💡First, I need to determine the speed of the stream that affects the boat's downstream travel.The boat's speed in still water is 30 km/hr. When moving downstream, the stream's speed adds to the boat's speed.The boat travels 70 km downstream in 2 hours. Using the formula:[text{Distance} = text{Speed} times text{Time}]I can set up the equation:[70 = (30 + v) times 2]Solving for ( v ):[70 = 60 + 2v][10 = 2v][v = 5]Therefore, the speed of the stream is 5 km/hr.

💡Okay, so I have this problem here with parametric equations of a line, and I need to find the angle of inclination. The parametric equations are given as:x = 1 + 3ty = 2 - √3 tAlright, so t is the parameter. I remember that parametric equations express x and y in terms of a third variable, which is t in this case. To find the angle of inclination, I think I need to find the slope of the line first because the angle is related to the slope.Let me recall, the angle of inclination is the angle that the line makes with the positive direction of the x-axis, measured counterclockwise. So, it's like the angle between the line and the x-axis. And the slope is the tangent of that angle. So, if I can find the slope, I can find the angle using the arctangent function.But how do I find the slope from these parametric equations? Hmm. I think I need to eliminate the parameter t to get the equation of the line in slope-intercept form, which is y = mx + b, where m is the slope.So, let's try to eliminate t. From the first equation, x = 1 + 3t, I can solve for t:x - 1 = 3tSo, t = (x - 1)/3Now, plug this expression for t into the second equation, which is y = 2 - √3 t:y = 2 - √3 * [(x - 1)/3]Let me simplify that:y = 2 - (√3 / 3)(x - 1)Distribute the (√3 / 3):y = 2 - (√3 / 3)x + (√3 / 3)Combine the constant terms:2 + (√3 / 3) is just a constant, so let's write that as:y = (-√3 / 3)x + [2 + (√3 / 3)]So, now the equation is in the form y = mx + b, where m is the slope. Therefore, the slope m is -√3 / 3.Alright, so the slope is negative, which means the line is decreasing from left to right. The angle of inclination is measured from the positive x-axis to the line, going counterclockwise. Since the slope is negative, the angle should be between 90° and 180°, right? Because a positive slope would be between 0° and 90°, and a negative slope would be in the second quadrant.Now, to find the angle α such that tan(α) = m, which is tan(α) = -√3 / 3. Hmm, I remember that tan(30°) is √3 / 3, but here it's negative. So, the reference angle would be 30°, but since the slope is negative, the angle is in the second quadrant.So, the angle would be 180° - 30° = 150°. Let me verify that. If I take tan(150°), what do I get?Well, tan(150°) is tan(180° - 30°) = -tan(30°) = -√3 / 3. Yes, that matches the slope. So, the angle of inclination is 150°.Wait, but let me think again. Sometimes, people define the angle of inclination as the smallest angle between the line and the x-axis, which would be acute or obtuse. Since the slope is negative, it's an obtuse angle, so 150° makes sense.Alternatively, if I think about the direction of the line, since the slope is negative, it's going downwards from left to right, so the angle is measured above the negative x-axis, which would be 150° from the positive x-axis.Let me also recall that the angle of inclination is always between 0° and 180°, so 150° is within that range.Just to make sure, let's recap:1. Start with parametric equations.2. Eliminate the parameter t to get y in terms of x.3. Find the slope, which is -√3 / 3.4. Recognize that tan(α) = -√3 / 3, so the reference angle is 30°.5. Since the slope is negative, the angle is in the second quadrant, so 180° - 30° = 150°.6. Confirm that tan(150°) = -√3 / 3.Yes, that all checks out. So, the angle of inclination is 150°, which corresponds to option D.**Final Answer**boxed{D}

💡Okay, so I have this problem about placing five distinct pawns on a 5x5 chessboard. The conditions are that each row and each column must have exactly one pawn, and no two pawns can attack each other like queens do in chess. Hmm, okay, let's break this down.First, I know that placing one pawn in each row and each column is similar to arranging queens on a chessboard so that none attack each other. That's the n-queens problem, right? For a 5x5 board, I think the number of solutions is known, but I'm not exactly sure what it is. Maybe I can figure it out.So, if I think about the first row, I have 5 choices for where to place the pawn. Once I place a pawn in the first row, say in column 1, then in the second row, I can't place a pawn in column 1, and also can't place it in columns that are diagonally adjacent to the first pawn. That would be columns 2 and 5, right? Wait, no, diagonally adjacent from column 1 in the first row would be columns 2 in the second row. So, actually, in the second row, I can't place a pawn in column 1 or 2. So that leaves columns 3, 4, and 5. So, 3 choices for the second row.Okay, moving on to the third row. Now, I have to avoid the columns already used in the first and second rows, and also avoid the diagonals from both pawns. This is getting a bit complicated. Maybe I should try to visualize it or draw it out.Alternatively, I remember that for the n-queens problem, the number of solutions increases factorially, but I think for n=5, it's a specific number. I think it's 10 solutions. Let me check that. If I recall correctly, the number of solutions for 5-queens is indeed 10. So, there are 10 ways to place the pawns so that they don't attack each other.But wait, the pawns are distinct. That means that not only do I need to count the number of ways to place them on the board without attacking each other, but also the number of ways to assign the distinct pawns to those positions. So, if there are 10 ways to place the pawns, and for each of those ways, I can arrange the 5 distinct pawns in 5! ways, which is 120.So, the total number of ways should be 10 multiplied by 120, which is 1200. That seems right. But let me make sure I'm not missing anything.Is there any overlap or double-counting? I don't think so because each solution to the n-queens problem is unique in terms of the positions, and since the pawns are distinct, each permutation of pawns in those positions is also unique. So, multiplying the number of arrangements by the number of permutations should give the correct total.Yeah, I think that's it. So, the final answer should be 1200.

💡Okay, so I need to figure out how many ways to color this hexagon figure. Let me try to visualize it first. There's an initial hexagon labeled R, which is red. Then, it's part of a grid that's arranged in a 5x3 layout, meaning there are 5 rows and 3 columns of hexagons. Each hexagon can be colored red, yellow, or green, but no two adjacent hexagons can share the same color. First, I should probably sketch out the structure or at least imagine it. The Asymptote code draws a bunch of hexagons, so I can assume it's a honeycomb-like structure. Each hexagon is connected to its neighbors on the sides, so each one has up to six neighbors, but in a grid, maybe each has four neighbors? Hmm, not sure, but the key is that adjacent hexagons can't have the same color.Starting with the initial red hexagon, which is fixed. Let's call this the first column. Then, a new column is added to the right, making it a 5x3 grid. So, the first column has one hexagon, the second column has two hexagons, and the third column has two hexagons? Wait, no, 5x3 grid—maybe it's five rows and three columns, so each column has five hexagons? Hmm, the Asymptote code draws multiple hexagons, but it's a bit hard to tell without seeing the image.Wait, the Asymptote code seems to draw a grid where each column has three hexagons, but arranged in a way that each subsequent column is offset. So, maybe it's a hexagonal grid where each column is offset by half a hexagon, creating a sort of honeycomb pattern. So, each hexagon in a column is adjacent to two hexagons in the next column.But the problem says it's arranged in a 5x3 grid. So, 5 rows and 3 columns. Each column has 5 hexagons? Or is it 5 columns and 3 rows? Wait, the problem says a new column is added to the right of the existing configuration, making the total layout arranged in a 5x3 grid. So, the original configuration was probably 5x2, and adding a column makes it 5x3.But the Asymptote code seems to draw more than 3 columns. Maybe I'm overcomplicating. Let me focus on the problem statement.We have a hexagon colored red. Each hexagon is colored red, yellow, or green, with no two adjacent hexagons sharing the same color. A new column is added to the right, making it a 5x3 grid. So, the initial configuration was 5x2, and adding a column makes it 5x3. So, each column has 5 hexagons, arranged vertically.Wait, but hexagons tiling in a grid usually have each column offset. So, each hexagon in a column is adjacent to two in the next column. So, in a 5x3 grid, each column has 5 hexagons, and each hexagon in a column is connected to two in the next column.But the initial hexagon is colored red. So, the first column has one hexagon, red. Then, the second column has two hexagons adjacent to it, which must be colored yellow and green. Then, the third column has two hexagons adjacent to each of those, so their colors are determined by the second column.Wait, maybe I should approach this step by step.1. Start with the first column: one hexagon, colored red.2. The second column has two hexagons adjacent to the first one. Since they can't be red, they must be yellow and green. There are two possibilities: top is yellow, bottom is green; or top is green, bottom is yellow.3. Now, moving to the third column. Each hexagon in the third column is adjacent to two hexagons in the second column. So, for each hexagon in the third column, its color is determined by the colors of the two hexagons it's adjacent to in the second column.Wait, but in a hexagonal grid, each hexagon in the third column is adjacent to two hexagons in the second column. So, for each hexagon in the third column, it can't be the same color as either of its two neighbors in the second column.But in the second column, we have two colors: yellow and green. So, if a hexagon in the third column is adjacent to a yellow and a green, it can't be yellow or green, so it has to be red.Wait, but red is already used in the first column. Is that allowed? Because the third column is two columns away from the first, so they don't share a side. So, yes, red is allowed.So, the middle hexagon in the third column must be red. Then, the top and bottom hexagons in the third column are adjacent to the top and bottom hexagons in the second column. So, if the second column's top is yellow, the third column's top can't be yellow. It can be red or green. But wait, the third column's top is also adjacent to the middle hexagon in the third column, which is red. So, it can't be red either. Therefore, it has to be green. Similarly, the bottom hexagon in the third column is adjacent to the bottom hexagon in the second column, which is green, so it can't be green. It's also adjacent to the middle red, so it can't be red. Therefore, it has to be yellow.Wait, so regardless of whether the second column was yellow on top and green on bottom or vice versa, the third column ends up with red in the middle, green on top, and yellow on bottom, or red in the middle, yellow on top, and green on bottom. So, the third column's colors are determined by the second column.But hold on, the third column is the new column added to the right, making it a 5x3 grid. So, does that mean we have three columns, each with five hexagons? Or is the third column just one hexagon? Hmm, the problem says a new column is added to the right of the existing configuration, making the total layout arranged in a 5x3 grid. So, the existing configuration was 5x2, and adding a column makes it 5x3.So, each column has five hexagons. So, starting from the first column, which has one hexagon, red. Then, the second column has two hexagons, adjacent to the first. Then, the third column has two hexagons, adjacent to the second. Wait, but 5x3 would mean 5 rows and 3 columns, each column having 5 hexagons. So, maybe each column is a vertical stack of five hexagons.But the Asymptote code seems to draw a more complex structure. Maybe it's a 5x3 grid in terms of hexagons, meaning 5 columns and 3 rows? Hmm, I'm confused.Wait, the Asymptote code draws a path for a hexagon and then shifts it multiple times. It seems like it's drawing a grid where each column is offset, creating a honeycomb pattern. So, each column has three hexagons, and there are five columns? Or maybe three columns and five rows? It's hard to tell without seeing the image.But the problem says a new column is added to the right, making it a 5x3 grid. So, the original was 5x2, adding a column makes it 5x3. So, 5 rows and 3 columns.Therefore, each column has five hexagons. So, the first column has five hexagons, the second column has five, and the third column has five. But the initial hexagon is colored red. So, is the first column only one hexagon, or is it five? Hmm.Wait, the Asymptote code draws a single hexagon labeled R, then adds columns to the right. So, maybe the initial configuration is a single hexagon, and adding a column to the right makes it a 5x3 grid? That doesn't quite make sense.Alternatively, maybe the initial configuration is a 5x2 grid, and adding a column makes it 5x3. So, each column has five hexagons.But regardless, the key is that the first hexagon is red, and each adjacent hexagon must be a different color. So, starting from red, the adjacent hexagons must be yellow or green. Then, their neighbors must be different from them, and so on.Given that, maybe the coloring propagates in a pattern. Since it's a grid, the coloring might follow a repeating pattern every two columns, so adding another column doesn't increase the number of possibilities beyond the initial choices.Wait, in the initial step, the first column has one hexagon, red. The second column has two hexagons, which can be yellow and green in two different orders. Then, the third column's colors are determined by the second column. So, for each choice in the second column, the third column's colors are fixed.Therefore, the total number of colorings is 2.But wait, let me think again. If the third column is determined by the second, and the second has two possibilities, then the total is 2. But if the grid is larger, maybe the number increases.Wait, the problem says a 5x3 grid. So, 5 rows and 3 columns. Each column has five hexagons. So, starting from the first column, which is all red? Or just one red hexagon?Wait, the initial hexagon is red, and it's part of the first column. So, maybe the first column has five hexagons, with the middle one red. Then, the second column has five hexagons, each adjacent to the first column's hexagons.But this is getting complicated. Maybe I should think in terms of graph coloring. Each hexagon is a vertex, adjacent to its neighbors. The graph is a hexagonal grid, which is a bipartite graph? Wait, no, hexagonal grids are tripartite because each hexagon has six neighbors, which can be colored with three colors.Wait, actually, hexagonal grids can be colored with two colors in a checkerboard pattern, but since each hexagon has six neighbors, which is even, it's bipartite. Wait, no, bipartite graphs can only have two colors, but hexagonal grids require three colors because each hexagon is adjacent to six others, which can't all be colored with two colors without conflict.Wait, actually, no. A hexagonal grid is a bipartite graph because it's a planar graph with no odd-length cycles. Wait, no, hexagonal grids have cycles of length six, which are even, so they are bipartite. Therefore, they can be colored with two colors. But the problem allows three colors, so maybe more possibilities.But in this problem, the initial hexagon is fixed as red. So, the coloring is constrained by that.Wait, maybe it's better to model this as a recurrence relation. Each column's coloring depends on the previous column. Since each hexagon in a column is adjacent to two hexagons in the previous column, the color of each hexagon in the current column is determined by the colors of the two hexagons it's adjacent to.But since we have three colors, the number of possibilities might multiply as we go along.Wait, let me think step by step.1. First column: one hexagon, red.2. Second column: two hexagons adjacent to the first. They can't be red, so each can be yellow or green. But they also can't be the same color as each other because they are adjacent? Wait, are the two hexagons in the second column adjacent to each other?In a hexagonal grid, each column is offset, so the two hexagons in the second column are not adjacent to each other. They are only adjacent to the first column's hexagon. So, they can be independently colored yellow or green, but since they are adjacent to the same red hexagon, they just need to be different from red. So, each can be yellow or green, but they don't need to be different from each other.Wait, but in a hexagonal grid, are the two hexagons in the second column adjacent? If the columns are offset, then the top hexagon in the second column is adjacent to the bottom hexagon in the first column, but in this case, the first column only has one hexagon. So, maybe the two hexagons in the second column are not adjacent to each other.Therefore, each can be independently yellow or green. So, for the second column, each hexagon has two choices, so total possibilities are 2 * 2 = 4.But wait, the problem says no two adjacent hexagons can have the same color. So, if the two hexagons in the second column are not adjacent, then they can be the same color. So, yes, 4 possibilities.But hold on, in the Asymptote code, it's drawing a more complex structure. Maybe the second column has three hexagons? Hmm, I'm not sure.Wait, the Asymptote code draws a hexagon, then shifts it multiple times. It seems like it's drawing a grid where each column has three hexagons, but arranged in a way that each subsequent column is offset. So, maybe each column has three hexagons, and there are five columns? Or three columns and five rows?I think I need to clarify the structure. The problem says a 5x3 grid, so 5 rows and 3 columns. Each column has five hexagons. So, starting from the first column, which has five hexagons, with the middle one colored red. Then, the second column has five hexagons, each adjacent to the first column's hexagons. Then, the third column has five hexagons, each adjacent to the second column's hexagons.But the initial hexagon is colored red. So, maybe the first column has one hexagon, red, and the rest are added as columns to the right, making it 5x3.Wait, this is getting too confusing. Maybe I should look for patterns or use graph theory.In a hexagonal grid, each hexagon is part of a honeycomb lattice, which is a bipartite graph. So, it can be divided into two sets where no two adjacent hexagons are in the same set. So, if we fix one hexagon as red, the adjacent ones must be from the other set, which can be colored yellow or green.But since we have three colors, maybe the number of colorings is related to the number of proper colorings with three colors for a hexagonal grid.However, the problem specifies that each hexagon is colored either red, yellow, or green, with no two adjacent the same. The initial hexagon is red, and we're adding a column to make it a 5x3 grid.Wait, maybe it's simpler. Since the initial hexagon is red, the adjacent ones must be yellow or green. Then, their neighbors must be different, and so on. Since it's a grid, the coloring propagates in a pattern.If I think of the first column as red, the second column must alternate between yellow and green. Then, the third column must alternate between red and the opposite color of the second column.Wait, let's try to visualize it.First column: RSecond column: Y, G (top to bottom)Third column: R, Y, G (but wait, each hexagon in the third column is adjacent to two in the second column. So, the top hexagon in the third column is adjacent to Y and something else? Wait, no, in a hexagonal grid, each hexagon in the third column is adjacent to two in the second column.Wait, maybe each hexagon in the third column is adjacent to one above and one below in the second column.So, if the second column is Y on top and G on bottom, then the third column's top hexagon is adjacent to Y and something else. Wait, no, in a hexagonal grid, each hexagon in column three is adjacent to two in column two.Wait, maybe it's better to think of it as a brick wall pattern, where each hexagon in column three is adjacent to one above and one below in column two.So, if column two has Y on top and G on bottom, then the top hexagon in column three is adjacent to Y and something else. Wait, but column two only has two hexagons, so the top hexagon in column three is adjacent to the top hexagon in column two (Y) and the middle hexagon in column two? Wait, no, column two has two hexagons, so the top hexagon in column three is adjacent to the top hexagon in column two and the middle hexagon in column two? But column two only has two hexagons, so maybe the top hexagon in column three is only adjacent to the top hexagon in column two.Wait, I'm getting confused. Maybe I should think of it as a grid where each column has three hexagons, arranged vertically, and each hexagon is adjacent to the hexagons in the previous column.So, column one: RColumn two: two hexagons, each adjacent to R. So, they can be Y and G in two different orders.Column three: each hexagon is adjacent to two hexagons in column two. So, for each hexagon in column three, it can't be the same as either of its two neighbors in column two.So, if column two is Y on top and G on bottom, then the top hexagon in column three is adjacent to Y and something else? Wait, no, in a hexagonal grid, each hexagon in column three is adjacent to one in column two above and one below.Wait, maybe each hexagon in column three is adjacent to two in column two: one directly to the left and one diagonally left.So, if column two has Y on top and G on bottom, then the top hexagon in column three is adjacent to Y (directly left) and something else. Wait, but column two only has two hexagons, so maybe the top hexagon in column three is only adjacent to Y.Wait, this is too confusing without seeing the image. Maybe I should think in terms of the number of colorings.If the first column has one hexagon, R. The second column has two hexagons, each can be Y or G, but they can't be R. So, 2 choices for each, but since they are not adjacent to each other, they can be the same or different. Wait, but in a hexagonal grid, are they adjacent? If the columns are offset, then the two hexagons in the second column are not adjacent to each other, so they can be independently colored Y or G. So, 2 * 2 = 4 possibilities.But wait, the problem says no two adjacent hexagons can have the same color. So, if the two hexagons in the second column are not adjacent, they can be the same color. So, 4 possibilities.Then, moving to the third column, each hexagon is adjacent to two in the second column. So, for each hexagon in the third column, it can't be the same as either of its two neighbors in the second column.So, for each hexagon in the third column, if the two neighbors in the second column are different, then the third column's hexagon has only one color left. If the two neighbors in the second column are the same, then the third column's hexagon has two choices.Wait, but in the second column, the two hexagons can be both Y, both G, or one Y and one G.Case 1: Second column is Y and Y.Then, each hexagon in the third column is adjacent to Y and Y, so they can't be Y. So, they can be R or G. But wait, the third column's hexagons are adjacent to the first column's hexagon, which is R. So, they can't be R either. Therefore, they must be G.Case 2: Second column is G and G.Similarly, third column's hexagons must be Y.Case 3: Second column is Y and G.Then, each hexagon in the third column is adjacent to Y and G, so they can't be Y or G. So, they must be R.Wait, but in the third column, the hexagons are adjacent to the first column's hexagon, which is R. So, they can't be R either. Wait, that's a problem.Wait, if the second column is Y and G, then the third column's hexagons are adjacent to Y and G, so they can't be Y or G. But they are also adjacent to the first column's R, so they can't be R either. That's a contradiction. So, that coloring is impossible.Wait, that can't be. So, if the second column is Y and G, the third column's hexagons can't be colored without violating the rules. Therefore, the second column can't have Y and G. So, the second column must be both Y or both G.Therefore, the second column has two possibilities: both Y or both G.Then, the third column's hexagons are forced to be G or Y, respectively.So, total colorings: 2.Wait, that makes sense. Because if the second column is both Y, the third column must be G. If the second column is both G, the third column must be Y. So, only two possibilities.But wait, the problem says a 5x3 grid. So, does that mean we have five columns and three rows? Or five rows and three columns? If it's five rows and three columns, then each column has five hexagons. So, the initial column has five hexagons, with the middle one red. Then, the second column has five hexagons, each adjacent to the first column's hexagons.But this complicates things because each hexagon in the second column is adjacent to two in the first column. So, the coloring would have to alternate.Wait, maybe it's similar to a chessboard, but with three colors. Since it's a hexagonal grid, which is bipartite, we can color it with two colors, but since we have three, maybe we can have more flexibility.But given that the initial hexagon is red, the adjacent ones must be yellow or green, and their adjacent ones must be the opposite.Wait, maybe the entire grid can be colored in two ways, alternating between yellow and green in the second column, and then red in the third, but since it's a 5x3 grid, it might wrap around or something.Wait, no, it's a finite grid. So, starting from red, the second column can be all yellow or all green. Then, the third column must be the opposite.But if it's a 5x3 grid, meaning five rows and three columns, each column has five hexagons. So, the first column has five hexagons, with the middle one red. The second column has five hexagons, each adjacent to the first column's hexagons. So, each hexagon in the second column is adjacent to two in the first column.Wait, but if the first column has five hexagons, each with two neighbors in the second column, then the second column must have five hexagons as well. So, each hexagon in the second column is adjacent to two in the first column.Therefore, the coloring of the second column is constrained by the first column. Since the first column has red in the middle, the second column's hexagons adjacent to red must be yellow or green. But the second column's hexagons are also adjacent to each other, so they can't be the same color.Wait, in a hexagonal grid, each hexagon in the second column is adjacent to two in the first column and two in the third column, but also adjacent to their neighbors in the same column.So, if the first column has red in the middle, the second column's hexagons adjacent to red must be yellow or green. But the second column's hexagons are also adjacent to each other, so they must alternate colors.Therefore, the second column must alternate between yellow and green. Since the first column's middle is red, the second column's middle is adjacent to red, so it must be yellow or green. Then, the hexagons above and below must alternate.But since the first column has five hexagons, the second column must have five as well, alternating between yellow and green. So, starting from the middle, if it's yellow, then above and below must be green, then yellow, etc. Similarly, if the middle is green, then above and below are yellow.Therefore, the second column has two possibilities: middle is yellow or middle is green.Then, the third column is adjacent to the second column. Each hexagon in the third column is adjacent to two in the second column. Since the second column alternates, each hexagon in the third column is adjacent to two different colors. Therefore, they must be red.But wait, the third column's hexagons are also adjacent to the first column's hexagons. The first column's hexagons are red, so the third column's hexagons can't be red. Wait, that's a problem.Wait, no, the third column's hexagons are adjacent to the second column's hexagons, which are yellow and green, so they can't be yellow or green. But they are also adjacent to the first column's hexagons, which are red, so they can't be red either. That's a contradiction. So, this coloring is impossible.Wait, that can't be right. So, maybe the third column's hexagons are only adjacent to the second column's hexagons, not the first column's. So, if the second column alternates between yellow and green, the third column's hexagons are adjacent to two different colors, so they must be red.But then, the third column's hexagons are red, which is allowed because they are not adjacent to the first column's red hexagons.Wait, but in a hexagonal grid, each hexagon in the third column is adjacent to two in the second column and two in the fourth column, but since we're only adding up to the third column, maybe they're only adjacent to the second column.Wait, I'm getting confused again. Maybe I should think of it as a linear chain.First column: RSecond column: Y, G, Y, G, Y (if starting with Y) or G, Y, G, Y, G (if starting with G)Third column: Each hexagon is adjacent to two in the second column. So, if the second column alternates Y and G, then each hexagon in the third column is adjacent to Y and G, so they must be R.But the third column's hexagons are also adjacent to the first column's hexagons, which are R. So, they can't be R. Therefore, this is impossible.Wait, so maybe the third column can't be colored without conflict. Therefore, the only way is to have the second column not alternate, but be all Y or all G.Wait, but if the second column is all Y, then each hexagon in the third column is adjacent to Y, so they must be G or R. But they are also adjacent to the first column's R, so they can't be R. Therefore, they must be G.Similarly, if the second column is all G, the third column must be Y.But in this case, the second column can't be all Y or all G because the hexagons in the second column are adjacent to each other. If they are all Y, then adjacent hexagons in the second column would be the same color, which is not allowed.Wait, so the second column must alternate between Y and G to satisfy the adjacency condition. But then, the third column can't be colored without conflict.This seems like a contradiction, meaning that the only way to color the grid is if the second column is all Y or all G, but that would cause adjacent hexagons in the second column to have the same color, which is not allowed.Therefore, the only way is that the second column alternates Y and G, and the third column is colored R, but that causes a conflict with the first column.Wait, maybe the third column is not adjacent to the first column. If the grid is 5x3, meaning five rows and three columns, then each column is only adjacent to the previous and next column, not wrapping around. So, the third column is only adjacent to the second column, not the first.Therefore, the third column's hexagons are only adjacent to the second column's hexagons, which are Y and G. So, they can be colored R, since they are not adjacent to the first column's R.Wait, that makes sense. So, the third column's hexagons are adjacent only to the second column, not the first. Therefore, they can be colored R.So, the third column is all R.But then, the fourth column would be adjacent to the third column's R, so they must be Y or G. But since we're only adding up to the third column, maybe that's it.Wait, no, the problem says a new column is added to the right, making it a 5x3 grid. So, starting from 5x2, adding a column to make it 5x3.So, the initial configuration was 5x2, with the first column having one R, and the second column having two hexagons. Then, adding a third column.Wait, this is getting too tangled. Maybe I should look for a pattern.If the first column is R, the second column must be Y and G in some order. Then, the third column is determined by the second column.If the second column is Y on top and G on bottom, the third column must be G on top and Y on bottom, because each hexagon in the third column is adjacent to two in the second column.Wait, no, if the second column is Y and G, then the third column's top hexagon is adjacent to Y and something else. Wait, maybe it's better to think that each hexagon in the third column is adjacent to two in the second column, so it must be the third color.But since the second column has Y and G, the third column must be R. But R is already in the first column, but they are not adjacent, so that's okay.Wait, but if the third column is R, then the fourth column would have to be Y and G again. But since we're only adding up to the third column, maybe that's it.Wait, the problem says a new column is added to the right, making it a 5x3 grid. So, starting from 5x2, adding a column to make it 5x3. So, the initial configuration was 5x2, with the first column having one R, and the second column having two hexagons. Then, adding a third column.But in that case, the third column is determined by the second column, which had two possibilities. Therefore, the total number of colorings is 2.Wait, that seems to make sense. Because the second column can be Y and G in two different orders, and the third column is determined by that. So, total colorings: 2.But wait, the problem says a 5x3 grid, so maybe each column has five hexagons. So, the first column has five hexagons, with the middle one R. The second column has five hexagons, each adjacent to the first column's hexagons. Then, the third column has five hexagons, each adjacent to the second column's hexagons.In this case, the second column must alternate between Y and G, starting with Y or G. So, two possibilities. Then, the third column must alternate between R and the opposite color. But since the third column is adjacent to the second column's alternating Y and G, they must be R. But R is already in the first column, but they are not adjacent, so that's okay.Wait, but the third column's hexagons are adjacent to the second column's hexagons, which are Y and G, so they must be R. But the third column's hexagons are also adjacent to each other, so they can't be the same color. But if they are all R, that's a problem.Wait, no, in a hexagonal grid, each hexagon in the third column is adjacent to two in the second column and two in the fourth column, but since we're only adding up to the third column, maybe they are only adjacent to the second column.Wait, this is too confusing. Maybe the answer is 2, as the initial choices determine the rest.

💡Okay, so I have this problem about a convex quadrilateral ABCD. Points P and Q are the midpoints of sides AB and CD, respectively. It says that the line PQ bisects the diagonal AC, and I need to prove that PQ also bisects the diagonal BD. Hmm, interesting.Alright, let me start by visualizing the quadrilateral. Since it's convex, all its interior angles are less than 180 degrees, and the diagonals AC and BD intersect each other inside the quadrilateral. Points P and Q are midpoints, so they're exactly halfway along AB and CD.I remember that in geometry, midpoints often lead to properties involving midlines or midsegments. The midline theorem says that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Maybe that can help here.Let me try to apply that theorem. If I consider triangle ABC, then P is the midpoint of AB, and if I can find another midpoint, maybe that would form a midline. But wait, Q is the midpoint of CD, not BC. Hmm, maybe that's not directly applicable.Alternatively, maybe I should consider the entire quadrilateral. Since P and Q are midpoints, perhaps the line PQ has some special properties. The problem states that PQ bisects AC, meaning it intersects AC at its midpoint. Let me denote the midpoint of AC as M. So, M is the point where PQ intersects AC.Now, I need to show that PQ also bisects BD. Let's denote the midpoint of BD as N. So, I need to prove that N lies on PQ.How can I relate these midpoints? Maybe using vectors or coordinate geometry could help. Let me try coordinate geometry because it allows me to assign coordinates to the points and compute the necessary intersections.Let's assign coordinates to the quadrilateral. Let me place point A at (0, 0) for simplicity. Let me denote point B as (2a, 0) so that the midpoint P is at (a, 0). Then, let me assign point C as (2b, 2c) so that the midpoint of AC is at (b, c). Similarly, let me assign point D as (2d, 2e), making the midpoint Q of CD at (b + d, c + e).Wait, hold on. If P is the midpoint of AB, which is from (0,0) to (2a,0), then P is indeed at (a,0). Similarly, Q is the midpoint of CD, which is from (2b,2c) to (2d,2e), so Q is at ((2b + 2d)/2, (2c + 2e)/2) = (b + d, c + e). That makes sense.Now, the diagonal AC goes from (0,0) to (2b,2c), so its midpoint M is at (b, c). The line PQ goes from (a,0) to (b + d, c + e). Since PQ bisects AC, point M must lie on PQ.So, let me write the equation of line PQ. The coordinates of P are (a, 0) and Q are (b + d, c + e). The slope of PQ is (c + e - 0)/(b + d - a) = (c + e)/(b + d - a). So, the equation of PQ is y = [(c + e)/(b + d - a)](x - a).Since point M (b, c) lies on PQ, substituting into the equation:c = [(c + e)/(b + d - a)](b - a)Multiplying both sides by (b + d - a):c(b + d - a) = (c + e)(b - a)Let me expand both sides:Left side: cb + cd - acRight side: cb - ac + eb - eaSo, set them equal:cb + cd - ac = cb - ac + eb - eaSubtract cb - ac from both sides:cd = eb - eaFactor out e on the right side:cd = e(b - a)So, e = cd / (b - a)Hmm, interesting. So, e is expressed in terms of c, d, and (b - a). I need to remember this relation.Now, I need to find the midpoint N of BD. Point B is at (2a, 0), and point D is at (2d, 2e). So, the midpoint N is at ((2a + 2d)/2, (0 + 2e)/2) = (a + d, e).So, N is at (a + d, e). I need to check if this point lies on PQ.The equation of PQ is y = [(c + e)/(b + d - a)](x - a). Let's plug in x = a + d and y = e into this equation:e = [(c + e)/(b + d - a)](a + d - a) = [(c + e)/(b + d - a)](d)So, e = [d(c + e)] / (b + d - a)Multiply both sides by (b + d - a):e(b + d - a) = d(c + e)Expand the left side:eb + ed - ea = dc + deSubtract de from both sides:eb - ea = dcFactor out e on the left:e(b - a) = dcBut from earlier, we had e = cd / (b - a). Let's substitute that into the left side:(cd / (b - a))(b - a) = cdWhich simplifies to cd = cd, which is true.So, this shows that point N lies on PQ. Therefore, PQ bisects BD.Wait, let me recap. I assigned coordinates to the quadrilateral, found expressions for midpoints, used the given condition that PQ bisects AC to find a relation between the coordinates, and then checked if the midpoint of BD lies on PQ. It turned out to satisfy the equation, so it does lie on PQ. Therefore, PQ bisects BD.I think that makes sense. Maybe I should try to think of another approach, like using vectors or affine geometry, but the coordinate method seems straightforward and worked out.Alternatively, maybe using the concept of homothety or affine transformations, but I think the coordinate approach is sufficient here.Yeah, I think I'm confident with this proof. So, PQ bisects both diagonals AC and BD.

💡First, I need to determine the perimeter of the given triangle by adding its side lengths: 6 cm + 10 cm + 11 cm = 27 cm.Next, since the equilateral triangle has the same perimeter, I know that all three sides of the equilateral triangle must add up to 27 cm. To find the length of one side, I divide the perimeter by 3: 27 cm ÷ 3 = 9 cm.Therefore, each side of the equilateral triangle is 9 cm long.