🧠:[<think>Okay, so I need to find the intersection point B between line l₁ and line l₂, and then calculate the distance from point A(1,2) to point B. Let me start by understanding what each line is like.Line l₁ is given in parametric form: x = 1 + 3t and y = 2 - 4t, where t is a parameter. That means as t varies, x and y change accordingly. Line l₂ is given by the equation 2x - 4y = 5, which is a standard linear equation. To find their intersection point B, I need to find the values of x and y that satisfy both equations. Since l₁ is in parametric form, maybe I can substitute x and y from l₁ into the equation of l₂ and solve for t. Once I have t, I can plug it back into the parametric equations to get the coordinates of B. Then, using the distance formula between points A and B, which are (1,2) and the coordinates of B, I can find the distance. Let's try that step by step.First, substituting x and y from l₁ into l₂'s equation. So, x from l₁ is 1 + 3t and y is 2 - 4t. Plugging these into 2x - 4y = 5 gives:2*(1 + 3t) - 4*(2 - 4t) = 5.Let me compute this left side. First term: 2*(1 + 3t) = 2*1 + 2*3t = 2 + 6t.Second term: -4*(2 - 4t) = -4*2 + (-4)*(-4t) = -8 + 16t.So combining these two terms:2 + 6t - 8 + 16t = (2 - 8) + (6t + 16t) = -6 + 22t.Set that equal to 5:-6 + 22t = 5.Now, solving for t:22t = 5 + 6 = 11.So t = 11 / 22 = 1/2.Okay, so t is 1/2. Now, plug this back into the parametric equations of l₁ to find the coordinates of B.For x: x = 1 + 3*(1/2) = 1 + 3/2 = 5/2 or 2.5.For y: y = 2 - 4*(1/2) = 2 - 2 = 0.So point B has coordinates (5/2, 0). Let me check if this point lies on l₂ as well, just to verify.Plugging x = 5/2 and y = 0 into 2x - 4y:2*(5/2) - 4*0 = 5 - 0 = 5. Which matches the right side of l₂'s equation. Great, so that's correct.Now, the distance between point A(1,2) and point B(5/2, 0). The distance formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2].So plugging in the coordinates:sqrt[(5/2 - 1)^2 + (0 - 2)^2] = sqrt[(3/2)^2 + (-2)^2] = sqrt[(9/4) + 4].Convert 4 to fourths: 4 = 16/4. So 9/4 + 16/4 = 25/4.Thus, sqrt(25/4) = 5/2. So the distance is 5/2.Wait, 5/2 is 2.5. Let me check if that's correct. The x-coordinate difference is from 1 to 5/2, which is 3/2. The y-coordinate difference is from 2 to 0, which is -2. Squaring those gives (9/4) and 4, sum is 25/4, square root is 5/2. Yes, that's correct. So the distance is 5/2.So the intersection point B is (5/2, 0) and the distance from A to B is 5/2. Let me just recap to make sure I didn't make any mistakes. The substitution into l₂'s equation gave me t = 1/2, which led to coordinates (5/2, 0). Checked in l₂, it works. Calculated the distance using the formula, correct steps. Looks good. Maybe double-check the arithmetic?Original substitution:2*(1 + 3t) -4*(2 -4t) = 5.Compute 2*1 = 2, 2*3t = 6t, -4*2 = -8, -4*(-4t) = 16t.So 2 + 6t -8 +16t = (2 -8) + (6t +16t) = -6 +22t =5. So 22t =11, t=0.5. Correct.Plugging t=0.5 into x: 1 +3*(0.5)=1 +1.5=2.5=5/2. Correct.Y: 2 -4*(0.5)=2 -2=0. Correct. Then distance sqrt((2.5-1)^2 + (0-2)^2)=sqrt(1.5^2 + (-2)^2)=sqrt(2.25 +4)=sqrt(6.25)=2.5=5/2. Correct.Yes, all steps check out. So the answer should be point B at (5/2, 0) and distance 5/2.Final AnswerThe coordinates of intersection point ( B ) are (boxed{left( dfrac{5}{2}, 0 right)}) and the distance from ( A ) to ( B ) is (boxed{dfrac{5}{2}}).</think>To find the coordinates of the intersection point ( B ) of line ( l_1 ) and line ( l_2 ), we start by substituting the parametric equations of ( l_1 ) into the equation of ( l_2 ).The parametric equations of ( l_1 ) are:[begin{cases}x = 1 + 3t y = 2 - 4tend{cases}]Substituting ( x ) and ( y ) into the equation of ( l_2 ) ( ( 2x - 4y = 5 ) ):[2(1 + 3t) - 4(2 - 4t) = 5]Simplifying the left-hand side:[2 + 6t - 8 + 16t = -6 + 22t]Setting this equal to 5:[-6 + 22t = 5 implies 22t = 11 implies t = frac{1}{2}]Using ( t = frac{1}{2} ) in the parametric equations of ( l_1 ):[x = 1 + 3 left( frac{1}{2} right) = 1 + frac{3}{2} = frac{5}{2}][y = 2 - 4 left( frac{1}{2} right) = 2 - 2 = 0]Thus, the coordinates of point ( B ) are ( left( frac{5}{2}, 0 right) ).Next, we calculate the distance between point ( A(1, 2) ) and point ( B left( frac{5}{2}, 0 right) ) using the distance formula:[text{Distance} = sqrt{left( frac{5}{2} - 1 right)^2 + (0 - 2)^2} = sqrt{left( frac{3}{2} right)^2 + (-2)^2} = sqrt{frac{9}{4} + 4} = sqrt{frac{25}{4}} = frac{5}{2}]The coordinates of intersection point ( B ) are (boxed{left( dfrac{5}{2}, 0 right)}) and the distance from ( A ) to ( B ) is (boxed{dfrac{5}{2}}).,<think>Okay, so I need to find the coordinates of point B where line l₁ and line l₂ intersect, and then calculate the distance from point A(1,2) to point B. Let me start by understanding the problem.First, line l₁ is given in parametric form. The equations are:x = 1 + 3ty = 2 - 4twhere t is a parameter. Line l₂ is given by the equation 2x - 4y = 5. To find their intersection point B, I need to find the values of x and y that satisfy both equations. That means substituting the parametric equations of l₁ into the equation of l₂ and solving for t. Once I have t, I can plug it back into the parametric equations to get the coordinates of B. Then, using the distance formula between points A(1,2) and B(x,y), I can find the distance.Let me try substituting x and y from l₁ into l₂'s equation. So, substituting x = 1 + 3t and y = 2 - 4t into 2x - 4y = 5:2(1 + 3t) - 4(2 - 4t) = 5Let me compute that step by step. First, multiply out the terms:2*1 + 2*3t - 4*2 + 4*4t = 52 + 6t - 8 + 16t = 5Now combine like terms:(6t + 16t) + (2 - 8) = 522t - 6 = 5Then add 6 to both sides:22t = 11Divide both sides by 22:t = 11/22 = 1/2Okay, so t is 1/2. Now, substitute t = 1/2 into the parametric equations for l₁ to find the coordinates of point B.For x:x = 1 + 3*(1/2) = 1 + 3/2 = 5/2 or 2.5For y:y = 2 - 4*(1/2) = 2 - 2 = 0So, the coordinates of point B are (5/2, 0). Let me verify that this point satisfies the equation of l₂.Plugging into 2x - 4y:2*(5/2) - 4*0 = 5 - 0 = 5Which matches the right-hand side of l₂'s equation, so that checks out. Good.Now, I need to find the distance between point A(1,2) and point B(5/2, 0). The distance formula is:distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]Plugging in the coordinates:distance = sqrt[(5/2 - 1)^2 + (0 - 2)^2]First compute the differences:For x: 5/2 - 1 = 5/2 - 2/2 = 3/2For y: 0 - 2 = -2Then square them:(3/2)^2 = 9/4(-2)^2 = 4Add them up:9/4 + 4 = 9/4 + 16/4 = 25/4Take the square root:sqrt(25/4) = 5/2So, the distance is 5/2 units.Wait a second, let me check again. The x-coordinate of B is 5/2, which is 2.5, and A's x-coordinate is 1. The difference is 1.5, which is 3/2. Squared is 9/4. Then y-coordinate difference is 0 - 2 = -2, squared is 4. Sum is 25/4, square root is 5/2. That's correct.So the intersection point B is at (5/2, 0) and the distance from A to B is 5/2. Let me just recap to make sure I didn't miss anything.1. Found t by substituting parametric equations into l₂.2. Solved for t, got 1/2.3. Plugged t back into l₁ to get B.4. Verified B lies on l₂.5. Calculated distance between A and B using distance formula.6. All steps seem to check out.I think that's all. Seems straightforward. Let me just write the final answers.Final AnswerThe coordinates of point ( B ) are (boxed{left( dfrac{5}{2}, 0 right)}) and the distance from ( A ) to ( B ) is (boxed{dfrac{5}{2}}).</think>To find the coordinates of the intersection point ( B ) of line ( l_1 ) and line ( l_2 ), we start by substituting the parametric equations of ( l_1 ) into the equation of ( l_2 ).The parametric equations for ( l_1 ) are:[begin{cases}x = 1 + 3t y = 2 - 4tend{cases}]Substituting ( x ) and ( y ) into the equation of ( l_2 ) ( 2x - 4y = 5 ):[2(1 + 3t) - 4(2 - 4t) = 5]Expanding and simplifying:[2 + 6t - 8 + 16t = 5 22t - 6 = 5 22t = 11 t = frac{1}{2}]Substituting ( t = frac{1}{2} ) back into the parametric equations of ( l_1 ):[x = 1 + 3 left( frac{1}{2} right) = 1 + frac{3}{2} = frac{5}{2}][y = 2 - 4 left( frac{1}{2} right) = 2 - 2 = 0]Thus, the coordinates of point ( B ) are ( left( frac{5}{2}, 0 right) ).Next, we calculate the distance between point ( A(1, 2) ) and point ( B left( frac{5}{2}, 0 right) ) using the distance formula:[text{distance} = sqrt{left( frac{5}{2} - 1 right)^2 + (0 - 2)^2}]Simplifying the differences:[left( frac{5}{2} - 1 right) = frac{3}{2} quad text{and} quad (0 - 2) = -2]Squaring and summing:[left( frac{3}{2} right)^2 + (-2)^2 = frac{9}{4} + 4 = frac{9}{4} + frac{16}{4} = frac{25}{4}]Taking the square root:[sqrt{frac{25}{4}} = frac{5}{2}] Final AnswerThe coordinates of point ( B ) are (boxed{left( dfrac{5}{2}, 0 right)}) and the distance from ( A ) to ( B ) is (boxed{dfrac{5}{2}}).]

🔑:The standard parametric equations of line l_{1} are begin{cases} x=1-frac{3}{5}t y=2+frac{4}{5}t end{cases}. Substitute them into 2x-4y=5, we get 2-frac{6}{5}t-4(2+frac{4}{5}t)=5. Solving for t, we obtain t=-frac{5}{2}.Therefore, the distance between points A and B is |AB|=|t|=boxed{frac{5}{2}}.

🧠:[<think>Okay, so I need to find the equation of the tangent line to the function f(x) = x ln x at the point (e, f(e)). Hmm, let me recall how to do this. From what I remember, the equation of a tangent line at a specific point involves two main things: the slope of the tangent line at that point and a point through which the tangent line passes. Since they already gave me the point (e, f(e)), I just need to find the slope of the tangent line at x = e and then use the point-slope form to write the equation.First, let me make sure I know how to find the slope. The slope of the tangent line to a function at a given point is just the derivative of the function evaluated at that point. So I need to compute f'(x) and then plug in x = e.The function here is f(x) = x ln x. To find f'(x), I need to use differentiation rules. Since this is a product of two functions, x and ln x, I should use the product rule. The product rule states that if you have a function h(x) = u(x)v(x), then the derivative h'(x) = u'(x)v(x) + u(x)v'(x). Let me apply that here. Let u(x) = x and v(x) = ln x. Then u'(x) = 1, since the derivative of x is 1. And v'(x) = 1/x, since the derivative of ln x is 1/x. Putting that into the product rule formula:f'(x) = u'(x)v(x) + u(x)v'(x) = (1)(ln x) + (x)(1/x)Simplify that. The first term is ln x, and the second term is x*(1/x) which simplifies to 1. So f'(x) = ln x + 1.Okay, so the derivative is ln x + 1. Therefore, the slope of the tangent line at x = e is f'(e) = ln e + 1. Since ln e = 1 (because e^1 = e), this simplifies to 1 + 1 = 2. So the slope, m, is 2.Alright, now that I have the slope, I need to write the equation of the tangent line. The standard way is using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the point of tangency. Here, the point is (e, f(e)). Let me compute f(e) first.f(e) = e * ln e. Again, ln e = 1, so f(e) = e * 1 = e. Therefore, the point is (e, e).Plugging into the point-slope form:y - e = 2(x - e)Now, I can write this in slope-intercept form if needed, but the problem doesn't specify. Let me check if they just want the equation in some standard form. Generally, either form is acceptable unless specified otherwise, but since the question says "the equation of the tangent line," it's probably okay to leave it in point-slope form. However, maybe they want it simplified. Let me do both steps.Starting with y - e = 2(x - e)Distribute the 2 on the right side: y - e = 2x - 2eThen, add e to both sides: y = 2x - 2e + eSimplify: y = 2x - eSo the equation is y = 2x - e.Let me verify this. To make sure, when x = e, y should equal e. Plugging in x = e into the tangent line equation: y = 2e - e = e. That checks out. Also, the derivative at x = e is 2, so the slope is correct. Seems good.Wait a second, let me double-check the derivative. f'(x) = ln x + 1. At x = e, ln e is indeed 1, so f'(e) = 1 + 1 = 2. Correct.And f(e) = e * ln e = e*1 = e. Correct. So the point (e, e) is on both the function and the tangent line. That's consistent.Therefore, the equation of the tangent line is y = 2x - e.Alternatively, if they prefer it in a different form, but usually this is acceptable. Let me think if there's another way to present it, but I think y = 2x - e is the simplest form.Wait, just to make sure, maybe I should present it as y = 2x - e. Yes, that's correct. Because when you simplify the point-slope form, that's exactly what you get. So yes, that's the equation.Final AnswerThe equation of the tangent line is boxed{y = 2x - e}.</think>To find the equation of the tangent line to the function ( f(x) = x ln x ) at the point ( (e, f(e)) ):1. Compute the derivative ( f'(x) ) using the product rule: [ f'(x) = frac{d}{dx}[x] cdot ln x + x cdot frac{d}{dx}[ln x] = 1 cdot ln x + x cdot frac{1}{x} = ln x + 1 ]2. Evaluate the derivative at ( x = e ) to find the slope of the tangent line: [ f'(e) = ln e + 1 = 1 + 1 = 2 ]3. Determine the coordinates of the point ( (e, f(e)) ): [ f(e) = e cdot ln e = e cdot 1 = e ] Thus, the point is ( (e, e) ).4. Use the point-slope form of the equation of a line: [ y - e = 2(x - e) ]5. Simplify to slope-intercept form: [ y - e = 2x - 2e implies y = 2x - e ]The equation of the tangent line is (boxed{y = 2x - e}).,<think>Okay, so I need to find the equation of the tangent line to the function f(x) = x ln x at the point (e, f(e)). Hmm, tangent lines... right. From what I remember, the equation of a tangent line at a specific point is given by the formula y = f(a) + f’(a)(x - a), where a is the x-coordinate of the point. In this case, the point is (e, f(e)), so a is e. That means I need to find f(e) and the derivative f’(e) to plug into the formula. Let me start by calculating f(e). Since f(x) = x ln x, substituting x = e gives f(e) = e * ln e. But ln e is 1 because the natural logarithm of e is 1. So, f(e) = e * 1 = e. That means the point (e, f(e)) is actually (e, e). Okay, that part was straightforward.Now, the next step is to find the derivative f’(x) so I can compute f’(e). The function is f(x) = x ln x. To find the derivative, I need to use the product rule because it's the product of two functions: u(x) = x and v(x) = ln x. The product rule states that (uv)’ = u’v + uv’. Let me compute u’(x) first. Since u(x) = x, the derivative u’(x) is 1. Then, v(x) = ln x, so the derivative v’(x) is 1/x. Applying the product rule: f’(x) = u’(x)v(x) + u(x)v’(x) = 1 * ln x + x * (1/x). Simplify that: ln x + (x/x) = ln x + 1. So, f’(x) = ln x + 1.Now, evaluate this derivative at x = e. So, f’(e) = ln e + 1. Again, ln e is 1, so f’(e) = 1 + 1 = 2. Therefore, the slope of the tangent line at the point (e, e) is 2.Putting this all into the tangent line equation formula: y = f(e) + f’(e)(x - e). Substituting the known values, that becomes y = e + 2(x - e). Let me simplify that. Distribute the 2: y = e + 2x - 2e. Then combine like terms: y = 2x - e. Wait, let me check that again. Starting with y = e + 2(x - e). Distribute the 2: 2x - 2e. Then add the e: 2x - 2e + e = 2x - e. Yep, that's correct. So the equation of the tangent line is y = 2x - e. Hmm, just to make sure I didn't make any mistakes. Let's verify each step. First, f(e) = e * ln e. Since ln e = 1, yes, f(e) = e. Correct. Then, the derivative. Product rule: derivative of x is 1, derivative of ln x is 1/x. So 1 * ln x + x * 1/x. Simplifies to ln x + 1. Correct. Then plugging in x = e: ln e + 1 = 1 + 1 = 2. That's right. Then the equation: y = e + 2(x - e). Expanding gives 2x - 2e + e = 2x - e. Yes, that's correct. Alternatively, maybe I can check using another method. Let me pick a point close to x = e and see if the slope is approximately 2. Let's take x = e + 0.001. Compute the slope between (e, e) and (e + 0.001, f(e + 0.001)). First, compute f(e + 0.001) = (e + 0.001) * ln(e + 0.001). Let me approximate ln(e + 0.001). Since ln(e + h) ≈ ln e + (h)/e when h is small. Here, h = 0.001. So ln(e + 0.001) ≈ 1 + 0.001/e. Therefore, f(e + 0.001) ≈ (e + 0.001)(1 + 0.001/e) = e*(1 + 0.001/e) + 0.001*(1 + 0.001/e). Distribute: e + 0.001 + 0.001 + 0.000001/e. Combine terms: e + 0.002 + something negligible. So approximately e + 0.002. Then the slope between (e, e) and (e + 0.001, e + 0.002) is ( (e + 0.002 - e ) / (0.001) ) = 0.002 / 0.001 = 2. Which matches the derivative we found. So that seems to check out. Alternatively, if I compute the derivative directly, it's 2, so the slope is 2. Therefore, the tangent line equation is y = 2x - e. Another way to check: let's see if the point (e, e) satisfies the equation y = 2x - e. Plugging x = e into the tangent line equation: y = 2e - e = e. Which matches, so the point (e, e) is indeed on the line. That's a good consistency check. I think that's thorough enough. So, the equation is y = 2x - e.Final AnswerThe equation of the tangent line is boxed{y = 2x - e}.</think>To find the equation of the tangent line to the function ( f(x) = x ln x ) at the point ( (e, f(e)) ):1. Calculate ( f(e) ): [ f(e) = e ln e = e cdot 1 = e ] So, the point is ( (e, e) ).2. Find the derivative ( f'(x) ) using the product rule. Let ( u(x) = x ) and ( v(x) = ln x ): [ f'(x) = u'(x)v(x) + u(x)v'(x) = 1 cdot ln x + x cdot frac{1}{x} = ln x + 1 ]3. Evaluate the derivative at ( x = e ): [ f'(e) = ln e + 1 = 1 + 1 = 2 ]4. Use the point-slope form of the tangent line equation ( y = f(e) + f'(e)(x - e) ): [ y = e + 2(x - e) ] Simplifying this: [ y = 2x - 2e + e = 2x - e ]Thus, the equation of the tangent line is (boxed{y = 2x - e}).]

🔑:Solution: Given f(x)=xln x, we find f'(x)=ln x+1, thus f'(e)=ln e+1=2, also f(e)=e, therefore the equation of the tangent line to the function f(x)=xln x at the point (e,f(e)) is y-e=2(x-e), which simplifies to 2x-y-e=0. Therefore, the answer is: 2x-y-e=0. By finding the derivative of the original function, we obtain the slope of the function at x=e, and then we derive the answer using the point-slope form of the line equation. This question examines the use of derivatives to study the equation of the tangent line at a certain point on the curve. The slope of the tangent line passing through a point on the curve is the derivative of the function at that point, which is a fundamental concept.boxed{2x-y-e=0}

🧠:[<think>Okay, so I need to find the remainder when a_83 is divided by 49. Given that a_n = 6^n + 8^n. Hmm, let's see. The problem is asking for a_83 mod 49. That is, compute (6^83 + 8^83) mod 49. First, I remember that when dealing with modular arithmetic, especially with exponents, using Euler's theorem or Fermat's little theorem can be helpful. But Fermat's little theorem applies when the modulus is prime, right? 49 is 7 squared, so it's not prime. Maybe Euler's theorem then? Euler's theorem says that if a and n are coprime, then a^φ(n) ≡ 1 mod n, where φ is Euler's totient function. Let's compute φ(49). Since 49 is 7^2, φ(7^2) = 7^2 - 7^1 = 49 - 7 = 42. So φ(49) is 42. Therefore, if 6 and 49 are coprime, which they are because 6 and 7 are coprime, then 6^42 ≡ 1 mod 49. Similarly, 8 and 49 are coprime (since 8 and 7 are coprime), so 8^42 ≡ 1 mod 49. So, if that's the case, then 6^42 ≡ 1 mod 49 and 8^42 ≡ 1 mod 49. Therefore, exponents can be reduced modulo 42. So, 6^83 is equivalent to 6^(42*2 + (-1))? Wait, 42*2 = 84, so 83 is 84 - 1, which is 42*2 -1. So, 6^83 ≡ 6^(42*2 -1) ≡ (6^42)^2 * 6^(-1) ≡ 1^2 * 6^(-1) ≡ 6^(-1) mod 49. Similarly, 8^83 ≡ 8^(-1) mod 49. Wait, so then 6^83 ≡ 6^{-1} mod 49 and 8^83 ≡ 8^{-1} mod 49. So, the total a_83 ≡ 6^{-1} + 8^{-1} mod 49. Therefore, I need to find the inverses of 6 and 8 modulo 49 and then add them together. So, first, find 6^{-1} mod 49. That is, find some integer x such that 6x ≡ 1 mod 49. Similarly, find y such that 8y ≡ 1 mod 49. Let me compute these inverses. For 6x ≡ 1 mod 49: Let's use the extended Euclidean algorithm. Find x such that 6x - 49k = 1 for some integer k. Let's perform the algorithm:49 divided by 6 is 8 with a remainder of 1. So, 49 = 6*8 + 1. Then, 1 = 49 - 6*8. Therefore, x is -8 mod 49. Since -8 mod 49 is 41, so 6*41 = 246. Let's check: 246 divided by 49 is 5*49=245, so 246-245=1. So yes, 6*41 ≡ 1 mod 49. So 6^{-1} ≡ 41 mod 49.Similarly for 8y ≡1 mod 49. Let's do the same. 49 divided by 8 is 6 with a remainder of 1, since 8*6=48, so 49 -48=1. So 1=49 -8*6. Therefore, y= -6 mod 49. Which is 43. Check: 8*43=344. 344 divided by 49 is 7*49=343, so 344-343=1. Therefore, 8*43 ≡1 mod 49. So 8^{-1} ≡43 mod49.Therefore, a_83 ≡41 +43 mod49. 41+43=84. 84 mod49: 49*1=49, 84-49=35. So 84≡35 mod49. Therefore, the remainder is 35. Wait, that seems straightforward, but let me verify. Let me check if my steps are correct. First, using Euler's theorem for modulus 49, φ(49)=42. So exponents cycle every 42. So 83 divided by 42 is 1 with remainder 41. Wait, hold on, 42*2=84, which is greater than 83, so 83=42*1 +41. So actually, 6^83 ≡6^{42 +41}≡6^{42}*6^{41}≡1*6^{41}≡6^{41} mod49. Similarly, 8^83≡8^{41} mod49. Hmm, so maybe my initial reasoning was wrong?Wait, I thought that 6^42 ≡1 mod49, so 6^{42k +r}≡1^k*6^r mod49. So 83 divided by 42 is 1*42=42, 83-42=41. So 83=42*1 +41. Therefore, 6^{83}=6^{42*1 +41}= (6^{42})^1 *6^{41}≡1*6^{41}≡6^{41} mod49. Similarly, 8^{83}≡8^{41} mod49. So my earlier reasoning was incorrect. I thought that 83=42*2 -1, but that's not correct because 42*2=84 which is larger than 83. So 83=42*1 +41. Therefore, the exponents reduce to 41 instead of -1. So my previous answer is wrong.So that was a mistake. So, how to resolve this?Therefore, the correct exponents are 6^{41} and 8^{41} mod49. Hmm. So maybe I need to compute 6^{41} +8^{41} mod49. Hmm, that's more involved. How to compute 6^{41} mod49 and 8^{41} mod49.Alternatively, perhaps notice that 6 and 7 are coprime, 8 and 7 are coprime. Maybe find cycles for 6 and 8 modulo 49. Let me see.Alternatively, perhaps notice that 6^42 ≡1 mod49, so 6^{41}=6^{-1} mod49, as 6^{42}=6^{41}*6≡1 mod49, so 6^{41}≡6^{-1} mod49. Similarly, 8^{42}≡1 mod49, so 8^{41}=8^{-1} mod49. Therefore, in fact, even though 83=42*1 +41, 6^{41} is 6^{-1} mod49, and 8^{41}=8^{-1} mod49. Therefore, the original reasoning holds. Therefore, 6^{83}≡6^{-1} mod49, and 8^{83}≡8^{-1} mod49. So adding them gives 6^{-1} +8^{-1}≡41 +43≡84≡35 mod49. Therefore, remainder is 35. But wait, why? Because 6^{42}≡1 mod49, so 6^{42 -1}=6^{41}≡6^{-1} mod49. Similarly for 8. So even though 41 is the remainder when 83 is divided by 42, 6^{41} can be expressed as 6^{-1} mod49. Since 41=42-1, so 6^{41}=6^{42 -1}=6^{42}*6^{-1}≡1*6^{-1}≡6^{-1}. Therefore, same for 8^{41}=8^{-1}. So the initial conclusion was correct. Therefore, the answer is 35. Wait, but let me verify with a smaller exponent. For example, take n=1. a1=6+8=14. 14 mod49=14. But 6^{-1}+8^{-1}=41+43=84≡35 mod49. But 14≠35. So in that case, for n=1, the formula doesn't hold. So that suggests that my reasoning is wrong.Wait, there's a problem here. For n=1, a1=14, which is 14 mod49. But according to the formula, if we compute 6^{-1} +8^{-1}=41+43=84≡35 mod49. Which is not equal to 14. So clearly, my previous reasoning is flawed. So, why is this discrepancy happening? Let me check again. The key is that Euler's theorem tells us that a^{φ(n)}≡1 mod n when a and n are coprime. So, since φ(49)=42, 6^{42}≡1 mod49 and 8^{42}≡1 mod49. Therefore, 6^{k} mod49 cycles every 42. So 6^{83}=6^{42*1 +41}= (6^{42})^1 *6^{41}≡1*6^{41}≡6^{41} mod49. But 6^{41}=6^{-1} because 6^{42}≡1, so 6^{41}≡6^{-1} mod49. Similarly, 8^{41}≡8^{-1} mod49. Therefore, 6^{83} +8^{83}≡6^{-1} +8^{-1} mod49. However, for n=1, 6^1 +8^1=14≡14 mod49, but 6^{-1} +8^{-1}≡35 mod49. So they are different. Therefore, this approach only works when n≡-1 mod42, which 83 is 83=42*2 -1=84-1. Wait, but 42*1 +41=83. So, 83=42*1 +41. So, 41 is the remainder. So, 6^{83}≡6^{41}≡6^{-1}. But why is this possible?Wait, perhaps the confusion is because 42 is the order for 6 and 8 modulo49? But actually, Euler's theorem gives that the multiplicative order divides φ(n). So, the multiplicative order of 6 modulo49 could be a divisor of 42. So maybe it's less than 42. Similarly for 8. So, if the order is less than 42, then 6^{order}≡1 mod49, but Euler's theorem just gives an upper bound. Therefore, perhaps my initial assumption that 6^{42}≡1 is correct, but maybe 6 has a lower order. Let's check. For example, compute the order of 6 modulo49. The order must divide φ(49)=42. So possible orders are divisors of 42: 1,2,3,6,7,14,21,42. Let's check the order of 6. Compute 6^k mod49 until it equals 1. 6^1=6 mod496^2=36 mod496^3=6*36=216 mod49. 216 divided by49: 4*49=196. 216-196=20. So 6^3≡20 mod496^6=(6^3)^2=20^2=400 mod49. 400 divided by49: 8*49=392. 400-392=8. So 6^6≡8 mod496^7=6^6*6=8*6=48≡-1 mod496^14=(6^7)^2=(-1)^2=1 mod49Ah! So 6^14≡1 mod49. Therefore, the order of 6 modulo49 is 14. Similarly, let's check for 8.Compute the order of 8 modulo49:8^1=8 mod498^2=64≡15 mod498^3=8*15=120≡120-2*49=120-98=22 mod498^6=(8^3)^2=22^2=484. 484-9*49=484-441=43 mod498^7=8^6*8=43*8=344≡344-7*49=344-343=1 mod49So 8^7≡1 mod49. Therefore, the order of 8 modulo49 is 7. Therefore, my previous assumption that the exponent cycles every 42 was wrong. The actual orders are 14 for 6 and 7 for 8. Therefore, to compute 6^83 and 8^83 mod49, we can use the orders. So for 6^83: since the order is 14, 83 divided by14 gives 83=14*5 +13. So 6^83=6^{14*5 +13}= (6^14)^5 *6^{13}≡1^5 *6^{13}≡6^{13} mod49. Similarly, 6^13. Let's compute 6^13. Since we know 6^7≡-1 mod49, then 6^13=6^{7+6}=6^7*6^6≡(-1)*8≡-8 mod49. -8 mod49 is 41. So 6^83≡41 mod49.Similarly, for 8^83: since the order is 7, 83 divided by7 is 83=7*11 +6. So 8^83= (8^7)^11 *8^6≡1^11 *8^6≡8^6 mod49. From earlier computation, 8^6≡43 mod49. So 8^83≡43 mod49.Therefore, a_83=6^83 +8^83≡41 +43=84 mod49. 84-49=35. So 84≡35 mod49. Therefore, the remainder is35. But wait, earlier when I tried n=1, the answer didn't match. But here, using orders, for n=1, 6^1 +8^1=6 +8=14≡14 mod49. If we follow the same method: for n=1, 1 divided by14 (order of6) is0 with remainder1. So 6^1≡6 mod49. For 8^1, 1 divided by7 (order of8) is0 with remainder1. 8^1≡8 mod49. So 6+8=14. So correct. So, in that case, the approach works. The initial mistake was not considering the actual orders but just using Euler's theorem. So the key was to find the orders of 6 and8 modulo49 and then reduce the exponents accordingly. Therefore, the correct answer is35. Let me confirm this with another approach. Let's compute 6^83 mod49 and8^83 mod49 using their orders. As found, order of6 is14. So 83 divided by14:14*5=70, 83-70=13. So exponent13. So compute 6^13. Since 6^7≡-1 mod49, so 6^13=6^(7+6)=6^7*6^6≡(-1)*8≡-8≡41 mod49. Similarly, order of8 is7. 83 divided by7 is11*7=77, remainder6. So exponent6. 8^6≡43 mod49 as before. Therefore, total is41+43=84≡35 mod49. Alternatively, let's compute 6^83 mod49 step by step, using exponentiation by squaring. Let's see:First, compute 6^2=36 mod496^4=(6^2)^2=36^2=1296. 1296 divided by49: 26*49=1274, 1296-1274=22. So 6^4≡22 mod496^8=(6^4)^2=22^2=484. 484-9*49=484-441=43. So 6^8≡43 mod496^16=(6^8)^2=43^2=1849. 1849 divided by49: 37*49=1813, 1849-1813=36. So 6^16≡36 mod496^32=(6^16)^2=36^2=1296≡22 mod49 (as before)6^64=(6^32)^2=22^2=484≡43 mod49Now, 83 in binary is 64 +16 +2 +1. So 83=64+16+2+1. Therefore, 6^83=6^64 *6^16 *6^2 *6^1. So compute 6^64≡43, 6^16≡36, 6^2≡36, 6^1≡6. Multiply them all together:43*36=1548. 1548 divided by49: 31*49=1519, 1548-1519=29. So 43*36≡29 mod49.29*36=1044. 1044 divided by49: 21*49=1029, 1044-1029=15. So 29*36≡15 mod49.15*6=90. 90-1*49=41. So 6^83≡41 mod49. Which matches previous result. Similarly, compute 8^83 mod49. 8^2=64≡15 mod498^4=(8^2)^2=15^2=225. 225-4*49=225-196=29 mod498^8=(8^4)^2=29^2=841. 841-17*49=841-833=8 mod498^16=(8^8)^2=8^2=15 mod498^32=(8^16)^2=15^2=225≡29 mod498^64=(8^32)^2=29^2=841≡8 mod49Now, 83 in binary is64 +16 +2 +1. So 8^83=8^64 *8^16 *8^2 *8^1. 8^64≡8, 8^16≡15, 8^2≡15, 8^1≡8. Multiply them:8*15=120. 120-2*49=120-98=22 mod4922*15=330. 330-6*49=330-294=36 mod4936*8=288. 288-5*49=288-245=43 mod49. So 8^83≡43 mod49. Adding 41 +43=84≡35 mod49. Therefore, confirms the answer is35. But to ensure I didn't make any miscalculations, let's check with another method. Let's use the fact that 49=7^2, and use the lifting the exponent lemma (LTE). Although LTE is typically for prime exponents, maybe here since 7 is prime, we can use it. But LTE applies to specific conditions. However, perhaps for 6 and8, which are both ≡6 mod7 and1 mod7, respectively. Wait, 6≡6 mod7, 8≡1 mod7. Let me see:But LTE states that for odd primes p, if p divides (a + b) and p divides a but not b, then v_p(a^n + b^n) = v_p(a + b) + v_p(n). But in this case, p=7. Let's check 6 +8=14≡0 mod7. So 7 divides 6 +8. Also, 7 doesn't divide6 or8. Wait, 7 divides6? No, 6≡6 mod7, 8≡1 mod7. So 7 doesn't divide6 or8. Therefore, according to LTE, v_7(6^n +8^n)=v_7(6 +8)+v_7(n)=v_7(14)+v_7(n)=1 +v_7(n). But 83 is not divisible by7, so v_7(83)=0. Therefore, v_7(a_83)=1 +0=1. Therefore, 7 divides a_83 exactly once. Therefore, a_83≡0 mod7, but a_83≡k mod49 where k≡0 mod7 but k≡? So, since a_83 is divisible by7 but not by49, so when divided by49, the remainder is a multiple of7 but not multiple of49, i.e.,7,14,...,42. But how does this help? Alternatively, since a_83 ≡0 mod7, but we need to find it mod49. Let's write a_83=7m, and find m mod7. Then, a_83 mod49=7*(m mod7). To compute m mod7, note that a_83=7m, so m=(a_83)/7. So, compute (6^83 +8^83)/7 mod7. Compute (6^83 +8^83)/7 mod7. Let's compute numerator first modulo49, but since a_83=7m, we can write:6^83 +8^83 ≡0 mod7, so let's compute (6^83 +8^83)/7 mod7. First, note that 6≡6 mod7, 8≡1 mod7. Therefore, 6^83≡6^{83 mod6} mod7 (by Fermat's little theorem, since φ(7)=6). 83 mod6=5, so 6^5≡6^(5) mod7. 6^1=6, 6^2=36≡1, 6^3=6, 6^4=1, 6^5=6 mod7. So 6^83≡6 mod7. Similarly,8≡1 mod7, so 8^83≡1^83≡1 mod7. Therefore, a_83=6 +1=7≡0 mod7. Then, (a_83)/7 ≡ (0)/7≡0 mod7? Wait, no. To compute m=(6^83 +8^83)/7 mod7, need to use the expansion. Alternatively, perhaps use the binomial theorem. Since 6=7-1, so 6= -1 mod7. But 6≡-1 mod7, so 6^83≡(-1)^83≡-1 mod7. Similarly,8≡1 mod7, so 8^83≡1 mod7. Then, a_83≡-1 +1=0 mod7. But how to lift this to mod49. Use the lifting the exponent lemma as above. Wait, LTE says that v_7(a_83)=1, so a_83=7*m where m is an integer not divisible by7. Therefore, a_83≡7*m mod49, and m≡(a_83)/7 mod7. To compute m mod7, note that:a_83 =6^{83} +8^{83}= (7 -1)^{83} + (7 +1)^{83}Expand both terms using binomial theorem:(7 -1)^{83} = Σ_{k=0}^{83} C(83,k)*7^{k}*(-1)^{83 -k}Similarly, (7 +1)^{83} = Σ_{k=0}^{83} C(83,k)*7^{k}*1^{83 -k}Adding these two:Σ_{k=0}^{83} C(83,k)*7^{k}*(-1)^{83 -k} + Σ_{k=0}^{83} C(83,k)*7^{k}*1^{83 -k}= Σ_{k=0}^{83} C(83,k)*7^{k}[ (-1)^{83 -k} +1^{83 -k} ]Note that when 83 -k is even, (-1)^{83 -k}=1, so [1 +1]=2. When 83 -k is odd, (-1)^{83 -k}= -1, so [-1 +1]=0. Therefore, only terms where83 -k is even contribute, i.e.,k is odd (since83 is odd, so if83 -k is even, thenk is odd). Therefore, the sum becomes Σ_{k odd} C(83,k)*7^{k}*2Therefore, a_83=2*Σ_{k odd} C(83,k)*7^{k}Therefore, divide by7:m=(a_83)/7=2*Σ_{k odd} C(83,k)*7^{k -1}Now, compute m mod7. Since k ≥1 (because k is odd and starts from k=1). Therefore, for each term in the sum, when k=1: C(83,1)*7^{0}=83*1=83 mod7. For k≥3, terms have7^{2} or higher, which are≡0 mod7. Therefore, m≡2*C(83,1)*7^{0}≡2*83≡2*(83 mod7) mod7. 83 divided by7 is11*7=77, remainder6. So83≡6 mod7. Therefore, m≡2*6≡12≡5 mod7. Therefore, m≡5 mod7. Therefore, a_83=7m≡7*5=35 mod49. Therefore, the remainder is35. This method also gives35. So this confirms the previous result. Therefore, the remainder whena_83 is divided by49 is35. Final AnswerThe remainder when (a_{83}) is divided by 49 is (boxed{35}).</think>To determine the remainder when (a_{83} = 6^{83} + 8^{83}) is divided by 49, we use modular arithmetic and properties of exponents.First, we note that Euler's theorem tells us (phi(49) = 42), so (6^{42} equiv 1 pmod{49}) and (8^{42} equiv 1 pmod{49}). However, the actual orders of 6 and 8 modulo 49 are 14 and 7, respectively.For (6^{83}):- The order of 6 modulo 49 is 14. We reduce 83 modulo 14: (83 = 14 cdot 5 + 13).- Thus, (6^{83} equiv 6^{13} pmod{49}).- Using the order, we find (6^7 equiv -1 pmod{49}), so (6^{13} equiv 6^{7+6} equiv (-1) cdot 6^6 pmod{49}).- Calculate (6^6 equiv 8 pmod{49}), thus (6^{13} equiv -8 equiv 41 pmod{49}).For (8^{83}):- The order of 8 modulo 49 is 7. We reduce 83 modulo 7: (83 = 7 cdot 11 + 6).- Thus, (8^{83} equiv 8^6 pmod{49}).- Calculate (8^6 equiv 43 pmod{49}).Adding the results:- (6^{83} + 8^{83} equiv 41 + 43 pmod{49}).- (41 + 43 = 84), and (84 equiv 35 pmod{49}).Thus, the remainder when (a_{83}) is divided by 49 is (boxed{35}).,<think>Alright, so I need to find the remainder when a_83 is divided by 49, where a_n is defined as 6^n + 8^n. Hmm, okay. Let me think about how to approach this. First, the problem is about modular arithmetic, specifically modulo 49. I remember that for such problems, Euler's theorem or Fermat's little theorem might be useful. But wait, Fermat's little theorem applies when the modulus is a prime number, right? 49 is 7 squared, so it's not prime. But Euler's theorem generalizes to any modulus as long as the base and modulus are coprime. So, Euler's theorem says that if a and n are coprime, then a^φ(n) ≡ 1 mod n, where φ(n) is Euler's totient function. For 49, which is 7^2, φ(49) = 49*(1 - 1/7) = 49*(6/7) = 42. So φ(49) = 42. That means that for any number a that is coprime with 49, a^42 ≡ 1 mod 49. Now, let's check if 6 and 8 are coprime with 49. Since 49 is 7^2, any number not divisible by 7 is coprime with 49. 6 and 8 are both not divisible by 7, so they are coprime with 49. Therefore, Euler's theorem applies. So, 6^42 ≡ 1 mod 49 and 8^42 ≡ 1 mod 49. Therefore, both 6^n and 8^n will have cycles of 42 modulo 49. So, their exponents can be reduced modulo 42. That is, 6^83 ≡ 6^(83 mod 42) mod 49, and similarly for 8^83. Let's compute 83 mod 42. 42*1 = 42, 42*2 = 84. So 83 - 42*1 = 41, so 83 mod 42 is 41. Therefore, 6^83 ≡ 6^41 mod 49 and 8^83 ≡ 8^41 mod 49. Therefore, a_83 = 6^83 + 8^83 ≡ 6^41 + 8^41 mod 49. Hmm, but computing 6^41 and 8^41 modulo 49 directly seems complicated. Maybe there's a shorter cycle? Sometimes, the order of an element divides φ(n), but could be smaller. Let me check if there's a smaller exponent for which 6^k ≡ 1 mod 49 or 8^k ≡ 1 mod 49. Let me first check for 6. Let's compute powers of 6 modulo 49:6^1 = 6 mod 496^2 = 36 mod 496^3 = 6*36 = 216 mod 49. Let's divide 216 by 49: 49*4=196, 216-196=20. So 6^3 ≡ 20 mod 49.6^4 = 6*20 = 120 mod 49. 49*2=98, 120-98=22. So 6^4 ≡ 22 mod 49.6^5 = 6*22 = 132 mod 49. 49*2=98, 132-98=34. 34 mod 49 is 34. So 6^5 ≡ 34 mod 49.6^6 = 6*34 = 204 mod 49. 49*4=196, 204-196=8. So 6^6 ≡ 8 mod 49.6^7 = 6*8 = 48 mod 49.6^8 = 6*48 = 288 mod 49. 49*5=245, 288-245=43. So 6^8 ≡ 43 mod 49.6^9 = 6*43 = 258 mod 49. 49*5=245, 258-245=13. 6^9 ≡13 mod49.6^10=6*13=78 mod49. 78-49=29. So 29.6^11=6*29=174 mod49. 49*3=147, 174-147=27. 27.6^12=6*27=162 mod49. 49*3=147, 162-147=15. 15.6^13=6*15=90 mod49. 90-49=41. 41.6^14=6*41=246 mod49. 49*5=245, 246-245=1. So 6^14 ≡1 mod49.Oh! So the order of 6 modulo 49 is 14. That means 6^14 ≡1 mod49. Therefore, the cycle length is 14. So exponents of 6 can be reduced modulo 14. Similarly, let's check for 8.Let me compute powers of 8 modulo 49:8^1=8 mod49.8^2=64-49=15 mod49.8^3=8*15=120 mod49. 120-2*49=120-98=22. 22.8^4=8*22=176 mod49. 49*3=147, 176-147=29. 29.8^5=8*29=232 mod49. 49*4=196, 232-196=36. 36.8^6=8*36=288 mod49. 288-5*49=288-245=43. 43.8^7=8*43=344 mod49. 49*7=343, 344-343=1. So 8^7 ≡1 mod49.Wow, so the order of 8 modulo 49 is 7. Therefore, 8^7 ≡1 mod49, so exponents can be reduced modulo7.So, now, since 6 has order 14 and 8 has order 7 modulo49. Therefore, to compute 6^41 mod49 and 8^41 mod49, we can reduce the exponents modulo14 and 7 respectively.For 6^41:41 divided by14: 14*2=28, 41-28=13. So 41≡13 mod14. Therefore, 6^41 ≡6^13 mod49.Similarly, for 8^41:41 divided by7: 7*5=35, 41-35=6. So 41≡6 mod7. Therefore, 8^41≡8^6 mod49.So now, we need to compute 6^13 mod49 and 8^6 mod49.Earlier, when I was computing the powers of 6, I had already calculated up to 6^14. Let me check back:6^1=66^2=366^3=206^4=226^5=346^6=86^7=486^8=436^9=136^10=296^11=276^12=156^13=416^14=1Yes, so 6^13 is 41 mod49.For 8^6, when I calculated before, 8^6 was 43 mod49.So, 6^13=41 mod49 and 8^6=43 mod49.Therefore, a_83=6^83 +8^83≡6^13 +8^6≡41 +43=84 mod49.Now, 84 divided by49 is 1*49=49, 84-49=35. So 84≡35 mod49. Therefore, the remainder is 35.Wait, is that correct? Let me verify.Wait, 41 +43=84. 84-49=35. So 84≡35 mod49. So the remainder is 35. Hmm. Let me check if I made a mistake in the calculations.First, confirming the orders:For 6 modulo49:We saw that 6^14≡1 mod49, and the previous powers didn't reach 1, so the order is indeed 14.For 8 modulo49:8^7≡1 mod49, and prior exponents didn't reach 1, so order is 7.Then, exponents 83:83 divided by14 is 5*14=70, 83-70=13, so 83≡13 mod14. Therefore, 6^83≡6^13≡41 mod49.Similarly, 83 divided by7 is 11*7=77, 83-77=6, so 83≡6 mod7. Therefore, 8^83≡8^6≡43 mod49.Adding 41 +43 gives 84, which is 35 mod49. So, 35 is the remainder. But let me cross-verify using another method. Maybe using binomial theorem?Alternatively, notice that 6 and 7 are involved. Maybe there's a way to express 6 and 8 in terms of 7 +/- something?Wait, 6 = 7 -1, and 8=7 +1. Hmm, interesting. So 6 ≡ -1 mod7 and 8≡1 mod7, but we're working mod49 here, not mod7. However, perhaps we can use the binomial theorem for expansions. Since 6=7-1 and 8=7+1, maybe raising them to powers and expanding. But 83 is a large exponent, but maybe we can use some properties.Alternatively, using the fact that for modulus 49, which is 7^2, we can use lifting the exponent lemma (LTE) or other techniques. Wait, LTE is usually for prime exponents, but perhaps...Alternatively, consider that 6 and 8 can be written as 7 -1 and 7 +1. Let's explore that.Let me write 6 = 7 -1. Then 6^n = (7 -1)^n. Similarly, 8 = 7 +1, so 8^n = (7 +1)^n. Then, expanding both using binomial theorem:(7 -1)^n = sum_{k=0}^n C(n,k) 7^{k} (-1)^{n -k}Similarly, (7 +1)^n = sum_{k=0}^n C(n,k) 7^{k} (1)^{n -k}But when we take modulo49, any term with 7^2 or higher will be 0 mod49. Therefore, we can truncate the expansions at k=1.Therefore:(7 -1)^n ≡ C(n,0) (-1)^n + C(n,1) 7 (-1)^{n -1} mod49Similarly,(7 +1)^n ≡ C(n,0) 1^n + C(n,1)7*1^{n -1} mod49Calculating:For (7 -1)^n:≡ (-1)^n + 7n (-1)^{n -1} mod49= (-1)^n -7n (-1)^{n} mod49= (-1)^n (1 -7n) mod49Similarly, for (7 +1)^n:≡1 +7n mod49Therefore, combining both:a_n = (7 -1)^n + (7 +1)^n ≡ (-1)^n (1 -7n) + (1 +7n) mod49= (-1)^n (1 -7n) +1 +7n mod49If n is even, then (-1)^n =1, so:= (1 -7n) +1 +7n = 1 -7n +1 +7n = 2 mod49If n is odd, (-1)^n =-1, so:= (-1)(1 -7n) +1 +7n = -1 +7n +1 +7n =14n mod49Wait, so according to this formula, if n is even, a_n ≡2 mod49, and if n is odd, a_n≡14n mod49.But wait, let me check that step again. If n is odd:(-1)^n = -1, so:(-1)(1 -7n) +1 +7n= -1*(1) + (-1)*(-7n) +1 +7n= -1 +7n +1 +7n= ( -1 +1 ) + (7n +7n)=0 +14n=14n mod49.Yes, that's right. Therefore, if n is even, a_n ≡2 mod49; if n is odd, a_n≡14n mod49.Wow, that's a really useful formula! So for a_n =6^n +8^n, modulo49, it's 2 if n even, 14n if n odd.But let's test this formula with small n to see if it works.Let's take n=1:a_1=6 +8=14. 14 mod49=14. According to formula, since n is odd, 14*1=14. Correct.n=2:a_2=6^2 +8^2=36 +64=100. 100 mod49: 100 -2*49=100-98=2. According to formula, n even, so 2. Correct.n=3:a_3=6^3 +8^3=216 +512=728. 728 divided by49: 49*14=686, 728-686=42. 42 mod49=42. According to formula, n odd, 14*3=42. Correct.n=4:a4=6^4 +8^4=1296 +4096=5392. 5392 divided by49: 49*110=5390, 5392-5390=2. So 2 mod49. Formula says even n, 2. Correct.n=5:a5=6^5 +8^5=7776 +32768=40544. 40544 divided by49: Let's see, 49*827=49*(800 +27)=49*800=39200 +49*27=1323. Total 39200+1323=40523. 40544-40523=21. So 21 mod49. Formula: n odd, 14*5=70≡70-49=21 mod49. Correct.So this formula seems to hold. Therefore, generalizing, for a_n=6^n +8^n, modulo49, the result is 2 if n is even, and 14n mod49 if n is odd.Since we need a_83, and 83 is odd, then according to the formula, a_83 ≡14*83 mod49.Compute 14*83. 14*80=1120, 14*3=42, so total 1120 +42=1162. Now, 1162 divided by49:49*23=1127. 1162 -1127=35. Therefore, 1162 ≡35 mod49. Therefore, the remainder is35.Therefore, the answer is35.Alternatively, using the formula: 14*83 mod49. Since14 and49 have a common factor of7, we can simplify:14*83 ≡7*2*83 ≡7*(2*83) mod49But 2*83=166. 166 mod49: 49*3=147, 166-147=19. So 166≡19 mod49. Then 7*19=133. 133 mod49: 49*2=98, 133-98=35. So 133≡35 mod49. So same result.Therefore, regardless of the method, the remainder is35.But let me verify this with another approach to be thorough. Let's compute a_83 mod49 using the formula a_n ≡14n mod49 when n is odd. Since83 is odd, it's 14*83. As above, 14*83=1162. 1162 divided by49: 49*23=1127, 1162-1127=35. So 35. So yes, same result.Alternatively, maybe using cyclicity.Earlier, we saw that 6^83 ≡6^(83 mod14). 83 divided by14: 14*5=70, 83-70=13. So 83≡13 mod14. So 6^83≡6^13≡41 mod49.Similarly, 8^83≡8^(83 mod7)=8^6≡43 mod49.41+43=84. 84 mod49=35. So same answer.Therefore, all methods converge to 35. So the remainder is35.Thus, the answer is boxed{35}.Final Answerboxed{35}</think>To determine the remainder when (a_{83} = 6^{83} + 8^{83}) is divided by 49, we use modular arithmetic and properties of exponents.First, we note that Euler's theorem tells us (phi(49) = 42), so any number coprime to 49 raised to the 42nd power is congruent to 1 modulo 49. However, we can find smaller cycles for 6 and 8 modulo 49.For 6 modulo 49, we find the order by computing powers:- The order of 6 modulo 49 is found to be 14, since (6^{14} equiv 1 mod 49).For 8 modulo 49, we similarly compute powers:- The order of 8 modulo 49 is found to be 7, since (8^{7} equiv 1 mod 49).Using these orders, we reduce the exponents:- (83 mod 14 = 13), so (6^{83} equiv 6^{13} mod 49).- (83 mod 7 = 6), so (8^{83} equiv 8^{6} mod 49).We compute the reduced powers:- (6^{13} mod 49) results in 41.- (8^{6} mod 49) results in 43.Adding these results:- (6^{83} + 8^{83} equiv 41 + 43 mod 49).- (41 + 43 = 84), and (84 mod 49 = 35).Thus, the remainder when (a_{83}) is divided by 49 is (boxed{35}).]

🔑:Solution 1Firstly, we try to find a relationship between the numbers we're provided with and 49. We notice that 49=7^2, and both 6 and 8 are greater or less than 7 by 1. Thus, expressing the numbers in terms of 7, we get a_{83} = (7-1)^{83}+(7+1)^{83}.Applying the [Binomial Theorem](https://artofproblemsolving.com/wiki/index.php/Binomial_Theorem), half of our terms cancel out and we are left with 2(7^{83}+3403cdot7^{81}+cdots + 83cdot7). We realize that all of these terms are divisible by 49 except the final term.After some quick division, our answer is boxed{035}.Solution 2Since phi(49) = 42 (see [Euler's totient function](https://artofproblemsolving.com/wiki/index.php/Euler%27s_totient_function)), [Euler's Totient Theorem](https://artofproblemsolving.com/wiki/index.php/Euler%27s_Totient_Theorem) tells us that a^{42} equiv 1 pmod{49} where text{gcd}(a,49) = 1. Thus 6^{83} + 8^{83} equiv 6^{2(42)-1}+8^{2(42)-1}equiv 6^{-1} + 8^{-1} equiv frac{8+6}{48} equiv frac{14}{-1}equiv boxed{035} pmod{49}.Alternatively, we could have noted that a^bequiv a^{bpmod{phi{(n)}}}pmod n. This way, we have 6^{83}equiv 6^{83pmod {42}}equiv 6^{-1}pmod {49}, and can finish the same way.Solution 3 (cheap and quick)As the value of a is obviously 6^{83}+8^{83} we look for a pattern with others. With a bit of digging, we discover that 6^n+6^m where m and n are odd is equal to boxed{35}:text{mod}:49-dragoonSolution 36^{83} + 8^{83} = (6+8)(6^{82}-6^{81}8+ldots-8^{81}6+8^{82})Becuase 7|(6+8), we only consider 6^{82}-6^{81}8+ldots-8^{81}6+8^{82} pmod{7}6^{82}-6^{81}8+ldots-8^{81}6+8^{82} equiv (-1)^{82} - (-1)^{81}+ ldots - (-1)^1 + 1 = 83 equiv 6 pmod{7}6^{83} + 8^{83} equiv 14 cdot 6 equiv boxed{035} pmod{49}Solution 4 last resort (bash)Repeat the steps of taking modulo 49 after reducing the exponents over and over again until you get a residue of 49, namely 35. This bashing takes a lot of time but it isn’t too bad. ~peelybonehead

🧠:[<think>Okay, let's see. I need to figure out the ages of the father, mother, older child, and younger child. There are four conditions given. Let me take them one by one and try to set up equations.First, the sum of all four ages is 124. So, if I let F be the father's age, M the mother's age, O the older child's age, and Y the younger child's age, then the equation would be:F + M + O + Y = 124. (1)Second, the father and mother together are three times as old as their two children combined. So that would be:F + M = 3*(O + Y) (2)Third, the mother is more than twice as old as the oldest child. So:M > 2*O (3)Fourth, the difference between the father's age and the mother's age is nine times the difference between the older child's and the younger child's age. So:F - M = 9*(O - Y) (4)Alright, so we have four equations here, but note that equation (3) is an inequality, not an equality. So we'll need to keep that in mind once we have potential solutions.Let me start by trying to express variables in terms of others to reduce the number of variables.From equation (2): F + M = 3*(O + Y). Let's call that equation (2). Also, equation (1) says that F + M + O + Y = 124. If we substitute F + M from equation (2) into equation (1), we get:3*(O + Y) + O + Y = 124Simplify that:3*(O + Y) + (O + Y) = 4*(O + Y) = 124So, 4*(O + Y) = 124 => O + Y = 124 / 4 = 31So, O + Y = 31. Therefore, from equation (2), F + M = 3*31 = 93.So now we know that F + M = 93 and O + Y = 31.Now, equation (4) says F - M = 9*(O - Y). Let me note that. So we have:F - M = 9*(O - Y) (4)So, if I can express F and M in terms of each other, and O and Y in terms of each other, maybe I can solve this.From O + Y = 31, we can express Y as 31 - O. Similarly, from F + M = 93, we can express M as 93 - F.But let's see equation (4):F - M = 9*(O - Y)Substitute M = 93 - F into the left side:F - (93 - F) = 9*(O - Y)Simplify left side:F - 93 + F = 2F - 93So, 2F - 93 = 9*(O - Y)But O - Y is O - (31 - O) = 2O - 31Therefore, equation (4) becomes:2F - 93 = 9*(2O - 31)Let me compute the right side:9*(2O - 31) = 18O - 279So, equation (4) is:2F - 93 = 18O - 279Let me rearrange this equation to solve for F:2F = 18O - 279 + 932F = 18O - 186Divide both sides by 2:F = 9O - 93So, F = 9O - 93. Therefore, since M = 93 - F, we can write:M = 93 - (9O - 93) = 93 - 9O + 93 = 186 - 9OSo, now we have expressions for F and M in terms of O. And since Y = 31 - O, we have all variables expressed in terms of O. So, now, let's write down the variables:F = 9O - 93M = 186 - 9OY = 31 - ONow, we need to check the remaining conditions. Let's recall that all ages must be positive integers (since they are given in full years). Also, equation (3) is M > 2O.So, first, let's ensure all variables are positive.Starting with Y = 31 - O. Since Y must be positive, 31 - O > 0 => O < 31. Also, since Y is the younger child, O must be greater than Y. Since Y = 31 - O, O > 31 - O => 2O > 31 => O > 15.5. Since age is in full years, O must be at least 16.So O is between 16 and 30 inclusive.Similarly, check F = 9O - 93 must be positive. So:9O - 93 > 0 => 9O > 93 => O > 10.333... But since O is at least 16, this is already satisfied. Let's confirm for O =16:F = 9*16 -93 = 144 -93=51. That's positive.For M =186 -9O must be positive:186 -9O >0 => 9O <186 => O <20.666... So O must be less than 20.666, so since O is an integer, O <=20.But earlier we had O >=16 and O <=20 (since O must be at least 16 and less than 20.666). So O can be 16,17,18,19,20.But we also have the condition that the mother's age M must be more than twice the oldest child's age, i.e., M >2O.So M =186 -9O >2OTherefore:186 -9O >2O186 >11OO <186/11186 divided by 11 is 16.909...Since O is an integer, O <=16. But earlier, O must be at least 16. So O can only be 16.Wait, but that's conflicting. Wait, let's check.Wait, O is between 16 and 20 (inclusive) because from Y =31 - O, we have O >=16 (since O >15.5) and O <=20 (from M =186 -9O >0). So O is 16,17,18,19,20.But then, the condition M >2O translates to 186 -9O >2O =>186 >11O =>O <186/11≈16.909. So O must be less than 16.909, so the maximum integer value O can take is 16. So O=16.Therefore, O=16. Let's check that.If O=16,Then Y=31 -16=15.F=9*16 -93=144 -93=51.M=186 -9*16=186 -144=42.So father is 51, mother is 42, older child is 16, younger child is15.Check the conditions:1. Sum: 51 +42 +16 +15=51+42=93, 16+15=31, 93+31=124. Correct.2. Father and mother's age sum:51+42=93. Children's sum:31. 93=3*31. Correct.3. Mother's age 42 >2*16=32. Yes, 42>32. Correct.4. Father - mother=51-42=9. Older child - younger child=16-15=1. 9=9*1. Correct.So all conditions are satisfied.Wait, but let's check if there could be any other possible values. According to the previous logic, O must be 16 because O must be less than 16.909. So O=16 is the only possible integer value. Let me check O=17.If O=17,Then Y=31 -17=14.F=9*17 -93=153-93=60.M=186 -9*17=186 -153=33.Check mother's age 33>2*17=34? No, 33<34. So that violates condition (3). Hence invalid.Similarly, O=17 gives M=33 which is less than 34, so invalid.Similarly, O=18:Y=13, F=9*18 -93=162-93=69, M=186-162=24. M=24>2*18=36? 24<36. Not valid.O=19:Y=12, F=9*19 -93=171-93=78, M=186-171=15. 15>38? No.O=20:Y=11, F=9*20 -93=180-93=87, M=186-180=6. 6>40? No.So all higher O values result in M being too young, violating condition (3). Similarly, O=16 is the only possible value.Therefore, the ages must be:Father:51, Mother:42, Older child:16, Younger child:15.Wait, but let's check if there's another possibility if O=16.Is there another way to get O=16? Let me see. Wait, maybe I made a mistake in assuming O must be integer. But in the problem statement, it says "full years of life," so all ages are integers. So O must be integer. So O=16 is the only solution.So the final answer should be Father:51, Mother:42, Older child:16, Younger child:15. Let me check once again all conditions:1. 51+42+16+15=124. Correct.2. 51+42=93, 16+15=31. 93=3*31. Correct.3. Mother is 42, which is more than twice 16 (32). Correct.4. Father - mother=9, Older - younger=1. 9=9*1. Correct.All conditions are satisfied. Therefore, this must be the solution.Final AnswerThe father is boxed{51} years old, the mother is boxed{42} years old, the older child is boxed{16} years old, and the younger child is boxed{15} years old.</think>Given the problem of determining the ages of a family consisting of a father, mother, and two children, we start by defining the variables as follows: ( F ) for the father's age, ( M ) for the mother's age, ( O ) for the older child's age, and ( Y ) for the younger child's age. We use the given conditions to set up equations and inequalities.1. The sum of all four ages is 124: [ F + M + O + Y = 124 ]2. The father and mother together are three times as old as their two children combined: [ F + M = 3(O + Y) ]3. The mother is more than twice as old as the oldest child: [ M > 2O ]4. The difference between the father's age and the mother's age is nine times the difference between the older child's and the younger child's age: [ F - M = 9(O - Y) ]Using equations (1) and (2), we substitute ( F + M ) from equation (2) into equation (1):[3(O + Y) + O + Y = 124 implies 4(O + Y) = 124 implies O + Y = 31]Thus, ( F + M = 3 times 31 = 93 ).Next, we express ( Y ) in terms of ( O ) and ( M ) in terms of ( F ):[Y = 31 - O quad text{and} quad M = 93 - F]Substituting these into equation (4):[F - (93 - F) = 9(O - (31 - O)) implies 2F - 93 = 9(2O - 31)]Simplifying, we get:[2F - 93 = 18O - 279 implies 2F = 18O - 186 implies F = 9O - 93]Substituting ( F = 9O - 93 ) into ( M = 93 - F ):[M = 93 - (9O - 93) = 186 - 9O]We check the validity of possible ages for ( O ) considering ( O ) must be an integer between 16 and 20 (due to constraints from positive ages and ( M > 2O )). We find that ( O = 16 ) is the only valid solution:- ( Y = 31 - 16 = 15 )- ( F = 9 times 16 - 93 = 51 )- ( M = 186 - 9 times 16 = 42 )Verifying all conditions:1. Sum: ( 51 + 42 + 16 + 15 = 124 )2. Parent sum: ( 51 + 42 = 93 ), Child sum: ( 16 + 15 = 31 ), ( 93 = 3 times 31 )3. Mother's age: ( 42 > 2 times 16 )4. Age difference: ( 51 - 42 = 9 ), ( 16 - 15 = 1 ), ( 9 = 9 times 1 )Thus, the ages are:[boxed{51} text{ (father)}, boxed{42} text{ (mother)}, boxed{16} text{ (older child)}, boxed{15} text{ (younger child)}],<think>Okay, let's see. We have a family of four: father, mother, and two children. We need to find their ages based on the given conditions. Let me break down each condition and see how to approach this.First, let's assign variables to each person's age to make it easier. Let's say:- Father's age = F- Mother's age = M- Older child's age = C1- Younger child's age = C2Now, the first condition is that the sum of all four ages is 124. So, that gives us the equation:1) F + M + C1 + C2 = 124Second condition: The father and mother together are three times as old as their two children combined. So, that translates to:2) F + M = 3*(C1 + C2)Third condition: The mother is more than twice as old as the oldest of the two children. So:3) M > 2*C1Fourth condition: The difference between the father's age and the mother's age is nine times the difference between the older child's and the younger child's age. So:4) F - M = 9*(C1 - C2)Alright, so we have four equations here, although the third one is an inequality. Let's see if we can solve these step by step.Starting with the first two equations. Since equation 2 gives F + M in terms of the children's ages, maybe we can substitute that into equation 1.From equation 2: F + M = 3*(C1 + C2)Substituting into equation 1: 3*(C1 + C2) + C1 + C2 = 124Simplify that: 3*(C1 + C2) + (C1 + C2) = 4*(C1 + C2) = 124So, 4*(C1 + C2) = 124 => C1 + C2 = 124 / 4 = 31Therefore, the sum of the children's ages is 31. Then, from equation 2, F + M = 3*31 = 93So now we know that the father and mother together are 93 years old, and the children together are 31.So, equations simplified:F + M = 93C1 + C2 = 31Now, moving to the fourth condition: F - M = 9*(C1 - C2)Let's denote D = F - M and d = C1 - C2. Then, D = 9d.But we can also use the two equations F + M = 93 and F - M = 9d to solve for F and M.Adding these two equations: (F + M) + (F - M) = 93 + 9d => 2F = 93 + 9d => F = (93 + 9d)/2 = 46.5 + 4.5dSimilarly, subtracting the two equations: (F + M) - (F - M) = 93 - 9d => 2M = 93 - 9d => M = (93 - 9d)/2 = 46.5 - 4.5dSo, F and M are expressed in terms of d, which is C1 - C2.But we also know from the children's ages that C1 + C2 = 31 and C1 - C2 = d. So, we can solve for C1 and C2 in terms of d:Adding C1 + C2 = 31 and C1 - C2 = d:2C1 = 31 + d => C1 = (31 + d)/2Subtracting the two equations:2C2 = 31 - d => C2 = (31 - d)/2So, now we have expressions for all four variables in terms of d.But we need to remember that all ages must be positive integers (since age is counted in full years). So, C1 and C2 must be integers, and F and M must also be integers. Therefore, (31 + d)/2 and (31 - d)/2 must be integers. That implies that 31 + d must be even, so d must be odd because 31 is odd. So, d is an odd integer.Also, since C1 > C2 (because C1 is the older child), d = C1 - C2 must be a positive integer. So, d ≥ 1, and since C2 must be positive, (31 - d)/2 > 0 => 31 - d > 0 => d < 31. So d is in 1, 3, 5, ..., 29.Similarly, F and M must be positive. Let's check:From F = 46.5 + 4.5d, since d is an integer, 4.5d = (9/2)d, so 46.5 + 4.5d = 93/2 + 9d/2 = (93 + 9d)/2. For this to be an integer, 93 + 9d must be even. 93 is odd, 9d is odd (since d is odd, 9d is 9*odd = odd). Odd + odd = even. So, yes, 93 +9d is even, so F is integer.Similarly, M = 46.5 - 4.5d = (93 - 9d)/2. 93 is odd, 9d is odd, so odd - odd = even. So (93 -9d) is even, so M is integer. So that's good.So, all variables will be integers as long as d is an odd integer between 1 and 29.Now, let's also remember the third condition: M > 2*C1We have expressions for M and C1 in terms of d:M = (93 - 9d)/2C1 = (31 + d)/2Therefore, the inequality becomes:(93 - 9d)/2 > 2*( (31 + d)/2 )Multiply both sides by 2 to eliminate denominators:93 - 9d > 2*(31 + d)Compute right side: 62 + 2dSo, 93 -9d > 62 + 2dSubtract 62 from both sides:31 -9d > 2dAdd 9d to both sides:31 > 11dTherefore, 11d < 31 => d < 31/11 ≈2.818Since d is a positive odd integer, d can be 1 or possibly 3, but 3*11=33, which is more than 31. Wait, 31/11 is approximately 2.818, so d must be less than that. Since d is a positive integer (and odd), the only possible value is d=1.Wait, let's verify:If d=1, then 11d=11 <31, which holds.If d=3, 11*3=33 >31, which does not hold.Therefore, the only possible value is d=1.So, d=1.Therefore, we can compute all the variables:C1 = (31 +1)/2=32/2=16C2=(31 -1)/2=30/2=15So, the children are 16 and 15 years old.Then, F = (93 +9d)/2=(93 +9*1)/2=102/2=51M=(93 -9d)/2=(93 -9)/2=84/2=42Wait, but check the mother's age: 42, and the oldest child is 16. Then, 42 > 2*16=32. 42>32, which is true. So condition 3 is satisfied.Also, check the fourth condition: F - M=51-42=9. The difference between the children's ages is 16-15=1. 9*1=9, so that's correct.Check the sum: 51+42+16+15=51+42=93, 16+15=31; 93+31=124, correct.All conditions seem to be satisfied.But let me double-check. Let's verify each condition:1) Sum: 51 + 42 +16 +15=124. Yes.2) Father and mother:51+42=93. Children:16+15=31. 93=3*31. Correct.3) Mother is 42, oldest child is 16. 42 >2*16=32. Yes, 42>32.4) Father - mother=9, older child - younger child=1. 9=9*1. Correct.Everything checks out.But wait, just to be thorough, let's check if d could be 3 or higher. Wait, according to the third condition, d must be less than 31/11≈2.818, so d=1 is the only possible odd integer. So, there's no other solution. Therefore, this must be the correct answer.So, the ages are:Father:51Mother:42Older child:16Younger child:15Final AnswerThe father is boxed{51} years old, the mother is boxed{42} years old, the older child is boxed{16} years old, and the younger child is boxed{15} years old.</think>Given the age information of a family (father, mother, and two children) with the following conditions:1. The sum of all four ages is 124.2. The father and mother together are three times as old as their two children combined.3. The mother is more than twice as old as the oldest child.4. The difference between the father's age and the mother's age is nine times the difference between the older child's and the younger child's age.We start by assigning variables to each family member's age:- Father's age: ( F )- Mother's age: ( M )- Older child's age: ( C_1 )- Younger child's age: ( C_2 )From the given conditions, we derive the following equations:1. ( F + M + C_1 + C_2 = 124 )2. ( F + M = 3(C_1 + C_2) )3. ( M > 2C_1 )4. ( F - M = 9(C_1 - C_2) )Using equations 1 and 2, we substitute ( F + M ) from equation 2 into equation 1:[ 3(C_1 + C_2) + C_1 + C_2 = 124 ][ 4(C_1 + C_2) = 124 ][ C_1 + C_2 = 31 ]Thus, ( F + M = 3 times 31 = 93 ).Next, we use equation 4. Let ( d = C_1 - C_2 ). Then:[ F - M = 9d ]We solve for ( F ) and ( M ) using the equations ( F + M = 93 ) and ( F - M = 9d ):[ F = frac{93 + 9d}{2} ][ M = frac{93 - 9d}{2} ]We also solve for ( C_1 ) and ( C_2 ):[ C_1 = frac{31 + d}{2} ][ C_2 = frac{31 - d}{2} ]Given that all ages must be integers, ( d ) must be an odd integer. From condition 3:[ M > 2C_1 ][ frac{93 - 9d}{2} > 2 times frac{31 + d}{2} ][ 93 - 9d > 62 + 2d ][ 31 > 11d ][ d < frac{31}{11} approx 2.818 ]The only possible odd integer value for ( d ) is 1. Substituting ( d = 1 ):[ C_1 = frac{31 + 1}{2} = 16 ][ C_2 = frac{31 - 1}{2} = 15 ][ F = frac{93 + 9 times 1}{2} = 51 ][ M = frac{93 - 9 times 1}{2} = 42 ]Verifying all conditions:1. ( 51 + 42 + 16 + 15 = 124 )2. ( 51 + 42 = 93 ) and ( 3 times (16 + 15) = 93 )3. ( 42 > 2 times 16 = 32 )4. ( 51 - 42 = 9 ) and ( 9 times (16 - 15) = 9 )All conditions are satisfied. The ages are:- Father: (boxed{51})- Mother: (boxed{42})- Older child: (boxed{16})- Younger child: (boxed{15})]

🔑:Due to (1) and (2), the quadruple sum of the ages of the two children is 124; therefore, the children together are 31 years old, and the parents together are 93 years old.Since the ages of the four individuals are given in whole years and since 31 is an odd number, one of the children's ages is even, and the other is odd. Therefore, the difference in the ages of the two children is an odd number.If this difference were 3 years or more, then the difference in the parents' ages would be 27 years or more. In that case, one child would be 17 years or older, the other child 14 years or younger, the mother 33 years or younger, and the father 60 years or older.Because 2 cdot 17 > 33 and (3), this possibility is ruled out. Therefore, the difference in the children's ages is 1 year, and the difference in the parents' ages is 9 years. From this, it follows:The father is 51 years old, the mother is 42 years old, the oldest child is 16 years old, and the other child is 15 years old.

🧠:[<think>Okay, let's see. The problem says that we have real numbers x, y, z such that x + y + z is not zero, and (x + y + z)^2 equals 3 plus 3 times (xy + yz + zx). Then we need to find the value of [x^3 + y^3 + z^3 - 3xyz] divided by (x + y + z). Hmm, alright.First, I remember that there's a formula for x^3 + y^3 + z^3 - 3xyz. Let me recall... I think it factors into (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). Yes, that sounds right. So if that's the case, then when we divide this by (x + y + z), assuming x + y + z is not zero (which the problem states), we get x^2 + y^2 + z^2 - xy - yz - zx. So the expression we need to compute simplifies to x^2 + y^2 + z^2 minus the sum of the products of pairs.So, the problem reduces to finding the value of x^2 + y^2 + z^2 - xy - yz - zx. Now, how can we relate this to the given equation?The given condition is (x + y + z)^2 = 3 + 3(xy + yz + zx). Let's expand the left-hand side. (x + y + z)^2 is equal to x^2 + y^2 + z^2 + 2(xy + yz + zx). So, substituting that in, we have:x^2 + y^2 + z^2 + 2(xy + yz + zx) = 3 + 3(xy + yz + zx)Let me write that down step by step:Left-hand side (LHS) = x² + y² + z² + 2(xy + yz + zx)Right-hand side (RHS) = 3 + 3(xy + yz + zx)So, subtract 2(xy + yz + zx) from both sides to get:x² + y² + z² = 3 + 3(xy + yz + zx) - 2(xy + yz + zx)Which simplifies to:x² + y² + z² = 3 + (xy + yz + zx)So, x² + y² + z² = 3 + (xy + yz + zx)Now, recall that the expression we need is x² + y² + z² - xy - yz - zx. Let's substitute the value of x² + y² + z² from the equation above:x² + y² + z² - xy - yz - zx = [3 + (xy + yz + zx)] - (xy + yz + zx) = 3Wait, that's just 3? So, the entire expression simplifies to 3?Let me verify that again. If x² + y² + z² = 3 + (xy + yz + zx), then subtracting (xy + yz + zx) from both sides gives x² + y² + z² - (xy + yz + zx) = 3. Yes, exactly. Therefore, the value of [x³ + y³ + z³ - 3xyz]/(x + y + z) is equal to 3.Hmm, that seems straightforward, but let me check if there's any step I might have overlooked.First, confirming the formula for x³ + y³ + z³ - 3xyz: yes, it's indeed (x + y + z)(x² + y² + z² - xy - yz - zx). Then, dividing by (x + y + z) gives us the second factor, which is x² + y² + z² - xy - yz - zx. Then, using the given equation, expanding (x + y + z)^2 and setting it equal to 3 + 3(xy + yz + zx). Then, solving for x² + y² + z², which gives 3 + (xy + yz + zx). Then subtracting (xy + yz + zx) from that gives 3. So, all steps check out.Therefore, the answer is 3. So, in boxed form, it's boxed{3}.Final Answerboxed{3}</think>Given real numbers (x, y, z) satisfy (x + y + z neq 0) and ((x + y + z)^2 = 3 + 3(xy + yz + zx)), we need to find the value of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}).First, recall the identity:[x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)]Thus, dividing by (x + y + z) (which is non-zero), we get:[frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z} = x^2 + y^2 + z^2 - xy - yz - zx]Next, use the given equation ((x + y + z)^2 = 3 + 3(xy + yz + zx)). Expanding the left-hand side:[(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)]Equating this to the right-hand side:[x^2 + y^2 + z^2 + 2(xy + yz + zx) = 3 + 3(xy + yz + zx)]Subtracting (2(xy + yz + zx)) from both sides:[x^2 + y^2 + z^2 = 3 + (xy + yz + zx)]Therefore, substituting this into the expression (x^2 + y^2 + z^2 - xy - yz - zx):[x^2 + y^2 + z^2 - xy - yz - zx = [3 + (xy + yz + zx)] - (xy + yz + zx) = 3]Thus, the value of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}) is (boxed{3}).,<think>Okay, let's see here. I need to find the value of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}) given that (x + y + z neq 0) and ((x + y + z)^2 = 3 + 3(xy + yz + zx)). Hmm, alright. Let me start by recalling some algebraic identities that might help here.First, I remember that the numerator (x^3 + y^3 + z^3 - 3xyz) can be factored using the formula (x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)). So if that's the case, then dividing by (x + y + z) would leave me with (x^2 + y^2 + z^2 - xy - yz - zx). Therefore, the expression we need to find is actually equal to (x^2 + y^2 + z^2 - xy - yz - zx). Cool, so maybe I can find the value of this expression using the given equation.The problem states that ((x + y + z)^2 = 3 + 3(xy + yz + zx)). Let me expand the left-hand side of that equation. Expanding ((x + y + z)^2), we get (x^2 + y^2 + z^2 + 2xy + 2yz + 2zx). So the equation becomes:(x^2 + y^2 + z^2 + 2xy + 2yz + 2zx = 3 + 3(xy + yz + zx)).Let me rearrange this equation to group like terms. Let's subtract (3(xy + yz + zx)) from both sides:(x^2 + y^2 + z^2 + 2xy + 2yz + 2zx - 3xy - 3yz - 3zx = 3).Simplifying the left-hand side:(x^2 + y^2 + z^2 - xy - yz - zx = 3).Wait a second! That expression on the left is exactly the same as the one I need for the final answer. So (x^2 + y^2 + z^2 - xy - yz - zx = 3), which means that the original expression (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}) is equal to 3. So the answer is 3?Let me double-check that to make sure I didn't make a mistake. Starting again, the key identity is (x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)). So when we divide by (x + y + z), which is non-zero, we indeed get (x^2 + y^2 + z^2 - xy - yz - zx). Then the given equation is ((x + y + z)^2 = 3 + 3(xy + yz + zx)), which expands to (x^2 + y^2 + z^2 + 2(xy + yz + zx) = 3 + 3(xy + yz + zx)). Subtracting (3(xy + yz + zx)) from both sides gives (x^2 + y^2 + z^2 - xy - yz - zx = 3), which is exactly the expression we need. Therefore, the value is indeed 3.Hmm, seems straightforward. Let me check if there's another way to approach this problem, just to confirm.Alternatively, maybe I can use the identity (x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)) directly. Since we need to compute (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}), that's equivalent to (x^2 + y^2 + z^2 - xy - yz - zx), which is the same as (frac{1}{2}[(x - y)^2 + (y - z)^2 + (z - x)^2]). But since the problem gives an equation involving ((x + y + z)^2) and (xy + yz + zx), we can use that equation to find (x^2 + y^2 + z^2 - xy - yz - zx) directly, as I did before.Another way might be to let (S = x + y + z), (P = xy + yz + zx), and (Q = xyz). Then, we know that (x^3 + y^3 + z^3 = S^3 - 3SP + 3Q). Therefore, (x^3 + y^3 + z^3 - 3xyz = S^3 - 3SP + 3Q - 3Q = S^3 - 3SP). Therefore, (frac{x^3 + y^3 + z^3 - 3xyz}{S} = frac{S^3 - 3SP}{S} = S^2 - 3P). So the expression simplifies to (S^2 - 3P).But from the given equation, (S^2 = 3 + 3P). Therefore, substituting (S^2) into the expression (S^2 - 3P) gives (3 + 3P - 3P = 3). So again, we get 3. That's consistent with the previous result. So whether I approach it by expanding the given equation or by using the identity with S and P, the answer is 3. That gives me more confidence that the answer is indeed correct.Is there any chance I might have missed something? Let's consider possible edge cases. The problem states (x + y + z neq 0), which is important because we are dividing by (x + y + z). So as long as that's satisfied, which it is, the division is valid. Also, the given equation must hold. Let's see if there are specific values of x, y, z that satisfy the equation and check the result.For example, suppose x = y = z. Let's test that case. Let x = y = z = a. Then, the left-hand side of the given equation is ((3a)^2 = 9a^2). The right-hand side is 3 + 3(3a^2) = 3 + 9a^2. So setting 9a^2 = 3 + 9a^2, we get 0 = 3, which is impossible. Therefore, there's no solution where all three variables are equal. That's interesting. So maybe x, y, z cannot all be equal.Alternatively, let's try a simple case where one variable is zero. Let z = 0. Then the given equation becomes ((x + y)^2 = 3 + 3xy). Let me set z = 0. Then, the numerator becomes (x^3 + y^3 - 0), divided by (x + y). So (frac{x^3 + y^3}{x + y} = x^2 - xy + y^2). According to the problem, this should equal 3. Let's check if the given equation holds. If z = 0, then the equation is ((x + y)^2 = 3 + 3xy). Expanding the left-hand side: (x^2 + 2xy + y^2 = 3 + 3xy). Rearranging: (x^2 - xy + y^2 = 3). Which is exactly equal to the numerator divided by (x + y). So in this case, the expression is 3, which matches the result. So this specific case works.Another test case: Let x = 1, y = 1, then solve for z. Let's see. Let x = 1, y = 1. Then the given equation is (1 + 1 + z)^2 = 3 + 3(1*1 + 1*z + z*1). Let's compute that:Left side: (2 + z)^2 = z^2 + 4z + 4.Right side: 3 + 3(1 + z + z) = 3 + 3(1 + 2z) = 3 + 3 + 6z = 6 + 6z.Set them equal: z^2 + 4z + 4 = 6 + 6z.Bring all terms to left: z^2 - 2z - 2 = 0.Solve for z: z = [2 ± sqrt(4 + 8)] / 2 = [2 ± sqrt(12)] / 2 = [2 ± 2*sqrt(3)] / 2 = 1 ± sqrt(3).So z = 1 + sqrt(3) or z = 1 - sqrt(3).Now compute the expression (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}).First, x + y + z = 2 + z. Let's take z = 1 + sqrt(3). Then x + y + z = 3 + sqrt(3).Compute numerator: 1 + 1 + z^3 - 3*1*1*z = 2 + z^3 - 3z.So let's compute z^3. z = 1 + sqrt(3):z^3 = (1 + sqrt(3))^3 = 1 + 3*sqrt(3) + 3*(sqrt(3))^2 + (sqrt(3))^3= 1 + 3*sqrt(3) + 9 + 3*3*sqrt(3)= 1 + 3*sqrt(3) + 9 + 9*sqrt(3)= 10 + 12*sqrt(3)So numerator: 2 + 10 + 12*sqrt(3) - 3*(1 + sqrt(3)) = 12 + 12*sqrt(3) - 3 - 3*sqrt(3) = 9 + 9*sqrt(3)Denominator: 3 + sqrt(3)So (frac{9 + 9sqrt{3}}{3 + sqrt{3}}). Let's rationalize the denominator:Multiply numerator and denominator by 3 - sqrt(3):Numerator: (9 + 9sqrt(3))(3 - sqrt(3)) = 9*(3 - sqrt(3)) + 9sqrt(3)*(3 - sqrt(3)) = 27 - 9sqrt(3) + 27sqrt(3) - 9*3 = 27 + 18sqrt(3) - 27 = 18sqrt(3)Denominator: (3 + sqrt(3))(3 - sqrt(3)) = 9 - 3 = 6So the expression is 18sqrt(3)/6 = 3sqrt(3). Wait, that's not 3. Wait, hold on, that contradicts our previous conclusion. What's wrong here?Wait, no. Wait, hold on. Wait, let's check this again. If x = 1, y = 1, z = 1 + sqrt(3), then compute the expression.Wait, first, the numerator was x^3 + y^3 + z^3 - 3xyz. So:x^3 = 1, y^3 = 1, z^3 = (1 + sqrt(3))^3 as calculated above: 10 + 12sqrt(3). So total x^3 + y^3 + z^3 = 1 + 1 + 10 + 12sqrt(3) = 12 + 12sqrt(3).Then, 3xyz = 3*1*1*(1 + sqrt(3)) = 3(1 + sqrt(3)) = 3 + 3sqrt(3).So numerator: 12 + 12sqrt(3) - (3 + 3sqrt(3)) = 9 + 9sqrt(3). Correct.Denominator: x + y + z = 1 + 1 + 1 + sqrt(3) = 3 + sqrt(3). Correct.So the expression is (9 + 9sqrt(3))/(3 + sqrt(3)). When we rationalize:Multiply numerator and denominator by (3 - sqrt(3)):Numerator: 9(3 - sqrt(3)) + 9sqrt(3)(3 - sqrt(3)) = 27 - 9sqrt(3) + 27sqrt(3) - 9*3 = 27 + 18sqrt(3) - 27 = 18sqrt(3)Denominator: 9 - 3 = 6.So 18sqrt(3)/6 = 3sqrt(3). Hmm, but according to our previous conclusion, it should be 3. But here, we get 3sqrt(3). That's a problem. So this seems contradictory.Wait, so either my test case is invalid, or I made a mistake in the calculation. Wait, but the test case was derived from the given equation. If x = 1, y = 1, z = 1 + sqrt(3), then does it satisfy the original equation?Let's check. Original equation: (x + y + z)^2 = 3 + 3(xy + yz + zx). So x + y + z = 3 + sqrt(3). Squaring that gives (3 + sqrt(3))^2 = 9 + 6sqrt(3) + 3 = 12 + 6sqrt(3).On the other hand, 3 + 3(xy + yz + zx). Compute xy + yz + zx:xy = 1*1 = 1yz = 1*(1 + sqrt(3)) = 1 + sqrt(3)zx = 1*(1 + sqrt(3)) = 1 + sqrt(3)Total: 1 + (1 + sqrt(3)) + (1 + sqrt(3)) = 3 + 2sqrt(3)Multiply by 3: 9 + 6sqrt(3). Then add 3: 12 + 6sqrt(3). Which matches the left-hand side. So the test case is valid.But then why is the expression evaluating to 3sqrt(3) instead of 3? That contradicts our previous conclusion. So there must be a mistake in my reasoning.Wait, wait, hold on. Wait, let me check the expression again. The problem states that (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}). In the test case, x = 1, y = 1, z = 1 + sqrt(3), we computed this as 3sqrt(3). But according to the problem's given equation and our derivation, it should be 3. So this is a contradiction. Therefore, my earlier conclusion must be wrong. But where?Wait, let's backtrack. Let me re-examine the key steps.First, the identity:(x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)). Correct. Then dividing by (x + y + z) gives (x^2 + y^2 + z^2 - xy - yz - zx). Then, from the given equation, ((x + y + z)^2 = 3 + 3(xy + yz + zx)). Expanding gives:(x^2 + y^2 + z^2 + 2(xy + yz + zx) = 3 + 3(xy + yz + zx)). Subtract (3(xy + yz + zx)) from both sides:(x^2 + y^2 + z^2 - xy - yz - zx = 3). So according to this, the expression is 3. However, in the test case, we get 3sqrt(3). Therefore, something is wrong here. How come?Wait, in the test case, we calculated (x^2 + y^2 + z^2 - xy - yz - zx). Let me compute that directly. x = 1, y = 1, z = 1 + sqrt(3):x^2 = 1, y^2 = 1, z^2 = (1 + sqrt(3))^2 = 1 + 2sqrt(3) + 3 = 4 + 2sqrt(3)So x^2 + y^2 + z^2 = 1 + 1 + 4 + 2sqrt(3) = 6 + 2sqrt(3)Now, compute xy + yz + zx:xy = 1*1 = 1yz = 1*(1 + sqrt(3)) = 1 + sqrt(3)zx = 1*(1 + sqrt(3)) = 1 + sqrt(3)So xy + yz + zx = 1 + (1 + sqrt(3)) + (1 + sqrt(3)) = 3 + 2sqrt(3)Therefore, (x^2 + y^2 + z^2 - xy - yz - zx = (6 + 2sqrt(3)) - (3 + 2sqrt(3)) = 3). Wait, that's 3. So according to this, the expression is 3. But when we computed the original expression, (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}), we got 3sqrt(3). That can't be. There's a contradiction here.Wait, so why is that? Let me check the calculation of the original expression again.x^3 + y^3 + z^3 - 3xyz:x^3 = 1y^3 = 1z^3 = (1 + sqrt(3))^3 = 1 + 3sqrt(3) + 3*(sqrt(3))^2 + (sqrt(3))^3= 1 + 3sqrt(3) + 9 + 3*3*sqrt(3)Wait, (sqrt(3))^3 is (sqrt(3))^2*sqrt(3) = 3*sqrt(3). So:= 1 + 3sqrt(3) + 9 + 3*3*sqrt(3)Wait, 3*3*sqrt(3)? Wait, no. Wait, the expansion is:(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. So here, a = 1, b = sqrt(3):= 1^3 + 3*1^2*sqrt(3) + 3*1*(sqrt(3))^2 + (sqrt(3))^3= 1 + 3sqrt(3) + 3*(3) + 3*sqrt(3)= 1 + 3sqrt(3) + 9 + 3sqrt(3)= 10 + 6sqrt(3)Therefore, z^3 = 10 + 6sqrt(3)Therefore, x^3 + y^3 + z^3 = 1 + 1 + 10 + 6sqrt(3) = 12 + 6sqrt(3)Then, 3xyz = 3*1*1*(1 + sqrt(3)) = 3*(1 + sqrt(3)) = 3 + 3sqrt(3)So numerator: 12 + 6sqrt(3) - (3 + 3sqrt(3)) = 9 + 3sqrt(3)Denominator: x + y + z = 1 + 1 + 1 + sqrt(3) = 3 + sqrt(3)Therefore, (frac{9 + 3sqrt(3)}{3 + sqrt(3)})Factor numerator: 3*(3 + sqrt(3))Denominator: 3 + sqrt(3)Therefore, (frac{3*(3 + sqrt(3))}{3 + sqrt(3)} = 3)Ah! There we go. Earlier, I must have miscalculated z^3. So actually, z^3 is 10 + 6sqrt(3), not 10 + 12sqrt(3). Therefore, the numerator is 12 + 6sqrt(3) - 3 - 3sqrt(3) = 9 + 3sqrt(3), which factors into 3*(3 + sqrt(3)), canceling with the denominator to give 3. So the expression does equal 3, which aligns with our previous conclusion. So my earlier mistake was in computing z^3 incorrectly. Phew! That was a close call.Therefore, the test case actually does result in 3, confirming our answer. Therefore, the initial conclusion was correct. The answer is 3.Another test case: Let me try two variables equal, say x = y = 1, z = ?From the equation (x + y + z)^2 = 3 + 3(xy + yz + zx)Let x = y = 1, then (2 + z)^2 = 3 + 3(1 + z + z)So (2 + z)^2 = 3 + 3(1 + 2z) = 3 + 3 + 6z = 6 + 6zExpanding left side: z^2 + 4z + 4 = 6 + 6zBring all terms to left: z^2 - 2z - 2 = 0Solutions: z = [2 ± sqrt(4 + 8)] / 2 = [2 ± sqrt(12)] / 2 = 1 ± sqrt(3). So same as before.Compute the expression for z = 1 + sqrt(3):As above, the expression is 3. Similarly, for z = 1 - sqrt(3):Let me check. z = 1 - sqrt(3). Then x + y + z = 1 + 1 + 1 - sqrt(3) = 3 - sqrt(3)Numerator: x^3 + y^3 + z^3 - 3xyz = 1 + 1 + z^3 - 3*1*1*zCompute z^3: (1 - sqrt(3))^3 = 1 - 3sqrt(3) + 3*(sqrt(3))^2 - (sqrt(3))^3= 1 - 3sqrt(3) + 9 - 3*3*sqrt(3)Wait, similar to before:= 1 - 3sqrt(3) + 9 - 9sqrt(3)= 10 - 12sqrt(3)Therefore, x^3 + y^3 + z^3 = 1 + 1 + 10 - 12sqrt(3) = 12 - 12sqrt(3)3xyz = 3*1*1*(1 - sqrt(3)) = 3 - 3sqrt(3)Numerator: 12 - 12sqrt(3) - 3 + 3sqrt(3) = 9 - 9sqrt(3)Denominator: 3 - sqrt(3)Expression: (9 - 9sqrt(3))/(3 - sqrt(3)) = 9*(1 - sqrt(3))/(3 - sqrt(3))Multiply numerator and denominator by (3 + sqrt(3)):Numerator: 9*(1 - sqrt(3))(3 + sqrt(3)) = 9*(3 + sqrt(3) - 3sqrt(3) - 3) = 9*(-2sqrt(3)) = -18sqrt(3)Denominator: (3 - sqrt(3))(3 + sqrt(3)) = 9 - 3 = 6So the expression is (-18sqrt(3))/6 = -3sqrt(3). Wait, but according to our formula, the expression should be 3. What's going on here?Wait, no. Wait, z = 1 - sqrt(3). Let's compute (x^2 + y^2 + z^2 - xy - yz - zx):x^2 + y^2 + z^2 = 1 + 1 + (1 - sqrt(3))^2 = 1 + 1 + 1 - 2sqrt(3) + 3 = 6 - 2sqrt(3)xy + yz + zx = 1 + (1*(1 - sqrt(3)) + 1*(1 - sqrt(3))) = 1 + (1 - sqrt(3) + 1 - sqrt(3)) = 1 + 2 - 2sqrt(3) = 3 - 2sqrt(3)Therefore, (x^2 + y^2 + z^2 - xy - yz - zx = (6 - 2sqrt(3)) - (3 - 2sqrt(3)) = 3). So according to this, the expression is 3, but when we compute the original fraction, we get -3sqrt(3). Hmm, so what's the discrepancy here?Wait, let's recalculate the original expression for z = 1 - sqrt(3):x = 1, y = 1, z = 1 - sqrt(3)Compute numerator: x^3 + y^3 + z^3 - 3xyz.x^3 = 1, y^3 = 1, z^3 = (1 - sqrt(3))^3.As above, z^3 = 10 - 12sqrt(3). Therefore, x^3 + y^3 + z^3 = 1 + 1 + 10 - 12sqrt(3) = 12 - 12sqrt(3).3xyz = 3*1*1*(1 - sqrt(3)) = 3 - 3sqrt(3).Thus, numerator = 12 - 12sqrt(3) - 3 + 3sqrt(3) = 9 - 9sqrt(3).Denominator = x + y + z = 3 - sqrt(3).So, (frac{9 - 9sqrt(3)}{3 - sqrt(3)}).Factor numerator: 9(1 - sqrt(3)).Denominator: 3 - sqrt(3).Therefore, (frac{9(1 - sqrt(3))}{3 - sqrt(3)} = frac{9(1 - sqrt(3))}{3 - sqrt(3)}).Multiply numerator and denominator by (3 + sqrt(3)):Numerator: 9(1 - sqrt(3))(3 + sqrt(3)) = 9[3 + sqrt(3) - 3sqrt(3) - 3] = 9[-2sqrt(3)] = -18sqrt(3).Denominator: 9 - 3 = 6.Thus, the expression is (-18sqrt(3)/6 = -3sqrt(3)). But according to our formula, it should be 3. This is a contradiction.But wait, hold on, we computed (x^2 + y^2 + z^2 - xy - yz - zx = 3). So according to the identity, the expression should be 3. But the actual calculation gives -3sqrt(3). Which one is correct?Wait, this suggests that there might be an error in the identity or in the given conditions. Wait, but the identity (x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)) is a standard algebraic identity. So that can't be wrong. So perhaps the mistake is in the test case.But the test case was derived from the given equation. Wait, if z = 1 - sqrt(3), then x + y + z = 3 - sqrt(3). Then, (x + y + z)^2 = (3 - sqrt(3))^2 = 9 - 6sqrt(3) + 3 = 12 - 6sqrt(3).On the other side, 3 + 3(xy + yz + zx). Compute xy + yz + zx = 1 + (1*(1 - sqrt(3)) + 1*(1 - sqrt(3))) = 1 + 2*(1 - sqrt(3)) = 1 + 2 - 2sqrt(3) = 3 - 2sqrt(3).Multiply by 3: 9 - 6sqrt(3). Add 3: 12 - 6sqrt(3). Which matches. So z = 1 - sqrt(3) is a valid solution. Therefore, this should satisfy the expression equals 3. But in reality, it's giving -3sqrt(3). Which is a problem.Wait, but according to the identity, since x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx), then dividing by x + y + z should give x^2 + y^2 + z^2 - xy - yz - zx, which we calculated as 3. Therefore, the expression should be 3. However, when we compute it numerically, we get -3sqrt(3). This suggests there's a mistake in the calculation.Wait, but x^2 + y^2 + z^2 - xy - yz - zx was calculated as 3, which is correct. So why is the expression evaluating to -3sqrt(3)? Let me check the calculation step by step.x = 1, y = 1, z = 1 - sqrt(3). So:x^3 = 1, y^3 = 1, z^3 = (1 - sqrt(3))^3.Compute (1 - sqrt(3))^3:= 1^3 - 3*1^2*sqrt(3) + 3*1*(sqrt(3))^2 - (sqrt(3))^3= 1 - 3sqrt(3) + 9 - 3*3*sqrt(3)Wait, hold on. (sqrt(3))^3 is (sqrt(3))^2 * sqrt(3) = 3*sqrt(3). Therefore:= 1 - 3sqrt(3) + 9 - 3*3*sqrt(3)= 1 - 3sqrt(3) + 9 - 9sqrt(3)= 10 - 12sqrt(3). Correct.So x^3 + y^3 + z^3 = 1 + 1 + 10 - 12sqrt(3) = 12 - 12sqrt(3). Correct.3xyz = 3*1*1*(1 - sqrt(3)) = 3*(1 - sqrt(3)) = 3 - 3sqrt(3). Correct.So numerator: 12 - 12sqrt(3) - 3 + 3sqrt(3) = 9 - 9sqrt(3). Correct.Denominator: x + y + z = 3 - sqrt(3). Correct.So (frac{9 - 9sqrt(3)}{3 - sqrt(3)}). Let's compute this again:Factor numerator: 9(1 - sqrt(3)).Denominator: 3 - sqrt(3).Multiply numerator and denominator by (3 + sqrt(3)):Numerator: 9(1 - sqrt(3))(3 + sqrt(3)) = 9[(1)(3) + 1*sqrt(3) - 3sqrt(3) - (sqrt(3))^2]= 9[3 + sqrt(3) - 3sqrt(3) - 3]= 9[-2sqrt(3)] = -18sqrt(3)Denominator: (3 - sqrt(3))(3 + sqrt(3)) = 9 - 3 = 6.Thus, result: -18sqrt(3)/6 = -3sqrt(3). But according to our identity, it should be 3. This is a contradiction. How?Wait, but according to the identity, (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z} = x^2 + y^2 + z^2 - xy - yz - zx = 3). But here, we get -3sqrt(3). So something is wrong.Wait, unless there's a miscalculation here. Wait, x^2 + y^2 + z^2 - xy - yz - zx was calculated as 3. Let's verify that.x = 1, y = 1, z = 1 - sqrt(3):x^2 = 1, y^2 = 1, z^2 = (1 - sqrt(3))^2 = 1 - 2sqrt(3) + 3 = 4 - 2sqrt(3)Thus, x^2 + y^2 + z^2 = 1 + 1 + 4 - 2sqrt(3) = 6 - 2sqrt(3)xy = 1*1 = 1yz = 1*(1 - sqrt(3)) = 1 - sqrt(3)zx = 1*(1 - sqrt(3)) = 1 - sqrt(3)Thus, xy + yz + zx = 1 + (1 - sqrt(3)) + (1 - sqrt(3)) = 3 - 2sqrt(3)Therefore, x^2 + y^2 + z^2 - xy - yz - zx = (6 - 2sqrt(3)) - (3 - 2sqrt(3)) = 3. So yes, that's correct.But according to the direct computation of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}), we get -3sqrt(3). This is a problem. So where is the mistake?Wait, no, there's no mistake. The key is that in the identity, (x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)). This identity holds regardless of the value of x + y + z, as long as the operations are defined. However, if x + y + z = 0, the identity still holds but division by zero would be undefined. However, in our case, x + y + z ≠ 0, so division is allowed.But here, even though the identity says the expression equals 3, when we compute it directly, it's giving -3sqrt(3). How is this possible?Wait, let me check once more for z = 1 - sqrt(3):x + y + z = 3 - sqrt(3)x^2 + y^2 + z^2 - xy - yz - zx = 3Therefore, according to the identity, (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z} = 3). Therefore, x^3 + y^3 + z^3 - 3xyz = 3*(x + y + z) = 3*(3 - sqrt(3)) = 9 - 3sqrt(3)But according to the direct computation, x^3 + y^3 + z^3 - 3xyz = 9 - 9sqrt(3). So 9 - 9sqrt(3) vs 9 - 3sqrt(3). Therefore, contradiction.This means that either the identity is wrong, which it isn't, or there's a miscalculation in the numerator.Wait, the identity is correct. Therefore, my calculation must be wrong.Wait, compute x^3 + y^3 + z^3 - 3xyz again:x^3 = 1y^3 = 1z^3 = (1 - sqrt(3))^3 = 1 - 3sqrt(3) + 3*(sqrt(3))^2 - (sqrt(3))^3Wait, (sqrt(3))^2 = 3, so 3*(sqrt(3))^2 = 3*3 = 9(sqrt(3))^3 = 3*sqrt(3)So, z^3 = 1 - 3sqrt(3) + 9 - 3*sqrt(3) = 10 - 6sqrt(3)Therefore, x^3 + y^3 + z^3 = 1 + 1 + 10 - 6sqrt(3) = 12 - 6sqrt(3)3xyz = 3*1*1*(1 - sqrt(3)) = 3 - 3sqrt(3)Therefore, numerator: 12 - 6sqrt(3) - 3 + 3sqrt(3) = 9 - 3sqrt(3)Denominator: 3 - sqrt(3)Thus, the expression is (frac{9 - 3sqrt(3)}{3 - sqrt(3)}). Factor numerator: 3*(3 - sqrt(3))/ (3 - sqrt(3)) = 3. Ah! There we go. I had miscalculated z^3 earlier. Instead of 10 - 12sqrt(3), it's 10 - 6sqrt(3). So the numerator is 12 - 6sqrt(3) - 3 + 3sqrt(3) = 9 - 3sqrt(3), which divided by denominator 3 - sqrt(3) gives 3. Therefore, the previous error was due to an incorrect computation of z^3. Therefore, the test case does indeed evaluate to 3, which matches our conclusion.Phew, that was a bit of a rollercoaster. So the key takeaway here is that the identity holds, and as long as the given equation is satisfied, the expression will evaluate to 3. Even in edge cases, as long as the given equation holds, the expression simplifies to 3. Therefore, the answer is indeed 3.Final Answerboxed{3}</think>Given the problem, we need to find the value of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}) under the conditions (x + y + z neq 0) and ((x + y + z)^2 = 3 + 3(xy + yz + zx)).First, we use the identity:[x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)]Dividing both sides by (x + y + z), we get:[frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z} = x^2 + y^2 + z^2 - xy - yz - zx]Next, we use the given equation ((x + y + z)^2 = 3 + 3(xy + yz + zx)). Expanding the left-hand side, we have:[(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)]Equating this to the right-hand side of the given equation:[x^2 + y^2 + z^2 + 2(xy + yz + zx) = 3 + 3(xy + yz + zx)]Subtracting (3(xy + yz + zx)) from both sides, we get:[x^2 + y^2 + z^2 - xy - yz - zx = 3]Thus, the value of (frac{x^3 + y^3 + z^3 - 3xyz}{x + y + z}) is:[boxed{3}]]

🔑:3

🧠:[<think>Okay, so I have this problem here with two parts, a) and b). Let me start by understanding what's being asked in part a). We have a continuous function f from the non-negative real numbers to the positive real numbers. So f: R≥0 → R>0, and it's continuous. We need to show that there exists a natural number n₀ and a sequence of positive real numbers (xₙ)ₙ>n₀ such that for all n > n₀, the equation n times the integral from 0 to xₙ of f(t) dt equals 1. So, n ∫₀^{xₙ} f(t) dt = 1. Alright, so the first thought is that since f is continuous and positive, the integral of f from 0 to x is going to be a strictly increasing function. Because f is positive, integrating over a larger interval should give a larger value. Since f is continuous, the integral function F(x) = ∫₀ˣ f(t) dt is differentiable, and its derivative is f(x), which is positive. Therefore, F(x) is strictly increasing and continuous (since differentiable functions are continuous). Since F is strictly increasing and continuous, it should be invertible on its image. The image of F is going to be from F(0) = 0 up to lim_{x→∞} F(x). But since f is positive, as x approaches infinity, the integral ∫₀ˣ f(t) dt might go to infinity if f doesn't decay too quickly. However, the problem doesn't specify whether f is integrable over [0, ∞) or not. Hmm, so wait. If f is integrable over [0, ∞), then the integral would converge to some finite limit as x approaches infinity. If it's not integrable, then the integral would go to infinity. But the problem states that f: R≥0 → R>0 is continuous. So f is always positive, but we don't know anything else about its behavior at infinity. But for the integral ∫₀ˣ f(t) dt, depending on f, it could be that it tends to infinity as x→infty or not. For example, if f(t) = 1/(1 + t²), then the integral from 0 to x would be arctan(x), which approaches π/2 as x→∞, so it's integrable. But if f(t) = 1, then the integral is x, which goes to infinity. But the problem doesn't specify, so we have to consider both possibilities. However, the equation n ∫₀^{xₙ} f(t) dt = 1 implies that ∫₀^{xₙ} f(t) dt = 1/n. So as n increases, 1/n decreases towards 0. Therefore, we need to find xₙ such that the integral up to xₙ is 1/n. But since F(x) = ∫₀ˣ f(t) dt is strictly increasing, if F is surjective onto [0, ∞), then for every 1/n > 0, there exists a unique xₙ such that F(xₙ) = 1/n. However, if F is bounded, meaning that the integral converges to some finite limit L as x→infty, then for 1/n < L, we can find xₙ such that F(xₙ) = 1/n. But if n is such that 1/n > L, then there is no solution. Therefore, we need to have 1/n < L, i.e., n > 1/L. But since L might be infinity or finite. If L is infinity (i.e., the integral diverges), then for every n, 1/n is positive, and since F(x) approaches infinity as x→infty, then for each n, there exists xₙ such that F(xₙ) = 1/n. Wait, but 1/n is going to zero, so as n increases, xₙ should decrease towards zero? Wait, but if F(x) is increasing, then as x increases, F(x) increases. Therefore, to get F(x) = 1/n, which is decreasing in n, xₙ must be decreasing as n increases. So as n increases, 1/n decreases, so xₙ would be the value such that the integral up to xₙ is 1/n, so xₙ must get smaller as n increases. But if F(x) is bounded, say by L, then once 1/n < L, which would require n > 1/L, but since L is finite, 1/L is a positive number. Therefore, for n > 1/L, we can define xₙ such that F(xₙ) = 1/n. However, if L is infinite, then for every n, no matter how large, 1/n is positive, so there exists xₙ such that F(xₙ) = 1/n. But in the problem statement, they just say "there exists a natural number n₀ and a sequence (xₙ) for n > n₀". So if the integral is bounded, then n₀ is the smallest integer greater than 1/L, but if the integral is unbounded, then n₀ can be 1, right? Because even for n=1, x₁ would exist. Wait, but the problem says "n₀ < ∀ n ∈ N", which I think is a typo and should be "for all n > n₀". So, in any case, whether F is bounded or not, if F is surjective onto [0, L) where L is possibly infinity, then as long as 1/n is in the image of F, which is [0, L), then xₙ exists. Therefore, if L is finite, then xₙ exists only when 1/n < L, which is equivalent to n > 1/L. So n₀ can be taken as the smallest integer greater than 1/L. If L is infinite, then n₀ can be 1. So in either case, such n₀ exists. Therefore, there exists n₀ such that for all n > n₀, xₙ exists. Hence, the sequence (xₙ) can be defined for n > n₀. Therefore, part a) is shown by considering the inverse of the integral function F(x). Since F is continuous and strictly increasing, it has an inverse function on its image. So xₙ = F^{-1}(1/n) for n > n₀, where n₀ is 1 if L is infinity, else n₀ is the smallest integer greater than 1/L. Okay, that makes sense. So part a) is essentially about the existence of an inverse function for the integral, which is guaranteed by the continuity and monotonicity of F(x). Moving on to part b). We need to prove that the sequence (n xₙ)_{n > n₀} is convergent and find its limit. So, we have xₙ = F^{-1}(1/n). We need to analyze the behavior of n xₙ as n approaches infinity. First, let's note that as n approaches infinity, 1/n approaches 0, so xₙ = F^{-1}(1/n) approaches F^{-1}(0). But F(x) = ∫₀ˣ f(t) dt, so F(0) = 0. Therefore, as 1/n approaches 0, xₙ approaches 0. Therefore, xₙ is approaching zero, but we are multiplying it by n, which is going to infinity. So it's a 0 * ∞ indeterminate form. We need to find the limit of n xₙ as n → ∞. To find this limit, let's recall that F(xₙ) = 1/n. So ∫₀^{xₙ} f(t) dt = 1/n. Assuming that f is continuous at 0, which it is, since f is continuous on R≥0, so in particular at 0. Therefore, near 0, f(t) is approximately f(0). So the integral ∫₀^{xₙ} f(t) dt ≈ f(0) xₙ when xₙ is small. So if xₙ is small, which it is as n becomes large, we can approximate the integral as f(0) xₙ. Then we have f(0) xₙ ≈ 1/n, so xₙ ≈ 1/(n f(0)). Therefore, n xₙ ≈ 1/f(0). So the limit should be 1/f(0). But to make this rigorous, we need to use some calculus. Perhaps using the Mean Value Theorem for integrals or L’Hospital’s Rule. Let me think. Let's denote F(x) = ∫₀ˣ f(t) dt. Then F(xₙ) = 1/n. We can write F(xₙ) = xₙ f(cₙ) for some cₙ in (0, xₙ) by the Mean Value Theorem for integrals. So 1/n = F(xₙ) = xₙ f(cₙ). Therefore, n xₙ = 1/f(cₙ). Now, as n → ∞, xₙ → 0, so cₙ is between 0 and xₙ, so cₙ also approaches 0. Since f is continuous at 0, f(cₙ) → f(0). Therefore, 1/f(cₙ) → 1/f(0). Hence, the limit of n xₙ is 1/f(0). Therefore, the sequence (n xₙ) converges to 1/f(0). But let me check if there are any gaps here. The Mean Value Theorem for integrals states that if f is continuous on [a, b], then there exists c in [a, b] such that ∫ₐᵇ f(t) dt = f(c)(b - a). So in our case, since f is continuous on [0, xₙ], we have ∫₀^{xₙ} f(t) dt = f(cₙ) xₙ for some cₙ in (0, xₙ). Therefore, indeed, 1/n = f(cₙ) xₙ, so n xₙ = 1/f(cₙ). As n → ∞, xₙ → 0, so cₙ → 0, and by continuity of f, f(cₙ) → f(0). Therefore, lim_{n→∞} n xₙ = 1/f(0). Therefore, part b) is proved, and the limit is 1/f(0). But maybe there's another way to see this using L’Hospital’s Rule. Let me see. Suppose we set y = 1/n, so as n → ∞, y → 0. Then xₙ = F^{-1}(y). So we can consider the inverse function. Let’s denote G(y) = F^{-1}(y), so that xₙ = G(1/n). Then we can write the limit as lim_{y→0} G(y)/y, since n xₙ = n G(1/n) = G(1/n)/(1/n). Let’s set y = 1/n, so as n → ∞, y → 0, and the limit becomes lim_{y→0} G(y)/y. But since G(y) is the inverse function of F, we have F(G(y)) = y. Let's differentiate both sides with respect to y. Then F’(G(y)) * G’(y) = 1. But F’(x) = f(x), so f(G(y)) * G’(y) = 1. Therefore, G’(y) = 1/f(G(y)). So, as y → 0, G(y) → 0, so f(G(y)) → f(0). Therefore, G’(y) → 1/f(0). Therefore, by the definition of the derivative, G(y) ≈ G(0) + G’(0) y as y → 0. But G(0) = 0, since F(0) = 0. Therefore, G(y) ≈ (1/f(0)) y. Therefore, G(y)/y ≈ 1/f(0), so lim_{y→0} G(y)/y = 1/f(0). Therefore, lim_{n→∞} n xₙ = 1/f(0). So both methods confirm that the limit is 1/f(0). Therefore, part b) is solved. But let me check if there are any conditions where this might not hold. For example, if f(0) = 0, but in the problem statement, f maps to R>0, so f(t) is positive for all t ≥ 0. Therefore, f(0) > 0, so 1/f(0) is defined and positive. Therefore, all steps are valid, and the conclusion is that the sequence (n xₙ) converges to 1/f(0).Final Answera) boxed{text{Such } n_0 text{ and sequence } (x_n) text{ exist}} b) boxed{dfrac{1}{f(0)}}</think>Given a continuous function ( f: mathbb{R}_{ge 0} longrightarrow mathbb{R}_{>0} ), we need to address the following parts: Part a)To show there exists a natural number ( n_0 ) and a sequence ( (x_n)_{n > n_0} ) such that ( n int_0^{x_n} f(t) , dt = 1 ) for all ( n > n_0 ):1. Integral Function Properties: Define ( F(x) = int_0^x f(t) , dt ). Since ( f ) is continuous and positive, ( F(x) ) is strictly increasing, continuous, and differentiable with ( F'(x) = f(x) ).2. Surjectivity: If ( F(x) ) is unbounded (i.e., ( lim_{x to infty} F(x) = infty )), then for every ( 1/n > 0 ), there exists ( x_n ) such that ( F(x_n) = 1/n ). If ( F(x) ) is bounded by some ( L ), then ( x_n ) exists for ( 1/n < L ), i.e., ( n > 1/L ).3. Existence of ( n_0 ): Choose ( n_0 ) such that ( 1/n_0 < L ) if ( F(x) ) is bounded, otherwise ( n_0 = 1 ). Thus, ( x_n = F^{-1}(1/n) ) exists for ( n > n_0 ). Part b)To prove ( (nx_n)_{n > n_0} ) converges and find its limit:1. Mean Value Theorem: For each ( x_n ), there exists ( c_n in (0, x_n) ) such that ( int_0^{x_n} f(t) , dt = f(c_n) x_n ). Hence, ( 1/n = f(c_n) x_n ) leading to ( nx_n = 1/f(c_n) ).2. Continuity at 0: As ( n to infty ), ( x_n to 0 ), so ( c_n to 0 ). By continuity of ( f ), ( f(c_n) to f(0) ).3. Limit Calculation: Thus, ( lim_{n to infty} nx_n = lim_{n to infty} 1/f(c_n) = 1/f(0) ). Final Answersa) boxed{text{Such } n_0 text{ and sequence } (x_n) text{ exist}} b) boxed{dfrac{1}{f(0)}},<think>Okay, let's tackle part a) first. So, we have a continuous function f from the non-negative real numbers to the positive real numbers. We need to show that there exists a natural number n₀ and a sequence of positive real numbers (xₙ) for n > n₀ such that n times the integral from 0 to xₙ of f(t)dt equals 1. Hmm. First, let me parse this. The function f is continuous and positive, right? So f(t) > 0 for all t ≥ 0. Also, since f is continuous, the integral of f from 0 to any x will be a continuous function in x. Let's define F(x) = ∫₀ˣ f(t)dt. Then F is an antiderivative of f, and since f is positive, F is strictly increasing. Because f(t) > 0, F(x) is strictly increasing. Also, as x approaches infinity, F(x) will tend to infinity because integrating a positive function over an ever-increasing interval. Similarly, as x approaches 0 from the right, F(x) approaches 0. So F: [0, ∞) → [0, ∞) is a strictly increasing, continuous bijection. Therefore, F is invertible, and its inverse function F⁻¹: [0, ∞) → [0, ∞) is also strictly increasing and continuous.Now, the equation given is n * F(xₙ) = 1. So, solving for xₙ, we get xₙ = F⁻¹(1/n). But wait, F maps [0, ∞) to [0, ∞), so for any n such that 1/n is in the range of F, which is [0, ∞). Since 1/n is a positive real number for all n ∈ ℕ, and F is surjective onto [0, ∞), there exists an xₙ for each n such that F(xₙ) = 1/n. Therefore, xₙ = F⁻¹(1/n). But the problem states that we need to show there exists a natural number n₀ and a sequence (xₙ) for n > n₀. Wait, why do we need an n₀? If F is defined for all x ≥ 0 and F(x) can take any positive value, then even for n = 1, F(x₁) = 1, which would require x₁ = F⁻¹(1). But maybe the issue is that for small n, 1/n might be too large? Wait, no, because as x increases, F(x) increases without bound. So for any n, 1/n is a positive real number, so F⁻¹(1/n) exists. So xₙ exists for all n ∈ ℕ. But the problem says "there exists a natural number n₀ and a sequence of positive real numbers (xₙ)_{n > n₀}" that satisfy the equation. Maybe they just want us to note that for sufficiently large n, xₙ will be small? Wait, but xₙ is defined for all n. Unless there's a problem when n is too small. Wait, let's check with an example. Suppose f(t) = 1, so F(x) = x. Then the equation is n * xₙ = 1, so xₙ = 1/n, which is defined for all n ≥ 1. So in this case, n₀ could be 0, but the problem wants n₀ ∈ ℕ. So maybe for some functions f(t), F(x) might not reach 1/n for small n? Wait, but f(t) is positive and continuous, so F(x) = ∫₀ˣ f(t)dt is strictly increasing from 0 to ∞. Therefore, for any positive y, there exists a unique x such that F(x) = y. So 1/n is positive for all n ∈ ℕ, so xₙ exists for all n ≥ 1. Therefore, the n₀ is just 1, and the sequence is defined for n > 1, which is all natural numbers. But the problem says n₀ < ∀n ∈ ℕ, so n must be greater than n₀. So maybe n₀ is 0? But n must be natural numbers. Wait, perhaps the problem is just requiring that we can find such a sequence for all sufficiently large n, but given that xₙ exists for all n, maybe the n₀ is redundant here. Maybe the problem is misstated, or perhaps there's a different consideration. Wait, maybe if f(t) is integrable over [0, ∞)? But the problem says f is defined on ℝ≥0 → ℝ>0, continuous, but doesn't say it's integrable. Wait, if f is not integrable, then F(x) tends to infinity as x tends to infinity, so F is surjective. So even if f is integrable, F(x) would approach some finite limit as x → ∞, so in that case, F would map [0, ∞) to [0, L), where L is the integral of f over [0, ∞). Then, if 1/n < L, then xₙ exists. So if f is integrable, then for n > 1/L, 1/n < L, so xₙ exists. Therefore, in this case, n₀ could be the smallest integer greater than 1/L. But if f is not integrable, then L is infinity, so n₀ can be 1. Therefore, depending on whether f is integrable or not, we might need to take n₀ such that 1/n < L for n > n₀. But the problem statement just says f is continuous and maps to positive reals. So f may or may not be integrable. Therefore, to cover both cases, we need to define n₀ as 1 if f is not integrable (i.e., L = ∞), or n₀ = ⎡1/L⎤ if f is integrable. Therefore, in either case, there exists an n₀ such that for all n > n₀, 1/n < L, hence xₙ exists. Wait, but if f is integrable, then L = ∫₀^∞ f(t)dt is finite, so 1/n < L requires n > 1/L. Therefore, n₀ can be taken as the ceiling of 1/L. But if f is not integrable, then L is infinity, so 1/n is always less than L for any n, so n₀ can be 1. Therefore, the problem is likely expecting us to consider that if f is integrable, then there exists some n₀ such that for n > n₀, 1/n is less than the total integral, allowing xₙ to exist. If f is not integrable, then xₙ exists for all n. But since the problem statement says "there exists a natural number n₀" without specifying anything else, we need to argue that regardless of f's integrability, such an n₀ exists. Therefore, the answer would involve defining n₀ as 1 if the integral of f is infinite, or n₀ = ceiling(1/L) + 1 if the integral is finite. But since the problem just states f is continuous and positive, we have to handle both cases. But maybe there's a simpler way. Since F(x) is continuous and strictly increasing from 0 to L (where L could be infinity), for each n, 1/n is in [0, L) if L is infinity, so for all n, 1/n is in [0, L). If L is finite, then for n > 1/L, 1/n < L. Therefore, in either case, there exists an n₀ such that for all n > n₀, 1/n < L. Therefore, xₙ exists for n > n₀. Therefore, the existence of xₙ is guaranteed for all sufficiently large n, i.e., n > n₀ where n₀ is such that 1/n₀ < L if L is finite, otherwise n₀ = 1. Hence, the required sequence exists. Therefore, the answer for part a) is that since F(x) is continuous, strictly increasing, and surjective onto [0, L) where L is the integral of f over [0, ∞), then for n sufficiently large such that 1/n < L (which is always possible since L > 0 and 1/n approaches 0), there exists xₙ = F⁻¹(1/n). Hence, such an n₀ exists, and the sequence (xₙ) can be defined for n > n₀.Moving on to part b). We need to prove that the sequence (n xₙ) is convergent and find its limit. Given that xₙ = F⁻¹(1/n), so n xₙ = n F⁻¹(1/n). Let's analyze the behavior as n approaches infinity. Since xₙ is F⁻¹(1/n), and as n increases, 1/n approaches 0, so xₙ approaches F⁻¹(0) = 0. Therefore, xₙ tends to 0 as n → ∞. So we have a sequence that tends to 0, multiplied by n. The question is whether n xₙ converges. To find the limit of n xₙ as n → ∞, we can consider substituting variables. Let’s set yₙ = xₙ. Then we have F(yₙ) = 1/n. Let’s write this as ∫₀^{yₙ} f(t) dt = 1/n. We need to relate yₙ to n. If f is continuous at 0, then near t = 0, f(t) ≈ f(0). So the integral ∫₀^{yₙ} f(t) dt ≈ f(0) yₙ. Therefore, if this approximation is valid, then f(0) yₙ ≈ 1/n, so yₙ ≈ 1/(n f(0)), and thus n yₙ ≈ 1/f(0). Therefore, the limit would be 1/f(0). But to make this rigorous, we need to use the Mean Value Theorem for integrals or Taylor expansion. Since f is continuous at 0, for any ε > 0, there exists a δ > 0 such that for |t| < δ, |f(t) - f(0)| < ε. Then, for yₙ sufficiently small (which it is as n → ∞), we can write:∫₀^{yₙ} f(t) dt = ∫₀^{yₙ} [f(0) + (f(t) - f(0))] dt = f(0) yₙ + ∫₀^{yₙ} (f(t) - f(0)) dt.The second integral can be bounded by:|∫₀^{yₙ} (f(t) - f(0)) dt| ≤ ∫₀^{yₙ} |f(t) - f(0)| dt ≤ ε yₙ.Therefore,∫₀^{yₙ} f(t) dt = f(0) yₙ + O(ε yₙ).But we know that ∫₀^{yₙ} f(t) dt = 1/n. Therefore,1/n = f(0) yₙ + O(ε yₙ).Dividing both sides by f(0):yₙ = 1/(n f(0)) + O(ε yₙ / f(0)).But since yₙ ~ 1/(n f(0)), the error term is O(ε / (n f(0)) ). Therefore,yₙ = 1/(n f(0)) + o(1/n).Multiplying both sides by n:n yₙ = 1/f(0) + o(1).Hence, as n → ∞, n yₙ → 1/f(0). Therefore, the limit is 1/f(0). But we need to ensure that this holds even if f isn't just continuous at 0 but continuous on [0, ∞). Since f is continuous at 0, the approximation f(t) ≈ f(0) near t=0 is valid, and the error term becomes negligible as yₙ → 0. Therefore, the limit of n xₙ is 1/f(0).Wait, but let's check with an example. Suppose f(t) = 1, then F(x) = x, so xₙ = 1/n, so n xₙ = 1, which is 1/f(0) = 1/1 = 1. That checks out. If f(t) = 2, then xₙ = 1/(2n), so n xₙ = 1/2 = 1/f(0). Another example: f(t) = e^t. Then F(x) = e^x - 1. So setting e^{xₙ} - 1 = 1/n → e^{xₙ} = 1 + 1/n → xₙ = ln(1 + 1/n). Then n xₙ = n ln(1 + 1/n). As n → ∞, ln(1 + 1/n) ≈ 1/n - 1/(2n²) + ..., so n ln(1 + 1/n) ≈ 1 - 1/(2n) + ... → 1. But f(0) = e^0 = 1, so 1/f(0) = 1. So the limit is 1, which matches. Another example: f(t) = 1 + t. Then F(x) = x + x²/2. Set x + x²/2 = 1/n. For large n, x is small, so x²/2 is negligible. So x ≈ 1/n. But let's do it properly. Let xₙ satisfy xₙ + xₙ²/2 = 1/n. Let's write xₙ = 1/n - xₙ²/2. If xₙ ≈ 1/n, then xₙ² ≈ 1/n², so xₙ ≈ 1/n - 1/(2n²). Then n xₙ ≈ 1 - 1/(2n) → 1. But f(0) = 1, so again the limit is 1/f(0) = 1. Wait, but what if f(0) is different? Let's take f(t) = 2 + t. Then f(0) = 2. Then F(x) = 2x + x²/2. Set 2x + x²/2 = 1/n. For large n, x is small, so x ≈ 1/(2n). Then n x ≈ 1/2. But 1/f(0) is 1/2, which matches. Let's compute it. xₙ = [sqrt(4 + 4*(1/n)) - 2]/1 = sqrt(4 + 4/n) - 2. Using the expansion sqrt(4 + 4/n) = 2 sqrt(1 + 1/n) ≈ 2(1 + 1/(2n) - 1/(8n²) + ...). Therefore, sqrt(4 + 4/n) - 2 ≈ 2(1 + 1/(2n)) - 2 = 1/n. So xₙ ≈ 1/n, but wait, that doesn't match the previous. Wait, let me solve 2x + x²/2 = 1/n. Let's rearrange:x² + 4x - 2/n = 0.Solutions: x = [-4 ± sqrt(16 + 8/n)]/2. Since x is positive, take the positive root:x = [-4 + sqrt(16 + 8/n)]/2 = [sqrt(16 + 8/n) - 4]/2.Multiply numerator and denominator by sqrt(16 +8/n) +4:x = [ (16 +8/n -16) ] / [ 2 (sqrt(16 +8/n) +4) ] = (8/n) / [2 (sqrt(16 +8/n) +4)] = (4/n) / (sqrt(16 +8/n) +4).For large n, sqrt(16 +8/n) ≈ 4 + (8/n)/(2*4) = 4 + 1/n. Therefore,x ≈ (4/n) / (4 +1/n +4) = (4/n)/(8 +1/n) ≈ (4/n)/8 = 1/(2n). Therefore, xₙ ≈1/(2n), so n xₙ ≈1/2 =1/f(0). Correct.Therefore, the general approach is that as n→infty, xₙ→0, and the integral ∫₀^{xₙ} f(t)dt ≈ f(0)xₙ. Therefore, setting this ≈1/n gives xₙ≈1/(n f(0)), so n xₙ ≈1/f(0). To make this rigorous, we can use the definition of the derivative. Since F(x) =∫₀ˣ f(t)dt, then F'(0) =f(0). Therefore, the inverse function theorem applies to F⁻¹ near 0. The derivative of F⁻¹ at 0 is 1/F'(F⁻¹(0)) =1/f(0). But F⁻¹(y) behaves like y/f(0) as y→0. Therefore, F⁻¹(1/n) ~ (1/n)/f(0), so n F⁻¹(1/n) ~1/f(0). Therefore, the limit is 1/f(0).Alternatively, using the substitution y = n xₙ. Let’s set yₙ =n xₙ. Then xₙ= yₙ /n. The equation becomes:n ∫₀^{yₙ/n} f(t)dt =1.Let’s change variable: let t = s/n. Then dt=ds/n, and the integral becomes:n ∫₀^{yₙ} f(s/n) * (ds/n) = ∫₀^{yₙ} f(s/n) ds =1.So we have ∫₀^{yₙ} f(s/n) ds =1.As n→infty, f(s/n) →f(0) uniformly on compact intervals (since f is continuous at 0). Therefore, ∫₀^{yₙ} f(s/n) ds ≈ ∫₀^{yₙ} f(0) ds =f(0) yₙ. Therefore, f(0) yₙ ≈1. Hence, yₙ≈1/f(0). To make this precise, we can use the Dominated Convergence Theorem. Since f is continuous at 0, for any ε>0, there exists N such that for n≥N, |f(s/n)-f(0)|<ε for s in [0, yₙ]. Since yₙ is supposed to converge to 1/f(0), we can bound yₙ in some interval [0, M]. Therefore, ∫₀^{yₙ} f(s/n) ds converges to ∫₀^{yₙ} f(0) ds =f(0) yₙ. Since the integral equals1, we have f(0)yₙ→1, hence yₙ→1/f(0). Therefore, the limit is1/f(0).Therefore, the sequence (n xₙ) converges to1/f(0).Final Answera) boxed{text{Such } n_0 text{ and sequence } (x_n) text{ exist}} b) boxed{dfrac{1}{f(0)}}</think>Given a continuous function ( f: mathbb{R}_{ge 0} longrightarrow mathbb{R}_{>0} ), we need to address the following parts: Part a)To show there exists a natural number ( n_0 ) and a sequence of positive real numbers ( (x_n)_{n > n_0} ) such that ( n int_0^{x_n} f(t) , dt = 1 ):1. Define ( F(x) = int_0^x f(t) , dt ). Since ( f ) is continuous and positive, ( F ) is strictly increasing and continuous.2. ( F ) maps ( [0, infty) ) to ( [0, L) ) where ( L ) is the integral of ( f ) over ( [0, infty) ). If ( L ) is finite, choose ( n_0 ) such that ( 1/n < L ) for all ( n > n_0 ). If ( L ) is infinite, ( n_0 = 1 ).3. For each ( n > n_0 ), ( 1/n ) is in the range of ( F ), so there exists ( x_n ) such that ( F(x_n) = 1/n ). Thus, ( x_n = F^{-1}(1/n) ).Therefore, such ( n_0 ) and sequence ( (x_n) ) exist. Part b)To prove ( (n x_n) ) is convergent and find its limit:1. As ( n to infty ), ( x_n to 0 ) because ( F(x_n) = 1/n to 0 ).2. Use the substitution ( t = s/n ) in the integral equation ( n int_0^{x_n} f(t) , dt = 1 ): [ int_0^{n x_n} fleft(frac{s}{n}right) , ds = 1 ]3. By the continuity of ( f ) at 0, ( fleft(frac{s}{n}right) to f(0) ) uniformly on compact intervals. Thus, [ int_0^{n x_n} f(0) , ds approx 1 implies f(0) cdot n x_n approx 1 implies n x_n to frac{1}{f(0)} ]Thus, the sequence ( (n x_n) ) converges to ( frac{1}{f(0)} ). Final Answersa) boxed{text{Such } n_0 text{ and sequence } (x_n) text{ exist}} b) boxed{dfrac{1}{f(0)}}]

🔑: Part (a)1. Existence of ( n_0 ) and ( x_n ): Given that ( f ) is a continuous function on ( mathbb{R}_{ge 0} ) and ( f(t) > 0 ) for all ( t ge 0 ), we can consider the integral ( int_0^1 f(t) , dt ). Since ( f ) is continuous and positive on ([0, 1]), this integral is positive: [ u = int_0^1 f(t) , dt > 0. ] Let ( n_0 ) be a natural number such that ( u > frac{1}{n_0} ). For any ( n ge n_0 ), we need to find ( x_n ) such that: [ n int_0^{x_n} f(t) , dt = 1. ] Define the function ( g(x) = int_0^x f(t) , dt ). Since ( f(t) > 0 ) for all ( t ge 0 ), ( g(x) ) is strictly increasing and continuous. Also, ( g(0) = 0 ) and ( g(1) = u ). For ( n ge n_0 ), we have: [ frac{1}{n} < u. ] By the Intermediate Value Theorem, for each ( n ge n_0 ), there exists ( x_n in (0, 1) ) such that: [ g(x_n) = frac{1}{n}. ] Therefore, we have: [ n int_0^{x_n} f(t) , dt = 1. ] Part (b)2. Convergence of ( nx_n ): By the Mean Value Theorem for integrals, there exists ( t_n in [0, x_n] ) such that: [ int_0^{x_n} f(t) , dt = x_n f(t_n). ] Substituting this into the given equation ( n int_0^{x_n} f(t) , dt = 1 ), we get: [ n x_n f(t_n) = 1. ] Solving for ( nx_n ), we have: [ nx_n = frac{1}{f(t_n)}. ] Since ( f ) is continuous and positive on ([0, 1]), let ( m = inf_{x in [0, 1]} f(x) ). Then: [ 0 < m le f(t_n). ] This implies: [ nx_n = frac{1}{f(t_n)} le frac{1}{m}. ] As ( n to infty ), ( x_n to 0 ) because ( g(x_n) = frac{1}{n} to 0 ). Since ( t_n in [0, x_n] ), ( t_n to 0 ) as well. By the continuity of ( f ) at ( 0 ), ( f(t_n) to f(0) ). Therefore: [ lim_{n to infty} nx_n = frac{1}{f(0)}. ]The final answer is ( boxed{ lim_{n to infty} nx_n = frac{1}{f(0)} } ).