💡Okay, so I have this inequality to prove: given positive real numbers x, y, z, I need to show that (xy + 2yz + 2zx)/(x² + y² + z²) is less than or equal to (1/4)(√33 + 1). Hmm, that seems a bit complicated, but let's break it down step by step.First, I remember that inequalities like this often involve using some standard inequalities like Cauchy-Schwarz or AM-GM. Maybe I can apply one of those here. Let me think about how to approach this.Looking at the numerator, it's a combination of products of variables: xy, 2yz, and 2zx. The denominator is the sum of squares. So, maybe I can relate the numerator to the denominator using some inequality that connects products and squares.I recall that the AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Maybe I can apply that here somehow. Let's see.Let me consider each term in the numerator separately. The term xy is straightforward, but the terms 2yz and 2zx are a bit more complex because of the coefficients. Maybe I can factor out the 2 or find a way to bound these terms.Wait, another thought: perhaps I can use the Cauchy-Schwarz inequality. The Cauchy-Schwarz inequality states that for vectors a and b, (a·b)² ≤ (a·a)(b·b). Maybe I can represent the numerator as a dot product and then apply Cauchy-Schwarz.Let me try that. Let me think of two vectors such that their dot product gives me the numerator. Let's see, if I take vector a as (x, y, z) and vector b as (y, 2z, 2x). Then, the dot product a·b would be x*y + y*2z + z*2x, which is exactly the numerator: xy + 2yz + 2zx.So, according to Cauchy-Schwarz, (a·b)² ≤ (a·a)(b·b). Let's compute both sides.First, a·a is x² + y² + z², which is our denominator. Then, b·b is y² + (2z)² + (2x)² = y² + 4z² + 4x². So, b·b is 4x² + y² + 4z².Therefore, by Cauchy-Schwarz, (xy + 2yz + 2zx)² ≤ (x² + y² + z²)(4x² + y² + 4z²).So, if I take square roots on both sides, I get:xy + 2yz + 2zx ≤ sqrt[(x² + y² + z²)(4x² + y² + 4z²)]But I need to find the maximum value of (xy + 2yz + 2zx)/(x² + y² + z²). So, let me denote S = x² + y² + z². Then, the inequality becomes:(xy + 2yz + 2zx)/S ≤ sqrt[(S)(4x² + y² + 4z²)] / SSimplifying the right-hand side:sqrt[(S)(4x² + y² + 4z²)] / S = sqrt[(4x² + y² + 4z²)/S]So, we have:(xy + 2yz + 2zx)/S ≤ sqrt[(4x² + y² + 4z²)/S]Hmm, that doesn't seem to directly give me the desired inequality. Maybe I need a different approach.Let me try using the AM-GM inequality on the numerator. The numerator is xy + 2yz + 2zx. Maybe I can bound each term separately.First, for the term xy, I know that by AM-GM, xy ≤ (x² + y²)/2.Similarly, for 2yz, I can write 2yz ≤ y² + z², again by AM-GM.Similarly, 2zx ≤ z² + x².So, adding these up:xy + 2yz + 2zx ≤ (x² + y²)/2 + y² + z² + z² + x²Simplify the right-hand side:= (x² + y²)/2 + y² + 2z² + x²= (x²/2 + y²/2) + y² + 2z² + x²= (x²/2 + x²) + (y²/2 + y²) + 2z²= (3x²/2) + (3y²/2) + 2z²So, we have:xy + 2yz + 2zx ≤ (3/2)x² + (3/2)y² + 2z²Therefore, the ratio becomes:(xy + 2yz + 2zx)/(x² + y² + z²) ≤ [(3/2)x² + (3/2)y² + 2z²]/(x² + y² + z²)Let me denote t = z²/(x² + y² + z²). Then, since x, y, z are positive, t is between 0 and 1.But maybe that's complicating things. Alternatively, let me factor out the denominator:= (3/2)(x² + y²) + 2z²)/(x² + y² + z²)Let me write this as:= (3/2)(x² + y² + z²) + (2z² - 3/2 z²)/(x² + y² + z²)Wait, that might not be helpful. Let me think differently.Let me consider the expression [(3/2)x² + (3/2)y² + 2z²]/(x² + y² + z²). Let me write this as:= (3/2)(x² + y²) + 2z²)/(x² + y² + z²)Let me denote A = x² + y² and B = z². Then, the expression becomes:(3/2 A + 2B)/(A + B)I can write this as:(3/2 A + 2B)/(A + B) = (3/2)(A + B) + (2B - 3/2 B)/(A + B)Wait, that might not be helpful. Alternatively, let me write it as:= (3/2)(A + B) + (2B - 3/2 B)/(A + B) = 3/2 + ( (2 - 3/2)B )/(A + B) = 3/2 + (1/2)B/(A + B)Hmm, since B/(A + B) ≤ 1, this gives:≤ 3/2 + 1/2 = 2But that's a very loose bound, and the desired bound is around (1/4)(√33 + 1) ≈ (1/4)(5.744 + 1) ≈ (1/4)(6.744) ≈ 1.686, which is less than 2. So, this approach is too crude.Maybe I need a better way to bound the numerator. Let me think about homogenizing the inequality. Since the inequality is homogeneous, I can assume that x² + y² + z² = 1. Then, I need to maximize xy + 2yz + 2zx under the constraint x² + y² + z² = 1.This is a constrained optimization problem. I can use Lagrange multipliers to find the maximum.Let me set up the Lagrangian:L = xy + 2yz + 2zx - λ(x² + y² + z² - 1)Taking partial derivatives with respect to x, y, z, and λ, and setting them equal to zero:∂L/∂x = y + 2z - 2λx = 0∂L/∂y = x + 2z - 2λy = 0∂L/∂z = 2y + 2x - 2λz = 0∂L/∂λ = -(x² + y² + z² - 1) = 0So, we have the system of equations:1) y + 2z = 2λx2) x + 2z = 2λy3) 2y + 2x = 2λz4) x² + y² + z² = 1Let me try to solve this system.From equation 3: 2y + 2x = 2λz ⇒ y + x = λz ⇒ z = (x + y)/λFrom equation 1: y + 2z = 2λxSubstitute z from equation 3 into equation 1:y + 2*(x + y)/λ = 2λxMultiply both sides by λ:λ y + 2(x + y) = 2λ² xSimilarly, from equation 2: x + 2z = 2λySubstitute z from equation 3 into equation 2:x + 2*(x + y)/λ = 2λyMultiply both sides by λ:λ x + 2(x + y) = 2λ² yNow, we have two equations:From equation 1 substitution:λ y + 2x + 2y = 2λ² x ⇒ (λ + 2) y + 2x = 2λ² xFrom equation 2 substitution:λ x + 2x + 2y = 2λ² y ⇒ (λ + 2) x + 2y = 2λ² yLet me rearrange both equations:From equation 1:(λ + 2) y = 2λ² x - 2x ⇒ y = [2(λ² - 1)]/(λ + 2) xFrom equation 2:(λ + 2) x = 2λ² y - 2y ⇒ x = [2(λ² - 1)]/(λ + 2) yWait, that's interesting. From equation 1, y is proportional to x, and from equation 2, x is proportional to y. Let me denote the proportionality constant.Let me denote k = [2(λ² - 1)]/(λ + 2)From equation 1: y = k xFrom equation 2: x = k ySubstituting y = k x into equation 2: x = k*(k x) ⇒ x = k² x ⇒ k² = 1 ⇒ k = ±1But since x, y, z are positive, k must be positive, so k = 1.Therefore, [2(λ² - 1)]/(λ + 2) = 1 ⇒ 2(λ² - 1) = λ + 2 ⇒ 2λ² - 2 = λ + 2 ⇒ 2λ² - λ - 4 = 0Solving this quadratic equation for λ:λ = [1 ± sqrt(1 + 32)]/(4) = [1 ± sqrt(33)]/4Since λ must be positive (as x, y, z are positive), we take the positive root:λ = [1 + sqrt(33)]/4So, λ = (1 + √33)/4Now, from equation 3: z = (x + y)/λBut from equation 1 and 2, we have y = k x = x, since k = 1. So, y = x.Therefore, z = (x + x)/λ = 2x/λSo, z = (2/λ) xNow, let's express everything in terms of x.Let me denote x = t, then y = t, and z = (2/λ) tNow, from the constraint x² + y² + z² = 1:t² + t² + (4/λ²) t² = 1 ⇒ 2t² + (4/λ²) t² = 1 ⇒ t²(2 + 4/λ²) = 1 ⇒ t² = 1/(2 + 4/λ²)So, t = 1/sqrt(2 + 4/λ²)Now, let's compute the numerator: xy + 2yz + 2zxSince x = y = t and z = (2/λ) t,xy = t*t = t²2yz = 2*t*(2/λ t) = 4/λ t²2zx = 2*(2/λ t)*t = 4/λ t²So, total numerator: t² + 4/λ t² + 4/λ t² = t² + 8/λ t² = t²(1 + 8/λ)Therefore, the ratio is:(xy + 2yz + 2zx)/(x² + y² + z²) = [t²(1 + 8/λ)] / 1 = t²(1 + 8/λ)But t² = 1/(2 + 4/λ²)So, the ratio becomes:[1/(2 + 4/λ²)] * (1 + 8/λ)Let me simplify this expression.First, let's compute 1 + 8/λ:1 + 8/λ = 1 + 8λ/λ² = (λ² + 8λ)/λ²Similarly, 2 + 4/λ² = (2λ² + 4)/λ²Therefore, the ratio is:[(λ² + 8λ)/λ²] / [(2λ² + 4)/λ²] = (λ² + 8λ)/(2λ² + 4)Simplify numerator and denominator:Factor numerator: λ(λ + 8)Denominator: 2(λ² + 2)So, ratio = λ(λ + 8)/(2(λ² + 2))Now, recall that λ = (1 + √33)/4Let me compute λ²:λ² = [(1 + √33)/4]^2 = (1 + 2√33 + 33)/16 = (34 + 2√33)/16 = (17 + √33)/8Now, compute numerator:λ(λ + 8) = [(1 + √33)/4] * [(1 + √33)/4 + 8] = [(1 + √33)/4] * [(1 + √33 + 32)/4] = [(1 + √33)/4] * [(33 + √33)/4]Multiply these:= (1 + √33)(33 + √33)/16Let me compute (1 + √33)(33 + √33):= 1*33 + 1*√33 + √33*33 + √33*√33= 33 + √33 + 33√33 + 33= 66 + 34√33So, numerator = (66 + 34√33)/16Denominator: 2(λ² + 2) = 2[(17 + √33)/8 + 2] = 2[(17 + √33 + 16)/8] = 2[(33 + √33)/8] = (33 + √33)/4Therefore, the ratio is:[(66 + 34√33)/16] / [(33 + √33)/4] = [(66 + 34√33)/16] * [4/(33 + √33)] = [(66 + 34√33)/4] / (33 + √33)Simplify numerator:66 + 34√33 = 2*(33 + 17√33)Wait, actually, let me factor 2 from numerator:= 2*(33 + 17√33)/4 = (33 + 17√33)/2So, the ratio becomes:(33 + 17√33)/2 divided by (33 + √33)= [(33 + 17√33)/2] / (33 + √33) = [33 + 17√33]/[2*(33 + √33)]Let me rationalize the denominator by multiplying numerator and denominator by (33 - √33):= [ (33 + 17√33)(33 - √33) ] / [2*(33 + √33)(33 - √33) ]Compute denominator first:(33 + √33)(33 - √33) = 33² - (√33)² = 1089 - 33 = 1056So, denominator becomes 2*1056 = 2112Now, numerator:(33 + 17√33)(33 - √33) = 33*33 - 33√33 + 17√33*33 - 17√33*√33= 1089 - 33√33 + 561√33 - 17*33= 1089 - 33√33 + 561√33 - 561= (1089 - 561) + (-33√33 + 561√33)= 528 + 528√33So, numerator = 528(1 + √33)Therefore, the ratio is:528(1 + √33)/2112 = (528/2112)(1 + √33) = (1/4)(1 + √33)So, the maximum value of (xy + 2yz + 2zx)/(x² + y² + z²) is indeed (1/4)(1 + √33), which is approximately 1.686.Therefore, we have proven that for positive real numbers x, y, z, the given inequality holds.

💡Okay, so I have this problem where two sides of a triangle are 10 and 12, and the median drawn to the third side is 5. I need to find the area of the triangle. Hmm, let me think about how to approach this.First, let me recall what a median is. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. So, in this case, the median is drawn to the third side, which is opposite the vertex where the two given sides meet. Let me denote the triangle as triangle ABC, where AB = 10, AC = 12, and the median from A to BC is 5. Wait, actually, the median is drawn to the third side, so if AB and AC are the two sides, then the median would be from A to BC. But the length of the median is 5. So, the median AM, where M is the midpoint of BC, is 5.Now, I need to find the area of triangle ABC. I know two sides and the median to the third side. I wonder if there's a formula that relates the sides of a triangle and the length of a median to the area. I remember there is a formula for the length of a median in terms of the sides of the triangle. Let me recall that.The formula for the length of a median (m_a) from vertex A is:m_a = (1/2) * sqrt(2b^2 + 2c^2 - a^2)Where a, b, c are the lengths of the sides of the triangle, and m_a is the median to side a.In this case, the median is 5, so:5 = (1/2) * sqrt(2*10^2 + 2*12^2 - a^2)Wait, let me make sure I get the notation right. If the median is from A to BC, then BC is the side opposite A, so BC is side a. So, the formula becomes:m_a = (1/2) * sqrt(2b^2 + 2c^2 - a^2)Where b and c are the lengths of the other two sides, which are AB = 10 and AC = 12. So, substituting the known values:5 = (1/2) * sqrt(2*(10)^2 + 2*(12)^2 - a^2)Let me compute the right-hand side step by step.First, compute 2*(10)^2:2*100 = 200Then, compute 2*(12)^2:2*144 = 288So, adding those together:200 + 288 = 488So, the equation becomes:5 = (1/2) * sqrt(488 - a^2)Multiply both sides by 2:10 = sqrt(488 - a^2)Now, square both sides:100 = 488 - a^2Solve for a^2:a^2 = 488 - 100 = 388So, a = sqrt(388)Simplify sqrt(388):388 divided by 4 is 97, so sqrt(388) = sqrt(4*97) = 2*sqrt(97)So, the length of side BC is 2*sqrt(97). Hmm, that seems a bit complicated, but okay.Now, I need to find the area of triangle ABC. I know two sides and the included angle? Wait, no, I don't know the included angle. Alternatively, maybe I can use Heron's formula, which requires knowing all three sides.Wait, I have sides AB = 10, AC = 12, and BC = 2*sqrt(97). Let me compute the semi-perimeter first.s = (a + b + c)/2 = (10 + 12 + 2*sqrt(97))/2 = (22 + 2*sqrt(97))/2 = 11 + sqrt(97)Then, Heron's formula is:Area = sqrt[s*(s - a)*(s - b)*(s - c)]So, plugging in the values:Area = sqrt[(11 + sqrt(97))*(11 + sqrt(97) - 10)*(11 + sqrt(97) - 12)*(11 + sqrt(97) - 2*sqrt(97))]Simplify each term inside the square root:First term: 11 + sqrt(97)Second term: 11 + sqrt(97) - 10 = 1 + sqrt(97)Third term: 11 + sqrt(97) - 12 = -1 + sqrt(97)Fourth term: 11 + sqrt(97) - 2*sqrt(97) = 11 - sqrt(97)So, the expression becomes:sqrt[(11 + sqrt(97))*(1 + sqrt(97))*(-1 + sqrt(97))*(11 - sqrt(97))]Hmm, this looks a bit messy, but maybe I can pair the terms to simplify.Notice that (11 + sqrt(97))*(11 - sqrt(97)) is a difference of squares:(11)^2 - (sqrt(97))^2 = 121 - 97 = 24Similarly, (1 + sqrt(97))*(-1 + sqrt(97)) is also a difference of squares:(sqrt(97))^2 - (1)^2 = 97 - 1 = 96So, now the expression inside the square root is:24 * 96 = 2304Therefore, the area is sqrt(2304) = 48Wait, that's a nice number. So, the area is 48.Alternatively, I could have used another method, like coordinate geometry. Let me try that to verify.Let me place the triangle in a coordinate system. Let me put point A at (0, 0). Let me denote point B at (x, y) and point C at (p, q). Since AB = 10, AC = 12, and the median from A to BC is 5.Wait, but maybe it's easier to place point A at (0, 0), and let the midpoint M of BC be at some point (h, k). Since the median AM is 5, the distance from A(0,0) to M(h,k) is 5, so sqrt(h^2 + k^2) = 5.Also, since M is the midpoint of BC, the coordinates of M are ((x + p)/2, (y + q)/2). So, h = (x + p)/2 and k = (y + q)/2.But this might get complicated. Maybe another approach.Alternatively, I remember that if you have two sides and the median, you can use the formula for the area in terms of two sides and the median. Wait, is there such a formula?I think so. Let me recall. The area can be found using the formula:Area = (4/3) * sqrt[s*(s - a)*(s - b)*(s - m)]Where s is the semi-perimeter of the triangle formed by sides a, b, and m. Wait, no, that might not be correct.Alternatively, maybe using vectors or coordinate geometry.Wait, another idea: using Apollonius's theorem. Apollonius's theorem states that in any triangle, the sum of the squares of two sides is equal to twice the square of the median to the third side plus twice the square of half the third side.So, Apollonius's theorem:AB^2 + AC^2 = 2AM^2 + 2BM^2Since M is the midpoint of BC, BM = MC = BC/2.So, plugging in the known values:10^2 + 12^2 = 2*(5)^2 + 2*(BC/2)^2Compute each term:100 + 144 = 2*25 + 2*(BC^2)/4Simplify:244 = 50 + (BC^2)/2Subtract 50 from both sides:194 = (BC^2)/2Multiply both sides by 2:388 = BC^2So, BC = sqrt(388) = 2*sqrt(97), which matches what I found earlier.Now, with all three sides known, I can use Heron's formula as I did before, which gave me an area of 48.Alternatively, maybe I can use the formula for the area in terms of two sides and the included angle. But I don't know the included angle. However, perhaps I can find it using the Law of Cosines.Let me denote angle at A as θ. Then, using the Law of Cosines on triangle ABC:BC^2 = AB^2 + AC^2 - 2*AB*AC*cosθWe know BC^2 = 388, AB = 10, AC = 12.So,388 = 100 + 144 - 2*10*12*cosθSimplify:388 = 244 - 240*cosθSubtract 244 from both sides:144 = -240*cosθDivide both sides by -240:cosθ = -144/240 = -3/5So, cosθ = -3/5. Therefore, sinθ = sqrt(1 - (9/25)) = sqrt(16/25) = 4/5Now, the area of triangle ABC is (1/2)*AB*AC*sinθ = (1/2)*10*12*(4/5) = (1/2)*120*(4/5) = 60*(4/5) = 48So, again, I get the area as 48.Therefore, the area of the triangle is 48.

💡Okay, so I have this dataset: 2, 1, 3, 2, 3. I need to figure out which of the statements A, B, C, or D is incorrect. Let's go through each one step by step.Starting with statement A: "The mean is 2.2." To find the mean, I remember that I need to add up all the numbers and then divide by how many numbers there are. So, adding them up: 2 + 1 + 3 + 2 + 3. Let's see, 2 plus 1 is 3, plus 3 is 6, plus 2 is 8, plus 3 is 11. So the total is 11. There are 5 numbers in the dataset, so the mean is 11 divided by 5, which is 2.2. Okay, so statement A seems correct.Next, statement B: "The variance is 4." Hmm, variance is a measure of how spread out the numbers are. I think the formula for variance is the average of the squared differences from the mean. First, I already know the mean is 2.2. So, I need to subtract the mean from each number, square the result, and then take the average of those squared differences.Let's do that step by step. The numbers are 2, 1, 3, 2, 3.First number: 2. Subtract the mean: 2 - 2.2 = -0.2. Square that: (-0.2)^2 = 0.04.Second number: 1. Subtract the mean: 1 - 2.2 = -1.2. Square that: (-1.2)^2 = 1.44.Third number: 3. Subtract the mean: 3 - 2.2 = 0.8. Square that: 0.8^2 = 0.64.Fourth number: 2. Subtract the mean: 2 - 2.2 = -0.2. Square that: (-0.2)^2 = 0.04.Fifth number: 3. Subtract the mean: 3 - 2.2 = 0.8. Square that: 0.8^2 = 0.64.Now, add up all these squared differences: 0.04 + 1.44 + 0.64 + 0.04 + 0.64. Let's see, 0.04 plus 1.44 is 1.48, plus 0.64 is 2.12, plus 0.04 is 2.16, plus 0.64 is 2.8. So the total of the squared differences is 2.8.Now, to find the variance, I need to divide this by the number of data points, which is 5. So, 2.8 divided by 5 is 0.56. Wait, that's not 4. So statement B says the variance is 4, but I calculated it as 0.56. That means statement B is incorrect.But let me double-check my calculations to make sure I didn't make a mistake. Maybe I messed up somewhere. So, let's go through the squared differences again:For 2: (2 - 2.2)^2 = (-0.2)^2 = 0.04.For 1: (1 - 2.2)^2 = (-1.2)^2 = 1.44.For 3: (3 - 2.2)^2 = (0.8)^2 = 0.64.For 2: (2 - 2.2)^2 = (-0.2)^2 = 0.04.For 3: (3 - 2.2)^2 = (0.8)^2 = 0.64.Adding them up: 0.04 + 1.44 + 0.64 + 0.04 + 0.64.0.04 + 1.44 is 1.48.1.48 + 0.64 is 2.12.2.12 + 0.04 is 2.16.2.16 + 0.64 is 2.8.Yes, that's correct. Then, 2.8 divided by 5 is indeed 0.56. So, variance is 0.56, not 4. So, statement B is definitely incorrect.Moving on to statement C: "The mode is 3 and 2." The mode is the number that appears most frequently in the dataset. Let's count how many times each number appears.Number 1 appears once.Number 2 appears twice.Number 3 appears twice.So, both 2 and 3 appear twice, which is more than any other number. Therefore, the modes are 2 and 3. So, statement C is correct.Finally, statement D: "The median is 2." The median is the middle number when the data is arranged in order. Let's arrange the dataset in ascending order: 1, 2, 2, 3, 3.There are 5 numbers, so the middle one is the third number, which is 2. So, the median is indeed 2. Therefore, statement D is correct.So, out of all the statements, only statement B is incorrect. Therefore, the answer should be B.

💡First, I need to determine the total value of the item based on the given taxes and fees.Let's denote the total value of the item as ( V ).1. **Import Tax**: This is 10% of the portion of ( V ) that exceeds 1,000. So, the import tax is ( 0.10 times (V - 1000) ).2. **Transportation Fee**: This is 5% of the portion of ( V ) that exceeds 2,000. Therefore, the transportation fee is ( 0.05 times (V - 2000) ).3. **Intermediary Fee**: This is 2% of the total value of the item, which is ( 0.02 times V ).The combined total of these fees and taxes is 359.50. So, I can set up the equation:[0.10(V - 1000) + 0.05(V - 2000) + 0.02V = 359.50]Next, I'll simplify and solve for ( V ):[0.10V - 100 + 0.05V - 100 + 0.02V = 359.50]Combine like terms:[0.17V - 200 = 359.50]Add 200 to both sides:[0.17V = 559.50]Finally, divide both sides by 0.17 to find ( V ):[V = frac{559.50}{0.17} = 3291.18]

💡Okay, so I'm trying to solve this problem where we have four positive integers p, q, r, and s. The problem gives us several greatest common divisors (gcds) between these numbers and tells us that the gcd of s and p is between 100 and 150. We need to figure out which of the given options must be a divisor of p. The options are 7, 11, 13, 17, and 19.First, let me write down the given information:1. gcd(p, q) = 402. gcd(q, r) = 453. gcd(r, s) = 604. 100 < gcd(s, p) < 150I think the key here is to analyze the prime factors of each gcd and see how they relate to the prime factors of p, q, r, and s. Let's start by factoring each of the given gcds into their prime factors.- 40 can be factored into 2^3 * 5.- 45 can be factored into 3^2 * 5.- 60 can be factored into 2^2 * 3 * 5.So, from the first gcd, gcd(p, q) = 40, we know that both p and q must have at least the prime factors 2^3 and 5. Similarly, from gcd(q, r) = 45, both q and r must have at least 3^2 and 5. From gcd(r, s) = 60, both r and s must have at least 2^2, 3, and 5.Let me try to express each variable in terms of their prime factors, considering the gcds.Starting with p: Since gcd(p, q) = 40, p must have at least 2^3 and 5. So, p can be written as 2^3 * 5 * m, where m is some positive integer that doesn't share any common factors with q beyond what's already in 40. Similarly, q must have 2^3, 3^2, and 5. So, q can be written as 2^3 * 3^2 * 5 * n, where n is another positive integer.Moving on to r: Since gcd(q, r) = 45, r must have 3^2 and 5. From gcd(r, s) = 60, r must also have 2^2, 3, and 5. So, combining these, r must have at least 2^2, 3^2, and 5. Therefore, r can be written as 2^2 * 3^2 * 5 * o, where o is a positive integer.Similarly, s must have at least 2^2, 3, and 5, so s can be written as 2^2 * 3 * 5 * p, where p is another positive integer. Wait, hold on, I just used p again, but p is already a variable here. Maybe I should use a different letter, like t. So, s = 2^2 * 3 * 5 * t, where t is a positive integer.Now, let's summarize the expressions:- p = 2^3 * 5 * m- q = 2^3 * 3^2 * 5 * n- r = 2^2 * 3^2 * 5 * o- s = 2^2 * 3 * 5 * tNow, we need to find gcd(s, p). Let's express both s and p in terms of their prime factors:- s = 2^2 * 3 * 5 * t- p = 2^3 * 5 * mThe gcd of s and p will be the product of the smallest powers of the common prime factors. So, let's see:- For prime 2: the minimum power between 2^2 (from s) and 2^3 (from p) is 2^2.- For prime 3: p doesn't have 3 as a factor, so the power is 3^0.- For prime 5: both have 5^1, so that's 5^1.- For any other primes, since m and t could have other factors, but the gcd will only consider the common ones.Wait, but hold on. If m and t have any common prime factors, those would also contribute to the gcd(s, p). So, actually, gcd(s, p) = 2^2 * 5 * gcd(m, t). Because m is part of p and t is part of s, so their gcd would multiply with the common prime factors we already have.So, gcd(s, p) = 2^2 * 5 * gcd(m, t) = 4 * 5 * gcd(m, t) = 20 * gcd(m, t).But the problem states that 100 < gcd(s, p) < 150. So, 100 < 20 * gcd(m, t) < 150. Let's divide all parts by 20 to find the range for gcd(m, t):100 / 20 = 5, and 150 / 20 = 7.5. So, 5 < gcd(m, t) < 7.5. Since gcd(m, t) must be an integer, the possible values are 6 or 7.But wait, let me double-check that. 20 * 5 = 100, which is not greater than 100, so gcd(m, t) must be greater than 5. 20 * 6 = 120, which is between 100 and 150. 20 * 7 = 140, which is also between 100 and 150. 20 * 8 = 160, which is above 150, so that's too big. So, gcd(m, t) can be 6 or 7.But wait, gcd(m, t) must be an integer that divides both m and t. So, if gcd(m, t) is 6, then 6 divides both m and t. Similarly, if it's 7, then 7 divides both m and t.But let's think about the prime factors. From earlier, we have:- p = 2^3 * 5 * m- s = 2^2 * 3 * 5 * tWe also have to consider that in the expressions for q and r, the variables n and o are multiplied, but they don't interfere with m and t because they are separate variables.But wait, actually, when we look at the gcd(s, p), we have to ensure that the factors in m and t don't introduce any new primes that would conflict with the previous gcds.Wait, for example, if m had a factor of 3, then p would have a factor of 3, but looking back at gcd(p, q) = 40, which is 2^3 * 5. If p had a factor of 3, then gcd(p, q) would also include 3, but it doesn't. So, p cannot have any prime factors other than 2, 5, and whatever is in m, but m cannot have any prime factors that q already has beyond what's in 40.Wait, q is 2^3 * 3^2 * 5 * n. So, q has primes 2, 3, 5, and whatever is in n. So, p is 2^3 * 5 * m, and since gcd(p, q) = 40, which is 2^3 * 5, that means m cannot have any prime factors that q has beyond 2 and 5. So, m cannot have 3 as a factor because q has 3^2, and if m had 3, then gcd(p, q) would include 3, which it doesn't. Similarly, m cannot have any other prime factors that n has because n is part of q.Similarly, for t, since s is 2^2 * 3 * 5 * t, and r is 2^2 * 3^2 * 5 * o, the gcd(r, s) is 60, which is 2^2 * 3 * 5. So, t cannot have any prime factors that r has beyond 2, 3, and 5. Since r has 2^2, 3^2, 5, and o, t cannot have any prime factors that o has beyond 2, 3, and 5. Wait, but actually, t is part of s, and r is 2^2 * 3^2 * 5 * o. So, the gcd(r, s) is 60, which is 2^2 * 3 * 5. Therefore, t cannot have any prime factors that would make the gcd(r, s) larger. So, t cannot have 3^2 because r already has 3^2, but the gcd only has 3^1, so t can have at most 3^1. Similarly, t can't have any other primes that o has because that would increase the gcd beyond 60.Wait, this is getting a bit complicated. Let me try to simplify.We know that:- p = 2^3 * 5 * m- q = 2^3 * 3^2 * 5 * n- r = 2^2 * 3^2 * 5 * o- s = 2^2 * 3 * 5 * tAnd we have:- gcd(p, q) = 40 = 2^3 * 5, so m and n must not share any common factors beyond 1, and m cannot have 3 as a factor because q has 3^2 and p doesn't.- gcd(q, r) = 45 = 3^2 * 5, so n and o must not share any common factors beyond 1, and o cannot have 2 as a factor because q has 2^3 and r has 2^2, but the gcd is only 3^2 * 5.- gcd(r, s) = 60 = 2^2 * 3 * 5, so o and t must not share any common factors beyond 1, and t cannot have 3^2 because r has 3^2 and s has 3^1, so the gcd is 3^1.Wait, so from gcd(r, s) = 60, which is 2^2 * 3 * 5, we can infer that:- The minimum power of 2 between r and s is 2^2, so s must have at least 2^2, which it does, and r has 2^2, so that's fine.- The minimum power of 3 is 3^1, so s has 3^1 and r has 3^2, so that's fine.- The minimum power of 5 is 5^1, which both have.But importantly, t cannot have any prime factors that would cause the gcd(r, s) to be higher. Since r has 3^2 and s has 3^1, the gcd is 3^1. So, t cannot have 3^2 because then the gcd would be 3^2. Similarly, t cannot have any other prime factors that o has because that would increase the gcd beyond 60.Similarly, looking back at p = 2^3 * 5 * m, since gcd(p, q) = 40, which is 2^3 * 5, m cannot have any prime factors that q has beyond 2 and 5. Since q has 3^2 and 5, m cannot have 3 as a factor because that would make the gcd(p, q) include 3, which it doesn't. So, m must be coprime with 3 and any other primes in q beyond 2 and 5.Similarly, for t, since s = 2^2 * 3 * 5 * t, and gcd(r, s) = 60, which is 2^2 * 3 * 5, t cannot have any prime factors that would cause the gcd(r, s) to be higher. Since r has 2^2, 3^2, and 5, t cannot have 3^2 or any other primes that o has. So, t must be coprime with o and cannot have 3^2.Now, going back to gcd(s, p) = 20 * gcd(m, t). We have 100 < 20 * gcd(m, t) < 150, so 5 < gcd(m, t) < 7.5. Therefore, gcd(m, t) can be 6 or 7.But wait, gcd(m, t) must be an integer, so possible values are 6 or 7.But let's think about the prime factors of m and t. From earlier, m cannot have 3 as a factor because p cannot have 3 (since gcd(p, q) = 40, which doesn't include 3). Similarly, t cannot have 3^2 because of gcd(r, s) = 60. However, t can have 3^1 because s has 3^1.Wait, but if m cannot have 3, then gcd(m, t) cannot have 3 as a factor because m doesn't have 3. So, if gcd(m, t) is 6, which is 2 * 3, but m cannot have 3, so that's impossible. Therefore, gcd(m, t) cannot be 6 because m doesn't have 3, so the gcd cannot include 3.Therefore, the only possible value for gcd(m, t) is 7, which is a prime number. So, gcd(m, t) = 7. Therefore, 7 must divide both m and t.Since m is part of p, which is 2^3 * 5 * m, and m must be divisible by 7, that means 7 divides p.Therefore, the answer must be 7, which is option A.Wait, let me double-check this reasoning.We have:- p = 2^3 * 5 * m- s = 2^2 * 3 * 5 * t- gcd(s, p) = 20 * gcd(m, t)Given that 100 < gcd(s, p) < 150, so 5 < gcd(m, t) < 7.5, so possible values are 6 or 7.But m cannot have 3 because p cannot have 3 (since gcd(p, q) = 40, which doesn't include 3). Therefore, m and t cannot have a gcd that includes 3, so gcd(m, t) cannot be 6. Therefore, the only possible value is 7.Thus, 7 must divide both m and t, so 7 divides p.Therefore, the answer is 7, which is option A.

💡First, I need to simplify the expression (frac{1}{a} + frac{1}{b} - frac{2a + b}{2ab}).I'll start by finding a common denominator for all the terms, which is (2ab). This allows me to rewrite each fraction with the same denominator:[frac{1}{a} = frac{2b}{2ab}, quad frac{1}{b} = frac{2a}{2ab}, quad text{and} quad frac{2a + b}{2ab} text{ remains the same}.]Next, I'll combine these fractions over the common denominator:[frac{2b}{2ab} + frac{2a}{2ab} - frac{2a + b}{2ab} = frac{2b + 2a - (2a + b)}{2ab}.]Now, I'll simplify the numerator by distributing the negative sign and combining like terms:[2b + 2a - 2a - b = b.]This leaves me with:[frac{b}{2ab}.]Finally, I'll simplify the fraction by canceling out the common factor of (b) in the numerator and the denominator:[frac{b}{2ab} = frac{1}{2a}.]So, the simplified form of the expression is (frac{1}{2a}).