💡Okay, so I need to figure out how much Tom will pay for the fruits he bought. Let's see, he bought 6 lemons, 4 papayas, and 2 mangos. The prices are 2 for a lemon, 1 for a papaya, and 4 for a mango. Also, there's a discount: for every 4 fruits bought, the store gives a 1 discount. First, I think I should calculate the total cost without any discounts. So, for the lemons, 6 lemons at 2 each would be 6 times 2, which is 12. For the papayas, 4 papayas at 1 each would be 4 times 1, which is 4. And for the mangos, 2 mangos at 4 each would be 2 times 4, which is 8. Now, adding those up: 12 for lemons plus 4 for papayas is 16, and then plus 8 for mangos makes a total of 24 before any discounts.Next, I need to calculate the discount. The store offers a 1 discount for every 4 fruits bought. So, I should find out how many fruits Tom bought in total. He bought 6 lemons, 4 papayas, and 2 mangos. Adding those together: 6 plus 4 is 10, and 10 plus 2 is 12 fruits in total.Since the discount is 1 for every 4 fruits, I need to see how many sets of 4 fruits are in 12. So, 12 divided by 4 is 3. That means Tom gets a 1 discount for each of those 3 sets, so the total discount is 3 times 1, which is 3.Now, subtracting the discount from the total cost before discount: 24 minus 3 equals 21. So, Tom will pay 21 in total.Wait a minute, let me double-check my calculations to make sure I didn't make any mistakes. For the lemons, 6 times 2 is indeed 12. Papayas are 4 times 1, which is 4. Mangos are 2 times 4, which is 8. Adding those up: 12 plus 4 is 16, plus 8 is 24. That seems correct.For the discount, total fruits are 6 plus 4 plus 2, which is 12. Divided by 4, that's 3 sets, so 3 times 1 discount is 3. Subtracting that from 24 gives 21. Yeah, that seems right.I think I've got it. Tom will pay 21 for all the fruits after the discount.

💡First, I need to calculate the product of 0.45 and 0.65. To simplify the multiplication, I'll express each decimal as a fraction multiplied by a power of ten. So, 0.45 becomes 45 times 10 to the power of -2, and 0.65 becomes 65 times 10 to the power of -2. Multiplying these together gives me (45 multiplied by 65) times 10 to the power of -4. Calculating 45 times 65 results in 2925, so the product is 2925 times 10 to the power of -4, which equals 0.2925.Next, I'll calculate the product of 0.1 and 0.2. Similarly, I'll express 0.1 as 1 times 10 to the power of -1 and 0.2 as 2 times 10 to the power of -1. Multiplying these gives (1 multiplied by 2) times 10 to the power of -2, which equals 2 times 10 to the power of -2, or 0.02.Finally, I'll add the two results together: 0.2925 plus 0.02 equals 0.3125. Therefore, the final answer is 0.3125.

💡Okay, so I have this continued fraction expression to solve: (3 + frac{9}{2 + frac{9}{3 + frac{9}{2 + cdots}}}), where the 2s and 3s alternate. Hmm, continued fractions can be tricky, but I think I remember that sometimes you can set them equal to a variable and solve an equation. Let me try that.Let me call the entire expression (y). So, (y = 3 + frac{9}{2 + frac{9}{3 + frac{9}{2 + cdots}}}). Now, looking at the structure, after the first 3, the pattern repeats with a 2 and then another similar fraction. Maybe I can express the inner part in terms of (y) as well.Wait, if I look at the denominator after the first 9, it's (2 + frac{9}{3 + frac{9}{2 + cdots}}). But that inner part (3 + frac{9}{2 + cdots}) looks similar to the original expression (y). So, maybe I can set that inner part equal to (y) as well. Let me write that down.So, let me denote the inner continued fraction as (y) again. Therefore, the expression becomes:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Okay, that seems manageable. Now, I can try to solve this equation for (y). Let me write it out:[ y = 3 + frac{9}{2 + frac{9}{y}} ]To simplify this, maybe I can let (z = 2 + frac{9}{y}), so that the equation becomes:[ y = 3 + frac{9}{z} ]and[ z = 2 + frac{9}{y} ]Now, I can substitute (z) from the second equation into the first one. So, replacing (z) in the first equation:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Hmm, that's the same as before. Maybe I should multiply both sides by the denominator to eliminate the fraction. Let's try that.Multiplying both sides by (2 + frac{9}{y}):[ y left(2 + frac{9}{y}right) = 3 left(2 + frac{9}{y}right) + 9 ]Wait, let me check that. If I multiply the entire equation ( y = 3 + frac{9}{2 + frac{9}{y}} ) by (2 + frac{9}{y}), I get:[ y left(2 + frac{9}{y}right) = 3 left(2 + frac{9}{y}right) + 9 ]Yes, that seems right. Now, let's simplify both sides.Starting with the left side:[ y left(2 + frac{9}{y}right) = 2y + 9 ]On the right side:[ 3 left(2 + frac{9}{y}right) + 9 = 6 + frac{27}{y} + 9 = 15 + frac{27}{y} ]So, putting it all together:[ 2y + 9 = 15 + frac{27}{y} ]Hmm, now I have an equation with (y) and (frac{1}{y}). Maybe I can multiply both sides by (y) to eliminate the fraction. Let's try that.Multiplying both sides by (y):[ 2y^2 + 9y = 15y + 27 ]Now, let's bring all terms to one side to form a quadratic equation:[ 2y^2 + 9y - 15y - 27 = 0 ][ 2y^2 - 6y - 27 = 0 ]Wait, let me double-check that. Starting from:[ 2y + 9 = 15 + frac{27}{y} ]Multiply both sides by (y):[ 2y^2 + 9y = 15y + 27 ]Subtract (15y + 27) from both sides:[ 2y^2 + 9y - 15y - 27 = 0 ][ 2y^2 - 6y - 27 = 0 ]Yes, that looks correct. Now, I have a quadratic equation:[ 2y^2 - 6y - 27 = 0 ]I can solve this using the quadratic formula. The quadratic formula is:[ y = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]where (a = 2), (b = -6), and (c = -27).Plugging in the values:[ y = frac{-(-6) pm sqrt{(-6)^2 - 4 cdot 2 cdot (-27)}}{2 cdot 2} ][ y = frac{6 pm sqrt{36 + 216}}{4} ][ y = frac{6 pm sqrt{252}}{4} ]Simplify (sqrt{252}):[ sqrt{252} = sqrt{36 cdot 7} = 6sqrt{7} ]So, substituting back:[ y = frac{6 pm 6sqrt{7}}{4} ][ y = frac{6(1 pm sqrt{7})}{4} ][ y = frac{3(1 pm sqrt{7})}{2} ]So, the solutions are:[ y = frac{3(1 + sqrt{7})}{2} ]and[ y = frac{3(1 - sqrt{7})}{2} ]Now, since the continued fraction starts with 3 and all the terms are positive, the value of (y) must be positive. Let's evaluate both solutions.First solution:[ y = frac{3(1 + sqrt{7})}{2} ]Since (sqrt{7}) is approximately 2.6458, this gives:[ y approx frac{3(1 + 2.6458)}{2} = frac{3 times 3.6458}{2} approx frac{10.9374}{2} approx 5.4687 ]Second solution:[ y = frac{3(1 - sqrt{7})}{2} ]This would be:[ y approx frac{3(1 - 2.6458)}{2} = frac{3 times (-1.6458)}{2} approx frac{-4.9374}{2} approx -2.4687 ]Since the continued fraction is clearly positive, we discard the negative solution. Therefore, the value of the continued fraction is approximately 5.4687. But let me see if this can be expressed more neatly.Wait, earlier when I set up the equation, I might have made a mistake in the substitution. Let me double-check my steps.Starting again, I set (y = 3 + frac{9}{2 + frac{9}{y}}). Then, I let (z = 2 + frac{9}{y}), so (y = 3 + frac{9}{z}) and (z = 2 + frac{9}{y}). Substituting (z) into the first equation gives (y = 3 + frac{9}{2 + frac{9}{y}}), which is correct.Then, multiplying both sides by (2 + frac{9}{y}) gives:[ y left(2 + frac{9}{y}right) = 3 left(2 + frac{9}{y}right) + 9 ]Simplifying:Left side: (2y + 9)Right side: (6 + frac{27}{y} + 9 = 15 + frac{27}{y})So, (2y + 9 = 15 + frac{27}{y})Then, multiplying both sides by (y):(2y^2 + 9y = 15y + 27)Bringing all terms to one side:(2y^2 - 6y - 27 = 0)Yes, that seems correct. So, the quadratic equation is correct, and the solutions are as above. Therefore, the positive solution is approximately 5.4687, which is (frac{3(1 + sqrt{7})}{2}).But wait, in the initial problem, the continued fraction alternates between 2 and 3. Maybe there's a different approach or perhaps I missed a step in setting up the equation. Let me think again.Alternatively, perhaps I should consider that the continued fraction repeats every two terms, so maybe I need to set up a system of equations. Let me try that.Let me denote the entire expression as (y), so:[ y = 3 + frac{9}{2 + frac{9}{3 + frac{9}{2 + cdots}}} ]Now, notice that after the first 3, the pattern is 2 + 9/(3 + 9/(2 + ...)). Let me denote the inner part as another variable, say (x). So, after the first 3, we have:[ y = 3 + frac{9}{2 + frac{9}{x}} ]But then, (x) itself is similar to the original expression but starting with 3. So, (x = 3 + frac{9}{2 + frac{9}{x}}). Wait, that's the same as (y). So, (x = y). Therefore, the equation remains:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Which brings me back to the same equation as before. So, my initial approach was correct.Alternatively, maybe I can express the continued fraction in terms of itself after two steps. Let me try that.Let me write the continued fraction as:[ y = 3 + frac{9}{2 + frac{9}{3 + frac{9}{2 + cdots}}} ]But the part after the first 3 is:[ 2 + frac{9}{3 + frac{9}{2 + cdots}} ]Which is similar to the original expression but starting with 2 instead of 3. Let me denote this as another variable, say (z). So:[ z = 2 + frac{9}{3 + frac{9}{2 + cdots}} ]But notice that the inner part (3 + frac{9}{2 + cdots}) is the original expression (y). Therefore, (z = 2 + frac{9}{y}).So, now I have:[ y = 3 + frac{9}{z} ]and[ z = 2 + frac{9}{y} ]This is a system of two equations with two variables. Let me solve this system.From the first equation:[ y = 3 + frac{9}{z} ]From the second equation:[ z = 2 + frac{9}{y} ]Let me substitute (z) from the second equation into the first one:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Which is the same equation as before. So, again, I end up with the quadratic equation:[ 2y^2 - 6y - 27 = 0 ]And the solutions are:[ y = frac{3(1 pm sqrt{7})}{2} ]Since (y) must be positive, we take the positive solution:[ y = frac{3(1 + sqrt{7})}{2} ]Approximately, this is about 5.4687. But let me check if this makes sense in the context of the continued fraction.If I plug this value back into the original expression, does it hold?Let me compute (2 + frac{9}{y}):[ 2 + frac{9}{frac{3(1 + sqrt{7})}{2}} = 2 + frac{18}{3(1 + sqrt{7})} = 2 + frac{6}{1 + sqrt{7}} ]Rationalizing the denominator:[ frac{6}{1 + sqrt{7}} times frac{1 - sqrt{7}}{1 - sqrt{7}} = frac{6(1 - sqrt{7})}{1 - 7} = frac{6(1 - sqrt{7})}{-6} = -(1 - sqrt{7}) = sqrt{7} - 1 ]So, (2 + frac{9}{y} = 2 + (sqrt{7} - 1) = 1 + sqrt{7})Now, plug this back into the expression for (y):[ y = 3 + frac{9}{1 + sqrt{7}} ]Again, rationalizing the denominator:[ frac{9}{1 + sqrt{7}} times frac{1 - sqrt{7}}{1 - sqrt{7}} = frac{9(1 - sqrt{7})}{1 - 7} = frac{9(1 - sqrt{7})}{-6} = -frac{3(1 - sqrt{7})}{2} = frac{3(sqrt{7} - 1)}{2} ]So, (y = 3 + frac{3(sqrt{7} - 1)}{2} = frac{6}{2} + frac{3sqrt{7} - 3}{2} = frac{6 + 3sqrt{7} - 3}{2} = frac{3 + 3sqrt{7}}{2} = frac{3(1 + sqrt{7})}{2})Which matches our earlier solution. So, that checks out.Therefore, the value of the continued fraction is (frac{3(1 + sqrt{7})}{2}), which is approximately 5.4687.But wait, in the initial problem, the user had a similar expression and got 6 as the answer. Did I make a mistake somewhere? Let me check.Looking back, the user's problem was:[ 3 + frac{9}{2 + frac{9}{3 + frac{9}{2 + cdots}}} ]And they set (y = 3 + frac{9}{2 + frac{9}{y}}), leading to a quadratic equation (y^2 - 3y - 18 = 0), which factors to ((y - 6)(y + 3) = 0), giving (y = 6) as the positive solution.Wait, in my approach, I ended up with a different quadratic equation: (2y^2 - 6y - 27 = 0), leading to (y = frac{3(1 pm sqrt{7})}{2}). So, why the discrepancy?Let me see. Maybe I made a mistake in setting up the equation. Let me go back.The user set (y = 3 + frac{9}{2 + frac{9}{y}}), then multiplied both sides by (2 + frac{9}{y}), leading to:[ y(2 + frac{9}{y}) = 3(2 + frac{9}{y}) + 9 ]Simplifying:Left side: (2y + 9)Right side: (6 + frac{27}{y} + 9 = 15 + frac{27}{y})So, (2y + 9 = 15 + frac{27}{y})Then, multiplying both sides by (y):(2y^2 + 9y = 15y + 27)Bringing all terms to one side:(2y^2 - 6y - 27 = 0)But the user's solution led to (y^2 - 3y - 18 = 0). So, perhaps they made a mistake in their algebra.Wait, let me see. If I take the user's equation:[ y = 3 + frac{9}{2 + frac{9}{y}} ]And multiply both sides by (2 + frac{9}{y}), I get:[ y(2 + frac{9}{y}) = 3(2 + frac{9}{y}) + 9 ]Which simplifies to:[ 2y + 9 = 6 + frac{27}{y} + 9 ][ 2y + 9 = 15 + frac{27}{y} ]Then, multiplying both sides by (y):[ 2y^2 + 9y = 15y + 27 ][ 2y^2 - 6y - 27 = 0 ]But the user's solution led to (y^2 - 3y - 18 = 0). So, perhaps they made a mistake in their algebra when multiplying through.Alternatively, maybe the user incorrectly set up the equation. Let me check their steps.The user wrote:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Then, they set (z = 2 + frac{9}{y}), so:[ y = 3 + frac{9}{z} ]and[ z = 2 + frac{9}{y} ]Then, substituting (z) into the first equation:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Which is correct. Then, they multiplied through by (y) and (2 + frac{9}{y}), leading to:[ y(2 + frac{9}{y}) = 3y + 9 ]Simplifying:[ 2y + 9 = 3y + 9 ]Which gives (y = 0), which is incorrect.Wait, that's a mistake. Because when they multiplied both sides by (y) and (2 + frac{9}{y}), they should have multiplied both sides by (2 + frac{9}{y}), not by (y) and (2 + frac{9}{y}) separately.So, the correct step is:[ y = 3 + frac{9}{2 + frac{9}{y}} ]Multiply both sides by (2 + frac{9}{y}):[ y(2 + frac{9}{y}) = 3(2 + frac{9}{y}) + 9 ]Which simplifies to:[ 2y + 9 = 6 + frac{27}{y} + 9 ][ 2y + 9 = 15 + frac{27}{y} ]Then, multiplying both sides by (y):[ 2y^2 + 9y = 15y + 27 ][ 2y^2 - 6y - 27 = 0 ]So, the user made a mistake in their multiplication step, leading them to an incorrect quadratic equation. Therefore, their solution of (y = 6) is incorrect.In conclusion, the correct value of the continued fraction is (frac{3(1 + sqrt{7})}{2}), which is approximately 5.4687.

💡Okay, so I have this system of equations involving sine functions of angles α, β, and γ. The equations are:1. sin α = sin(α + β + γ) + 12. sin β = 3 sin(α + β + γ) + 23. sin γ = 5 sin(α + β + γ) + 3And I need to find all possible values of the product sin α · sin β · sin γ.Hmm, let's see. All three equations have sin(α + β + γ) in them. Maybe I can let’s denote θ = α + β + γ to simplify things. So, θ is the sum of the three angles. Then, the equations become:1. sin α = sin θ + 12. sin β = 3 sin θ + 23. sin γ = 5 sin θ + 3Alright, so now I have expressions for sin α, sin β, and sin γ in terms of sin θ. Maybe I can express everything in terms of sin θ and then find its value.Let me denote x = sin θ. Then, the equations become:1. sin α = x + 12. sin β = 3x + 23. sin γ = 5x + 3Now, since the sine function has a range between -1 and 1, each of these expressions must also lie within that interval. So, let's write down the constraints:For sin α = x + 1:-1 ≤ x + 1 ≤ 1Subtracting 1: -2 ≤ x ≤ 0For sin β = 3x + 2:-1 ≤ 3x + 2 ≤ 1Subtracting 2: -3 ≤ 3x ≤ -1Dividing by 3: -1 ≤ x ≤ -1/3For sin γ = 5x + 3:-1 ≤ 5x + 3 ≤ 1Subtracting 3: -4 ≤ 5x ≤ -2Dividing by 5: -4/5 ≤ x ≤ -2/5So, combining all these constraints:From sin α: x ∈ [-2, 0]From sin β: x ∈ [-1, -1/3]From sin γ: x ∈ [-4/5, -2/5]The intersection of these intervals is x ∈ [-4/5, -2/5] ∩ [-1, -1/3] = [-4/5, -2/5]. So, x must be between -4/5 and -2/5.Now, let's see if we can find x such that θ = α + β + γ. But θ is the sum of the angles, so maybe we can relate sin θ with sin α, sin β, and sin γ.Wait, but I don't know the relationship between θ and the individual angles. Maybe I can use the sine addition formula or some identity.Alternatively, since I have expressions for sin α, sin β, sin γ in terms of x, maybe I can use the fact that α, β, γ are angles such that their sum is θ, and perhaps use some trigonometric identities involving the sum of angles.But this seems a bit complicated. Maybe another approach is to consider the sum of the equations.Let me try adding the first two equations:sin α + sin β = (x + 1) + (3x + 2) = 4x + 3Similarly, from the third equation, sin γ = 5x + 3.So, sin α + sin β = 4x + 3, and sin γ = 5x + 3.Hmm, interesting. So, sin α + sin β = sin γ - x. Because sin γ = 5x + 3, so sin γ - x = 5x + 3 - x = 4x + 3, which matches the left side.So, sin α + sin β = sin γ - x.But x is sin θ, which is sin(α + β + γ). So, sin α + sin β = sin γ - sin(α + β + γ).Hmm, maybe I can use the sine addition formula on sin(α + β + γ). Let me recall that sin(A + B + C) can be expanded, but it's a bit messy. Alternatively, maybe I can use sum-to-product identities.Let me consider sin α + sin β. The sum-to-product identity says that sin α + sin β = 2 sin[(α + β)/2] cos[(α - β)/2].Similarly, sin γ - sin(α + β + γ) can be written using the difference identity: sin γ - sin(α + β + γ) = 2 cos[(γ + α + β + γ)/2] sin[(γ - (α + β + γ))/2] = 2 cos[(α + β + 2γ)/2] sin[(-α - β)/2].Wait, that seems complicated. Let me write it step by step.Using the identity sin A - sin B = 2 cos[(A + B)/2] sin[(A - B)/2], so sin γ - sin(α + β + γ) = 2 cos[(γ + α + β + γ)/2] sin[(γ - (α + β + γ))/2] = 2 cos[(α + β + 2γ)/2] sin[(-α - β)/2].Simplify sin[(-α - β)/2] = -sin[(α + β)/2], so we have:sin γ - sin(α + β + γ) = -2 cos[(α + β + 2γ)/2] sin[(α + β)/2].So, putting it all together:sin α + sin β = -2 cos[(α + β + 2γ)/2] sin[(α + β)/2]But from earlier, sin α + sin β = 2 sin[(α + β)/2] cos[(α - β)/2].So, equating the two expressions:2 sin[(α + β)/2] cos[(α - β)/2] = -2 cos[(α + β + 2γ)/2] sin[(α + β)/2]Assuming sin[(α + β)/2] ≠ 0, we can divide both sides by 2 sin[(α + β)/2]:cos[(α - β)/2] = -cos[(α + β + 2γ)/2]Hmm, interesting. So, cos[(α - β)/2] = -cos[(α + β + 2γ)/2].Recall that cos A = -cos B implies that A = π - B + 2πk or A = π + B + 2πk for some integer k.So, let's write:(α - β)/2 = π - (α + β + 2γ)/2 + 2πkor(α - β)/2 = π + (α + β + 2γ)/2 + 2πkLet me solve the first case:(α - β)/2 = π - (α + β + 2γ)/2 + 2πkMultiply both sides by 2:α - β = 2π - (α + β + 2γ) + 4πkSimplify:α - β = 2π - α - β - 2γ + 4πkBring all terms to the left:α - β + α + β + 2γ - 2π - 4πk = 0Simplify:2α + 2γ - 2π - 4πk = 0Divide by 2:α + γ - π - 2πk = 0So,α + γ = π + 2πkSimilarly, for the second case:(α - β)/2 = π + (α + β + 2γ)/2 + 2πkMultiply both sides by 2:α - β = 2π + α + β + 2γ + 4πkSimplify:α - β - α - β - 2γ - 2π - 4πk = 0Simplify:-2β - 2γ - 2π - 4πk = 0Divide by -2:β + γ + π + 2πk = 0So,β + γ = -π - 2πkBut angles are typically considered modulo 2π, so we can adjust k accordingly. However, since sine is periodic with period 2π, maybe we can consider principal values.So, from the first case, α + γ = π + 2πk, and from the second case, β + γ = -π - 2πk.But let's consider the first case first: α + γ = π + 2πk.Since we're dealing with sines, which have a period of 2π, we can set k=0 without loss of generality, so α + γ = π.Similarly, for the second case, β + γ = -π, but adding 2π gives β + γ = π.So, in both cases, we can have either α + γ = π or β + γ = π.Wait, but in the first case, it's α + γ = π, and in the second case, it's β + γ = π.So, depending on which case we are in, either α + γ = π or β + γ = π.Let me consider these two cases separately.**Case 1: α + γ = π**If α + γ = π, then γ = π - α.Substituting back into the third equation:sin γ = 5x + 3But γ = π - α, so sin γ = sin(π - α) = sin α.So, sin α = 5x + 3.But from the first equation, sin α = x + 1.Therefore,x + 1 = 5x + 3Solving for x:x + 1 = 5x + 3Subtract x:1 = 4x + 3Subtract 3:-2 = 4xDivide by 4:x = -2/4 = -1/2So, x = -1/2.Now, let's check if this x satisfies the earlier constraints.From sin α = x + 1 = -1/2 + 1 = 1/2From sin β = 3x + 2 = 3*(-1/2) + 2 = -3/2 + 2 = 1/2From sin γ = 5x + 3 = 5*(-1/2) + 3 = -5/2 + 3 = 1/2So, sin α = sin β = sin γ = 1/2.But wait, earlier we had that α + γ = π, so sin γ = sin α = 1/2.So, α and γ are angles whose sine is 1/2, and their sum is π.So, possible solutions are:α = π/6, γ = 5π/6orα = 5π/6, γ = π/6Similarly, sin β = 1/2, so β = π/6 or 5π/6.But let's check θ = α + β + γ.If α = π/6, γ = 5π/6, then α + γ = π. So, θ = π + β.But θ = α + β + γ = π + β.But θ is also such that sin θ = x = -1/2.So, sin θ = -1/2, which implies θ = 7π/6 or 11π/6.But θ = π + β, so:If θ = 7π/6, then β = θ - π = 7π/6 - π = π/6If θ = 11π/6, then β = 11π/6 - π = 5π/6So, in both cases, β is either π/6 or 5π/6, which matches sin β = 1/2.So, the solutions are consistent.Therefore, in this case, sin α = sin β = sin γ = 1/2.Thus, the product sin α · sin β · sin γ = (1/2) * (1/2) * (1/2) = 1/8.**Case 2: β + γ = π**Similarly, if β + γ = π, then γ = π - β.Substituting back into the third equation:sin γ = 5x + 3But γ = π - β, so sin γ = sin(π - β) = sin β.So, sin β = 5x + 3.But from the second equation, sin β = 3x + 2.Therefore,3x + 2 = 5x + 3Solving for x:3x + 2 = 5x + 3Subtract 3x:2 = 2x + 3Subtract 3:-1 = 2xDivide by 2:x = -1/2So, again, x = -1/2.Now, let's find sin α, sin β, sin γ.From the first equation, sin α = x + 1 = -1/2 + 1 = 1/2From the second equation, sin β = 3x + 2 = 3*(-1/2) + 2 = -3/2 + 2 = 1/2From the third equation, sin γ = 5x + 3 = 5*(-1/2) + 3 = -5/2 + 3 = 1/2So, again, sin α = sin β = sin γ = 1/2.But now, β + γ = π, so similar to before, β and γ are angles whose sine is 1/2 and their sum is π.So, possible solutions are:β = π/6, γ = 5π/6orβ = 5π/6, γ = π/6Similarly, sin α = 1/2, so α = π/6 or 5π/6.Now, θ = α + β + γ.If β = π/6, γ = 5π/6, then β + γ = π. So, θ = α + π.But θ is such that sin θ = x = -1/2.So, θ = 7π/6 or 11π/6.Thus, α = θ - π.If θ = 7π/6, then α = 7π/6 - π = π/6If θ = 11π/6, then α = 11π/6 - π = 5π/6So, again, consistent with sin α = 1/2.Therefore, in this case as well, the product sin α · sin β · sin γ = (1/2)^3 = 1/8.Wait, so both cases lead to the same product value of 1/8.But earlier, when I considered the constraints on x, I found that x must be between -4/5 and -2/5, which is approximately between -0.8 and -0.4. But in both cases, x = -1/2 = -0.5, which is within that interval. So, that's consistent.But wait, is there another possibility? Because when I considered the equation cos[(α - β)/2] = -cos[(α + β + 2γ)/2], I assumed that sin[(α + β)/2] ≠ 0. What if sin[(α + β)/2] = 0?So, let's consider that case.**Case 3: sin[(α + β)/2] = 0**This implies that (α + β)/2 = kπ, so α + β = 2kπ.So, α + β is a multiple of 2π.Given that, let's substitute back into the equations.From the first equation, sin α = x + 1.From the second equation, sin β = 3x + 2.From the third equation, sin γ = 5x + 3.Also, since α + β = 2kπ, then γ = θ - α - β = θ - 2kπ.But θ = α + β + γ, so θ = 2kπ + γ.Therefore, sin θ = sin(2kπ + γ) = sin γ.So, x = sin θ = sin γ.But from the third equation, sin γ = 5x + 3.So, x = 5x + 3Solving for x:x = 5x + 3Subtract 5x:-4x = 3Divide by -4:x = -3/4So, x = -3/4.Now, let's find sin α, sin β, sin γ.From the first equation, sin α = x + 1 = -3/4 + 1 = 1/4From the second equation, sin β = 3x + 2 = 3*(-3/4) + 2 = -9/4 + 8/4 = (-9 + 8)/4 = -1/4From the third equation, sin γ = 5x + 3 = 5*(-3/4) + 3 = -15/4 + 12/4 = (-15 + 12)/4 = -3/4So, sin α = 1/4, sin β = -1/4, sin γ = -3/4.Now, let's check if these values satisfy the constraints.From sin α = 1/4: valid since 1/4 is within [-1, 1].From sin β = -1/4: valid.From sin γ = -3/4: valid.Also, x = -3/4 is within the earlier constraint of x ∈ [-4/5, -2/5], since -4/5 = -0.8 and -2/5 = -0.4, and -3/4 = -0.75 is between -0.8 and -0.4.So, this is a valid solution.Now, let's check if α + β = 2kπ.Given that sin α = 1/4 and sin β = -1/4, we can find possible angles.Let’s denote α = arcsin(1/4) and β = -arcsin(1/4) + 2πk, since sin(-θ) = -sin θ.So, α + β = arcsin(1/4) - arcsin(1/4) + 2πk = 0 + 2πk = 2πk, which satisfies α + β = 2kπ.Similarly, γ = θ - α - β = θ - 2kπ.But θ = α + β + γ = 2kπ + γ, so sin θ = sin γ = -3/4, which is consistent with x = -3/4.Therefore, this case is valid.So, in this case, the product sin α · sin β · sin γ = (1/4) * (-1/4) * (-3/4) = (1/4)*(3/16) = 3/64.So, in this case, the product is 3/64.Therefore, we have two possible values for the product: 1/8 and 3/64.Wait, but earlier in Case 1 and 2, we had the same product value of 1/8, and in Case 3, we have 3/64.So, are these all the possible cases?Yes, because when we considered the equation cos[(α - β)/2] = -cos[(α + β + 2γ)/2], we had two possibilities: either sin[(α + β)/2] = 0 or the cosine terms equate to negative of each other, leading to the cases where either α + γ = π or β + γ = π.And in the case where sin[(α + β)/2] = 0, we get another solution.Therefore, the possible values of sin α · sin β · sin γ are 1/8 and 3/64.But let me double-check if there are any other possibilities.Wait, in the first case, when we had α + γ = π, we got x = -1/2, leading to sin α = sin β = sin γ = 1/2, and the product 1/8.In the second case, β + γ = π, same result.In the third case, sin[(α + β)/2] = 0, leading to x = -3/4, and the product 3/64.So, these are the only possible cases.Therefore, the possible values are 1/8 and 3/64.

💡Okay, so I have this geometry problem here. Let me try to understand it step by step. We have triangle ABC with sides AB = 20, AC = 34, and BC = 42. There are two semicircles, ω₁ and ω₂, with diameters AB and AC respectively, erected outward from the triangle. Then, there's a common external tangent ℓ to these two semicircles. Next, we're told that the line through point A perpendicular to BC intersects ℓ at point X and BC at point Y. We need to find the length of XY, which can be written in the form m + √n, where m and n are positive integers, and then compute 100m + n.Alright, let's break this down. First, maybe it's helpful to draw a diagram. Since I can't actually draw it, I'll try to visualize it and describe the steps.1. **Setting up the triangle:** Triangle ABC has sides AB = 20, AC = 34, and BC = 42. Let me check if this is a valid triangle. The sum of AB and AC is 54, which is greater than BC = 42. Similarly, the other triangle inequalities hold, so it's a valid triangle.2. **Coordinates:** Maybe assigning coordinates to the points will help. Let me place point B at (0, 0) and point C at (42, 0). Then, point A is somewhere above the x-axis. To find the coordinates of A, I can use the distance formula. Let’s denote A as (x, y). Then, the distance from A to B is 20, so: √[(x - 0)² + (y - 0)²] = 20 ⇒ x² + y² = 400. Similarly, the distance from A to C is 34, so: √[(x - 42)² + (y - 0)²] = 34 ⇒ (x - 42)² + y² = 1156. Subtracting the first equation from the second: (x - 42)² + y² - (x² + y²) = 1156 - 400 ⇒ x² - 84x + 1764 + y² - x² - y² = 756 ⇒ -84x + 1764 = 756 ⇒ -84x = 756 - 1764 ⇒ -84x = -1008 ⇒ x = 12. Plugging x = 12 back into x² + y² = 400: 144 + y² = 400 ⇒ y² = 256 ⇒ y = 16 (since it's above the x-axis). So, coordinates are: - B: (0, 0) - C: (42, 0) - A: (12, 16)3. **Semicircles ω₁ and ω₂:** - ω₁ has diameter AB, so its center is the midpoint of AB. Midpoint of AB is ((0 + 12)/2, (0 + 16)/2) = (6, 8). Radius is half of AB, which is 10. - ω₂ has diameter AC, so its center is the midpoint of AC. Midpoint of AC is ((12 + 42)/2, (16 + 0)/2) = (27, 8). Radius is half of AC, which is 17.4. **Common external tangent ℓ:** We need to find the common external tangent to ω₁ and ω₂. The centers of ω₁ and ω₂ are at (6, 8) and (27, 8) respectively, both with the same y-coordinate. The distance between the centers is 27 - 6 = 21 units. The radii are 10 and 17. For external tangent, the formula for the length of the tangent between two circles is √[d² - (r₁ + r₂)²], but wait, actually, that's for internal tangent. For external tangent, it's √[d² - (r₁ - r₂)²]. Wait, no, actually, I need to recall the correct formula. Let me recall: For two circles with centers separated by distance d, radii r₁ and r₂, the length of the external tangent is √[d² - (r₁ - r₂)²], and the length of the internal tangent is √[d² - (r₁ + r₂)²]. But wait, actually, it's the other way around. Let me double-check. Wait, no. The external tangent doesn't cross between the circles, so the formula should involve the difference of radii. The internal tangent crosses between them, so it involves the sum. So, yes, external tangent length is √[d² - (r₁ - r₂)²], and internal tangent is √[d² - (r₁ + r₂)²]. But in our case, since the semicircles are erected outward, the external tangent would be the one that doesn't cross the line connecting the centers. Wait, but in our case, the centers are aligned horizontally at y = 8. So, the external tangent would be above or below this line. Since the semicircles are erected outward, which direction is that? The semicircles are on the outside of the triangle, so probably above the triangle. Hmm, maybe I need to think differently. Maybe it's better to compute the equations of the external tangent. Let me denote the centers as D(6, 8) and E(27, 8). The radii are r₁ = 10 and r₂ = 17. The external tangent will touch ω₁ at some point and ω₂ at another point, and the tangent line will be at a certain slope. Let me denote the tangent line as y = mx + c. The condition for a line to be tangent to a circle is that the distance from the center to the line equals the radius. So, for ω₁, the distance from D(6, 8) to the line y = mx + c is |m*6 - 8 + c| / √(m² + 1) = 10. Similarly, for ω₂, the distance from E(27, 8) to the line y = mx + c is |m*27 - 8 + c| / √(m² + 1) = 17. So, we have two equations: |6m - 8 + c| = 10√(m² + 1) ...(1) |27m - 8 + c| = 17√(m² + 1) ...(2) Since the tangent is external, the signs of the expressions inside the absolute value should be the same. Let's assume they are both positive. So, we can drop the absolute value: 6m - 8 + c = 10√(m² + 1) ...(1a) 27m - 8 + c = 17√(m² + 1) ...(2a) Subtracting (1a) from (2a): (27m - 8 + c) - (6m - 8 + c) = 17√(m² + 1) - 10√(m² + 1) Simplify: 21m = 7√(m² + 1) Divide both sides by 7: 3m = √(m² + 1) Square both sides: 9m² = m² + 1 ⇒ 8m² = 1 ⇒ m² = 1/8 ⇒ m = ±1/(2√2) = ±√2/4 Since the tangent is above the centers (assuming external tangent above the line DE), the slope should be positive. So, m = √2/4. Now, plug m back into (1a): 6*(√2/4) - 8 + c = 10√((√2/4)² + 1) Simplify: (3√2/2) - 8 + c = 10√( (2/16) + 1 ) = 10√( (1/8) + 1 ) = 10√(9/8) = 10*(3)/(2√2) = (30)/(2√2) = 15/√2 = (15√2)/2 So, (3√2/2) - 8 + c = (15√2)/2 Subtract (3√2/2) from both sides: -8 + c = (15√2)/2 - (3√2)/2 = (12√2)/2 = 6√2 So, c = 6√2 + 8 Therefore, the equation of the tangent line ℓ is: y = (√2/4)x + 8 + 6√25. **Line through A perpendicular to BC:** BC is the base of the triangle from (0, 0) to (42, 0), so it's along the x-axis. Therefore, the line perpendicular to BC through A is a vertical line, since BC is horizontal. Point A is at (12, 16). A vertical line through A is x = 12. This line intersects ℓ at point X and BC at point Y.6. **Finding point Y:** Y is the intersection of x = 12 and BC. Since BC is along y = 0, Y is (12, 0).7. **Finding point X:** X is the intersection of x = 12 and ℓ. Plug x = 12 into ℓ's equation: y = (√2/4)*12 + 8 + 6√2 = 3√2 + 8 + 6√2 = 8 + 9√2 So, X is at (12, 8 + 9√2).8. **Calculating XY:** Since both X and Y are on x = 12, the distance XY is the difference in y-coordinates. Y is at (12, 0), and X is at (12, 8 + 9√2). So, XY = |8 + 9√2 - 0| = 8 + 9√2 Wait, but the problem states that XY can be written as m + √n. However, 8 + 9√2 is already in the form m + k√n, where k is an integer. But the problem specifies m + √n, implying that the coefficient of the square root is 1. Hmm, maybe I made a mistake. Let me double-check my calculations. Wait, in step 4, when I found the equation of the tangent line, I assumed the slope was positive. But maybe the external tangent is below the line DE? Let me check. If I take m = -√2/4, would that make sense? Let's try m = -√2/4. Then, plugging into (1a): 6*(-√2/4) - 8 + c = 10√( ( (-√2/4)^2 ) + 1 ) Simplify: (-3√2/2) - 8 + c = 10√( (2/16) + 1 ) = same as before, 15√2/2 So, (-3√2/2) - 8 + c = 15√2/2 Adding 3√2/2 to both sides: -8 + c = 15√2/2 + 3√2/2 = 18√2/2 = 9√2 So, c = 9√2 + 8 Then, the equation of ℓ would be y = (-√2/4)x + 8 + 9√2 Now, the line x = 12 intersects ℓ at: y = (-√2/4)*12 + 8 + 9√2 = -3√2 + 8 + 9√2 = 8 + 6√2 So, point X would be at (12, 8 + 6√2), and Y is at (12, 0). Then, XY = 8 + 6√2. Wait, but this is still in the form m + k√n, not m + √n. Hmm. Maybe I made a mistake in the earlier steps. Let me check the calculation of the tangent line. Alternatively, perhaps the length of the external tangent is different. Let me recast the problem. The centers are at (6, 8) and (27, 8), with radii 10 and 17. The distance between centers is 21. The formula for the length of the external tangent is √[d² - (r₁ + r₂)²]. Wait, no, that's for internal tangent. Wait, actually, I think I got it wrong earlier. Let me recall: For external tangent, the formula is √[d² - (r₁ - r₂)²], and for internal tangent, it's √[d² - (r₁ + r₂)²]. But in this case, since the semicircles are on the same side (both erected outward from the triangle), the external tangent would be the one that doesn't cross the line connecting the centers. Wait, but in our case, the centers are aligned horizontally, so the external tangent would be above or below. Since the semicircles are erected outward, which direction is that? If the triangle is above BC, then outward would mean the semicircles are above the triangle. So, the external tangent would be above the line DE. Wait, but in my earlier calculation, when I took m positive, I got point X at (12, 8 + 9√2), which is quite high up. When I took m negative, I got X at (12, 8 + 6√2). But the problem says the tangent is erected outward, so maybe the external tangent is above. However, regardless, the length XY is either 8 + 9√2 or 8 + 6√2. But the problem states it's in the form m + √n, so perhaps I need to rationalize or find another approach. Alternatively, maybe my initial coordinate system is complicating things. Let me try a different approach. Instead of placing B at (0, 0) and C at (42, 0), maybe it's better to place Y at (0, 0) since Y is the foot of the perpendicular from A to BC. Let me try that. Let me denote Y as (0, 0). Then, since AY is perpendicular to BC, and Y is on BC, we can set up coordinates accordingly. Let me denote: - Y: (0, 0) - A: (0, h), where h is the height from A to BC. - B: (-b, 0) - C: (c, 0) Then, AB = 20, AC = 34, BC = 42. So, the distance from A to B is √[( -b - 0 )² + (0 - h )²] = √(b² + h²) = 20 ⇒ b² + h² = 400. Similarly, distance from A to C is √[(c - 0 )² + (0 - h )²] = √(c² + h²) = 34 ⇒ c² + h² = 1156. Distance from B to C is |c - (-b)| = c + b = 42 ⇒ b + c = 42. So, we have: 1. b + c = 42 2. b² + h² = 400 3. c² + h² = 1156 Subtracting equation 2 from equation 3: c² - b² = 1156 - 400 ⇒ (c - b)(c + b) = 756 But c + b = 42, so: (c - b)*42 = 756 ⇒ c - b = 756 / 42 = 18. So, we have: c + b = 42 c - b = 18 Adding these: 2c = 60 ⇒ c = 30 ⇒ b = 42 - 30 = 12. So, b = 12, c = 30. Then, from equation 2: b² + h² = 400 ⇒ 144 + h² = 400 ⇒ h² = 256 ⇒ h = 16. So, coordinates are: - Y: (0, 0) - A: (0, 16) - B: (-12, 0) - C: (30, 0) This seems better because now the altitude from A to BC is along the y-axis. Now, semicircles ω₁ and ω₂: - ω₁ has diameter AB, so midpoint is at (-6, 8), radius 10. - ω₂ has diameter AC, so midpoint is at (15, 8), radius 17. So, centers at D(-6, 8) and E(15, 8), radii 10 and 17. Distance between centers DE is 15 - (-6) = 21. Now, to find the common external tangent. The formula for the length of the external tangent between two circles is √[d² - (r₁ - r₂)²]. Wait, but in this case, since the circles are on the same horizontal line, the external tangent will be above or below. Wait, but in our case, the semicircles are erected outward from the triangle, which is above BC. So, the semicircles are above BC, so the external tangent would be below the line DE? Or above? Wait, actually, since both semicircles are above BC, the external tangent would be above the line DE. Wait, but in our coordinate system, DE is at y = 8, and the semicircles are above BC, which is at y = 0. So, the external tangent would be above y = 8. Wait, but let me think. The external tangent doesn't cross between the circles, so since the semicircles are both above BC, the external tangent would be above them. Alternatively, maybe it's better to compute the equations again. Let me denote the tangent line as y = mx + c. The distance from D(-6, 8) to the line is |m*(-6) - 8 + c| / √(m² + 1) = 10. Similarly, the distance from E(15, 8) to the line is |m*15 - 8 + c| / √(m² + 1) = 17. So, we have: | -6m - 8 + c | = 10√(m² + 1) ...(1) |15m - 8 + c | = 17√(m² + 1) ...(2) Assuming the tangent is above the line DE, which is at y = 8, so the y-intercept c should be greater than 8. Also, the slope m could be positive or negative. Let me assume the tangent is above DE, so the expressions inside the absolute value are positive. So, -6m - 8 + c = 10√(m² + 1) ...(1a) 15m - 8 + c = 17√(m² + 1) ...(2a) Subtracting (1a) from (2a): (15m - 8 + c) - (-6m - 8 + c) = 17√(m² + 1) - 10√(m² + 1) Simplify: 21m = 7√(m² + 1) Divide both sides by 7: 3m = √(m² + 1) Square both sides: 9m² = m² + 1 ⇒ 8m² = 1 ⇒ m² = 1/8 ⇒ m = ±1/(2√2) = ±√2/4 Since the tangent is above DE, and given the positions of the centers, the slope should be positive. So, m = √2/4. Now, plug m into (1a): -6*(√2/4) - 8 + c = 10√( (√2/4)^2 + 1 ) Simplify: (-3√2/2) - 8 + c = 10√( (2/16) + 1 ) = 10√(9/8) = 10*(3)/(2√2) = 15/√2 = (15√2)/2 So, (-3√2/2) - 8 + c = (15√2)/2 Adding 3√2/2 to both sides: -8 + c = (15√2)/2 + (3√2)/2 = (18√2)/2 = 9√2 So, c = 9√2 + 8 Therefore, the equation of the tangent line ℓ is: y = (√2/4)x + 8 + 9√2 Now, the line through A perpendicular to BC is the vertical line x = 0 (since A is at (0, 16) and BC is horizontal). Wait, no, in this coordinate system, A is at (0, 16), and the line through A perpendicular to BC is the y-axis, x = 0. Wait, but in the problem statement, the line through A perpendicular to BC intersects ℓ at X and BC at Y. So, Y is the foot of the perpendicular from A to BC, which is (0, 0). And X is the intersection of x = 0 with ℓ. So, plugging x = 0 into ℓ's equation: y = (√2/4)*0 + 8 + 9√2 = 8 + 9√2 Therefore, X is at (0, 8 + 9√2), and Y is at (0, 0). So, the distance XY is |8 + 9√2 - 0| = 8 + 9√2. Wait, but the problem states that XY can be written as m + √n. However, 8 + 9√2 is in the form m + k√n, where k is 9. So, maybe I need to rationalize or express it differently. Alternatively, perhaps I made a mistake in the direction of the tangent. Let me check if the tangent is below DE. If I take m = -√2/4, then: From (1a): -6*(-√2/4) - 8 + c = 10√( ( (-√2/4)^2 ) + 1 ) Simplify: (3√2/2) - 8 + c = 10√(9/8) = 15√2/2 So, 3√2/2 - 8 + c = 15√2/2 Subtract 3√2/2: -8 + c = 12√2/2 = 6√2 So, c = 6√2 + 8 Then, the equation of ℓ is y = (-√2/4)x + 8 + 6√2 Now, the line x = 0 intersects ℓ at y = 8 + 6√2. So, X is at (0, 8 + 6√2), Y is at (0, 0). Then, XY = 8 + 6√2. Still, this is in the form m + k√n, not m + √n. Hmm. Wait, maybe the problem expects the length in terms of the external tangent length. Earlier, I thought the length of the external tangent is √[d² - (r₁ - r₂)²] = √[21² - (17 - 10)²] = √[441 - 49] = √392 = 14√2. So, the length of the external tangent is 14√2. But how does this relate to XY? Wait, in the coordinate system where Y is at (0, 0), and X is at (0, 8 + 9√2), the distance XY is 8 + 9√2. But if the external tangent length is 14√2, maybe there's a relation. Alternatively, perhaps I need to consider the distance from A to ℓ. Since A is at (0, 16), and ℓ is y = (√2/4)x + 8 + 9√2, the distance from A to ℓ is |(√2/4)*0 - 16 + 8 + 9√2| / √( (√2/4)^2 + 1 ) = | -8 + 9√2 | / √( (2/16) + 1 ) = | -8 + 9√2 | / √(9/8) = | -8 + 9√2 | * (2√2)/3. But I'm not sure if this helps. Alternatively, maybe the length XY is equal to the length of the external tangent plus something. Wait, XY is along the y-axis from (0, 0) to (0, 8 + 9√2), which is 8 + 9√2. But the external tangent length is 14√2. Maybe 8 + 9√2 is the correct answer, but the problem says it can be written as m + √n. So, perhaps I need to express 8 + 9√2 as m + √n, but that's not possible unless 9√2 is √n, which would mean n = 81*2 = 162. So, 8 + √162. But 162 = 81*2, so √162 = 9√2. So, yes, 8 + 9√2 = 8 + √162. Therefore, m = 8, n = 162, so 100m + n = 800 + 162 = 962. Wait, but earlier I thought the external tangent length was 14√2, which is about 19.798, while 8 + 9√2 is about 8 + 12.727 = 20.727. These are different. So, maybe my initial assumption about the tangent line was incorrect. Alternatively, perhaps the length XY is equal to the external tangent length plus the distance from A to Y. Wait, AY is the altitude, which is 16. So, XY = AY + something? No, because X is above A. Wait, in the coordinate system where Y is at (0, 0), A is at (0, 16), and X is at (0, 8 + 9√2). So, XY is from (0, 0) to (0, 8 + 9√2), which is 8 + 9√2. But AY is from (0, 16) to (0, 0), which is 16. So, the distance from X to A is |8 + 9√2 - 16| = | -8 + 9√2 |. But that's not directly relevant. Alternatively, maybe I need to consider similar triangles or coordinate geometry differently. Wait, perhaps I made a mistake in the sign when solving for c. Let me check again. When I took m = √2/4, I got c = 8 + 9√2. When I took m = -√2/4, I got c = 8 + 6√2. Both are valid, but which one is the correct external tangent? Since the semicircles are erected outward from the triangle, which is above BC, the external tangent should be above the semicircles. So, the tangent line should be above the line DE (y = 8). Therefore, the y-intercept c should be greater than 8. Both c = 8 + 9√2 and c = 8 + 6√2 are greater than 8, but which one is the correct external tangent. Wait, actually, the external tangent can be on either side, but in this case, since the semicircles are above BC, the external tangent that is relevant is the one above them, which would have a higher y-intercept. So, c = 8 + 9√2 is the correct one. Therefore, XY = 8 + 9√2, which can be written as 8 + √(81*2) = 8 + √162. So, m = 8, n = 162, and 100m + n = 800 + 162 = 962. But wait, earlier I thought the external tangent length was 14√2, which is approximately 19.798, while 8 + 9√2 is approximately 20.727. These are close but not the same. So, maybe I need to reconcile this. Alternatively, perhaps the length XY is equal to the external tangent length plus the distance from A to the tangent line. Wait, the distance from A to ℓ is |(√2/4)*0 - 16 + 8 + 9√2| / √( (√2/4)^2 + 1 ) = | -8 + 9√2 | / √(9/8) = | -8 + 9√2 | * (2√2)/3. Let me compute this: | -8 + 9√2 | = 9√2 - 8 (since 9√2 ≈ 12.727 > 8) So, (9√2 - 8) * (2√2)/3 = (9√2 * 2√2)/3 - (8 * 2√2)/3 = (18*2)/3 - (16√2)/3 = 12 - (16√2)/3 ≈ 12 - 7.542 ≈ 4.458 But this doesn't seem to relate directly to XY. Alternatively, maybe I need to consider the coordinates again. Since X is at (0, 8 + 9√2) and Y is at (0, 0), XY is 8 + 9√2. But the problem states it's in the form m + √n, so 8 + 9√2 = 8 + √(81*2) = 8 + √162. Therefore, m = 8, n = 162, and 100m + n = 800 + 162 = 962. However, earlier I thought the external tangent length was 14√2, which is about 19.798, while 8 + 9√2 is about 20.727. These are different, so I must have made a mistake somewhere. Wait, perhaps the length of the external tangent is indeed 14√2, and XY is equal to that plus something. Let me think. The external tangent length is the distance between the two tangent points on ω₁ and ω₂. But in our case, the line ℓ is the external tangent, and X is the intersection of ℓ with the vertical line through A. So, XY is not the length of the external tangent, but rather the distance from Y to X along the vertical line. Therefore, XY is 8 + 9√2, which is approximately 20.727, while the external tangent length is 14√2 ≈ 19.798. So, they are different. Therefore, the answer should be 8 + √162, so 100m + n = 962. But wait, let me check if 8 + 9√2 can be written as m + √n. Since 9√2 = √(81*2) = √162, yes, so 8 + 9√2 = 8 + √162. Therefore, m = 8, n = 162, and 100m + n = 800 + 162 = 962. However, I recall that in the initial calculation, the length of the external tangent was 14√2, which is different from XY. So, perhaps the problem is expecting the length of the external tangent, but no, the problem clearly states XY. Alternatively, maybe I made a mistake in the coordinate system. Let me try to calculate the distance from X to Y again. In the coordinate system where Y is at (0, 0), X is at (0, 8 + 9√2). Therefore, XY is simply 8 + 9√2, which is 8 + √162. So, m = 8, n = 162, and 100m + n = 962. But wait, let me check if 8 + 9√2 is indeed the correct answer. Alternatively, perhaps I need to consider the length from X to Y as the length from (0, 8 + 9√2) to (0, 0), which is 8 + 9√2. So, yes, that's correct. Therefore, the final answer is 8 + √162, so 100m + n = 800 + 162 = 962. Wait, but I think I made a mistake in the calculation of the tangent line. Let me double-check. When I solved for m, I got m = √2/4, and c = 8 + 9√2. So, the equation of ℓ is y = (√2/4)x + 8 + 9√2. The line x = 0 intersects ℓ at y = 8 + 9√2, so X is at (0, 8 + 9√2). Y is at (0, 0). Therefore, XY is 8 + 9√2. Yes, that seems correct. So, the answer is 8 + √162, which is 8 + 9√2, so m = 8, n = 162, and 100m + n = 962. However, I'm a bit confused because earlier I thought the external tangent length was 14√2, but that's a different length. XY is the distance from Y to X along the vertical line, which is indeed 8 + 9√2. Therefore, I think the correct answer is 962.

💡Alright, so I have this problem where I need to find constants P, Q, and R such that the fraction (5x + 1) divided by (x - 4)(x - 2)^2 can be expressed as P over (x - 4) plus Q over (x - 2) plus R over (x - 2)^2. Hmm, okay, this looks like a partial fraction decomposition problem. I remember that partial fractions are used to break down complicated rational expressions into simpler ones that are easier to work with, especially for integration or other operations.First, I need to recall the general method for partial fractions. When the denominator factors into linear terms, some of which are repeated, we can express the fraction as a sum of simpler fractions with denominators being each of these factors. In this case, the denominator is (x - 4)(x - 2)^2, which means we have a linear term (x - 4) and a repeated linear term (x - 2)^2. So, the decomposition should have terms for each of these factors.The given expression is:(5x + 1) / [(x - 4)(x - 2)^2] = P/(x - 4) + Q/(x - 2) + R/(x - 2)^2Okay, so the next step is to get rid of the denominators by multiplying both sides of the equation by the common denominator, which is (x - 4)(x - 2)^2. This should simplify things because it will eliminate the fractions.Multiplying both sides:(5x + 1) = P(x - 2)^2 + Q(x - 4)(x - 2) + R(x - 4)Now, I have an equation without denominators. The goal is to find the constants P, Q, and R. To do this, I can use specific values of x that simplify the equation by making some terms zero. This is often called the "cover-up" method.First, let's choose x = 4. Plugging x = 4 into the equation:5(4) + 1 = P(4 - 2)^2 + Q(4 - 4)(4 - 2) + R(4 - 4)Simplify:20 + 1 = P(2)^2 + Q(0)(2) + R(0)21 = 4PSo, solving for P:P = 21 / 4Alright, that gives me P. Now, let's find R. For R, I can choose x = 2 because that will make the terms with (x - 2) zero, which should help isolate R.Plugging x = 2 into the equation:5(2) + 1 = P(2 - 2)^2 + Q(2 - 4)(2 - 2) + R(2 - 4)Simplify:10 + 1 = P(0)^2 + Q(-2)(0) + R(-2)11 = 0 + 0 - 2RSo, solving for R:-2R = 11R = -11 / 2Great, now I have P and R. The only remaining constant is Q. To find Q, I can choose another value for x, but since x = 4 and x = 2 already gave me P and R, I need to pick another value. Alternatively, I can expand the equation and compare coefficients.Let me try expanding the right-hand side:P(x - 2)^2 + Q(x - 4)(x - 2) + R(x - 4)First, expand each term:P(x^2 - 4x + 4) + Q(x^2 - 6x + 8) + R(x - 4)Wait, hold on, let me double-check that expansion.(x - 2)^2 is x^2 - 4x + 4, correct.(x - 4)(x - 2) is x^2 - 6x + 8, yes.And (x - 4) is just x - 4.So, substituting back:P(x^2 - 4x + 4) + Q(x^2 - 6x + 8) + R(x - 4)Now, distribute P, Q, and R:P*x^2 - 4P*x + 4P + Q*x^2 - 6Q*x + 8Q + R*x - 4RNow, combine like terms:(P + Q)x^2 + (-4P - 6Q + R)x + (4P + 8Q - 4R)So, the right-hand side simplifies to:(P + Q)x^2 + (-4P - 6Q + R)x + (4P + 8Q - 4R)Now, the left-hand side is 5x + 1, which can be written as:0x^2 + 5x + 1Since these two expressions are equal for all x (except x = 4 and x = 2 where the original function is undefined), their corresponding coefficients must be equal.So, set up equations by equating the coefficients:1. Coefficient of x^2: P + Q = 02. Coefficient of x: -4P - 6Q + R = 53. Constant term: 4P + 8Q - 4R = 1Now, we already know P and R, so we can substitute those values into these equations to solve for Q.From equation 1:P + Q = 0We know P = 21/4, so:21/4 + Q = 0Solving for Q:Q = -21/4Wait, hold on, that seems conflicting because earlier when I thought about it, I thought Q was 15. Maybe I made a mistake here.Wait, no, let's see. Let's go step by step.We have:1. P + Q = 02. -4P - 6Q + R = 53. 4P + 8Q - 4R = 1Given P = 21/4 and R = -11/2.Let's plug these into equation 1:21/4 + Q = 0So, Q = -21/4But that contradicts my initial thought. Hmm, maybe I made a mistake in the expansion.Wait, let me check the expansion again.Original equation after multiplying both sides:5x + 1 = P(x - 2)^2 + Q(x - 4)(x - 2) + R(x - 4)I expanded P(x - 2)^2 as P(x^2 - 4x + 4), which is correct.Q(x - 4)(x - 2) is Q(x^2 - 6x + 8), correct.R(x - 4) is R*x - 4R, correct.So, when I combine them:P*x^2 - 4P*x + 4P + Q*x^2 - 6Q*x + 8Q + R*x - 4RCombine like terms:(P + Q)x^2 + (-4P - 6Q + R)x + (4P + 8Q - 4R)Yes, that seems correct.So, equating to left-hand side:0x^2 + 5x + 1Therefore:1. P + Q = 02. -4P - 6Q + R = 53. 4P + 8Q - 4R = 1Given P = 21/4 and R = -11/2, let's plug into equation 1:21/4 + Q = 0 => Q = -21/4But that seems odd because when I plug into equation 2:-4*(21/4) -6*(-21/4) + (-11/2) = ?Calculate each term:-4*(21/4) = -21-6*(-21/4) = 126/4 = 31.5-11/2 = -5.5So, total: -21 + 31.5 -5.5 = (-21 -5.5) + 31.5 = (-26.5) + 31.5 = 5Which matches equation 2.Similarly, equation 3:4*(21/4) + 8*(-21/4) -4*(-11/2) = ?Calculate each term:4*(21/4) = 218*(-21/4) = -42-4*(-11/2) = 22So, total: 21 -42 +22 = (21 +22) -42 = 43 -42 = 1Which matches equation 3.So, actually, Q is -21/4, not 15. Wait, so where did I go wrong earlier?Looking back at my initial solution, I think I made a mistake in the expansion step. Let me check.In the initial solution, I had:5x + 1 = (21/4)(x - 2)^2 + Q(x - 4)(x - 2) - (11/2)(x - 4)Then, expanding:(21/4)(x^2 -4x +4) - (11/2)(x -4) = (21/4)x^2 -10x +29/2Wait, let's compute that again.(21/4)(x^2 -4x +4) = (21/4)x^2 - (21/4)*4x + (21/4)*4 = (21/4)x^2 -21x +21Similarly, - (11/2)(x -4) = - (11/2)x + 22So, combining these:(21/4)x^2 -21x +21 - (11/2)x +22Combine like terms:(21/4)x^2 + (-21x - (11/2)x) + (21 +22)Convert -21x to halves: -42/2 x, so total x terms: (-42/2 -11/2)x = (-53/2)xConstant terms: 43So, the expanded form is:(21/4)x^2 - (53/2)x +43But the left-hand side is 5x +1, so:(21/4)x^2 - (53/2)x +43 + Q(x -4)(x -2) = 5x +1Wait, no, I think I messed up the initial substitution.Wait, in the initial solution, I had:5x +1 = (21/4)(x -2)^2 + Q(x -4)(x -2) - (11/2)(x -4)But when I expanded, I only expanded the terms with P and R, but forgot to include the Q term.So, actually, the correct expansion should include all three terms: P(x -2)^2 + Q(x -4)(x -2) + R(x -4)So, in my initial solution, I only expanded P and R, but forgot to expand Q, which led to an incorrect conclusion.Therefore, in the initial solution, I incorrectly concluded Q =15, but actually, Q is -21/4.So, that was a mistake in the initial solution.Therefore, the correct values are P =21/4, Q = -21/4, and R = -11/2.Wait, but let me verify this.So, if P =21/4, Q = -21/4, R = -11/2.Let me plug back into the original equation:(5x +1) = P(x -2)^2 + Q(x -4)(x -2) + R(x -4)Substitute P, Q, R:5x +1 = (21/4)(x -2)^2 + (-21/4)(x -4)(x -2) + (-11/2)(x -4)Let me compute each term:First term: (21/4)(x -2)^2 = (21/4)(x^2 -4x +4) = (21/4)x^2 -21x +21Second term: (-21/4)(x -4)(x -2) = (-21/4)(x^2 -6x +8) = (-21/4)x^2 + (126/4)x - (168/4) = (-21/4)x^2 +31.5x -42Third term: (-11/2)(x -4) = (-11/2)x +22Now, add all three terms together:First term: (21/4)x^2 -21x +21Second term: (-21/4)x^2 +31.5x -42Third term: (-11/2)x +22Combine like terms:x^2 terms: (21/4 -21/4)x^2 = 0x^2x terms: (-21x +31.5x -5.5x) = (-21 +31.5 -5.5)x = (5)xConstant terms: 21 -42 +22 = 1So, total: 0x^2 +5x +1, which matches the left-hand side.Therefore, the correct values are P =21/4, Q = -21/4, R = -11/2.Wait, but in the initial solution, I had Q =15, which was incorrect because I forgot to include the Q term in the expansion.Therefore, the correct ordered triple is (21/4, -21/4, -11/2).But let me double-check the initial method.Alternatively, another way is to plug in another value for x, say x =0, and solve for Q.Let me try that.Original equation after multiplying both sides:5x +1 = P(x -2)^2 + Q(x -4)(x -2) + R(x -4)Let x =0:5(0) +1 = P(0 -2)^2 + Q(0 -4)(0 -2) + R(0 -4)Simplify:1 = P(4) + Q(4)(2) + R(-4)1 = 4P +8Q -4RWe know P =21/4, R = -11/2.Plug in:1 = 4*(21/4) +8Q -4*(-11/2)Simplify:1 =21 +8Q +221 =43 +8QSo, 8Q =1 -43 = -42Q = -42 /8 = -21/4Which matches our previous result.Therefore, Q is indeed -21/4.So, the initial solution had an error in the expansion step, leading to an incorrect Q value. The correct Q is -21/4.Therefore, the ordered triple is (21/4, -21/4, -11/2).But wait, let me check if this makes sense.If I plug x =3 into the original equation, let's see:Left-hand side: (5*3 +1)/[(3 -4)(3 -2)^2] = (15 +1)/[(-1)(1)^2] =16/(-1) = -16Right-hand side: P/(3 -4) + Q/(3 -2) + R/(3 -2)^2 = P/(-1) + Q/1 + R/1 = -P + Q + RSubstitute P =21/4, Q =-21/4, R=-11/2:-21/4 + (-21/4) + (-11/2) = (-21/4 -21/4) -11/2 = (-42/4) -11/2 = (-21/2) -11/2 = (-32/2) = -16Which matches the left-hand side.Therefore, the values are correct.So, in conclusion, the constants are P =21/4, Q =-21/4, and R =-11/2.**Final Answer**The ordered triple ((P, Q, R)) is boxed{left(frac{21}{4}, -frac{21}{4}, -frac{11}{2}right)}).