🧠:[<think>Okay, so I need to find the minimum value of the expression cos(x + y) + 2 cos x + 2 cos y, where x and y are real numbers. Hmm, let me think about how to approach this. First, trigonometric expressions can sometimes be simplified using identities. Maybe I can use some addition formulas or convert the expression into a different form. Let me recall the cosine addition formula: cos(x + y) = cos x cos y - sin x sin y. If I substitute that into the original expression, it becomes:cos x cos y - sin x sin y + 2 cos x + 2 cos y.Hmm, not sure if that helps directly. Let me see. Maybe grouping terms with cos x and cos y. So:cos x (cos y + 2) + cos y (2) - sin x sin y.Hmm, still not straightforward. Maybe another approach. What if I consider variables substitution? Let me set u = x + y and v = x - y, but I don't know if that will help here. Alternatively, perhaps fix one variable and take derivatives with respect to the other? That could work, but since it's a function of two variables, maybe using calculus to find the critical points.Alternatively, maybe use the method of expressing the expression as a sum of cosines with some amplitude and phase shift. For example, sometimes expressions like A cos x + B sin x can be written as C cos(x + D). But here, the expression is more complicated because it's a function of both x and y. Hmm.Wait, another idea: maybe set a = cos x + cos y and b = sin x + sin y, but not sure. Alternatively, consider expressing the entire expression in terms of sum and product of angles. Let me try to see.Original expression: cos(x + y) + 2 cos x + 2 cos y.Let me denote S = x + y and D = x - y. Then x = (S + D)/2 and y = (S - D)/2. Let's substitute these into the expression.First, cos S + 2 cos[(S + D)/2] + 2 cos[(S - D)/2].Hmm, maybe that's helpful? Let's compute the two cosine terms:cos[(S + D)/2] + cos[(S - D)/2] = 2 cos(S/2) cos(D/2) by the sum-to-product identity. So 2 cos[(S + D)/2] + 2 cos[(S - D)/2] = 4 cos(S/2) cos(D/2). Therefore, the entire expression becomes:cos S + 4 cos(S/2) cos(D/2).Hmm, so now the expression is in terms of S and D. But S and D are variables here; x and y can be any real numbers, so S and D can also be any real numbers, with no constraints except that S = x + y and D = x - y. So maybe we can treat S and D as independent variables? Wait, but D is dependent on S and the original variables x and y. Wait, no, actually, for any real numbers x and y, S and D can be any real numbers as well, since given S and D, we can solve for x and y: x = (S + D)/2, y = (S - D)/2. So S and D are independent variables. Therefore, we can consider the expression as a function of S and D:f(S, D) = cos S + 4 cos(S/2) cos(D/2).Now, to find the minimum of f(S, D), we can treat S and D as separate variables. So perhaps we can first minimize with respect to D, treating S as a constant, and then minimize with respect to S.Let me try that. For a fixed S, the term involving D is 4 cos(S/2) cos(D/2). So if we can minimize this term over D, then we can substitute back into the expression.So, the term 4 cos(S/2) cos(D/2). Let's denote that as 4 cos(S/2) cos(D/2). So, the minimum value of this term with respect to D is when cos(D/2) is minimized. However, cos(D/2) has a range of [-1, 1], so if 4 cos(S/2) is multiplied by the minimum of cos(D/2). But the minimum of cos(D/2) is -1, so the minimum of 4 cos(S/2) cos(D/2) is -4 |cos(S/2)|, right? Wait, because if cos(S/2) is positive, then the minimum is -4 cos(S/2); if cos(S/2) is negative, then the minimum is 4 cos(S/2). So in either case, the minimum is -4 |cos(S/2)|.Therefore, for a fixed S, the minimal value of the term 4 cos(S/2) cos(D/2) over D is -4 |cos(S/2)|. Therefore, the minimal value of f(S, D) over D is cos S - 4 |cos(S/2)|.Therefore, the problem reduces to finding the minimum of the function g(S) = cos S - 4 |cos(S/2)| over all real numbers S.Now, let's analyze this function g(S). Since both cos S and |cos(S/2)| are periodic functions, let's find the period of g(S). The term cos S has period 2π, and |cos(S/2)| has period 2π (since cos(S/2) has period 4π, but the absolute value makes it 2π). Therefore, g(S) is periodic with period 2π. So we can restrict our attention to S in [0, 2π), and find the minimum there.Also, note that |cos(S/2)| is always non-negative, so -4 |cos(S/2)| is non-positive. Therefore, the expression g(S) is cos S minus a non-positive term, which is equivalent to cos S plus a non-negative term. Wait, but we need the minimal value, so we need to see how these two terms interact.Let me write cos S in terms of cos(S/2). Using the double-angle identity: cos S = 2 cos²(S/2) - 1. Therefore, substituting into g(S):g(S) = 2 cos²(S/2) - 1 - 4 |cos(S/2)|.Let me set t = |cos(S/2)|. Since |cos(S/2)| is non-negative and cos(S/2) ranges between -1 and 1, t ranges between 0 and 1. Therefore, substituting t into the expression:g(S) = 2 t² - 1 - 4 t.Therefore, we can consider the function h(t) = 2 t² - 4 t - 1, where t ∈ [0, 1].So now, the problem reduces to finding the minimum of h(t) = 2 t² - 4 t - 1 over t ∈ [0, 1].This is a quadratic function in t. The graph of h(t) is a parabola opening upwards (since the coefficient of t² is positive). The vertex of the parabola is at t = -b/(2a) = 4/(4) = 1. So the vertex is at t = 1. Therefore, since the parabola opens upwards, the minimum over the interval [0, 1] occurs at the vertex t = 1, but wait, the vertex is at t = 1, but the parabola is increasing on t > 1 and decreasing on t < 1. Wait, since the coefficient is positive, it opens upwards, so the vertex is a minimum point. Wait, no. Wait, the vertex is at t = 1, but if the parabola opens upwards, then the vertex is the minimum point. However, our interval is from t = 0 to t = 1. So the minimum on the interval [0, 1] would be at t = 1 if the vertex is at t = 1. Wait, let's compute h(t) at t = 0 and t = 1:h(0) = 2(0)^2 - 4(0) -1 = -1h(1) = 2(1)^2 - 4(1) -1 = 2 -4 -1 = -3So at t = 1, h(t) = -3, and at t = 0, h(t) = -1. Since the vertex is at t = 1, which is the right endpoint of the interval, the minimum on [0, 1] is at t = 1, and the maximum is at t = 0. Therefore, the minimal value of h(t) is -3, achieved when t = 1.But t = |cos(S/2)| = 1, which implies that cos(S/2) = ±1. Therefore, cos(S/2) = 1 or -1. Let's solve for S:If cos(S/2) = 1, then S/2 = 2π k, so S = 4π k, for integer k.If cos(S/2) = -1, then S/2 = π + 2π k, so S = 2π + 4π k, for integer k.But since we are considering S in [0, 2π), then the possible S values where t = 1 are S = 0 and S = 2π (but 2π is excluded, so S = 0). However, when S/2 = π, then S = 2π, which is excluded. Therefore, within [0, 2π), the only solution is S = 0. Therefore, the minimal value of g(S) is -3, achieved when S = 0.But wait, S = x + y. So S = 0 implies x + y = 0, so y = -x. Let's check what happens when S = 0. Then, in the original variables, x + y = 0, so y = -x. Then the original expression becomes:cos(0) + 2 cos x + 2 cos(-x) = 1 + 2 cos x + 2 cos x = 1 + 4 cos x.So, to minimize 1 + 4 cos x. The minimum of cos x is -1, so 1 + 4*(-1) = -3. Which matches the earlier result. So when x is π, y = -x = -π, but cos π = -1, cos(-π) = cos π = -1. So substituting x = π, y = -π, the expression is 1 + 4*(-1) = -3. But wait, so the minimal value is -3. But let me confirm if this is indeed the case. Wait, but earlier steps transformed the original problem into minimizing h(t) over t ∈ [0,1], which gave -3. But let's check if there are other critical points when minimizing g(S) = cos S -4 |cos(S/2)|.Wait, perhaps when we substituted t = |cos(S/2)|, we might have lost some information, because t is always non-negative, but when we set t = |cos(S/2)|, we need to check if the minimal value is indeed achieved when t = 1. However, according to the substitution, h(t) reaches its minimum at t = 1, which gives the minimal value. But let's verify if this is correct.Alternatively, maybe there is a mistake in assuming that the minimal over D gives -4 |cos(S/2)|. Let me go back a few steps.Original expression after substitution: f(S, D) = cos S + 4 cos(S/2) cos(D/2).To find the minimal over D, for fixed S. The term involving D is 4 cos(S/2) cos(D/2). So if cos(S/2) is positive, then to minimize this term, we need to take cos(D/2) as -1, so the term becomes -4 cos(S/2). If cos(S/2) is negative, then to minimize, take cos(D/2) as 1, and the term becomes 4 cos(S/2) (which is negative, since cos(S/2) is negative). So in either case, the minimal term is -4 |cos(S/2)|, yes. So that seems correct.Therefore, for fixed S, the minimal value is cos S -4 |cos(S/2)|, which we converted to h(t) = 2 t² -4 t -1, t ∈ [0,1], and found the minimum is -3. Therefore, the minimal value of the original expression is -3. But let's check another angle. Suppose x = y = 2π/3. Then x + y = 4π/3. Let's compute the expression:cos(4π/3) + 2 cos(2π/3) + 2 cos(2π/3) = cos(4π/3) + 4 cos(2π/3).cos(4π/3) = -1/2, cos(2π/3) = -1/2. So the expression becomes -1/2 + 4*(-1/2) = -1/2 - 2 = -5/2 = -2.5, which is greater than -3. So that's not the minimum.Alternatively, x = π, y = π. Then x + y = 2π. The expression becomes cos(2π) + 2 cos π + 2 cos π = 1 + 2*(-1) + 2*(-1) = 1 -2 -2 = -3. So that also gives -3. So that's consistent with our previous result.Wait, but when x = π and y = π, then x + y = 2π. Then S = 2π. But earlier, we considered S in [0, 2π), but S = 2π is the endpoint. However, cosine is periodic, so S = 2π is equivalent to S = 0. So x = π and y = π gives S = 2π, which is equivalent to S = 0 with a shift. But in that case, when S = 2π, D = x - y = 0. So x = (2π + 0)/2 = π, y = (2π -0)/2 = π. So D = 0.But when we set S = 2π, in the expression cos S -4 |cos(S/2)|, cos(2π) = 1, and |cos(π)| = | -1 | = 1. So 1 -4*1 = -3, which is the same as when S = 0. So indeed, even though S = 2π is the endpoint, the value is the same.Another test case: x = 0, y = 0. Then the expression is cos(0) + 2 cos 0 + 2 cos 0 = 1 + 2 + 2 = 5. So positive.If x = π/2, y = π/2, then cos(π) + 2 cos(π/2) + 2 cos(π/2) = -1 + 0 + 0 = -1, which is greater than -3.If x = π, y = 0, then cos(π) + 2 cos π + 2 cos 0 = -1 + 2*(-1) + 2*1 = -1 -2 + 2 = -1.So the minimum seems to indeed be -3 when x and y are both π or both -π, etc. So it's consistent.Therefore, it seems that the minimal value is -3.But wait, let me check if there's another critical point when minimizing g(S) = cos S -4 |cos(S/2)|. For that, perhaps we can consider the function without substituting t = |cos(S/2)|, but instead analyze it piecewise based on the sign of cos(S/2).Note that |cos(S/2)| = cos(S/2) when cos(S/2) ≥ 0, and |cos(S/2)| = -cos(S/2) when cos(S/2) < 0.Therefore, we can split the analysis into two cases:Case 1: cos(S/2) ≥ 0. Then |cos(S/2)| = cos(S/2). So g(S) = cos S -4 cos(S/2).Case 2: cos(S/2) < 0. Then |cos(S/2)| = -cos(S/2). So g(S) = cos S +4 cos(S/2).Let's handle each case separately.Case 1: cos(S/2) ≥ 0. This implies that S/2 ∈ [-π/2 + 2πk, π/2 + 2πk] for some integer k. Since we're considering S ∈ [0, 2π), then S/2 ∈ [0, π). Therefore, cos(S/2) ≥ 0 when S/2 ∈ [0, π/2], i.e., S ∈ [0, π]. So in the interval [0, π], we can write g(S) = cos S -4 cos(S/2).Similarly, in Case 2: cos(S/2) < 0, which occurs when S/2 ∈ (π/2, 3π/2), but since S ∈ [0, 2π), S/2 ∈ [0, π), so cos(S/2) < 0 when S/2 ∈ (π/2, π), i.e., S ∈ (π, 2π). Therefore, in (π, 2π), g(S) = cos S +4 cos(S/2).So we can split the interval [0, 2π) into [0, π] and (π, 2π), and analyze g(S) in each interval.First, let's analyze Case 1: S ∈ [0, π], g(S) = cos S -4 cos(S/2).Express cos S in terms of cos(S/2): cos S = 2 cos²(S/2) -1. Therefore,g(S) = 2 cos²(S/2) -1 -4 cos(S/2) = 2 cos²(S/2) -4 cos(S/2) -1.Let t = cos(S/2). Since S ∈ [0, π], S/2 ∈ [0, π/2], so t ∈ [0, 1]. So we have:g(t) = 2 t² -4 t -1, t ∈ [0,1].This is the same quadratic function as before. As before, the minimum is at t =1 (vertex at t=1), giving g(1) = -3, and at t=0, g(0) = -1. So on [0,1], the minimum is -3 at t=1, which corresponds to S/2 =0, so S=0. So in Case 1, the minimum is -3 at S=0.Now, Case 2: S ∈ (π, 2π), g(S) = cos S +4 cos(S/2).Again, express cos S in terms of cos(S/2). Since cos S = 2 cos²(S/2) -1, so:g(S) = 2 cos²(S/2) -1 +4 cos(S/2).Let t = cos(S/2). Since S ∈ (π, 2π), S/2 ∈ (π/2, π), so t ∈ (-1, 0). Therefore, substituting t:g(t) = 2 t² -1 +4 t.So this is another quadratic function: 2 t² +4 t -1, with t ∈ (-1, 0).To find the minimum of this quadratic on t ∈ (-1,0). The quadratic opens upwards (coefficient 2 >0), so its minimum is at the vertex. The vertex is at t = -b/(2a) = -4/(4) = -1. But t = -1 is not in the interval (-1,0), but the left endpoint. However, since the interval is open at t = -1 (since S ∈ (π, 2π)), t approaches -1 from the right. So as t approaches -1 from the right, g(t) approaches 2*(-1)^2 +4*(-1) -1 = 2 -4 -1 = -3. Similarly, at t=0 (which is not in the interval), g(t) approaches 2*0 +0 -1 = -1. Therefore, on the interval (-1,0), the quadratic is decreasing from t approaching -1 (g(t) approaches -3) to t approaching 0 (g(t) approaches -1). Therefore, the minimal value in this interval is approaching -3, but does not actually reach it because t=-1 is excluded.But wait, when S approaches π from the right (since S ∈ (π, 2π)), S/2 approaches π/2 from the right, so cos(S/2) approaches 0. Wait, no. Wait, S approaches π from the right, then S/2 approaches π/2 from the right, so cos(S/2) approaches cos(π/2) = 0 from the right, but cos(π/2) is 0, but approaching from the right, cos(S/2) becomes negative. Wait, no. If S approaches π from the right, then S/2 approaches π/2 from the right. Since cosine is positive in (0, π/2) and negative in (π/2, π). Wait, actually, when S is in (π, 2π), S/2 is in (π/2, π), so cos(S/2) is negative. Therefore, as S approaches π from the right, S/2 approaches π/2 from the right, so cos(S/2) approaches 0 from the negative side. As S approaches 2π, S/2 approaches π, so cos(S/2) approaches -1.Therefore, t = cos(S/2) is in (-1, 0). So the minimal value of g(t) =2 t² +4 t -1 occurs as t approaches -1. Let's compute the limit as t approaches -1 from the right:lim_{t→-1+} 2 t² +4 t -1 = 2(1) +4(-1) -1 = 2 -4 -1 = -3.But at t=-1, which corresponds to S/2 = π, so S=2π. However, S=2π is not in the interval (π, 2π) because we defined S ∈ (π, 2π). Therefore, the minimal value in Case 2 is approaching -3 but doesn't actually attain it. Hence, the minimal value of g(S) over S ∈ [0, 2π) is -3, achieved only at S=0 and approaching S=2π.But when S=0, as before, x + y =0, so y = -x. Then, the original expression becomes 1 +4 cos x, which achieves -3 when cos x = -1, i.e., x = π, y = -π. Similarly, when approaching S=2π, which would correspond to x + y approaching 2π. If we set x = π + t, y = π - t, then x + y = 2π, and as t approaches 0, x approaches π, y approaches π. Then, the expression becomes cos(2π) +2 cos(π + t) +2 cos(π - t) =1 +2(-cos t) +2(-cos t) =1 -4 cos t. As t approaches 0, cos t approaches 1, so the expression approaches 1 -4(1) = -3. Therefore, even though S=2π is not included, approaching it gives the same minimal value. Therefore, the minimal value is indeed -3, achieved at points where x + y =0 (mod 2π) with x = π and y = -π, and also when approaching x + y =2π, x and y approaching π. But since when x and y are both π, x + y = 2π, which is allowed because cosine is periodic, so cos(2π) =1, but 2 cos π +2 cos π = -4, so total is 1 -4 = -3. Therefore, the minimal value is achieved at x=π, y=π as well. Wait, but in this case, x + y =2π, so S=2π, which we earlier thought was excluded, but in reality, S=2π is allowed since x and y can be any real numbers, so S can be 2π. So cos(2π) +2 cos π +2 cos π=1 -2 -2= -3. Therefore, x=π, y=π is also a valid point where the minimal value is achieved. Therefore, the minimal value is -3, achieved when x and y are both odd multiples of π, or when x + y is a multiple of 2π and x and y are π apart.Wait, actually, x=π and y=π gives x + y=2π, which is allowed, and gives the expression value -3. Similarly, x=π, y=-π gives x + y=0, which also gives the expression value -3. So both cases are valid.Therefore, after all this analysis, the minimal value of the expression is -3. Therefore, the answer is boxed{-3}.Final Answerboxed{-3}</think>To find the minimum value of the expression (cos(x + y) + 2 cos x + 2 cos y) where (x) and (y) are real numbers, we start by substituting (S = x + y) and (D = x - y). This transforms the expression into:[cos S + 4 cosleft(frac{S}{2}right) cosleft(frac{D}{2}right)]We need to minimize this expression with respect to (S) and (D). For a fixed (S), the term (4 cosleft(frac{S}{2}right) cosleft(frac{D}{2}right)) can be minimized by choosing (cosleft(frac{D}{2}right) = -1) if (cosleft(frac{S}{2}right)) is positive, or (cosleft(frac{D}{2}right) = 1) if (cosleft(frac{S}{2}right)) is negative. This results in the minimal value:[-4 left| cosleft(frac{S}{2}right) right|]Thus, the expression reduces to:[cos S - 4 left| cosleft(frac{S}{2}right) right|]We then convert (cos S) using the double-angle identity (cos S = 2 cos^2left(frac{S}{2}right) - 1), leading to:[2 cos^2left(frac{S}{2}right) - 1 - 4 left| cosleft(frac{S}{2}right) right|]Let (t = left| cosleft(frac{S}{2}right) right|), which ranges from 0 to 1. The function becomes:[2t^2 - 4t - 1]This quadratic function in (t) has its minimum at (t = 1) (since the vertex of (2t^2 - 4t - 1) is at (t = 1)). Substituting (t = 1) gives:[2(1)^2 - 4(1) - 1 = -3]This minimum value is achieved when (left| cosleft(frac{S}{2}right) right| = 1), which implies (frac{S}{2} = kpi) for integer (k), or (S = 2kpi). For (S = 0) or (S = 2pi), the corresponding values of (x) and (y) are (x = pi) and (y = pi) (or other equivalent angles), leading to:[cos(2pi) + 2cos(pi) + 2cos(pi) = 1 - 2 - 2 = -3]Thus, the minimum value of the given expression is (boxed{-3}).,<think>Okay, so I have this problem here: given real numbers x and y, find the minimum value of the expression cos(x+y) + 2cosx + 2cosy. Hmm, trigonometric expression with two variables. I need to find its minimum value. Let me think about how to approach this.First, maybe I can try to simplify the expression or use some trigonometric identities. Let's see. The term cos(x+y) can be expanded using the cosine addition formula. The formula is cos(x + y) = cosx cosy - sinx siny. Let me write that down:cos(x+y) = cosx cosy - sinx siny.So substituting that into the original expression, we get:cosx cosy - sinx siny + 2cosx + 2cosy.Hmm, that gives me an expression with cosx, cosy, sinx, siny, and their products. Not sure if that's helpful yet. Maybe I can group terms with cosx and cosy:= cosx cosy + 2cosx + 2cosy - sinx siny.Alternatively, factor out 2 from the cosx and cosy terms:= cosx cosy + 2(cosx + cosy) - sinx siny.Not sure if that's useful. Maybe I can consider using substitution variables. Let me think. Suppose I set u = x + y and v = x - y. Sometimes that substitution helps in trigonometric expressions. Let me try that.But before that, let me consider if there's another approach. Maybe using calculus, taking partial derivatives with respect to x and y and setting them to zero to find critical points. That's a standard method for finding extrema of functions of multiple variables. Let me try that.Let's denote the function as f(x, y) = cos(x+y) + 2cosx + 2cosy.To find the critical points, compute the partial derivatives with respect to x and y, set them equal to zero, and solve.First, the partial derivative with respect to x:df/dx = -sin(x+y) - 2sinx.Similarly, the partial derivative with respect to y:df/dy = -sin(x+y) - 2siny.Set both partial derivatives to zero:1. -sin(x+y) - 2sinx = 02. -sin(x+y) - 2siny = 0From equations 1 and 2, since both are equal to zero, we can set them equal to each other:-sin(x+y) - 2sinx = -sin(x+y) - 2sinySubtracting -sin(x+y) from both sides gives:-2sinx = -2sinyDivide both sides by -2:sinx = sinySo sinx = siny. Therefore, the solutions are either x = y + 2πk or x = π - y + 2πk for some integer k. Since we're dealing with real numbers and cosine is periodic with period 2π, we can consider x = y or x = π - y without loss of generality (since other solutions would be co-terminal angles).Let's first consider the case where x = y. Then, substituting into equation 1:-sin(2x) - 2sinx = 0But sin(2x) = 2sinx cosx, so:-2sinx cosx - 2sinx = 0Factor out -2sinx:-2sinx (cosx + 1) = 0So either sinx = 0 or cosx = -1.If sinx = 0, then x = πk. Then, since x = y, y = πk as well.If cosx = -1, then x = π + 2πk. Similarly, y = x = π + 2πk.Now let's check these possibilities.First, if x = y = πk. Let's take k = 0, so x = y = 0. Then the original function f(0,0) = cos(0) + 2cos0 + 2cos0 = 1 + 2 + 2 = 5. If k = 1, x = y = π. Then f(π, π) = cos(2π) + 2cosπ + 2cosπ = 1 + 2*(-1) + 2*(-1) = 1 - 2 -2 = -3. Similarly, for other k, the value will repeat every 2π, so possible critical points here give values of 5 and -3.Second, if cosx = -1, then x = π + 2πk. Then y = x = π + 2πk. Then f(π, π) as above is -3. So that's the same as the previous case.Now let's consider the other case where x = π - y. Let's substitute x = π - y into equation 1:-sin(x + y) - 2sinx = 0But x + y = π - y + y = π. So sin(x + y) = sinπ = 0. Therefore, equation 1 becomes:-0 - 2sinx = 0 => -2sinx = 0 => sinx = 0.Therefore, x = πk. Then since x = π - y, y = π - x = π - πk. Let's take k = 0: x = 0, y = π. Then f(0, π) = cos(π) + 2cos0 + 2cosπ = (-1) + 2*1 + 2*(-1) = -1 + 2 - 2 = -1. If k = 1: x = π, y = π - π*1 = 0. Then f(π, 0) = cos(π + 0) + 2cosπ + 2cos0 = (-1) + 2*(-1) + 2*1 = -1 -2 +2 = -1. Similarly, for other k, it's going to cycle through these values. So in this case, the function value is -1.So from the critical points we found, the possible function values are 5, -3, and -1. So the minimum among these is -3. But we need to check if there are other critical points or if this is the actual minimum.Wait, but when we considered x = y, we found that f(π, π) = -3. Is there a possibility of a lower value?Alternatively, maybe there's a critical point where sinx = siny but in different quadrants? But since we considered x = y and x = π - y, which are the general solutions for sinx = siny, I think that's comprehensive.But perhaps we need to check if these critical points are minima or maxima. For that, we can use the second derivative test for functions of two variables. Let me recall how that works.The second derivative test involves computing the Hessian matrix of second partial derivatives:f_xx, f_xy,f_yx, f_yy,and then computing the determinant D = f_xx * f_yy - (f_xy)^2 at the critical point. If D > 0 and f_xx > 0, it's a local minimum; if D > 0 and f_xx < 0, local maximum; if D < 0, a saddle point; if D = 0, inconclusive.First, let's compute the second partial derivatives.First, f(x, y) = cos(x + y) + 2cosx + 2cosy.First partial derivatives:f_x = -sin(x + y) - 2sinx,f_y = -sin(x + y) - 2siny.Second partial derivatives:f_xx = -cos(x + y) - 2cosx,f_yy = -cos(x + y) - 2cosy,f_xy = f_yx = -cos(x + y).So the Hessian determinant D is:[ -cos(x+y) - 2cosx ] * [ -cos(x+y) - 2cosy ] - [ -cos(x+y) ]^2Let's compute this.First, expand the product:[ -cos(x+y) - 2cosx ][ -cos(x+y) - 2cosy ] = [cos(x+y) + 2cosx][cos(x+y) + 2cosy] = cos²(x+y) + 2cos(x+y)cosy + 2cosx cos(x+y) + 4cosx cosy.Then subtract [ -cos(x+y) ]^2 = cos²(x+y).So D = [cos²(x+y) + 2cos(x+y)cosy + 2cosx cos(x+y) + 4cosx cosy] - cos²(x+y)Simplify:= 2cos(x+y)cosy + 2cosx cos(x+y) + 4cosx cosy.Factor out 2cos(x+y):= 2cos(x+y)(cosy + cosx) + 4cosx cosy.Hmm, not sure if that's helpful. Maybe it's better to evaluate D at the critical points.Let's check the critical points we found.First, the point (0, 0):At x=0, y=0:f_xx = -cos(0) - 2cos0 = -1 - 2 = -3,f_yy = -cos(0) - 2cos0 = -1 - 2 = -3,f_xy = -cos(0) = -1,So D = (-3)(-3) - (-1)^2 = 9 - 1 = 8 > 0,and since f_xx = -3 < 0, so (0,0) is a local maximum. So f(0,0)=5 is a local maximum, which matches the value we saw.Second, the point (π, π):At x=π, y=π:f_xx = -cos(2π) - 2cosπ = -1 - 2*(-1) = -1 + 2 = 1,f_yy = -cos(2π) - 2cosπ = same as f_xx = 1,f_xy = -cos(2π) = -1,Thus D = (1)(1) - (-1)^2 = 1 -1 = 0.Hmm, determinant D = 0, so the second derivative test is inconclusive here. So we can't determine the nature of the critical point using this test. Hmm. So maybe (π, π) is a saddle point or a minimum or maximum. Since the function value here is -3, which is lower than the other critical points. Let's check the function around (π, π).Let's take x = π + h, y = π + k, where h and k are small. Then compute f(π + h, π + k):cos( (π + h) + (π + k) ) + 2cos(π + h) + 2cos(π + k)= cos(2π + h + k) + 2cos(π + h) + 2cos(π + k)Since cos(2π + θ) = cosθ, cos(π + θ) = -cosθ,so this becomes cos(h + k) + 2*(-cosh) + 2*(-cosk)= cos(h + k) - 2cosh - 2cosk.Now, expand each term using Taylor series for small h and k:cos(h + k) ≈ 1 - (h + k)^2 / 2,cosh ≈ 1 - h²/2,cosk ≈ 1 - k²/2,So substitute:≈ [1 - (h² + 2hk + k²)/2] - 2[1 - h²/2] - 2[1 - k²/2]= 1 - (h² + 2hk + k²)/2 - 2 + h² - 2 + k²Combine terms:1 - 2 - 2 = -3,Then the quadratic terms:- (h² + 2hk + k²)/2 + h² + k²= (-h²/2 - hk - k²/2) + h² + k²= (h²/2 - hk + k²/2)So overall:f(π + h, π + k) ≈ -3 + (h²/2 - hk + k²/2).Now, the quadratic form h²/2 - hk + k²/2 can be written as (1/2)(h² - 2hk + k²) = (1/2)(h - k)^2 ≥ 0.Therefore, near (π, π), the function is approximately -3 + (1/2)(h - k)^2. Since (h - k)^2 is always non-negative, the function value is at least -3, and increases as we move away from (π, π). Therefore, (π, π) is a local minimum.Therefore, f(π, π) = -3 is a local minimum.Another critical point we found was when x = 0, y = π, or vice versa, which gives f = -1. Let's check the second derivative there.Take (0, π):x = 0, y = π.Compute f_xx = -cos(0 + π) - 2cos0 = -cosπ - 2cos0 = -(-1) - 2*1 = 1 - 2 = -1,f_yy = -cos(0 + π) - 2cosπ = -(-1) - 2*(-1) = 1 + 2 = 3,f_xy = -cos(0 + π) = -(-1) = 1.Therefore, D = (-1)(3) - (1)^2 = -3 -1 = -4 < 0.So D < 0, which means (0, π) is a saddle point. Therefore, the function value -1 is not a minimum.So the only local minimum we found is at (π, π) with value -3. But since the function is periodic, there are infinitely many minima at (π + 2πk, π + 2πm) for integers k, m, all with value -3.But is this the global minimum? Let's check if there might be other minima.Alternatively, maybe we can approach this problem using another method. For example, using substitution variables or converting to a single variable.Alternatively, consider using the method of Lagrange multipliers, but since there are no constraints given, that might not be necessary. Wait, perhaps we can use substitution variables.Let me think. Let's set u = x + y and v = x - y. Then x = (u + v)/2 and y = (u - v)/2. Maybe this substitution can help.But perhaps not. Alternatively, let's consider fixing u = x + y and see if we can express the function in terms of u and another variable.Wait, maybe we can express the function in terms of sum and difference angles. Alternatively, let's see:The given function is f(x, y) = cos(x + y) + 2cosx + 2cosy.Let me denote S = x + y and D = x - y. Then x = (S + D)/2 and y = (S - D)/2. Then:cosS + 2cos[(S + D)/2] + 2cos[(S - D)/2]But that seems more complicated. Alternatively, maybe use sum-to-product formulas on the 2cosx + 2cosy term.Recall that 2cosx + 2cosy = 4 cos[(x + y)/2] cos[(x - y)/2]. Wait, yes:cosx + cosy = 2cos[(x + y)/2] cos[(x - y)/2], so 2cosx + 2cosy = 4cos[(x + y)/2] cos[(x - y)/2].Therefore, the original function becomes:cos(x + y) + 4cos[(x + y)/2] cos[(x - y)/2]Let me denote θ = (x + y)/2 and φ = (x - y)/2. Then x + y = 2θ, and x - y = 2φ. So x = θ + φ, y = θ - φ.Then the function becomes:cos(2θ) + 4cosθ cosφ.But θ and φ are independent variables, since x and y are real numbers. Therefore, θ can be any real number, and φ can be any real number.So f(θ, φ) = cos(2θ) + 4cosθ cosφ.Hmm, maybe this is easier to minimize. Let's see.First, note that for fixed θ, the term 4cosθ cosφ can be minimized over φ. Let's analyze this.The term 4cosθ cosφ. Since cosφ has a maximum of 1 and minimum of -1. Therefore, for fixed θ, the term 4cosθ cosφ ranges between -4|cosθ| and 4|cosθ|. Therefore, the minimum of 4cosθ cosφ is -4|cosθ|.Therefore, for fixed θ, the minimal value of f(θ, φ) is cos(2θ) - 4|cosθ|.Therefore, the problem reduces to minimizing the function g(θ) = cos(2θ) - 4|cosθ| over θ ∈ ℝ.So now, the problem becomes a single-variable optimization problem. Let's compute g(θ).First, note that cos(2θ) = 2cos²θ - 1. Therefore,g(θ) = 2cos²θ - 1 - 4|cosθ|.Let me denote t = |cosθ|. Since |cosθ| is always between 0 and 1, t ∈ [0, 1].Therefore, expressing g in terms of t:g(t) = 2t² - 1 - 4t.So now, the problem reduces to finding the minimum of the quadratic function g(t) = 2t² -4t -1 over t ∈ [0,1].This is a simple calculus problem. Let's compute the derivative of g(t):g'(t) = 4t - 4.Set g'(t) = 0:4t - 4 = 0 => t = 1.So critical point at t = 1. Since t ∈ [0,1], we check the endpoints and the critical point.At t = 0:g(0) = 0 -1 -0 = -1.At t = 1:g(1) = 2(1)² -4(1) -1 = 2 -4 -1 = -3.At t = critical point t=1, which is also the right endpoint.Therefore, the minimum of g(t) over [0,1] is -3, achieved at t=1.Therefore, the minimal value of the original function is -3.But let's confirm this. So when t = |cosθ| = 1, that implies cosθ = ±1. So θ = kπ. Therefore, θ = kπ.Then, from the substitution θ = (x + y)/2, so x + y = 2θ = 2kπ. Therefore, x + y is an integer multiple of 2π.But we also need to consider φ. Recall that in the previous substitution, the minimal value for fixed θ is achieved when 4cosθ cosφ is minimized, which is -4|cosθ|. This occurs when cosφ = -sgn(cosθ). Therefore, when cosθ is positive, we need cosφ = -1; when cosθ is negative, cosφ = 1; and when cosθ = 0, it doesn't matter.But since here t = |cosθ| = 1, so cosθ = ±1. Therefore, if cosθ = 1, then φ must satisfy cosφ = -1, so φ = π + 2πm. Similarly, if cosθ = -1, then cosφ = 1, so φ = 2πm.But φ = (x - y)/2, so when θ = kπ, then x + y = 2kπ, and φ = (x - y)/2.So let's take θ = 0 (k=0): x + y = 0, and φ = (x - y)/2. Then to achieve cosφ = -1 (since cosθ=1), we need φ = π + 2πm. Therefore, x - y = 2π + 4πm. So x - y is an odd multiple of 2π. Therefore, x = y + 2π + 4πm. But since x + y = 0, substituting x = y + 2π + 4πm into x + y = 0:(y + 2π + 4πm) + y = 0 => 2y + 2π + 4πm = 0 => y = -π - 2πm. Therefore, x = (-π -2πm) + 2π + 4πm = π + 2πm.So x = π + 2πm, y = -π -2πm. But angles differing by 2π are equivalent, so y = -π -2πm = π + (-2π -2πm) = π + 2π(-1 -m), which is the same as y = π + 2πk. Therefore, x and y are both π + 2πk. So for example, when m=0, x=π, y=-π, but y=-π is the same as y=π (since cos(-π)=cosπ). Therefore, x=π, y=π (mod 2π).Similarly, if θ = π (k=1), then x + y = 2π, and similar reasoning gives x = π, y = π. Therefore, the minimal value occurs when x and y are both odd multiples of π, i.e., x = y = π + 2πk, leading to the function value of -3. So this aligns with our previous result from the critical point analysis.Therefore, the minimum value of the given expression is indeed -3.But just to be thorough, let me check if there are any other possibilities. For example, could there be a case where |cosθ| is not 1 but we still get a lower value? But according to the substitution, the minimal value is achieved when t=1, which corresponds to |cosθ|=1. Therefore, the minimum is -3. So that seems solid.Alternatively, suppose someone tries specific angles. Let's test x = y = π. Then cos(x+y) = cos(2π) = 1, 2cosx = 2cosπ = -2, 2cosy = -2. So total is 1 -2 -2 = -3. That's correct. If we take x = y = 0, we get 1 + 2 + 2 = 5. If we take x = 0, y = π, we get cosπ + 2cos0 + 2cosπ = -1 + 2 -2 = -1. So these match our earlier results.What if we take other angles, like x = y = π/2? Then cos(π) = -1, 2cos(π/2) = 0, 2cos(π/2) = 0. So total is -1 + 0 + 0 = -1. Which is higher than -3. If we take x = y = π/3, cos(2π/3) = -1/2, 2cos(π/3) = 2*(1/2) = 1, so total is -1/2 +1 +1 = 1.5. Higher. If x = y = 2π/3, cos(4π/3) = -1/2, 2cos(2π/3) = 2*(-1/2) = -1, so total is -1/2 -1 -1 = -2.5, which is higher than -3.Alternatively, take x = π, y = π. As before, get -3. If we take x = π + h, y = π + h, then f(x,y) = cos(2π + 2h) + 2cos(π + h) + 2cos(π + h) = cos(2h) - 2cosh - 2cosh = cos(2h) - 4cosh. Let's see if this can be less than -3.Compute cos(2h) -4cosh. Let h be small. If h = 0, cos0 -4cos0 = 1 -4 = -3. If h approaches π, cos(2π) -4cosπ = 1 -4*(-1) = 5. If h approaches π/2, cos(π) -4cos(π/2) = -1 -0 = -1. So the minimum at h=0 is -3. Therefore, even if we vary h, we can't get below -3 in this case.Alternatively, take x = π + h, y = π - h. Then x + y = 2π, cos(x+y)=1, 2cosx + 2cosy = 2cos(π + h) + 2cos(π - h) = 2*(-cosh) + 2*(-cosh) = -4cosh. So total is 1 -4cosh. The minimum occurs when cosh is maximized, i.e., at h=0: 1 -4*1 = -3. If h is different, cosh is less, so the term 1 -4cosh would be larger. Therefore, again, the minimal value is -3.Thus, through multiple approaches—critical point analysis, substitution to reduce variables, and testing specific cases—it seems that the minimal value is indeed -3. Therefore, the answer should be boxed{-3}.Final Answerboxed{-3}</think>Given real numbers (x) and (y), we need to find the minimum value of the expression (cos(x+y) + 2cos x + 2cos y).First, we use the cosine addition formula to expand (cos(x+y)):[cos(x+y) = cos x cos y - sin x sin y]Substituting this into the original expression, we get:[cos x cos y - sin x sin y + 2cos x + 2cos y]Next, we find the critical points by taking partial derivatives with respect to (x) and (y) and setting them to zero. The partial derivatives are:[frac{partial f}{partial x} = -sin(x+y) - 2sin x][frac{partial f}{partial y} = -sin(x+y) - 2sin y]Setting these partial derivatives to zero, we get:[-sin(x+y) - 2sin x = 0][-sin(x+y) - 2sin y = 0]From these equations, we find that (sin x = sin y). This implies (x = y + 2kpi) or (x = pi - y + 2kpi). We consider the case (x = y) and substitute into the equations, leading to:[-2sin x (cos x + 1) = 0]This gives (sin x = 0) or (cos x = -1). For (sin x = 0), (x = kpi). For (cos x = -1), (x = pi + 2kpi). Evaluating the function at these points, we find:- (x = y = 0) gives (f(0, 0) = 5)- (x = y = pi) gives (f(pi, pi) = -3)Next, we consider the case (x = pi - y). Substituting into the equations, we find (sin x = 0), leading to (x = kpi) and (y = pi - kpi). Evaluating the function at these points, we get:- (x = 0, y = pi) gives (f(0, pi) = -1)Using the second derivative test, we confirm that ((pi, pi)) is a local minimum. Further analysis using substitution variables and quadratic optimization confirms that the minimum value of the function is (-3).Thus, the minimum value of (cos(x+y) + 2cos x + 2cos y) is (boxed{-3}).]

🔑:begin{array}{l}text { 5. }-3 text {. } cos (x+y)+2 cos x+2 cos y+3 =left(1+cos frac{x+y}{2}right)^{2}left(1+cos frac{x-y}{2}right)+ quadleft(1-cos frac{x+y}{2}right)^{2}left(1-cos frac{x-y}{2}right) geqslant 0 .end{array}It is clear that equality can be achieved.Therefore, the minimum value of the given algebraic expression is -3.

🧠:[<think>Okay, so I need to figure out the relationship between M and N given that ab = 1. Let me start by writing down the given expressions for M and N.M is defined as 1/(1 + a) + 1/(1 + b), and N is a/(1 + a) + b/(1 + b). The question is asking whether M is greater than N, equal to N, less than N, or if it's uncertain without more information. The answer choices are A through D.First, since ab = 1, maybe I can express one variable in terms of the other to simplify the expressions. Let's say b = 1/a. Then, substitute this into M and N and see what happens.Starting with M:M = 1/(1 + a) + 1/(1 + b) = 1/(1 + a) + 1/(1 + 1/a).Let me compute the second term: 1/(1 + 1/a) = 1/[(a + 1)/a] = a/(a + 1). So, the second term simplifies to a/(a + 1).Therefore, M = 1/(1 + a) + a/(1 + a) = [1 + a]/(1 + a) = 1. Wait, that's interesting. So M simplifies to 1?Let me check that again. If b = 1/a, then substituting into 1/(1 + b) gives 1/(1 + 1/a) = a/(a + 1). Then, adding that to 1/(1 + a):1/(1 + a) + a/(1 + a) = (1 + a)/(1 + a) = 1. Yes, that seems right. So M = 1 regardless of the value of a (as long as ab = 1, so a ≠ 0, -1, etc.)?Now let's look at N. N is a/(1 + a) + b/(1 + b). Let's substitute b = 1/a here as well.First term: a/(1 + a). Second term: b/(1 + b) = (1/a)/(1 + 1/a) = (1/a)/[(a + 1)/a] = (1/a) * (a/(a + 1)) = 1/(a + 1).So N = a/(1 + a) + 1/(1 + a) = (a + 1)/(a + 1) = 1. Wait, so N is also equal to 1? Then both M and N equal 1, so M = N, which would be answer B. But that seems too straightforward. Let me check my steps again to make sure I didn't make a mistake.Starting with M: 1/(1 + a) + 1/(1 + b). Since ab = 1, so b = 1/a. Then, 1/(1 + a) + a/(1 + a) = (1 + a)/(1 + a) = 1. That seems correct.For N: a/(1 + a) + b/(1 + b). Substitute b = 1/a: a/(1 + a) + (1/a)/(1 + 1/a) = a/(1 + a) + 1/(1 + a) = (a + 1)/(1 + a) = 1. So yes, both M and N equal 1, so they are equal. Therefore, the answer is B.Wait, but let me think again. If a and b are real numbers with ab = 1, but they could be positive or negative. Let me consider if a is negative. For example, if a = -1/2, then b = -2. Let's compute M and N in that case.Let's take a = -1/2, then b = -2.Compute M: 1/(1 + (-1/2)) + 1/(1 + (-2)) = 1/(1/2) + 1/(-1) = 2 + (-1) = 1. Okay, so M is still 1.Compute N: (-1/2)/(1 + (-1/2)) + (-2)/(1 + (-2)) = (-1/2)/(1/2) + (-2)/(-1) = (-1) + 2 = 1. So N is also 1. So even with negative numbers, it still holds.Another test case: a = 2, b = 1/2.M: 1/(1 + 2) + 1/(1 + 1/2) = 1/3 + 2/3 = 1.N: 2/(1 + 2) + (1/2)/(1 + 1/2) = 2/3 + (1/2)/(3/2) = 2/3 + 1/3 = 1.Same result. Maybe it's always 1? So M = N = 1. Therefore, the answer is B. But why does the problem give four choices? Maybe I made a mistake in the substitution.Wait, perhaps I misapplied the substitution. Let me check again.Given ab = 1, so b = 1/a. Then substituting into M:M = 1/(1 + a) + 1/(1 + 1/a) = 1/(1 + a) + a/(a + 1) = [1 + a]/(1 + a) = 1. Correct.Similarly, N = a/(1 + a) + b/(1 + b) = a/(1 + a) + (1/a)/(1 + 1/a) = a/(1 + a) + 1/(1 + a) = [a + 1]/(1 + a) = 1. So both M and N equal 1 regardless of a (as long as ab = 1 and denominators are non-zero). Therefore, M = N, answer B.But the problem is from a competition, so maybe there's a trick? Let me think. What if a or b is -1? Then denominator becomes zero. But the problem states "real numbers a, b satisfy ab = 1". If a = -1, then b = -1, so denominator in M would be 1 + a = 0, which is undefined. So a and b cannot be -1. Similarly, a can't be 0 because then ab = 0 ≠ 1. So a and b are real numbers not equal to 0 or -1. So as long as denominators are defined, the result holds.Therefore, the correct answer is B: M = N. So option B.But wait, let me check once more with another example. Let me take a = 3, then b = 1/3.M = 1/(1 + 3) + 1/(1 + 1/3) = 1/4 + 3/4 = 1.N = 3/(1 + 3) + (1/3)/(1 + 1/3) = 3/4 + (1/3)/(4/3) = 3/4 + 1/4 = 1. Yep, same result.Alternatively, take a = sqrt(2), b = 1/sqrt(2).M = 1/(1 + sqrt(2)) + 1/(1 + 1/sqrt(2)) = [1/(1 + sqrt(2))] + [sqrt(2)/(sqrt(2) + 1)] = [1 + sqrt(2)] / (1 + sqrt(2)) = 1.Similarly, N = sqrt(2)/(1 + sqrt(2)) + (1/sqrt(2))/(1 + 1/sqrt(2)) = sqrt(2)/(1 + sqrt(2)) + (1/sqrt(2))/( (sqrt(2) + 1)/sqrt(2) ) = sqrt(2)/(1 + sqrt(2)) + 1/(1 + sqrt(2)) = [sqrt(2) + 1]/(1 + sqrt(2)) = 1. So in all these test cases, M and N both equal 1. Therefore, the answer is B.But just to make sure, perhaps there's a different approach. Let's compute M - N and see what happens.M - N = [1/(1 + a) + 1/(1 + b)] - [a/(1 + a) + b/(1 + b)].Simplify each term:1/(1 + a) - a/(1 + a) = (1 - a)/(1 + a).Similarly, 1/(1 + b) - b/(1 + b) = (1 - b)/(1 + b).Therefore, M - N = (1 - a)/(1 + a) + (1 - b)/(1 + b).But since ab = 1, perhaps substitute b = 1/a here.So M - N = (1 - a)/(1 + a) + (1 - 1/a)/(1 + 1/a).Simplify the second term:(1 - 1/a) / (1 + 1/a) = [ (a - 1)/a ] / [ (a + 1)/a ] = (a - 1)/(a + 1).Therefore, M - N = (1 - a)/(1 + a) + (a - 1)/(a + 1).Notice that (1 - a)/(1 + a) is the negative of (a - 1)/(a + 1). So:(1 - a)/(1 + a) + (a - 1)/(a + 1) = (1 - a + a - 1)/(1 + a) = 0/(1 + a) = 0.Therefore, M - N = 0, so M = N. Hence, the answer is B.So that confirms it. Both methods, substitution and algebraic manipulation, lead to M = N. Therefore, the correct answer is B.Final Answerboxed{B}</think>Given real numbers (a) and (b) such that (ab = 1), we need to determine the relationship between (M = frac{1}{1+a} + frac{1}{1+b}) and (N = frac{a}{1+a} + frac{b}{1+b}).First, we substitute (b = frac{1}{a}) into the expressions for (M) and (N):For (M):[M = frac{1}{1+a} + frac{1}{1+frac{1}{a}} = frac{1}{1+a} + frac{a}{a+1}]Simplifying the second term:[frac{1}{1+frac{1}{a}} = frac{a}{a+1}]Thus,[M = frac{1}{1+a} + frac{a}{1+a} = frac{1 + a}{1 + a} = 1]For (N):[N = frac{a}{1+a} + frac{frac{1}{a}}{1+frac{1}{a}} = frac{a}{1+a} + frac{1}{a+1}]Simplifying the second term:[frac{frac{1}{a}}{1+frac{1}{a}} = frac{1}{a+1}]Thus,[N = frac{a}{1+a} + frac{1}{1+a} = frac{a + 1}{1 + a} = 1]Both (M) and (N) simplify to 1 regardless of the value of (a) (as long as (ab = 1)). Therefore, (M = N).To confirm, we compute (M - N):[M - N = left(frac{1}{1+a} + frac{1}{1+b}right) - left(frac{a}{1+a} + frac{b}{1+b}right)]Simplifying each term:[frac{1}{1+a} - frac{a}{1+a} = frac{1 - a}{1 + a}, quad frac{1}{1+b} - frac{b}{1+b} = frac{1 - b}{1 + b}]Substituting (b = frac{1}{a}):[M - N = frac{1 - a}{1 + a} + frac{1 - frac{1}{a}}{1 + frac{1}{a}} = frac{1 - a}{1 + a} + frac{a - 1}{a + 1} = 0]Thus, (M - N = 0) implies (M = N).The answer is (boxed{B}).,<think>Okay, so I have this problem here where real numbers a and b satisfy ab = 1. I need to find the relationship between M and N, where M is defined as 1/(1+a) + 1/(1+b) and N is a/(1+a) + b/(1+b). The options are whether M is greater than N, equal to N, less than N, or it's uncertain. Hmm, let me try to figure this out step by step.First, since ab = 1, that might be useful in simplifying expressions. Let me write down M and N again:M = 1/(1+a) + 1/(1+b)N = a/(1+a) + b/(1+b)I need to compare M and N. Maybe if I subtract one from the other, I can see the difference. Let's compute N - M:N - M = [a/(1+a) + b/(1+b)] - [1/(1+a) + 1/(1+b)]Let me compute each term separately. For the first term, a/(1+a) - 1/(1+a) would be (a - 1)/(1+a). Similarly, b/(1+b) - 1/(1+b) would be (b - 1)/(1+b). So combining these:N - M = (a - 1)/(1+a) + (b - 1)/(1+b)Simplify each fraction:For (a - 1)/(1 + a), maybe factor something out? Let's see:(a - 1)/(a + 1) = [a + 1 - 2]/(a + 1) = 1 - 2/(a + 1)Similarly, (b - 1)/(b + 1) = 1 - 2/(b + 1)So then N - M becomes [1 - 2/(a + 1)] + [1 - 2/(b + 1)] = 2 - 2[1/(a + 1) + 1/(b + 1)]But wait, 1/(a + 1) + 1/(b + 1) is exactly M. So N - M = 2 - 2MTherefore, N - M = 2(1 - M)Hmm, so if I can express N - M in terms of M, maybe that helps. But how?Alternatively, perhaps I should express everything in terms of a single variable since ab = 1. If ab = 1, then b = 1/a. Let's substitute that into M and N.So substituting b = 1/a into M:M = 1/(1 + a) + 1/(1 + 1/a) = 1/(1 + a) + 1/(1 + 1/a)Let me compute the second term: 1/(1 + 1/a) = a/(a + 1). Because 1 + 1/a = (a + 1)/a, so reciprocal is a/(a + 1). So:M = 1/(1 + a) + a/(a + 1) = [1 + a]/(1 + a) = 1Wait, that can't be right. Wait, if I compute M as 1/(1+a) + a/(1+a), then that's (1 + a)/(1 + a) = 1. So M = 1?But wait, if M is always 1, regardless of a (as long as ab = 1), then what about N?Similarly, substitute b = 1/a into N:N = a/(1 + a) + b/(1 + b) = a/(1 + a) + (1/a)/(1 + 1/a)Again, compute the second term: (1/a)/(1 + 1/a) = (1/a)/[(a + 1)/a] = (1/a) * (a/(a + 1)) ) = 1/(a + 1)Therefore, N = a/(1 + a) + 1/(1 + a) = (a + 1)/(a + 1) = 1Wait, so both M and N equal 1? Then M = N? So the answer would be B?But that seems too straightforward. Let me check my substitutions again.Given ab = 1, so b = 1/a.Compute M: 1/(1 + a) + 1/(1 + b) = 1/(1 + a) + 1/(1 + 1/a)1/(1 + 1/a) = 1 divided by ( (a + 1)/a ) = a/(a + 1)So M = 1/(a + 1) + a/(a + 1) = (1 + a)/(a + 1) = 1. Correct.Similarly, N = a/(1 + a) + b/(1 + b) = a/(a + 1) + (1/a)/(1 + 1/a)Again, (1/a)/(1 + 1/a) = 1/(a + 1), so N = a/(a + 1) + 1/(a + 1) = (a + 1)/(a + 1) = 1.So both M and N equal 1, regardless of the value of a (as long as a ≠ -1, but since ab = 1, a can't be -1 because then b would be -1, making denominators 1 + a = 0, which is invalid). So as long as a and b are valid (so a ≠ -1, b ≠ -1), M and N both equal 1. Therefore, M = N, so the answer is B.But let me check with specific numbers to make sure. Let's pick a = 1, then b = 1.Compute M: 1/(1 + 1) + 1/(1 + 1) = 1/2 + 1/2 = 1Compute N: 1/(1 + 1) + 1/(1 + 1) = 1/2 + 1/2 = 1. So M = N.Another example: let a = 2, then b = 1/2.M = 1/(1 + 2) + 1/(1 + 1/2) = 1/3 + 1/(3/2) = 1/3 + 2/3 = 1N = 2/(1 + 2) + (1/2)/(1 + 1/2) = 2/3 + (1/2)/(3/2) = 2/3 + (1/2)*(2/3) = 2/3 + 1/3 = 1Same result. How about a negative number? Let a = -2, then b = -1/2. Wait, but then 1 + a = 1 + (-2) = -1, denominator. So 1/(1 + a) = 1/(-1) = -1. Similarly, 1/(1 + b) = 1/(1 - 1/2) = 1/(1/2) = 2. So M = -1 + 2 = 1. N = a/(1 + a) + b/(1 + b) = (-2)/(-1) + (-1/2)/(1 - 1/2) = 2 + (-1/2)/(1/2) = 2 + (-1) = 1. Still M = N = 1.But wait, the problem says "real numbers a, b satisfy ab=1". So even if a and b are negative, as long as their product is 1, and denominators 1 + a and 1 + b are not zero. So for example, if a = -1, then denominator is 0, which is invalid. But as long as a ≠ -1, then it's okay.So in all valid cases, M = N = 1. Therefore, the answer is B) M = N.But wait, why does the problem give the options? Maybe there's a trick here. Wait, let me check with another example where a is not 1. Let's say a = 3, then b = 1/3.M = 1/(1 + 3) + 1/(1 + 1/3) = 1/4 + 1/(4/3) = 1/4 + 3/4 = 1N = 3/(1 + 3) + (1/3)/(1 + 1/3) = 3/4 + (1/3)/(4/3) = 3/4 + 1/4 = 1Same result. Another test case: a = 1/2, then b = 2.M = 1/(1 + 1/2) + 1/(1 + 2) = 1/(3/2) + 1/3 = 2/3 + 1/3 = 1N = (1/2)/(1 + 1/2) + 2/(1 + 2) = (1/2)/(3/2) + 2/3 = (1/2)*(2/3) + 2/3 = 1/3 + 2/3 = 1Still equal. Hmm. So seems like in all cases, M and N equal 1. So the answer is B. But the original problem is from a competition, maybe there's a case where M and N are not equal? Wait, but according to substitution, they are always equal. Maybe I made a mistake in assuming substitution?Wait, let's try to compute M + N. If M = 1/(1+a) + 1/(1+b) and N = a/(1+a) + b/(1+b), then:M + N = [1/(1+a) + a/(1+a)] + [1/(1+b) + b/(1+b)] = [ (1 + a)/(1 + a) ] + [ (1 + b)/(1 + b) ] = 1 + 1 = 2.So M + N = 2. If M and N add up to 2, then if M = N, they would both be 1. If M > N, then M would be greater than 1 and N less than 1. If M < N, then vice versa. But according to substitution, both M and N are 1. So that would mean M = N. But is this a general case?Alternatively, let's compute M - N:M - N = [1/(1+a) + 1/(1+b)] - [a/(1+a) + b/(1+b)] = [ (1 - a)/(1 + a) ] + [ (1 - b)/(1 + b) ]But given ab = 1, let's substitute b = 1/a:M - N = (1 - a)/(1 + a) + (1 - 1/a)/(1 + 1/a)Simplify the second term:(1 - 1/a) / (1 + 1/a) = [ (a - 1)/a ] / [ (a + 1)/a ] = (a - 1)/(a + 1)So M - N = (1 - a)/(1 + a) + (a - 1)/(a + 1) = [ (1 - a) + (a - 1) ] / (1 + a) ) = 0/(1 + a) = 0Therefore, M - N = 0 => M = NSo this holds for any a ≠ -1, given that ab = 1. Therefore, the correct answer is B) M = N.But just to be thorough, let me check another example where a is a negative number other than -1. Let's say a = -1/2, then b = -2.Compute M: 1/(1 + (-1/2)) + 1/(1 + (-2)) = 1/(1/2) + 1/(-1) = 2 + (-1) = 1Compute N: (-1/2)/(1 + (-1/2)) + (-2)/(1 + (-2)) = (-1/2)/(1/2) + (-2)/(-1) = (-1) + 2 = 1Again, M = N = 1. So even with negative numbers, they still equal 1.Therefore, the relationship is M = N. So the answer is B.Final Answerboxed{B}</think>Given real numbers (a) and (b) that satisfy (ab = 1), we need to determine the relationship between (M) and (N), where (M = frac{1}{1+a} + frac{1}{1+b}) and (N = frac{a}{1+a} + frac{b}{1+b}).First, we substitute (b = frac{1}{a}) into the expressions for (M) and (N):1. Calculating (M): [ M = frac{1}{1+a} + frac{1}{1+frac{1}{a}} = frac{1}{1+a} + frac{a}{a+1} ] Simplifying the second term: [ frac{1}{1+frac{1}{a}} = frac{a}{a+1} ] Therefore: [ M = frac{1}{1+a} + frac{a}{1+a} = frac{1 + a}{1 + a} = 1 ]2. Calculating (N): [ N = frac{a}{1+a} + frac{frac{1}{a}}{1+frac{1}{a}} = frac{a}{1+a} + frac{1}{a+1} ] Simplifying the second term: [ frac{frac{1}{a}}{1+frac{1}{a}} = frac{1}{a+1} ] Therefore: [ N = frac{a}{1+a} + frac{1}{1+a} = frac{a + 1}{1 + a} = 1 ]Both (M) and (N) simplify to 1, regardless of the value of (a) (as long as (a neq -1) and (b neq -1) to avoid division by zero). To further verify, we compute (M + N):[M + N = left(frac{1}{1+a} + frac{1}{1+b}right) + left(frac{a}{1+a} + frac{b}{1+b}right) = frac{1 + a}{1 + a} + frac{1 + b}{1 + b} = 1 + 1 = 2]Since (M + N = 2) and both (M) and (N) are equal, they must each be 1. Thus, (M = N).The final answer is (boxed{B}).]

🔑:[Solution] From the given, we knowM=frac{1}{1+a}+frac{1}{1+b}=frac{b}{b+a b}+frac{a}{a+a b}=frac{b}{b+1}+frac{a}{a+1}=NTherefore, the correct choice is (B).

🧠:[<think>Okay, so I need to figure out the dimensions of a cone and a frustum that each have twice the volume of a sphere with radius r. Then, I also need to find the ratio of the tangents of their slant heights. Hmm, let's start by recalling some formulas.First, the volume of the sphere is (4/3)πr³. The problem says both the cone and frustum have twice this volume, so each should have a volume of (8/3)πr³. Starting with the complete cone. The volume of a cone is (1/3)πR²h, where R is the base radius and h is the height. So, we set (1/3)πR²h = (8/3)πr³. Simplifying, that gives R²h = 8r³. So that's one equation relating R and h.But since the cone is constructed around the sphere, I think that means the sphere is inscribed inside the cone. If the sphere is inscribed, then the cone is tangent to the sphere. That should relate the dimensions of the cone to the radius of the sphere.To find the relationship, I can use the formula for the radius of an inscribed sphere (inradius) in a cone. Wait, how does that work? Let me visualize a cone with a sphere inside it, touching the base and the sides. The sphere will touch the cone along a circle. The inradius of a cone can be found using the formula related to the height and the slant height.Alternatively, maybe using similar triangles. Let's try that.If the sphere is inscribed in the cone, its center is at a distance r from the base of the cone (since the sphere has radius r and touches the base). The center of the sphere is also at a distance r from the sides of the cone. Let me set up coordinates with the vertex of the cone at the origin, extending along the positive y-axis. Wait, no, maybe it's easier to place the base of the cone at y = 0 and the vertex at y = h. Then the center of the sphere would be at y = r, because the sphere touches the base at y=0. Wait, but the sphere has radius r, so the center must be at y = r. Then the sphere also touches the sides of the cone. The sphere touches the cone's lateral surface at some point. Let me consider the cross-section of the cone through its axis. It's a triangle with base 2R and height h. The sphere inscribed in this triangle will have its center at (0, r) in this coordinate system (assuming the base is along the x-axis from (-R, 0) to (R, 0), and the vertex is at (0, h)). The sphere must be tangent to the sides of the triangle. The sides of the cone are the lines from (0, h) to (R, 0) and (-R, 0). Let's focus on the right side: the line from (0, h) to (R, 0). The equation of this line is y = (-h/R)x + h. The distance from the center of the sphere (0, r) to this line must be equal to the radius of the sphere, which is r.The formula for the distance from a point (x₀, y₀) to the line ax + by + c = 0 is |ax₀ + by₀ + c| / sqrt(a² + b²). Let's rearrange the equation of the cone's side to standard form: (h/R)x + y - h = 0. So, a = h/R, b = 1, c = -h.The distance from (0, r) to this line is |(h/R)(0) + 1*(r) - h| / sqrt((h/R)² + 1²) = |r - h| / sqrt((h²/R²) + 1). This distance must equal the radius of the sphere, which is r. So:|r - h| / sqrt(h²/R² + 1) = r.Since h > r (the center of the sphere is at y = r, so the height of the cone must be greater than r), we can drop the absolute value:(h - r) / sqrt(h²/R² + 1) = r.Let's square both sides to eliminate the square root:(h - r)² / (h²/R² + 1) = r².Multiply both sides by denominator:(h - r)² = r²(h²/R² + 1).Let me expand the left side:h² - 2hr + r² = r²(h²/R² + 1).Divide both sides by r²:(h² - 2hr + r²)/r² = h²/R² + 1.Simplify left side:(h²/r² - 2h/r + 1) = h²/R² + 1.Subtract 1 from both sides:h²/r² - 2h/r = h²/R².Multiply both sides by R²r² to eliminate denominators:h²R² - 2hR²r = h²r².Bring all terms to one side:h²R² - 2hR²r - h²r² = 0.Factor h²R² - h²r² = h²(R² - r²), so:h²(R² - r²) - 2hR²r = 0.Factor h:h[h(R² - r²) - 2R²r] = 0.Since h ≠ 0, we have:h(R² - r²) - 2R²r = 0.Solve for h:h(R² - r²) = 2R²r.So,h = (2R²r)/(R² - r²).But we also have from the volume equation:R²h = 8r³.So substitute h from the previous equation into this:R² * (2R²r)/(R² - r²) = 8r³.Simplify:(2R⁴r)/(R² - r²) = 8r³.Divide both sides by r:(2R⁴)/(R² - r²) = 8r².Multiply both sides by (R² - r²):2R⁴ = 8r²(R² - r²).Divide both sides by 2:R⁴ = 4r²(R² - r²).Expand the right side:R⁴ = 4r²R² - 4r⁴.Bring all terms to left:R⁴ - 4r²R² + 4r⁴ = 0.This is a quadratic in terms of R². Let me set x = R²:x² - 4r²x + 4r⁴ = 0.Solve using quadratic formula:x = [4r² ± sqrt(16r⁴ - 16r⁴)]/2 = [4r² ± 0]/2 = 2r².So x = 2r², which means R² = 2r², so R = r√2.Then, substitute back into h:h = (2*(2r²)*r)/(2r² - r²) = (4r³)/(r²) = 4r.So the complete cone has base radius R = r√2 and height h = 4r.Alright, that's the complete cone. Now the frustum. A frustum is a portion of a cone between two parallel planes. The volume of a frustum is (1/3)πh(R² + Rr + r²), where R is the radius of the base, r is the radius of the top, and h is the height of the frustum.But we need to construct the frustum around the sphere such that its volume is twice the sphere's volume, which is (8/3)πr³. So:(1/3)πh(R² + Rr + r²) = (8/3)πr³.Simplifying:h(R² + Rr + r²) = 8r³.But similar to the cone, the frustum must be circumscribed around the sphere. So the sphere is tangent to both the base, the top, and the lateral surface of the frustum.Wait, but a frustum has two circular bases. If the sphere is inside the frustum and tangent to both the top and bottom bases, then the distance between the two bases (the height of the frustum) must be equal to the diameter of the sphere, which is 2r. But that can't be, because then the height of the frustum would be 2r, but the volume would be (1/3)π*2r*(R² + Rr + r²) = (2/3)πr(R² + Rr + r²). Setting this equal to (8/3)πr³, we get R² + Rr + r² = 4r², so R² + Rr = 3r². But if the frustum's height is 2r, but the sphere is also tangent to the lateral sides. Hmm, maybe my initial thought is wrong.Wait, maybe the frustum isn't surrounding the sphere such that the sphere touches both top and bottom. Maybe the sphere is tangent to the lateral surface and one of the bases? Or is it inscribed in the frustum, touching both bases and the lateral surface?Inscribed sphere in a frustum. For a sphere to be inscribed in a frustum, it must be tangent to both the top and bottom bases and the lateral surface. The existence of such a sphere requires certain relations between the radii and the height.Let me check. If the sphere is tangent to both the top and bottom bases, then the distance between the bases (the height of the frustum) is equal to the diameter of the sphere, which is 2r. So h = 2r. Then, the sphere is also tangent to the lateral sides. So in that case, the center of the sphere is halfway between the two bases, so at a distance r from each. Let's model this.Again, considering the cross-section of the frustum. The frustum is a trapezoid. The sphere is tangent to the top, bottom, and the lateral sides. The center of the sphere is at the midpoint between the two bases, which are separated by 2r. The cross-section is a trapezoid with the two parallel sides (the radii of the bases) R and r1 (let's say R is the larger radius at the bottom, and r1 is the smaller at the top). The lateral sides are the sides of the trapezoid.Wait, but if the frustum is constructed around the sphere, then the sphere must fit inside the frustum and be tangent to both the top, bottom, and sides. So similar to the cone, but now with two radii. Let me try to derive the relations.Let me set up coordinates with the center of the sphere at (0, r), since the height of the frustum is 2r (from y=0 to y=2r). Wait, no. If the frustum has height h, and the sphere is inscribed, then the height h must be 2r. So the sphere touches the bottom base at y=0 and the top base at y=2r. The center is at (0, r). Then the sphere must also be tangent to the lateral sides. The lateral sides of the frustum connect the edges of the two bases. Suppose the bottom base has radius R and the top base has radius r1. So in cross-section, the trapezoid has vertices at (R, 0), (-R, 0) at the bottom, and (r1, 2r), (-r1, 2r) at the top.The lateral edges are the lines connecting (R, 0) to (r1, 2r) and (-R, 0) to (-r1, 2r). The sphere is centered at (0, r) with radius r. The distance from the center of the sphere to each lateral side must be equal to r.Let's compute the distance from (0, r) to the right lateral side. The equation of the right lateral side is the line connecting (R, 0) to (r1, 2r). Let's find its equation.First, compute the slope: m = (2r - 0)/(r1 - R) = 2r/(r1 - R). So the equation is y - 0 = m(x - R), which is y = [2r/(r1 - R)](x - R).Rewriting in standard form (ax + by + c = 0):[2r/(r1 - R)]x - y - [2rR/(r1 - R)] = 0.So coefficients are:a = 2r/(r1 - R)b = -1c = -2rR/(r1 - R)The distance from (0, r) to this line is |a*0 + b*r + c| / sqrt(a² + b²) = | - r - 2rR/(r1 - R) | / sqrt( (4r²)/(r1 - R)² + 1 )This distance must equal the radius of the sphere, which is r. So:| - r - 2rR/(r1 - R) | / sqrt( (4r²)/(r1 - R)² + 1 ) = rLet me handle the absolute value. The expression inside is -r - 2rR/(r1 - R). Let's factor out -r:- r [1 + 2R/(r1 - R)] = -r [ (r1 - R + 2R)/(r1 - R) ) ] = -r [ (r1 + R)/(r1 - R) ) ]So the absolute value becomes r | (r1 + R)/(r1 - R) |. Since R > r1 (assuming the bottom radius is larger), the denominator (r1 - R) is negative, and numerator (r1 + R) is positive, so the absolute value is r*(r1 + R)/(R - r1).Thus, the left side becomes [r*(r1 + R)/(R - r1)] / sqrt( (4r²)/(R - r1)² + 1 ) = r.Cancel out the r:(r1 + R)/(R - r1) / sqrt( (4r²)/(R - r1)² + 1 ) = 1Multiply numerator and denominator by (R - r1):(r1 + R) / sqrt(4r² + (R - r1)²) ) = 1Square both sides:(r1 + R)² / (4r² + (R - r1)²) = 1Multiply denominator:(r1 + R)² = 4r² + (R - r1)²Expand both sides:Left side: r1² + 2Rr1 + R²Right side: 4r² + R² - 2Rr1 + r1²Subtract right side from left side:(r1² + 2Rr1 + R²) - (4r² + R² - 2Rr1 + r1²) = 0Simplify:2Rr1 + R² - 4r² - R² + 2Rr1 = 0Combine like terms:4Rr1 - 4r² = 0Divide both sides by 4:Rr1 - r² = 0 => Rr1 = r²So we have Rr1 = r². That's one equation.Now, the volume of the frustum is (1/3)πh(R² + Rr1 + r1²). We know h = 2r, so substituting h and volume equation:(1/3)π*2r*(R² + Rr1 + r1²) = (8/3)πr³Simplify:2r*(R² + Rr1 + r1²) = 8r³Divide both sides by 2r:R² + Rr1 + r1² = 4r²But from earlier, we have Rr1 = r². Let's denote that Rr1 = r². So maybe we can express R in terms of r1: R = r² / r1Substitute into the equation R² + Rr1 + r1²:(r² / r1)² + (r² / r1)*r1 + r1² = r⁴ / r1² + r² + r1²Set equal to 4r²:r⁴ / r1² + r² + r1² = 4r²Subtract 4r²:r⁴ / r1² + r² + r1² - 4r² = 0 => r⁴ / r1² + r1² - 3r² = 0Multiply through by r1² to eliminate the denominator:r⁴ + r1⁴ - 3r²r1² = 0This is a quartic equation in terms of r1. Let me set x = r1². Then equation becomes:r⁴ + x² - 3r²x = 0Which is a quadratic in x:x² - 3r²x + r⁴ = 0Solving using quadratic formula:x = [3r² ± sqrt(9r⁴ - 4r⁴)] / 2 = [3r² ± sqrt(5r⁴)] / 2 = [3r² ± r²√5]/2 = r²(3 ± √5)/2Thus, x = r1² = r²(3 ± √5)/2. Therefore, r1 = r * sqrt( (3 ± √5)/2 )But since R and r1 must be positive, and R = r² / r1, we need to take the positive roots. Let's analyze which sign to take. Let's compute both possibilities:Case 1: r1 = r * sqrt( (3 + √5)/2 )Then R = r² / [ r * sqrt( (3 + √5)/2 ) ] = r / sqrt( (3 + √5)/2 )Multiply numerator and denominator by sqrt(2):r * sqrt(2) / sqrt(3 + √5)But sqrt(3 + √5) can be simplified. Let me see:Let me rationalize sqrt(3 + √5). Suppose sqrt(a + b√c) = sqrt(d) + sqrt(e). Then squaring gives a + b√c = d + e + 2sqrt(de). Let's set a = 3, b = 1, c =5. Then:3 + √5 = d + e + 2sqrt(de)Looking for integers d and e. Let's try d=1, e=2: 1+2=3, 2sqrt(2) ≈ 2.828, not equal to √5≈2.236. Not working.Alternatively, perhaps fractions. Let me suppose that sqrt(3 + √5) = sqrt(x) + sqrt(y). Then:3 + √5 = x + y + 2sqrt(xy)Comparing, x + y = 3 and 2sqrt(xy) = √5. Then sqrt(xy) = √5 / 2, so xy = 5/4. So we have x + y =3 and xy=5/4. The solutions are the roots of t² - 3t + 5/4 =0. Discriminant: 9 -5=4. So t=(3±2)/2. So t=5/2 or t=1/2. Therefore, x=5/2 and y=1/2. Thus,sqrt(3 + √5) = sqrt(5/2) + sqrt(1/2) = (sqrt(10) + sqrt(2))/2Therefore, sqrt(3 + √5) = (sqrt(10)+sqrt(2))/2Thus, R = r * sqrt(2) / [ (sqrt(10)+sqrt(2))/2 ] = r * 2sqrt(2) / (sqrt(10) + sqrt(2)) )Multiply numerator and denominator by (sqrt(10)-sqrt(2)):R = r * 2sqrt(2)(sqrt(10)-sqrt(2)) / [ (sqrt(10)+sqrt(2))(sqrt(10)-sqrt(2)) ] = r * 2sqrt(2)(sqrt(10)-sqrt(2)) / (10 - 2) = r * 2sqrt(2)(sqrt(10)-sqrt(2))/8 = r * sqrt(2)(sqrt(10)-sqrt(2))/4Simplify sqrt(10)-sqrt(2) multiplied by sqrt(2):sqrt(10)*sqrt(2) - sqrt(2)*sqrt(2) = sqrt(20) - 2 = 2sqrt(5) - 2Therefore, R = r*(2sqrt(5) - 2)/4 = r*(sqrt(5)-1)/2So R = r*(sqrt(5)-1)/2 ≈ r*(2.236 -1)/2≈ r*0.618, which is the golden ratio conjugate.Similarly, r1 was r * sqrt( (3 + √5)/2 ). Let me compute sqrt( (3 + √5)/2 ):From earlier, (3 + √5)/2 is (sqrt(5)+1)^2 / 4. Wait, (sqrt(5)+1)/2 squared is (6 + 2sqrt(5))/4 = (3 + sqrt(5))/2. So sqrt( (3 + sqrt(5))/2 ) = (sqrt(5) +1)/2 ≈ (2.236 +1)/2≈1.618, the golden ratio.So r1 = r*(sqrt(5)+1)/2 ≈1.618rBut wait, but we had R = r*(sqrt(5)-1)/2 ≈0.618r. But in a frustum, R is the larger radius and r1 is the smaller? Wait, but here we have R ≈0.618r and r1≈1.618r? That would mean the top radius is larger than the bottom radius. That contradicts the assumption unless the frustum is inverted. But since the sphere is inscribed, the frustum should be such that the sphere is inside. If the frustum is upright, with the larger radius at the bottom, but according to this, r1 (the top radius) is larger. Hmm. So perhaps my initial assumption that R is the bottom radius and r1 is the top was incorrect. Wait, but in the coordinate system, the bottom is at y=0 and the top at y=2r. If the radius at y=0 is R and at y=2r is r1. If R is smaller than r1, the frustum would be expanding upwards. But in that case, how is the sphere inscribed? Let me check.Wait, in our equations, we found Rr1 = r². If R = r*(sqrt(5)-1)/2 ≈0.618r and r1 = r*(sqrt(5)+1)/2≈1.618r, then Rr1 ≈0.618*1.618≈1, which is indeed r². So that works. But geometrically, if the frustum is oriented with the smaller radius at the bottom and larger at the top, then the sphere is inside. But does the sphere being tangent to both bases and the sides make sense in this orientation?Wait, the height is 2r. So the sphere with radius r is centered at y=r. The bottom base is at y=0 with radius R≈0.618r, and the top base is at y=2r with radius≈1.618r. The sphere of radius r is centered at (0, r). The distance from the center to the bottom base (y=0) is r, which is equal to the radius. Similarly, to the top base (y=2r), the distance is also r. So the sphere touches both bases. Then the sides of the frustum, which flare out from R≈0.618r to r1≈1.618r over a height of 2r, must also be tangent to the sphere. So the sphere is inside the frustum, touching both ends and the sides. That seems possible. So even though the frustum is expanding upwards, it's possible for the sphere to fit inside. So R is the bottom radius, r1 is the top radius, with R < r1. So the frustum is like a truncated cone that's wider at the top.Alternatively, maybe the problem allows the frustum to be oriented either way, as long as it's around the sphere. So the solution is R = (sqrt(5)-1)/2 * r and r1 = (sqrt(5)+1)/2 * r.Alternatively, if we take the other root when solving for x = r1², which was x = r²(3 - sqrt(5))/2. Then r1 = r*sqrt( (3 - sqrt(5))/2 ). Compute sqrt( (3 - sqrt(5))/2 ):Similarly, (3 - sqrt(5))/2 = (sqrt(5)-1)^2 / 4. So sqrt( (3 - sqrt(5))/2 ) = (sqrt(5)-1)/2 ≈ (2.236 -1)/2≈0.618. Then R = r² / r1 = r / [ sqrt( (3 - sqrt(5))/2 ) ] = r / [ (sqrt(5)-1)/2 ] = 2r / (sqrt(5)-1). Rationalize denominator:2r*(sqrt(5)+1)/ ( (sqrt(5)-1)(sqrt(5)+1) ) = 2r*(sqrt(5)+1)/4 = r*(sqrt(5)+1)/2≈1.618r. So in this case, R =1.618r and r1=0.618r. So this would correspond to the frustum being narrower at the top. But then, similar to the previous case, but swapping R and r1. So which one is correct? Depending on which root we take. But in the problem statement, it's mentioned that we have a frustum of a cone. A frustum can be created by cutting a cone with a plane, resulting in two circular bases. The original cone could have been either expanding upwards or downwards. However, since the sphere is inside the frustum, both possibilities are valid. But maybe the problem expects both radii to be in terms of the original cone. Wait, but perhaps we need to check which one satisfies the volume equation.Wait, both solutions for R and r1 would satisfy the volume equation because they are reciprocal. Since in the volume equation, R and r1 are symmetric except for the substitution. However, in the frustum volume formula, R is the larger base and r1 is the smaller, but in our problem, depending on the solution, they can be either. So perhaps both solutions are valid, but since the problem doesn't specify orientation, we can present both. Wait, but maybe not. The problem says "around a sphere", so perhaps the frustum is symmetrical in some way. But actually, the frustum must contain the sphere, which is fixed in size. So either orientation is possible, but since the problem mentions "radii of the bases of the frustum and the top of the frustum", it might just want the two radii regardless of order. However, in the context of a frustum, typically R is the base radius and r1 is the top radius, so maybe we need to specify which is which. But from our calculation, we have two possibilities:Case 1: R = (sqrt(5)-1)/2 * r and r1 = (sqrt(5)+1)/2 * rCase 2: R = (sqrt(5)+1)/2 * r and r1 = (sqrt(5)-1)/2 * rBut given that Rr1 = r², they are reciprocal. So depending on which one is considered the base. If we take R as the base radius (larger one), then Case 2 gives R = golden ratio * r, and r1 as the smaller. If we take R as the smaller, then Case 1. However, in the frustum formula, R is the larger base. Wait, no, actually, the frustum can be either way. The problem doesn't specify, so maybe we can just present the two radii as R and r1 where R = [(sqrt(5)+1)/2]r and r1 = [(sqrt(5)-1)/2]r, or vice versa. Wait, but let's check which one makes sense with the tangent.In the cross-section, if the frustum is oriented with the larger radius at the bottom (Case 2), then the lateral side slopes outward. The sphere centered at (0, r) with radius r must be tangent to this lateral side. Similarly, if the frustum is oriented with the smaller radius at the bottom (Case 1), the lateral side slopes inward. Wait, but in that case, the frustum would not contain the sphere if it's narrowing upwards. Wait, no, actually, if the frustum is constructed around the sphere, whether it's expanding or narrowing depends on the orientation. Wait, if R is the bottom radius and r1 is the top radius, then for Case 1: R ≈0.618r, r1≈1.618r. So the frustum expands from bottom to top. The sphere is inside, touching both bases and the sides. For Case 2: R≈1.618r, r1≈0.618r, frustum narrows from bottom to top. The sphere is inside, touching both bases and the sides. Both are possible. But in the problem statement, it just says "around a sphere", so both configurations are possible. However, in the case of the frustum being part of a cone, if the original cone was cut to form the frustum, then the frustum could be part of a larger cone or a smaller cone. But since the problem mentions constructing a frustum around the sphere, likely it's similar to the cone case, where the frustum is such that the sphere is inscribed within it. So the problem doesn't specify orientation, so perhaps we need to note both radii as the two solutions.But let's check the problem statement again: "Calculate the height and the radius of the base of the complete cone, as well as the radii of the bases of the frustum and the top of the frustum, and finally determine the ratio of the tangents of the slant heights."So the problem mentions "radii of the bases of the frustum and the top of the frustum". So "bases" here might refer to the two circular bases, so we need to specify both radii. It doesn't matter which is which as long as both are given. So we can present them as [(sqrt(5)+1)/2]r and [(sqrt(5)-1)/2]r. So both radii are (sqrt(5)±1)/2 times r.So in summary:For the complete cone: R = r√2, h =4r.For the frustum: radii R = [ (sqrt(5)+1)/2 ]r and r1 = [ (sqrt(5)-1)/2 ]r (or vice versa), height h=2r.Now, the last part is to determine the ratio of the tangents of the slant heights. First, we need to find the slant heights of the cone and the frustum, then find the tangent of those angles, then take their ratio.For the complete cone: The slant height l is the distance from the base to the vertex along the lateral surface. For a cone with radius R and height h, the slant height is sqrt(R² + h²). So for our cone, R = r√2, h=4r. So l_cone = sqrt( (r√2)² + (4r)² ) = sqrt(2r² + 16r²) = sqrt(18r²)= 3r√2.The slant angle θ (angle between the slant height and the base) has tanθ = h/R = (4r)/(r√2) = 4/√2 = 2√2. So tanθ_cone = 2√2.For the frustum: The slant height is the distance between the two edges along the lateral side. For a frustum with height h and radii R and r1, the slant height l_frustum is sqrt( (R - r1)² + h² ). Here, R and r1 are [(sqrt(5)+1)/2]r and [(sqrt(5)-1)/2]r. Let's compute R - r1:[(sqrt(5)+1)/2 - (sqrt(5)-1)/2]r = [ (sqrt(5)+1 - sqrt(5)+1)/2 ]r = (2/2)r = r.So the difference in radii is r. The height h of the frustum is 2r. Therefore, slant height l_frustum = sqrt( r² + (2r)² ) = sqrt(5r²) = r√5.The slant angle φ (angle between the slant height and the base) can be found by tanφ = h / (R - r1) = 2r / r = 2. So tanφ_frustum = 2.Wait, but hold on. The tangent of the slant angle for the frustum is the ratio of the height of the frustum to the difference in radii. But in the frustum, the "slant angle" is the angle between the lateral side and the base. Alternatively, for the original cone before truncation, the slant angle would be different. But in the problem statement, it says "the ratio of the tangents of the slant heights". Wait, maybe it's the ratio of the tangents of the angles that the slant heights make with the base. So for the cone, tanθ_cone = 2√2, for the frustum, tanφ_frustum = 2. Therefore, the ratio is tanθ_cone / tanφ_frustum = (2√2)/2 = √2. Alternatively, depending on how the problem defines the ratio. If it's frustum tan over cone tan, then it's 2 / (2√2) = 1/√2. But the problem says "the ratio of the tangents of the slant heights". It's not specified which order, but in the problem statement, the cone and frustum are both constructed around the sphere, so likely comparing their respective angles. But since they ask for the ratio, we need to see if it's cone to frustum or vice versa.Wait, let me check. The problem says: "the ratio of the tangents of the slant heights". Since both the cone and frustum have slant heights, but the problem doesn't specify which one is numerator or denominator. But in mathematical terms, usually, unless specified, the ratio is presented in the order mentioned. The problem mentions first the cone and then the frustum: "construct a complete cone and a frustum of a cone". So maybe the ratio is cone's tangent over frustum's tangent. Which would be 2√2 / 2 = √2. So the ratio is √2:1, or simply √2.But let me verify. For the cone, we calculated tanθ = 2√2. For the frustum, tanφ = 2. Therefore, the ratio tanθ / tanφ = (2√2)/2 = √2. So the ratio is √2.Alternatively, if they meant the frustum's tangent over cone's tangent, it would be 1/√2, but since the problem mentions "the ratio of the tangents of the slant heights", and both are mentioned in the problem statement (the cone and the frustum), but the question is to "determine the ratio", perhaps they want the ratio of the cone's tangent to the frustum's tangent. So likely √2.Therefore, the final ratio is √2.So to recap:Complete cone:- Radius R = r√2- Height h =4rFrustum:- Radii R1 = [(sqrt(5)+1)/2]r and R2 = [(sqrt(5)-1)/2]r- Height h =2rRatio of tangents: √2Final AnswerThe complete cone has height boxed{4r} and base radius boxed{rsqrt{2}}. The frustum has radii boxed{frac{(sqrt{5} + 1)r}{2}} and boxed{frac{(sqrt{5} - 1)r}{2}}, and the ratio of the tangents of the slant heights is boxed{sqrt{2}}.</think>To solve the problem of constructing a complete cone and a frustum around a sphere of radius ( r ) such that each has twice the volume of the sphere, we start by noting the volume of the sphere is ( frac{4}{3}pi r^3 ). Therefore, each shape must have a volume of ( frac{8}{3}pi r^3 ). Complete Cone1. Volume Equation: The volume of a cone is ( frac{1}{3}pi R^2 h ). Setting this equal to ( frac{8}{3}pi r^3 ), we get: [ R^2 h = 8r^3 ]2. Inscribed Sphere: The sphere is inscribed in the cone, leading to the relationship derived from the distance from the sphere's center to the cone's side. This results in: [ h = frac{2R^2 r}{R^2 - r^2} ]3. Solving for ( R ) and ( h ): Substituting ( h ) into the volume equation and solving, we find: [ R = rsqrt{2} quad text{and} quad h = 4r ] Frustum of a Cone1. Volume Equation: The volume of a frustum is ( frac{1}{3}pi h (R^2 + Rr_1 + r_1^2) ). Setting this equal to ( frac{8}{3}pi r^3 ), we get: [ h(R^2 + Rr_1 + r_1^2) = 8r^3 ]2. Inscribed Sphere: The sphere is inscribed in the frustum, leading to the relationships: [ Rr_1 = r^2 quad text{and} quad h = 2r ]3. Solving for ( R ) and ( r_1 ): Substituting ( h = 2r ) into the volume equation and solving, we find: [ R = frac{(sqrt{5} + 1)r}{2} quad text{and} quad r_1 = frac{(sqrt{5} - 1)r}{2} ] Ratio of Tangents of Slant Heights1. Cone Slant Height: The slant height ( l ) of the cone is ( sqrt{R^2 + h^2} ). The tangent of the slant angle is: [ tan theta = frac{h}{R} = 2sqrt{2} ]2. Frustum Slant Height: The slant height ( l ) of the frustum is ( sqrt{(R - r_1)^2 + h^2} ). The tangent of the slant angle is: [ tan phi = frac{h}{R - r_1} = 2 ]3. Ratio: The ratio of the tangents is: [ frac{tan theta}{tan phi} = sqrt{2} ] Final Answer- Complete Cone: Height ( boxed{4r} ) and base radius ( boxed{rsqrt{2}} ).- Frustum: Radii ( boxed{frac{(sqrt{5} + 1)r}{2}} ) and ( boxed{frac{(sqrt{5} - 1)r}{2}} ).- Ratio of Tangents: ( boxed{sqrt{2}} ).,<think>Okay, so I have this problem where I need to construct a complete cone and a frustum of a cone around a sphere of radius r. The volume of each should be twice the volume of the sphere. Then, I need to calculate various dimensions: height and base radius for the complete cone, radii for the bases of the frustum, and the ratio of the tangents of their slant heights. Hmm, okay. Let me start by recalling some formulas and visualizing the problem.First, the sphere has radius r, so its volume is (4/3)πr³. The problem states that the volume of the cone and the frustum each should be twice that, so their volumes are 2*(4/3)πr³ = (8/3)πr³. Got that.Starting with the complete cone. The volume of a cone is (1/3)πR²H, where R is the base radius and H is the height. So, we have (1/3)πR²H = (8/3)πr³. Simplifying, that gives R²H = 8r³. So, equation one: R²H = 8r³.But there's another condition here: the cone is constructed "around" the sphere. That must mean the sphere is inscribed inside the cone. So, the sphere touches the cone's base and its lateral surface. That gives some geometric relationships.For a sphere inscribed in a cone, there's a formula relating the radius of the sphere, the radius of the cone's base, and its height. Let me recall that. If a sphere is inscribed in a cone, the relationship can be found using similar triangles or trigonometric relationships. Let me try to derive it.Imagine a cross-section of the cone and the sphere. The sphere touches the cone's side and the base. The center of the sphere will be at a distance r from the base (since the sphere's radius is r). The sphere also touches the lateral side of the cone. Let me denote the apex of the cone as point A, the center of the sphere as O, the point where the sphere touches the cone's lateral surface as T, and the base's center as B.The triangle formed by the apex, the center of the sphere, and the point of tangency should be a right triangle. Wait, maybe not. Let's think.The radius of the sphere to the point of tangency is perpendicular to the cone's lateral surface. So, OT is perpendicular to the cone's lateral surface. The cone's lateral surface makes an angle θ with the axis of the cone. Let's denote θ as the semi-vertical angle of the cone. Then, the line OT is perpendicular to the lateral surface, so the angle between OT and the axis is θ. The length OT is r (the radius of the sphere). The distance from the apex to the center of the sphere is AO. Let's see.But maybe it's better to use coordinates. Let's place the apex of the cone at the origin (0,0), and the axis of the cone along the positive y-axis. The base of the cone is then at y = H, with radius R. The sphere is inscribed in the cone, so its center is at some point along the axis. Since the sphere touches the base of the cone, the center of the sphere must be at a distance r from the base. Therefore, the center is at y = H - r. The sphere has radius r, so it extends from y = H - 2r to y = H. Wait, but the apex is at (0,0), and the sphere is inside the cone. Wait, if the sphere is touching the base (which is at y = H), then the center is at y = H - r. But the sphere must also touch the sides of the cone. So, the sphere must be tangent to the cone's lateral surface.The cone's lateral surface in this coordinate system is a line from the origin (0,0) to the point (R, H). So, the slope of the lateral surface is (H)/R. The equation of the lateral surface is y = (H/R)x. The sphere is centered at (0, H - r) with radius r. The distance from the center of the sphere to the lateral surface must be equal to r. The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²). Let's write the equation of the cone's lateral surface in standard form. The line is y - (H/R)x = 0. So, a = -H/R, b = 1, c = 0.The distance from the center (0, H - r) to this line is |(-H/R)(0) + 1*(H - r) + 0| / sqrt((H/R)² + 1) = |H - r| / sqrt((H²/R²) + 1). This distance must equal the radius of the sphere, which is r. So,|H - r| / sqrt((H²/R²) + 1) = rSince H > r (the sphere is inside the cone), we can drop the absolute value:(H - r) / sqrt((H²/R²) + 1) = rLet me square both sides to eliminate the square root:(H - r)² / ((H²/R²) + 1) = r²Multiply both sides by ((H²/R²) + 1):(H - r)² = r²*(H²/R² + 1)Expand the left side:H² - 2Hr + r² = r²*(H²/R² + 1)Divide both sides by r²:(H² - 2Hr + r²)/r² = H²/R² + 1Simplify left side:(H/r)² - 2(H/r) + 1 = H²/R² + 1Subtract 1 from both sides:(H/r)² - 2(H/r) = H²/R²Multiply both sides by R²:R²[(H/r)² - 2(H/r)] = H²Let me denote k = H/r, so that H = kr. Then substituting:R²[(k²) - 2k] = (k r)²Simplify RHS:R²(k² - 2k) = k² r²Divide both sides by r²:(R²/r²)(k² - 2k) = k²Let me denote m = R/r, so R = m r. Then:(m²)(k² - 2k) = k²So,m² k² - 2 m² k - k² = 0Factor out k²:k²(m² - 1) - 2 m² k = 0This is a quadratic in k:k²(m² - 1) - 2 m² k = 0Factor out k:k [k(m² - 1) - 2 m²] = 0So, solutions are k = 0 (which is invalid because H = kr, and H must be positive) and k(m² - 1) - 2 m² = 0So,k(m² - 1) = 2 m²Thus,k = (2 m²)/(m² - 1)But remember that H = kr and R = m r, so H = (2 m²/(m² - 1)) rAlso, from the volume equation, R² H = 8 r³Substituting R = m r and H = (2 m²/(m² - 1)) r,(m r)² * (2 m²/(m² - 1)) r = 8 r³Simplify:m² r² * 2 m²/(m² - 1) * r = 8 r³Multiply terms:2 m⁴ / (m² - 1) * r³ = 8 r³Divide both sides by r³:2 m⁴ / (m² - 1) = 8Multiply both sides by (m² - 1):2 m⁴ = 8(m² - 1)Divide both sides by 2:m⁴ = 4(m² - 1)Bring all terms to left side:m⁴ - 4 m² + 4 = 0This is a quadratic in m²:Let n = m²Then:n² - 4n + 4 = 0Which factors as (n - 2)² = 0So, n = 2Thus, m² = 2 => m = sqrt(2)Therefore, m = sqrt(2), so R = sqrt(2) rThen, k = (2 m²)/(m² - 1) = (2*2)/(2 - 1) = 4/1 = 4So, H = 4 rSo, for the cone, the base radius R = sqrt(2) r and height H = 4 r.Okay, that seems good. Let me check with the volume:Volume is (1/3)πR²H = (1/3)π*(2 r²)*(4 r) = (1/3)π*8 r³ = (8/3)πr³, which is twice the sphere's volume. Perfect.So, the complete cone has height 4r and base radius sqrt(2) r.Now, moving on to the frustum. The frustum is also around the sphere, so the sphere is inscribed in the frustum. The volume of the frustum should also be twice the sphere's volume, which is (8/3)πr³.A frustum of a cone can be thought of as a portion of the cone between two parallel planes. The volume of a frustum is (1/3)πh(R² + Rr + r²), where h is the height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base. Wait, but in this case, since the frustum is around the sphere, the sphere must be tangent to both the top and bottom bases of the frustum and also tangent to the lateral surface.Wait, but if it's a frustum, the original cone would have been larger, and the frustum is a part of it. But since the frustum is constructed around the sphere, perhaps similar to the cone, but with two radii now. Let me think.Let me visualize the frustum. The sphere is inside the frustum, touching both the top and bottom bases and the lateral surface. So, the sphere is tangent to both bases and the lateral side. Therefore, the center of the sphere must lie along the axis of the frustum, equidistant from both bases? Wait, no. If the sphere is touching both bases, then the distance from the center to each base is r (since the radius is r). Therefore, the height of the frustum must be 2r? Wait, but that would mean the distance between the two bases is 2r. But is that necessarily the case?Wait, if the sphere is tangent to both the top and bottom bases, then the center of the sphere is at the midpoint between the two bases, right? Because the distance from the center to each base is r, so the total height is 2r. So, the height h of the frustum is 2r. Wait, but the volume of the frustum is (1/3)πh(R² + Rr + r²) = (1/3)π*2r*(R² + Rr + r²) = (2/3)πr(R² + Rr + r²). And this needs to equal (8/3)πr³. So,(2/3)πr(R² + Rr + r²) = (8/3)πr³Divide both sides by (2/3)πr:R² + Rr + r² = 4r²Therefore,R² + Rr + r² - 4r² = 0Simplify:R² + Rr - 3r² = 0This is a quadratic equation in R:R² + Rr - 3r² = 0Using quadratic formula:R = [-r ± sqrt(r² + 12r²)] / 2 = [-r ± sqrt(13r²)] / 2 = [-r ± r*sqrt(13)] / 2Since radius cannot be negative,R = [ -r + r*sqrt(13) ] / 2 = r( -1 + sqrt(13) ) / 2 ≈ ... well, we can leave it as is.So, R = r( sqrt(13) - 1 ) / 2But wait, let me check if the height of the frustum is indeed 2r. That comes from the sphere touching both bases. So, the sphere is inside the frustum, touching the top and bottom. So, the center of the sphere is at the midpoint of the frustum. Since the sphere's radius is r, the distance from the center to each base is r, so the total height is 2r. That seems correct.But is that the case? Wait, if the frustum is around the sphere, the sphere must fit snugly, touching both the top, bottom, and the sides. But in a frustum, the sides are inclined. So, similar to the cone, the sphere must be tangent to the lateral face. So, similar to the cone, but now in a frustum.Wait a minute, maybe the height of the frustum isn't 2r? Because in the cone, the sphere was tangent to the base and the lateral surface, but the height was 4r. Here, for the frustum, the sphere is tangent to both the top and bottom bases, and the lateral surface. So, the center of the sphere is midway between the two bases, so height h = 2r. But also, the sphere is tangent to the lateral surface, which will give another condition relating the radii R and r (of the frustum's bases) and the height h.Wait, so in the frustum case, we have two radii: let's denote R1 as the radius of the lower base and R2 as the radius of the upper base. Then, the height h is 2r, since the sphere is tangent to both bases. The sphere's center is at h/2 = r from each base. Now, the sphere must also be tangent to the lateral surface of the frustum. So, similar to the cone, we need to relate R1, R2, and h using the tangency condition.Let me model the frustum in coordinates. Let me place the center of the sphere at the origin (0,0) for simplicity. Then, the lower base is at z = -r, with radius R1, and the upper base is at z = r, with radius R2. The lateral surface connects the edge of the lower base (R1, -r) to the edge of the upper base (R2, r). The sphere is centered at the origin with radius r, so it touches both bases at z = -r and z = r. The sphere must also touch the lateral surface.The equation of the lateral surface can be found by connecting the points (R1, -r) and (R2, r). Let's parametrize this line. The slope of the line is (r - (-r))/(R2 - R1) = (2r)/(R2 - R1). So, the equation is z - (-r) = (2r)/(R2 - R1)(x - R1). Simplifying, z + r = (2r)/(R2 - R1)(x - R1). Rearranged: z = (2r)/(R2 - R1)(x - R1) - r.But the distance from the center of the sphere (origin) to this lateral surface must be equal to the sphere's radius, which is r. Wait, but the sphere is already touching the lateral surface, so the distance from the center to the lateral surface is equal to the radius. So, we can use the formula for the distance from a point to a plane (in 3D), but since we're in 2D (considering a cross-section), it's the distance from a point to a line.The distance from (0,0) to the line defined by z = (2r)/(R2 - R1)(x - R1) - r.First, let's write the line in standard form ax + bz + c = 0.Starting from z = (2r)/(R2 - R1)x - (2r R1)/(R2 - R1) - rBring all terms to left-hand side:z - (2r)/(R2 - R1)x + (2r R1)/(R2 - R1) + r = 0Multiply through by (R2 - R1) to eliminate denominators:(R2 - R1)z - 2r x + 2r R1 + r(R2 - R1) = 0Simplify:-2r x + (R2 - R1)z + 2r R1 + r R2 - r R1 = 0Combine like terms:-2r x + (R2 - R1)z + r R1 + r R2 = 0So, the equation is:-2r x + (R2 - R1)z + r(R1 + R2) = 0Now, the distance from (0,0) to this line is:| -2r*0 + (R2 - R1)*0 + r(R1 + R2) | / sqrt( (-2r)^2 + (R2 - R1)^2 )Which simplifies to:| r(R1 + R2) | / sqrt(4r² + (R2 - R1)^2 )Since all terms are positive, this is:r(R1 + R2) / sqrt(4r² + (R2 - R1)^2 )This distance must equal the sphere's radius, r. Therefore:r(R1 + R2) / sqrt(4r² + (R2 - R1)^2 ) = rDivide both sides by r:(R1 + R2) / sqrt(4r² + (R2 - R1)^2 ) = 1Multiply both sides by the denominator:R1 + R2 = sqrt(4r² + (R2 - R1)^2 )Square both sides:(R1 + R2)^2 = 4r² + (R2 - R1)^2Expand both sides:Left side: R1² + 2 R1 R2 + R2²Right side: 4r² + R2² - 2 R1 R2 + R1²Set equal:R1² + 2 R1 R2 + R2² = 4r² + R2² - 2 R1 R2 + R1²Subtract R1² and R2² from both sides:2 R1 R2 = 4r² - 2 R1 R2Bring all terms to left side:2 R1 R2 + 2 R1 R2 - 4r² = 04 R1 R2 - 4r² = 0Divide by 4:R1 R2 = r²So, the product of the radii of the frustum's bases is r².Additionally, we already have from the volume equation. The volume of the frustum is (1/3)πh(R1² + R1 R2 + R2²) = (8/3)πr³.Given that h = 2r, substituting h:(1/3)π*2r*(R1² + R1 R2 + R2²) = (8/3)πr³Multiply left side:(2/3)πr(R1² + R1 R2 + R2²) = (8/3)πr³Divide both sides by (2/3)πr:R1² + R1 R2 + R2² = 4r²But we already found that R1 R2 = r². So, substitute R1 R2 = r² into the above equation:R1² + r² + R2² = 4r²Thus,R1² + R2² = 3r²Now, we have two equations:1. R1 R2 = r²2. R1² + R2² = 3r²We can solve these equations for R1 and R2.Let me denote S = R1 + R2 and P = R1 R2 = r². Then,R1² + R2² = (R1 + R2)² - 2 R1 R2 = S² - 2P = 3r²So,S² - 2P = 3r²But P = r², so,S² - 2r² = 3r² => S² = 5r² => S = sqrt(5) rTherefore, R1 + R2 = sqrt(5) r, and R1 R2 = r².So, the quadratic equation for R1 and R2 is x² - Sx + P = 0 => x² - sqrt(5) r x + r² = 0Solving this quadratic:x = [sqrt(5) r ± sqrt(5 r² - 4 r²)] / 2 = [sqrt(5) r ± sqrt(r²)] / 2 = [sqrt(5) r ± r] / 2 = r [sqrt(5) ± 1]/2Therefore, the radii are:R1 = r [sqrt(5) + 1]/2 and R2 = r [sqrt(5) - 1]/2Or vice versa. Depending on which is the larger radius. Since R1 + R2 = sqrt(5) r, and R1 is the larger one, so R1 = [sqrt(5) + 1]/2 r and R2 = [sqrt(5) - 1]/2 rLet me check if R1 R2 = ([sqrt(5) + 1]/2 r)([sqrt(5) - 1]/2 r) = [(5 -1)/4] r² = (4/4)r² = r². Correct.And R1² + R2²: let's compute:([sqrt(5) + 1]/2 r)² + ([sqrt(5) - 1]/2 r)² = [ (6 + 2sqrt(5))/4 + (6 - 2sqrt(5))/4 ] r² = (12 /4 ) r² = 3r². Correct.So, the radii of the frustum's bases are R1 = [sqrt(5) + 1]/2 r and R2 = [sqrt(5) - 1]/2 r.Therefore, the frustum has lower base radius ([sqrt(5) + 1]/2) r and upper base radius ([sqrt(5) - 1]/2) r, with height 2r.Now, the last part is to determine the ratio of the tangents of the slant heights of the complete cone and the frustum.Wait, the problem says: "the ratio of the tangents of the slant heights". Hmm. Slant heights. For the complete cone, the slant height L is sqrt(R² + H²). For the frustum, the slant height is sqrt( (R1 - R2)^2 + h^2 ). The tangent of the angle of the slant height... Wait, the tangent of the angle that the slant height makes with the base? Or is it the angle of the lateral surface with the axis?Wait, in cones, the slant height makes an angle with the axis, whose tangent is (R)/H. Similarly, for the frustum, the slant height is the same as the original cone's slant height. Wait, but the frustum is part of a cone. Wait, perhaps the tangent of the angle for the frustum is (R1 - R2)/h. Because the frustum's lateral side is a trapezoid, and the "slope" would be (R1 - R2)/h. So, the tangent of the angle that the lateral side makes with the axis is (R1 - R2)/h.Similarly, for the complete cone, the tangent of the angle is R/H.Therefore, the ratio would be [ (R1 - R2)/h ] / [ R/H ].But let me confirm.For the complete cone, the semi-vertical angle θ has tanθ = R/H. So, the tangent of the angle between the slant height and the axis is R/H.For the frustum, which is part of a cone, the original cone would have a certain semi-vertical angle θ. The frustum's slant height makes the same angle θ with the axis. Therefore, tanθ is the same as for the original cone. However, since the frustum is a portion of the cone, the tangent of the angle can also be calculated from the difference in radii and the height of the frustum. Specifically, if you consider the frustum as a truncated cone, the difference in radii (R1 - R2) over the height h gives the same tanθ. Because, in the original cone, if you take a section of height h, the change in radius would be (R1 - R2), so tanθ = (R1 - R2)/h. However, in the original complete cone, tanθ = R/H. So, if the frustum is part of a larger cone, then the tanθ for both the frustum and the original cone would be the same. But in our problem, the frustum is a separate entity, constructed around the sphere. Wait, but the frustum is not part of the complete cone we calculated earlier. It's a separate frustum. So, perhaps the angles could be different.Wait, but in our calculation for the frustum, the height is h = 2r, R1 = [sqrt(5)+1]/2 r, R2 = [sqrt(5)-1]/2 r. So, R1 - R2 = [sqrt(5)+1 - sqrt(5)+1]/2 r = (2)/2 r = r. Therefore, tanθ for the frustum's lateral side is (R1 - R2)/h = r / 2r = 1/2.For the complete cone, tanθ = R/H = sqrt(2) r / 4 r = sqrt(2)/4.Therefore, the ratio of the tangents is (1/2) / (sqrt(2)/4) = (1/2)*(4/sqrt(2)) = 2/sqrt(2) = sqrt(2). So, the ratio is sqrt(2).Wait, but the problem says "the ratio of the tangents of the slant heights". Depending on which one is numerator and denominator. The problem doesn't specify which one first. But in the problem statement, it says "the ratio of the tangents of the slant heights". It might refer to the frustum's tangent over the cone's tangent, or vice versa. But in my calculation, the frustum's tanθ is 1/2, the cone's tanθ is sqrt(2)/4. So, 1/2 divided by sqrt(2)/4 is sqrt(2), as above.Alternatively, if we take the other way, it would be 1/sqrt(2). But since the problem says "the ratio", and given that the frustum is mentioned after the cone, maybe it's frustum's tangent over cone's tangent. So, the ratio is sqrt(2).But let me verify the frustum's tanθ. The frustum has upper radius R2, lower radius R1, height h. The slant height is sqrt( (R1 - R2)^2 + h^2 ). The angle θ is the angle between the slant height and the axis. Then, tanθ is the "opposite" over "adjacent", which would be (R1 - R2)/h. Since R1 > R2, this is positive. So, tanθ = (R1 - R2)/h. For our frustum, R1 - R2 = [ (sqrt(5) + 1)/2 - (sqrt(5) - 1)/2 ] r = [2/2] r = r. So, tanθ = r / 2r = 1/2.For the complete cone, tanθ = R / H = sqrt(2)r / 4r = sqrt(2)/4. So, the ratio is (1/2)/(sqrt(2)/4) = (1/2)*(4/sqrt(2)) = 2/sqrt(2) = sqrt(2). So, the ratio is sqrt(2).Therefore, the ratio of the tangents is sqrt(2).So, summarizing:Complete cone: Height H = 4r, Base radius R = sqrt(2) r.Frustum: Lower base radius R1 = (sqrt(5) + 1)/2 r, Upper base radius R2 = (sqrt(5) - 1)/2 r, Height h = 2r.Ratio of tangents of the slant heights: sqrt(2).Let me double-check all the steps to ensure there are no mistakes.For the cone:- Volume equation correct: R²H = 8r³.- Derived the relationship from the inscribed sphere, leading to H = 4r and R = sqrt(2) r. Checked the volume, which matches.For the frustum:- Assumed height h = 2r because the sphere is tangent to both bases. Then used the volume equation and the tangency condition to find R1 and R2. Got R1 R2 = r² and R1² + R2² = 3r². Solved to find R1 and R2 in terms of sqrt(5). Checked the product and sum of squares, which are correct.- Calculated the ratio of the tangents, which turned out to be sqrt(2). The key step was recognizing that tanθ for the frustum is (R1 - R2)/h = 1/2, and for the cone, sqrt(2)/4. Their ratio is sqrt(2).Everything seems consistent. I think this should be the correct answer.Final AnswerThe complete cone has height (boxed{4r}) and base radius (boxed{sqrt{2}r}). The frustum has radii (boxed{frac{sqrt{5} + 1}{2}r}) and (boxed{frac{sqrt{5} - 1}{2}r}), and the ratio of the tangents is (boxed{sqrt{2}}).</think>Given a sphere of radius ( r ), we need to construct a complete cone and a frustum of a cone each with twice the volume of the sphere. We also need to determine the ratio of the tangents of their slant heights. Complete Cone:1. Volume Calculation: - Volume of the sphere: ( frac{4}{3}pi r^3 ) - Volume of the cone: ( 2 times frac{4}{3}pi r^3 = frac{8}{3}pi r^3 ) - Volume formula for a cone: ( frac{1}{3}pi R^2 H = frac{8}{3}pi r^3 ) - Solving for ( R ) and ( H ): ( R^2 H = 8r^3 )2. Inscribed Sphere Condition: - The sphere is tangent to the base and lateral surface of the cone. - Using geometric relationships and solving the quadratic equation, we find: - ( H = 4r ) - ( R = sqrt{2}r ) Frustum of a Cone:1. Volume Calculation: - Volume of the frustum: ( frac{1}{3}pi h (R_1^2 + R_1 R_2 + R_2^2) = frac{8}{3}pi r^3 ) - Height of the frustum ( h = 2r ) (since the sphere is tangent to both bases).2. Inscribed Sphere Condition: - The sphere is tangent to both bases and the lateral surface. - Solving the system of equations for ( R_1 ) and ( R_2 ): - ( R_1 R_2 = r^2 ) - ( R_1^2 + R_2^2 = 3r^2 ) - Results: - ( R_1 = frac{sqrt{5} + 1}{2}r ) - ( R_2 = frac{sqrt{5} - 1}{2}r ) Ratio of Tangents of Slant Heights:1. Complete Cone: - Tangent of the angle: ( tan theta = frac{R}{H} = frac{sqrt{2}r}{4r} = frac{sqrt{2}}{4} )2. Frustum: - Tangent of the angle: ( tan phi = frac{R_1 - R_2}{h} = frac{r}{2r} = frac{1}{2} )3. Ratio: - Ratio of tangents: ( frac{tan phi}{tan theta} = frac{frac{1}{2}}{frac{sqrt{2}}{4}} = sqrt{2} ) Final Answer:- Complete cone: Height ( boxed{4r} ), Base radius ( boxed{sqrt{2}r} )- Frustum: Radii ( boxed{frac{sqrt{5} + 1}{2}r} ) and ( boxed{frac{sqrt{5} - 1}{2}r} )- Ratio of tangents: ( boxed{sqrt{2}} )]

🔑:Let the radius of the base of the cone be AD = x, its height CD = m, and AC touches the sphere at E. According to the problem,frac{8 r^{3} pi}{3} = frac{x^{2} pi m}{3}butx = frac{m r}{CE} = frac{m r}{sqrt{m^{2} - 2 m r}}Substituting this into (1):m^{2} - 8 r m = -16 r^{2}from whichm = 4 rand thusx = r sqrt{2}If, furthermore, the radius of the base of the frustum is z = AB, the radius of the top is DC = v, and AD touches the sphere at E, then according to the problem:frac{8 r^{2} pi}{3} = frac{2 r pi}{3} (z^{2} + vz + v^{2})or4 r^{2} = z^{2} + vz + v^{2}but(z - v)^{2} + 4 r^{2} = (v + z)^{2}from whichvz = r^{2}Substituting this into (2):z^{2} + v^{2} = 3 r^{2}From (3) and (4) we get thatz = frac{r}{2} (sqrt{5} + 1) text{ and } frac{r}{2} (sqrt{5} - 1)If the angles of inclination of the slant heights are denoted by alpha and beta, then for the conetan alpha = frac{m}{x} = frac{4 r}{r sqrt{2}} = 2 sqrt{2}and for the frustumtan beta = frac{2 r}{z - v} = frac{2 r}{r} = 2sotan alpha : tan beta = sqrt{2} : 1(Krisztián György.)The problem was also solved by: Czank K., Frank J., Kerekes T., Kiss A., Lukhaub Gy., Lupsa Gy., Obláth R., Sasvári G., Weisz J.

🧠:[<think>Okay, so I need to figure out this math problem: "The sum of two prime numbers is an odd number, one of the numbers must be ____." Hmm, let's break this down step by step. First, I remember that prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. So primes are like 2, 3, 5, 7, 11, and so on. Now, the question is about the sum of two primes being odd. Let me recall that when you add two numbers, the result is odd only if one number is even and the other is odd. Because even + even = even, odd + odd = even, and even + odd = odd. So, for the sum to be odd, one prime has to be even and the other has to be odd.But wait, primes are mostly odd numbers, right? Except for the number 2. Because any other even number would be divisible by 2, so they can't be prime. So 2 is the only even prime number. All other primes are odd. So, if the sum of two primes is odd, one of them must be 2 because that's the only even prime. The other one would be an odd prime. Let me check this with examples.For example, let's take 2 and 3. Their sum is 5, which is odd. If I take 2 and 5, the sum is 7, also odd. What if I try two odd primes? Like 3 and 5. Their sum is 8, which is even. So that matches the rule. If both primes are odd, their sum is even. So to get an odd sum, one of them has to be even, which can only be 2. Therefore, the missing word must be 2.Wait, but let me think again. What if someone says, "But 2 is a prime, but can there be another even prime?" No, because any even number greater than 2 is divisible by 2, so it can't be prime. Therefore, 2 is the only even prime. So yes, in order for the sum of two primes to be odd, one of them has to be 2. Let me test another example. Let's say 2 and 7. Sum is 9, which is odd. If I take 2 and 11, sum is 13, odd. If I take two odd primes again, like 7 and 11, sum is 18, which is even. So the pattern holds. So the answer must be 2. But just to be thorough, what if someone suggests 1? Wait, 1 is not considered a prime number. The definition of a prime number is a number greater than 1 that has no divisors other than 1 and itself. So 1 is excluded. Therefore, 2 is the only even prime. Another edge case: If someone says, "What if one of the primes is 2 and the other is also 2?" Then 2 + 2 = 4, which is even. So in that case, the sum is even, which doesn't satisfy the condition. So in the problem statement, since the sum is given as odd, if one of the primes is 2, the other must be an odd prime. That would result in an odd sum. So in all cases where the sum is odd, one of the primes must be 2. Therefore, the answer is 2. Let me just make sure there's no other possibility. Suppose there's a prime that's even but not 2. But as I thought earlier, any even number greater than 2 is divisible by 2, hence not prime. So 2 is the only one. Therefore, the blank must be filled with 2.</think>The sum of two prime numbers is odd only if one of the primes is even (since an odd sum requires one even and one odd number). The only even prime number is 2, as all other even numbers are divisible by 2 and thus not prime. Therefore, if the sum is odd, one of the primes must be 2.boxed{2},<think>Okay, so I need to figure out this math problem: "The sum of two prime numbers is an odd number, one of the numbers must be ____." Hmm, let me think. Alright, primes are numbers greater than 1 that have no divisors other than 1 and themselves. The question is about adding two primes and getting an odd number. So, first, I should remember that when you add two numbers, the result is odd only if one number is even and the other is odd. Because even + even = even, and odd + odd = even, right? So to get an odd sum, one has to be even and the other odd.But wait, primes are mostly odd numbers. The only even prime number is 2. All other primes are odd because any even number greater than 2 can be divided by 2, so it's not prime. So if we're adding two primes and getting an odd number, one of them must be the even prime, which is 2, and the other must be an odd prime. Let me check that logic again. If both primes are odd, adding them would give an even number. If one is even and the other is odd, then the sum is odd. Since the only even prime is 2, that must be one of the numbers. So the answer should be 2.Wait, let me test some examples to be sure. Let's take 2 plus 3. That's 5, which is odd. Okay, that works. What about 2 plus 5? That's 7, still odd. How about another example: 2 plus 7 is 9, which is odd. If I take two odd primes, like 3 plus 5, that's 8, which is even. Yeah, that's even. So if the sum is odd, one of them must definitely be 2. Otherwise, both are odd, and their sum is even. So the blank should be filled with 2. Let me see if there are any exceptions. Suppose one prime is 2 and the other is an even prime. But wait, 2 is the only even prime. So the other prime has to be odd. Therefore, whenever the sum of two primes is odd, one of them must be 2. That makes sense.Is there any case where adding 2 and another prime gives an even number? Let's see. 2 plus 2 is 4, which is even, but 2 is prime. Wait, but 4 is even. So if we have 2 and 2, the sum is even. But the question says the sum is odd, so in that case, 2 and 2 wouldn't be a valid example. Therefore, in the cases where the sum is odd, one of the primes must be 2, and the other must be an odd prime. So the answer is 2. That seems solid. I can't think of any other even primes, so unless there's a prime I'm forgetting, which I don't think so. Yeah, 2 is the only even prime. So the answer has to be 2.Final AnswerThe one of the numbers must be boxed{2}.</think>To determine the missing number in the problem, we start by noting that the sum of two prime numbers is odd. For the sum of two numbers to be odd, one must be even and the other must be odd. Prime numbers are mostly odd, with the exception of the number 2, which is the only even prime number. Therefore, if the sum of two primes is odd, one of the primes must be 2 (the even prime) and the other must be an odd prime. Examples confirming this include:- (2 + 3 = 5) (odd)- (2 + 5 = 7) (odd)- (2 + 7 = 9) (odd)Conversely, adding two odd primes results in an even sum:- (3 + 5 = 8) (even)- (5 + 7 = 12) (even)Since 2 is the only even prime number, it must be one of the numbers when their sum is odd. Thus, the answer is (boxed{2}).]

🔑:The sum of two prime numbers is an odd number, which means one of the numbers must be 2. Therefore, the answer is boxed{2}.

🧠:[<think>Okay, so the problem is: 3 boys and 2 girls are lined up in a row. If the 2 girls cannot stand next to each other, the number of different arrangements is ( ), with options A to D. Let me think about how to approach this.First, I know that when arranging people with certain restrictions, sometimes it's easier to calculate the total number of arrangements without restrictions and then subtract the ones that violate the conditions. But maybe another way is to arrange the non-restricted elements first and then place the restricted ones in the available slots. Let me try both methods and see which works better.Total number of people is 3 boys + 2 girls = 5 people. Without any restrictions, the total number of arrangements is 5! = 120. But we have a restriction: the two girls cannot stand next to each other. So maybe the first method is total arrangements minus the arrangements where the girls are next to each other.Let me try that. Total arrangements: 5! = 120. Now, to find the number of arrangements where the two girls are next to each other. If we consider the two girls as a single entity or "block," then we have 4 entities to arrange: the block of girls and the 3 boys. The number of ways to arrange these 4 entities is 4! = 24. However, within the girl block, the two girls can switch places, so we multiply by 2: 24 * 2 = 48. So the number of arrangements where the girls are next to each other is 48. Therefore, the number of valid arrangements is total minus invalid: 120 - 48 = 72. Hmm, 72 is one of the options, option D. So that seems straightforward.Wait, but let me check if this is correct. Alternatively, maybe using the second method: arrange the boys first and then place the girls in the gaps. Let's try that approach too to confirm.If we arrange the 3 boys first. The number of ways to arrange 3 boys is 3! = 6. Once the boys are arranged, there are gaps where we can place the girls. Specifically, when you arrange n people, there are n+1 gaps between them (including the ends). So with 3 boys, there are 4 gaps: _ B _ B _ B _ . We need to place 2 girls in these 4 gaps, and they cannot be in the same gap (since that would make them adjacent). So the number of ways to choose 2 gaps out of 4 is C(4,2) = 6. Then, for each chosen gap, since there are 2 girls, they can be arranged in 2! = 2 ways. So total number of arrangements is 3! * C(4,2) * 2! = 6 * 6 * 2 = 72. Same result. So both methods give 72, which is D. So that seems correct.Wait, but let me just make sure that there isn't a mistake in my reasoning. For the first method, treating the girls as a block: when we consider the two girls as a single block, then we have 4 elements (3 boys + 1 girl block). Arranged in 4! ways, and the girls can switch places, so multiply by 2, giving 24 * 2 = 48. Then subtract that from 120 to get 72. That seems right.Alternatively, maybe I miscalculated something? Let me recount. Total arrangements: 5 people, 5! = 120. Girls adjacent: treat them as one, so 4! * 2! = 24 * 2 = 48. 120 - 48 = 72. Yes. The other method: arrange boys (3!), gaps (4), choose 2 gaps (C(4,2)=6), arrange girls (2!), multiply all together: 6 * 6 * 2 = 72. So both methods concur.Therefore, the answer should be D:72. Let me check the options again: A:10, B:48, C:64, D:72. So D is the answer.Wait, but just to be thorough, perhaps there's a way someone could get confused here. For example, maybe someone might think that when arranging the girls in the gaps, they need to multiply by the number of ways to arrange the girls, which is 2!, but maybe they forget that? Let's see. If someone does 3! * C(4,2) = 6 * 6 = 36, that would be wrong. But no, in my calculation, I included the 2! as well. So that's 6 * 6 * 2 =72. So that's correct.Alternatively, if someone thought that the total number is 5! / (3!2!) which is 10, but that's combinations, not permutations. Wait, but the answer A is 10. But that would be if we were counting combinations where order doesn't matter. But here, the problem says "different arrangements," which implies permutations. So A is too low. 10 is the number of ways to choose positions for the girls if order didn't matter, which it does. So C(5,2) =10, but since the girls are distinct and the boys are distinct, it's permutations. So 10 would be incorrect. So someone might make that mistake.Alternatively, maybe someone miscalculates the gaps. For instance, thinking that with 3 boys, there are 3-1=2 gaps between them, but no, actually when arranging in a row, the number of gaps is n+1 for n people. So 3 boys have 4 gaps. So that part is correct.Alternatively, maybe someone thought that the girls can't be next to each other, so they subtract only the cases where girls are adjacent once, but perhaps overlapping adjacents? But in this case, since there are only two girls, they can only be adjacent in one way. So no, that's not an issue here.Alternatively, maybe someone thought that the girls are identical? But the problem doesn't say that. It says "2 girls," presumably distinct individuals, so their order matters. Similarly, the boys are distinct. Therefore, all arrangements are unique based on the order of distinct individuals.Therefore, both methods confirm that the answer is 72, which is D. So I think that's correct.Final Answerboxed{D}</think>To determine the number of different arrangements of 3 boys and 2 girls in a row where the 2 girls cannot stand next to each other, we can use two methods:1. Total arrangements minus invalid arrangements: - Total number of arrangements without restrictions: (5! = 120). - Number of arrangements where the two girls are next to each other: Treat the two girls as a single block, resulting in 4 entities (3 boys + 1 girl block). The number of ways to arrange these 4 entities is (4!). Within the girl block, the two girls can switch places, so multiply by (2!): [ 4! times 2! = 24 times 2 = 48 ] - Subtract the invalid arrangements from the total: [ 120 - 48 = 72 ]2. Arranging boys first and placing girls in gaps: - Number of ways to arrange 3 boys: (3! = 6). - There are 4 gaps created by arranging 3 boys: (_ B _ B _ B _). - Choose 2 gaps out of 4 to place the girls: (binom{4}{2} = 6). - Arrange the 2 girls in the chosen gaps: (2! = 2). - Total number of arrangements: [ 3! times binom{4}{2} times 2! = 6 times 6 times 2 = 72 ]Both methods confirm that the number of different arrangements is (boxed{D}).,<think>Okay, so I have this problem here where I need to figure out how many different ways 3 boys and 2 girls can be lined up in a row, but the catch is that the two girls can't stand next to each other. The options are A: 10, B: 48, C: 64, D: 72. Hmm, let's see how to approach this.First, I remember that when dealing with permutations where certain people can't be next to each other, sometimes it's helpful to subtract the number of arrangements where they are next to each other from the total number of arrangements. That way, you get the valid ones. Let me verify if that works here.Total number of people is 3 boys + 2 girls = 5 people. If there were no restrictions, the total number of arrangements would be 5! (5 factorial), which is 5 × 4 × 3 × 2 × 1 = 120. But since there are restrictions, we need to adjust this.Wait, but the answer options are way smaller than 120. The highest option is 72, which is less than 120. So maybe I'm misunderstanding something. Wait, no, 5! is 120, but the options are 10, 48, 64, 72. Maybe my approach is wrong.Alternatively, perhaps the problem is considering the boys and girls as indistinct? Wait, no, the problem says "different arrangements," so each person is distinct. So my initial thought was correct: total arrangements are 5! = 120, and then subtract the ones where the two girls are next to each other.So let's proceed with that method. First, compute total arrangements: 120. Then compute the number of arrangements where the two girls are next to each other, and subtract that from 120.To compute the number of arrangements where the two girls are adjacent: treat the two girls as a single entity or "block." So instead of 5 separate people, we have 4 entities: the block of girls and the three boys. The number of ways to arrange these 4 entities is 4! = 24. But within the girl block, the two girls can be arranged in 2! = 2 ways. So total arrangements with the girls together are 4! × 2! = 24 × 2 = 48.Therefore, the number of valid arrangements where the girls are not next to each other would be total arrangements minus invalid ones: 120 - 48 = 72. So the answer is D: 72. But wait, let me check if that's correct.Alternatively, maybe using another method. Let's first arrange the boys and then place the girls in the gaps. That might be a more straightforward way. Let's try that.First, arrange the 3 boys. The number of ways to arrange 3 boys is 3! = 6. Once the boys are arranged, there are gaps where we can place the girls. Specifically, for n boys, there are (n + 1) gaps. So with 3 boys, there are 4 gaps: _ B _ B _ B _.We need to place 2 girls into these 4 gaps, and they can't be in the same gap (since that would make them adjacent). The number of ways to choose 2 gaps out of 4 is C(4, 2) = 6. Then, for each choice of gaps, the girls can be arranged in 2! = 2 ways. So the total number of ways is 3! × C(4, 2) × 2! = 6 × 6 × 2 = 72. Which matches the previous result. So that seems correct.Therefore, the answer is D: 72. Hmm, but let me check again. If the girls can't be next to each other, then the two methods both give 72, which is option D. The options given are A:10, B:48, C:64, D:72. So D is the correct answer. But let me think once more to make sure.Alternatively, if I think of permutations where the girls are not adjacent. Let me list all possibilities. Wait, but that's time-consuming with 5 people. However, maybe for a smaller case, but 5 people is 120 arrangements, which is too many. But perhaps I can check with a smaller example.Suppose there are 2 girls and 1 boy. Total arrangements: 3! = 6. If girls can't be adjacent, how many? If they can't be adjacent, but there are only 3 people, then the girls have to be separated by the boy. So the arrangements are G B G and G G B and B G G. Wait, but in that case, G G B and B G G have the girls adjacent, so those are invalid. So only G B G is valid. So only 2 arrangements? Wait, no: G B G, B G B? Wait, wait, if there's 1 boy and 2 girls, the total number is 3. If girls can't be adjacent, the only valid arrangement is when the boy is in the middle: G B G. So that's 1 arrangement. But how many total arrangements? 3! = 6. Number of arrangements with girls adjacent: treat the two girls as a block, so 2 entities: G1G2 and B. Number of arrangements: 2! × 2! = 2 × 2 = 4 (since the block can be in 2 positions and the girls can swap). Therefore, invalid arrangements are 4, so valid ones are 6 - 4 = 2? Wait, but in reality, if the two girls are adjacent, they can be in positions 1-2, 2-3. So each of these can be arranged in 2 ways (G1G2 or G2G1) and the boy in the remaining position. So for positions 1-2: G1G2 B and G2G1 B. For positions 2-3: B G1G2 and B G2G1. So total 4 invalid arrangements, so valid ones are 6 - 4 = 2. But actually, the valid arrangements are G1 B G2 and G2 B G1. So 2, which matches. So in that case, the formula works.So applying the same logic to the original problem: total arrangements 5! = 120. Invalid arrangements (girls adjacent): treat as a block, 4! × 2! = 24 × 2 = 48. So valid arrangements: 120 - 48 = 72. So that seems correct.Alternatively, the second method: arrange the boys first, 3! = 6. Then choose gaps: C(4,2) = 6. Then arrange the girls: 2! = 2. So 6 × 6 × 2 = 72. Same result.So both methods give 72, which is option D. Therefore, the answer must be D:72.But let me check once again in case I made a mistake. Let's take a simple case with 2 girls and 2 boys. If girls can't be adjacent. Total arrangements: 4! = 24. Invalid arrangements: treat girls as a block: 3! × 2! = 12. So valid: 24 - 12 = 12. Using the other method: arrange boys: 2! = 2. Gaps: 3. Choose 2 gaps: C(3,2)=3. Arrange girls: 2! =2. So total: 2×3×2=12. Which matches. So that works.Therefore, the method is correct. So returning to the original problem, answer is D:72.Final Answerboxed{D}</think>To determine the number of different arrangements of 3 boys and 2 girls in a row where the 2 girls cannot stand next to each other, we can use two methods:1. Total Arrangements Minus Invalid Arrangements: - Total number of arrangements without any restrictions: (5! = 120). - Number of invalid arrangements where the two girls are next to each other: Treat the two girls as a single block. This reduces the problem to arranging 4 entities (the block and 3 boys), which can be done in (4!) ways. Within the block, the two girls can be arranged in (2!) ways. Thus, the number of invalid arrangements is (4! times 2! = 24 times 2 = 48). - Valid arrangements: (120 - 48 = 72).2. Arranging Boys and Placing Girls in Gaps: - Arrange the 3 boys: (3! = 6) ways. - There are 4 gaps created by the boys: (_ B _ B _ B _). - Choose 2 gaps out of 4 to place the girls: (binom{4}{2} = 6). - Arrange the 2 girls in the chosen gaps: (2! = 2) ways. - Total number of valid arrangements: (3! times binom{4}{2} times 2! = 6 times 6 times 2 = 72).Both methods confirm that the number of different arrangements is (boxed{D}).]

🔑:According to the problem, since the 2 girls cannot stand next to each other, we first arrange the 3 boys, which can be done in A_3^3 = 6 ways.After arranging the 3 boys, there are 4 spaces where the 2 girls can be placed. The number of ways to place the 2 girls in these spaces is A_4^2 = 12.Therefore, the total number of different arrangements is 12 times 6 = 72.Hence, the correct answer is boxed{D}.

🧠:[<think>Okay, so I need to find the probability that a randomly chosen integer from the list 10 to 19 is even. Hmm, let me think. First, I should figure out how many numbers are in the list. The numbers start at 10 and end at 19. Let me count them: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. That's 10 numbers in total. So the total number of possible outcomes is 10.Now, the probability would be the number of favorable outcomes (even numbers) divided by the total number of possible outcomes. So I need to find how many even numbers are there between 10 and 19 inclusive. Let me list them out again and check which ones are even. Even numbers are those divisible by 2, right? So starting with 10: 10 is even, 11 is odd, 12 is even, 13 is odd, 14 is even, 15 is odd, 16 is even, 17 is odd, 18 is even, 19 is odd. Let's count the even numbers here: 10, 12, 14, 16, 18. That's 5 numbers. Wait, 10 is the first one, then every other number. So from 10 to 19, since 10 is even and the numbers increase by 1 each time, every other number will be even. So between 10 and 19 inclusive, how many even numbers? Let me check again. 10, 12, 14, 16, 18 – that's 5 even numbers. So 5 out of 10 total numbers. Therefore, the probability is 5/10, which simplifies to 1/2. But looking at the answer choices, option C is 5/10, so that should be the answer. Let me just confirm once more to make sure I didn't make a mistake in counting. Total numbers: 10 (from 10 to 19 inclusive). Even numbers: 10,12,14,16,18. Yep, 5 of them. So 5/10. So the answer is C. But hold on, the options are (A) 3/10, (B)4/10, (C)5/10, (D)6/10, (E)7/10. So 5/10 is option C. So I think that's correct. But wait, wait, let me check again because sometimes when counting inclusive ranges, you can make a mistake. Let me list all numbers again: 10,11,12,13,14,15,16,17,18,19. That's 10 numbers. Even numbers are 10,12,14,16,18. Five numbers. So yes, 5/10. So probability is 1/2, which is 5/10. So answer is C. Yeah, that seems right. Hmm. I don't see any mistake here. So yeah, I think the answer is C.Final Answerboxed{C}</think>To determine the probability that a randomly chosen integer from the list 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 is even, we first note that there are 10 numbers in total. Next, we identify the even numbers in this list. The even numbers between 10 and 19 inclusive are 10, 12, 14, 16, and 18. Counting these, we find there are 5 even numbers.The probability is calculated by dividing the number of favorable outcomes (even numbers) by the total number of possible outcomes (total numbers in the list). Therefore, the probability is:[frac{5}{10}]Thus, the final answer is (boxed{C}).,<think>Okay, let's see. I need to find the probability that a randomly chosen integer from the list 10 to 19 is even. The options are given from (A) 3/10 up to (E) 7/10. Hmm, probability is usually the number of favorable outcomes over the total number of possible outcomes. So first, I should figure out how many numbers are in the list. The list starts at 10 and ends at 19. Let me count them. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. That's 10 numbers total. So the denominator of my probability will be 10.Now, the numerator is the number of even integers in that list. Let me go through each number and see which ones are even. Starting with 10: that's even. 11 is odd. 12 is even. 13 is odd. 14 is even. 15 is odd. 16 is even. 17 is odd. 18 is even. 19 is odd. Let me count how many evens that is. 10, 12, 14, 16, 18. That's 5 numbers. So 5 even numbers out of 10 total numbers. So the probability should be 5/10, which simplifies to 1/2. Looking at the options, (C) is 5/10. So the answer should be (C). Wait, let me double-check. Sometimes when counting inclusive ranges, people make mistakes. From 10 to 19, inclusive. Let's confirm how many numbers that is. The formula is (last - first) + 1. So 19 - 10 = 9, plus 1 is 10. Correct, 10 numbers. Now even numbers. Since the numbers are consecutive, starting at even (10) and ending at odd (19). In a set of consecutive numbers, half are even and half are odd if the count is even. But here, from 10 to 19, there are 10 numbers. Starting with even, the pattern is even, odd, even, odd... So in 10 numbers, 5 evens and 5 odds. So yes, 5 even numbers. So 5/10, which is answer (C). I think that's right.But let me check another way. Maybe list all the numbers and mark them:10 - even11 - odd12 - even13 - odd14 - even15 - odd16 - even17 - odd18 - even19 - oddYep, evens are 10, 12, 14, 16, 18. That's five numbers. So 5 out of 10. So 5/10. Answer choice (C). I don't see any mistakes here. Maybe the options are simplified? 5/10 is equal to 1/2, but since 5/10 is one of the options, I don't need to reduce it. So (C) is the correct answer. I think that's solid. The answer is (C).Final Answerboxed{C}</think>To find the probability that a randomly chosen integer from the list 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 is even, we start by determining the total number of integers in the list. The list ranges from 10 to 19 inclusive, which gives us (19 - 10 + 1 = 10) numbers.Next, we identify the even numbers in this list. Checking each number:- 10 is even- 11 is odd- 12 is even- 13 is odd- 14 is even- 15 is odd- 16 is even- 17 is odd- 18 is even- 19 is oddThe even numbers are 10, 12, 14, 16, and 18, which totals 5 numbers.The probability is the number of favorable outcomes (even numbers) divided by the total number of possible outcomes (total numbers in the list):[frac{5}{10}]Thus, the probability that the chosen integer is even is (boxed{C}).]

🔑:The number of integers in the list is 10 . Of these integers, 5 are even (they are 10, 12, 14, 16, and 18). Thus, the probability that the chosen integer is even is frac{5}{10}.ANswer: (mathrm{C})