2. Substitute ( T = 2S - 3 ) into the second equation: [ 2(2S - 3) - S = 39 ] Simplify and solve for ( S ): [ 4S - 6 - S = 39 3S - 6 = 39 3S = 45 S = 15 ]3. Substitute ( S = 15 ) back into ( T = 2S - 3 ): [ T = 2(15) - 3 = 30 - 3 = 27 ]Both conditions are verified to be satisfied with these values. Therefore, the teacher is (boxed{27}) years old.]

🔑:Let's denote the age difference between the teacher and the student as x. Thus, the student's current age is x + 3, and the teacher's current age is 2x + 3.From the equation 2x + 3 + x = 39, we get:3x = 36x = 12Therefore, the teacher's current age is 2x + 3 = 2 times 12 + 3 = 27.Answer: The teacher is boxed{27} years old.To solve this problem, we assume the age difference between the teacher and the student is x. The student's current age is x + 3, and the teacher's current age is 2x + 3. Based on the statement "When you are as old as I am now, I will be 39 years old," we can establish a relationship: the teacher's current age plus the age difference equals 39. From this, we can solve for the age difference and then easily find the teacher's current age.The key to solving this problem is to recognize that the age difference remains constant. The difficulty lies in understanding the quantitative relationship implied in the two age comparisons.

🧠:[<think>Okay, let's see. I need to find the sum of the values of the polynomial (x^{5} - 1.7x^{3} + 2.5) at (x = 19.1) and (x = -19.1). Hmm, so basically, I have to compute the polynomial at both these points and then add the results together. But wait, maybe there's a smarter way than just plugging in each value separately? Let me think about the properties of polynomials and odd or even functions. First, let's recall that if a function is even, then (f(-x) = f(x)), and if it's odd, then (f(-x) = -f(x)). The polynomial given here is (x^5 - 1.7x^3 + 2.5). Let's check the parity of each term:- The term (x^5) is an odd function because 5 is an odd exponent.- The term (-1.7x^3) is also an odd function since 3 is odd.- The constant term (2.5) is an even function because constants don't change with (x).So, combining these, the polynomial is a combination of odd terms and an even term. Specifically, (x^5 - 1.7x^3) is odd, and (2.5) is even. Therefore, when we evaluate the polynomial at (-x), we get:[f(-x) = (-x)^5 - 1.7(-x)^3 + 2.5 = -x^5 + 1.7x^3 + 2.5]Comparing this to (f(x)), which is (x^5 - 1.7x^3 + 2.5), we can see that the odd terms change sign, while the constant term remains the same. So, if we compute (f(x) + f(-x)), the odd terms should cancel out, and the even term should double. Let's verify this:[f(x) + f(-x) = [x^5 - 1.7x^3 + 2.5] + [-x^5 + 1.7x^3 + 2.5] = (x^5 - x^5) + (-1.7x^3 + 1.7x^3) + (2.5 + 2.5) = 0 + 0 + 5 = 5]Wait, that's interesting! The sum of the polynomial evaluated at (x) and (-x) is always 5, regardless of the value of (x). That simplifies things a lot! Therefore, instead of calculating each value separately and then adding, I can directly say that the sum is 5. But let me make sure I didn't make a mistake here. Let's test this with a simpler value, say (x = 1). Compute (f(1)):(1^5 - 1.7 times 1^3 + 2.5 = 1 - 1.7 + 2.5 = 1.8)Compute (f(-1)):((-1)^5 - 1.7 times (-1)^3 + 2.5 = -1 + 1.7 + 2.5 = 3.2)Sum: (1.8 + 3.2 = 5). Perfect, it works for (x = 1). Let's try another value, maybe (x = 0):(f(0) = 0 - 0 + 2.5 = 2.5)(f(-0) = f(0) = 2.5)Sum: (2.5 + 2.5 = 5). Wait, but (x = 0) is a special case because (-0 = 0), so of course the sum is double the constant term, which is 5. So that checks out.Another test case: (x = 2).(f(2) = 32 - 1.7 times 8 + 2.5 = 32 - 13.6 + 2.5 = 20.9)(f(-2) = -32 - 1.7 times (-8) + 2.5 = -32 + 13.6 + 2.5 = -15.9)Sum: (20.9 + (-15.9) = 5). Yep, works again. So this seems to hold.Therefore, applying this general result, the sum of the polynomial evaluated at (x = 19.1) and (x = -19.1) is 5. Therefore, the answer should be 5. But just to be thorough, let me compute it directly, at least approximately, to check if there's any mistake. Let's compute (f(19.1)) and (f(-19.1)) separately and sum them.First, (x = 19.1):Compute (x^5): (19.1^5). That's a huge number. Let me see, 19^5 is 19*19*19*19*19. 19^2=361, 19^3=6859, 19^4=130321, 19^5=2476099. But 19.1 is slightly larger than 19, so 19.1^5 is a bit more than 2,476,099. Let me compute 19.1^5.Alternatively, maybe use logarithm or approximate? Wait, but maybe this is too time-consuming. Alternatively, note that when we compute (f(19.1) + f(-19.1)), the odd terms cancel, as established before, so the sum is 2*2.5 = 5. So even though each individual term is massive, their sum is small. But that's counterintuitive. Let me check again.Wait, for the polynomial (x^5 -1.7x^3 + 2.5), when we plug in x and -x:(x^5 -1.7x^3 + 2.5 + (-x^5) +1.7x^3 + 2.5 = 5).Yes, the x^5 terms cancel, the -1.7x^3 and +1.7x^3 cancel, and the constants add up. So regardless of how big x is, the sum is 5. Therefore, even if x is 19.1, which is a large number, the fifth powers are massive, but they cancel each other out. Similarly, the cubic terms as well. Therefore, the answer is indeed 5. But just to be absolutely sure, maybe compute (f(19.1)) and (f(-19.1)) numerically.First, compute (x = 19.1):Compute (x^5):19.1^5. Let's compute step by step:19.1^2 = (19 + 0.1)^2 = 19^2 + 2*19*0.1 + 0.1^2 = 361 + 3.8 + 0.01 = 364.8119.1^3 = 19.1 * 364.81. Let's compute:364.81 * 20 = 7,296.2Subtract 364.81 * 0.9 = 328.329So, 7,296.2 - 328.329 = 6,967.871But wait, that's for 19.1 * 364.81. Wait, 19.1 is 20 - 0.9, so 364.81 * 19.1 = 364.81*(20 - 0.9) = 364.81*20 - 364.81*0.9 = 7,296.2 - 328.329 = 6,967.871. So 19.1^3 = 6,967.871.Then 19.1^4 = 19.1 * 6,967.871. Again, let's approximate:6,967.871 * 20 = 139,357.42Subtract 6,967.871 * 0.9 = 6,271.0839So, 139,357.42 - 6,271.0839 ≈ 133,086.3361Then 19.1^5 = 19.1 * 133,086.3361Again, 133,086.3361 * 20 = 2,661,726.722Subtract 133,086.3361 * 0.9 = 119,777.7025Thus, 2,661,726.722 - 119,777.7025 ≈ 2,541,949.02So, approximately, 19.1^5 ≈ 2,541,949.02Now, compute -1.7x^3. We already found x^3 ≈ 6,967.871, so:-1.7 * 6,967.871 ≈ -1.7 * 6,967.871 ≈ -11,845.3807Then, the constant term is +2.5Therefore, f(19.1) ≈ 2,541,949.02 -11,845.3807 + 2.5 ≈ 2,541,949.02 -11,845.3807 = 2,530,103.6393 + 2.5 ≈ 2,530,106.1393That's a very large number. Now, compute f(-19.1):First, (-19.1)^5 = -19.1^5 ≈ -2,541,949.02Then, -1.7*(-19.1)^3. We have (-19.1)^3 = -6,967.871, so:-1.7*(-6,967.871) = 1.7*6,967.871 ≈ 11,845.3807Then, the constant term is +2.5Therefore, f(-19.1) ≈ -2,541,949.02 + 11,845.3807 + 2.5 ≈ -2,541,949.02 + 11,845.3807 = -2,530,103.6393 + 2.5 ≈ -2,530,101.1393Now, adding f(19.1) + f(-19.1):2,530,106.1393 + (-2,530,101.1393) = 5.0Exactly 5. So even though each value is enormous, their sum cancels out all the variable terms and leaves only the doubled constant term. Therefore, the answer is indeed 5. So regardless of the x value, as long as we evaluate the polynomial at x and -x, the sum will always be 5. Therefore, the problem is designed to test understanding of function symmetry, allowing us to avoid cumbersome calculations by recognizing the odd and even components of the polynomial. This was a good exercise in recognizing function properties to simplify computations. At first glance, calculating such high exponents for large x values seems intimidating, but by breaking down the polynomial into its odd and even parts, we can find the answer without any heavy computation. It's a neat trick that saves a lot of time and potential calculation errors.Final Answerboxed{5}</think>To find the sum of the values of the polynomial (x^5 - 1.7x^3 + 2.5) at (x = 19.1) and (x = -19.1), we can utilize the properties of odd and even functions. The polynomial (x^5 - 1.7x^3 + 2.5) consists of odd terms ((x^5) and (-1.7x^3)) and an even term (the constant 2.5). When evaluating the polynomial at (x) and (-x), the odd terms will cancel out, and the constant term will double. For any (x), we have:[f(x) + f(-x) = (x^5 - 1.7x^3 + 2.5) + (-x^5 + 1.7x^3 + 2.5)]Simplifying this expression:[f(x) + f(-x) = (x^5 - x^5) + (-1.7x^3 + 1.7x^3) + (2.5 + 2.5) = 0 + 0 + 5 = 5]Thus, the sum of the values of the polynomial at (x = 19.1) and (x = -19.1) is:[boxed{5}],<think>Alright, let's see. I need to find the sum of the values of the polynomial (x^{5} - 1.7x^{3} + 2.5) at (x = 19.1) and (x = -19.1). Hmm, okay. So, basically, I have to calculate the polynomial at both these points and then add them together. But maybe there's a smarter way to do this without calculating each term separately? Let me think.First, let's write out the polynomial again: (f(x) = x^5 - 1.7x^3 + 2.5). If I plug in (x = 19.1) and (x = -19.1), then add the results, I need to compute (f(19.1) + f(-19.1)). Let's express that:(f(19.1) + f(-19.1) = (19.1)^5 - 1.7(19.1)^3 + 2.5 + (-19.1)^5 - 1.7(-19.1)^3 + 2.5)Now, let me analyze each term. For the fifth power, (x^5) when x is positive and negative: Since 5 is an odd exponent, ((-19.1)^5 = - (19.1)^5). Similarly, the cubic term: 3 is also odd, so ((-19.1)^3 = - (19.1)^3). The constant term is 2.5 in both cases, so we have two constants.Let me substitute these into the expression:= ( (19.1)^5 - 1.7(19.1)^3 + 2.5 + (- (19.1)^5) - 1.7(- (19.1)^3) + 2.5 )Simplify term by term:1. The ( (19.1)^5 ) and ( - (19.1)^5 ) will cancel each other out.2. Next, the cubic terms: ( -1.7(19.1)^3 ) and ( -1.7(- (19.1)^3) ). Let's compute that. The first term is negative, the second term is -1.7 multiplied by a negative, which becomes positive. So:= ( -1.7(19.1)^3 + 1.7(19.1)^3 )These also cancel each other out.3. Then, the constants: 2.5 + 2.5 = 5.0.So, putting it all together:(f(19.1) + f(-19.1) = 0 + 0 + 5.0 = 5.0).Wait, is that right? Let me check again. Because the odd-powered terms cancel when adding f(x) and f(-x), but the even-powered terms would add. However, in this polynomial, the highest exponent is 5 (odd), then 3 (odd), and then the constant term. There are no even-powered terms except the constant, which is degree 0 (even). So, when we add f(x) + f(-x), all the odd terms cancel, and the even terms double. But in our case, the only even term is the constant 2.5, so when we add f(x) + f(-x), the constants add up to 5.0, and all other terms cancel out. Therefore, the sum is indeed 5.0.Therefore, regardless of the value of x, as long as we plug in x and -x into this polynomial, the sum will just be 2 * 2.5 = 5.0. That's pretty neat. So, even though the numbers 19.1 and -19.1 are quite large, their fifth powers and cubes are huge, but when added together, they cancel out, leaving only the sum of the constants.So, the answer should be 5.0. Let me verify with a simpler example to make sure I'm not making a mistake here. Let's take x = 1 and x = -1.Compute f(1) = 1 - 1.7*1 + 2.5 = 1 - 1.7 + 2.5 = 1.8.f(-1) = (-1)^5 -1.7*(-1)^3 + 2.5 = -1 -1.7*(-1) + 2.5 = -1 + 1.7 + 2.5 = 3.2.Sum is 1.8 + 3.2 = 5.0. Yep, same result. So that checks out.Another test: x = 2 and x = -2.f(2) = 32 - 1.7*8 + 2.5 = 32 - 13.6 + 2.5 = 20.9f(-2) = (-32) -1.7*(-8) + 2.5 = -32 +13.6 +2.5 = -15.9Sum: 20.9 + (-15.9) = 5.0. Perfect, same result. So the logic holds. Therefore, regardless of the x, the sum of f(x) + f(-x) for this polynomial is always 5.0. Therefore, the answer is 5.0.But since the problem might expect the answer in a box as boxed{5} or boxed{5.0}? Well, 5.0 is 5, so both are correct, but in polynomial contexts, if it's an exact value, maybe 5 is better, but since the coefficients have decimals, perhaps 5.0 is acceptable. However, in the problem statement, the coefficients are given with one decimal (1.7 and 2.5), so maybe the answer should also have one decimal. Hmm, but 5.0 is exact here. Let me check with the initial calculation.Wait, if I actually compute f(19.1) + f(-19.1), considering that 19.1 is a large number, maybe there's a mistake in assuming that the terms cancel? Let's compute f(19.1) and f(-19.1) numerically to confirm.First, compute f(19.1):(x = 19.1)(x^5 = 19.1^5). Let's compute that. 19.1^5. That's a huge number. Let me see if I can compute it step by step.First, 19.1 squared: 19.1 * 19.1 = 364.81Then, 19.1 cubed: 364.81 * 19.1. Let's compute 364.81 * 20 = 7,296.2, subtract 364.81 * 0.9 = 328.329, so 7,296.2 - 328.329 = 6,967.871. Wait, actually, no:Wait, 364.81 * 19.1 = 364.81*(20 - 0.9) = 364.81*20 - 364.81*0.9 = 7,296.2 - 328.329 = 6,967.871.So, 19.1^3 ≈ 6,967.871.Then, 19.1^5 = (19.1^2) * (19.1^3) = 364.81 * 6,967.871. That's going to be a massive number. Let's approximate:364.81 * 6,967.871 ≈ 365 * 7,000 ≈ 2,555,000. But more precisely, 364.81 * 6,967.871. Let's compute 364.81 * 6,967.871:First, 364.81 * 6,000 = 2,188,860364.81 * 900 = 328,329364.81 * 67.871 ≈ Let's compute 364.81 * 60 = 21,888.6; 364.81 * 7.871 ≈ 364.81 * 7 = 2,553.67; 364.81 * 0.871 ≈ 317.72. So total for 67.871: 21,888.6 + 2,553.67 + 317.72 ≈ 24,760.Adding all together: 2,188,860 + 328,329 + 24,760 ≈ 2,541,949.So, approximately 2,541,949. So, 19.1^5 ≈ 2,541,949.Then, 1.7x^3 = 1.7 * 6,967.871 ≈ 1.7 * 6,967.871. Let's compute 6,967.871 * 1 = 6,967.871; 6,967.871 * 0.7 ≈ 4,877.5097. So total ≈ 6,967.871 + 4,877.5097 ≈ 11,845.3807.Therefore, the polynomial at x=19.1:≈ 2,541,949 - 11,845.3807 + 2.5 ≈ 2,541,949 - 11,845.3807 = 2,530,103.6193 + 2.5 ≈ 2,530,106.1193.Similarly, compute f(-19.1):(-19.1)^5 = -19.1^5 ≈ -2,541,949.1.7*(-19.1)^3 = 1.7*(-6,967.871) ≈ -11,845.3807.So, f(-19.1) ≈ -2,541,949 - (-11,845.3807) + 2.5 ≈ -2,541,949 + 11,845.3807 + 2.5 ≈ -2,530,101.1193.Now, add f(19.1) + f(-19.1):≈ 2,530,106.1193 + (-2,530,101.1193) = 5.0.Wow, that actually worked out. Despite the massive numbers, the difference between them is exactly 5.0. So, my initial reasoning was correct. The odd-powered terms cancel out when adding f(x) and f(-x), leaving only twice the constant term, which is 2.5*2=5.0.Therefore, the answer is 5.0, which can be written as boxed{5} or boxed{5.0}. Since the original polynomial has decimal coefficients, maybe 5.0 is more precise here, but in mathematical notation, both are acceptable. However, in many math problems, unless specifying decimal places, 5 might be sufficient. Let me check the problem statement again: it says "the sum of the values of the polynomial...". The polynomial has decimal coefficients, so the answer is likely a decimal. But since 5.0 is an exact value here, not an approximation, writing boxed{5} is also correct because 5.0 is equal to 5. But to maintain decimal consistency, maybe 5.0.Wait, in the polynomial, the constants are 2.5, so 2.5 + 2.5 = 5.0. So, 5.0 is exact. However, in many math competitions or contexts, 5 is acceptable. Let me check the problem statement once more. It says "put your final answer within boxed{}". It doesn't specify the format. Given that the original polynomial has decimal coefficients, but the result is an exact integer, 5, perhaps they expect boxed{5}. Alternatively, in some contexts, if calculations involve decimals, you might put the decimal. But since 5.0 is equal to 5, and in the polynomial the constants are 2.5, which when doubled gives 5.0, but mathematically, 5.0 is 5. So, both are correct. But since in the problem statement, numbers are written with decimals (19.1, 1.7, 2.5), it might be safer to put boxed{5.0} to show that it's a decimal result. Alternatively, if the problem expects an exact answer, boxed{5} is also correct.But let's check with the example I did earlier with x=1 and x=-1: sum was 5.0, but written as 5. Similarly, with x=2 and x=-2, sum was 5.0, which is 5. So, in the context of the problem, even though the intermediate steps have decimals, the final result is an integer. Therefore, the answer is 5. So, I think boxed{5} is appropriate here.Final Answerboxed{5}</think>To find the sum of the values of the polynomial (x^5 - 1.7x^3 + 2.5) at (x = 19.1) and (x = -19.1), we start by evaluating the polynomial at these points and then adding the results.The polynomial is (f(x) = x^5 - 1.7x^3 + 2.5). We need to compute (f(19.1) + f(-19.1)).First, consider the polynomial evaluated at (x = 19.1):[f(19.1) = (19.1)^5 - 1.7(19.1)^3 + 2.5]Next, consider the polynomial evaluated at (x = -19.1):[f(-19.1) = (-19.1)^5 - 1.7(-19.1)^3 + 2.5]When we add these two expressions, we analyze each term:- The fifth power terms: ((19.1)^5 + (-19.1)^5 = 0) because ((-19.1)^5 = - (19.1)^5).- The cubic terms: (-1.7(19.1)^3 + 1.7(19.1)^3 = 0) because (-1.7(-19.1)^3 = 1.7(19.1)^3).- The constant terms: (2.5 + 2.5 = 5.0).Thus, the sum of the polynomial values at (x = 19.1) and (x = -19.1) is:[f(19.1) + f(-19.1) = 0 + 0 + 5.0 = 5.0]Therefore, the final answer is (boxed{5}).]

🔑:11. Solution. If the sign of (x) is changed to the opposite, the given polynomial will take the form: (-x^{5}+1.7 x^{3}+2.5). The sum of the given polynomial and the obtained one is 5.

🧠:[<think>Okay, let's see. The problem says that the terminal side of angle α passes through the point P(-1, 2). We need to find sin α, and there are four options given. Hmm, let me recall how to find trigonometric function values when given a point on the terminal side.First, I remember that for any angle in standard position (where the vertex is at the origin and the initial side is along the positive x-axis), if the terminal side passes through a point (x, y), then the sine of that angle is y divided by the radius r, where r is the distance from the origin to the point. The formula is sin α = y / r. Similarly, cosine is x / r. So, I need to find r first.The point given is P(-1, 2), so the coordinates are x = -1 and y = 2. To find r, which is the distance from the origin to the point P, I can use the Pythagorean theorem. The formula is r = sqrt(x² + y²). Let me compute that.Calculating x squared: (-1)² = 1. Then y squared is 2² = 4. Adding those together: 1 + 4 = 5. Therefore, r = sqrt(5). So, the radius is the square root of 5.Now, sin α is y / r. Here, y is 2, so sin α = 2 / sqrt(5). But wait, usually we rationalize the denominator, right? So, multiplying numerator and denominator by sqrt(5), that becomes (2 sqrt(5)) / 5. So, that would be 2 sqrt(5) over 5. Let me check the answer choices.Option B is 2 sqrt(5)/5, so that's one of the choices. But hold on, let me make sure I didn't make a mistake. The point is in the second quadrant because x is negative and y is positive, so angle α is in the second quadrant. In the second quadrant, sine is positive because sine corresponds to the y-coordinate. So, sin α should indeed be positive, which matches option B. The other options with negative signs (A and D) can be eliminated because sine is positive here.Just to double-check, maybe I miscalculated r. Let's confirm: x is -1, y is 2. So, sqrt((-1)^2 + 2^2) = sqrt(1 + 4) = sqrt(5). That's correct. Then sin α is 2/sqrt(5) which rationalizes to 2 sqrt(5)/5. Yep, that's definitely option B. The other options: option C is sqrt(5)/5, which would be if x was involved, but no, sine is y over r. So, C is probably a distractor if someone confuses x and y. But no, we have to use y here. So, yeah, B is the correct answer.Final Answerboxed{B}</think>To find (sin alpha) when the terminal side of angle (alpha) passes through the point (P(-1, 2)):1. Identify the coordinates: The point (P(-1, 2)) gives (x = -1) and (y = 2).2. Calculate the radius (r) using the Pythagorean theorem: [ r = sqrt{x^2 + y^2} = sqrt{(-1)^2 + 2^2} = sqrt{1 + 4} = sqrt{5} ]3. Determine (sin alpha) using the formula (sin alpha = frac{y}{r}): [ sin alpha = frac{2}{sqrt{5}} = frac{2sqrt{5}}{5} ]4. Verify the sign: Since the point (P(-1, 2)) is in the second quadrant, where sine is positive, the calculated value is correct.Thus, the correct answer is (boxed{B}).,<think>Okay, so I have this problem here where I need to find the sine of angle α, given that the terminal side of α passes through the point P(-1, 2). The options are A through D, some involving square roots and fractions. Let me think about how to approach this.First, I remember that when dealing with trigonometric functions and a point on the terminal side of an angle in standard position, we can use the coordinates of the point to find the values of the trigonometric functions. Specifically, for any point (x, y) on the terminal side, the sine of the angle is equal to y divided by the radius r, where r is the distance from the origin to the point (x, y). Similarly, cosine is x over r, and tangent is y over x. So, in this case, since the point is (-1, 2), x is -1 and y is 2. So, the formula for sine should be sin α = y / r. Therefore, I need to find the value of r. To find r, which is the distance from the origin to the point P(-1, 2), I can use the Pythagorean theorem. The formula for r is sqrt(x² + y²). Let me compute that.Plugging in the values, x is -1 and y is 2. Squaring x gives (-1)² = 1, and squaring y gives 2² = 4. Adding those together gives 1 + 4 = 5. Therefore, r = sqrt(5). So now, sin α should be y / r, which is 2 / sqrt(5). But wait, the answer choices don't have that. Instead, they have options with sqrt(5) in the numerator and multiplied by 2 or 1. Maybe I need to rationalize the denominator? Let me check. Yes, if I rationalize 2 / sqrt(5), I multiply numerator and denominator by sqrt(5), which gives (2 * sqrt(5)) / (sqrt(5) * sqrt(5)) = (2 sqrt(5)) / 5. So that's equal to 2√5 / 5. Looking at the options, that's option B. But wait, the point is in the second quadrant because x is negative and y is positive. But sine is positive in the second quadrant, right? Since sine corresponds to the y-coordinate, which is positive here. So 2 is positive, so sin α should be positive. The options B is positive, and D is negative. So that makes sense. So answer should be B. Wait, let me verify once again to make sure I didn't make a mistake. The coordinates are (-1, 2), so x is negative, y is positive. The radius is sqrt((-1)^2 + 2^2) = sqrt(1 + 4) = sqrt(5). So r is sqrt(5). Then sin α is y / r = 2 / sqrt(5). Rationalizing gives 2√5 / 5, which is option B. And since the y-coordinate is positive, sine is positive. So yes, option B.Just to be thorough, let's check the other options. Option A is -√5 / 5, which is negative, so probably not. Option C is √5 /5, which is approximately 0.447, while 2√5 /5 is approximately 0.894. Since 2 / sqrt(5) is roughly 0.894, which is larger, so option B. Option D is -2√5 /5, negative. So definitely, the answer is B. I think that's correct. Wait, let me make sure I didn't mix up sine and cosine. Because cosine would be x / r, which is -1 / sqrt(5), which is -√5 /5, which is option A. So cosine would be A, but sine is y / r, which is 2 / sqrt(5) rationalized to B. So yes, the answer is B. I don't see any mistakes in the reasoning. So the answer should be B.Final Answerboxed{B}</think>Given that the terminal side of angle (alpha) passes through the point (P(-1, 2)), we need to find (sin alpha).1. Identify the coordinates: The point (P) has coordinates ((-1, 2)), so (x = -1) and (y = 2).2. Calculate the radius (r) using the Pythagorean theorem: [ r = sqrt{x^2 + y^2} = sqrt{(-1)^2 + 2^2} = sqrt{1 + 4} = sqrt{5} ]3. Determine (sin alpha): The sine of an angle in standard position is given by (sin alpha = frac{y}{r}): [ sin alpha = frac{2}{sqrt{5}} ]4. Rationalize the denominator: [ sin alpha = frac{2}{sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{2sqrt{5}}{5} ]5. Check the sign: Since the point (P(-1, 2)) is in the second quadrant where sine values are positive, the result is positive.Thus, the correct answer is (boxed{B}).]

🔑:Given that x = -1 and y = 2, we can find the radius r as follows: r = sqrt{(-1)^2 + 2^2} = sqrt{5}. Therefore, sinalpha = frac{y}{r} = frac{2}{sqrt{5}} = frac{2sqrt{5}}{5}. Hence, the correct option is boxed{B}. This problem can be solved by using the definition of trigonometric functions for any angle, which tests the student's computational ability and is considered a basic question.