💡First, I recognize that the cotangent of an angle is the reciprocal of the tangent of that angle. So, to find (cot 30^circ), I need to determine (tan 30^circ) first.I recall that (tan 30^circ) is a known value from trigonometric tables. Specifically, (tan 30^circ = frac{1}{sqrt{3}}).Now, taking the reciprocal of (tan 30^circ) gives me (cot 30^circ):[cot 30^circ = frac{1}{tan 30^circ} = frac{1}{frac{1}{sqrt{3}}} = sqrt{3}]Additionally, I can verify this result using the definitions of sine and cosine. I know that:[cos 30^circ = frac{sqrt{3}}{2} quad text{and} quad sin 30^circ = frac{1}{2}]Using the identity (cot theta = frac{cos theta}{sin theta}), I substitute the known values:[cot 30^circ = frac{cos 30^circ}{sin 30^circ} = frac{frac{sqrt{3}}{2}}{frac{1}{2}} = sqrt{3}]Both methods confirm that (cot 30^circ = sqrt{3}).

💡First, I'll denote the number of boxes Stan has as ( S ).John has 20% more boxes than Jules. Since John has 30 boxes, I can set up the equation:[text{John} = 1.20 times text{Jules}][30 = 1.20 times text{Jules}]Solving for Jules:[text{Jules} = frac{30}{1.20} = 25 text{ boxes}]Next, Jules has 5 more boxes than Joseph:[text{Joseph} = text{Jules} - 5 = 25 - 5 = 20 text{ boxes}]Joseph has 80% fewer boxes than Stan, which means Joseph has 20% of Stan's boxes:[text{Joseph} = 0.20 times S][20 = 0.20 times S]Solving for ( S ):[S = frac{20}{0.20} = 100]Therefore, Stan has 100 boxes.

💡Okay, so I've got this integral to solve:int frac{sqrt[5]{left(1+sqrt[5]{x^{4}}right)^{4}}}{x^{2} cdot sqrt[25]{x^{11}}} , dxHmm, that looks pretty complicated with all those roots and exponents. Let me try to break it down step by step.First, I need to simplify the integrand. The numerator is a fifth root of something raised to the fourth power, and the denominator has x squared times the 25th root of x to the 11th power. Maybe rewriting everything with fractional exponents will make it easier.Starting with the numerator:sqrt[5]{left(1+sqrt[5]{x^{4}}right)^{4}} = left(1 + x^{4/5}right)^{4/5}Okay, that seems manageable. Now the denominator:x^{2} cdot sqrt[25]{x^{11}} = x^{2} cdot x^{11/25}To combine these, I add the exponents:2 + frac{11}{25} = frac{50}{25} + frac{11}{25} = frac{61}{25}So the denominator simplifies to x^{61/25}. Putting it all together, the integral becomes:int frac{left(1 + x^{4/5}right)^{4/5}}{x^{61/25}} , dxHmm, that still looks a bit messy. Maybe I can rewrite it as:int x^{-61/25} left(1 + x^{4/5}right)^{4/5} , dxNow, I need to figure out a substitution that can simplify this expression. Let me think about the exponents. The term inside the parentheses is 1 + x^{4/5}, and it's raised to the 4/5 power. The rest of the integrand has x^{-61/25}.I recall that when dealing with integrals of the form x^m (a + x^n)^p, sometimes substitution can help, especially if the exponents relate in a certain way. Let me check if this fits a pattern where substitution is feasible.Let me denote:m = -frac{61}{25}, quad n = frac{4}{5}, quad p = frac{4}{5}I remember that if frac{m + 1}{n} + p is an integer, then the integral might be manageable. Let me compute that:frac{m + 1}{n} + p = frac{-frac{61}{25} + 1}{frac{4}{5}} + frac{4}{5} = frac{-frac{36}{25}}{frac{4}{5}} + frac{4}{5}Simplifying the division:frac{-frac{36}{25}}{frac{4}{5}} = -frac{36}{25} times frac{5}{4} = -frac{9}{5}So,-frac{9}{5} + frac{4}{5} = -1Which is indeed an integer. That’s a good sign; it suggests that substitution might work here.Now, I need to choose a substitution. Let me set:z = sqrt[5]{x^{-4/5} + 1}Wait, let me think about that. Alternatively, maybe I should set:u = 1 + x^{4/5}But then, du = frac{4}{5} x^{-1/5} dx, which doesn't directly appear in the integrand. Hmm, maybe that's not the best substitution.Alternatively, considering the form of the integrand, perhaps a substitution that can rationalize the expression. Let me try:Let u = x^{4/5}, then du = frac{4}{5} x^{-1/5} dx, which still doesn't directly match the integrand.Wait, maybe I need a substitution that can handle both the x^{-61/25} and the (1 + x^{4/5})^{4/5} terms. Let me think about the exponents again.Looking back, since frac{m + 1}{n} + p = -1, which is an integer, perhaps I can use a substitution where z = (1 + x^{4/5})^{k} for some exponent k.Alternatively, maybe I can set z = x^{a} for some a to simplify the exponents. Let me try that.Let me set z = x^{b}, then dz = b x^{b - 1} dx. I need to choose b such that the substitution simplifies the integral.Looking at the exponents in the integrand: x^{-61/25} and (1 + x^{4/5})^{4/5}. If I set z = x^{4/5}, then dz = frac{4}{5} x^{-1/5} dx. Hmm, but in the integrand, I have x^{-61/25}, which is x^{-2.44}, and x^{-1/5} is x^{-0.2}. They don't directly relate, but maybe I can express x^{-61/25} in terms of z.Wait, let me compute x^{-61/25} in terms of z = x^{4/5}:Since z = x^{4/5}, then x = z^{5/4}. Therefore,x^{-61/25} = left(z^{5/4}right)^{-61/25} = z^{-61/25 times 5/4} = z^{-61/20}Hmm, that might complicate things further. Maybe another substitution.Let me think about the entire expression inside the integral:x^{-61/25} (1 + x^{4/5})^{4/5}If I let u = 1 + x^{4/5}, then du = frac{4}{5} x^{-1/5} dx, which is similar to the derivative of x^{4/5}. But in the integrand, I have x^{-61/25}, which is x^{-2.44}, and x^{-1/5} is x^{-0.2}. They are not directly related, but perhaps I can manipulate the substitution.Alternatively, maybe I can express x^{-61/25} as x^{-2 - 11/25}, which is x^{-2} cdot x^{-11/25}. Wait, that's how the original denominator was written. Maybe that's not helpful.Wait, another approach: Let me consider the substitution t = x^{1/5}, so that x = t^5 and dx = 5t^4 dt. Let's see if that helps.Substituting, the integral becomes:int frac{left(1 + (t^5)^{4/5}right)^{4/5}}{(t^5)^2 cdot sqrt[25]{(t^5)^{11}}} cdot 5t^4 dtSimplify each part:First, (t^5)^{4/5} = t^4, so the numerator becomes:left(1 + t^4right)^{4/5}The denominator:(t^5)^2 = t^{10}sqrt[25]{(t^5)^{11}} = (t^5)^{11/25} = t^{11/5}So the denominator is:t^{10} cdot t^{11/5} = t^{10 + 11/5} = t^{50/5 + 11/5} = t^{61/5}Putting it all together, the integral becomes:int frac{left(1 + t^4right)^{4/5}}{t^{61/5}} cdot 5t^4 dt = 5 int frac{left(1 + t^4right)^{4/5} t^4}{t^{61/5}} dtSimplify the exponents:t^4 / t^{61/5} = t^{4 - 61/5} = t^{20/5 - 61/5} = t^{-41/5}So the integral is:5 int left(1 + t^4right)^{4/5} t^{-41/5} dtHmm, that still looks complicated, but maybe it's easier to handle in terms of t. Let me see if I can find a substitution here.Let me set u = 1 + t^4, then du = 4t^3 dt. But in the integrand, I have t^{-41/5}, which is t^{-8.2}. That doesn't directly relate to t^3. Maybe another substitution.Alternatively, perhaps I can write t^{-41/5} as t^{-8 - 1/5} = t^{-8} cdot t^{-1/5}. Hmm, not sure if that helps.Wait, maybe I can express t^{-41/5} as t^{-8.2} = t^{-8 - 0.2} = t^{-8} cdot t^{-0.2}. Still not helpful.Alternatively, maybe I can factor out t^4 from the expression inside the integral:5 int left(1 + t^4right)^{4/5} t^{-41/5} dt = 5 int t^{-41/5} left(1 + t^4right)^{4/5} dtLet me write t^{-41/5} as t^{-8 - 1/5} = t^{-8} cdot t^{-1/5}. So,5 int t^{-8} t^{-1/5} left(1 + t^4right)^{4/5} dt = 5 int t^{-8 - 1/5} left(1 + t^4right)^{4/5} dtHmm, not sure. Maybe another substitution. Let me try u = t^4, then du = 4t^3 dt. But in the integrand, I have t^{-8 - 1/5}, which is t^{-8.2}. Not directly related.Wait, maybe I can write t^{-8.2} = t^{-8 - 0.2} = t^{-8} cdot t^{-0.2}. If I set u = t^4, then t^{-0.2} = u^{-1/20}, but I'm not sure.Alternatively, maybe I can write the entire integrand in terms of u = 1 + t^4.Let me try that:Let u = 1 + t^4, then du = 4t^3 dt, so t^3 dt = du/4.But in the integrand, I have t^{-8.2} cdot (1 + t^4)^{4/5} dt. Let me express t^{-8.2} in terms of u.Since u = 1 + t^4, then t^4 = u - 1, so t = (u - 1)^{1/4}. Therefore,t^{-8.2} = left((u - 1)^{1/4}right)^{-8.2} = (u - 1)^{-8.2/4} = (u - 1)^{-2.05}Hmm, that seems messy. Maybe not the best approach.Wait, perhaps I need to consider a substitution that can handle both the t^{-41/5} and the (1 + t^4)^{4/5} terms. Let me think about the exponents again.The integrand is:5 int t^{-41/5} (1 + t^4)^{4/5} dtLet me write t^{-41/5} as t^{-8 - 1/5} = t^{-8} cdot t^{-1/5}. So,5 int t^{-8} t^{-1/5} (1 + t^4)^{4/5} dtHmm, maybe I can set v = t^{-1/5}, then dv = -frac{1}{5} t^{-6/5} dt. But I have t^{-8} t^{-1/5} = t^{-8 - 1/5} = t^{-41/5}, which is v^5 cdot t^{-8}. Not sure.Alternatively, maybe I can set w = t^4, then dw = 4t^3 dt, but again, not directly related.Wait, perhaps I can write the integrand as:5 int t^{-41/5} (1 + t^4)^{4/5} dt = 5 int t^{-41/5} (1 + t^4)^{4/5} dtLet me factor out t^4 from the (1 + t^4) term:(1 + t^4)^{4/5} = t^{16/5} left(1 + frac{1}{t^4}right)^{4/5}So the integrand becomes:5 int t^{-41/5} cdot t^{16/5} left(1 + frac{1}{t^4}right)^{4/5} dt = 5 int t^{-25/5} left(1 + frac{1}{t^4}right)^{4/5} dt = 5 int t^{-5} left(1 + frac{1}{t^4}right)^{4/5} dtHmm, that simplifies a bit. Now, let me set u = 1 + frac{1}{t^4}, then du = -frac{4}{t^5} dt. So,du = -frac{4}{t^5} dt implies dt = -frac{t^5}{4} duBut in the integrand, I have t^{-5} cdot u^{4/5} dt. Let's substitute:5 int t^{-5} u^{4/5} dt = 5 int t^{-5} u^{4/5} cdot left(-frac{t^5}{4}right) du = -frac{5}{4} int u^{4/5} duThat's much better! Now, integrating with respect to u:-frac{5}{4} int u^{4/5} du = -frac{5}{4} cdot frac{u^{9/5}}{9/5} + C = -frac{5}{4} cdot frac{5}{9} u^{9/5} + C = -frac{25}{36} u^{9/5} + CNow, substituting back u = 1 + frac{1}{t^4}:-frac{25}{36} left(1 + frac{1}{t^4}right)^{9/5} + CBut remember, t = x^{1/5}, so t^4 = x^{4/5}. Therefore,1 + frac{1}{t^4} = 1 + frac{1}{x^{4/5}} = frac{x^{4/5} + 1}{x^{4/5}} = frac{1 + x^{4/5}}{x^{4/5}}So,left(1 + frac{1}{t^4}right)^{9/5} = left(frac{1 + x^{4/5}}{x^{4/5}}right)^{9/5} = left(frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}right)^{9/5}Therefore, the integral becomes:-frac{25}{36} left(frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}right)^{9/5} + CSimplifying the expression inside the parentheses:frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}} = sqrt[5]{frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}} = sqrt[5]{frac{1 + sqrt[5]{x^4}}{x^{4/5}}}But actually, it's already simplified as frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}, so the final answer is:-frac{25}{36} left(frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}right)^{9/5} + COr, written with roots:-frac{25}{36} left(sqrt[5]{frac{1 + sqrt[5]{x^4}}{sqrt[5]{x^4}}}right)^9 + CI think that's the final answer. Let me just double-check my steps to make sure I didn't make any mistakes.1. I simplified the integrand correctly by rewriting the roots as exponents.2. I considered substitution and found that frac{m + 1}{n} + p was an integer, which suggested a substitution might work.3. I tried substituting t = x^{1/5}, which transformed the integral into a more manageable form.4. After substitution, I manipulated the integrand to factor out terms and found a suitable substitution u = 1 + frac{1}{t^4}.5. The substitution simplified the integral to a straightforward power function, which I integrated.6. Finally, I substituted back in terms of x to get the final answer.Everything seems to check out. I think I did it right!

💡Alright, so I have this problem where I need to find the minimum value of (a^2 + b^2) given that the quartic equation (x^4 + a x^3 + b x^2 + a x + 1 = 0) has at least one real root. Hmm, quartic equations can be tricky, but maybe there's a way to simplify this.First, I notice that the equation is palindromic. That means the coefficients read the same forwards and backwards. Specifically, the coefficients of (x^4) and the constant term are both 1, and the coefficients of (x^3) and (x) are both (a). This symmetry might be useful.I remember that for palindromic polynomials of even degree, like this quartic, we can make a substitution to reduce the degree. The substitution is usually (y = x + frac{1}{x}). Let me try that.If I let (y = x + frac{1}{x}), then (y^2 = x^2 + 2 + frac{1}{x^2}), and (y^3 = x^3 + 3x + frac{3}{x} + frac{1}{x^3}). But maybe I don't need to go that far. Let me see.Starting with the original equation:[x^4 + a x^3 + b x^2 + a x + 1 = 0]Divide both sides by (x^2) (assuming (x neq 0), which is fine because (x = 0) isn't a root):[x^2 + a x + b + frac{a}{x} + frac{1}{x^2} = 0]Now, this can be rewritten as:[left(x^2 + frac{1}{x^2}right) + a left(x + frac{1}{x}right) + b = 0]Notice that (x^2 + frac{1}{x^2}) is equal to (left(x + frac{1}{x}right)^2 - 2). So substituting (y = x + frac{1}{x}), we get:[y^2 - 2 + a y + b = 0]Simplifying:[y^2 + a y + (b - 2) = 0]So now, the quartic equation reduces to a quadratic in terms of (y). For the original equation to have a real root (x), the quadratic in (y) must have a real root (y) such that (y = x + frac{1}{x}) is real. But (y) must satisfy (|y| geq 2) because (x + frac{1}{x}) is minimized at 2 when (x = 1) or (x = -1).Therefore, the quadratic equation (y^2 + a y + (b - 2) = 0) must have at least one real root with (|y| geq 2).Now, I need to find the minimum value of (a^2 + b^2) such that this condition is satisfied.Let me denote the quadratic equation as:[y^2 + a y + c = 0]where (c = b - 2). So, we have:[y^2 + a y + c = 0]We need this quadratic to have at least one real root with (|y| geq 2). Let's denote the roots of this quadratic as (y_1) and (y_2). By Vieta's formulas, we have:[y_1 + y_2 = -a][y_1 y_2 = c = b - 2]We need at least one of (y_1) or (y_2) to satisfy (|y| geq 2).To ensure that, we can consider two cases:1. The quadratic has a root (y) such that (y geq 2).2. The quadratic has a root (y) such that (y leq -2).But since (a) and (b) are real numbers, and we are looking for the minimum value of (a^2 + b^2), perhaps it's symmetric whether we consider positive or negative roots. Let me assume without loss of generality that (a geq 0). If (a) is negative, we can consider (-a) and adjust accordingly.So, let's focus on the case where the quadratic has a root (y geq 2).For the quadratic (y^2 + a y + c = 0) to have a root (y geq 2), the following must hold:1. The discriminant must be non-negative: (a^2 - 4c geq 0).2. The quadratic evaluated at (y = 2) must be less than or equal to zero: (4 + 2a + c leq 0).Similarly, for the quadratic to have a root (y leq -2), we need:1. The discriminant must be non-negative: (a^2 - 4c geq 0).2. The quadratic evaluated at (y = -2) must be less than or equal to zero: (4 - 2a + c leq 0).But since we assumed (a geq 0), let's focus on the first case where (y geq 2).So, we have two conditions:1. (a^2 - 4c geq 0)2. (4 + 2a + c leq 0)But (c = b - 2), so substituting back:1. (a^2 - 4(b - 2) geq 0)2. (4 + 2a + (b - 2) leq 0)Simplify the second condition:(4 + 2a + b - 2 leq 0)Which simplifies to:(2 + 2a + b leq 0)Or:(b leq -2a - 2)So, now we have:1. (a^2 - 4b + 8 geq 0)2. (b leq -2a - 2)Our goal is to minimize (a^2 + b^2) subject to these constraints.Let me write down the constraints:1. (a^2 - 4b + 8 geq 0)2. (b leq -2a - 2)3. (a geq 0)We can substitute (b) from the second inequality into the first inequality.From the second inequality, (b leq -2a - 2). Let's substitute (b = -2a - 2) into the first inequality to see if it holds:(a^2 - 4(-2a - 2) + 8 geq 0)Simplify:(a^2 + 8a + 8 + 8 geq 0)Wait, that's:(a^2 + 8a + 16 geq 0)Which is:((a + 4)^2 geq 0)Which is always true. So, the equality holds when (a = -4), but since we assumed (a geq 0), the minimum occurs at (a = 0). But wait, if (a = 0), then from the second inequality, (b leq -2(0) - 2 = -2). Then, substituting into the first inequality:(0^2 - 4b + 8 geq 0)Which is:(-4b + 8 geq 0)So:(-4b geq -8)Multiply both sides by (-1), reversing the inequality:(4b leq 8)(b leq 2)But from the second inequality, (b leq -2). So, (b) must be less than or equal to -2.Therefore, when (a = 0), (b) must be (leq -2). Then, (a^2 + b^2 = 0 + b^2), which is minimized when (b = -2), giving (a^2 + b^2 = 4).But is this the minimum? Maybe not, because perhaps for some (a > 0), (b) can be closer to zero, resulting in a smaller (a^2 + b^2).So, let's consider the case where (b = -2a - 2), as per the second inequality. Then, substitute this into the expression (a^2 + b^2):(a^2 + (-2a - 2)^2 = a^2 + 4a^2 + 8a + 4 = 5a^2 + 8a + 4)Now, we need to minimize this quadratic in terms of (a):(5a^2 + 8a + 4)To find the minimum, take the derivative with respect to (a) and set it to zero:(d/da (5a^2 + 8a + 4) = 10a + 8 = 0)Solving for (a):(10a + 8 = 0)(a = -8/10 = -4/5)But we assumed (a geq 0), so this critical point is not in our domain. Therefore, the minimum occurs at the boundary of the domain, which is (a = 0). As we saw earlier, at (a = 0), (b = -2), giving (a^2 + b^2 = 4).Wait, but earlier I thought maybe there's a smaller value. Maybe I need to consider the other case where the quadratic has a root (y leq -2). Let's explore that.If the quadratic has a root (y leq -2), then similar to before, we have:1. Discriminant (a^2 - 4c geq 0)2. Quadratic evaluated at (y = -2) is (leq 0):((-2)^2 + a(-2) + c leq 0)Which is:(4 - 2a + c leq 0)Again, (c = b - 2), so:(4 - 2a + b - 2 leq 0)Simplify:(2 - 2a + b leq 0)Or:(b leq 2a - 2)So, now we have:1. (a^2 - 4(b - 2) geq 0)2. (b leq 2a - 2)3. (a geq 0)Again, substituting (b = 2a - 2) into the first inequality:(a^2 - 4(2a - 2 - 2) geq 0)Wait, (c = b - 2 = (2a - 2) - 2 = 2a - 4). So, substituting into the first inequality:(a^2 - 4(2a - 4) geq 0)Simplify:(a^2 - 8a + 16 geq 0)Which is:((a - 4)^2 geq 0)Again, always true. So, the equality holds when (a = 4). But we are looking for the minimum (a^2 + b^2).So, substituting (b = 2a - 2) into (a^2 + b^2):(a^2 + (2a - 2)^2 = a^2 + 4a^2 - 8a + 4 = 5a^2 - 8a + 4)Now, minimize this quadratic in terms of (a):(5a^2 - 8a + 4)Take the derivative:(d/da (5a^2 - 8a + 4) = 10a - 8 = 0)Solving for (a):(10a - 8 = 0)(a = 8/10 = 4/5)Since (a geq 0), this is valid. So, at (a = 4/5), (b = 2*(4/5) - 2 = 8/5 - 10/5 = -2/5).Therefore, (a^2 + b^2 = (4/5)^2 + (-2/5)^2 = 16/25 + 4/25 = 20/25 = 4/5).So, this is smaller than the previous case where (a = 0) and (b = -2), which gave (a^2 + b^2 = 4).Therefore, the minimum value of (a^2 + b^2) is (4/5).But wait, let me verify if this is indeed the case. Let me check if with (a = 4/5) and (b = -2/5), the original quartic equation has a real root.So, substituting (a = 4/5) and (b = -2/5) into the quartic:(x^4 + (4/5)x^3 + (-2/5)x^2 + (4/5)x + 1 = 0)Let me see if this equation has a real root. Maybe I can try (x = 1):(1 + 4/5 - 2/5 + 4/5 + 1 = 1 + 1 + (4/5 - 2/5 + 4/5) = 2 + (6/5) = 16/5 neq 0)Not zero. How about (x = -1):(1 - 4/5 - 2/5 - 4/5 + 1 = 1 + 1 - (4/5 + 2/5 + 4/5) = 2 - 10/5 = 2 - 2 = 0)Oh, so (x = -1) is a root. Therefore, the quartic equation does have a real root when (a = 4/5) and (b = -2/5), and (a^2 + b^2 = 4/5).Therefore, the minimum value is indeed (4/5).**Final Answer**The minimum value of (a^2 + b^2) is boxed{dfrac{4}{5}}.

💡First, I need to ensure that both edge lengths are in the same unit. Since one cube has an edge length in inches and the other in feet, I'll convert the feet measurement to inches.Next, I'll calculate the volume of each cube using the formula for the volume of a cube, which is the edge length raised to the power of three.Finally, I'll find the ratio of the two volumes by dividing the volume of the smaller cube by the volume of the larger cube.

💡Okay, so I have this problem where I'm given two sequences of real numbers, (a_1, a_2, ldots, a_n) and (b_1, b_2, ldots, b_n). I need to prove that there exists an integer (k) between 1 and (n) such that the sum of the absolute differences between each (a_i) and (a_k) is less than or equal to the sum of the absolute differences between each (b_i) and (a_k). Hmm, let me try to understand what this means. Essentially, I need to find a point (a_k) such that the total distance from all (a_i) to (a_k) is not greater than the total distance from all (b_i) to (a_k). That sounds a bit like comparing the "central tendency" of the two sets of numbers. Maybe something related to medians or means?I remember that the sum of absolute deviations is minimized at the median. So, if I think about the median of the (a_i)s and the median of the (b_i)s, maybe that could help. But wait, the problem doesn't specify anything about the order of the (a_i)s and (b_i)s. Maybe I need to consider ordering them?Let me try to assume without loss of generality that both sequences are sorted in non-increasing order. So, (a_1 geq a_2 geq ldots geq a_n) and (b_1 geq b_2 geq ldots geq b_n). I think this might simplify things because it allows me to compare corresponding elements more easily.Now, if I consider the case where (n = 1), it's trivial because there's only one element in each sequence, and the inequality holds by default. So, the base case is fine.For the inductive step, suppose the statement is true for (n - 1). I need to show it holds for (n). Let me consider two cases:**Case 1: (b_n leq a_n)**If the smallest (b_i) is less than or equal to the smallest (a_i), then by the induction hypothesis, there exists some (k) between 1 and (n - 1) such that the sum of absolute differences for the first (n - 1) elements holds. Since (b_n leq a_n leq a_k), the absolute difference (|a_n - a_k|) is less than or equal to (|b_n - a_k|). Adding these to the existing sums, the inequality still holds for (n).**Case 2: (a_1 leq b_1)**Similarly, if the largest (a_i) is less than or equal to the largest (b_i), then by induction, there exists some (k) between 2 and (n) such that the sum of absolute differences for the last (n - 1) elements holds. Since (a_k leq a_1 leq b_1), the absolute difference (|a_1 - a_k|) is less than or equal to (|b_1 - a_k|). Adding these to the existing sums, the inequality still holds for (n).**Case 3: All (a_i) are interleaved with (b_i)**This is the trickier part. If neither of the above cases holds, it means that for all (i), (a_n leq b_i leq a_1). In this situation, I need to consider the functions (f(x) = sum_{i=1}^{n} |a_i - x|) and (g(x) = sum_{i=1}^{n} |b_i - x|). I recall that these functions are piecewise linear and convex. The key insight here is that if (f(a_1) > g(a_1)) and (f(a_n) > g(a_n)), then adding these would lead to a contradiction because the total sum of deviations from (a_1) and (a_n) should be equal for both functions. Therefore, there must be some (k) where (f(a_k) leq g(a_k)).Putting it all together, in all possible cases, there exists an integer (k) such that the required inequality holds. This completes the proof.**Final Answer**boxed{k}