💡Okay, so I have this problem here where I need to find the modulus of a complex number z, which is given as z = m + ni. The condition provided is that m/(1 + i) equals 1 - ni. Hmm, let me try to figure this out step by step.First, I remember that modulus of a complex number z = a + bi is calculated as |z| = sqrt(a² + b²). So, in this case, once I find the values of m and n, I can plug them into this formula to get the modulus. But before that, I need to solve for m and n using the given equation.The equation is m/(1 + i) = 1 - ni. I think I should start by simplifying this equation. Maybe I can get rid of the denominator by multiplying both sides by (1 + i). Let me try that.Multiplying both sides by (1 + i):m = (1 - ni)(1 + i)Alright, now I need to expand the right-hand side. Let me recall how to multiply complex numbers. It's like the distributive property, right? So, I'll multiply each term in the first parenthesis by each term in the second.So, (1)(1) + (1)(i) + (-ni)(1) + (-ni)(i)Calculating each term:1 * 1 = 11 * i = i(-ni) * 1 = -ni(-ni) * i = -n i²Wait, I remember that i² is equal to -1. So, -n i² becomes -n*(-1) = n.Putting it all together:1 + i - ni + nNow, let me combine like terms. The real parts are 1 and n, and the imaginary parts are i and -ni.So, real part: 1 + nImaginary part: (1 - n)iTherefore, the equation becomes:m = (1 + n) + (1 - n)iBut wait, m is a real number, right? So, the imaginary part on the right-hand side must be zero because m has no imaginary component. That gives me two equations:1. The real part: m = 1 + n2. The imaginary part: 1 - n = 0Let me solve the second equation first because it directly gives me n.From 1 - n = 0, adding n to both sides gives 1 = n, so n = 1.Now that I know n is 1, I can substitute this back into the first equation to find m.From m = 1 + n, substituting n = 1:m = 1 + 1 = 2So, m is 2 and n is 1. Therefore, the complex number z is z = m + ni = 2 + i.Now, to find the modulus |z|, I use the formula |z| = sqrt(m² + n²). Plugging in m = 2 and n = 1:|z| = sqrt(2² + 1²) = sqrt(4 + 1) = sqrt(5)So, the modulus of z is sqrt(5). Let me just double-check my steps to make sure I didn't make any mistakes.Starting from m/(1 + i) = 1 - ni, I multiplied both sides by (1 + i) correctly, expanded the right-hand side, separated real and imaginary parts, solved for n first, then found m, and finally calculated the modulus. Everything seems to check out.I think that's it. The modulus should be sqrt(5).**Final Answer**The modulus of the complex number ( z ) is boxed{sqrt{5}}.

💡Okay, so I need to find a complex number ( z ) that satisfies the equation ( 3z - 4ioverline{z} = -8 + 4i ). Hmm, complex numbers can sometimes be tricky, especially when they involve conjugates. Let me break this down step by step.First, I remember that any complex number ( z ) can be written as ( a + bi ), where ( a ) and ( b ) are real numbers. The conjugate of ( z ), denoted ( overline{z} ), is ( a - bi ). So, I can substitute these into the equation to make it easier to handle.Let me write that out:( z = a + bi )( overline{z} = a - bi )Now, substituting these into the equation:( 3z - 4ioverline{z} = 3(a + bi) - 4i(a - bi) )Let me expand this expression. First, distribute the 3 into ( (a + bi) ):( 3a + 3bi )Next, distribute the ( -4i ) into ( (a - bi) ):( -4i cdot a + (-4i) cdot (-bi) )Simplify each term:( -4ai + 4b i^2 )Wait, since ( i^2 = -1 ), this becomes:( -4ai + 4b(-1) = -4ai - 4b )So, putting it all together, the entire left side of the equation is:( 3a + 3bi - 4ai - 4b )Now, let's combine like terms. The real parts are ( 3a ) and ( -4b ), and the imaginary parts are ( 3bi ) and ( -4ai ).So, grouping them:Real part: ( 3a - 4b )Imaginary part: ( (3b - 4a)i )Therefore, the equation becomes:( (3a - 4b) + (3b - 4a)i = -8 + 4i )Since two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal, I can set up a system of equations:1. ( 3a - 4b = -8 ) (equating the real parts)2. ( 3b - 4a = 4 ) (equating the imaginary parts)Alright, now I have a system of two linear equations with two variables, ( a ) and ( b ). I need to solve this system to find the values of ( a ) and ( b ).Let me write the equations again:1. ( 3a - 4b = -8 )2. ( -4a + 3b = 4 )Hmm, I can solve this using either substitution or elimination. Let me try elimination because the coefficients seem manageable.First, I'll label the equations for clarity:Equation (1): ( 3a - 4b = -8 )Equation (2): ( -4a + 3b = 4 )I need to eliminate one of the variables. Let's try to eliminate ( a ). To do that, I'll make the coefficients of ( a ) in both equations equal in magnitude. The coefficients are 3 and -4. The least common multiple of 3 and 4 is 12. So, I'll multiply Equation (1) by 4 and Equation (2) by 3.Multiplying Equation (1) by 4:( 4 times (3a - 4b) = 4 times (-8) )Which gives:( 12a - 16b = -32 ) (Equation 3)Multiplying Equation (2) by 3:( 3 times (-4a + 3b) = 3 times 4 )Which gives:( -12a + 9b = 12 ) (Equation 4)Now, I'll add Equation (3) and Equation (4) to eliminate ( a ):( (12a - 16b) + (-12a + 9b) = -32 + 12 )Simplify:( 12a - 12a -16b + 9b = -20 )Which simplifies to:( -7b = -20 )Divide both sides by -7:( b = frac{20}{7} )Okay, so ( b = frac{20}{7} ). Now, I need to find ( a ). I'll substitute this value of ( b ) back into one of the original equations. Let's use Equation (1):( 3a - 4b = -8 )Substitute ( b = frac{20}{7} ):( 3a - 4 times frac{20}{7} = -8 )Calculate ( 4 times frac{20}{7} ):( frac{80}{7} )So, the equation becomes:( 3a - frac{80}{7} = -8 )Add ( frac{80}{7} ) to both sides:( 3a = -8 + frac{80}{7} )Convert -8 to sevenths to add:( -8 = -frac{56}{7} )So,( 3a = -frac{56}{7} + frac{80}{7} = frac{24}{7} )Now, divide both sides by 3:( a = frac{24}{7} times frac{1}{3} = frac{8}{7} )So, ( a = frac{8}{7} ) and ( b = frac{20}{7} ).Therefore, the complex number ( z ) is:( z = a + bi = frac{8}{7} + frac{20}{7}i )Let me double-check my work to make sure I didn't make any mistakes.Starting with the original equation:( 3z - 4ioverline{z} = -8 + 4i )Substitute ( z = frac{8}{7} + frac{20}{7}i ) and ( overline{z} = frac{8}{7} - frac{20}{7}i ):Calculate ( 3z ):( 3 times left( frac{8}{7} + frac{20}{7}i right) = frac{24}{7} + frac{60}{7}i )Calculate ( -4ioverline{z} ):( -4i times left( frac{8}{7} - frac{20}{7}i right) = -4i times frac{8}{7} + (-4i) times left( -frac{20}{7}i right) )Simplify each term:First term: ( -frac{32}{7}i )Second term: ( frac{80}{7}i^2 ). Since ( i^2 = -1 ), this becomes ( -frac{80}{7} )So, combining these:( -frac{32}{7}i - frac{80}{7} )Now, add ( 3z ) and ( -4ioverline{z} ):( left( frac{24}{7} + frac{60}{7}i right) + left( -frac{80}{7} - frac{32}{7}i right) )Combine like terms:Real parts: ( frac{24}{7} - frac{80}{7} = -frac{56}{7} = -8 )Imaginary parts: ( frac{60}{7}i - frac{32}{7}i = frac{28}{7}i = 4i )So, the result is ( -8 + 4i ), which matches the right side of the original equation. Therefore, my solution is correct.I think I've got it. The complex number ( z ) that satisfies the equation is ( frac{8}{7} + frac{20}{7}i ).**Final Answer**The complex number ( z ) is boxed{dfrac{8}{7} + dfrac{20}{7}i}.

💡First, I need to determine the total number of baseball cards bought by Nancy, Melanie, Mary, and Alyssa. Since each of them bought 540 baseball cards and there are four individuals, the total number of baseball cards is 4 multiplied by 540.Next, I know that the total number of packs they have is 108. To find out how many baseball cards are in each pack, I should divide the total number of baseball cards by the total number of packs.So, I will divide 2160 (the total number of baseball cards) by 108 (the total number of packs) to find the number of baseball cards in each pack.Finally, performing the division will give me the number of baseball cards per pack.

💡Alright, so I have this problem about the function ( f(x) = x ln x ). It has two parts. Let me try to tackle them one by one.Starting with part (1): Find the equation of the tangent line of the function ( f(x) ) at the point ( (1, 0) ).Okay, I remember that to find the equation of a tangent line at a specific point, I need two things: the slope of the tangent line at that point and a point through which the tangent line passes. The point is already given as ( (1, 0) ), so I just need to find the slope.The slope of the tangent line at a point is given by the derivative of the function evaluated at that point. So, I need to find ( f'(x) ) and then plug in ( x = 1 ).Let me compute the derivative ( f'(x) ). The function is ( f(x) = x ln x ). This is a product of two functions: ( u(x) = x ) and ( v(x) = ln x ). So, I should use the product rule for differentiation, which states that ( (uv)' = u'v + uv' ).Calculating the derivatives of ( u(x) ) and ( v(x) ):- ( u'(x) = 1 ) because the derivative of ( x ) with respect to ( x ) is 1.- ( v'(x) = frac{1}{x} ) because the derivative of ( ln x ) with respect to ( x ) is ( frac{1}{x} ).Applying the product rule:[f'(x) = u'(x)v(x) + u(x)v'(x) = 1 cdot ln x + x cdot frac{1}{x} = ln x + 1]So, ( f'(x) = ln x + 1 ).Now, I need to evaluate this derivative at ( x = 1 ) to find the slope of the tangent line at that point:[f'(1) = ln 1 + 1 = 0 + 1 = 1]So, the slope ( m ) of the tangent line at ( (1, 0) ) is 1.Now that I have the slope and a point, I can write the equation of the tangent line using the point-slope form:[y - y_1 = m(x - x_1)]Plugging in the values:[y - 0 = 1(x - 1)]Simplifying:[y = x - 1]So, the equation of the tangent line is ( y = x - 1 ).Alright, that seems straightforward. Let me double-check my steps:1. Found the derivative using the product rule.2. Evaluated the derivative at ( x = 1 ) to get the slope.3. Used point-slope form to write the equation of the tangent line.Everything seems correct. I think part (1) is done.Moving on to part (2): If for any ( x > 0 ), ( x ln(ax) geq x - a ) holds, find the range of positive values for ( a ).Hmm, okay. So, I need to find all positive real numbers ( a ) such that the inequality ( x ln(ax) geq x - a ) holds for any ( x > 0 ).Let me rewrite the inequality to make it easier to handle:[x ln(ax) geq x - a]I can subtract ( x ) from both sides to get:[x ln(ax) - x geq -a]Then, adding ( a ) to both sides:[x ln(ax) - x + a geq 0]Let me define a function ( g(x) = x ln(ax) - x + a ). So, the inequality becomes ( g(x) geq 0 ) for all ( x > 0 ).To ensure that ( g(x) geq 0 ) for all ( x > 0 ), I need to analyze the behavior of ( g(x) ). Maybe I can find its minimum value and ensure that the minimum is greater than or equal to zero.First, let's find the derivative of ( g(x) ) to find critical points.Compute ( g'(x) ):[g(x) = x ln(ax) - x + a]Let me differentiate term by term:- The derivative of ( x ln(ax) ) can be found using the product rule. Let ( u = x ) and ( v = ln(ax) ). - ( u' = 1 ) - ( v' = frac{1}{ax} cdot a = frac{1}{x} ) (since the derivative of ( ln(ax) ) with respect to ( x ) is ( frac{1}{ax} cdot a )) - So, the derivative is ( u'v + uv' = 1 cdot ln(ax) + x cdot frac{1}{x} = ln(ax) + 1 )- The derivative of ( -x ) is ( -1 )- The derivative of ( a ) (a constant) is 0Putting it all together:[g'(x) = ln(ax) + 1 - 1 = ln(ax)]So, ( g'(x) = ln(ax) ).To find critical points, set ( g'(x) = 0 ):[ln(ax) = 0]Solving for ( x ):[ax = e^0 = 1 implies x = frac{1}{a}]So, the critical point is at ( x = frac{1}{a} ).Now, let's analyze the behavior of ( g(x) ) around this critical point to determine if it's a minimum or maximum.Looking at the derivative ( g'(x) = ln(ax) ):- When ( x < frac{1}{a} ), ( ax < 1 ), so ( ln(ax) < 0 ). Therefore, ( g'(x) < 0 ), meaning ( g(x) ) is decreasing.- When ( x > frac{1}{a} ), ( ax > 1 ), so ( ln(ax) > 0 ). Therefore, ( g'(x) > 0 ), meaning ( g(x) ) is increasing.This indicates that ( g(x) ) has a minimum at ( x = frac{1}{a} ).To ensure ( g(x) geq 0 ) for all ( x > 0 ), the minimum value of ( g(x) ) must be greater than or equal to zero. So, let's compute ( g ) at ( x = frac{1}{a} ):[gleft(frac{1}{a}right) = frac{1}{a} lnleft(a cdot frac{1}{a}right) - frac{1}{a} + a]Simplify the argument of the logarithm:[a cdot frac{1}{a} = 1]So,[gleft(frac{1}{a}right) = frac{1}{a} ln(1) - frac{1}{a} + a]Since ( ln(1) = 0 ), this simplifies to:[gleft(frac{1}{a}right) = 0 - frac{1}{a} + a = a - frac{1}{a}]So, the minimum value of ( g(x) ) is ( a - frac{1}{a} ).For ( g(x) geq 0 ) for all ( x > 0 ), we require:[a - frac{1}{a} geq 0]Let's solve this inequality for ( a ).Multiply both sides by ( a ) (since ( a > 0 ), the inequality sign doesn't change):[a^2 - 1 geq 0]This simplifies to:[a^2 geq 1]Taking square roots (and considering ( a > 0 )):[a geq 1]So, the range of positive values for ( a ) is ( [1, infty) ).Let me recap the steps to ensure I didn't make a mistake:1. Rewrote the inequality ( x ln(ax) geq x - a ) as ( g(x) = x ln(ax) - x + a geq 0 ).2. Found the derivative ( g'(x) = ln(ax) ).3. Determined the critical point at ( x = frac{1}{a} ).4. Analyzed the behavior of ( g(x) ) around the critical point to find that it's a minimum.5. Evaluated ( g ) at the critical point to get ( a - frac{1}{a} ).6. Set this minimum value greater than or equal to zero and solved for ( a ), resulting in ( a geq 1 ).Everything seems to check out. I think I've got the correct range for ( a ).**Final Answer**(1) The equation of the tangent line is boxed{y = x - 1}.(2) The range of positive values for ( a ) is boxed{[1, +infty)}.

💡Okay, so I've got this problem with two lines, l₁ and l₂, and I need to find some values based on certain conditions. Let me try to break it down step by step.First, the equations of the lines are given as:- l₁: ax - by - 1 = 0- l₂: (a + 2)x + y + a = 0And a and b are non-zero constants. There are two parts to this problem.**Part (1): If b = 0 and l₁ is perpendicular to l₂, find the value of the real number a.**Alright, so starting with part (1). They tell us that b = 0. Let me substitute b = 0 into the equation of l₁.So, l₁ becomes: ax - 0*y - 1 = 0, which simplifies to ax - 1 = 0. That can be rewritten as x = 1/a. Hmm, so this is a vertical line because it's in the form x = constant.Now, l₂ is given by (a + 2)x + y + a = 0. Let me rewrite this in slope-intercept form (y = mx + c) to find its slope.Subtracting (a + 2)x and a from both sides: y = -(a + 2)x - a.So, the slope of l₂ is -(a + 2).Since l₁ is a vertical line, its slope is undefined. But for two lines to be perpendicular, one has to be vertical and the other horizontal, right? Wait, no, actually, if one line is vertical, the other just needs to be horizontal to be perpendicular. But in this case, l₂ is not horizontal because its slope is -(a + 2), which is not zero unless a + 2 = 0.Wait, maybe I'm overcomplicating. Let me think again. If l₁ is vertical (slope undefined), then for l₂ to be perpendicular to it, l₂ must be horizontal. A horizontal line has a slope of 0.So, the slope of l₂ must be 0. Therefore, -(a + 2) = 0.Solving that: -(a + 2) = 0 ⇒ a + 2 = 0 ⇒ a = -2.Wait, that seems straightforward. So, a is -2.But let me double-check. If a = -2, then l₁ becomes -2x - 1 = 0 ⇒ x = -1/2, which is a vertical line. And l₂ becomes (-2 + 2)x + y - 2 = 0 ⇒ 0x + y - 2 = 0 ⇒ y = 2, which is a horizontal line. Yes, they are perpendicular. That makes sense.**Part (2): If b = 2 and l₁ is parallel to l₂, find the distance between the lines l₁ and l₂.**Alright, moving on to part (2). Now, b = 2. So, substituting b = 2 into l₁'s equation:l₁: ax - 2y - 1 = 0.Let me write both lines in slope-intercept form to find their slopes.Starting with l₁: ax - 2y - 1 = 0.Solving for y: -2y = -ax + 1 ⇒ y = (a/2)x - 1/2.So, the slope of l₁ is a/2.Now, l₂ is given by (a + 2)x + y + a = 0.Solving for y: y = -(a + 2)x - a.So, the slope of l₂ is -(a + 2).Since l₁ is parallel to l₂, their slopes must be equal. Therefore:a/2 = -(a + 2).Let me solve for a.Multiply both sides by 2: a = -2(a + 2).Expanding the right side: a = -2a - 4.Bring all terms to one side: a + 2a + 4 = 0 ⇒ 3a + 4 = 0 ⇒ 3a = -4 ⇒ a = -4/3.So, a is -4/3.Now, I need to find the distance between the two parallel lines l₁ and l₂.First, let me write both lines with a = -4/3 and b = 2.For l₁: ax - 2y - 1 = 0 ⇒ (-4/3)x - 2y - 1 = 0.Multiply both sides by 3 to eliminate the fraction: -4x - 6y - 3 = 0 ⇒ 4x + 6y + 3 = 0.For l₂: (a + 2)x + y + a = 0 ⇒ (-4/3 + 2)x + y - 4/3 = 0.Simplify (-4/3 + 2): 2 is 6/3, so 6/3 - 4/3 = 2/3. So, l₂ becomes (2/3)x + y - 4/3 = 0.Multiply both sides by 3 to eliminate fractions: 2x + 3y - 4 = 0.Wait, but I need both lines in the same form to apply the distance formula. Let me check again.Wait, l₁ after substitution is 4x + 6y + 3 = 0.l₂ after substitution is (2/3)x + y - 4/3 = 0. If I multiply by 3, it becomes 2x + 3y - 4 = 0.But to use the distance formula, both lines should have the same coefficients for x and y. So, I can write l₂ as 4x + 6y - 8 = 0 by multiplying both sides by 2.Wait, let me see:Original l₂ after substitution: 2x + 3y - 4 = 0.If I multiply by 2, it becomes 4x + 6y - 8 = 0.Yes, that's correct because 2*(2x + 3y - 4) = 4x + 6y - 8.So now, l₁ is 4x + 6y + 3 = 0 and l₂ is 4x + 6y - 8 = 0.The distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by:Distance = |C₂ - C₁| / sqrt(A² + B²)So, here, A = 4, B = 6, C₁ = 3, C₂ = -8.Wait, hold on. The formula is |C₂ - C₁| / sqrt(A² + B²). But I think it's actually |C₁ - C₂| / sqrt(A² + B²). Let me confirm.Yes, the distance formula is |C₁ - C₂| / sqrt(A² + B²). So, it's the absolute difference of the constants divided by the square root of the sum of squares of the coefficients.So, plugging in the values:Distance = |3 - (-8)| / sqrt(4² + 6²) = |3 + 8| / sqrt(16 + 36) = |11| / sqrt(52).Simplify sqrt(52): sqrt(4*13) = 2*sqrt(13).So, distance = 11 / (2*sqrt(13)).But we can rationalize the denominator:11 / (2*sqrt(13)) = (11*sqrt(13)) / (2*13) = (11*sqrt(13)) / 26.So, the distance is 11√13 / 26.Let me double-check my steps.1. Found a = -4/3 correctly by equating slopes.2. Substituted a into both lines and converted them to standard form with same coefficients.3. Applied the distance formula correctly, making sure to take the absolute difference of the constants.4. Simplified sqrt(52) correctly to 2√13.5. Rationalized the denominator properly.Everything seems correct. So, the distance is indeed 11√13 / 26.

💡Alright, so I've got this problem here about arrangeable 5-tuples and a 2017-tuple arranged in a circle. Let me try to unpack this step by step.First, the problem defines an "arrangeable" 5-tuple of integers. It means that if you label the elements (a, b, c, d, e) in some order, then (a - b + c - d + e = 29). So, for any five consecutive numbers in the circle, they can be rearranged such that this equation holds.Now, the task is to determine all 2017-tuples of integers (n_1, n_2, ldots, n_{2017}) arranged in a circle such that any five consecutive numbers form an arrangeable 5-tuple. That means, no matter where you start on the circle, the next five numbers can be labeled in some order to satisfy that equation.Hmm, okay. So, first, maybe I should think about what constraints this imposes on the entire 2017-tuple. Since every set of five consecutive numbers must satisfy this condition, there must be some regularity or pattern in the numbers.Let me consider the equation (a - b + c - d + e = 29). If I can find a way to express this in terms of the original sequence, maybe I can derive some relationships between the numbers.Wait, perhaps I can think of this as a kind of recurrence relation. If every five consecutive numbers must satisfy this equation, maybe the numbers follow a linear pattern or something similar.But before jumping into that, let me try a substitution. Let me define (m_i = n_i - 29). Then, substituting into the equation, we get:[(a - 29) - (b - 29) + (c - 29) - (d - 29) + (e - 29) = 29]Simplifying this:[a - b + c - d + e - 5 times 29 = 29]Which simplifies to:[a - b + c - d + e = 29 + 5 times 29 = 164]Wait, that doesn't seem right. Let me check my substitution again.Oh, no, actually, substituting (n_i = m_i + 29) into the equation (a - b + c - d + e = 29), we get:[(m_a + 29) - (m_b + 29) + (m_c + 29) - (m_d + 29) + (m_e + 29) = 29]Simplifying:[m_a - m_b + m_c - m_d + m_e + (29 - 29 + 29 - 29 + 29) = 29]Which simplifies to:[m_a - m_b + m_c - m_d + m_e + 29 = 29]So, subtracting 29 from both sides:[m_a - m_b + m_c - m_d + m_e = 0]Ah, that's better. So, the condition becomes that for any five consecutive (m_i), their alternating sum is zero. So, (m_i - m_{i+1} + m_{i+2} - m_{i+3} + m_{i+4} = 0) for all (i), considering the circular arrangement.Okay, so now we have this condition on the (m_i). Let's see if we can find a pattern or a recurrence here.Let me write down the condition for a few consecutive terms to see if a pattern emerges.For (i):[m_i - m_{i+1} + m_{i+2} - m_{i+3} + m_{i+4} = 0]For (i+1):[m_{i+1} - m_{i+2} + m_{i+3} - m_{i+4} + m_{i+5} = 0]Hmm, if I subtract the first equation from the second, I get:[(m_{i+1} - m_{i+2} + m_{i+3} - m_{i+4} + m_{i+5}) - (m_i - m_{i+1} + m_{i+2} - m_{i+3} + m_{i+4})) = 0]Simplifying:[m_{i+1} - m_{i+2} + m_{i+3} - m_{i+4} + m_{i+5} - m_i + m_{i+1} - m_{i+2} + m_{i+3} - m_{i+4} = 0]Wait, that seems messy. Maybe I should group like terms.Let's see:- (m_{i+1}) appears twice: (2m_{i+1})- (-m_{i+2}) appears twice: (-2m_{i+2})- (m_{i+3}) appears twice: (2m_{i+3})- (-m_{i+4}) appears twice: (-2m_{i+4})- (m_{i+5}) appears once: (m_{i+5})- (-m_i) appears once: (-m_i)So, putting it all together:[2m_{i+1} - 2m_{i+2} + 2m_{i+3} - 2m_{i+4} + m_{i+5} - m_i = 0]Hmm, that's a fifth-order linear recurrence relation. Maybe this is getting too complicated. Perhaps there's a simpler approach.Wait, since the condition must hold for all five consecutive terms, maybe the sequence has a periodicity or all terms are equal.Let me test the simplest case: suppose all (m_i) are equal. Let (m_i = k) for some constant (k).Then, substituting into the condition:[k - k + k - k + k = k = 0]So, (k = 0). Therefore, all (m_i = 0), which implies all (n_i = 29).Is this the only solution? Let me see.Suppose not all (m_i) are zero. Maybe there's a non-constant solution.But given the circular arrangement and the recurrence relation, it's likely that the only solution is the constant sequence.Wait, let me think about the system of equations. For each (i), we have:[m_i - m_{i+1} + m_{i+2} - m_{i+3} + m_{i+4} = 0]This is a system of 2017 equations with 2017 variables. Since the system is cyclic, it's a circulant matrix. The determinant of such a matrix can be analyzed using the discrete Fourier transform, but that might be too advanced for me right now.Alternatively, maybe I can consider the sum of all equations.Summing all equations:[sum_{i=1}^{2017} (m_i - m_{i+1} + m_{i+2} - m_{i+3} + m_{i+4}) = 0]But since it's a circular arrangement, each (m_j) appears exactly once in each position:- (m_i) appears once as (m_i)- (-m_{i+1}) appears once as (-m_{i+1})- (m_{i+2}) appears once as (m_{i+2})- (-m_{i+3}) appears once as (-m_{i+3})- (m_{i+4}) appears once as (m_{i+4})So, summing all equations:[sum_{i=1}^{2017} m_i - sum_{i=1}^{2017} m_{i+1} + sum_{i=1}^{2017} m_{i+2} - sum_{i=1}^{2017} m_{i+3} + sum_{i=1}^{2017} m_{i+4} = 0]But since the indices are cyclic, each sum is just the sum of all (m_i):[sum_{i=1}^{2017} m_i - sum_{i=1}^{2017} m_i + sum_{i=1}^{2017} m_i - sum_{i=1}^{2017} m_i + sum_{i=1}^{2017} m_i = 0]Simplifying:[(1 - 1 + 1 - 1 + 1) sum_{i=1}^{2017} m_i = 0]Which is:[1 times sum_{i=1}^{2017} m_i = 0]So, the total sum of all (m_i) is zero.But earlier, we saw that if all (m_i) are equal, they must be zero. So, the total sum being zero is consistent with that.But does this imply that all (m_i) must be zero? Not necessarily, but in combination with the recurrence relation, it might.Let me consider the recurrence relation again:[m_i = m_{i+1} - m_{i+2} + m_{i+3} - m_{i+4}]Wait, that's a fourth-order recurrence. Maybe I can write it in terms of previous terms.But since it's a circular system, it's tricky. Maybe I can assume a solution of the form (m_i = r^i), where (r) is some constant.Substituting into the recurrence:[r^i = r^{i+1} - r^{i+2} + r^{i+3} - r^{i+4}]Dividing both sides by (r^i) (assuming (r neq 0)):[1 = r - r^2 + r^3 - r^4]So, the characteristic equation is:[r^4 - r^3 + r^2 - r + 1 = 0]Hmm, solving this quartic equation might be complex, but perhaps it factors.Let me try to factor it. Maybe it's a cyclotomic polynomial.Indeed, (r^4 - r^3 + r^2 - r + 1) is the 10th cyclotomic polynomial, which factors as ((r^2 + r + 1)(r^2 - r + 1)). Wait, no, let me check:Wait, actually, (r^4 - r^3 + r^2 - r + 1) is irreducible over the rationals. So, it doesn't factor into polynomials with integer coefficients.Therefore, the roots are complex. So, the general solution would involve complex exponentials or sines and cosines.But since we're dealing with integers, the only solution is the trivial one where all (m_i = 0).Therefore, the only solution is (m_i = 0) for all (i), which implies (n_i = 29) for all (i).So, the only 2017-tuple that satisfies the condition is the constant tuple where every number is 29.**Final Answer**boxed{29}boxed{n_1 = n_2 = cdots = n_{2017} = 29}