💡Okay, so I have this problem about two circles, C and D. Circle C is given by the equation x² + y² - 2x - 4y + m = 0, and circle D is given by (x + 2)² + (y + 2)² = 1. The problem says that circle C has three common tangents with circle D, and I need to find the value of m.Hmm, first, I remember that the number of common tangents between two circles depends on their positions relative to each other. If two circles intersect, they have two common tangents. If they are separate, they have four common tangents. If they are externally tangent, meaning they touch at exactly one point from outside, they have three common tangents. Similarly, if they are internally tangent, they also have three common tangents but one is inside the other. But in this case, since both circles are given with positive radii, I think it's external tangency.So, the key here is that since there are three common tangents, the circles must be externally tangent. That means the distance between their centers is equal to the sum of their radii.Alright, let me start by finding the centers and radii of both circles.For circle C: x² + y² - 2x - 4y + m = 0. I need to rewrite this equation in standard form by completing the square.Let me group the x terms and y terms:(x² - 2x) + (y² - 4y) = -mNow, complete the square for x:x² - 2x. Take half of -2, which is -1, square it, which is 1. So, add and subtract 1.Similarly, for y² - 4y. Take half of -4, which is -2, square it, which is 4. So, add and subtract 4.So, the equation becomes:(x² - 2x + 1 - 1) + (y² - 4y + 4 - 4) = -mWhich simplifies to:(x - 1)² - 1 + (y - 2)² - 4 = -mCombine constants:(x - 1)² + (y - 2)² - 5 = -mBring -5 to the right side:(x - 1)² + (y - 2)² = 5 - mSo, the center of circle C is (1, 2), and the radius is sqrt(5 - m). Wait, radius can't be negative, so 5 - m must be positive. Therefore, m < 5.Now, circle D is given by (x + 2)² + (y + 2)² = 1. So, its center is (-2, -2), and the radius is 1.Alright, now I need to find the distance between the centers of circle C and circle D.Center of C: (1, 2)Center of D: (-2, -2)Distance formula: sqrt[(x2 - x1)² + (y2 - y1)²]So, plugging in the values:sqrt[(-2 - 1)² + (-2 - 2)²] = sqrt[(-3)² + (-4)²] = sqrt[9 + 16] = sqrt[25] = 5.So, the distance between centers is 5 units.Since the circles are externally tangent, the distance between centers is equal to the sum of their radii.Radius of circle C: sqrt(5 - m)Radius of circle D: 1So, sum of radii: sqrt(5 - m) + 1This should equal the distance between centers, which is 5.Therefore:sqrt(5 - m) + 1 = 5Subtract 1 from both sides:sqrt(5 - m) = 4Square both sides:5 - m = 16Subtract 5 from both sides:-m = 11Multiply both sides by -1:m = -11Wait, let me check if this makes sense. If m = -11, then the radius of circle C is sqrt(5 - (-11)) = sqrt(16) = 4. So, radius of C is 4, radius of D is 1, distance between centers is 5. So, 4 + 1 = 5, which matches the distance. So, yes, they are externally tangent, which means there are three common tangents. That seems correct.But just to be thorough, let me consider if there's another scenario where they could have three common tangents. I remember that if one circle is inside another and they are tangent, they also have three common tangents. But in this case, circle D has radius 1, and circle C would have radius 4 if m = -11. The distance between centers is 5, which is equal to 4 + 1, so they are externally tangent. If they were internally tangent, the distance between centers would be equal to the difference of radii. So, 4 - 1 = 3. But the distance is 5, which is more than 4, so they can't be internally tangent. So, only external tangency is possible here.Therefore, m must be -11.**Final Answer**The value of ( m ) is boxed{-11}.

💡Okay, so I have this problem where I need to place 5 points inside an equilateral triangle with side length 1, and I have to show that there are at least two points that are at a distance less than or equal to 0.5. Hmm, interesting. I remember something about the pigeonhole principle from my math class, maybe that could help here.First, let me visualize the problem. An equilateral triangle with side length 1. If I place 5 points inside it, I need to prove that at least two of them are within 0.5 units of each other. So, how can I approach this?Maybe I can divide the triangle into smaller regions. If I can divide it into four smaller regions, then by the pigeonhole principle, since I have five points and four regions, at least one region must contain at least two points. Then, if I can show that the maximum distance between any two points within a region is at most 0.5, that would solve the problem.Alright, how do I divide the triangle into four smaller regions? Well, an equilateral triangle can be divided into four smaller equilateral triangles by connecting the midpoints of each side. Let me think about that. If I connect the midpoints, each side of the original triangle is divided into two segments of length 0.5. So, each smaller triangle will have sides of length 0.5.Let me draw this in my mind. The original triangle ABC, with each side of length 1. Let me mark the midpoints: M on AB, N on BC, and P on AC. Then, connecting these midpoints M, N, P will form four smaller equilateral triangles inside ABC. Each of these smaller triangles has sides of length 0.5.So, now I have four regions, each of which is an equilateral triangle with side length 0.5. If I place five points inside the original triangle, by the pigeonhole principle, at least one of these smaller triangles must contain at least two points.Now, what's the maximum distance between two points within one of these smaller triangles? Since each smaller triangle is equilateral with side length 0.5, the maximum distance between any two points inside it would be 0.5, right? Because the farthest apart two points can be is the length of the side of the triangle.Therefore, if two points are inside the same smaller triangle, they must be at most 0.5 units apart. So, that means there must be at least two points among the five that are within 0.5 units of each other.Wait, let me make sure I didn't skip any steps. I divided the triangle into four smaller ones, each with side length 0.5. Placed five points, so at least two points are in the same small triangle. The maximum distance in that small triangle is 0.5, so those two points are at most 0.5 apart. That seems to cover it.Is there another way to think about this? Maybe using coordinates? Let me try assigning coordinates to the triangle to see if that helps.Let's place the original triangle ABC with point A at (0, 0), point B at (1, 0), and point C at (0.5, sqrt(3)/2). The midpoints M, N, P would then be at (0.5, 0), (0.75, sqrt(3)/4), and (0.25, sqrt(3)/4), respectively.So, the four smaller triangles are:1. Triangle AMP: points A, M, P2. Triangle BMN: points B, M, N3. Triangle CNP: points C, N, P4. Triangle MNP: points M, N, PEach of these smaller triangles has side length 0.5. If I place five points in the original triangle, at least two must be in the same smaller triangle. The maximum distance between two points in a smaller triangle is 0.5, so those two points are within 0.5 units of each other.Hmm, that seems consistent with my earlier reasoning. I think this works. But let me double-check if the maximum distance in the smaller triangle is indeed 0.5.In an equilateral triangle, the maximum distance between any two points is the length of the side. So, if each smaller triangle has sides of length 0.5, then yes, the maximum distance between two points inside it is 0.5. Therefore, if two points are in the same smaller triangle, they must be at most 0.5 apart.Is there a possibility that two points could be placed in such a way that they are exactly 0.5 apart? Well, yes, if they are placed at the endpoints of a side of the smaller triangle. But the problem says "less than or equal to 0.5," so that's acceptable.Wait, what if all five points are placed exactly at the vertices of the original triangle? But the original triangle only has three vertices, so placing five points would require at least two points to be inside the triangle or on the edges. But even if two points are on the same edge, the distance between them could be up to 1, which is more than 0.5. Hmm, so maybe my initial approach isn't sufficient?Wait, no. Because if I divide the triangle into four smaller triangles, each with side length 0.5, then any two points within the same smaller triangle are at most 0.5 apart. But if points are placed on the edges or vertices, they might be in multiple smaller triangles. Hmm, maybe I need to consider the regions more carefully.Alternatively, perhaps I should use the concept of dividing the triangle into four regions, each of which has a diameter (maximum distance between any two points) of 0.5. Then, by the pigeonhole principle, two points must be within the same region, hence within 0.5 of each other.Yes, that makes sense. The key is that each region has a diameter of 0.5, so any two points in the same region are guaranteed to be within 0.5 of each other.So, to summarize, by dividing the original triangle into four smaller equilateral triangles each with side length 0.5, and then applying the pigeonhole principle with five points, we can conclude that at least two points must be within 0.5 units of each other.I think that's a solid argument. I don't see any flaws in this reasoning. It uses the pigeonhole principle appropriately and the geometric division ensures that the maximum distance within each sub-triangle is controlled.

💡To solve for ( x ) in the equation ( 3 oslash x = 625 ) where the operation ( a oslash b = (sqrt{3a + 2b})^4 ) is defined, follow these steps:1. **Substitute the values into the operation:** [ 3 oslash x = (sqrt{3(3) + 2x})^4 ] Simplifying inside the square root: [ (sqrt{9 + 2x})^4 = 625 ]2. **Take the fourth root of both sides to simplify:** [ sqrt{9 + 2x} = sqrt[4]{625} ] Since ( sqrt[4]{625} = 5 ): [ sqrt{9 + 2x} = 5 ]3. **Square both sides to eliminate the square root:** [ 9 + 2x = 5^2 ] [ 9 + 2x = 25 ]4. **Solve for ( x ):** [ 2x = 25 - 9 ] [ 2x = 16 ] [ x = frac{16}{2} ] [ x = 8 ]Therefore, the value of ( x ) is (boxed{8}).

💡To determine how many pairs of shoes Daniela buys, follow these steps:1. **Calculate the discounted price for the shoes:** - Original price per pair of shoes: 50 - Discount percentage: 40% - Discount amount: 40% of 50 = 0.40 × 50 = 20 - Discounted price per pair of shoes: 50 - 20 = 302. **Calculate the discounted price for the dress:** - Original price of the dress: 100 - Discount percentage: 20% - Discount amount: 20% of 100 = 0.20 × 100 = 20 - Discounted price of the dress: 100 - 20 = 803. **Determine the total amount spent on shoes:** - Total amount spent: 140 - Amount spent on the dress: 80 - Amount spent on shoes: 140 - 80 = 604. **Calculate the number of pairs of shoes bought:** - Discounted price per pair of shoes: 30 - Number of pairs: 60 ÷ 30 = 2 pairs

💡First, I recognize that the function ( f ) is defined in terms of ( x+1 ). To find ( f(1) ), I need to determine the value of ( x ) that makes ( x+1 = 1 ).Solving ( x+1 = 1 ) gives ( x = 0 ).Next, I substitute ( x = 0 ) into the given equation ( f(x+1) = 2x - 1 ):[f(0+1) = 2 times 0 - 1]Simplifying this, I find:[f(1) = -1]

💡First, I need to determine which pairs of angles have the same terminal side. Angles with the same terminal side differ by an integer multiple of (2pi).**Option A:** (dfrac{20pi}{3}) and (dfrac{87pi}{9})- Simplify (dfrac{20pi}{3}): (dfrac{20pi}{3} = 6pi + dfrac{2pi}{3}) This places the angle in the second quadrant.- Simplify (dfrac{87pi}{9}): (dfrac{87pi}{9} = 9pi + dfrac{2pi}{3}) This places the angle in the fourth quadrant.Since they are in different quadrants, their terminal sides are different.**Option B:** (-dfrac{pi}{3}) and (dfrac{22pi}{3})- (-dfrac{pi}{3}) is in the fourth quadrant.- Simplify (dfrac{22pi}{3}): (dfrac{22pi}{3} = 7pi + dfrac{pi}{3}) This places the angle in the third quadrant.Since they are in different quadrants, their terminal sides are different.**Option C:** (dfrac{3pi}{2}) and (-dfrac{3pi}{2})- (dfrac{3pi}{2}) is on the negative y-axis.- (-dfrac{3pi}{2}) is on the positive y-axis.Since they are on different axes, their terminal sides are different.**Option D:** (-dfrac{7pi}{9}) and (-dfrac{25pi}{9})- (-dfrac{7pi}{9}) is in the third quadrant.- Simplify (-dfrac{25pi}{9}): (-dfrac{25pi}{9} = -2pi - dfrac{7pi}{9}) This places the angle in the same position as (-dfrac{7pi}{9}).Since they are in the same quadrant and differ by an integer multiple of (2pi), their terminal sides are the same.