answer：First, I need to determine the equal distance traveled at each of the three speeds: 3 km/hr, 4 km/hr, and 5 km/hr. Let's denote this equal distance as ( d ) kilometers.Next, I'll calculate the time taken to travel each segment of the journey. The time taken to travel ( d ) kilometers at 3 km/hr is ( frac{d}{3} ) hours. Similarly, the time taken at 4 km/hr is ( frac{d}{4} ) hours, and at 5 km/hr is ( frac{d}{5} ) hours.The total time for the entire journey is given as 47 minutes. To work with consistent units, I'll convert 47 minutes into hours by dividing by 60: ( frac{47}{60} ) hours.Now, I'll set up the equation for the total time:[frac{d}{3} + frac{d}{4} + frac{d}{5} = frac{47}{60}]To solve for ( d ), I'll find a common denominator for the fractions on the left side of the equation. The least common multiple of 3, 4, and 5 is 60. Rewriting the equation with this common denominator:[frac{20d}{60} + frac{15d}{60} + frac{12d}{60} = frac{47}{60}]Combining the fractions on the left side:[frac{47d}{60} = frac{47}{60}]Solving for ( d ) by multiplying both sides of the equation by ( frac{60}{47} ):[d = 1]So, the distance ( d ) traveled at each speed is 1 kilometer. Since the person travels this distance at each of the three speeds, the total distance traveled is:[3d = 3 times 1 = 3 text{ km}]Therefore, the total distance traveled is 3 kilometers.

answer：First, I need to evaluate the numerator of the complex number ( z = frac{(i - 1)^2 + 4}{i + 1} ).I'll start by expanding ( (i - 1)^2 ):[(i - 1)^2 = i^2 - 2i + 1 = -1 - 2i + 1 = -2i]Adding 4 to this result gives:[-2i + 4]Now, the expression for ( z ) becomes:[z = frac{-2i + 4}{i + 1}]To simplify this fraction, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is ( 1 - i ):[z = frac{(-2i + 4)(1 - i)}{(i + 1)(1 - i)}]Expanding the numerator:[(-2i + 4)(1 - i) = -2i + 4 - (-2i^2 + 4i) = -2i + 4 + 2 - 4i = 6 - 6i]Expanding the denominator:[(i + 1)(1 - i) = i(1 - i) + 1(1 - i) = i - i^2 + 1 - i = 1 + 1 = 2]So, the expression simplifies to:[z = frac{6 - 6i}{2} = 3 - 3i]The imaginary part of ( z ) is the coefficient of ( i ), which is (-3).

answer：First, I need to calculate (3 star 15) using the given operation definition (A star B = frac{A + B}{3}).[3 star 15 = frac{3 + 15}{3} = frac{18}{3} = 6]Next, I will calculate (6 star 2) using the same operation.[6 star 2 = frac{6 + 2}{3} = frac{8}{3}]Now, I will use the results from the previous steps to find ((3 star 15) star (6 star 2)).[(3 star 15) star (6 star 2) = 6 star left(frac{8}{3}right) = frac{6 + frac{8}{3}}{3} = frac{frac{18}{3} + frac{8}{3}}{3} = frac{frac{26}{3}}{3} = frac{26}{9}]Therefore, the final result is (boxed{frac{26}{9}}).

answer：Okay, so I have this problem here where I need to find the smallest possible value of ( n ) such that there exist real numbers ( x_1, x_2, ldots, x_n ) each with absolute value less than 1, and they satisfy the equation:[|x_1| + |x_2| + ldots + |x_n| = 25 + |x_1 + x_2 + ldots + x_n|.]Hmm, let me try to unpack this step by step.First, I know that each ( |x_i| < 1 ). So, the sum ( |x_1| + |x_2| + ldots + |x_n| ) is less than ( n ) because each term is less than 1, and there are ( n ) terms. So, the left-hand side (LHS) of the equation is less than ( n ).On the right-hand side (RHS), we have 25 plus the absolute value of the sum of all ( x_i ). The absolute value ( |x_1 + x_2 + ldots + x_n| ) is always non-negative, so the RHS is at least 25. Therefore, for the equation to hold, the LHS must be at least 25. But since the LHS is less than ( n ), it must be that ( n > 25 ). So, ( n ) must be at least 26. That gives me a starting point: the smallest possible ( n ) is 26 or higher.But wait, is 26 actually possible? I need to check if there's a configuration of ( x_i )s that satisfies the equation when ( n = 26 ).Let me think about how the equation can be satisfied. The equation is:[sum_{i=1}^{n} |x_i| = 25 + left| sum_{i=1}^{n} x_i right|.]I can rearrange this equation to:[sum_{i=1}^{n} |x_i| - left| sum_{i=1}^{n} x_i right| = 25.]This difference equals 25. I wonder what this difference represents. I recall that for real numbers, the triangle inequality tells us that:[left| sum_{i=1}^{n} x_i right| leq sum_{i=1}^{n} |x_i|.]So, the difference ( sum |x_i| - left| sum x_i right| ) is always non-negative. In our case, this difference is 25, which is quite large. To maximize this difference, we need to maximize ( sum |x_i| ) while minimizing ( left| sum x_i right| ).How can I minimize ( left| sum x_i right| )? If I have some positive ( x_i )s and some negative ( x_i )s, their sum can be smaller, even zero, if they cancel each other out. That might help in maximizing the difference.So, suppose I split the ( x_i )s into two groups: some positive and some negative. Let's say ( k ) of them are positive and ( n - k ) are negative. Then, the sum ( sum x_i ) would be ( k cdot a - (n - k) cdot b ), where ( a ) and ( b ) are the magnitudes of the positive and negative terms, respectively.But since each ( |x_i| < 1 ), both ( a ) and ( b ) are less than 1. To maximize ( sum |x_i| ), I should set ( a ) and ( b ) as close to 1 as possible. Let's set ( a = b = c ), where ( c ) is just less than 1, say ( c = 1 - epsilon ) for a very small ( epsilon > 0 ).If I set ( k ) terms to ( c ) and ( n - k ) terms to ( -c ), then the sum ( sum x_i ) becomes ( k cdot c - (n - k) cdot c = (2k - n) cdot c ). The absolute value of this sum is ( |2k - n| cdot c ).The total sum of absolute values is ( k cdot c + (n - k) cdot c = n cdot c ).So, plugging into the equation:[n cdot c = 25 + |2k - n| cdot c.]Let me solve for ( |2k - n| ):[n cdot c - |2k - n| cdot c = 25 c (n - |2k - n|) = 25 n - |2k - n| = frac{25}{c}]Since ( c ) is close to 1, ( frac{25}{c} ) is slightly larger than 25. Therefore, ( n - |2k - n| ) must be slightly larger than 25.But ( n - |2k - n| ) is equal to ( min(2k, 2(n - k)) ). Because:- If ( 2k leq n ), then ( |2k - n| = n - 2k ), so ( n - (n - 2k) = 2k ).- If ( 2k > n ), then ( |2k - n| = 2k - n ), so ( n - (2k - n) = 2(n - k) ).Therefore, ( n - |2k - n| = 2 min(k, n - k) ).So, we have:[2 min(k, n - k) = frac{25}{c}]Since ( c ) is close to 1, ( frac{25}{c} ) is slightly more than 25. Therefore, ( 2 min(k, n - k) ) must be slightly more than 25.But ( min(k, n - k) ) must be an integer because ( k ) is an integer (number of terms). So, ( 2 min(k, n - k) ) must be at least 26 because it's slightly more than 25. Therefore, ( min(k, n - k) geq 13 ).This implies that ( k geq 13 ) or ( n - k geq 13 ). So, in order to have ( 2 min(k, n - k) geq 26 ), we need ( min(k, n - k) geq 13 ).Therefore, ( k geq 13 ) or ( n - k geq 13 ). So, if ( n = 26 ), then ( k = 13 ) and ( n - k = 13 ). So, ( min(k, n - k) = 13 ), which gives ( 2 times 13 = 26 ). Therefore, ( 26 = frac{25}{c} ), which implies ( c = frac{25}{26} ).So, if I set ( c = frac{25}{26} ), which is less than 1, as required, and set ( k = 13 ), then I have 13 terms equal to ( frac{25}{26} ) and 13 terms equal to ( -frac{25}{26} ).Let me check if this works.First, the sum of absolute values:[13 times frac{25}{26} + 13 times frac{25}{26} = 26 times frac{25}{26} = 25.]The sum of the terms:[13 times frac{25}{26} + 13 times left(-frac{25}{26}right) = 0.]So, the absolute value of the sum is 0.Therefore, the equation becomes:[25 = 25 + 0,]which is true.So, with ( n = 26 ), it's possible to satisfy the given condition. Therefore, the minimal ( n ) is 26.Wait, but let me just make sure that ( c = frac{25}{26} ) is indeed less than 1. Yes, because ( 25 < 26 ), so ( frac{25}{26} < 1 ). So, each ( |x_i| = frac{25}{26} < 1 ), which satisfies the condition.Is there a way to have ( n = 25 )? Let's see.If ( n = 25 ), then the sum of absolute values is less than 25, because each term is less than 1. But the RHS is 25 plus something non-negative, so the RHS is at least 25. Therefore, the LHS must be at least 25, but since ( n = 25 ), the LHS is less than 25. Therefore, it's impossible for ( n = 25 ).Hence, the minimal ( n ) is indeed 26.**Final Answer**The smallest possible value of ( n ) is boxed{26}.

answer：First, I need to expand the expression ((1 - i)^2). I'll use the binomial theorem for this expansion.Applying the binomial theorem:[(1 - i)^2 = 1^2 - 2 cdot 1 cdot i + i^2][= 1 - 2i + i^2]Next, I recall that (i^2 = -1). Substituting this value into the equation:[1 - 2i + (-1) = 1 - 2i - 1][= -2i]Therefore, the result of ((1 - i)^2) is (-2i).

answer：Alright, so I have this problem about a circle and a chord passing through a specific point. Let me try to understand what's being asked here.We have a circle defined by the equation ( x^2 + y^2 = r^2 ), where ( r > 0 ). There's a fixed point ( P(1,1) ), and we're told that there's exactly one chord passing through this point with a length of 2. We need to find the possible values of ( r ).First, I need to recall some properties of circles and chords. I know that the length of a chord in a circle is related to the radius and the distance from the center of the circle to the chord. Specifically, the formula for the length of a chord is:[ text{Length} = 2 sqrt{r^2 - d^2} ]where ( d ) is the perpendicular distance from the center to the chord.In this problem, the length of the chord is given as 2. So, plugging that into the formula:[ 2 = 2 sqrt{r^2 - d^2} ]Dividing both sides by 2:[ 1 = sqrt{r^2 - d^2} ]Squaring both sides:[ 1 = r^2 - d^2 ][ d^2 = r^2 - 1 ][ d = sqrt{r^2 - 1} ]So, the distance from the center of the circle to the chord is ( sqrt{r^2 - 1} ).Now, the chord passes through the point ( P(1,1) ). Since the chord passes through this point, the distance from the center of the circle to the chord must be less than or equal to the distance from the center to the point ( P ). Otherwise, the chord wouldn't pass through ( P ).The center of the circle is at the origin ( (0,0) ). The distance from the center to the point ( P(1,1) ) is:[ sqrt{(1-0)^2 + (1-0)^2} = sqrt{1 + 1} = sqrt{2} ]So, the distance from the center to the chord ( d ) must satisfy:[ d leq sqrt{2} ]From earlier, we have ( d = sqrt{r^2 - 1} ). Therefore:[ sqrt{r^2 - 1} leq sqrt{2} ]Squaring both sides:[ r^2 - 1 leq 2 ][ r^2 leq 3 ][ r leq sqrt{3} ]But we also know that ( d ) must be a real number, so ( r^2 - 1 geq 0 ), which implies:[ r geq 1 ]So, combining these two inequalities, we have:[ 1 leq r leq sqrt{3} ]But the problem states that there is exactly one chord passing through ( P(1,1) ) with a length of 2. This suggests that there's only one such chord, which geometrically means that the point ( P ) lies on the circle of radius ( d ) centered at the origin. Wait, that might not be the right way to think about it.Alternatively, if there's exactly one chord through ( P ) with length 2, it means that the chord is tangent to some circle related to ( P ). Maybe the set of all chords of length 2 through ( P ) forms a circle, and for there to be exactly one such chord, this circle must be tangent to the original circle.Let me think about that. The set of all chords of length 2 through ( P(1,1) ) can be thought of as points on a circle with radius 1 centered at ( P ). So, the equation of this circle would be:[ (x - 1)^2 + (y - 1)^2 = 1 ]Now, for there to be exactly one chord of length 2 through ( P ), this circle must be tangent to the original circle ( x^2 + y^2 = r^2 ). So, the distance between the centers of these two circles must be equal to the sum or difference of their radii.The center of the original circle is at ( (0,0) ), and the center of the circle of chords is at ( (1,1) ). The distance between these centers is ( sqrt{(1-0)^2 + (1-0)^2} = sqrt{2} ).For tangency, the distance between centers must be equal to ( r pm 1 ). So:[ sqrt{2} = r + 1 quad text{or} quad sqrt{2} = |r - 1| ]Solving these:1. ( sqrt{2} = r + 1 ) leads to ( r = sqrt{2} - 1 approx 0.414 ). But earlier, we found ( r geq 1 ), so this solution is invalid.2. ( sqrt{2} = |r - 1| ). This gives two cases: a. ( sqrt{2} = r - 1 ) leads to ( r = sqrt{2} + 1 approx 2.414 ). But earlier, ( r leq sqrt{3} approx 1.732 ), so this is also invalid. b. ( sqrt{2} = 1 - r ). Since ( r > 0 ), this would imply ( r = 1 - sqrt{2} ), which is negative, so invalid.Hmm, this approach seems to lead to contradictions with the earlier inequalities. Maybe my assumption about the set of chords forming a circle is incorrect.Let me try another approach. Suppose the chord passes through ( P(1,1) ) and has length 2. Let's denote the midpoint of the chord as ( M ). Since the chord is of length 2, the distance from ( M ) to each end of the chord is 1.Also, the line from the center of the circle ( O(0,0) ) to the midpoint ( M ) is perpendicular to the chord. So, triangle ( OMP ) is a right triangle, where ( OM = d ), ( MP ) is the distance from ( M ) to ( P ), and ( OP = sqrt{2} ).Using the Pythagorean theorem:[ OM^2 + MP^2 = OP^2 ][ d^2 + MP^2 = (sqrt{2})^2 ][ d^2 + MP^2 = 2 ]But we also know that ( MP ) is related to the length of the chord. Since ( M ) is the midpoint, the distance from ( M ) to each end is 1. So, in triangle ( MPP' ), where ( P' ) is one end of the chord, we have:[ MP^2 + 1^2 = (PP')^2 ]But ( PP' ) is the distance from ( P ) to the end of the chord, which isn't directly helpful. Maybe I need a different relation.Wait, since ( M ) is the midpoint, the vector from ( O ) to ( M ) is along the line perpendicular to the chord. So, if I can express ( M ) in terms of coordinates, maybe I can find a relation.Let me denote ( M ) as ( (h, k) ). Since ( M ) lies on the line from ( O ) to ( M ), which is perpendicular to the chord. The chord passes through ( P(1,1) ), so the line from ( M ) to ( P ) is along the chord.The slope of ( OM ) is ( frac{k}{h} ), and the slope of the chord (which is perpendicular to ( OM )) is ( -frac{h}{k} ).Since ( P(1,1) ) lies on the chord, the line connecting ( M(h,k) ) and ( P(1,1) ) has slope ( -frac{h}{k} ). So, the slope between ( M ) and ( P ) is:[ frac{1 - k}{1 - h} = -frac{h}{k} ]Cross-multiplying:[ (1 - k)k = -h(1 - h) ][ k - k^2 = -h + h^2 ][ h^2 + k^2 - h - k = 0 ]But ( M(h,k) ) is the midpoint of the chord, so it must satisfy the equation of the circle with radius ( d ):[ h^2 + k^2 = d^2 ]And from earlier, ( d^2 = r^2 - 1 ). So,[ h^2 + k^2 = r^2 - 1 ]Also, from the relation above:[ h^2 + k^2 - h - k = 0 ][ (r^2 - 1) - h - k = 0 ][ h + k = r^2 - 1 ]So, we have:1. ( h + k = r^2 - 1 )2. ( h^2 + k^2 = r^2 - 1 )Let me square equation 1:[ (h + k)^2 = (r^2 - 1)^2 ][ h^2 + 2hk + k^2 = r^4 - 2r^2 + 1 ]But from equation 2, ( h^2 + k^2 = r^2 - 1 ). Substituting:[ (r^2 - 1) + 2hk = r^4 - 2r^2 + 1 ][ 2hk = r^4 - 3r^2 + 2 ][ hk = frac{r^4 - 3r^2 + 2}{2} ]Now, I need another equation to relate ( h ) and ( k ). From the slope condition earlier:[ frac{1 - k}{1 - h} = -frac{h}{k} ][ (1 - k)k = -h(1 - h) ][ k - k^2 = -h + h^2 ][ h^2 + k^2 - h - k = 0 ]But we already used this to get ( h + k = r^2 - 1 ). Maybe I need to express ( h ) in terms of ( k ) or vice versa.From ( h + k = r^2 - 1 ), let's express ( h = r^2 - 1 - k ).Substitute into ( h^2 + k^2 = r^2 - 1 ):[ (r^2 - 1 - k)^2 + k^2 = r^2 - 1 ][ (r^4 - 2r^2 + 1) - 2(r^2 - 1)k + k^2 + k^2 = r^2 - 1 ][ r^4 - 2r^2 + 1 - 2(r^2 - 1)k + 2k^2 = r^2 - 1 ]This seems complicated. Maybe there's a simpler way.Let me consider that for there to be exactly one such chord, the system of equations must have exactly one solution. That would mean the discriminant of the resulting quadratic equation is zero.From earlier, we have:1. ( h + k = r^2 - 1 )2. ( h^2 + k^2 = r^2 - 1 )Let me subtract equation 1 squared from equation 2:[ h^2 + k^2 - (h + k)^2 = (r^2 - 1) - (r^2 - 1)^2 ][ -2hk = (r^2 - 1) - (r^4 - 2r^2 + 1) ][ -2hk = r^2 - 1 - r^4 + 2r^2 - 1 ][ -2hk = -r^4 + 3r^2 - 2 ][ 2hk = r^4 - 3r^2 + 2 ][ hk = frac{r^4 - 3r^2 + 2}{2} ]This is the same as before. So, if I consider ( h ) and ( k ) as roots of a quadratic equation, then:Let ( h ) and ( k ) be roots of ( x^2 - (h + k)x + hk = 0 )From above:[ x^2 - (r^2 - 1)x + frac{r^4 - 3r^2 + 2}{2} = 0 ]For there to be exactly one solution, the discriminant must be zero:[ (r^2 - 1)^2 - 4 cdot 1 cdot frac{r^4 - 3r^2 + 2}{2} = 0 ][ (r^4 - 2r^2 + 1) - 2(r^4 - 3r^2 + 2) = 0 ][ r^4 - 2r^2 + 1 - 2r^4 + 6r^2 - 4 = 0 ][ -r^4 + 4r^2 - 3 = 0 ][ r^4 - 4r^2 + 3 = 0 ]Let me set ( u = r^2 ):[ u^2 - 4u + 3 = 0 ][ (u - 1)(u - 3) = 0 ][ u = 1 quad text{or} quad u = 3 ][ r^2 = 1 quad text{or} quad r^2 = 3 ][ r = 1 quad text{or} quad r = sqrt{3} ]So, the possible values of ( r ) are 1 and ( sqrt{3} ).But earlier, I had the inequality ( 1 leq r leq sqrt{3} ). So both values are within this range.Therefore, the possible values of ( r ) are 1 and ( sqrt{3} ).