🧠:[<think>Okay, so I need to figure out the minimal number of pairs of selected cells on an 8x8 chessboard where 16 cells are selected, such that the number of pairs sharing the same row or column is as small as possible. Hmm, okay. Let me start by understanding the problem properly.First, the chessboard is 8x8, so there are 64 cells in total. We're selecting 16 cells. Now, the question is about the minimal number of pairs that are in the same row or the same column. Each pair of cells in the same row or column contributes to this count. So, if two cells are in the same row, that's one pair, and if they're in the same column, that's another pair. Wait, no, actually, if two cells are in the same row and the same column, that would mean they are the same cell, which isn't possible. So, a pair of cells can only be in the same row or in the same column, not both. So, each pair can be counted at most once, either for a row or for a column. Hmm, but how exactly is the count done? Is it the number of pairs that share a row plus the number of pairs that share a column? Or is it the number of pairs that share either a row or a column? Let me check.The problem says "pairs of selected cells in the same row or column". So, each pair is counted once if they are in the same row, once if they are in the same column. Wait, but if two cells are in the same row and same column, they would be the same cell, so that's impossible. Therefore, each pair of cells can be in the same row, in the same column, or in neither. So, the total number of conflicting pairs is the number of pairs in the same row plus the number of pairs in the same column. But wait, each pair is unique, so if two cells are in the same row, they are not in the same column, and vice versa. So, actually, each pair can only be counted in one of the two categories. Therefore, the total number of conflicting pairs is equal to the sum over all rows of the number of pairs in that row, plus the sum over all columns of the number of pairs in that column. That is, for each row, if there are k selected cells in that row, then the number of pairs in that row is C(k, 2) = k(k-1)/2. Similarly for each column. So, the total number of conflicting pairs is the sum over all rows of C(k_i, 2) plus the sum over all columns of C(l_j, 2), where k_i is the number of selected cells in row i, and l_j is the number in column j.Therefore, the problem reduces to selecting 16 cells on the chessboard such that the sum of C(k_i, 2) over all rows plus the sum of C(l_j, 2) over all columns is minimized. So, to find the minimal possible number of such pairs. Hmm, okay. So, the question is equivalent to minimizing the total number of pairs in rows and columns. To minimize this sum, we need to distribute the 16 cells as evenly as possible across the rows and columns. Because the function C(k, 2) is convex, right? So, by Jensen's inequality, the minimal sum is achieved when the distribution is as uniform as possible.So, for rows: we have 8 rows. If we place 16 cells into 8 rows, the most even distribution is 2 cells per row. Because 16 divided by 8 is 2. Similarly, for columns: we have 8 columns. So, 16 cells over 8 columns is 2 per column. Therefore, if we can arrange the 16 cells such that each row and each column contains exactly 2 cells, then the total number of pairs would be 8 * C(2, 2) + 8 * C(2, 2) = 8 * 1 + 8 * 1 = 16. But wait, is such an arrangement possible? That would be a 2-regular bipartite graph between rows and columns, right? Essentially, a 2-regular graph in a bipartite sense, which would mean that it's a union of cycles covering all rows and columns, each cycle alternating between rows and columns. Since the graph is 2-regular, each node (row or column) has exactly two edges. Wait, but in an 8x8 bipartite graph, arranging such a 2-regular graph is possible? For example, arranging two perfect matchings. If we can find two perfect matchings between rows and columns that don't overlap, then their union would be a 2-regular graph. However, in an 8x8 chessboard, arranging two non-overlapping perfect matchings is possible. For example, the first perfect matching could pair row 1 with column 1, row 2 with column 2, etc., and the second perfect matching could pair row 1 with column 2, row 2 with column 3, ..., row 8 with column 1. Then, their union would form a 2-regular graph where each row and column has exactly two cells selected. However, we need to check if these two perfect matchings intersect. If they do, then the union would have overlapping cells, which is not allowed since each cell can be selected only once.Wait, actually, perfect matchings in a bipartite graph are sets of edges without shared vertices. So, two perfect matchings in the same bipartite graph can share edges. So, if we take two different perfect matchings, their union would give a 2-regular graph only if they are edge-disjoint. So, to have two edge-disjoint perfect matchings, the bipartite graph must be such that it's possible. In the case of a complete bipartite graph K_{8,8}, it's certainly possible, as it's a well-known result that a regular bipartite graph has a perfect matching. So, in fact, by König's theorem or Hall's theorem, we can decompose the 2-regular bipartite graph into two perfect matchings. Therefore, such an arrangement is possible. Therefore, we can indeed arrange 16 cells on the chessboard with exactly 2 per row and 2 per column, resulting in 16 conflicting pairs. Therefore, is 16 the minimal number?Wait, but wait a second. Let me verify. Suppose that we have 2 cells in each row and each column. Then, for each row, the number of pairs is C(2, 2) = 1. There are 8 rows, so 8. Similarly, for each column, also 1, so 8 columns, 8. Total 16. So, that seems correct.But is this the minimal possible? Let's think. Suppose that instead of distributing 2 per row and column, we have some rows with 1 cell and others with 3. Then, the number of pairs in those rows would be C(1, 2) = 0 and C(3, 2) = 3. If we have two such rows, replacing two 2s with a 1 and a 3, the total pairs would go from 2*1 = 2 to 0 + 3 = 3, which is an increase. Similarly for columns. So, by the convexity of the function C(k, 2), any deviation from the uniform distribution will increase the total sum. Therefore, the minimal total number of conflicting pairs is indeed achieved when the cells are as evenly distributed as possible across rows and columns, which in this case is exactly 2 per row and 2 per column.Therefore, the minimal number is 16. But wait, hold on. Let me check an example. If we have 2 cells in each row and each column, that's a biadjacency matrix of a 2-regular bipartite graph, which is equivalent to a 2x2 block diagonal matrix with 4x4 permutation matrices? Wait, no, perhaps it's a union of two permutation matrices. For example, arranging the cells such that in each row and column, exactly two cells are selected, with no overlap. So, for instance, in the first row, select columns 1 and 2; in the second row, columns 3 and 4; and so on. But then, in the columns, column 1 would have only the first row selected, column 2 only the first row, column 3 only the second row, etc. So each column would only have 1 cell selected, which contradicts the requirement of having 2 per column. Therefore, that approach doesn't work.Wait, so perhaps arranging it like two non-overlapping permutation matrices. For example, one permutation matrix where each row and column has exactly one cell selected, and another permutation matrix that doesn't overlap with the first. Then, the union would have two cells per row and column. For example, place rooks in such a way that no two attack each other (first permutation), then place another set of rooks in a different permutation. Since permutations are bijections, if they are disjoint, then each row and column will have exactly two cells selected. For example, in an 8x8 chessboard, a permutation could be the identity matrix, placing a rook at (1,1), (2,2), ..., (8,8). Then another permutation could be shifting by one, placing rooks at (1,2), (2,3), ..., (8,1). These two permutations only overlap if any diagonal element is adjacent in the shift. In this case, the first permutation has (1,1), and the second has (1,2), so no overlap. Similarly, (2,2) vs (2,3), etc. So, no overlapping cells. Therefore, the union of these two permutations would give a configuration where each row and column has exactly two cells, and no two cells are in the same row or column within each permutation, so in total, each row and column has two cells. Therefore, such a configuration is possible, resulting in 16 conflicting pairs. Therefore, 16 is achievable.But the problem says "the minimal number of pairs". So, is there a way to get fewer than 16? Let's see. Suppose that we arrange the cells such that some rows have 1 cell and others have 3. For example, if we have four rows with 3 cells and four rows with 1 cell. Then, the number of row pairs would be 4*C(3,2) + 4*C(1,2) = 4*3 + 4*0 = 12. Similarly, for columns, if we can arrange the columns such that each column also has 2 cells, but given the row distribution, it's tricky. Wait, the total number of cells is 16. If we have four rows with 3 cells (total 12) and four rows with 1 cell (total 4), summing to 16. Then, distributing these 16 cells into columns. If we try to distribute the column counts as evenly as possible, perhaps some columns would have 2 cells, others more or less. But due to the uneven row distribution, the column distribution might be forced to be uneven as well, leading to a higher total number of column pairs.Alternatively, maybe arranging 3 cells in a row and 1 in another affects the column counts. Let me try to calculate.Suppose we have four rows with 3 cells and four rows with 1 cell. Let's assume that in the columns, the cells are as evenly distributed as possible. Since there are 16 cells, and 8 columns, the average per column is 2. If we can arrange that each column has exactly 2 cells, then the total column pairs would be 8*C(2,2)=8. Then total pairs would be row pairs (12) + column pairs (8) = 20, which is higher than 16. But is such a column distribution possible? If in the rows with 3 cells, the columns are spread out such that each column gets 2 cells. However, if four rows have 3 cells each, then each of these rows must place 3 cells in different columns. If the four rows with 3 cells are placing their cells in the columns, to have each column end up with exactly 2 cells, we need 8 columns * 2 cells = 16 cells. Since four rows with 3 cells contribute 12 cells, and four rows with 1 cell contribute 4 cells, totaling 16. Therefore, the 12 cells from the four rows with 3 cells must be spread over the columns such that each column has 2 - (number of cells from the single-cell rows). Since the single-cell rows contribute 1 cell each, spread over four rows. Wait, actually, the total cells from the single-cell rows is 4. If each single-cell row's cell is in a different column, then those 4 columns would have 1 cell each from the single-cell rows, and the remaining 4 columns have 0 from the single-cell rows. Then, the four rows with 3 cells must contribute 2 cells to the first 4 columns and 2 cells to the remaining 4 columns. Wait, this is getting complicated.Alternatively, maybe it's not possible to have the column counts all be 2 in this case. Let's see. Each of the four rows with 3 cells needs to place 3 cells in columns. Suppose we have four rows each placing 3 cells. If we try to distribute these 12 cells into 8 columns, ideally 12/8 = 1.5 per column. But since we can't have half cells, some columns will have 1 and some 2. Then, the four single-cell rows can place their cells into the columns that have 1 cell from the 3-cell rows, thus making those columns have 2, and the other columns would have 2 from the 3-cell rows. Wait, but 12 cells over 8 columns: let's say 4 columns receive 2 cells each from the 3-cell rows, and 4 columns receive 1 cell each. Then, the four single-cell rows can place their 1 cell each into the 4 columns that only have 1 cell from the 3-cell rows, resulting in those columns having 2, and the other 4 columns remaining at 2 from the 3-cell rows. So, all columns would have 2 cells. Therefore, in this case, the column counts would all be 2, leading to column pairs being 8. Then, total pairs would be 12 (row pairs) + 8 (column pairs) = 20. But in the uniform distribution, we have 16 total pairs. Therefore, this uneven distribution in rows leads to a higher total. Similarly, if we try to have more uneven distributions, the total pairs would be even higher. Therefore, deviating from uniform distribution in rows or columns increases the total number of pairs. Therefore, the minimal total pairs is achieved when both rows and columns are as evenly distributed as possible, which in this case is 2 per row and 2 per column.But let me test another configuration. Suppose we have one row with 4 cells and the rest with 2. Wait, 1*4 + 7*2 = 4 + 14 = 18, which is more than 16. Not useful. Alternatively, two rows with 3 cells and six rows with 2 cells: 2*3 + 6*2 = 6 + 12 = 18. Still too much. To get 16, it's necessary to have all rows with 2, or some combination of 1, 2, and 3. For example, 8 rows with 2 each gives 16. Any other combination would need to compensate with lower numbers. For example, 1 row with 0, but then we have to distribute 16 over 7 rows, which would require some rows with more than 2. But 16 divided by 7 is about 2.28, so likely some rows with 3 and others with 2. But this would lead to a higher number of row pairs. For instance, 5 rows with 3 and 2 rows with 0: 5*3 + 2*0 = 15, but we need 16. So maybe 6 rows with 3 and 2 rows with -1? That doesn't make sense. So perhaps 4 rows with 3 and 4 rows with 1: 4*3 + 4*1 = 12 + 4 = 16. Which we considered earlier. The row pairs here would be 4*3 + 4*0 = 12, and column pairs if columns are all 2 would be 8, total 20. Worse than 16.Alternatively, if we arrange the 16 cells in a way that some columns have 1 cell and others have 3. Let's see. For example, if we have four columns with 3 cells and four columns with 1 cell, then column pairs would be 4*3 + 4*0 = 12, and if the rows are all 2, then row pairs would be 8*1 = 8. Total pairs 20 again. So regardless, deviating from uniform distribution in rows or columns increases the total.Therefore, it seems that the minimal total is indeed 16, achievable by 2 per row and 2 per column. So, the answer should be 16. Wait, but wait a second. Let me check again.Suppose we have a different configuration where some rows have 2 cells, some have 1 or 3, but arrange the columns cleverly. For example, maybe overlapping pairs in rows and columns could somehow cancel out? But I don't think so. Because each pair is counted once if they are in the same row or column. Wait, but if two cells are in the same row and same column, which is impossible, but two cells can be in the same row or column, but each pair is counted in at most one category. So, the total number of conflicting pairs is equal to the sum over all rows of C(k_i, 2) plus the sum over all columns of C(l_j, 2). So, these are entirely separate counts. Therefore, there's no overlap in counts. Therefore, the total number is indeed the sum of row pairs and column pairs, with no subtraction. Therefore, the minimal total is achieved when both row and column distributions are as uniform as possible.Therefore, if we have 2 per row and 2 per column, the total is 8*1 + 8*1 = 16. If we deviate, both row and column counts would increase. For example, if we have one row with 3 and one with 1, then the row pairs increase by C(3,2) + C(1,2) - 2*C(2,2) = 3 + 0 - 2 = 1. So, an increase of 1. If the columns are kept at 2, then total would be 16 + 1 = 17. But if the columns also have to compensate, maybe they also have some deviations, leading to further increases.Therefore, the minimal is indeed 16. However, I need to confirm if such an arrangement is possible. Let's see. If we can arrange the 16 cells such that each row and each column has exactly two cells, that's equivalent to a 2-regular bipartite graph between rows and columns. As mentioned earlier, such a graph is a union of two perfect matchings. For an 8x8 chessboard, it's possible to have two disjoint perfect matchings. For example, first, place rooks on the main diagonal: (1,1), (2,2), ..., (8,8). Then, place another set on a shifted diagonal, say (1,2), (2,3), ..., (7,8), (8,1). These two sets don't overlap, and each row and column has exactly two cells selected. Therefore, this configuration is possible. Therefore, 16 conflicting pairs is achievable.Therefore, the minimal number is 16.Wait, but wait a moment. Let me check with a small example. Suppose a 2x2 chessboard, selecting 4 cells. Wait, but that's the entire board. Then, conflicting pairs would be rows: 2 rows with C(2,2)=1 each, total 2. Columns: similarly 2. Total 4. But if you select all cells, every pair is conflicting. There are C(4,2)=6 pairs. Wait, but according to the formula, rows: each row has 2 cells, so 2*1=2. Columns: same, total 4. But actually, each pair is in a row or column. So, total conflicting pairs should be 6. But according to the formula, it's 4. There's a discrepancy here. Wait, so maybe my reasoning is flawed.Wait, in the 2x2 chessboard, selecting all 4 cells. The number of pairs in rows: each row has C(2,2)=1 pair. Two rows, so 2. Similarly, columns: two columns, each with C(2,2)=1 pair, total 2. So total conflicting pairs 4. However, the actual number of pairs is C(4,2)=6. Each pair is either in the same row or same column. Let's count: in each row, there's 1 pair. Each column, 1 pair. But the pairs in the rows are the same as the pairs in the columns? Wait, no. For example, in the 2x2 grid, the cells are (1,1), (1,2), (2,1), (2,2). The pairs in rows: (1,1)-(1,2), (2,1)-(2,2). The pairs in columns: (1,1)-(2,1), (1,2)-(2,2). So, total 4 pairs. The other two pairs are (1,1)-(2,2) and (1,2)-(2,1), which are diagonal and don't share a row or column. Therefore, total conflicting pairs are 4, which matches the formula. So, the formula is correct. Therefore, the formula counts only the pairs that are in the same row or column, not all pairs. Therefore, in the 8x8 case, the total conflicting pairs are the sum over rows of C(k_i, 2) plus sum over columns of C(l_j, 2). So, the answer of 16 is correct for the problem as stated.But in the 2x2 case, the minimal number is 4 when all cells are selected, but if we select fewer cells, say 2 cells, then the minimal conflicting pairs would be 0 if they are placed on different rows and columns. So, the formula works there. So, returning to the original problem, with 16 cells on 8x8, distributing 2 per row and 2 per column gives 16 conflicting pairs, which seems minimal. Therefore, I think the answer is 16.However, let me check another aspect. Suppose that we arrange the 16 cells in such a way that some rows and columns have more than 2, but the overall sum is less than 16. Is that possible? Let's suppose that in some rows, we have 1 cell, and in others 3, but arrange the columns such that the column pairs are minimized. Wait, but as we saw earlier, the total tends to increase.Wait, here's another thought. The total number of pairs in rows is equal to sum_{i=1}^8 C(k_i, 2). Similarly for columns. We have sum_{i=1}^8 k_i = 16. The minimal sum of C(k_i, 2) is achieved when k_i are as equal as possible, which is 2 each. Then, the sum is 8*1 = 8. Similarly for columns. Therefore, total minimal pairs is 8 + 8 = 16.Alternatively, using the formula for the sum of combinations: sum_{i=1}^n C(k_i, 2) = C(total, 2) - sum_{i=1}^n C(total - k_i, 2). Wait, maybe not. Alternatively, we can use the formula that sum_{i=1}^n C(k_i, 2) = (sum_{i=1}^n k_i^2 - sum_{i=1}^n k_i)/2. Since sum_{i=1}^n k_i = 16. Therefore, sum_{i=1}^8 C(k_i, 2) = (sum k_i^2 - 16)/2. Similarly for columns. Therefore, the total conflicting pairs is (sum k_i^2 - 16)/2 + (sum l_j^2 - 16)/2 = (sum k_i^2 + sum l_j^2 - 32)/2. So, to minimize this, we need to minimize sum k_i^2 + sum l_j^2.But we know from the Cauchy-Schwarz inequality that sum k_i^2 >= (sum k_i)^2 / n = 16^2 /8 = 256 /8 = 32. Similarly, sum l_j^2 >= 32. Therefore, sum k_i^2 + sum l_j^2 >= 64. Equality holds when all k_i are equal and all l_j are equal, which is when each k_i = 2 and each l_j = 2. Therefore, the minimal total conflicting pairs is (64 - 32)/2 = 32/2 = 16. Therefore, this mathematical approach confirms that the minimal total is indeed 16.Therefore, regardless of how you arrange the 16 cells, the minimal number of conflicting pairs (same row or column) is 16, achieved when each row and each column contains exactly 2 cells.Therefore, the answer is boxed{16}.Final Answerboxed{16}</think>To determine the minimal number of pairs of selected cells in the same row or column when 16 cells are selected on an 8x8 chessboard, we need to minimize the sum of pairs in rows and columns. 1. Understanding the Problem: - Each pair of cells in the same row or column contributes to the count. - The number of pairs in a row with ( k ) cells is ( binom{k}{2} ), and similarly for columns. - The goal is to minimize the total number of such pairs.2. Key Insight: - The function ( binom{k}{2} ) is convex, so the minimal sum is achieved with the most uniform distribution of cells across rows and columns. - For 16 cells on an 8x8 board, the most uniform distribution is 2 cells per row and 2 cells per column.3. Calculation: - Each row with 2 cells contributes ( binom{2}{2} = 1 ) pair. - There are 8 rows, so row pairs sum to ( 8 times 1 = 8 ). - Similarly, each column with 2 cells contributes ( binom{2}{2} = 1 ) pair. - There are 8 columns, so column pairs sum to ( 8 times 1 = 8 ). - Total pairs = row pairs + column pairs = ( 8 + 8 = 16 ).4. Verification: - Using the Cauchy-Schwarz inequality, the sum of squares of the number of cells per row and column is minimized when each row and column has exactly 2 cells. - This confirms the minimal total number of pairs is indeed 16.Thus, the minimal number of pairs of selected cells in the same row or column is (boxed{16}).,<think>Alright, let's tackle this problem. So, the question is: If we select 16 cells on an 8x8 chessboard, what's the minimal number of pairs of selected cells that are in the same row or column? Hmm, okay. So, we need to arrange 16 cells such that as few as possible pairs share the same row or column. Let me think.First, let me make sure I understand the problem correctly. We have a standard chessboard, 8 rows and 8 columns. We need to choose 16 cells. Each pair of selected cells that are in the same row or column counts towards our total. We need to minimize this total. So, the minimal number of such pairs. Got it.So, this is similar to some kind of combinatorial optimization problem. Maybe related to distributing objects (in this case, the selected cells) in such a way that we minimize collisions in rows and columns. Hmm. Let me think of similar problems. It's a bit like the pigeonhole principle. If we distribute 16 cells across 8 rows, how do we minimize the number of pairs in each row? Similarly for columns.Wait, but the problem is about pairs in the same row OR column. So, if two cells are in the same row, that's a pair, same column, that's a pair. So, we need to count all such pairs. The total number of pairs in the same row plus the total number in the same column. So, the total would be the sum over each row of combinations of selected cells in that row taken two at a time, plus the same for columns. So, for each row, if there are k selected cells, that contributes C(k,2) pairs. Similarly for each column. Then, we sum all these up. Our goal is to arrange the 16 cells so that this sum is as small as possible.So, the question becomes: how to place 16 cells on an 8x8 grid so that the sum over all rows and columns of C(k,2) is minimized. That is, we need to distribute the 16 cells as evenly as possible across the rows and columns to minimize the number of collisions.Wait, but how exactly? Let's think. If we spread out the cells so that each row and each column has as close to the same number of cells as possible, then the number of pairs per row and per column will be minimized. Because the number of pairs C(k,2) is minimized when the distribution is as even as possible. For example, if you have 16 cells in 8 rows, the most even distribution is 2 cells per row (since 16 divided by 8 is 2). Similarly, if we can arrange 2 cells in each row and 2 cells in each column, then each row contributes C(2,2)=1 pair, so 8 rows contribute 8*1=8 pairs. Similarly, each column would contribute 1 pair, so 8 columns contribute another 8 pairs. Total would be 16 pairs. But is this possible? Wait, arranging 2 cells per row and 2 per column on an 8x8 board? That's essentially a Latin square or a block design. Wait, if we can arrange two cells in each row and column, like a 2-regular graph in both rows and columns. Hmm.But wait, actually, arranging 2 cells in each row and each column would require that the total number of cells is 2*8 = 16, which is exactly our case. So, this seems possible. So, if we can arrange 2 cells in each row and 2 in each column, then the total number of pairs would be 8 rows * C(2,2) + 8 columns * C(2,2) = 8*1 + 8*1 = 16. So, total 16 pairs.Is this the minimal possible? Let's check. Suppose that in some row, we have 3 cells. Then, the number of pairs in that row would be C(3,2)=3. So, if instead of having two rows with 2 cells each, we have one row with 3 and another with 1, the total pairs for those two rows would be 3 + 0 = 3, whereas if they both had 2, it would be 1 + 1 = 2. So, having an uneven distribution increases the total number of pairs. Similarly for columns. Therefore, to minimize the total number of pairs, we need to distribute the cells as evenly as possible across rows and columns.Therefore, the minimal total number of pairs would be when each row and each column has exactly 2 cells. If that's possible, then the total number of pairs is 16. So, is this possible?Yes, such a configuration is possible. For example, arrange the cells in a way that each row and column has exactly two cells. This is similar to placing rooks on a chessboard such that each rook has another in its row and column, but without attacking each other. Wait, but rooks attack in straight lines, so two rooks in the same row or column attack each other. But in our case, we just need to count the number of attacking pairs. Wait, but the problem is not about non-attacking positions, but rather about minimizing the number of attacking pairs. However, in our case, if we can arrange 2 non-attacking rooks in each row and column, but wait, no. If you have two rooks in the same row, they attack each other. So, to have two rooks in each row and column, but arranged such that they don't attack each other, that's impossible. Wait, no. Wait, actually, if you place two rooks in each row and column, they will necessarily attack each other. Wait, each rook in a row attacks the other in the same row. So, in order to have two rooks in each row and column, you would have to have them not attacking each other? That's not possible. Wait, no. If you have two rooks in the same row, they are attacking each other. So, if you place two rooks in each row, each pair in a row is an attacking pair, similarly for columns. So, in that case, the number of attacking pairs would be 8 rows * C(2,2) + 8 columns * C(2,2) = 16. But in reality, if you place two rooks in each row and two in each column, each rook is part of one attacking pair in its row and one in its column. But the total number of attacking pairs would actually be the same as the total number of pairs counted in rows and columns. Wait, but each attacking pair is counted once in the row and once in the column. Wait, no. Wait, if two rooks are in the same row, that's one attacking pair. Similarly, if they are in the same column, that's another attacking pair. So, two rooks that are in the same row and column (but that can't happen unless they are at the intersection, i.e., the same cell). So, two rooks in the same row contribute one attacking pair, and two rooks in the same column contribute another attacking pair, but if two rooks are in the same row and column, that's impossible unless they are the same rook.Wait, so in our problem, we are to count all pairs that are in the same row OR in the same column. So, if two rooks are in the same row, that's a pair. If two rooks are in the same column, that's another pair. So, two rooks that are in the same row and column would be the same cell, which is not allowed. So, in our problem, the total number of pairs is the number of pairs in the same rows plus the number in the same columns. So, for example, if two rooks are in the same row, that's one pair, and if they are in the same column, that's another pair. So, two distinct rooks can contribute at most two pairs (if they are in the same row and column, which is impossible). Wait, actually, two distinct rooks can be in the same row or same column but not both. So, each pair of rooks can be either in the same row, same column, or neither. So, the total number of pairs is the number of pairs in the same rows plus the number in the same columns. Since each pair is counted at most once in rows and once in columns, but actually, if two rooks are in the same row, they are not in the same column (unless they are the same cell, which they aren't). Therefore, the total number of pairs is the number of row pairs plus the number of column pairs. So, the problem is to minimize this total.Therefore, if we can arrange the 16 rooks such that each row and each column has exactly two rooks, then the number of row pairs is 8*C(2,2)=8, and the number of column pairs is also 8*C(2,2)=8. So, total 16 pairs. So, is this possible? Because arranging two rooks in each row and two in each column seems like a possible configuration, but we have to make sure that such a configuration exists.Wait, this is similar to a 2-regular bipartite graph between rows and columns. Each row is connected to two columns, and each column is connected to two rows. Such a graph would consist of a union of even-length cycles. For an 8x8 bipartite graph, a 2-regular graph would decompose into cycles of even length. For example, we can have two cycles of length 8, or four cycles of length 4, etc. So, for example, if we create a permutation where each row is connected to two columns, forming cycles. Let me try to construct such a configuration.Let's label the rows as R1 to R8 and columns as C1 to C8.One way to arrange this is to have a grid where each row has two cells selected in such a way that each column also has two cells selected. For example, we can divide the chessboard into 2x2 blocks. In each 2x2 block, place two cells. Wait, but 8x8 divided into 2x2 blocks would give 16 blocks, each with two cells. But that would result in 32 cells, which is more than 16. Hmm, maybe not.Alternatively, arrange in a pattern where each row has two cells, offset appropriately so that columns also have two. For example, a diagonal pattern with two cells per row and column. Wait, let's try constructing such a configuration.Suppose in the first row, we select columns 1 and 2. In the second row, columns 3 and 4. Third row, 5 and 6. Fourth row, 7 and 8. Fifth row, 1 and 2. Sixth row, 3 and 4. Seventh row, 5 and 6. Eighth row, 7 and 8. Then, each column would have two cells selected. For example, column 1 has rows 1 and 5, column 2 has rows 1 and 5, column 3 has rows 2 and 6, etc. So, in this case, each column has two cells. Each row also has two cells. So, this configuration would result in 8 rows with two cells each and 8 columns with two cells each, leading to 8*1 + 8*1 = 16 pairs. So, this works.But wait, in this configuration, some columns have two cells but in different rows. Wait, no, columns 1 and 2 have two cells each, but in rows 1 and 5. So, each column has two cells. So, in this case, yes, each column and each row has exactly two cells. Therefore, the total number of pairs would be 8 (from rows) + 8 (from columns) = 16. Therefore, this configuration gives 16 pairs. So, is this the minimal possible?Wait, but let me check. Suppose that instead of arranging two cells per row and column, we allow some rows or columns to have three cells, but compensate with others having one. Would that result in a lower total? For instance, if in one row we have three cells and in another row we have one, then the total number of pairs for those two rows would be C(3,2) + C(1,2) = 3 + 0 = 3, compared to 1 + 1 = 2 if they both have two. So, the total increases. Similarly, for columns. Therefore, any deviation from the uniform distribution would increase the total number of pairs. Therefore, the minimal total is achieved when the distribution is as uniform as possible. Since 16 divided by 8 is exactly 2, we can have exactly two cells per row and two per column. Therefore, the minimal total is 16.But wait, the problem is stated as "the minimal number of pairs of selected cells in the same row or column." So, the answer is 16? But let me think again. Wait, but maybe there is a different way to arrange the cells where some pairs are counted both in a row and a column? Wait, but each pair is either in the same row or the same column, not both. Because two distinct cells cannot be in the same row and same column unless they are the same cell. So, each pair is counted either in a row or in a column, but not both. Therefore, the total number of pairs is the sum of pairs in rows plus pairs in columns. So, if you have two cells in each row and two in each column, the total is 8 + 8 = 16. So, 16 pairs.Wait, but let me confirm this with a small example. Suppose we have a 2x2 chessboard, and we select 2 cells. What's the minimal number of pairs? If we select two cells in the same row, that's one pair. If we select two cells in different rows and different columns, then there are zero pairs. So, the minimal is zero. But according to the formula, if we have two cells, each in their own row and column, then the number of row pairs is 0 (since each row has one cell) and column pairs is 0 (each column has one cell). So, total 0. Which is correct. So, in that case, the formula works.Another example: 3x3 chessboard, selecting 3 cells. If we place one in each row and column (a permutation matrix), then the number of pairs in rows is 0 and columns is 0. So total 0. If we place two in one row and one in another, then row pairs would be C(2,2)=1 and column pairs would depend on the columns. If the two in the same row are in different columns, then each column has one cell except for the column with the single cell. So, column pairs would be 0. So total pairs is 1. But the minimal is 0. So, the formula holds.Therefore, in the original problem, if we can arrange 16 cells with two per row and two per column, then the total number of pairs is 16. So, is this possible? The configuration I thought of earlier, where each pair of rows selects the same pair of columns offset by four rows, seems to work. For example:Rows 1-4: select columns 1-2, 3-4, 5-6, 7-8 respectively.Rows 5-8: repeat the same pattern as rows 1-4.So, in this case, each column (1-8) will have exactly two cells selected: columns 1 and 2 are selected in rows 1 and 5, columns 3 and 4 in rows 2 and 6, etc. So, each column has two cells. Each row has two cells. Therefore, the number of row pairs is 8 rows * C(2,2) = 8*1=8. The number of column pairs is 8 columns * C(2,2)=8. So, total 16 pairs. So, this seems to check out.But let me verify with a smaller version. Let's take a 4x4 chessboard and try to place 8 cells (since 4x4 has 16 cells, but 8 is analogous to 16 in 8x8). Wait, 8 cells on a 4x4. If we place 2 cells per row and 2 per column, the number of pairs would be 4 rows *1 +4 columns *1=8 pairs. Alternatively, if arranged differently, say, 3 in one row and 1 in another, etc., the total pairs would be higher. For example, 3 in a row contributes 3 pairs, 1 in another row contributes 0, and the remaining two rows with 2 each contribute 1 each. So total row pairs: 3+0+1+1=5. Columns would be variable depending on placement. Suppose in columns, we have 3 in one column and spread out the rest. It might get complicated, but the total pairs would probably be higher than 8. So, the minimal is indeed achieved by the even distribution.Therefore, returning to the original problem, it's 16. But wait, the answer feels low. Wait, let's think again. The problem is asking for the minimal number of pairs. If we have 16 cells, the total number of possible pairs is C(16,2)=120. Out of these, how many are in the same row or column. We need to minimize this number. So, the minimal possible is 16. That seems quite small. Wait, but 120 total pairs, 16 of them are in same row or column. So, 104 pairs are neither in the same row nor column. Hmm.Alternatively, maybe my approach is wrong. Let me think differently. Let's model this as a graph. Each cell is a vertex, and edges connect cells that are in the same row or column. Then, the problem reduces to finding a set of 16 vertices with the minimal number of edges between them. So, we need an induced subgraph with 16 vertices and as few edges as possible.In an 8x8 grid, each cell is connected to 7 others in its row and 7 in its column, so 14 edges per cell. But in the induced subgraph of 16 cells, the number of edges would be the number of pairs in the same row or column. So, our goal is to minimize that.Alternatively, perhaps the problem is equivalent to minimizing conflicts in a scheduling problem, where you don't want two events in the same time slot (row) or same room (column). But I need to get back.Wait, another way to compute the total number of pairs in rows and columns: for each cell, count how many other cells are in its row plus how many in its column, then divide by 2 (since each pair is counted twice). So, total pairs = (sum over all cells (number of other cells in row + number in column)) / 2.Let me see. If each cell has r_i cells in its row and c_i cells in its column, then for each cell, the number of pairs it contributes to is (r_i - 1) + (c_i - 1). So, sum over all cells: sum_{i=1 to 16} (r_i + c_i - 2). Then, divide by 2 to get the total number of pairs.But wait, each pair is counted twice, once for each cell in the pair. Therefore, total pairs = [sum_{cells} (number in row + number in column - 2)] / 2.Alternatively, note that sum_{cells} (number in row) = sum_{rows} (number in row)^2, because for each row, the number in row is added as many times as the number of cells in that row. Similarly, sum_{cells} (number in column) = sum_{columns} (number in column)^2. Therefore, total pairs = [sum_{rows} (k_r)^2 + sum_{columns} (k_c)^2 - 2*16 ] / 2.Since each of the 16 cells contributes -2, so total -32. Then, divided by 2.Therefore, total pairs = [sum_{rows} (k_r)^2 + sum_{columns} (k_c)^2 - 32 ] / 2.So, our goal is to minimize this quantity. Which is equivalent to minimizing sum_{rows} (k_r)^2 + sum_{columns} (k_c)^2.Given that sum_{rows} k_r = 16, and sum_{columns} k_c =16.We need to distribute 16 indistinct items into 8 rows (k_r) and 8 columns (k_c) such that sum(k_r) =16 and sum(k_c)=16, and minimize sum(k_r^2) + sum(k_c^2).From the Cauchy-Schwarz inequality, the sum of squares is minimized when the distribution is as even as possible. Therefore, for the rows, since 16 divided by 8 is 2, the minimal sum of squares is 8*(2^2)=32. Similarly for columns. So, sum(k_r^2) + sum(k_c^2) =32 +32=64. Therefore, total pairs = (64 -32)/2= 32/2=16. Therefore, 16 pairs. Therefore, this confirms the previous result. Therefore, the minimal number of pairs is 16.Therefore, the answer is 16. Therefore, the minimal number is 16, achieved by arranging two cells in each row and two in each column.But wait, let me make sure. Let's plug the numbers into the formula.sum_{rows} (k_r)^2 = 8*(2)^2 =32sum_{columns} (k_c)^2=8*(2)^2=32Total sum =64Total pairs = (64 -32)/2=16. Yep, that's correct.Alternatively, if the distribution is uneven, say one row has 3 cells and another has 1, keeping the rest at 2. Then, sum_{rows} (k_r)^2 = (3^2 +1^2 +6*2^2)=9+1+24=34. Similarly, columns might still be 32 if we adjust accordingly. So, total sum becomes 34 +32=66. Total pairs=(66-32)/2=34/2=17. Which is higher than 16. So, indeed, uneven distributions lead to higher total pairs. Therefore, the minimal is 16.Therefore, the answer is 16. So, boxed{16}.Final Answerboxed{16}</think>To solve the problem of finding the minimal number of pairs of selected cells in the same row or column when selecting 16 cells on an 8x8 chessboard, we need to minimize the total number of such pairs. First, we note that the number of pairs of selected cells in the same row or column can be calculated by summing the combinations of selected cells in each row and each column. The formula for combinations ( binom{k}{2} ) (which is ( frac{k(k-1)}{2} )) is used to calculate the number of pairs in each row and column. To minimize the total number of pairs, we should distribute the 16 cells as evenly as possible across the rows and columns. The most even distribution is 2 cells per row and 2 cells per column, since 16 divided by 8 is 2. This distribution ensures that each row and each column contributes the minimum number of pairs.For each row with 2 cells, the number of pairs is ( binom{2}{2} = 1 ). There are 8 rows, so the total number of pairs in rows is ( 8 times 1 = 8 ). Similarly, for each column with 2 cells, the number of pairs is ( binom{2}{2} = 1 ). There are 8 columns, so the total number of pairs in columns is ( 8 times 1 = 8 ).Adding these together, the total number of pairs is ( 8 + 8 = 16 ).We confirmed this result using the formula for the total number of pairs, which involves summing the squares of the number of cells in each row and column, and then applying the Cauchy-Schwarz inequality to ensure the distribution is as even as possible. This method also yielded a total of 16 pairs.Thus, the minimal number of pairs of selected cells in the same row or column is (boxed{16}).]

🔑:1. Understanding the Problem: We need to find the minimal number of pairs of selected cells in the same row or column when 16 cells are selected on an (8 times 8) chessboard. This is equivalent to finding the minimum number of pairs of rooks attacking each other when 16 rooks are placed on the chessboard.2. Mathematical Representation: Consider an (8 times 8) matrix where each cell can either be empty (0) or contain a rook (1). The total number of rooks is 16. We need to count the number of pairs of rooks that are in the same row or column.3. Counting Pairs in Rows and Columns: Let (r_i) be the number of rooks in the (i)-th row and (c_j) be the number of rooks in the (j)-th column. The number of pairs of rooks in the same row (i) is given by (binom{r_i}{2} = frac{r_i(r_i - 1)}{2}). Similarly, the number of pairs of rooks in the same column (j) is (binom{c_j}{2} = frac{c_j(c_j - 1)}{2}).4. Summing Over All Rows and Columns: The total number of pairs of rooks attacking each other is: [ sum_{i=1}^{8} binom{r_i}{2} + sum_{j=1}^{8} binom{c_j}{2} ]5. Using Jensen's Inequality: By the convexity of the function (binom{x}{2}), we can apply Jensen's inequality. For a convex function (f), Jensen's inequality states that: [ fleft(frac{1}{n} sum_{i=1}^{n} x_iright) leq frac{1}{n} sum_{i=1}^{n} f(x_i) ] Here, (f(x) = binom{x}{2}). Since the sum of the rooks in all rows and columns is 16, we have: [ sum_{i=1}^{8} r_i = 16 quad text{and} quad sum_{j=1}^{8} c_j = 16 ]6. Applying Jensen's Inequality: Applying Jensen's inequality to the sums: [ sum_{i=1}^{8} binom{r_i}{2} geq 8 binom{frac{16}{8}}{2} = 8 binom{2}{2} = 8 cdot 1 = 8 ] [ sum_{j=1}^{8} binom{c_j}{2} geq 8 binom{frac{16}{8}}{2} = 8 binom{2}{2} = 8 cdot 1 = 8 ]7. Combining the Results: Therefore, the total number of pairs of rooks attacking each other is at least: [ sum_{i=1}^{8} binom{r_i}{2} + sum_{j=1}^{8} binom{c_j}{2} geq 8 + 8 = 16 ]8. Constructing an Example: To achieve exactly 16 pairs, we can place the rooks such that each row and each column contains exactly 2 rooks. One possible configuration is placing the rooks on two diagonals: - Place rooks at ((i, i)) and ((i, 8-i+1)) for (i = 1, 2, ldots, 8). This configuration ensures that each row and each column has exactly 2 rooks, resulting in exactly 16 pairs of rooks attacking each other.(blacksquare)The final answer is ( boxed{ 16 } ).

🧠:[<think>Okay, so I need to find the function f: R → R that satisfies two conditions: first, f(0) = 1, and second, for any real numbers x and y, the equation f(xy + 1) = f(x)f(y) - f(y) - x + 2 holds. Hmm, functional equations can be tricky, but maybe I can figure this out by plugging in specific values for x and y to find a pattern or figure out the form of f.Let me start by plugging in some simple values. Let's see, if I set x = 0, what happens? Then the left side becomes f(0*y + 1) = f(1). The right side becomes f(0)f(y) - f(y) - 0 + 2. Since f(0) is given as 1, this simplifies to 1*f(y) - f(y) - 0 + 2 = f(y) - f(y) + 2 = 2. So, f(1) = 2. That's a good start. Now I know f(0) = 1 and f(1) = 2.Next, maybe try setting y = 0. Then the left side becomes f(x*0 + 1) = f(1) = 2. The right side is f(x)f(0) - f(0) - x + 2. Plugging in f(0) = 1, this becomes f(x)*1 - 1 - x + 2 = f(x) -1 -x + 2 = f(x) - x + 1. So we have 2 = f(x) - x + 1, which simplifies to f(x) = x + 1. Wait, does that work? If f(x) = x + 1, let's check if this satisfies the original equation.Let me verify. Let's substitute f(x) = x + 1 into the given functional equation. Left side: f(xy + 1) = (xy + 1) + 1 = xy + 2. Right side: f(x)f(y) - f(y) - x + 2. Let's compute that. f(x)f(y) = (x + 1)(y + 1) = xy + x + y + 1. Then subtract f(y): (xy + x + y + 1) - (y + 1) = xy + x. Then subtract x: xy + x - x = xy. Then add 2: xy + 2. So the right side is xy + 2, which matches the left side. And f(0) = 0 + 1 = 1, which is also satisfied. Wait, so f(x) = x + 1 works? But the problem seems to present this as a question, so maybe that's the answer? Let me check again, maybe I made a mistake.Wait, when I set y = 0, I derived that f(x) = x + 1. Then when substituting back into the original equation, it seems to hold. So is that the solution? That seems straightforward, but maybe I should check another value to be sure. Let's pick another pair for x and y. For example, let x = 1 and y = 1. Then left side: f(1*1 + 1) = f(2) = 2 + 1 = 3. Right side: f(1)f(1) - f(1) -1 + 2 = (2)(2) - 2 -1 + 2 = 4 - 2 -1 +2 = 3. That matches. How about x = 2, y = 3. Left side: f(2*3 +1) = f(7) = 7 +1 = 8. Right side: f(2)f(3) - f(3) -2 +2. f(2) = 3, f(3)=4, so 3*4 -4 -2 +2 =12 -4 -2 +2 =8. That works too. Hmm, seems consistent.Wait, but functional equations can sometimes have non-linear solutions, so maybe I should check if there's a quadratic or exponential function that might also work. For instance, maybe f(x) is quadratic. Let's suppose f(x) = ax² + bx + c. Then we can plug into the equation and solve for a, b, c. But since when we set y=0, we already derived that f(x) must be x + 1, which is linear. So unless there's a contradiction, that seems unique. Let me see.Alternatively, suppose that f is linear, so f(x) = mx + b. Then we can use the given conditions to solve for m and b. Let's try this approach as well. If f(x) = mx + b, then f(0) = b =1, so b=1. So f(x) = mx +1. Then let's plug into the functional equation. Left side: f(xy +1) = m(xy +1) +1 = m xy + m +1. Right side: f(x)f(y) - f(y) -x +2. Let's compute f(x)f(y): (mx +1)(my +1) = m²xy + mx + my +1. Then subtract f(y): m²xy + mx + my +1 - (my +1) = m²xy + mx. Then subtract x: m²xy + mx - x. Then add 2: m²xy + mx -x +2. So the right side is m²xy + (m -1)x +2. Comparing to left side: m xy + m +1. So equate coefficients:For the xy term: m² = m => m² - m =0 => m(m -1)=0 => m=0 or m=1.For the x term: On left side, there's no x term. On right side, coefficient is (m -1). But left side has no x term, so (m -1) must be 0. Therefore, m -1=0 => m=1.For the constants: On left side, m +1. On right side, 2. Since m=1, left side is 1 +1=2, which matches the right side. So yes, m=1 and b=1. Therefore, f(x)=x+1. So the linear solution works, and we already saw that plugging in specific values works. Therefore, f(x)=x+1 is the solution.But just to be thorough, let's suppose that there exists another function that's not linear. For example, maybe f(x) has a quadratic term. Suppose f(x)=ax² +bx +c. We know f(0)=1, so c=1. Then f(x)=ax² +bx +1. Let's try plugging into the functional equation. Left side: f(xy +1)=a(xy +1)² +b(xy +1) +1 = a(x²y² + 2xy +1) +bxy +b +1 = a x² y² + 2a xy +a + bxy +b +1. Right side: f(x)f(y) -f(y) -x +2. Compute f(x)f(y): (ax² +bx +1)(ay² +by +1). Let's expand that:= a²x²y² + a b x² y + a x² + a b x y² + b² x y + b x + a y² + b y +1.Then subtract f(y): - (a y² + b y +1).So subtract that: a²x²y² + a b x² y + a x² + a b x y² + b² x y + b x + a y² + b y +1 -a y² -b y -1.Simplify: a²x²y² + a b x² y + a x² + a b x y² + b² x y + b x.Then subtract x: -x.Then add 2: +2.So altogether, right side becomes:a²x²y² + a b x² y + a x² + a b x y² + b² x y + b x -x +2.Compare to left side: a x² y² + 2a xy +a + bxy +b +1.So equate coefficients for corresponding terms:1. x² y² term: Left side has a, right side has a². Therefore, a = a² ⇒ a² -a =0 ⇒ a(a -1)=0 ⇒ a=0 or a=1.2. x² y term: Left side has 0, right side has a b. So 0 = a b. Similarly, x y² term: Left side has 0, right side has a b. So same equation 0 = a b.3. x² term: Left side has 0, right side has a. So 0 = a.But from the first equation, a=0 or 1. However, third equation says a=0. Therefore, a=0. Then from second equation, 0 = 0*b, which is always true. So a=0, then f(x)=0 x² +bx +1 = bx +1. So back to linear function. Then let's proceed with a=0. Then left side becomes f(xy +1)=0 +0 +0 +b*xy +b +1. Wait, wait. Wait, if a=0, then left side was computed as a x² y² + 2a xy +a + bxy +b +1. Since a=0, that becomes 0 +0 +0 +bxy +b +1 = bxy +b +1. The right side, with a=0, becomes 0 +0 +0 +0 + b² x y +b x -x +2. Wait, let's recompute the right side with a=0.Original right side after expanding and simplifying: a²x²y² + a b x² y + a x² + a b x y² + b² x y + b x -x +2. If a=0, this becomes 0 +0 +0 +0 +0 +b x -x +2 = (b -1)x +2.But left side is bxy +b +1. So equate left and right:bxy +b +1 = (b -1)x +2.But this must hold for all x and y. However, the left side has a term with xy, while the right side has only x terms and constants. The only way this can be true for all x and y is if the coefficients of the xy term on the left is zero. Therefore, coefficient of xy: b =0. So b=0.Then, left side becomes 0*xy +0 +1 =1. Right side becomes (0 -1)x +2 = -x +2. So equation becomes 1 = -x +2 for all x. But that's impossible unless x is fixed, which it isn't. Therefore, contradiction. Therefore, a=0 leads to no solution.Therefore, the only possible solution is a=1. Wait, but earlier with a=1, from the third equation (x² term coefficient), we had 0 = a. But if a=1, then 0=1, which is a contradiction. Wait, that seems conflicting. Wait, no. Wait, when we considered the quadratic case, the x² term on the left side is 0 (since left side is f(xy +1) which when expanded as quadratic gives terms up to x² y², but when looking at x² term, in the left side, there is no x² term. Wait, maybe I made a mistake here. Wait, let's clarify.Wait, in the left side, after expanding f(xy +1) as a quadratic, we have:a(xy +1)^2 + b(xy +1) +1 = a(x² y² + 2xy +1) +bxy +b +1 = a x² y² + 2a xy +a +bxy +b +1. So the x² y² term is a x² y², the xy term is (2a +b)xy, and the constants are a +b +1.On the right side, after expansion with a quadratic f(x), we had:a² x² y² + a b x² y +a x² + a b x y² + b² x y +b x -x +2. So the x² y² term is a² x² y², the x² y term is a b x² y, the x y² term is a b x y², the x² term is a x², the x y term is b² x y, the x term is (b -1)x, and the constant term is 2.Therefore, equating coefficients:1. x² y²: a = a² ⇒ a=0 or a=1.2. x² y and x y²: coefficients are a b on the right, 0 on the left. So a b=0.3. x² term: coefficient is a on the right, 0 on the left ⇒ a=0.4. x y term: (2a +b) on the left, b² on the right ⇒ 2a +b = b².5. x term: coefficient is (b -1) on the right, 0 on the left ⇒ b -1 =0 ⇒ b=1.6. Constants: a +b +1 = 2 ⇒ a +b =1.So from equation 3: a=0. Then from equation 5: b=1. Then from equation 6: 0 +1=1, which is okay. Then equation 2: a b =0*1=0, which is okay. Equation 4: 2*0 +1 =1 = (1)^2=1, which holds. So even though when a=0 and b=1, we get f(x)=0 x² +1 x +1= x +1. So even in the quadratic assumption, we end up with the linear solution. So even if we assume quadratic, the only solution is linear. Therefore, the only solution is f(x)=x+1. Therefore, that must be the answer.Wait, but in the previous check when a=0, we had a conflict. Wait, that was when we considered the equation after setting a=0 and led to a contradiction, but when we considered the equations step by step, we found that a=0, b=1 satisfies all equations. So maybe my previous check was flawed. Let's check again with a=0 and b=1. Then f(x)=x +1.Left side: f(xy +1)=xy +1 +1=xy +2. Right side: f(x)f(y) -f(y) -x +2. Which is (x +1)(y +1) - (y +1) -x +2. Let's compute: (xy +x +y +1) - y -1 -x +2= xy +x +y +1 -y -1 -x +2= xy +0 +0 +0 +2=xy +2. Which matches left side. So even in quadratic case, the only solution is linear. So no higher degree polynomials work. Therefore, f(x)=x +1 is the unique solution.Therefore, after thorough checking, the answer is f(x)=x +1.Final Answerboxed{x + 1}</think>Given the function ( f: mathbf{R} rightarrow mathbf{R} ) that satisfies ( f(0) = 1 ) and the functional equation ( f(xy + 1) = f(x)f(y) - f(y) - x + 2 ) for any ( x, y in mathbf{R} ), we need to determine ( f(x) ).First, we set ( x = 0 ). This gives:[f(1) = f(0)f(y) - f(y) - 0 + 2]Since ( f(0) = 1 ), this simplifies to:[f(1) = 1 cdot f(y) - f(y) + 2 implies f(1) = 2]Next, we set ( y = 0 ). This gives:[f(1) = f(x)f(0) - f(0) - x + 2]Substituting ( f(0) = 1 ) and ( f(1) = 2 ), we get:[2 = f(x) cdot 1 - 1 - x + 2 implies 2 = f(x) - x + 1 implies f(x) = x + 1]To verify, we substitute ( f(x) = x + 1 ) back into the original functional equation. The left-hand side becomes:[f(xy + 1) = (xy + 1) + 1 = xy + 2]The right-hand side becomes:[f(x)f(y) - f(y) - x + 2 = (x + 1)(y + 1) - (y + 1) - x + 2]Expanding and simplifying:[(xy + x + y + 1) - (y + 1) - x + 2 = xy + x + y + 1 - y - 1 - x + 2 = xy + 2]Both sides match, confirming that ( f(x) = x + 1 ) satisfies the equation.Further checks for quadratic or other non-linear forms lead to contradictions or reduce to the linear solution ( f(x) = x + 1 ). Thus, the unique solution is:[boxed{x + 1}],<think>Okay, let's see. I have this functional equation problem here. The function f maps real numbers to real numbers, starts with f(0) = 1. And for any real numbers x and y, the equation f(xy + 1) = f(x)f(y) - f(y) - x + 2 holds. I need to find f(x). Hmm, functional equations can be tricky, but maybe I can find some specific values or guess the form of the function.First, since f(0) is given as 1, that's a starting point. Let me try plugging in some specific values for x and y to see if I can get more information.Let's try x = 0. If I set x = 0, then the left side becomes f(0 * y + 1) = f(1). The right side becomes f(0)f(y) - f(y) - 0 + 2. Simplify that: [1 * f(y) - f(y)] - 0 + 2 = 0 + 2 = 2. So, f(1) = 2. Okay, that's another value.Now, maybe try y = 0. Then the left side is f(x * 0 + 1) = f(1) which we already know is 2. The right side is f(x)f(0) - f(0) - x + 2. Plugging in f(0) = 1, this becomes f(x)*1 - 1 - x + 2 = f(x) -1 -x +2 = f(x) -x +1. So, setting left side equal to right side: 2 = f(x) - x +1. Therefore, rearranged: f(x) = 2 + x -1 = x +1. Wait, does this hold?Wait a minute, if y = 0 gives us f(1) = 2, which we already knew, and then the equation becomes 2 = f(x) - x +1, so f(x) = x +1. So, does this mean that f(x) is linear, just x +1? Let me check if that works in the original equation.Let's test f(x) = x +1. Then the left side of the equation is f(xy +1) = (xy +1) +1 = xy + 2.The right side is f(x)f(y) - f(y) -x +2. Let's compute each part:f(x) = x +1, f(y) = y +1. So f(x)f(y) = (x +1)(y +1) = xy +x + y +1.Then subtract f(y): xy +x + y +1 - (y +1) = xy +x.Then subtract x: xy +x -x = xy.Then add 2: xy + 2.So the right side is xy + 2, which matches the left side. So f(x) = x +1 seems to satisfy the equation. And f(0) = 0 +1 =1, which is correct. Also, f(1) =1 +1=2, which we found earlier. So maybe that's the solution?But wait, did we jump to the conclusion too quickly? Because when we set y =0, we derived f(x) = x +1, but functional equations sometimes require checking if that's the only solution or if there are other possibilities.Alternatively, maybe we can suppose that f is linear. Let's assume f(x) = ax + b. Then try to find a and b.Given f(0) =1, so b =1. So f(x) = ax +1.Now plug into the equation: f(xy +1) = a(xy +1) +1 = axy + a +1.On the other side, f(x)f(y) -f(y) -x +2 = (ax +1)(ay +1) - (ay +1) -x +2.Let's expand that:(ax * ay) + ax + ay +1 - ay -1 -x +2= a²xy + ax + ay +1 - ay -1 -x +2Simplify:a²xy + ax -x + (ay - ay) + (1 -1) + 2= a²xy + (a -1)x + 2So set this equal to the left side:axy + a +1 = a²xy + (a -1)x + 2Now, equate coefficients of like terms.First, the coefficient of xy: a must equal a². So a² = a, which implies a(a -1)=0. Therefore, a=0 or a=1.Then, the coefficient of x: On the left side, there is no x term, so coefficient is 0. On the right side, coefficient is (a -1). Therefore, (a -1) must equal 0. So a -1 =0 => a=1.Constant terms: On left side, a +1. On right side, 2. So a +1 =2 => a=1.Therefore, a=1 is the solution. Therefore, f(x)=1x +1=x +1. Which matches our previous result. So yes, that works.Therefore, the solution is f(x)=x +1. Let me just verify once again with another value. Let's pick x=1, y=1. Then left side: f(1*1 +1)=f(2)=2 +1=3.Right side: f(1)f(1) -f(1) -1 +2= (2)(2) -2 -1 +2=4 -2 -1 +2=3. So it works.Another test: x=2, y=3. Left side: f(2*3 +1)=f(7)=7 +1=8.Right side: f(2)f(3)-f(3)-2 +2. f(2)=3, f(3)=4. So 3*4 -4 -2 +2=12 -4 -2 +2=8. Correct.Seems consistent. Therefore, the answer should be f(x)=x +1.Final AnswerThe function is boxed{x + 1}.</think>Given the function ( f: mathbf{R} rightarrow mathbf{R} ) with ( f(0) = 1 ), and the functional equation ( f(xy + 1) = f(x)f(y) - f(y) - x + 2 ) for any ( x, y in mathbf{R} ), we need to find ( f(x) ).1. Substitute ( x = 0 ): [ f(0 cdot y + 1) = f(1) = f(0)f(y) - f(y) - 0 + 2 ] Simplifying the right side: [ f(1) = 1 cdot f(y) - f(y) + 2 = 2 ] Thus, ( f(1) = 2 ).2. Substitute ( y = 0 ): [ f(x cdot 0 + 1) = f(1) = f(x)f(0) - f(0) - x + 2 ] Simplifying with ( f(0) = 1 ) and ( f(1) = 2 ): [ 2 = f(x) cdot 1 - 1 - x + 2 ] Solving for ( f(x) ): [ 2 = f(x) - x + 1 implies f(x) = x + 1 ]3. Verification: - Check if ( f(x) = x + 1 ) satisfies the original equation. - Left side: ( f(xy + 1) = (xy + 1) + 1 = xy + 2 ). - Right side: [ f(x)f(y) - f(y) - x + 2 = (x + 1)(y + 1) - (y + 1) - x + 2 ] Simplifying: [ (xy + x + y + 1) - (y + 1) - x + 2 = xy + 2 ] - Both sides match, confirming ( f(x) = x + 1 ).4. Linear Function Check: - Assume ( f(x) = ax + b ). Given ( f(0) = 1 ), ( b = 1 ). - Substitute ( f(x) = ax + 1 ) into the equation and equate coefficients, leading to ( a = 1 ).Thus, the function is (boxed{x + 1}).]

🔑:For any x, y in mathbf{R}, we havef(x y+1)=f(x) f(y)-f(y)-x+2,so we also have quad f(y x+1)=f(y) f(x)-f(x)-y+2.Therefore, quad f(x) f(y)-f(y)-x+2=f(y) f(x)-f(x)-y+2.This simplifies to f(x)+y=f(y)+x.Let y=0, we get f(x)=x+1.

🧠:[<think>Okay, so I need to find the value of tanφ given that cos(π/2 - φ) equals √3/2 and that the absolute value of φ is less than π/2. Let me start by recalling some trigonometric identities. Hmm, I remember that cosine of (π/2 - φ) is equal to sine φ. Is that right? Let me verify. Yes, because cos(π/2 - x) = sin x. So that means cos(π/2 - φ) = sin φ = √3/2. So, sin φ is √3/2. Now, I need to find tan φ. To find tan φ, I know that tan φ is sin φ over cos φ. So, if I can find cos φ, then I can compute tan φ. Since sin φ is given as √3/2, I can use the Pythagorean identity to find cos φ. The Pythagorean identity states that sin²φ + cos²φ = 1. So, substituting sin φ into the equation:(√3/2)² + cos²φ = 1 (3/4) + cos²φ = 1 cos²φ = 1 - 3/4 cos²φ = 1/4 Therefore, cos φ = ±√(1/4) = ±1/2.But here, we have the condition that |φ| < π/2. That means φ is in the range (-π/2, π/2), which is the principal value range for the arctangent function. In this interval, cosine is positive because in the first and fourth quadrants, cosine is positive. So, cos φ = 1/2.Therefore, tan φ = sin φ / cos φ = (√3/2) / (1/2) = √3. Wait, but let me check the answer choices again. The options are A. -√3/3, B. √3/3, C. -√3, D. √3. So, according to my calculation, the answer is D. √3.But hold on, let me double-check. Maybe I made a mistake in interpreting the angle. Let's see. The original equation is cos(π/2 - φ) = √3/2. So, π/2 - φ is an angle whose cosine is √3/2. The cosine of π/6 is √3/2, right? So, π/2 - φ = π/6 + 2πk or π/2 - φ = -π/6 + 2πk, where k is any integer. Solving for φ:First solution:π/2 - φ = π/6 So, φ = π/2 - π/6 = π/3 ≈ 60 degrees.Second solution:π/2 - φ = -π/6 So, φ = π/2 + π/6 = 2π/3 ≈ 120 degrees.But wait, the condition is |φ| < π/2. π/2 is 90 degrees. So, 2π/3 is 120 degrees, which is outside the given range. Therefore, only the first solution φ = π/3 is valid. So φ is π/3, which is 60 degrees. Then tan φ = tan(π/3) = √3. That matches my previous result, so the answer should be D. √3.But let me check again if there's another angle where cosine is √3/2. Cosine is positive in the first and fourth quadrants. So, angles are π/6 and -π/6. Wait, but in the equation cos(π/2 - φ) = √3/2, the solutions for π/2 - φ would be π/6 and -π/6, plus multiples of 2π. But since φ is restricted between -π/2 and π/2, let's see:If π/2 - φ = π/6, then φ = π/2 - π/6 = π/3 ≈ 60 degrees.If π/2 - φ = -π/6, then φ = π/2 + π/6 = 2π/3 ≈ 120 degrees. But 2π/3 is more than π/2 (which is 90 degrees), so it's out of range. So the only valid solution is φ = π/3.Therefore, tan φ = tan(π/3) = √3. So answer D.Wait a second, but why is the answer not B? Because sometimes when solving trigonometric equations, there might be multiple angles, but here due to the restriction |φ| < π/2, only π/3 is valid, which is in the first quadrant, so tan is positive. Therefore, answer D is correct. The options given include D as √3, which is correct. So I think that's it.But wait, let me check if there's another approach. Let's use co-function identities. Since cos(π/2 - φ) = sin φ, so sin φ = √3/2. Then, φ = π/3 or φ = 2π/3. But 2π/3 is 120 degrees, which is greater than π/2 (90 degrees), so excluded. So only φ = π/3, which gives tan φ = √3. So yes, D is the answer. Alternatively, maybe the problem is designed to test knowledge of co-functions and the range restriction. So, yes, seems like D is the correct answer. Therefore, the answer is D.Final Answerboxed{D}</think>Given (cosleft(frac{pi}{2} - varphiright) = frac{sqrt{3}}{2}) and (|varphi| < frac{pi}{2}), we need to find the value of (tanvarphi).First, we use the co-function identity (cosleft(frac{pi}{2} - varphiright) = sinvarphi). This gives us:[sinvarphi = frac{sqrt{3}}{2}]Next, we determine the possible values of (varphi). The solutions for (sinvarphi = frac{sqrt{3}}{2}) are (varphi = frac{pi}{3} + 2kpi) or (varphi = frac{2pi}{3} + 2kpi) for integer (k). However, given the restriction (|varphi| < frac{pi}{2}), we check these solutions:- (varphi = frac{pi}{3}) is within the range (-frac{pi}{2} < varphi < frac{pi}{2}).- (varphi = frac{2pi}{3}) is outside this range.Thus, the valid solution is (varphi = frac{pi}{3}).To find (tanvarphi), we use (tanvarphi = tanleft(frac{pi}{3}right)). Since (tanleft(frac{pi}{3}right) = sqrt{3}), we have:[tanvarphi = sqrt{3}]Thus, the final answer is (boxed{D}).,<think>Okay, let's see. The problem is about finding the value of tanφ given that cos(π/2 - φ) = √3/2 and that |φ| < π/2. The answer choices are A to D, which are all different tan values. Alright, let me think step by step.First, I remember that cosine of an angle equals adjacent over hypotenuse in a right triangle. But here, the angle is π/2 - φ. Also, there's a trigonometric identity that relates cos(π/2 - φ) to sine. Let me recall... Oh, right! cos(π/2 - x) is equal to sin(x). So, cos(π/2 - φ) = sinφ. That simplifies things because then the equation becomes sinφ = √3/2.So, sinφ = √3/2. Now, I need to find tanφ. To find tanφ, I can use the identity tanφ = sinφ / cosφ. But first, I need to find cosφ. Since we know sinφ, we can use the Pythagorean identity sin²φ + cos²φ = 1.Let's compute cosφ. If sinφ = √3/2, then sin²φ = (√3/2)^2 = 3/4. Plugging that into the identity: 3/4 + cos²φ = 1. Therefore, cos²φ = 1 - 3/4 = 1/4. Taking the square root of both sides, cosφ = ±1/2. But here, we need to consider the quadrant where φ is located. The problem states that |φ| < π/2, which means φ is in the first or fourth quadrant. However, since sinφ = √3/2 is positive, and sine is positive in the first and second quadrants. But φ is restricted to |φ| < π/2, so φ must be in the first quadrant because the second quadrant would require φ to be between π/2 and π, but |φ| < π/2 means φ is between -π/2 and π/2. So, φ is in the first quadrant (if φ is positive) or fourth quadrant (if φ is negative). But sinφ is positive, so φ must be in the first quadrant. Therefore, cosφ is positive. So cosφ = 1/2.Therefore, tanφ = sinφ / cosφ = (√3/2) / (1/2) = √3. Hmm, but the answer choices are A. -√3/3, B. √3/3, C. -√3, D. √3. So that would be D. √3. But wait, let me double-check.Wait, hold on. Wait, φ is in the first quadrant because |φ| < π/2 and sinφ is positive. So φ is between 0 and π/2. Therefore, tanφ is positive. So tanφ = √3. So answer D. But let me check again.But let me check the original equation again. The equation is cos(π/2 - φ) = √3/2. If we use the identity cos(π/2 - φ) = sinφ, so sinφ = √3/2. Then φ = π/3 or 2π/3. But since |φ| < π/2, which is approximately 1.57, and 2π/3 is about 2.094, which is more than π/2, so only φ = π/3. Therefore, tanφ = tan(π/3) = √3. So answer D.Wait, but the answer choices are A: -√3/3, B: √3/3, C: -√3, D: √3. So D is correct. But let me check again because maybe I made a mistake.Alternatively, maybe I did something wrong. Let's see. Let's suppose that φ is negative. If φ is negative, but |φ| < π/2, so φ is between -π/2 and 0. Then sinφ would be negative. But in the problem, sinφ = √3/2 which is positive. So φ must be in the first quadrant. So φ is π/3. So tanφ is √3. Therefore, answer D. So D is the correct answer. Hmm.Wait, but maybe I made a mistake in the identity. Let me check again. The original equation is cos(π/2 - φ) = √3/2. The identity is cos(π/2 - x) = sinx. So yes, that is correct. So sinφ = √3/2. So φ is π/3 plus period. But restricted to |φ| < π/2, so only π/3 is in that interval? Wait, π/3 is approximately 1.047, which is less than π/2 (≈1.5708). So yes, φ = π/3. So tan(π/3) = √3. So D is correct.Alternatively, maybe the question is in degrees? Wait, no, π/2 is in radians, so the problem is in radians. So φ is π/3. Therefore, tanφ is √3. So answer D. Therefore, the correct answer is D.But wait, the answer choices include both positive and negative options. So maybe there's a negative solution? Let's see. If sinφ = √3/2, then φ could be π/3 or 2π/3, but 2π/3 is greater than π/2, which is not allowed. If φ is negative, then sinφ would be negative. So φ must be π/3. So tanφ is √3. Therefore, D. So I think that is correct.But let me check again using another method. Let's solve the equation cos(π/2 - φ) = √3/2. Let's find all angles θ where cosθ = √3/2. The solutions for θ are π/6 and 11π/6 in the interval [0, 2π). But since cosine is even, θ can also be -π/6 + 2π k. But in our case, θ is π/2 - φ. So:π/2 - φ = π/6 + 2π k, or π/2 - φ = 11π/6 + 2π k, where k is any integer.Solving for φ:Case 1: π/2 - φ = π/6 + 2π kThen φ = π/2 - π/6 - 2π k = π/3 - 2π kCase 2: π/2 - φ = 11π/6 + 2π kThen φ = π/2 - 11π/6 - 2π k = (3π/6 - 11π/6) - 2π k = (-8π/6) - 2π k = -4π/3 - 2π kNow, we need φ such that |φ| < π/2. Let's check the possible values.For case 1: φ = π/3 - 2π k. Let's take k=0: φ=π/3≈1.047, which is less than π/2≈1.570. So that's valid. k=1: φ=π/3 - 2π≈-5.759, which has absolute value more than π/2. Similarly, other k's will take φ out of the interval. So only k=0 gives a valid solution in case 1.Case 2: φ = -4π/3 - 2π k. Let's take k=0: φ=-4π/3≈-4.188, which absolute value is more than π/2. k=-1: φ=-4π/3 -2π*(-1)= -4π/3 +2π= 2π/3≈2.094, which is more than π/2. k=1: φ=-4π/3 -2π= -10π/3≈-10.471, which is still more than π/2. So no solutions in case 2 satisfy |φ|<π/2.Therefore, the only solution is φ=π/3. Therefore, tanφ=tan(π/3)=√3. So the answer is D.√3.Therefore, I think the correct answer is D.Final Answerboxed{D}</think>Given (cosleft(frac{pi}{2} - varphiright) = frac{sqrt{3}}{2}) and (|varphi| < frac{pi}{2}), we need to find the value of (tanvarphi).First, we use the trigonometric identity (cosleft(frac{pi}{2} - varphiright) = sinvarphi). This transforms the given equation into:[sinvarphi = frac{sqrt{3}}{2}]Next, we determine the possible values of (varphi). Since (sinvarphi = frac{sqrt{3}}{2}), the principal solutions are (varphi = frac{pi}{3}) or (varphi = frac{2pi}{3}). However, considering the constraint (|varphi| < frac{pi}{2}), we check which of these solutions fall within the interval (-frac{pi}{2} < varphi < frac{pi}{2}).- (frac{pi}{3} approx 1.047) is within the interval (-frac{pi}{2} < varphi < frac{pi}{2}).- (frac{2pi}{3} approx 2.094) is outside this interval.Thus, the valid solution is (varphi = frac{pi}{3}).To find (tanvarphi), we calculate:[tanleft(frac{pi}{3}right) = sqrt{3}]Therefore, the value of (tanvarphi) is (boxed{D}).]

🔑:Because it's given that cosleft(frac{pi}{2} - varphiright) = frac{sqrt{3}}{2}, by using the cosine of a difference formula we can express this as:cosfrac{pi}{2}cosvarphi + sinfrac{pi}{2}sinvarphi = frac{sqrt{3}}{2}.Since cosfrac{pi}{2} = 0 and sinfrac{pi}{2} = 1, the equation simplifies to:0 cdot cosvarphi + 1 cdot sinvarphi = frac{sqrt{3}}{2},which means:sinvarphi = frac{sqrt{3}}{2}.Furthermore, since |varphi| < frac{pi}{2}, we know the sine value must be positive, which implies that varphi is an acute angle. Thus, we have varphi = frac{pi}{3}.With this angle, the tangent is:tanvarphi = tanfrac{pi}{3} = sqrt{3}.Therefore, the correct answer is boxed{D}.

🧠:[<think>Okay, so I need to find the minimum value of the maximum among these three absolute values: |2x + y|, |x - y|, and |1 + y|, where x and y are real numbers. Hmm, let me try to break this down step by step.First, the problem is asking for the smallest possible value of the largest of these three expressions. So, I need to minimize the maximum of these three. That sounds like an optimization problem where I have to consider all possible x and y such that the highest of these three absolute values is as small as possible.Let me visualize this. Suppose I set each of these three expressions equal to some value k, and I want to find the smallest k such that all three inequalities are satisfied. So, the idea would be:1. |2x + y| ≤ k2. |x - y| ≤ k3. |1 + y| ≤ kAnd I need to find the smallest k for which there exist real numbers x and y that satisfy all three inequalities. Then, that smallest k would be the minimum value of the maximum of those three expressions.Alright, let's write down these inequalities as:- -k ≤ 2x + y ≤ k- -k ≤ x - y ≤ k- -k ≤ 1 + y ≤ kStarting with the third inequality: |1 + y| ≤ k. That simplifies to -k ≤ 1 + y ≤ k. Subtracting 1 from all parts gives: -k -1 ≤ y ≤ k -1. So, y must be between -k -1 and k -1.Similarly, the second inequality: |x - y| ≤ k, which means x is between y - k and y + k.The first inequality: |2x + y| ≤ k. Let's see if we can express x and y in terms of k.Maybe I can express x from the second inequality in terms of y. From |x - y| ≤ k, we have x ∈ [y - k, y + k]. Let's take x = y + t, where t ∈ [-k, k]. Then, substitute this into the first inequality.So, substituting x = y + t into |2x + y| gives |2(y + t) + y| = |3y + 2t| ≤ k.So, now we have |3y + 2t| ≤ k. Also, from the third inequality, we have y ∈ [-k -1, k -1].So, combining these, maybe we can express y in terms of k and t, but this might get complicated. Alternatively, maybe we can find relationships between these variables.Alternatively, perhaps consider that we need to find x and y such that all three expressions are ≤ k, and find the minimal k.Alternatively, think of this geometrically. Each of these inequalities represents a region in the xy-plane, and the intersection of these regions would give the feasible (x, y) for a given k. The minimal k is the smallest such that the intersection is non-empty.Alternatively, perhaps use Lagrange multipliers or some optimization technique. But since we're dealing with absolute values, maybe breaking into different cases based on the signs.But that might be too cumbersome. Let's think of it as a linear programming problem, but with the maximum function. The goal is to minimize k such that:2x + y ≤ k,-2x - y ≤ k,x - y ≤ k,-x + y ≤ k,1 + y ≤ k,-1 - y ≤ k.So, these six inequalities define the feasible region. To minimize k, we can set up this as a linear program. But maybe there's a smarter way.Alternatively, let's consider that we need to make all three expressions as small as possible. The third expression, |1 + y|, suggests that y should be as close to -1 as possible. But then, the other expressions depend on x and y. Let's see.Suppose we set y = -1, then |1 + y| = 0, which is the smallest possible. Then, we have to minimize the maximum of |2x -1| and |x - (-1)| = |x + 1|.So, in this case, we can set y = -1, then find x such that both |2x -1| and |x + 1| are minimized. But how?We need to minimize the maximum of |2x -1| and |x +1|.Let me set these two equal to each other. Suppose |2x -1| = |x +1|. Then, 2x -1 = x +1 → x = 2, or 2x -1 = -x -1 → 3x = 0 → x=0.At x=2, |2(2) -1| = |3| = 3, |2 +1| = 3. So the maximum is 3.At x=0, |2(0) -1| = 1, |0 +1| =1. So maximum is 1.So when y = -1, and x=0, the maximum of |2x + y| = |0 -1|=1, |x - y|=|0 - (-1)|=1, and |1 + y|=0. So the maximum is 1. Wait, but if we set x=0 and y=-1, then all three expressions are 1, 1, 0. So the maximum is 1. So that seems like a candidate.But is this the minimum possible?Wait, but perhaps we can find a smaller k. Let's see.Suppose we set y slightly above -1. For example, let's assume that y is a bit more than -1. Then |1 + y| would be something small, but maybe we can adjust x to make the other two expressions smaller.Alternatively, maybe there's a balance between |2x + y|, |x - y|, and |1 + y|. Let's try to set them equal. Let's suppose that |2x + y| = |x - y| = |1 + y| = k. Then, solving for x and y such that all three equal to k.But since they can be positive or negative, we need to consider the signs. Let's assume all three expressions are equal to k. Then:2x + y = k,x - y = k,1 + y = k.So, solving these equations:From the third equation: y = k -1.Substitute into the second equation: x - (k -1) = k → x = 2k -1.Substitute x into the first equation: 2(2k -1) + (k -1) = k → 4k -2 + k -1 = k → 5k -3 = k → 4k = 3 → k = 3/4.So, k = 3/4. Let's check if these satisfy the equations.Then y = 3/4 -1 = -1/4,x = 2*(3/4) -1 = 3/2 -1 = 1/2.Check:|2x + y| = |2*(1/2) + (-1/4)| = |1 - 1/4| = 3/4 = k.|x - y| = |1/2 - (-1/4)| = |3/4| = 3/4 = k.|1 + y| = |1 + (-1/4)| = |3/4| = 3/4 = k.So all three are equal to 3/4. So this gives us a solution where the maximum of the three is 3/4. So that's better than the previous case where k was 1. So maybe 3/4 is the minimal value.But let's check if this is indeed the minimum. Is there a way to get a smaller k?Alternatively, suppose we set two expressions equal to k and the third less than k. But maybe the minimal k is achieved when all three are equal, as in the case above. Because if one of them is less, then maybe we can reduce k further.Wait, let's see. Suppose we can find x and y such that all three expressions are less than 3/4. Let's suppose such x and y exist.But since in the previous case, when k=3/4, all three are exactly 3/4. So if you make k smaller, then the constraints would require all three expressions to be less than k, which would contradict the previous solution. So perhaps 3/4 is the minimal possible.But to verify, let's try to see if there's a way to have a lower k.Suppose we take k=1/2. Then, we need |1 + y| ≤ 1/2, so y ∈ [-3/2, -1/2]. Then, |x - y| ≤ 1/2 implies x ∈ [y -1/2, y +1/2]. Also, |2x + y| ≤1/2.Let's see if it's possible. Let me pick y in the middle of [-3/2, -1/2], say y=-1. Then, x would be in [-1 -1/2, -1 +1/2] = [-3/2, -1/2]. Then, |2x + y| = |2x -1|. Let's check the minimum possible |2x -1| when x ∈ [-3/2, -1/2].If x=-1/2, then |2*(-1/2) -1| = |-1 -1| = 2, which is greater than 1/2. If x=-3/2, |2*(-3/2) -1| = |-3 -1| = 4, which is way larger. So even the minimal value here is 2, which is greater than 1/2. So k=1/2 is impossible.Alternatively, pick y=-3/4. Then, |1 + y| = |1 -3/4|=1/4. So k needs to be at least 1/4. Then, |x - y| ≤ k → x ∈ [y -k, y +k] = [-3/4 -k, -3/4 +k]. And |2x + y| ≤k.Let's set k=1/2. Then y=-1/2 (since |1 + y|=1/2). Then x ∈ [-1/2 -1/2, -1/2 +1/2] = [-1, 0]. Then |2x + (-1/2)| ≤1/2. So |2x -1/2| ≤1/2. Which means 2x -1/2 ∈ [-1/2, 1/2]. So 2x ∈ [0,1]. So x ∈ [0, 1/2]. But x ∈ [-1,0], so the intersection is x=0. At x=0, |2*0 -1/2|=1/2 ≤1/2. Also, |0 - (-1/2)|=1/2 ≤1/2. So in this case, both |2x + y| and |x - y| are exactly 1/2, and |1 + y|=1/2. So maximum is 1/2. Wait, but earlier we had a case where k=3/4. But here, if k=1/2, then all three expressions are 1/2. Wait, but this contradicts the previous solution. What's going on here?Wait, if y=-1/2, then |1 + y|=1/2. Then x=0, so |2x + y|=|0 + (-1/2)|=1/2. And |x - y|=|0 - (-1/2)|=1/2. So all three expressions are 1/2, so the maximum is 1/2. But this contradicts the earlier solution where k=3/4. Wait, but this seems possible. Am I making a mistake here?Wait, let's check the calculations again.If y = -1/2, then |1 + y| = |1 - 1/2| = 1/2.Then x is supposed to be in [y -k, y +k] = [-1/2 -1/2, -1/2 +1/2] = [-1, 0]. So x ∈ [-1,0].To satisfy |2x + y| ≤1/2. So |2x -1/2| ≤1/2. Which implies:-1/2 ≤ 2x -1/2 ≤1/2Adding 1/2 to all parts:0 ≤ 2x ≤1Divide by 2:0 ≤ x ≤1/2But x must be in [-1,0]. The intersection of [0,1/2] and [-1,0] is just x=0. So x=0 is the only solution. Then, check the three expressions:|2*0 + (-1/2)| = 1/2,|0 - (-1/2)| = 1/2,|1 + (-1/2)| =1/2.So all three are 1/2, so the maximum is 1/2. Wait, but this contradicts the previous case where we thought k=3/4 was the minimum. But here, we have a solution with k=1/2. What's wrong here?Wait a minute, maybe I made a mistake in assuming that y can be set to -1/2 and x=0. Let me check the constraints again.Wait, if k=1/2, then y must satisfy |1 + y| ≤1/2, so y ∈ [-3/2, -1/2]. So y=-1/2 is allowed. Then, x must satisfy |x - y| ≤1/2. If y=-1/2, then x ∈ [-1,0]. Then, |2x + y| = |2x -1/2|. If x=0, then |2*0 -1/2|=1/2, which is okay. So x=0 is in the interval. So this seems valid. Then why in the previous case when we set all three expressions equal to k we got k=3/4? Because in that case, we assumed that all three expressions are equal to k, but here, we have a different configuration where all three are still equal to k=1/2.Wait, but this can't be right. If there is a solution with k=1/2, then that would be the minimum. But this seems conflicting.Wait, let's verify the calculations again. If x=0, y=-1/2:|2x + y| = |0 + (-1/2)| = 1/2,|x - y| = |0 - (-1/2)| = 1/2,|1 + y| = |1 - 1/2| =1/2.So yes, all three are exactly 1/2. So the maximum is 1/2, which is less than 3/4. So why did the previous method give 3/4?Ah, I see the mistake. In the previous approach, I assumed that all three expressions are equal to k, but that's not the only way to achieve the maximum. In this case, when we set y=-1/2 and x=0, all three expressions are equal to 1/2. So that's a valid solution. Therefore, the minimum k should be 1/2. But wait, this contradicts the previous result where solving the equations gave k=3/4. There must be an error here.Wait, no. Let me check again. Let's see:If we set |2x + y| = |x - y| = |1 + y| = k.But when we solved for k=3/4, we found x=1/2, y=-1/4. But here, when k=1/2, x=0, y=-1/2. So why did solving the equations give a higher k?Wait, because when I set all three expressions equal to k, the solution is k=3/4. But if we allow different signs, maybe there's another solution where all three expressions are equal to k with some signs.Wait, in the previous case, we assumed all three were positive. But perhaps if some are negative, we can have different solutions.Wait, but absolute values are always non-negative, so setting |a|=k, |b|=k, |c|=k doesn't depend on the sign inside. So maybe the previous system of equations assumes all three are equal in absolute value. But here, even though we didn't set them equal, but found a solution where all three are equal, so both solutions are valid. But why the discrepancy?Wait, no. When we set all three expressions equal to k, we are essentially looking for a point where all three reach the maximum simultaneously. However, in the case of k=1/2, it's also a point where all three expressions reach k, so that should be a solution to the system. Therefore, perhaps there was a mistake in the previous solution.Wait, let's redo the previous approach where we set all three expressions equal to k.Assume:2x + y = k,x - y = k,1 + y = k.Solving:From third equation: y = k -1.Substitute into second equation: x - (k -1) =k → x = 2k -1.Substitute into first equation: 2*(2k -1) + (k -1) =k → 4k -2 +k -1 =k →5k -3 =k →4k=3 →k=3/4. So that's correct.But when we take k=1/2, and set:2x + y =1/2,x - y=1/2,1 + y=1/2.Solving:From third equation: y= -1/2.From second equation: x - (-1/2)=1/2 →x +1/2=1/2 →x=0.From first equation: 2*0 + (-1/2)= -1/2. But | -1/2 | =1/2. So yes, this also satisfies 2x + y = -1/2, but its absolute value is 1/2. So the system I set up earlier required 2x + y =k, but in reality, it's |2x + y|=k, so it could be -k as well. So in this case, if we set 2x + y = -k, then:2x + y = -k,x - y =k,1 + y =k.Then solving:From third equation: y=k -1.From second equation: x = y +k = (k -1) +k =2k -1.From first equation: 2*(2k -1) + (k -1)= -k →4k -2 +k -1= -k →5k -3= -k →6k=3 →k=1/2.So there we go. That's the other case where instead of 2x + y =k, it's 2x + y = -k. So solving that system gives k=1/2. So that's the solution where x=0, y=-1/2. So depending on the sign chosen for the first equation, we get different k. Therefore, the minimal k is 1/2.But why did the previous approach give k=3/4? Because in that case, we assumed all three expressions were positive k, but here, allowing one to be negative gives a smaller k. Therefore, the minimal k is indeed 1/2.Wait, but this seems conflicting. So which one is correct? Let's check the two solutions.First solution: k=3/4, x=1/2, y=-1/4.Check:|2x + y| = |1 + (-1/4)| = |3/4|=3/4,|x - y| = |1/2 - (-1/4)| = |3/4|=3/4,|1 + y| = |1 -1/4|=3/4.All three are 3/4.Second solution: k=1/2, x=0, y=-1/2.Check:|2x + y| = |0 -1/2|=1/2,|x - y| = |0 - (-1/2)|=1/2,|1 + y| = |1 -1/2|=1/2.All three are 1/2.So both are valid. Therefore, the minimal k is 1/2. But this contradicts the initial assumption where I thought the minimal k was 3/4. So where is the mistake here?Wait, the problem says "the minimum value of the maximum number among |2x + y|, |x - y|, |1 + y|". So the answer should be the minimal possible maximum of these three. In the first case, we have a maximum of 3/4, and in the second case, a maximum of 1/2. Since 1/2 is smaller than 3/4, the minimal value should be 1/2. Therefore, the correct answer is 1/2.But why did the system of equations assuming positive k give 3/4? Because in that case, we assumed that 2x + y =k, but if we allow 2x + y = -k, we can get a smaller k. Therefore, the minimal k is indeed 1/2.Wait, but how can this be? If the answer is 1/2, then why in the first approach with y=-1, x=0 gave k=1, but here with y=-1/2, x=0 gives k=1/2.Therefore, the minimal k is 1/2.But I need to confirm this.Alternatively, perhaps there is a mistake in assuming that both cases are valid. Let me check the second solution more carefully.If x=0, y=-1/2:Compute each expression:1. |2x + y| = |0 + (-1/2)| = 1/2.2. |x - y| = |0 - (-1/2)| = 1/2.3. |1 + y| = |1 - 1/2| =1/2.Yes, all three are exactly 1/2. So the maximum is 1/2. So this is a valid solution. Therefore, the minimal possible value is 1/2.Therefore, the answer should be 1/2, or in boxed form, boxed{dfrac{1}{2}}.But wait, now I'm confused because two different approaches give different results. Let me verify once more.In the first approach, solving the system assuming 2x + y =k gives k=3/4. In the second approach, solving the system assuming 2x + y =-k gives k=1/2. Both are valid, but the second gives a smaller k. Therefore, the minimal k is 1/2.Therefore, the correct answer should be 1/2.But let me check another way. Suppose we set y=-1/2, x=0. Then, as above, all three expressions are 1/2. So that's valid. So the minimum possible maximum is at most 1/2. Is there a way to get it lower than 1/2?Suppose we try k=1/4. Then:|1 + y| ≤1/4 implies y ∈ [-5/4, -3/4].|x - y| ≤1/4 implies x ∈ [y -1/4, y +1/4].Then, for |2x + y| ≤1/4.Let's pick y=-1 (which is not in [-5/4, -3/4], but closer). Wait, y has to be in [-5/4, -3/4]. Let's take y=-1. Then, x ∈ [-1 -1/4, -1 +1/4] = [-5/4, -3/4]. Then, compute |2x + y|.Take x=-3/4, y=-1:|2*(-3/4) + (-1)| = |-3/2 -1| = |-5/2|=5/2 >1/4. So even the minimal here is 5/2, which is way larger than 1/4. So k=1/4 is impossible.Similarly, take y=-5/4, which is the lower end. Then x ∈ [-5/4 -1/4, -5/4 +1/4] = [-3/2, -1]. Compute |2x + y| for x=-1:|2*(-1) + (-5/4)| = |-2 -5/4|=|-13/4|=13/4 >1/4. So again, impossible.Therefore, k=1/2 is achievable, and lower k's are not. So the minimal k is indeed 1/2.But wait, the previous solution where x=0, y=-1/2 gives all three expressions as 1/2. Therefore, the minimal value is 1/2. Therefore, the answer is 1/2.But why did the first method give 3/4? Because in that case, I set all three expressions equal to k with 2x + y=k, but if we allow 2x + y=-k, we get a smaller k. Therefore, the correct answer is 1/2.Therefore, the minimum value is boxed{dfrac{1}{2}}.But wait, I need to verify this once again. Let's take x=0 and y=-1/2. Then:|2x + y| = |0 + (-1/2)| = 1/2,|x - y| = |0 - (-1/2)| = 1/2,|1 + y| = |1 - 1/2| = 1/2.So yes, all three are 1/2. Therefore, the maximum is 1/2. Therefore, the minimal possible maximum is 1/2. Therefore, the answer is 1/2.But I must have made a mistake earlier when I thought it was 3/4. The key was considering that the equations can be set with negative signs as well, which allows a lower k. Therefore, the correct answer is 1/2.</think>You will be given a problem. Please reason step by step, and put your final answer within boxed{}:6. If x, y are real numbers, then the minimum value of the maximum number among |2 x+y|,|x-y|,|1+y| isThe problem requires finding the minimum value of the maximum among the three absolute values: (|2x + y|), (|x - y|), and (|1 + y|). To solve this, we need to minimize the maximum value of these expressions.1. Setting up the inequalities: [ |2x + y| leq k, quad |x - y| leq k, quad |1 + y| leq k ] We need to find the smallest (k) such that there exist real numbers (x) and (y) satisfying all three inequalities.2. Analyzing the inequalities: - From (|1 + y| leq k), we get (y in [-k - 1, k - 1]). - From (|x - y| leq k), we get (x in [y - k, y + k]). - Substituting (x = y + t) (where (t in [-k, k])) into (|2x + y| leq k) gives (|3y + 2t| leq k).3. Equating all three expressions to (k): - Solving the system: [ 2x + y = pm k, quad x - y = pm k, quad 1 + y = pm k ] - Considering the case (2x + y = -k), (x - y = k), and (1 + y = k): - From (1 + y = k), we get (y = k - 1). - From (x - y = k), we get (x = 2k - 1). - Substituting (x = 2k - 1) and (y = k - 1) into (2x + y = -k): [ 2(2k - 1) + (k - 1) = -k implies 5k - 3 = -k implies 6k = 3 implies k = frac{1}{2} ]4. Verification: - Substituting (k = frac{1}{2}), we get (y = -frac{1}{2}) and (x = 0). - Checking the values: [ |2(0) + (-frac{1}{2})| = frac{1}{2}, quad |0 - (-frac{1}{2})| = frac{1}{2}, quad |1 + (-frac{1}{2})| = frac{1}{2} ] - All three expressions equal (frac{1}{2}), confirming the solution.Thus, the minimum value of the maximum number among (|2x + y|), (|x - y|), and (|1 + y|) is (boxed{dfrac{1}{2}}).,<think>Okay, so I need to find the minimum value of the maximum among these three absolute expressions: |2x + y|, |x - y|, and |1 + y|, where x and y are real numbers. Hmm. Let me try to visualize this.First, the problem is asking for the smallest possible value that the largest of these three absolute values can take. In other words, I need to choose x and y such that the biggest among |2x + y|, |x - y|, and |1 + y| is as small as possible. That sounds like an optimization problem where I want to minimize the maximum of these three expressions.Let me denote M as the maximum of those three, so M = max{ |2x + y|, |x - y|, |1 + y| }, and I need to find the minimum possible M. I remember that such problems can sometimes be approached by setting all the expressions equal to each other, but since they are absolute values, maybe there's a way to set them equal in a way that their positive or negative values balance out. Let me try to set up equations where |2x + y| = |x - y| = |1 + y| = M and solve for x and y such that these equalities hold. If I can find such x and y, then that M would be a candidate for the minimum value.Starting with the first two expressions: |2x + y| = |x - y|. Let's square both sides to eliminate the absolute values. That gives (2x + y)^2 = (x - y)^2. Expanding both sides:4x² + 4xy + y² = x² - 2xy + y²Subtract x² - 2xy + y² from both sides:3x² + 6xy = 0Factor out 3x:3x(x + 2y) = 0So either x = 0 or x + 2y = 0.Case 1: x = 0.If x = 0, then substitute into the other expressions. Let's check |x - y| = | -y | = |y|, and |2x + y| = |y|. So both |y| and |1 + y| need to be equal to M. Therefore, we need |y| = |1 + y| = M.But when does |y| = |1 + y|? Let's solve this. Squaring both sides:y² = (1 + y)² → y² = 1 + 2y + y² → 0 = 1 + 2y → y = -1/2.So if y = -1/2, then |y| = 1/2 and |1 + y| = |1 - 1/2| = 1/2. So M = 1/2 here. Also, since x=0, |2x + y| = | -1/2 | = 1/2, and |x - y| = |0 - (-1/2)| = 1/2. So all three expressions are equal to 1/2. Therefore, in this case, the maximum M is 1/2. So that's a candidate for the minimum value.But wait, before concluding, let's check the other case where x + 2y = 0.Case 2: x + 2y = 0 ⇒ x = -2y.Substitute x = -2y into the three expressions:First, |2x + y| = |2*(-2y) + y| = |-4y + y| = |-3y| = 3|y|Second, |x - y| = |-2y - y| = |-3y| = 3|y|Third, |1 + y|.So now, the maximum of 3|y| and |1 + y|. We want this maximum to be as small as possible. Let's set 3|y| = |1 + y| = M.So we have 3|y| = |1 + y|. Let's solve for y.Case a: y ≥ 0. Then |y| = y and |1 + y| = 1 + y. So 3y = 1 + y ⇒ 2y = 1 ⇒ y = 1/2. Then check if y = 1/2 ≥ 0, which it is. Then x = -2*(1/2) = -1. Let's compute the three expressions:|2x + y| = |2*(-1) + 1/2| = |-2 + 1/2| = |-3/2| = 3/2|x - y| = |-1 - 1/2| = |-3/2| = 3/2|1 + y| = |1 + 1/2| = 3/2So here, M = 3/2. But in the previous case, we had M = 1/2, which is smaller. So this is worse.Case b: y < 0. Then |y| = -y and |1 + y|. If y < 0, then 1 + y could be positive or negative. Let's see. Let me denote z = y, so z < 0.We have 3*(-z) = |1 + z|. Since z is negative, let's write z = -t where t > 0. Then the equation becomes 3t = |1 - t|.So |1 - t| = 3t. Since t > 0, 1 - t can be positive or negative.Subcase 1: 1 - t ≥ 0 ⇒ t ≤ 1. Then |1 - t| = 1 - t. So 1 - t = 3t ⇒ 1 = 4t ⇒ t = 1/4. Then z = -t = -1/4. So y = -1/4. Check if this satisfies t ≤ 1, which it does.Then x = -2y = -2*(-1/4) = 1/2. Now compute the three expressions:|2x + y| = |2*(1/2) + (-1/4)| = |1 - 1/4| = 3/4|x - y| = |1/2 - (-1/4)| = |3/4| = 3/4|1 + y| = |1 + (-1/4)| = |3/4| = 3/4So here, M = 3/4. That's better than 3/2 but worse than 1/2 from the first case. So 3/4 is another candidate.Subcase 2: If 1 - t < 0 ⇒ t > 1. Then |1 - t| = t - 1. So equation becomes t - 1 = 3t ⇒ -1 = 2t ⇒ t = -1/2. But t was defined as positive, so this is impossible. Therefore, no solution here.Therefore, in case 2, the best we can do is M = 3/4 when y = -1/4 and x = 1/2. However, in case 1, we have M = 1/2. So 1/2 is better.Wait, but in case 1, we set x = 0 and found y = -1/2. Then all three expressions become 1/2, which is indeed better. Therefore, 1/2 is a lower value. But is that correct? Let me check again.Wait, when x = 0 and y = -1/2:Compute |2x + y| = |0 + (-1/2)| = 1/2|x - y| = |0 - (-1/2)| = 1/2|1 + y| = |1 - 1/2| = 1/2So all three are exactly 1/2. That's perfect, all equal. So the maximum is 1/2. So that's a valid solution.But why in case 2, when x = -2y, we got 3/4. So that's higher. Therefore, the minimal value is 1/2? But wait, the problem says "the minimum value of the maximum number among...". So that would be 1/2. But let me check if there isn't a case where M is even smaller.Wait, perhaps there's a way to make all three expressions less than 1/2. Is that possible? Suppose we try to set all three expressions to be less than 1/2. Then:|2x + y| < 1/2|x - y| < 1/2|1 + y| < 1/2But |1 + y| < 1/2 implies that -1/2 < 1 + y < 1/2 ⇒ subtract 1: -3/2 < y < -1/2.So y must be between -3/2 and -1/2.Now, let's see if we can find x and y such that all three inequalities hold.From |x - y| < 1/2, since y is between -3/2 and -1/2, then x must be within (-3/2 - 1/2, -1/2 + 1/2) = (-2, 0). So x is between -2 and 0.From |2x + y| < 1/2. Let's express 2x + y. Since x is between -2 and 0, 2x is between -4 and 0. y is between -3/2 and -1/2. So 2x + y is between (-4) + (-3/2) = -5.5 and 0 + (-1/2) = -0.5. So 2x + y is between -5.5 and -0.5. The absolute value |2x + y| would then be between 0.5 and 5.5. Wait, but we need |2x + y| < 1/2. But 2x + y is between -5.5 and -0.5, so |2x + y| is between 0.5 and 5.5. So the minimal |2x + y| is 0.5. Therefore, |2x + y| cannot be less than 0.5. Hence, it's impossible to have all three expressions less than 1/2. Therefore, the minimal possible M is 1/2.But wait, in case 1, when x=0 and y=-1/2, we have |2x + y| = 1/2, |x - y| = 1/2, and |1 + y| = 1/2. So all three are exactly 1/2. Therefore, that's the minimal possible M, since we can't go below 1/2 because |2x + y| can't be smaller than 1/2 given the constraints on y from |1 + y|. Therefore, the minimum value is 1/2.But let me check if there's a way to have two of the expressions equal to M and the third less than M, which might allow M to be smaller. For example, maybe |1 + y| is less than M, but |2x + y| and |x - y| are equal to M. Could that allow a smaller M?Suppose |1 + y| < M, and |2x + y| = |x - y| = M. Then, similar to before, solving |2x + y| = |x - y| = M. Let's try to set up equations. Let's suppose that 2x + y = M and x - y = M. But since they are absolute values, they could also be negative. Let me consider the case where both are positive. So 2x + y = M and x - y = M. Solving these two equations:From the second equation: x = M + y.Substitute into first equation: 2(M + y) + y = M ⇒ 2M + 2y + y = M ⇒ 3y = -M ⇒ y = -M/3.Then x = M + y = M - M/3 = (2M)/3.Now, |1 + y| = |1 - M/3| < M. So we need |1 - M/3| < M.This gives two cases:1. 1 - M/3 < M and 1 - M/3 > -M.First inequality: 1 < M + M/3 = (4M)/3 ⇒ 3/4 < M.Second inequality: 1 - M/3 > -M ⇒ 1 > -M + M/3 ⇒ 1 > -2M/3 ⇒ 2M/3 > -1 ⇒ M > -3/2. But since M is a maximum of absolute values, M is non-negative. So this inequality is automatically satisfied.Therefore, combining these, we have M > 3/4. But we also have to check if |1 + y| < M. So |1 - M/3| < M. Let's compute |1 - M/3| < M.Case 1: 1 - M/3 ≥ 0 ⇒ M ≤ 3. Then |1 - M/3| = 1 - M/3 < M ⇒ 1 < M + M/3 ⇒ 1 < (4M)/3 ⇒ M > 3/4.Case 2: 1 - M/3 < 0 ⇒ M > 3. Then |1 - M/3| = M/3 - 1 < M ⇒ M/3 -1 < M ⇒ -1 < 2M/3 ⇒ -3/2 < M. Which is always true since M >3.But since we found that M > 3/4 from the first case, and in the second case M >3. But since we are trying to minimize M, the relevant case is M >3/4. But then, if M must be greater than 3/4, but previously we found a case where M=1/2. So this suggests that if we allow |1 + y| < M, then M must be at least 3/4. But since we already have a solution with M=1/2, which is smaller, this approach doesn't give a better result.Alternatively, perhaps if we set 2x + y = -M and x - y = -M. Let's try that.So 2x + y = -M and x - y = -M.Solving these equations:From the second equation: x = -M + y.Substitute into the first equation: 2(-M + y) + y = -M ⇒ -2M + 2y + y = -M ⇒ 3y = M ⇒ y = M/3.Then x = -M + y = -M + M/3 = -2M/3.Now, |1 + y| = |1 + M/3| < M.So |1 + M/3| < M.Again, two cases:1. 1 + M/3 ≥ 0 ⇒ M ≥ -3, which is always true. So |1 + M/3| = 1 + M/3 < M ⇒ 1 < M - M/3 ⇒ 1 < (2M)/3 ⇒ M > 3/2.So in this case, M has to be greater than 3/2, which is worse than our previous candidates. Therefore, this doesn't help.Therefore, trying to have two expressions at M and the third less than M only gives us M > 3/4 or M > 3/2, which are worse than 1/2. Hence, the minimal M is indeed 1/2.But to be thorough, let's check if there are other scenarios where all three expressions could be equal but with different signs.For example, maybe |2x + y| = |x - y| = |1 + y| = M, but some are positive and some are negative.But since we're taking absolute values, the signs inside don't matter. So maybe we can have 2x + y = M, x - y = -M, and 1 + y = -M. Let's try that.Set 2x + y = M,x - y = -M,1 + y = -M.From the third equation: y = -M -1.Substitute into the second equation: x - (-M -1) = -M ⇒ x + M +1 = -M ⇒ x = -2M -1.Substitute y and x into the first equation: 2*(-2M -1) + (-M -1) = M ⇒ -4M -2 -M -1 = M ⇒ -5M -3 = M ⇒ -6M =3 ⇒ M = -0.5. But M must be non-negative since it's the maximum of absolute values. So this is invalid.Alternatively, try 2x + y = -M, x - y = M, 1 + y = -M.From third equation: y = -M -1.From second equation: x - y = M ⇒ x = M + y = M + (-M -1) = -1.Substitute x = -1 and y = -M -1 into first equation: 2*(-1) + (-M -1) = -M ⇒ -2 -M -1 = -M ⇒ -3 -M = -M ⇒ -3 = 0. Contradiction. So no solution.Another combination: 2x + y = M, x - y = M, 1 + y = -M.From third equation: y = -M -1.From second equation: x - y = M ⇒ x = M + y = M + (-M -1) = -1.Substitute into first equation: 2*(-1) + (-M -1) = M ⇒ -2 -M -1 = M ⇒ -3 -M = M ⇒ -3 = 2M ⇒ M = -3/2. Again, invalid.Another possibility: 2x + y = -M, x - y = -M, 1 + y = M.From third equation: y = M -1.From second equation: x - y = -M ⇒ x = -M + y = -M + M -1 = -1.Substitute into first equation: 2*(-1) + (M -1) = -M ⇒ -2 + M -1 = -M ⇒ M -3 = -M ⇒ 2M =3 ⇒ M= 3/2.Check |1 + y| = |1 + (3/2 -1)| = |1 +1/2| = 3/2. So M=3/2. Then x = -1, y=3/2 -1=1/2.Compute the other expressions:|2x + y| = |2*(-1) +1/2| = |-2 +0.5|=1.5=3/2,|x - y| = |-1 -0.5|=1.5=3/2,|1 + y|=3/2.So M=3/2. But this is larger than our previous candidate of 1/2. So not helpful.Therefore, all combinations where two are positive and one negative (in their inside expressions) either lead to contradictions or higher M. So it seems the only valid solution where all three are equal is when they are all positive and M=1/2.Therefore, the minimal value is 1/2.But let me verify once more with another approach. Maybe using geometry. The expressions |2x + y|, |x - y|, and |1 + y| can be seen as distances from lines in the plane. We need to find the point (x, y) such that the maximum of these three distances is minimized. This is equivalent to finding the smallest M such that all three expressions are ≤ M, and at least one of them is equal to M. The intersection of the regions |2x + y| ≤ M, |x - y| ≤ M, and |1 + y| ≤ M forms a polygon, and we need to find the smallest M where this intersection is non-empty. The minimal such M would be the minimal value we are looking for.To find this M, we can set up inequalities:1. |2x + y| ≤ M2. |x - y| ≤ M3. |1 + y| ≤ MWe need to find the smallest M where these three inequalities have a common solution (x, y).From the third inequality: -M ≤ 1 + y ≤ M ⇒ -M -1 ≤ y ≤ M -1.From the first inequality: -M ≤ 2x + y ≤ M ⇒ -M - y ≤ 2x ≤ M - y ⇒ (-M - y)/2 ≤ x ≤ (M - y)/2.From the second inequality: -M ≤ x - y ≤ M ⇒ -M + y ≤ x ≤ M + y.So the solution set is the intersection of these regions. The minimal M is when these regions just touch each other.Alternatively, we can set up the problem as a linear programming problem where we minimize M subject to:- M ≤ 2x + y ≤ M- M ≤ x - y ≤ M- M ≤ 1 + y ≤ MBut since it's non-linear due to the absolute values, it's a bit more complex. However, since we found a solution where all three are equal to 1/2, that should be the minimal M. The geometrical interpretation would be that the three regions touch at a common point, which is (x, y) = (0, -1/2). Let me check that point.At x=0, y=-1/2:|2x + y| = |-1/2| =1/2,|x - y| = |0 - (-1/2)| =1/2,|1 + y| = |1 -1/2| =1/2.Yes, all three are exactly 1/2. So this point is the intersection of the three boundaries. Therefore, this is indeed the minimal M.Therefore, after thorough analysis, I conclude that the minimal value is 1/2.</think>The problem asks us to find the minimum value of the maximum among three absolute expressions: ( |2x + y| ), ( |x - y| ), and ( |1 + y| ) for real numbers ( x ) and ( y ). This translates to minimizing ( M = max{ |2x + y|, |x - y|, |1 + y| } ). Key Steps:1. Equating Expressions: We start by setting ( |2x + y| = |x - y| = |1 + y| = M ) and solve for ( x ) and ( y ). - Case 1: ( x = 0 ) - Substituting ( x = 0 ), we solve ( |y| = |1 + y| ). This gives ( y = -frac{1}{2} ). - Check: ( |2x + y| = frac{1}{2} ), ( |x - y| = frac{1}{2} ), ( |1 + y| = frac{1}{2} ). All equal ( M = frac{1}{2} ). - Case 2: ( x = -2y ) - Substituting ( x = -2y ), we solve ( 3|y| = |1 + y| ). This gives ( y = -frac{1}{4} ) (with ( x = frac{1}{2} )), leading to ( M = frac{3}{4} ), which is larger than ( frac{1}{2} ).2. Verification: - For ( x = 0 ) and ( y = -frac{1}{2} ), all three expressions equal ( frac{1}{2} ), confirming ( M = frac{1}{2} ). - Attempting to find ( M < frac{1}{2} ) leads to contradictions due to constraints from ( |1 + y| ).3. Geometric Interpretation: - The solution corresponds to the intersection point of the regions defined by ( |2x + y| leq M ), ( |x - y| leq M ), and ( |1 + y| leq M ). The point ( (0, -frac{1}{2}) ) satisfies all three conditions with ( M = frac{1}{2} ). Conclusion:The minimum value of the maximum among ( |2x + y| ), ( |x - y| ), and ( |1 + y| ) is (boxed{frac{1}{2}}).]

🔑:Answer: frac{1}{2}Analysis: max {|2 x+y|,|x-y|,|1+y|} geq frac{1}{6}(|2 x+y|+2 cdot|x-y|+3 cdot|1+y|), geq frac{1}{6} cdot|(2 x+y)-2(x-y)-3(1+y)|=frac{1}{2}, which is achieved when and only when x=0, y=-frac{1}{2}.

🧠:[<think>Okay, let's see. I need to find the probability that a randomly thrown point M into the quadrilateral AA'B'B falls within the triangle FA'B'. So, first, I need to figure out the areas of both the quadrilateral and the triangle, then divide the area of the triangle by the area of the quadrilateral to get the probability. Given the parabola y² = 2px with p > 0. The focus F of this parabola is at (p/2, 0) because for a parabola y² = 4ax, the focus is at (a, 0). Wait, but here the equation is y² = 2px. Comparing to y² = 4a x, that would mean 4a = 2p, so a = p/2. Hence, the focus F is at (a, 0) = (p/2, 0). The directrix l is the line x = -a = -p/2. A line passes through the focus F and intersects the parabola at points A and B, with the distance between A and B being 3p. The projections of A and B onto the directrix l are A' and B', respectively. The quadrilateral AA'B'B is formed by these points, and we need to find the probability that a random point in this quadrilateral lies within triangle FA'B'.First, let me visualize this. The directrix is a vertical line at x = -p/2. The parabola opens to the right. The line through F intersects the parabola at A and B. The projections of A and B onto the directrix would be points directly to the left (or right, but since directrix is left, probably left) horizontally from A and B onto the line x = -p/2. So, if A is (x1, y1), then A' would be (-p/2, y1), similarly for B' being (-p/2, y2).Quadrilateral AA'B'B would then be a four-sided figure with vertices at A, A', B', B. Since A' and B' are on the directrix, which is vertical, the sides AA' and BB' are horizontal lines? Wait, no. If you project a point onto the directrix, which is vertical, the projection would be along the horizontal line (since directrix is vertical, the projection is horizontal). So yes, AA' is a horizontal line from A to the left to A', and BB' is a horizontal line from B to the left to B'. Then the sides A'B' is the segment on the directrix between A' and B', and side AB is the original chord. So the quadrilateral is a trapezoid? Wait, AA' and BB' are both horizontal, and A'B' is vertical? Wait, no. The directrix is vertical, so A' and B' are points on the directrix with the same y-coordinates as A and B. So if A is (x1, y1), then A' is (-p/2, y1), and B is (x2, y2), so B' is (-p/2, y2). Therefore, A'B' is a vertical segment if y1 and y2 are different? Wait, no, A' and B' are both on the directrix, which is vertical, so their x-coordinates are the same (-p/2), but their y-coordinates are y1 and y2. So A'B' is a vertical segment only if y1 and y2 are aligned vertically, but in general, it's a line segment between (-p/2, y1) and (-p/2, y2), which is vertical if y1 ≠ y2. Wait, but if y1 and y2 are different, then A'B' is a vertical line segment. So the quadrilateral AA'B'B has two horizontal sides (AA' and BB') and two sides: AB and A'B'. But AB is the chord of the parabola, and A'B' is vertical. So the shape is like a trapezoid but with one side being a slant (AB) and another side vertical (A'B'). Hmm, not a standard trapezoid.To compute the area of quadrilateral AA'B'B, maybe it's easier to use coordinates. Let me parametrize the points.Let’s denote the coordinates:- Focus F: (p/2, 0)- Directrix l: x = -p/2The line passing through F can be any line passing through (p/2, 0). Let's consider a general line through F. Let's parametrize this line. Since the problem doesn't specify the slope, perhaps I need to consider the general case, but given that |AB| = 3p, which might mean that there is a specific line, or maybe the probability is the same regardless of the line? Hmm, but parabola is symmetric, so maybe regardless of the slope, the ratio of areas remains the same? Not sure. Maybe I need to find in terms of p.Alternatively, perhaps the problem is set such that for any such line with |AB| = 3p, the probability is the same. So I need to find the areas in terms of p.Alternatively, maybe we can choose a coordinate system where the line through F is at a particular angle to make the computation easier. For example, take the line to be horizontal. Wait, but a horizontal line through F would intersect the parabola at points symmetric about the x-axis. Let me check.Wait, the parabola y² = 2px. If we take a horizontal line y = k, it would intersect the parabola at y = k, x = k²/(2p). So for y = 0, the line is the x-axis, which is the axis of symmetry, and it intersects the parabola at x = 0, but since the parabola is y² = 2px, when y=0, x=0. Wait, but the focus is at (p/2, 0). So the x-axis passes through the focus and intersects the parabola at the vertex (0,0). But the distance from F to the vertex is p/2. So if the line is the x-axis, then points A and B would be the vertex (0,0) and maybe another point? Wait, but substituting y=0 into the parabola equation gives x=0, so only one point. Therefore, a horizontal line through F (the x-axis) is tangent to the parabola at the vertex. So that's a degenerate case where the line is tangent, so only one intersection point. Therefore, to have two points A and B, the line cannot be horizontal. So maybe we need to take a line with some slope.Alternatively, maybe take a line with slope 1. Let's suppose the line through F has slope m. Let me define the line parametrically. Let’s denote the line as passing through F (p/2, 0) with slope m. Then the equation is y = m(x - p/2).We need to find the points A and B where this line intersects the parabola y² = 2px. Substitute y from the line equation into the parabola equation:[m(x - p/2)]² = 2pxm²(x - p/2)² = 2pxExpand the left side:m²(x² - p x + p²/4) = 2p xBring all terms to one side:m² x² - m² p x + m² p²/4 - 2p x = 0Combine like terms:m² x² - (m² p + 2p) x + (m² p²)/4 = 0Factor out p where possible:m² x² - p(m² + 2) x + (m² p²)/4 = 0Divide both sides by p (since p ≠ 0):(m²/p) x² - (m² + 2) x + (m² p)/4 = 0Wait, maybe not necessary. Alternatively, let's treat this as a quadratic in x:m² x² - [p(m² + 2)] x + (m² p²)/4 = 0Let’s denote coefficients as:A = m²B = -p(m² + 2)C = m² p² / 4Then the quadratic equation is Ax² + Bx + C = 0.The solutions are:x = [p(m² + 2) ± sqrt( p²(m² + 2)² - 4 * m² * (m² p² /4) ) ] / (2 m²)Simplify discriminant:sqrt( p²(m² + 2)² - 4 * m² * (m² p² /4) )= sqrt( p²(m² + 2)² - m^4 p² )= p sqrt( (m² + 2)^2 - m^4 )Expand (m² + 2)^2 = m^4 + 4m² + 4Thus:sqrt( m^4 + 4m² + 4 - m^4 ) = sqrt(4m² + 4) = 2 sqrt(m² + 1)Therefore, x = [p(m² + 2) ± 2p sqrt(m² + 1) ] / (2 m²)Factor out p from numerator:x = p [ (m² + 2) ± 2 sqrt(m² + 1) ] / (2 m² )Therefore, two x-coordinates:x1 = p [ (m² + 2) + 2 sqrt(m² + 1) ] / (2 m² )x2 = p [ (m² + 2) - 2 sqrt(m² + 1) ] / (2 m² )Then, the corresponding y-coordinates are:y1 = m(x1 - p/2 )y2 = m(x2 - p/2 )Therefore, points A and B are (x1, y1) and (x2, y2). Then |AB| = 3p.We need to compute the distance between A and B and set it equal to 3p. Then, perhaps solve for m? Then proceed to find the coordinates of A, B, A', B', compute the areas, and find the probability.But this seems complicated. Maybe there is a smarter way.Alternatively, since the problem is about a general parabola and line through the focus with |AB| = 3p, maybe the probability is constant regardless of the slope. If that's the case, maybe choosing a specific slope would simplify calculations. For example, take the line to be the x-axis. Wait, but earlier we saw that the x-axis is tangent at the vertex, so it would only intersect at one point. So that doesn't work. Then, maybe take a vertical line through F. Wait, a vertical line through F (p/2, 0) is x = p/2. Substitute into parabola equation y² = 2p*(p/2) = p². Therefore, y = ±p. So points A and B would be (p/2, p) and (p/2, -p). Then |AB| = 2p. But in the problem, |AB| = 3p. So that's not the case. So vertical line gives AB length 2p. So 3p is longer. Therefore, maybe take a line with some other slope.Alternatively, consider a line making 45 degrees with the x-axis. Let’s set m = 1. Then the line is y = x - p/2.Substitute into parabola equation:(x - p/2)^2 = 2p xExpand left side:x² - p x + p²/4 = 2p xBring all terms to left:x² - 3p x + p²/4 = 0Solutions:x = [3p ± sqrt(9p² - p²)] / 2 = [3p ± sqrt(8p²)] / 2 = [3p ± 2p√2]/2 = p(3 ± 2√2)/2So x1 = p(3 + 2√2)/2, x2 = p(3 - 2√2)/2Then y1 = x1 - p/2 = p(3 + 2√2)/2 - p/2 = p(2 + 2√2)/2 = p(1 + √2)Similarly, y2 = x2 - p/2 = p(3 - 2√2)/2 - p/2 = p(2 - 2√2)/2 = p(1 - √2)Therefore, points A and B are (p(3 + 2√2)/2, p(1 + √2)) and (p(3 - 2√2)/2, p(1 - √2))Compute |AB|:Difference in x-coordinates: [p(3 + 2√2)/2 - p(3 - 2√2)/2] = p[ (3 + 2√2 - 3 + 2√2)/2 ] = p[4√2 / 2] = 2√2 pDifference in y-coordinates: [p(1 + √2) - p(1 - √2)] = p[2√2]Therefore, distance AB is sqrt( (2√2 p)^2 + (2√2 p)^2 ) = sqrt(8p² + 8p²) = sqrt(16p²) = 4p. But in the problem, |AB| = 3p. So 4p ≠ 3p. So this slope gives AB = 4p, which is longer. So this isn't the case either.Therefore, to get |AB| = 3p, we need to find the appropriate slope m.Wait, maybe parametrize the line in terms of angle θ, find m = tanθ, then compute |AB| in terms of θ, set to 3p, solve for θ, then proceed. But this might be complex. Alternatively, perhaps use parametric equations for the parabola.Another approach: for a parabola, the focal length is p/2. The length of the focal chord (a chord passing through the focus) can be found in terms of the slope. The length of any focal chord of the parabola y² = 4ax is 4a(c + 1/c), where c is the slope of the chord. Wait, maybe not. Wait, in standard parabola y² = 4ax, the length of focal chord making an angle θ with the x-axis is 4a / sin²θ. Let me verify.Alternatively, parametrize the parabola. For parabola y² = 2px, a parametric equation is (x, y) = (pt²/2, pt). Wait, let's see: if we set x = (pt²)/2, then y² = 2p*(pt²/2) = p² t², so y = ±pt. So parametric coordinates can be written as ( (p t²)/2, p t ). So a parameter t corresponds to the point ( (p t²)/2, p t ). Now, the focus is at (p/2, 0). Let’s consider a focal chord passing through the focus. So, points A and B on the parabola such that the line AB passes through the focus (p/2, 0). Let’s use the parametric coordinates. Let’s let point A correspond to parameter t1, so coordinates ( (p t1²)/2, p t1 ). Similarly, point B corresponds to parameter t2, coordinates ( (p t2²)/2, p t2 ). The line passing through A and B must pass through the focus (p/2, 0).First, find the equation of line AB. The slope of AB is [p t2 - p t1]/[ (p t2²)/2 - (p t1²)/2 ] = [ p(t2 - t1) ] / [ p(t2² - t1²)/2 ] = [ (t2 - t1) ] / [ (t2² - t1²)/2 ] = 2 / (t1 + t2 )So the slope m = 2 / (t1 + t2 )The equation of the line can be written as:y - p t1 = [2/(t1 + t2)] (x - (p t1²)/2 )This line passes through (p/2, 0). Therefore, substituting x = p/2, y = 0:0 - p t1 = [2/(t1 + t2)] ( p/2 - (p t1²)/2 )Multiply both sides by (t1 + t2):- p t1 (t1 + t2) = 2 * [ p/2 - (p t1²)/2 ] = 2*( p(1 - t1²)/2 ) = p(1 - t1²)Thus:- p t1(t1 + t2) = p(1 - t1² )Divide both sides by p:- t1(t1 + t2) = 1 - t1²Expand left side:- t1² - t1 t2 = 1 - t1²Add t1² to both sides:- t1 t2 = 1Thus, t1 t2 = -1So for any focal chord of the parabola y² = 2px, the product of the parameters t1 and t2 is -1. Therefore, if t1 is the parameter for point A, then t2 = -1/t1.Therefore, points A and B are given by t1 and t2 = -1/t1. So coordinates:A: ( (p t1²)/2, p t1 )B: ( (p t2²)/2, p t2 ) = ( (p (1/t1²))/2, -p / t1 )Now, compute the distance AB.Coordinates of A: ( (p t1²)/2, p t1 )Coordinates of B: ( p/(2 t1²), -p / t1 )Difference in x-coordinates: (p t1²)/2 - p/(2 t1² ) = p/2 ( t1² - 1/t1² )Difference in y-coordinates: p t1 - (-p/t1 ) = p(t1 + 1/t1 )Therefore, distance AB:sqrt [ (p/2 ( t1² - 1/t1² ))^2 + (p(t1 + 1/t1 ))^2 ]Factor out p^2:p * sqrt [ (1/4 (t1² - 1/t1² ))^2 + (t1 + 1/t1 )^2 ]Let me compute the expression inside the sqrt:First term: (1/4 (t1² - 1/t1² ))^2 = (1/16)(t1^4 - 2 + 1/t1^4 )Second term: (t1 + 1/t1 )^2 = t1^2 + 2 + 1/t1^2Therefore, total inside sqrt:1/16(t1^4 - 2 + 1/t1^4 ) + t1^2 + 2 + 1/t1^2Let’s write t1^4 + 1/t1^4 as (t1^2 + 1/t1^2 )^2 - 2. So:t1^4 - 2 + 1/t1^4 = (t1^2 + 1/t1^2 )^2 - 4But perhaps expanding is better.Alternatively, let’s compute term by term:First term: 1/16 t1^4 - 1/8 + 1/(16 t1^4 )Second term: t1^2 + 2 + 1/t1^2So combining:1/16 t1^4 + 1/(16 t1^4 ) + t1^2 + 1/t1^2 - 1/8 + 2= 1/16(t1^4 + 1/t1^4 ) + (t1^2 + 1/t1^2 ) + (2 - 1/8)= 1/16(t1^4 + 1/t1^4 ) + (t1^2 + 1/t1^2 ) + 15/8Hmm, this seems complicated. Maybe substitute u = t1 + 1/t1. Let’s note that t1 + 1/t1 = u, so t1^2 + 1/t1^2 = u² - 2, and t1^4 + 1/t1^4 = (t1^2 + 1/t1^2 )² - 2 = (u² - 2)^2 - 2 = u^4 -4u² + 2.Therefore, the expression becomes:1/16(u^4 -4u² + 2) + (u² - 2) + 15/8= (1/16)u^4 - (1/4)u² + 1/8 + u² - 2 + 15/8= (1/16)u^4 + ( -1/4 + 1 )u² + (1/8 - 2 + 15/8 )Simplify coefficients:-1/4 +1 = 3/41/8 - 2 + 15/8 = (1 + 15)/8 - 2 = 16/8 - 2 = 2 - 2 = 0Therefore, expression simplifies to:(1/16)u^4 + (3/4)u²Factor out u²:u² [ (1/16)u² + 3/4 ]= u² [ (u² + 12)/16 ]Therefore, the total expression under sqrt is u² (u² + 12)/16So the distance AB is:p * sqrt[ u² (u² + 12)/16 ] = p * ( |u| / 4 ) * sqrt(u² + 12 )But u = t1 + 1/t1, and since t1 can be positive or negative, but squaring takes care of signs. However, since we have |u|, but since u = t1 + 1/t1, depending on t1, u can be positive or negative, but since we are squaring, maybe it's okay.But perhaps instead of dealing with u, let's note that AB is given to be 3p, so:AB = p * sqrt[ u² (u² + 12)/16 ] = 3pDivide both sides by p:sqrt[ u² (u² + 12)/16 ] = 3Square both sides:u² (u² + 12)/16 = 9Multiply both sides by 16:u² (u² + 12) = 144Let’s let v = u², then:v(v + 12) = 144v² + 12v - 144 = 0Solve for v:v = [-12 ± sqrt(144 + 576)] / 2 = [-12 ± sqrt(720)] / 2 = [-12 ± 12 sqrt(5)] / 2 = -6 ± 6 sqrt(5)But since v = u² ≥ 0, discard the negative root:v = -6 + 6 sqrt(5) = 6(sqrt(5) -1 )Thus, u² = 6(sqrt(5) -1 )But u = t1 + 1/t1, so u² = t1² + 2 + 1/t1² = 6(sqrt(5) -1 )Thus, t1² + 1/t1² = 6(sqrt(5) -1 ) - 2 = 6 sqrt(5) -6 -2 = 6 sqrt(5) -8Hmm, this seems messy. Maybe there is another way.Alternatively, maybe use coordinates.Given that t1 t2 = -1, and |AB| = 3p. From the parametric coordinates:Coordinates of A: ( (p t1²)/2, p t1 )Coordinates of B: ( p/(2 t1²), -p/t1 )Distance AB squared:[ ( (p t1²)/2 - p/(2 t1²) )^2 + ( p t1 + p/t1 )^2 ] = (3p)^2 = 9p²Compute:First term: (p/2 ( t1² - 1/t1² ))^2 = p²/4 ( t1² - 1/t1² )²Second term: ( p ( t1 + 1/t1 ) )^2 = p² ( t1 + 1/t1 )²Thus:p²/4 ( t1² - 1/t1² )² + p² ( t1 + 1/t1 )² = 9p²Divide both sides by p²:[ ( t1² - 1/t1² )² /4 + ( t1 + 1/t1 )² ] = 9Let’s set u = t1 + 1/t1. Then, note that t1² + 1/t1² = u² - 2. Also, t1² - 1/t1² = (t1 - 1/t1)(t1 + 1/t1 ) = (sqrt(u² - 4)) * u? Wait, not exactly. Let me compute t1² - 1/t1².(t1 - 1/t1)^2 = t1² - 2 + 1/t1² = (t1² + 1/t1² ) - 2 = (u² - 2) - 2 = u² - 4. Therefore, t1 - 1/t1 = ± sqrt(u² -4). Therefore, t1² - 1/t1² = (t1 - 1/t1)(t1 + 1/t1 ) = ± sqrt(u² -4) * u.But since t1 can be positive or negative, perhaps we need to take absolute value, but let's proceed.Therefore, t1² - 1/t1² = ± u sqrt(u² -4 )But given that we have (t1² - 1/t1² )², the square will eliminate the sign. Therefore:[ ( t1² - 1/t1² )² /4 + ( t1 + 1/t1 )² ] = [ ( u² (u² -4 ) ) /4 + u² ] = [ (u^4 -4u²)/4 + u² ] = (u^4 -4u² +4u²)/4 = u^4 /4Therefore, equation becomes:u^4 /4 =9 => u^4 =36 => u²=6 (since u² is non-negative) => u=±√6But u = t1 +1/t1, which can be positive or negative. Let's take positive u for simplicity, so u=√6. Then, t1 +1/t1=√6.Let’s solve for t1:t1 +1/t1=√6Multiply both sides by t1:t1² +1=√6 t1Thus:t1² -√6 t1 +1=0Solutions:t1 = [√6 ± sqrt(6 -4)] / 2 = [√6 ± sqrt(2)] /2Therefore, t1 = (√6 +√2)/2 or t1=(√6 -√2)/2Therefore, two possible values for t1. Let's choose t1=(√6 +√2)/2. Then t2= -1/t1= -2/(√6 +√2 ). Rationalize denominator:Multiply numerator and denominator by (√6 -√2 ):t2= -2(√6 -√2 ) / [ (√6 +√2)(√6 -√2 ) ] = -2(√6 -√2 ) / (6 -2 )= -2(√6 -√2 ) /4= -(√6 -√2 )/2= (-√6 +√2 )/2= (√2 -√6 )/2Therefore, points A and B have parameters t1=(√6 +√2)/2 and t2=(√2 -√6 )/2.Now, let's compute coordinates of A and B.Point A:x1 = (p t1²)/2Compute t1²:t1=(√6 +√2)/2, so t1²=(6 +2 + 2√12)/4=(8 +4√3)/4=2 +√3Thus, x1 = p*(2 +√3)/2 = p(1 + (√3)/2 )Wait, wait:Wait, t1 squared is ((√6 +√2)/2)^2 = (6 + 2 + 2√12)/4 = (8 + 4√3)/4 = 2 + √3.Yes, correct. Therefore, x1 = p*(2 +√3)/2= p(1 + √3/2 )Wait, no: x1=(p t1²)/2= p*(2 +√3)/2= p*(1 + (√3)/2 )Similarly, y1= p t1= p*(√6 +√2)/2Similarly, point B:x2=(p t2²)/2t2=(√2 -√6)/2, so t2²=(2 +6 -2√12)/4=(8 -4√3)/4=2 -√3Therefore, x2= p*(2 -√3)/2= p*(1 -√3/2 )y2= p t2= p*(√2 -√6)/2Therefore, coordinates:A: ( p(1 + √3/2 ), p( (√6 +√2)/2 ) )B: ( p(1 - √3/2 ), p( (√2 -√6)/2 ) )Now, compute projections A' and B' onto the directrix l: x=-p/2.The projection of a point (x,y) onto the vertical line x=-p/2 is the point (-p/2, y). Therefore:A'= (-p/2, y1 )= (-p/2, p( (√6 +√2 )/2 )B'= (-p/2, y2 )= (-p/2, p( (√2 -√6 )/2 )Now, quadrilateral AA'B'B has vertices:A: ( p(1 + √3/2 ), p( (√6 +√2 )/2 )A': (-p/2, p( (√6 +√2 )/2 )B': (-p/2, p( (√2 -√6 )/2 )B: ( p(1 - √3/2 ), p( (√2 -√6 )/2 )Now, need to compute the area of quadrilateral AA'B'B and the area of triangle FA'B'.First, let's compute the area of quadrilateral AA'B'B.Since AA' and BB' are horizontal lines (projections onto the directrix, which is vertical), the quadrilateral can be considered as a trapezoid with bases AA' and BB', but since the sides A'B' and AB are not parallel, it's not a trapezoid. Alternatively, we can use the shoelace formula to compute the area.List the coordinates of the quadrilateral in order:A: (x1, y1) = ( p(1 + √3/2 ), p( (√6 +√2 )/2 )A': (-p/2, y1 )B': (-p/2, y2 )B: (x2, y2 )Then back to A.Using shoelace formula:Area = 1/2 | (x1*y1 + (-p/2)*y1 + (-p/2)*y2 + x2*y2 ) - ( y1*(-p/2) + y1*(-p/2) + y2*x2 + y2*x1 ) | Wait, perhaps better to list all coordinates in order and apply the formula.Order of points: A, A', B', B.Coordinates:A: (x1, y1) = ( p(1 + (√3)/2 ), p(√6 +√2)/2 )A': (-p/2, y1 )B': (-p/2, y2 )B: (x2, y2 ) = ( p(1 - (√3)/2 ), p(√2 -√6)/2 )Shoelace formula:Area = 1/2 |x1*y1 + (-p/2)*y1 + (-p/2)*y2 + x2*y2 - [ y1*(-p/2) + y1*(-p/2) + y2*x2 + y2*x1 ] |Let me compute each term step by step.First, compute the terms in the first sum:1. x1*y1: [ p(1 + √3/2 ) ] * [ p(√6 +√2)/2 ] = p² (1 + √3/2 )(√6 +√2 ) /22. (-p/2)*y1: (-p/2) * [ p(√6 +√2 )/2 ] = -p² (√6 +√2 ) /43. (-p/2)*y2: (-p/2) * [ p(√2 -√6 )/2 ] = -p² (√2 -√6 ) /44. x2*y2: [ p(1 - √3/2 ) ] * [ p(√2 -√6 )/2 ] = p² (1 - √3/2 )(√2 -√6 ) /2Now, compute the terms in the second sum:1. y1*(-p/2): [ p(√6 +√2 )/2 ]*(-p/2 ) = -p² (√6 +√2 ) /42. y1*(-p/2): Same as above, another term: -p² (√6 +√2 ) /43. y2*x2: [ p(√2 -√6 )/2 ] * [ p(1 - √3/2 ) ] = p² (√2 -√6 )(1 - √3/2 ) /24. y2*x1: [ p(√2 -√6 )/2 ] * [ p(1 + √3/2 ) ] = p² (√2 -√6 )(1 + √3/2 ) /2Now, compute the first total sum (sum1):sum1 = term1 + term2 + term3 + term4= p² [ (1 + √3/2 )(√6 +√2 ) /2 - (√6 +√2 ) /4 - (√2 -√6 ) /4 + (1 - √3/2 )(√2 -√6 ) /2 ]Similarly, compute the second total sum (sum2):sum2 = term1 + term2 + term3 + term4= p² [ - (√6 +√2 ) /4 - (√6 +√2 ) /4 + (√2 -√6 )(1 - √3/2 ) /2 + (√2 -√6 )(1 + √3/2 ) /2 ]Simplify sum1 and sum2.First, let's handle sum1:sum1 = p² /2 [ (1 + √3/2 )(√6 +√2 ) ] - p²/4 [ (√6 +√2 ) + (√2 -√6 ) ] + p² /2 [ (1 - √3/2 )(√2 -√6 ) ]Notice that (√6 +√2 ) + (√2 -√6 ) = 2√2, so the middle term becomes:- p²/4 * 2√2 = - p² √2 /2Now, compute the first and third terms:First term:(1 + √3/2 )(√6 +√2 )Multiply out:1*(√6 +√2 ) + (√3/2)(√6 +√2 )= √6 + √2 + (√18 + √6 )/2= √6 + √2 + (3√2 + √6 )/2= (2√6 + 2√2 +3√2 + √6 )/2= (3√6 +5√2 )/2Third term:(1 - √3/2 )(√2 -√6 )Multiply out:1*(√2 -√6 ) - (√3/2)(√2 -√6 )= √2 - √6 - (√6 - √18 )/2= √2 - √6 - (√6 - 3√2 )/2= √2 - √6 - √6 /2 + 3√2 /2= (√2 + 3√2/2 ) + (-√6 - √6 /2 )= (5√2 /2 ) - (3√6 /2 )Thus, sum1 becomes:sum1 = p² /2 * (3√6 +5√2 )/2 - p² √2 /2 + p² /2 * (5√2 /2 - 3√6 /2 )= p² [ (3√6 +5√2 )/4 - √2 /2 + (5√2 -3√6 )/4 ]Combine the terms:(3√6 +5√2 )/4 + (5√2 -3√6 )/4 = (10√2 )/4 = (5√2)/2Then subtract √2 /2:= (5√2)/2 - √2 /2 = (4√2)/2 = 2√2Therefore, sum1 = p² * 2√2Now compute sum2:sum2 = p² [ - (√6 +√2 ) /4 - (√6 +√2 ) /4 + (√2 -√6 )(1 - √3/2 ) /2 + (√2 -√6 )(1 + √3/2 ) /2 ]Simplify:First two terms: - (√6 +√2 ) /4 - (√6 +√2 ) /4 = - (√6 +√2 ) /2Last two terms:[(√2 -√6 )(1 - √3/2 ) + (√2 -√6 )(1 + √3/2 ) ] /2Factor out (√2 -√6 ):(√2 -√6 ) [ (1 - √3/2 +1 + √3/2 ) ] /2= (√2 -√6 ) [ 2 ] /2= (√2 -√6 )Thus, sum2 becomes:- (√6 +√2 ) /2 + (√2 -√6 )= (-√6 -√2 + 2√2 -2√6 ) /2Wait, no. Wait, sum2 is:sum2 = p² [ - (√6 +√2 ) /2 + (√2 -√6 ) ]= p² [ - (√6 +√2 ) /2 + √2 -√6 ]Convert to common denominator:= p² [ - (√6 +√2 ) /2 + 2√2 /2 - 2√6 /2 ]= p² [ ( -√6 -√2 + 2√2 -2√6 ) /2 ]= p² [ ( ( -√6 -2√6 ) + ( -√2 +2√2 ) ) /2 ]= p² [ ( -3√6 + √2 ) /2 ]Therefore, sum2 = p² ( -3√6 + √2 ) /2Now, compute the area:Area = 1/2 | sum1 - sum2 | = 1/2 | p² *2√2 - p² ( -3√6 + √2 ) /2 | = (1/2) p² | 2√2 + (3√6 -√2 )/2 | Convert to common denominator:= (1/2) p² | (4√2 +3√6 -√2 ) /2 | = (1/2) p² | (3√6 +3√2 ) /2 | = (1/2) p² * (3(√6 +√2 )) /2 = (3 p² /4 )(√6 +√2 )Therefore, area of quadrilateral AA'B'B is (3 p² /4 )(√6 +√2 )Now, compute the area of triangle FA'B'.Points F, A', B':F: (p/2, 0 )A': (-p/2, y1 ) = (-p/2, p(√6 +√2 )/2 )B': (-p/2, y2 ) = (-p/2, p(√2 -√6 )/2 )So, triangle FA'B' has vertices at (p/2, 0 ), (-p/2, y1 ), (-p/2, y2 )The area can be computed using the formula for the area of a triangle with coordinates.Alternatively, since two points are on the vertical line x=-p/2 (A' and B'), and F is at (p/2, 0), the base of the triangle is the distance between A' and B', which is |y1 - y2 |, and the height is the horizontal distance from F to the line x=-p/2, which is |p/2 - (-p/2)| = p.Therefore, area of triangle FA'B' is (1/2)*base*height = (1/2)* |y1 - y2 | * pCompute |y1 - y2 |:y1 = p(√6 +√2 )/2y2 = p(√2 -√6 )/2y1 - y2 = p(√6 +√2 -√2 +√6 )/2 = p(2√6 )/2 = p√6Therefore, area = (1/2)* p√6 * p = (1/2)*p²√6Thus, area of triangle FA'B' is (p²√6 )/2Now, compute the probability: area of triangle FA'B' divided by area of quadrilateral AA'B'B.Probability = [ (p²√6 /2 ) ] / [ (3 p² /4 )(√6 +√2 ) ] = ( √6 /2 ) / ( (3 /4 )(√6 +√2 ) ) = ( √6 /2 ) * (4 /3 ) / (√6 +√2 ) = ( 2√6 /3 ) / (√6 +√2 )Simplify the expression:Multiply numerator and denominator by (√6 -√2 ) to rationalize the denominator:= (2√6 /3 )(√6 -√2 ) / [ (√6 +√2 )(√6 -√2 ) ] = (2√6 /3 )(√6 -√2 ) / (6 -2 ) = (2√6 /3 )(√6 -√2 ) /4 = (2√6 /3 )(√6 -√2 ) /4Simplify:= (√6 /6 )(√6 -√2 )Multiply out:= ( (√6 *√6 ) - √6*√2 ) /6 = (6 - √12 ) /6 = (6 - 2√3 ) /6 = (3 - √3 ) /3 = 1 - (√3)/3Rationalizing:= 1 - (√3)/3 = (3 -√3)/3Thus, the probability is (3 -√3)/3 = 1 - (√3)/3 ≈ 1 - 0.577 ≈ 0.423Therefore, the probability is (3 -√3)/3, which simplifies to 1 - √3/3, but perhaps the answer is expressed as (3 -√3)/3. To check.Alternatively, since (3 -√3)/3 = 1 - √3/3, both are correct, but let's rationalize.But let's see:(3 -√3)/3 = 3/3 - √3/3 = 1 - √3/3.But which form is preferable? The problem might expect rationalized form or the fraction with numerator. Let's compute (3 -√3)/3.But to check the calculation:We had:Probability = [ (√6 /2 ) *4 /3 ] / (√6 +√2 ) = (2√6 /3 ) / (√6 +√2 )Then multiply numerator and denominator by (√6 -√2 ):= (2√6 (√6 -√2 )) / [ 3*(6 -2) ] = (2√6 (√6 -√2 )) / 12 = (2√6 (√6 -√2 )) /12 = (√6 (√6 -√2 )) /6= (6 -√12 ) /6 = (6 - 2√3 ) /6 = (3 -√3 ) /3Yes, so it's (3 -√3 ) /3, which is approximately 0.4226, which is 1 - √3/3 ≈1 - 1.732/3≈1 -0.577≈0.423.So the probability is (3 -√3)/3, which can be written as boxed{dfrac{3 - sqrt{3}}{3}}.Final AnswerThe probability is boxed{dfrac{3 - sqrt{3}}{3}}.</think>Given the parabola ( y^2 = 2px ) with focus ( F ) at ( left( frac{p}{2}, 0 right) ) and directrix ( l ) at ( x = -frac{p}{2} ), a line through ( F ) intersects the parabola at points ( A ) and ( B ) such that ( |AB| = 3p ). The projections of ( A ) and ( B ) onto the directrix are ( A' ) and ( B' ), respectively. We need to find the probability that a randomly thrown point ( M ) into the quadrilateral ( AA'B'B ) falls within the triangle ( FA'B' ).1. Coordinates of Points: - The line through ( F ) intersects the parabola at points ( A ) and ( B ). Using parametric equations for the parabola and solving for the parameters, we find the coordinates of ( A ) and ( B ). - Projections ( A' ) and ( B' ) onto the directrix ( x = -frac{p}{2} ) have the same ( y )-coordinates as ( A ) and ( B ), respectively.2. Area of Quadrilateral ( AA'B'B ): - Using the shoelace formula, the area of quadrilateral ( AA'B'B ) is calculated to be ( frac{3p^2}{4} (sqrt{6} + sqrt{2}) ).3. Area of Triangle ( FA'B' ): - The area of triangle ( FA'B' ) is calculated using the base (distance between ( A' ) and ( B' )) and height (horizontal distance from ( F ) to the directrix), resulting in ( frac{p^2 sqrt{6}}{2} ).4. Probability Calculation: - The probability is the ratio of the area of triangle ( FA'B' ) to the area of quadrilateral ( AA'B'B ): [ text{Probability} = frac{frac{p^2 sqrt{6}}{2}}{frac{3p^2}{4} (sqrt{6} + sqrt{2})} = frac{2sqrt{6}}{3(sqrt{6} + sqrt{2})} ] - Rationalizing the denominator, we find the probability simplifies to ( frac{3 - sqrt{3}}{3} ).The final answer is boxed{dfrac{3 - sqrt{3}}{3}}.,<think>Okay, let me try to solve this problem step by step. First, I need to understand the problem statement clearly. So, we have a parabola defined by the equation ( y^2 = 2px ) where ( p > 0 ). The focus of this parabola is ( F ), and the directrix is the line ( l ). There's a line passing through the focus ( F ) that intersects the parabola at points ( A ) and ( B ), and the distance between ( A ) and ( B ) is ( 3p ). Then, we project points ( A ) and ( B ) onto the directrix ( l ), resulting in points ( A' ) and ( B' ), respectively. The question is about finding the probability that a randomly thrown point ( M ) into the quadrilateral ( AA'B'B ) will fall within the triangle ( FA'B' ). So, the probability would be the area of triangle ( FA'B' ) divided by the area of quadrilateral ( AA'B'B ).First, let me recall some properties of a parabola. For the standard parabola ( y^2 = 4ax ), the focus is at ( (a, 0) ) and the directrix is ( x = -a ). But in our case, the equation is ( y^2 = 2px ). Comparing this with ( y^2 = 4ax ), we can see that ( 4a = 2p ), so ( a = p/2 ). Therefore, the focus ( F ) is at ( (a, 0) = (p/2, 0) ), and the directrix ( l ) is the line ( x = -a = -p/2 ).So, focus ( F ) is at ( (p/2, 0) ), directrix is ( x = -p/2 ).Next, there's a line passing through ( F ) that intersects the parabola at points ( A ) and ( B ), with ( |AB| = 3p ). We need to find the projections of ( A ) and ( B ) onto the directrix, which are ( A' ) and ( B' ). Then, we need to compute the areas of quadrilateral ( AA'B'B ) and triangle ( FA'B' ), and find their ratio.To approach this, I think we should first find the coordinates of points ( A ) and ( B ). Since the line passes through ( F ), which is ( (p/2, 0) ), we can parametrize the line with some slope or angle. Let me consider parametric equations for the line.Let me denote the line passing through ( F ) as having a slope ( m ). Then, the equation of the line is ( y = m(x - p/2) ).This line intersects the parabola ( y^2 = 2px ). So, substituting ( y = m(x - p/2) ) into the parabola's equation:( [m(x - p/2)]^2 = 2px )Expanding the left side:( m^2(x - p/2)^2 = 2px )( m^2(x^2 - px + p^2/4) = 2px )Bring all terms to one side:( m^2x^2 - m^2p x + m^2p^2/4 - 2px = 0 )This is a quadratic in ( x ):( m^2x^2 - (m^2p + 2p)x + (m^2p^2)/4 = 0 )Let me factor out ( p ) where possible:( m^2x^2 - p(m^2 + 2)x + (m^2p^2)/4 = 0 )Let me denote this quadratic equation as ( ax^2 + bx + c = 0 ), where:- ( a = m^2 )- ( b = -p(m^2 + 2) )- ( c = m^2p^2 / 4 )The solutions for ( x ) will give the x-coordinates of points ( A ) and ( B ). Let's compute the roots using the quadratic formula:( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Plugging in the values:( x = frac{p(m^2 + 2) pm sqrt{[p(m^2 + 2)]^2 - 4 cdot m^2 cdot (m^2p^2 / 4)}}{2m^2} )Simplify the discriminant:First, compute the square term:( [p(m^2 + 2)]^2 = p^2(m^2 + 2)^2 )Then compute ( 4ac ):( 4 cdot m^2 cdot (m^2p^2 / 4) = m^4p^2 )Therefore, the discriminant becomes:( p^2(m^2 + 2)^2 - m^4p^2 = p^2[(m^2 + 2)^2 - m^4] )Expand ( (m^2 + 2)^2 ):( m^4 + 4m^2 + 4 )So, subtracting ( m^4 ):( m^4 + 4m^2 + 4 - m^4 = 4m^2 + 4 = 4(m^2 + 1) )Therefore, the discriminant is:( p^2 cdot 4(m^2 + 1) = 4p^2(m^2 + 1) )Thus, the roots are:( x = frac{p(m^2 + 2) pm sqrt{4p^2(m^2 + 1)}}{2m^2} )Simplify the square root:( sqrt{4p^2(m^2 + 1)} = 2psqrt{m^2 + 1} )Therefore, substituting back:( x = frac{p(m^2 + 2) pm 2psqrt{m^2 + 1}}{2m^2} )Factor out ( p ):( x = frac{p[(m^2 + 2) pm 2sqrt{m^2 + 1}]}{2m^2} )So, the two x-coordinates of points ( A ) and ( B ) are:( x_1 = frac{p[(m^2 + 2) + 2sqrt{m^2 + 1}]}{2m^2} )and( x_2 = frac{p[(m^2 + 2) - 2sqrt{m^2 + 1}]}{2m^2} )Corresponding to these x-coordinates, the y-coordinates can be found from the line equation ( y = m(x - p/2) ).So, for ( x_1 ):( y_1 = mleft( frac{p[(m^2 + 2) + 2sqrt{m^2 + 1}]}{2m^2} - frac{p}{2} right) )Similarly for ( x_2 ):( y_2 = mleft( frac{p[(m^2 + 2) - 2sqrt{m^2 + 1}]}{2m^2} - frac{p}{2} right) )Let me simplify ( y_1 ):First, compute the term inside the parentheses:( frac{p[(m^2 + 2) + 2sqrt{m^2 + 1}]}{2m^2} - frac{p}{2} )Factor out ( frac{p}{2} ):( frac{p}{2} left[ frac{(m^2 + 2) + 2sqrt{m^2 + 1}}{m^2} - 1 right] )Simplify the expression inside the brackets:( frac{(m^2 + 2) + 2sqrt{m^2 + 1} - m^2}{m^2} = frac{2 + 2sqrt{m^2 + 1}}{m^2} = frac{2(1 + sqrt{m^2 + 1})}{m^2} )Therefore, ( y_1 ) becomes:( m cdot frac{p}{2} cdot frac{2(1 + sqrt{m^2 + 1})}{m^2} = frac{p(1 + sqrt{m^2 + 1})}{m} )Similarly, for ( y_2 ):Following similar steps:( frac{p[(m^2 + 2) - 2sqrt{m^2 + 1}]}{2m^2} - frac{p}{2} )Factor out ( frac{p}{2} ):( frac{p}{2} left[ frac{(m^2 + 2) - 2sqrt{m^2 + 1}}{m^2} - 1 right] )Simplify:( frac{(m^2 + 2) - 2sqrt{m^2 + 1} - m^2}{m^2} = frac{2 - 2sqrt{m^2 + 1}}{m^2} = frac{2(1 - sqrt{m^2 + 1})}{m^2} )Thus, ( y_2 ):( m cdot frac{p}{2} cdot frac{2(1 - sqrt{m^2 + 1})}{m^2} = frac{p(1 - sqrt{m^2 + 1})}{m} )So now, we have coordinates for points ( A ) and ( B ):- ( A ): ( left( frac{p[(m^2 + 2) + 2sqrt{m^2 + 1}]}{2m^2}, frac{p(1 + sqrt{m^2 + 1})}{m} right) )- ( B ): ( left( frac{p[(m^2 + 2) - 2sqrt{m^2 + 1}]}{2m^2}, frac{p(1 - sqrt{m^2 + 1})}{m} right) )Now, we need to compute the distance between ( A ) and ( B ), which is given as ( |AB| = 3p ).So, let's compute the distance between these two points.Let me denote the coordinates of ( A ) as ( (x_A, y_A) ) and ( B ) as ( (x_B, y_B) ). Then,( |AB| = sqrt{(x_A - x_B)^2 + (y_A - y_B)^2} )First, compute ( x_A - x_B ):From the expressions above:( x_A - x_B = frac{p[(m^2 + 2) + 2sqrt{m^2 + 1}]}{2m^2} - frac{p[(m^2 + 2) - 2sqrt{m^2 + 1}]}{2m^2} )Simplify:The ( (m^2 + 2) ) terms cancel out, and we have:( x_A - x_B = frac{p[2sqrt{m^2 + 1} - (-2sqrt{m^2 + 1})]}{2m^2} = frac{p[4sqrt{m^2 + 1}]}{2m^2} = frac{2psqrt{m^2 + 1}}{m^2} )Similarly, compute ( y_A - y_B ):( y_A - y_B = frac{p(1 + sqrt{m^2 + 1})}{m} - frac{p(1 - sqrt{m^2 + 1})}{m} = frac{p[1 + sqrt{m^2 + 1} - 1 + sqrt{m^2 + 1}]}{m} = frac{2psqrt{m^2 + 1}}{m} )Therefore, the distance ( |AB| ):( sqrt{left( frac{2psqrt{m^2 + 1}}{m^2} right)^2 + left( frac{2psqrt{m^2 + 1}}{m} right)^2 } )Compute the squares:First term: ( left( frac{2psqrt{m^2 + 1}}{m^2} right)^2 = frac{4p^2(m^2 + 1)}{m^4} )Second term: ( left( frac{2psqrt{m^2 + 1}}{m} right)^2 = frac{4p^2(m^2 + 1)}{m^2} )Add them together:( frac{4p^2(m^2 + 1)}{m^4} + frac{4p^2(m^2 + 1)}{m^2} = frac{4p^2(m^2 + 1)}{m^4} + frac{4p^2(m^2 + 1)m^2}{m^4} )Factor out ( frac{4p^2(m^2 + 1)}{m^4} ):( frac{4p^2(m^2 + 1)}{m^4} (1 + m^2) = frac{4p^2(m^2 + 1)^2}{m^4} )Therefore, the distance is:( sqrt{ frac{4p^2(m^2 + 1)^2}{m^4} } = frac{2p(m^2 + 1)}{m^2} )According to the problem, this distance is equal to ( 3p ):( frac{2p(m^2 + 1)}{m^2} = 3p )Divide both sides by ( p ):( frac{2(m^2 + 1)}{m^2} = 3 )Multiply both sides by ( m^2 ):( 2(m^2 + 1) = 3m^2 )Simplify:( 2m^2 + 2 = 3m^2 )Subtract ( 2m^2 ):( 2 = m^2 )Therefore, ( m = sqrt{2} ) or ( m = -sqrt{2} ). Since the slope can be positive or negative, but the problem doesn't specify direction, we can take either one, as the configuration would be symmetric. Let's take ( m = sqrt{2} ) for simplicity.So, the slope ( m = sqrt{2} ). Now, we can substitute ( m = sqrt{2} ) back into the coordinates of ( A ) and ( B ).First, compute ( m^2 = (sqrt{2})^2 = 2 ). Then, ( sqrt{m^2 + 1} = sqrt{2 + 1} = sqrt{3} ).Thus, coordinates of ( A ):( x_A = frac{p[(2 + 2) + 2sqrt{3}]}{2 cdot 2} = frac{p[4 + 2sqrt{3}]}{4} = frac{p(2 + sqrt{3})}{2} )( y_A = frac{p(1 + sqrt{3})}{sqrt{2}} )Coordinates of ( B ):( x_B = frac{p[(2 + 2) - 2sqrt{3}]}{4} = frac{p[4 - 2sqrt{3}]}{4} = frac{p(2 - sqrt{3})}{2} )( y_B = frac{p(1 - sqrt{3})}{sqrt{2}} )Therefore, points ( A ) and ( B ) are:- ( A left( frac{p(2 + sqrt{3})}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( B left( frac{p(2 - sqrt{3})}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )Next, we need to find the projections of ( A ) and ( B ) onto the directrix ( l ), which is ( x = -p/2 ). The projection of a point onto a vertical line ( x = c ) is simply the point with the same y-coordinate and x-coordinate equal to ( c ). Therefore, points ( A' ) and ( B' ) will have coordinates:- ( A' left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( B' left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )Now, we need to compute the area of quadrilateral ( AA'B'B ) and the area of triangle ( FA'B' ).First, let's outline the coordinates of all four points:- ( A ): ( left( frac{p(2 + sqrt{3})}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( A' ): ( left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( B' ): ( left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )- ( B ): ( left( frac{p(2 - sqrt{3})}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )Quadrilateral ( AA'B'B ) is a trapezoid since both ( AA' ) and ( BB' ) are vertical lines (projections onto the directrix). However, let me verify the shape.Actually, since ( A' ) and ( B' ) are both on the directrix ( x = -p/2 ), the sides ( AA' ) and ( BB' ) are horizontal lines? Wait, no. Wait, the projection is vertical. Since the directrix is a vertical line, the projection of a point onto the directrix is along the horizontal line (since projection onto a vertical line is horizontal). Wait, actually, projection onto a vertical line is horizontal. So, to project a point ( (x, y) ) onto the vertical line ( x = -p/2 ), we move horizontally to that line, so the projection is ( (-p/2, y) ). Therefore, the projections ( A' ) and ( B' ) have the same y-coordinates as ( A ) and ( B ), respectively, but x-coordinate ( -p/2 ).Therefore, quadrilateral ( AA'B'B ) has vertices at:- ( A ): ( left( frac{p(2 + sqrt{3})}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( A' ): ( left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )- ( B' ): ( left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )- ( B ): ( left( frac{p(2 - sqrt{3})}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )So, connecting these points in order: ( A ) to ( A' ) is a horizontal line to the left; ( A' ) to ( B' ) is a vertical line downward; ( B' ) to ( B ) is a horizontal line to the right; ( B ) to ( A ) is a line connecting back. Hmm, actually, the quadrilateral is a trapezoid because both ( AA' ) and ( BB' ) are horizontal lines (left and right sides), but actually, no. Wait, since ( A ) and ( B ) are on the parabola, and ( A' ) and ( B' ) are on the directrix. So, the sides are ( AA' ), ( A'B' ), ( B'B ), and ( BA ). Wait, but ( A'B' ) is a vertical line segment on the directrix, and ( BA ) is the chord of the parabola. Wait, perhaps it's not a trapezoid but a quadrilateral with two vertical sides and two slant sides.Wait, perhaps plotting the coordinates would help, but since I can't draw, I need to compute the area using coordinates.The area of a quadrilateral can be found using the shoelace formula. Let's apply that.First, list the coordinates of the quadrilateral in order. Let's list them as ( A ), ( A' ), ( B' ), ( B ).So, in order:1. ( A left( frac{p(2 + sqrt{3})}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )2. ( A' left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) )3. ( B' left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )4. ( B left( frac{p(2 - sqrt{3})}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )To apply the shoelace formula, let's denote these points as ( (x_1, y_1) ), ( (x_2, y_2) ), ( (x_3, y_3) ), ( (x_4, y_4) ).Shoelace formula:Area = ( frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - y_1x_2 - y_2x_3 - y_3x_4 - y_4x_1| )Plugging in the coordinates:First, compute each term:1. ( x_1y_2 = frac{p(2 + sqrt{3})}{2} cdot frac{p(1 + sqrt{3})}{sqrt{2}} )2. ( x_2y_3 = -frac{p}{2} cdot frac{p(1 - sqrt{3})}{sqrt{2}} )3. ( x_3y_4 = -frac{p}{2} cdot frac{p(1 - sqrt{3})}{sqrt{2}} )4. ( x_4y_1 = frac{p(2 - sqrt{3})}{2} cdot frac{p(1 + sqrt{3})}{sqrt{2}} )Then subtract:5. ( y_1x_2 = frac{p(1 + sqrt{3})}{sqrt{2}} cdot -frac{p}{2} )6. ( y_2x_3 = frac{p(1 + sqrt{3})}{sqrt{2}} cdot -frac{p}{2} )7. ( y_3x_4 = frac{p(1 - sqrt{3})}{sqrt{2}} cdot frac{p(2 - sqrt{3})}{2} )8. ( y_4x_1 = frac{p(1 - sqrt{3})}{sqrt{2}} cdot frac{p(2 + sqrt{3})}{2} )This seems quite complicated, but let's compute each term step by step. Let's factor out ( frac{p^2}{sqrt{2}} ) from all terms to simplify.First, compute terms 1-4:1. ( x_1y_2 = frac{p(2 + sqrt{3})}{2} cdot frac{p(1 + sqrt{3})}{sqrt{2}} = frac{p^2}{sqrt{2}} cdot frac{(2 + sqrt{3})(1 + sqrt{3})}{2} )2. ( x_2y_3 = -frac{p}{2} cdot frac{p(1 - sqrt{3})}{sqrt{2}} = -frac{p^2}{sqrt{2}} cdot frac{(1 - sqrt{3})}{2} )3. ( x_3y_4 = -frac{p}{2} cdot frac{p(1 - sqrt{3})}{sqrt{2}} = -frac{p^2}{sqrt{2}} cdot frac{(1 - sqrt{3})}{2} )4. ( x_4y_1 = frac{p(2 - sqrt{3})}{2} cdot frac{p(1 + sqrt{3})}{sqrt{2}} = frac{p^2}{sqrt{2}} cdot frac{(2 - sqrt{3})(1 + sqrt{3})}{2} )Now, compute terms 5-8:5. ( y_1x_2 = frac{p(1 + sqrt{3})}{sqrt{2}} cdot -frac{p}{2} = -frac{p^2}{sqrt{2}} cdot frac{(1 + sqrt{3})}{2} )6. ( y_2x_3 = frac{p(1 + sqrt{3})}{sqrt{2}} cdot -frac{p}{2} = -frac{p^2}{sqrt{2}} cdot frac{(1 + sqrt{3})}{2} )7. ( y_3x_4 = frac{p(1 - sqrt{3})}{sqrt{2}} cdot frac{p(2 - sqrt{3})}{2} = frac{p^2}{sqrt{2}} cdot frac{(1 - sqrt{3})(2 - sqrt{3})}{2} )8. ( y_4x_1 = frac{p(1 - sqrt{3})}{sqrt{2}} cdot frac{p(2 + sqrt{3})}{2} = frac{p^2}{sqrt{2}} cdot frac{(1 - sqrt{3})(2 + sqrt{3})}{2} )Now, let's compute each bracket:First, the sum of terms 1-4:Term1: ( frac{(2 + sqrt{3})(1 + sqrt{3})}{2} )Multiply ( (2 + sqrt{3})(1 + sqrt{3}) ):= ( 2 cdot 1 + 2 cdot sqrt{3} + sqrt{3} cdot 1 + sqrt{3} cdot sqrt{3} )= ( 2 + 2sqrt{3} + sqrt{3} + 3 )= ( 5 + 3sqrt{3} )Therefore, Term1: ( frac{5 + 3sqrt{3}}{2} )Term2 and Term3:Both are ( -frac{(1 - sqrt{3})}{2} ), so adding them together:= ( -frac{(1 - sqrt{3})}{2} - frac{(1 - sqrt{3})}{2} = - (1 - sqrt{3}) )= ( -1 + sqrt{3} )Term4: ( frac{(2 - sqrt{3})(1 + sqrt{3})}{2} )Multiply ( (2 - sqrt{3})(1 + sqrt{3}) ):= ( 2 cdot 1 + 2 cdot sqrt{3} - sqrt{3} cdot 1 - sqrt{3} cdot sqrt{3} )= ( 2 + 2sqrt{3} - sqrt{3} - 3 )= ( (2 - 3) + (2sqrt{3} - sqrt{3}) )= ( -1 + sqrt{3} )Therefore, Term4: ( frac{-1 + sqrt{3}}{2} )Thus, sum of terms 1-4:= ( frac{5 + 3sqrt{3}}{2} + (-1 + sqrt{3}) + frac{-1 + sqrt{3}}{2} )First, combine the fractions:= ( frac{5 + 3sqrt{3} -1 + sqrt{3}}{2} + (-1 + sqrt{3}) )= ( frac{4 + 4sqrt{3}}{2} + (-1 + sqrt{3}) )= ( 2 + 2sqrt{3} -1 + sqrt{3} )= ( 1 + 3sqrt{3} )Now, the sum of terms 5-8:Term5 and Term6:Both are ( -frac{(1 + sqrt{3})}{2} ), so sum:= ( -frac{(1 + sqrt{3})}{2} - frac{(1 + sqrt{3})}{2} = - (1 + sqrt{3}) )Term7: ( frac{(1 - sqrt{3})(2 - sqrt{3})}{2} )Multiply ( (1 - sqrt{3})(2 - sqrt{3}) ):= ( 1 cdot 2 + 1 cdot (-sqrt{3}) - sqrt{3} cdot 2 + sqrt{3} cdot sqrt{3} )= ( 2 - sqrt{3} - 2sqrt{3} + 3 )= ( 5 - 3sqrt{3} )Therefore, Term7: ( frac{5 - 3sqrt{3}}{2} )Term8: ( frac{(1 - sqrt{3})(2 + sqrt{3})}{2} )Multiply ( (1 - sqrt{3})(2 + sqrt{3}) ):= ( 1 cdot 2 + 1 cdot sqrt{3} - sqrt{3} cdot 2 - sqrt{3} cdot sqrt{3} )= ( 2 + sqrt{3} - 2sqrt{3} - 3 )= ( -1 - sqrt{3} )Therefore, Term8: ( frac{-1 - sqrt{3}}{2} )Therefore, sum of terms 5-8:= ( - (1 + sqrt{3}) + frac{5 - 3sqrt{3}}{2} + frac{-1 - sqrt{3}}{2} )Combine fractions:= ( frac{5 - 3sqrt{3} -1 - sqrt{3}}{2} - (1 + sqrt{3}) )= ( frac{4 - 4sqrt{3}}{2} - 1 - sqrt{3} )= ( 2 - 2sqrt{3} -1 - sqrt{3} )= ( 1 - 3sqrt{3} )Therefore, the area is:( frac{1}{2} left| (1 + 3sqrt{3}) - (1 - 3sqrt{3}) right| cdot frac{p^2}{sqrt{2}} )Simplify inside the absolute value:( (1 + 3sqrt{3} -1 + 3sqrt{3}) = 6sqrt{3} )Therefore, area:( frac{1}{2} cdot 6sqrt{3} cdot frac{p^2}{sqrt{2}} = 3sqrt{3} cdot frac{p^2}{sqrt{2}} = frac{3sqrt{6} p^2}{2} )Wait, but this seems a bit complicated. Let me check if there's a mistake in the calculations. Let me recast the shoelace formula:Wait, actually, the shoelace formula for quadrilaterals is a bit tricky because the order of the points matters. Let me confirm the order.The quadrilateral is ( A ), ( A' ), ( B' ), ( B ). So when connecting these points, it should form a non-intersecting quadrilateral. Let's confirm the coordinates again.Alternatively, maybe it's simpler to compute the area by breaking the quadrilateral into two triangles or a rectangle and two triangles? Let's try that.Looking at the coordinates:- The vertical line from ( A' ) to ( B' ) is along the directrix ( x = -p/2 ), with y-coordinates from ( frac{p(1 + sqrt{3})}{sqrt{2}} ) to ( frac{p(1 - sqrt{3})}{sqrt{2}} ). The length of this segment is ( frac{p(1 + sqrt{3})}{sqrt{2}} - frac{p(1 - sqrt{3})}{sqrt{2}} = frac{p(2sqrt{3})}{sqrt{2}} = frac{2psqrt{3}}{sqrt{2}} = psqrt{6} ).Then, the horizontal distance from ( A ) to ( A' ) is the difference in x-coordinates:From ( A ) at ( frac{p(2 + sqrt{3})}{2} ) to ( A' ) at ( -p/2 ). The horizontal distance is ( frac{p(2 + sqrt{3})}{2} - (-frac{p}{2}) = frac{p(2 + sqrt{3} + 1)}{2} = frac{p(3 + sqrt{3})}{2} ). Wait, no. Wait, the horizontal distance is ( |x_A - x_{A'}| = left| frac{p(2 + sqrt{3})}{2} - (-frac{p}{2}) right| = frac{p(2 + sqrt{3} + 1)}{2} = frac{p(3 + sqrt{3})}{2} ). Wait, but actually:Wait, ( x_A = frac{p(2 + sqrt{3})}{2} ), ( x_{A'} = -p/2 ). So the horizontal distance is ( frac{p(2 + sqrt{3})}{2} - (-p/2) = frac{p(2 + sqrt{3} + 1)}{2} = frac{p(3 + sqrt{3})}{2} ). Hmm, but actually, since ( AA' ) is a horizontal line segment, the length is the horizontal difference. Similarly for ( BB' ).But perhaps another approach: the quadrilateral ( AA'B'B ) can be considered as a trapezoid with the two parallel sides being ( AA' ) and ( BB' ), and the height being the horizontal distance between the directrix and the parabola. Wait, but actually, ( AA' ) and ( BB' ) are horizontal lines, but since ( A ) and ( B ) are on the parabola and ( A' ), ( B' ) are on the directrix, which is vertical. Wait, no, ( AA' ) is horizontal, moving from ( A ) to the directrix. Similarly, ( BB' ) is horizontal. Therefore, the two horizontal sides are ( AA' ) and ( BB' ), and the other two sides are ( A'B' ) (vertical) and ( AB ) (the chord).But trapezoid requires two sides to be parallel. Here, ( AA' ) and ( BB' ) are both horizontal, hence parallel. The other two sides ( A'B' ) and ( AB ). ( A'B' ) is vertical, and ( AB ) is slant. So they are not parallel. Therefore, it's a trapezoid only if two sides are parallel. Since only ( AA' ) and ( BB' ) are parallel, then yes, it is a trapezoid. So, trapezoid with bases ( AA' ) and ( BB' ), and legs ( A'B' ) and ( AB ).The formula for the area of a trapezoid is ( frac{1}{2} times ( text{sum of the lengths of the two bases } ) times text{height} ). However, in this case, the height would be the distance between the two bases. Wait, but in a trapezoid with horizontal bases, the height is the vertical distance between them. However, here, the two bases ( AA' ) and ( BB' ) are horizontal but located at different y-coordinates. Wait, but no, actually, ( AA' ) and ( BB' ) are both horizontal line segments, but they are at different y-coordinates. Therefore, the height of the trapezoid would be the vertical distance between the lines ( y = y_A ) and ( y = y_B ).Wait, but since ( AA' ) is horizontal from ( A ) to ( A' ) at ( y = y_A ), and ( BB' ) is horizontal from ( B ) to ( B' ) at ( y = y_B ). The vertical distance between these two horizontal lines is ( |y_A - y_B| ).The lengths of the bases ( AA' ) and ( BB' ):- ( AA' ): horizontal distance from ( A ) to ( A' ), which is ( |x_A - x_{A'}| = frac{p(2 + sqrt{3})}{2} - (-p/2) = frac{p(2 + sqrt{3} + 1)}{2} = frac{p(3 + sqrt{3})}{2} ). Wait, hold on:Wait, ( x_A = frac{p(2 + sqrt{3})}{2} ), ( x_{A'} = -p/2 ). Therefore, ( |x_A - x_{A'}| = frac{p(2 + sqrt{3})}{2} + frac{p}{2} = frac{p(2 + sqrt{3} + 1)}{2} ). Wait, no:Wait, ( 2 + sqrt{3} + 1 = 3 + sqrt{3} ), so yes, the length is ( frac{p(3 + sqrt{3})}{2} ).Similarly, length of ( BB' ):( |x_B - x_{B'}| = frac{p(2 - sqrt{3})}{2} - (-p/2) = frac{p(2 - sqrt{3} + 1)}{2} = frac{p(3 - sqrt{3})}{2} ).Therefore, the lengths of the two bases are:- ( AA' ): ( frac{p(3 + sqrt{3})}{2} )- ( BB' ): ( frac{p(3 - sqrt{3})}{2} )The vertical distance between the two bases is ( |y_A - y_B| ). From earlier, ( y_A = frac{p(1 + sqrt{3})}{sqrt{2}} ), ( y_B = frac{p(1 - sqrt{3})}{sqrt{2}} ). Therefore, the vertical distance is:( |y_A - y_B| = left| frac{p(1 + sqrt{3})}{sqrt{2}} - frac{p(1 - sqrt{3})}{sqrt{2}} right| = frac{p}{sqrt{2}} | (1 + sqrt{3}) - (1 - sqrt{3}) | = frac{p}{sqrt{2}} |2sqrt{3}| = frac{2psqrt{3}}{sqrt{2}} = psqrt{6} ).Therefore, the height of the trapezoid is ( psqrt{6} ).Thus, the area of the trapezoid is:( frac{1}{2} times left( frac{p(3 + sqrt{3})}{2} + frac{p(3 - sqrt{3})}{2} right) times psqrt{6} )Simplify the sum inside the parentheses:( frac{p(3 + sqrt{3} + 3 - sqrt{3})}{2} = frac{p(6)}{2} = 3p )Therefore, the area is:( frac{1}{2} times 3p times psqrt{6} = frac{3p^2sqrt{6}}{2} )Which matches the result from the shoelace formula. So, the area of quadrilateral ( AA'B'B ) is ( frac{3sqrt{6}}{2} p^2 ).Now, moving on to compute the area of triangle ( FA'B' ).Point ( F ) is the focus at ( (p/2, 0) ). Points ( A' ) and ( B' ) are on the directrix ( x = -p/2 ), with coordinates ( (-p/2, y_A) ) and ( (-p/2, y_B) ). So, triangle ( FA'B' ) has vertices at ( (p/2, 0) ), ( (-p/2, y_A) ), and ( (-p/2, y_B) ).To compute the area of triangle ( FA'B' ), we can use the formula for the area of a triangle given three vertices. Let's denote the points as ( F left( frac{p}{2}, 0 right) ), ( A' left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) ), and ( B' left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) ).The area can be computed using the formula:( text{Area} = frac{1}{2} |x_F(y_{A'} - y_{B'}) + x_{A'}(y_{B'} - y_F) + x_{B'}(y_F - y_{A'})| )Plugging in the coordinates:( x_F = frac{p}{2} ), ( y_F = 0 )( x_{A'} = x_{B'} = -frac{p}{2} )( y_{A'} = frac{p(1 + sqrt{3})}{sqrt{2}} )( y_{B'} = frac{p(1 - sqrt{3})}{sqrt{2}} )Substitute into the formula:( text{Area} = frac{1}{2} | frac{p}{2} left( frac{p(1 + sqrt{3})}{sqrt{2}} - frac{p(1 - sqrt{3})}{sqrt{2}} right) + (-frac{p}{2}) left( frac{p(1 - sqrt{3})}{sqrt{2}} - 0 right) + (-frac{p}{2}) left( 0 - frac{p(1 + sqrt{3})}{sqrt{2}} right) | )Simplify term by term:First term:( frac{p}{2} cdot frac{p}{sqrt{2}} [ (1 + sqrt{3}) - (1 - sqrt{3}) ] = frac{p^2}{2sqrt{2}} (2sqrt{3}) = frac{p^2 sqrt{3}}{sqrt{2}} )Second term:( -frac{p}{2} cdot frac{p(1 - sqrt{3})}{sqrt{2}} = -frac{p^2(1 - sqrt{3})}{2sqrt{2}} )Third term:( -frac{p}{2} cdot left( - frac{p(1 + sqrt{3})}{sqrt{2}} right) = frac{p^2(1 + sqrt{3})}{2sqrt{2}} )Therefore, sum all terms:( frac{p^2 sqrt{3}}{sqrt{2}} - frac{p^2(1 - sqrt{3})}{2sqrt{2}} + frac{p^2(1 + sqrt{3})}{2sqrt{2}} )Combine the second and third terms:= ( frac{p^2 sqrt{3}}{sqrt{2}} + frac{p^2}{2sqrt{2}} [ - (1 - sqrt{3}) + (1 + sqrt{3}) ] )= ( frac{p^2 sqrt{3}}{sqrt{2}} + frac{p^2}{2sqrt{2}} [ -1 + sqrt{3} + 1 + sqrt{3} ] )= ( frac{p^2 sqrt{3}}{sqrt{2}} + frac{p^2}{2sqrt{2}} (2sqrt{3}) )= ( frac{p^2 sqrt{3}}{sqrt{2}} + frac{p^2 sqrt{3}}{sqrt{2}} )= ( frac{2p^2 sqrt{3}}{sqrt{2}} )Therefore, the area is:( frac{1}{2} cdot frac{2p^2 sqrt{3}}{sqrt{2}} = frac{p^2 sqrt{3}}{sqrt{2}} = frac{p^2 sqrt{6}}{2} )So, the area of triangle ( FA'B' ) is ( frac{sqrt{6}}{2} p^2 ).Now, the probability that a randomly thrown point ( M ) falls within triangle ( FA'B' ) is the ratio of the area of ( FA'B' ) to the area of quadrilateral ( AA'B'B ):Probability = ( frac{text{Area of } FA'B'}{text{Area of } AA'B'B} = frac{frac{sqrt{6}}{2} p^2}{frac{3sqrt{6}}{2} p^2} = frac{1}{3} )Wait, so the probability is ( frac{1}{3} ). But let me check again to make sure.The area of the quadrilateral is ( frac{3sqrt{6}}{2} p^2 ), and the area of the triangle is ( frac{sqrt{6}}{2} p^2 ). The ratio is ( frac{1}{3} ), which seems straightforward.Alternatively, is there a simpler way to see this without so much computation?Given that the focus is at ( (p/2, 0) ), and the directrix is at ( x = -p/2 ). The line through the focus intersects the parabola at ( A ) and ( B ), with ( AB = 3p ). Then, projecting ( A ) and ( B ) onto the directrix gives ( A' ) and ( B' ). The quadrilateral ( AA'B'B ) is a trapezoid with area we computed as ( frac{3sqrt{6}}{2} p^2 ), and the triangle ( FA'B' ) has area ( frac{sqrt{6}}{2} p^2 ). Therefore, the probability is ( 1/3 ).But perhaps there's a more elegant approach using properties of parabolas. For instance, the distance from the focus to the directrix is ( p/2 - (-p/2) = p ). But not sure if that helps here.Alternatively, note that projections of ( A ) and ( B ) onto the directrix are ( A' ) and ( B' ), so the triangle ( FA'B' ) is a triangle with base ( A'B' ) and height the horizontal distance from ( F ) to the directrix, which is ( p/2 - (-p/2) = p ). Wait, but actually, ( F ) is at ( (p/2, 0) ), and the directrix is at ( x = -p/2 ). The triangle ( FA'B' ) is formed by points ( F ), ( A' ), ( B' ). Since ( A' ) and ( B' ) are on the directrix, their x-coordinates are both ( -p/2 ). Therefore, the line segment ( A'B' ) is vertical, with length equal to the difference in y-coordinates of ( A' ) and ( B' ), which we computed earlier as ( psqrt{6} ).The distance from ( F ) to the line ( A'B' ) (which is vertical at ( x = -p/2 )) is the horizontal distance between ( F ) at ( (p/2, 0) ) and ( x = -p/2 ), which is ( p/2 - (-p/2) = p ).Therefore, the area of triangle ( FA'B' ) is ( frac{1}{2} times text{base} times text{height} = frac{1}{2} times psqrt{6} times p = frac{p^2 sqrt{6}}{2} ). Wait, but this contradicts our previous result. Wait, no:Wait, the base is ( A'B' ), which is vertical with length ( psqrt{6} ). The height is the horizontal distance from ( F ) to ( A'B' ), which is ( p ). However, in the formula for the area of a triangle, if the base is a vertical segment, the height is the horizontal distance from the third point to that vertical line. However, in reality, triangle ( FA'B' ) is a triangle with vertices at ( F ), ( A' ), ( B' ). Since ( A' ) and ( B' ) are on the directrix (vertical line ( x = -p/2 )), and ( F ) is at ( (p/2, 0) ), the triangle is formed by connecting ( F ) to ( A' ) and ( B' ).To compute the area, since ( A'B' ) is a vertical segment of length ( psqrt{6} ), and the horizontal distance from ( F ) to this line is ( p ), the area should indeed be ( frac{1}{2} times p times psqrt{6} = frac{p^2 sqrt{6}}{2} ). Which matches our previous computation.Similarly, the area of the trapezoid ( AA'B'B ) can be considered as the average of the two bases ( AA' ) and ( BB' ) multiplied by the height between them. Wait, but the two bases ( AA' ) and ( BB' ) are horizontal lines at different y-coordinates. The height between them is the vertical distance between ( y_A ) and ( y_B ), which is ( psqrt{6} ). Then, the lengths of the bases are:- ( AA' = frac{p(3 + sqrt{3})}{2} )- ( BB' = frac{p(3 - sqrt{3})}{2} )Therefore, the average of the bases is ( frac{ frac{p(3 + sqrt{3})}{2} + frac{p(3 - sqrt{3})}{2} }{2} = frac{3p}{2} ). Then, multiply by the height ( psqrt{6} ):Area = ( frac{3p}{2} times psqrt{6} = frac{3p^2 sqrt{6}}{2} ), which matches our previous result.So, the area of the quadrilateral is three times the area of the triangle. Therefore, the probability is ( 1/3 ).Hence, the answer should be ( boxed{dfrac{1}{3}} ).Final Answerboxed{dfrac{1}{3}}</think>Given the parabola ( y^2 = 2px ) with focus ( F ) at ( left( frac{p}{2}, 0 right) ) and directrix ( x = -frac{p}{2} ), a line through ( F ) intersects the parabola at points ( A ) and ( B ) such that ( |AB| = 3p ). The projections of ( A ) and ( B ) onto the directrix are ( A' ) and ( B' ), respectively. We need to find the probability that a point ( M ) randomly thrown into the quadrilateral ( AA'B'B ) falls within the triangle ( FA'B' ).1. Finding coordinates of ( A ) and ( B ): - The line through ( F ) with slope ( sqrt{2} ) intersects the parabola at ( A ) and ( B ). - Coordinates of ( A ): ( left( frac{p(2 + sqrt{3})}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) ) - Coordinates of ( B ): ( left( frac{p(2 - sqrt{3})}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )2. Projections ( A' ) and ( B' ): - ( A' left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) ) - ( B' left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) )3. Area of quadrilateral ( AA'B'B ): - Using the trapezoid area formula with bases ( AA' ) and ( BB' ): - Length of ( AA' ): ( frac{p(3 + sqrt{3})}{2} ) - Length of ( BB' ): ( frac{p(3 - sqrt{3})}{2} ) - Vertical distance (height) between ( AA' ) and ( BB' ): ( psqrt{6} ) - Area: ( frac{1}{2} times left( frac{p(3 + sqrt{3})}{2} + frac{p(3 - sqrt{3})}{2} right) times psqrt{6} = frac{3sqrt{6}}{2} p^2 )4. Area of triangle ( FA'B' ): - Vertices: ( F left( frac{p}{2}, 0 right) ), ( A' left( -frac{p}{2}, frac{p(1 + sqrt{3})}{sqrt{2}} right) ), ( B' left( -frac{p}{2}, frac{p(1 - sqrt{3})}{sqrt{2}} right) ) - Base ( A'B' ): vertical segment of length ( psqrt{6} ) - Height: horizontal distance from ( F ) to directrix ( p ) - Area: ( frac{1}{2} times psqrt{6} times p = frac{sqrt{6}}{2} p^2 )5. Probability calculation: - Probability = ( frac{text{Area of } triangle FA'B'}{text{Area of quadrilateral } AA'B'B} = frac{frac{sqrt{6}}{2} p^2}{frac{3sqrt{6}}{2} p^2} = frac{1}{3} )The final answer is (boxed{dfrac{1}{3}}).]

🔑:Answer frac{1}{3}.AnalysisThe required probability =frac{S_{triangle A^{prime} B^{prime} F}}{S_{text {trapezoid } A A^{prime} B^{prime} B}}=frac{left|A^{prime} B^{prime}right| cdot p}{left|A^{prime} B^{prime}right| cdotleft(left|A A^{prime}right|+left|B B^{prime}right|right)}=frac{left|A^{prime} B^{prime}right| cdot p}{left|A^{prime} B^{prime}right| cdot(|A F|+|B F|)}=frac{left|A^{prime} B^{prime}right| cdot p}{left|A^{prime} B^{prime}right| cdot 3 p}=frac{1}{3}

🧠:[<think>Okay, let's see. The problem is about estimating how long a jet would take to fly around the Earth at the equator. The radius of the Earth is given as approximately 4000 miles, and the jet's speed is 500 miles per hour. The options are from 8 to 100 hours, so I need to calculate the circumference of the Earth first and then divide that by the jet's speed to find the time. Hmm, let's start.First, the Earth's radius at the equator is 4000 miles. To find the circumference, which is the distance the jet has to fly, the formula is C = 2 * π * r. Plugging in 4000 for r, that would be C = 2 * π * 4000. Let me compute that. 2 times 4000 is 8000, so the circumference is 8000π miles. Now, π is approximately 3.1416, so multiplying 8000 by 3.1416. Let me do that step by step.First, 8000 * 3 = 24,000. Then, 8000 * 0.1416. Let's see, 8000 * 0.1 is 800, 8000 * 0.04 is 320, and 8000 * 0.0016 is 12.8. So adding those together: 800 + 320 is 1120, plus 12.8 is 1132.8. So total circumference is 24,000 + 1,132.8 = 25,132.8 miles. So approximately 25,133 miles.Wait, but maybe the problem expects me to use a simpler approximation of π. Sometimes, in estimation problems, they use π ≈ 3 to make the math easier. Let me check both possibilities. If I use π ≈ 3, then the circumference is 8000 * 3 = 24,000 miles. If I use π ≈ 22/7, which is about 3.1429, that would be closer to the 25,132 figure. Hmm, but maybe they expect us to use π ≈ 3.14? Let's see.But wait, 25,133 divided by 500 miles per hour would give the time. Let me calculate both scenarios. If it's 24,000 miles, then 24,000 / 500 = 48 hours. If it's 25,133 miles, then 25,133 / 500. Let's see, 25,000 / 500 = 50, and 133 / 500 is approximately 0.266, so total about 50.266 hours, so approximately 50 hours. The options given are 8, 25, 50, 75, 100. So 50 is one of the options, choice C.But wait, maybe I should check if there's a trick here. The problem says "relative to Earth." Wait, does that mean the jet is flying with or against the Earth's rotation? Wait, no, when they say relative to Earth, I think it just means that the speed is measured relative to the Earth's surface. So the jet's speed is 500 mph relative to the ground. Therefore, the calculation is straightforward: distance divided by speed. So circumference divided by 500. So if circumference is approximately 25,000 miles (since 2πr ≈ 25,000 when r=4000; because π*4000 is about 12,566, times 2 is 25,132), so 25,132 divided by 500 is indeed approximately 50.26 hours. So the closest answer is 50, which is option C.Wait, but wait another thought. Is the radius 4000 miles? Let me confirm. The radius of the Earth is about 3,959 miles on average, but the problem says approximately 4000 miles, so that's a rounded figure. So that's okay. Then the circumference is 2π*4000 ≈ 25,132 miles as I had. Divided by 500 mph gives 50.264 hours. So approximately 50 hours, which is answer choice C. So that's the answer.But let me check why the options include 25. Maybe someone could confuse radius with diameter. If someone thought the diameter is 4000, then the radius is 2000, circumference would be π*4000 ≈ 12,566, which divided by 500 is about 25.13 hours. But the problem says radius is 4000, so that's not the case. So perhaps someone might miscalculate. But the problem clearly states radius, so 4000 is the radius, so circumference is 8000π. So 25.132 thousand miles. Divided by 500 is 50.26 hours. So answer C. Therefore, the best estimate is 50 hours.Final Answerboxed{C}</think>The radius of the Earth at the equator is given as approximately 4000 miles. To find the circumference, we use the formula ( C = 2 pi r ). Plugging in the given radius:[C = 2 pi times 4000 = 8000 pi]Using the approximation (pi approx 3.1416), the circumference is:[8000 times 3.1416 approx 25132.8 text{ miles}]This value can be approximated to 25,133 miles. To find the time it takes for the jet to fly around the Earth, we divide the circumference by the jet's speed of 500 miles per hour:[frac{25133}{500} approx 50.266 text{ hours}]Rounding this to the nearest whole number gives approximately 50 hours. Therefore, the best estimate is:[boxed{C}],<think>Okay, so I need to figure out how long it would take a jet flying around the equator at 500 miles per hour to circle the Earth once. The Earth's radius at the equator is given as approximately 4000 miles. Let me start by recalling the formula for the circumference of a circle because the jet is flying around the Earth's equator, which is a circle. The formula for circumference is C = 2πr, where r is the radius. Given the radius is 4000 miles, I can plug that into the formula. Let me calculate that. So, 2 times π times 4000. Hmm, π is approximately 3.1416. Let me compute 2 * 4000 first, which is 8000. Then multiply by π. 8000 * 3.1416. Let me do that step by step. 8000 * 3 is 24,000. 8000 * 0.1416 is... 0.1 times 8000 is 800, so 0.1416 is 800 + (0.0416 * 8000). 0.0416 is approximately 1/24, so 8000 divided by 24 is approximately 333.33. So adding that to the previous 800, that's 800 + 333.33 = 1133.33. So total circumference would be 24,000 + 1,133.33 = 25,133.33 miles. Wait, but maybe I should use a better approximation for π. Let me check. If π is 3.1416, then 8000 * 3.1416 = 8000 * 3 + 8000 * 0.1416. 8000*3 is 24,000, 8000*0.1 is 800, 8000*0.04 is 320, 8000*0.0016 is 12.8. So adding those: 800 + 320 + 12.8 = 1132.8. So total circumference is 24,000 + 1,132.8 = 25,132.8 miles. Let me round that to 25,133 miles. Alternatively, maybe I remember that the equatorial circumference of the Earth is approximately 24,901 miles. Wait, but maybe the problem is using a simplified value? The problem states that the radius is approximately 4000 miles, so maybe they expect us to use that approximation. Let me confirm. If radius is 4000, then circumference is 2π*4000 = 8000π. If we approximate π as 3, then it's 24,000 miles, but that's a rough approximation. If we use π ≈ 22/7, which is about 3.142857, then 8000 * 22/7 = (8000/7)*22. 8000 divided by 7 is approximately 1142.857. Multiply by 22: 1142.857 * 20 = 22,857.14, 1142.857 * 2 = 2,285.714. Adding them gives 22,857.14 + 2,285.714 ≈ 25,142.85 miles. So approximately 25,143 miles. That's pretty close to the actual circumference. But maybe the problem is expecting a simpler calculation? Let me see the answer choices. The options are 8, 25, 50, 75, 100. These are all orders of magnitude different. So even if my circumference is 25,000 miles, which is a common approximation, then time = distance/speed = 25,000 / 500 = 50 hours. That would be answer choice C. But if I use the given radius of 4000 miles and compute circumference as 2π*4000 ≈ 25,133, then 25,133 / 500 ≈ 50.266 hours, which is still approximately 50 hours. So the answer should be C, 50. But let me check why one of the options is 25. Maybe someone might think, well, radius is 4000, so diameter is 8000, and circumference is π*diameter, so π*8000 ≈ 25,133. But maybe someone divides 4000 by 500? 4000 / 500 = 8. But that's radius, not circumference. So if someone forgot to multiply by 2π and just did 4000 / 500, they'd get 8, which is option A. But that's incorrect. Alternatively, maybe someone could think that the distance is 2π*4000, which is roughly 25,000, then divide by 500, getting 50. So 50 is the correct answer. Let me confirm again. So if radius is 4000, circumference is 2πr = 2*π*4000 ≈ 2*3.1416*4000 ≈ 25,133. Divide by speed 500 mph: 25,133 / 500 ≈ 50.266, so approximately 50 hours. Therefore, the answer should be C. But wait, the answer choice B is 25. Maybe someone miscalculates. Let me see. If you take circumference as π*diameter, which is correct. So diameter is 8000 miles, circumference is π*8000. If you mistakenly use radius 4000 as diameter, then circumference would be π*4000 ≈ 12,566, which divided by 500 is about 25.12, so 25 hours. But that's wrong because radius is 4000, not diameter. So if someone confuses radius with diameter, they might get 25. But the problem clearly states radius. So the correct answer is C, 50. Let me check the problem again. "Radius of Earth at the equator is approximately 4000 miles." So yes, radius. So circumference is 2π*4000. So 50 hours. Therefore, answer is C.Final Answerboxed{C}</think>To determine how long it would take a jet flying around the equator at 500 miles per hour to circle the Earth once, we start by calculating the circumference of the Earth using the given radius of 4000 miles. The formula for the circumference of a circle is ( C = 2pi r ).Given the radius ( r = 4000 ) miles, we calculate the circumference:[C = 2 pi times 4000 approx 2 times 3.1416 times 4000 approx 25,133 text{ miles}]Next, we need to find the time it takes to travel this distance at a speed of 500 miles per hour. The time ( t ) is given by the formula:[t = frac{text{distance}}{text{speed}} = frac{25,133}{500} approx 50.266 text{ hours}]Approximating this value, we get around 50 hours. Therefore, the best estimate of the number of hours of flight is:[boxed{C}]]

🔑:We want the number of hours that it takes the jet to fly the length of the circumference. frac{8000pi}{500}=16pi. The best estimate of that is 50Rightarrow mathrm{(C)}