answer：Okay, so I have this problem where a point P(x, y) is moving on the circle defined by the equation x² + (y - 1)² = 1. There are two parts to the problem: first, I need to find the maximum and minimum values of k, and second, I need to find the maximum and minimum values of 2x + y. Hmm, I'm not entirely sure what k represents here, but maybe it's the slope of a line or something. Let me think.Starting with part (1): Find the maximum and minimum values of k. Since the problem mentions k, and it's related to a point moving on a circle, I think k might be the slope of a line connecting point P(x, y) to another fixed point. Maybe it's the slope from P to the origin or another specific point. Wait, the user didn't specify, but in the initial problem statement, it was mentioned as "the slope of the line connecting point P(x, y) and point (2, 1)". Oh, okay, so k is the slope between P(x, y) and (2, 1). That makes sense.So, if k is the slope between P(x, y) and (2, 1), then k = (y - 1)/(x - 2). But since P lies on the circle x² + (y - 1)² = 1, we can substitute y from the circle equation into the expression for k. Wait, maybe it's better to approach this using calculus or geometry.Alternatively, I remember that the maximum and minimum slopes from a point outside a circle can be found by considering the tangent lines from that point to the circle. So, if I consider the point (2, 1), which is outside the circle x² + (y - 1)² = 1, then the lines from (2, 1) tangent to the circle will have the maximum and minimum slopes. That sounds like a plan.First, let's find the equation of the circle: x² + (y - 1)² = 1. The center is at (0, 1) and the radius is 1. The point (2, 1) is 2 units away from the center along the x-axis. Since the radius is 1, the point is outside the circle.To find the tangent lines from (2, 1) to the circle, I can use the formula for the tangent from an external point to a circle. The general equation for a tangent line from point (x₁, y₁) to the circle (x - a)² + (y - b)² = r² is given by:(y - b) = m(x - a) ± r√(1 + m²)But in this case, the point is (2, 1), and the circle is centered at (0, 1) with radius 1. Alternatively, I can set up the equation of the line passing through (2, 1) with slope k: y - 1 = k(x - 2). This line should be tangent to the circle x² + (y - 1)² = 1.To find the condition for tangency, the distance from the center of the circle to the line should be equal to the radius. The distance from a point (a, b) to the line Ax + By + C = 0 is |Aa + Bb + C| / sqrt(A² + B²). Let's rewrite the line equation in standard form.Starting with y - 1 = k(x - 2), we can rearrange it to:kx - y + ( -2k + 1) = 0So, A = k, B = -1, C = -2k + 1.The center of the circle is (0, 1). Plugging into the distance formula:|k*0 + (-1)*1 + (-2k + 1)| / sqrt(k² + 1) = radius = 1Simplify the numerator:|0 - 1 - 2k + 1| = |-2k| = | -2k | = 2|k|So, the distance is 2|k| / sqrt(k² + 1) = 1Therefore, 2|k| = sqrt(k² + 1)Square both sides to eliminate the square root:(2|k|)² = (sqrt(k² + 1))²4k² = k² + 1Subtract k² from both sides:3k² = 1So, k² = 1/3Therefore, k = ±1/√3 = ±√3/3Wait, that's different from the initial thought. But wait, in the initial problem, the user mentioned k = ±√2. Hmm, maybe I made a mistake.Wait, let's check the calculation again.Starting from the distance formula:| -1 - 2k + 1 | = | -2k | = 2|k|So, 2|k| / sqrt(k² + 1) = 1So, 2|k| = sqrt(k² + 1)Square both sides:4k² = k² + 13k² = 1k² = 1/3k = ±1/√3 ≈ ±0.577But the initial thought was k = ±√2 ≈ ±1.414. So, which one is correct?Wait, maybe I messed up the standard form of the line.Let me double-check:Starting with y - 1 = k(x - 2)Bring all terms to one side:kx - y - 2k + 1 = 0So, A = k, B = -1, C = -2k + 1Then, the distance from (0,1) to this line is |k*0 + (-1)*1 + (-2k + 1)| / sqrt(k² + 1)Which is | -1 - 2k + 1 | = | -2k | = 2|k|So, 2|k| / sqrt(k² + 1) = 1Yes, that's correct. So, 2|k| = sqrt(k² + 1)Square both sides: 4k² = k² + 1 => 3k² = 1 => k² = 1/3 => k = ±1/√3Hmm, so the initial thought of k = ±√2 was incorrect. So, the correct maximum and minimum values of k are ±1/√3.But wait, maybe I misinterpreted the problem. The user initially mentioned k as the slope, but perhaps k is something else. Wait, the problem says "Find the maximum and minimum values of k". It doesn't specify what k is. Maybe in the original problem, k was defined differently?Wait, looking back, the user wrote:"Given a point P(x, y) moves on the circle x² + (y - 1)² = 1.(1) Find the maximum and minimum values of k;(2) Find the maximum and minimum values of 2x + y."So, it's not specified what k is. Hmm, that's confusing. Maybe in the original context, k was defined as the slope from (2,1). But since the user didn't specify, maybe I need to assume.Alternatively, perhaps k is the y-coordinate or something else. Wait, but in the initial problem, the user wrote:"Let k be the slope of the line connecting point P(x, y) and point (2, 1). When the line y - 1 = k(x - 2) is tangent to the circle, k reaches its maximum and minimum values. By solving x² + (y - 1)² = 1, we get k = ±√2, thus the maximum value of k is √2, and the minimum value of k is -√2."Wait, so in the initial problem, the user had k as the slope between P and (2,1), and found k = ±√2. But in my calculation, I got k = ±1/√3. So, which one is correct?Wait, perhaps I made a mistake in the distance formula. Let me recast the problem.Alternatively, maybe using parametric equations for the circle would help.The circle is x² + (y - 1)² = 1, so parametric equations can be written as:x = cosθy = 1 + sinθWhere θ is the parameter varying from 0 to 2π.Then, the slope k between P(x, y) and (2,1) is:k = (y - 1)/(x - 2) = (1 + sinθ - 1)/(cosθ - 2) = sinθ / (cosθ - 2)So, k = sinθ / (cosθ - 2)We need to find the maximum and minimum values of k as θ varies.To find the extrema, we can take the derivative of k with respect to θ and set it to zero.Let me compute dk/dθ:dk/dθ = [cosθ (cosθ - 2) - sinθ (-sinθ)] / (cosθ - 2)²Simplify numerator:cosθ (cosθ - 2) + sin²θ = cos²θ - 2cosθ + sin²θBut cos²θ + sin²θ = 1, so numerator becomes:1 - 2cosθThus, dk/dθ = (1 - 2cosθ) / (cosθ - 2)²Set derivative equal to zero:1 - 2cosθ = 0 => cosθ = 1/2So, θ = π/3 or 5π/3.Now, compute k at these θ values.For θ = π/3:sinθ = √3/2, cosθ = 1/2k = (√3/2) / (1/2 - 2) = (√3/2) / (-3/2) = -√3/3 ≈ -0.577For θ = 5π/3:sinθ = -√3/2, cosθ = 1/2k = (-√3/2) / (1/2 - 2) = (-√3/2) / (-3/2) = √3/3 ≈ 0.577So, the maximum value of k is √3/3 and the minimum is -√3/3.Wait, that's consistent with my earlier calculation. So, why did the initial thought say k = ±√2?Maybe the initial problem had a different setup. Alternatively, perhaps the user made a mistake in their initial thought.Alternatively, maybe k is not the slope but something else. Wait, if k is the slope, then it's ±√3/3. If k is something else, like the distance or another parameter, it could be different.Wait, but the user wrote:"Let k be the slope of the line connecting point P(x, y) and point (2, 1). When the line y - 1 = k(x - 2) is tangent to the circle, k reaches its maximum and minimum values. By solving x² + (y - 1)² = 1, we get k = ±√2, thus the maximum value of k is √2, and the minimum value of k is -√2."But according to my calculations, it's ±√3/3. So, perhaps the user made a mistake.Alternatively, maybe the point is not (2,1) but another point. Wait, in the initial problem, it's P(x,y) on the circle x² + (y - 1)² =1. The line connecting P and (2,1). So, the point is (2,1). So, the distance from (2,1) to the center (0,1) is 2 units, and the radius is 1. So, the tangent lines from (2,1) to the circle will form a right triangle with sides 2, 1, and sqrt(3). So, the angle θ satisfies sinθ = 1/2, so θ = 30 degrees. Therefore, the slope would be tan(60 degrees) = sqrt(3), but wait, that would be for the angle with the x-axis.Wait, maybe I'm confusing things. Let me think again.The distance from (2,1) to the center is 2. The radius is 1. So, the length of the tangent from (2,1) to the circle is sqrt(2² - 1²) = sqrt(3). So, the tangent length is sqrt(3).The slope of the tangent lines can be found using similar triangles. The triangle formed by the center, the external point, and the point of tangency is a right triangle with sides 1, sqrt(3), and 2.So, the angle between the line connecting (2,1) to the center (0,1) and the tangent line is arcsin(1/2) = 30 degrees.Therefore, the slope of the tangent lines would be tan(60 degrees) = sqrt(3) and tan(-60 degrees) = -sqrt(3). Wait, but that would mean the slopes are ±sqrt(3). But earlier, I calculated ±sqrt(3)/3. Hmm, which is correct?Wait, tan(60 degrees) is sqrt(3), but in our case, the angle between the x-axis and the tangent line is 60 degrees from the line connecting (2,1) to the center, which is along the x-axis. So, the actual slope would be tan(60 degrees) = sqrt(3). But wait, that's not considering the direction.Wait, the line connecting (2,1) to the center is along the x-axis from (2,1) to (0,1). The tangent lines make a 60-degree angle above and below this line. So, the slope would be tan(60 degrees) = sqrt(3) and tan(-60 degrees) = -sqrt(3). But that would mean the slopes are ±sqrt(3). But according to my earlier calculation using the parametric equations, it was ±sqrt(3)/3.Wait, I'm confused now. Let me try to visualize.The center is at (0,1), and the external point is at (2,1). The line connecting them is horizontal. The tangent lines from (2,1) to the circle will form a 60-degree angle with this horizontal line because the triangle is 1-2-sqrt(3), so the angle at (2,1) is 30 degrees, making the angle between the tangent and the horizontal line 60 degrees.Therefore, the slope of the tangent lines would be tan(60 degrees) = sqrt(3) and tan(-60 degrees) = -sqrt(3). So, the slopes are ±sqrt(3). But wait, that contradicts my earlier calculation of ±sqrt(3)/3.Wait, perhaps I made a mistake in the parametric approach. Let me re-examine that.Using parametric equations:x = cosθy = 1 + sinθSlope k = (y - 1)/(x - 2) = sinθ / (cosθ - 2)Then, dk/dθ = [cosθ (cosθ - 2) + sin²θ] / (cosθ - 2)²Wait, in my earlier calculation, I had:dk/dθ = [cosθ (cosθ - 2) + sin²θ] / (cosθ - 2)²But let's compute the numerator:cosθ (cosθ - 2) + sin²θ = cos²θ - 2cosθ + sin²θWhich is (cos²θ + sin²θ) - 2cosθ = 1 - 2cosθSo, dk/dθ = (1 - 2cosθ) / (cosθ - 2)²Setting numerator to zero: 1 - 2cosθ = 0 => cosθ = 1/2 => θ = π/3 or 5π/3So, at θ = π/3:sinθ = sqrt(3)/2, cosθ = 1/2k = (sqrt(3)/2) / (1/2 - 2) = (sqrt(3)/2) / (-3/2) = -sqrt(3)/3At θ = 5π/3:sinθ = -sqrt(3)/2, cosθ = 1/2k = (-sqrt(3)/2) / (1/2 - 2) = (-sqrt(3)/2) / (-3/2) = sqrt(3)/3So, the maximum k is sqrt(3)/3 and minimum is -sqrt(3)/3.But according to the geometric approach, the slopes should be ±sqrt(3). So, which one is correct?Wait, perhaps I made a mistake in the geometric approach. Let me recast it.The line from (2,1) to the circle is tangent at some point (x, y). The slope of this tangent line is k. The slope of the radius at the point of tangency is perpendicular to the tangent line. So, the slope of the radius is (y - 1)/x, and the slope of the tangent line is k. Therefore, their product should be -1:[(y - 1)/x] * k = -1But y - 1 = sinθ, x = cosθ, so:(sinθ / cosθ) * k = -1 => tanθ * k = -1 => k = -cotθBut from earlier, k = sinθ / (cosθ - 2)So, -cotθ = sinθ / (cosθ - 2)=> -cosθ / sinθ = sinθ / (cosθ - 2)Cross-multiplying:-cosθ (cosθ - 2) = sin²θ=> -cos²θ + 2cosθ = sin²θBut sin²θ = 1 - cos²θ, so:-cos²θ + 2cosθ = 1 - cos²θSimplify:-cos²θ + 2cosθ = 1 - cos²θAdd cos²θ to both sides:2cosθ = 1=> cosθ = 1/2So, θ = π/3 or 5π/3, which matches the earlier result.Therefore, k = sinθ / (cosθ - 2)At θ = π/3:k = (sqrt(3)/2) / (1/2 - 2) = (sqrt(3)/2) / (-3/2) = -sqrt(3)/3At θ = 5π/3:k = (-sqrt(3)/2) / (1/2 - 2) = (-sqrt(3)/2) / (-3/2) = sqrt(3)/3So, the maximum and minimum values of k are sqrt(3)/3 and -sqrt(3)/3.But earlier, using the geometric approach, I thought the slopes were ±sqrt(3). So, where did I go wrong?Wait, perhaps I confused the angle. The angle between the tangent line and the x-axis is not 60 degrees, but rather the angle between the tangent line and the line connecting (2,1) to the center is 60 degrees. So, the actual slope would be tan(60 degrees) relative to the line connecting (2,1) to the center, which is along the x-axis.So, if the angle between the tangent and the x-axis is 60 degrees, then the slope would be tan(60 degrees) = sqrt(3). But according to the parametric approach, the slope is sqrt(3)/3. So, which one is correct?Wait, perhaps I'm miscalculating the angle. Let me think.The line connecting (2,1) to the center is along the x-axis. The tangent line makes a 60-degree angle with this line. So, the slope of the tangent line is tan(60 degrees) = sqrt(3). But wait, in reality, the tangent line is below the x-axis, so the slope would be negative. Wait, no, the tangent lines are on both sides.Wait, actually, the angle between the tangent line and the x-axis is 60 degrees above and below. So, the slopes would be tan(60 degrees) = sqrt(3) and tan(-60 degrees) = -sqrt(3). But according to the parametric approach, the slopes are ±sqrt(3)/3.This is confusing. Let me try to plot it.Imagine the circle centered at (0,1) with radius 1. The point (2,1) is to the right of the circle. The tangent lines from (2,1) will touch the circle at points where the radius is perpendicular to the tangent line.The slope of the radius at the point of tangency is (y - 1)/x. The slope of the tangent line is k. Since they are perpendicular, their slopes multiply to -1:[(y - 1)/x] * k = -1But y - 1 = sinθ, x = cosθ, so:(sinθ / cosθ) * k = -1 => k = -cosθ / sinθ = -cotθBut from the line equation, k = (y - 1)/(x - 2) = sinθ / (cosθ - 2)So, -cotθ = sinθ / (cosθ - 2)=> -cosθ / sinθ = sinθ / (cosθ - 2)Cross-multiplying:-cosθ (cosθ - 2) = sin²θ=> -cos²θ + 2cosθ = sin²θBut sin²θ = 1 - cos²θ, so:-cos²θ + 2cosθ = 1 - cos²θSimplify:2cosθ = 1 => cosθ = 1/2 => θ = π/3 or 5π/3So, at θ = π/3:k = sin(π/3) / (cos(π/3) - 2) = (sqrt(3)/2) / (1/2 - 2) = (sqrt(3)/2) / (-3/2) = -sqrt(3)/3At θ = 5π/3:k = sin(5π/3) / (cos(5π/3) - 2) = (-sqrt(3)/2) / (1/2 - 2) = (-sqrt(3)/2) / (-3/2) = sqrt(3)/3So, the maximum k is sqrt(3)/3 and the minimum is -sqrt(3)/3.Therefore, the initial thought of k = ±sqrt(2) was incorrect. The correct values are ±sqrt(3)/3.Now, moving on to part (2): Find the maximum and minimum values of 2x + y.This is a linear expression, so its maximum and minimum on the circle can be found using the method of Lagrange multipliers or by recognizing that the maximum and minimum occur where the line 2x + y = m is tangent to the circle.Let me use the tangent line approach.The expression 2x + y = m represents a family of lines with slope -2. The maximum and minimum values of m occur when the line is tangent to the circle x² + (y - 1)² = 1.To find the condition for tangency, the distance from the center of the circle to the line should equal the radius.The center is (0,1), and the line is 2x + y - m = 0.The distance from (0,1) to this line is |2*0 + 1 - m| / sqrt(2² + 1²) = |1 - m| / sqrt(5)This distance must equal the radius, which is 1:|1 - m| / sqrt(5) = 1 => |1 - m| = sqrt(5)Therefore, 1 - m = sqrt(5) or 1 - m = -sqrt(5)Solving for m:Case 1: 1 - m = sqrt(5) => m = 1 - sqrt(5)Case 2: 1 - m = -sqrt(5) => m = 1 + sqrt(5)Therefore, the maximum value of 2x + y is 1 + sqrt(5), and the minimum value is 1 - sqrt(5).Alternatively, using parametric equations:x = cosθ, y = 1 + sinθSo, 2x + y = 2cosθ + 1 + sinθWe can write this as 2cosθ + sinθ + 1To find the maximum and minimum of 2cosθ + sinθ, we can use the amplitude method.The expression A cosθ + B sinθ has a maximum of sqrt(A² + B²) and a minimum of -sqrt(A² + B²).Here, A = 2, B = 1, so the maximum of 2cosθ + sinθ is sqrt(4 + 1) = sqrt(5), and the minimum is -sqrt(5).Therefore, 2x + y = 2cosθ + sinθ + 1 has a maximum of 1 + sqrt(5) and a minimum of 1 - sqrt(5).So, the maximum value is 1 + sqrt(5), and the minimum is 1 - sqrt(5).In summary:(1) The maximum and minimum values of k are sqrt(3)/3 and -sqrt(3)/3.(2) The maximum and minimum values of 2x + y are 1 + sqrt(5) and 1 - sqrt(5).But wait, in the initial problem, the user wrote:"Let k be the slope of the line connecting point P(x, y) and point (2, 1). When the line y - 1 = k(x - 2) is tangent to the circle, k reaches its maximum and minimum values. By solving x² + (y - 1)² = 1, we get k = ±√2, thus the maximum value of k is √2, and the minimum value of k is -√2."But according to my calculations, it's ±sqrt(3)/3. So, the user's initial thought was incorrect. Similarly, for part (2), the user wrote:"Let 2x + y = m, then m represents the y-intercept of the line 2x + y = m. When this line is tangent to the circle, m reaches its maximum and minimum values. By solving x² + (y - 1)² = 1, we get m = 1 ± √5, thus the maximum value of 2x + y is 1 + √5, and the minimum value of 2x + y is 1 - √5."Which is correct. So, the user made a mistake in part (1) but was correct in part (2).Therefore, the correct answers are:(1) Maximum k = sqrt(3)/3, Minimum k = -sqrt(3)/3(2) Maximum 2x + y = 1 + sqrt(5), Minimum 2x + y = 1 - sqrt(5)But the user initially thought k = ±sqrt(2). So, perhaps they made a mistake in their calculation.In conclusion, after careful consideration and multiple methods, the correct maximum and minimum values are as above.

answer：Okay, so I have this problem about a sequence defined by some initial terms and a recurrence relation. Let me try to understand what it's asking and figure out how to approach it step by step.First, the sequence is given as {a_n} with a₁ = 2, a₂ = 3, and for all natural numbers n, a_{n+2} = 3a_{n+1} - 2a_n. So, it's a second-order linear recurrence relation. I remember that these can often be solved by finding the characteristic equation, but maybe I don't need to solve it directly for this problem.Part (1) asks me to define d_n = a_{n+1} - a_n and prove that {d_n} is a geometric progression. Hmm, okay. So, I need to show that the ratio d_{n+1}/d_n is constant for all n. Let me write down what d_{n+1} and d_n are in terms of the sequence {a_n}.So, d_n = a_{n+1} - a_n, and d_{n+1} = a_{n+2} - a_{n+1}. Now, using the recurrence relation given, a_{n+2} = 3a_{n+1} - 2a_n. Let me substitute that into the expression for d_{n+1}:d_{n+1} = a_{n+2} - a_{n+1} = (3a_{n+1} - 2a_n) - a_{n+1} = 2a_{n+1} - 2a_n = 2(a_{n+1} - a_n) = 2d_n.Oh, so d_{n+1} = 2d_n. That means the ratio d_{n+1}/d_n = 2, which is constant. Therefore, {d_n} is a geometric progression with common ratio 2. That seems straightforward. I should also check the initial term to make sure.Given a₁ = 2 and a₂ = 3, so d₁ = a₂ - a₁ = 3 - 2 = 1. So, the first term of the GP is 1, and the common ratio is 2. Therefore, d_n = 1 * 2^{n-1} = 2^{n-1}. That makes sense.Alright, part (1) seems done. Now, moving on to part (2). It says that if {1/a_n} is a sequence whose sum of the first n terms is S_n, prove that S_n < 3/2.So, I need to consider the series S_n = 1/a₁ + 1/a₂ + ... + 1/a_n and show that this sum is always less than 3/2, regardless of how large n gets.First, maybe I should find an expression for a_n. Since I have the recurrence relation and the initial terms, perhaps I can solve the recurrence to find a closed-form expression for a_n.The recurrence is a_{n+2} = 3a_{n+1} - 2a_n. This is a linear homogeneous recurrence relation with constant coefficients. To solve it, I can find the characteristic equation.The characteristic equation is r² = 3r - 2, which simplifies to r² - 3r + 2 = 0. Let's factor this: (r - 1)(r - 2) = 0, so the roots are r = 1 and r = 2.Therefore, the general solution to the recurrence is a_n = A(1)^n + B(2)^n, where A and B are constants determined by the initial conditions.Let me plug in the initial terms to find A and B.For n = 1: a₁ = 2 = A(1) + B(2) => A + 2B = 2.For n = 2: a₂ = 3 = A(1)^2 + B(2)^2 => A + 4B = 3.Now, I have a system of equations:1) A + 2B = 22) A + 4B = 3Subtracting equation 1 from equation 2: (A + 4B) - (A + 2B) = 3 - 2 => 2B = 1 => B = 1/2.Substituting B = 1/2 into equation 1: A + 2*(1/2) = 2 => A + 1 = 2 => A = 1.So, the general solution is a_n = 1 + (1/2)(2^n) = 1 + 2^{n-1}.Let me verify this with the initial terms:For n = 1: a₁ = 1 + 2^{0} = 1 + 1 = 2. Correct.For n = 2: a₂ = 1 + 2^{1} = 1 + 2 = 3. Correct.For n = 3: a₃ = 3a₂ - 2a₁ = 3*3 - 2*2 = 9 - 4 = 5. Using the formula: 1 + 2^{2} = 1 + 4 = 5. Correct.Good, so the closed-form expression is a_n = 1 + 2^{n-1}.Therefore, 1/a_n = 1/(1 + 2^{n-1}).Now, the sum S_n = sum_{k=1}^{n} 1/(1 + 2^{k-1}).I need to show that S_n < 3/2 for all n.Let me compute the first few terms to get an idea:For n = 1: S₁ = 1/2 = 0.5 < 1.5. True.For n = 2: S₂ = 1/2 + 1/3 ≈ 0.5 + 0.333 ≈ 0.833 < 1.5. True.For n = 3: S₃ = 1/2 + 1/3 + 1/5 ≈ 0.5 + 0.333 + 0.2 ≈ 1.033 < 1.5. True.For n = 4: S₄ = previous + 1/9 ≈ 1.033 + 0.111 ≈ 1.144 < 1.5. True.For n = 5: S₅ ≈ 1.144 + 1/17 ≈ 1.144 + 0.0588 ≈ 1.2028 < 1.5. True.It seems like the sum is increasing, but very slowly, and it's bounded above by 1.5.I need to show that the infinite series sum_{k=1}^{∞} 1/(1 + 2^{k-1}) converges to a value less than 3/2.Alternatively, since the partial sums S_n are increasing and bounded above by 3/2, they must converge to a limit less than or equal to 3/2.But I need to show that S_n < 3/2 for all n, not just in the limit.Let me consider the infinite series:Sum_{k=1}^{∞} 1/(1 + 2^{k-1}) = 1/2 + 1/3 + 1/5 + 1/9 + 1/17 + ... I can try to compare this series to a geometric series.Note that 1/(1 + 2^{k-1}) < 1/2^{k-1} for k ≥ 2.Because 1 + 2^{k-1} > 2^{k-1}, so their reciprocals satisfy 1/(1 + 2^{k-1}) < 1/2^{k-1}.Therefore, for k ≥ 2, 1/(1 + 2^{k-1}) < 1/2^{k-1}.So, the sum from k=2 to ∞ of 1/(1 + 2^{k-1}) < sum from k=2 to ∞ of 1/2^{k-1}.The sum from k=2 to ∞ of 1/2^{k-1} is a geometric series with first term 1/2 and ratio 1/2, so it sums to (1/2)/(1 - 1/2) = 1.Therefore, sum from k=2 to ∞ of 1/(1 + 2^{k-1}) < 1.Adding the first term, which is 1/2, the total sum is less than 1/2 + 1 = 3/2.Therefore, the infinite series converges to a value less than 3/2. Hence, all partial sums S_n are less than 3/2.Wait, but let me make sure that this is rigorous. I compared the tail of the series starting from k=2 to a geometric series, and showed that the tail is less than 1. Then, adding the first term 1/2, the total sum is less than 3/2.Yes, that seems correct.Alternatively, I can write S_n = 1/2 + sum_{k=2}^{n} 1/(1 + 2^{k-1}).Since each term 1/(1 + 2^{k-1}) < 1/2^{k-1}, the sum from k=2 to n is less than sum_{k=2}^{n} 1/2^{k-1}.The sum from k=2 to n of 1/2^{k-1} is a finite geometric series with first term 1/2 and ratio 1/2, so it sums to (1/2)(1 - (1/2)^{n-1}) / (1 - 1/2) ) = (1/2)(1 - (1/2)^{n-1}) / (1/2) ) = 1 - (1/2)^{n-1}.Therefore, sum_{k=2}^{n} 1/(1 + 2^{k-1}) < 1 - (1/2)^{n-1}.Thus, S_n = 1/2 + sum_{k=2}^{n} 1/(1 + 2^{k-1}) < 1/2 + 1 - (1/2)^{n-1} = 3/2 - (1/2)^{n-1} < 3/2.Since (1/2)^{n-1} is positive, 3/2 - (1/2)^{n-1} is less than 3/2.Therefore, S_n < 3/2 for all n.This seems to confirm the result.Alternatively, another approach could be to observe that the sequence {1/a_n} is decreasing and positive, so the partial sums are increasing and bounded above, hence convergent. But since we need to show that each partial sum is less than 3/2, the comparison with the geometric series is sufficient.I think this should be a solid proof.

answer：Alright, so I've got these two math problems to solve. They look like they involve some square roots and basic operations. Let me take them one at a time and see if I can figure them out.Starting with the first problem: (sqrt{5}+sqrt{3})(sqrt{5}-sqrt{3})+(sqrt{3}-2)^{2}. Hmm, okay, so there are two parts here: the product of two binomials and then a squared binomial. I remember something about the difference of squares from algebra. Maybe that can help simplify the first part.So, for the first part, (sqrt{5}+sqrt{3})(sqrt{5}-sqrt{3}), if I recall correctly, this is a difference of squares. The formula is (a+b)(a-b) = a^2 - b^2. Applying that here, it should be (sqrt{5})^2 - (sqrt{3})^2. Let's compute that: (sqrt{5})^2 is 5, and (sqrt{3})^2 is 3. So, 5 - 3 equals 2. Okay, so the first part simplifies to 2.Now, the second part is (sqrt{3}-2)^2. I think I need to expand this using the formula (a - b)^2 = a^2 - 2ab + b^2. Let's apply that: (sqrt{3})^2 - 2*sqrt{3}*2 + 2^2. Calculating each term: (sqrt{3})^2 is 3, 2*sqrt(3)*2 is 4*sqrt(3), and 2^2 is 4. So, putting it all together: 3 - 4*sqrt(3) + 4. Combining like terms, 3 + 4 is 7, so it becomes 7 - 4*sqrt(3).Now, adding the two parts together: the first part was 2, and the second part is 7 - 4*sqrt(3). So, 2 + 7 is 9, and then we have -4*sqrt(3). So, the entire expression simplifies to 9 - 4*sqrt(3). That seems straightforward.Moving on to the second problem: sqrt{18} - 4sqrt{frac{1}{2}} + sqrt{24} ÷ sqrt{3}. Okay, this one has square roots and division. Let me break it down term by term.First, sqrt{18}. I know that 18 can be broken down into 9*2, and since 9 is a perfect square, this simplifies to sqrt{9*2} = sqrt{9}*sqrt{2} = 3sqrt{2}.Next, -4sqrt{frac{1}{2}}. Hmm, square roots of fractions can be tricky. I think sqrt{frac{1}{2}} is the same as frac{sqrt{1}}{sqrt{2}} = frac{1}{sqrt{2}}. But it's often rationalized to get rid of the square root in the denominator. So, multiplying numerator and denominator by sqrt(2), we get frac{sqrt{2}}{2}. Therefore, -4sqrt{frac{1}{2}} = -4*frac{sqrt{2}}{2} = -2sqrt{2}.Now, the last term is sqrt{24} ÷ sqrt{3}. I remember that dividing square roots can be simplified by dividing the numbers under the roots first. So, sqrt{24} ÷ sqrt{3} = sqrt{frac{24}{3}} = sqrt{8}. Simplifying sqrt(8), since 8 is 4*2, we get sqrt{4*2} = sqrt{4}*sqrt{2} = 2sqrt{2}.Putting it all together: the first term is 3sqrt{2}, the second term is -2sqrt{2}, and the third term is 2sqrt{2}. So, adding them up: 3sqrt{2} - 2sqrt{2} + 2sqrt{2}. Combining like terms, 3sqrt{2} - 2sqrt{2} is sqrt{2}, and then adding 2sqrt{2} gives 3sqrt{2}. So, the entire expression simplifies to 3sqrt{2}.Wait, let me double-check to make sure I didn't make any mistakes. For the first problem, the difference of squares gave me 2, and expanding the squared binomial gave me 7 - 4sqrt{3}. Adding them together gives 9 - 4sqrt{3}, which seems right.For the second problem, breaking down sqrt(18) into 3sqrt{2} is correct. Then, handling the -4sqrt{frac{1}{2}} by rationalizing the denominator to get -2sqrt{2} makes sense. Finally, simplifying sqrt(24)/sqrt(3) to sqrt(8) and then to 2sqrt{2} is accurate. Adding all those terms together: 3sqrt{2} - 2sqrt{2} + 2sqrt{2} equals 3sqrt{2}. Yeah, that seems correct.I think I've got both problems solved correctly. It's all about recognizing the patterns and applying the right formulas step by step. Square roots can be tricky, but breaking them down into simpler parts really helps.

answer：Okay, so I have this problem where I need to find a point P inside triangle ABC that minimizes the expression (BC/PD) + (CA/PE) + (AB/PF), where PD, PE, and PF are the perpendicular distances from P to the sides BC, CA, and AB respectively. Hmm, interesting. I remember something about minimizing expressions with distances in triangles, maybe related to the centroid or incenter? Let me think.First, let me recall some basic properties. The area of triangle ABC can be expressed in terms of the base and height. For example, the area is (1/2)*BC*height from A, right? Similarly, it can be expressed using other sides and their corresponding heights. Maybe this can help me relate PD, PE, and PF to the area.Let me denote BC as a, CA as b, and AB as c. Let the area of triangle ABC be P. Then, the area can also be written as (1/2)*a*PD + (1/2)*b*PE + (1/2)*c*PF. So, if I multiply both sides by 2, I get 2P = a*PD + b*PE + c*PF. That's a useful relation.Now, the expression I need to minimize is (a/PD) + (b/PE) + (c/PF). I wonder if there's an inequality that relates these terms. Cauchy-Schwarz comes to mind. The Cauchy-Schwarz inequality states that for any real numbers, (x1y1 + x2y2 + x3y3)^2 ≤ (x1^2 + x2^2 + x3^2)(y1^2 + y2^2 + y3^2). Maybe I can apply this here.Let me set x1 = sqrt(a), x2 = sqrt(b), x3 = sqrt(c), and y1 = sqrt(a)/PD, y2 = sqrt(b)/PE, y3 = sqrt(c)/PF. Then, the left side of Cauchy-Schwarz would be (a/PD + b/PE + c/PF), which is exactly the expression I need. The right side would be (a + b + c) times (a/(PD)^2 + b/(PE)^2 + c/(PF)^2). Hmm, not sure if that helps directly.Wait, maybe another approach. Let's consider the product of (a*PD + b*PE + c*PF) and (a/PD + b/PE + c/PF). From the Cauchy-Schwarz inequality, this product should be greater than or equal to (a + b + c)^2. Since I know that a*PD + b*PE + c*PF = 2P, then multiplying by (a/PD + b/PE + c/PF) gives me 2P*(expression) ≥ (a + b + c)^2. Therefore, the expression is greater than or equal to (a + b + c)^2 / (2P). So, the minimum value of the expression is (a + b + c)^2 / (2P).But when does equality hold in Cauchy-Schwarz? It holds when the ratios are equal, meaning a/PD = b/PE = c/PF. So, PD = PE = PF. That means the point P is equidistant from all three sides of the triangle. In a triangle, the point that is equidistant from all three sides is the incenter. So, P must be the incenter of triangle ABC.Let me verify this. If P is the incenter, then PD, PE, PF are the inradius. The inradius r is given by P = r*s, where s is the semi-perimeter, s = (a + b + c)/2. So, r = P/s. Then, PD = PE = PF = r. Plugging back into the expression, we get (a + b + c)/r. Since r = P/s, this becomes (a + b + c)/(P/s) = s*(a + b + c)/P. But s = (a + b + c)/2, so this becomes (a + b + c)^2 / (2P), which matches the earlier result.So, it seems that the minimal value is indeed (a + b + c)^2 / (2P), achieved when P is the incenter of triangle ABC. That makes sense because the incenter is the point that minimizes the sum of distances to the sides in some sense, and here we're dealing with a sum involving reciprocals of those distances.I think I've got it. The point P that minimizes the given expression is the incenter, and the minimal value is (a + b + c)^2 divided by twice the area of the triangle.

answer：Okay, so I have this equation here: (3x^2 - 12y^2 = 0). I need to solve for (x) and (y). Hmm, let me think about how to approach this. First, I notice that both terms have a common factor of 3. Maybe I can factor that out to simplify the equation. Let's try that:(3x^2 - 12y^2 = 0)Factoring out the 3:(3(x^2 - 4y^2) = 0)Now, since 3 isn't zero, I can divide both sides by 3 to simplify further:(x^2 - 4y^2 = 0)Alright, now I have (x^2 - 4y^2 = 0). This looks like a difference of squares because (x^2) is a square and (4y^2) is also a square (since (4y^2 = (2y)^2)). The difference of squares formula is (a^2 - b^2 = (a - b)(a + b)). Let me apply that here.So, (x^2 - (2y)^2 = (x - 2y)(x + 2y)). Therefore, the equation becomes:((x - 2y)(x + 2y) = 0)Now, for the product of two factors to be zero, at least one of the factors must be zero. So, I can set each factor equal to zero and solve for (x) in terms of (y).First factor:(x - 2y = 0)Adding (2y) to both sides:(x = 2y)Second factor:(x + 2y = 0)Subtracting (2y) from both sides:(x = -2y)So, the solutions are (x = 2y) and (x = -2y). Let me check the answer choices to see which one matches.A) (x = y) and (x = -y) – Hmm, that's not what I got.B) (x = 2y) and (x = -y) – Close, but the second part is (x = -y), which isn't what I found.C) (x = 2y) and (x = -2y) – Yes, this matches my solutions.D) (x = 3y) and (x = -3y) – No, that's not correct.So, the correct answer should be option C.

answer：Alright, so I have this problem with two parts, (a) and (b). Both parts are about residues modulo ( mk ) when multiplying sequences of integers. Let me try to understand what each part is asking.Starting with part (a): It says that if ( (m, k) = 1 ), which means ( m ) and ( k ) are coprime, then there exist integers ( a_1, a_2, ldots, a_m ) and ( b_1, b_2, ldots, b_k ) such that each product ( a_i b_j ) gives a different residue when divided by ( mk ). So, essentially, I need to find sequences ( a ) and ( b ) such that all their pairwise products are distinct modulo ( mk ).Hmm, okay. Since ( m ) and ( k ) are coprime, maybe I can use the Chinese Remainder Theorem here. The Chinese Remainder Theorem tells us that if two moduli are coprime, then the system of congruences has a unique solution modulo the product. Maybe I can construct the sequences ( a_i ) and ( b_j ) in such a way that their products cover all residues modulo ( mk ).Let me think about how to construct such sequences. Maybe if I choose ( a_i ) such that they are distinct modulo ( m ) and ( b_j ) such that they are distinct modulo ( k ). Then, their products might cover all residues modulo ( mk ).Wait, but how exactly? Let me try to formalize this. Suppose I set ( a_i = i ) modulo ( m ) and ( b_j = j ) modulo ( k ). Then, their products ( a_i b_j ) would be ( ij ) modulo ( mk ). But I'm not sure if this guarantees that all products are distinct.Alternatively, maybe I can use the fact that since ( m ) and ( k ) are coprime, I can find integers that are invertible modulo ( m ) and ( k ). Perhaps setting ( a_i ) to be multiples of ( k ) plus some invertible element modulo ( m ), and similarly for ( b_j ).Wait, that might work. Let me try to define ( a_i ) as ( ik + 1 ) and ( b_j ) as ( jm + 1 ). Then, ( a_i ) modulo ( m ) would be 1, and ( b_j ) modulo ( k ) would be 1. But their products would be ( (ik + 1)(jm + 1) ). Let me compute this:( (ik + 1)(jm + 1) = ijk m + ik + jm + 1 ).Since we're working modulo ( mk ), the term ( ijk m ) is 0 modulo ( mk ). So, we have ( ik + jm + 1 ) modulo ( mk ). Hmm, but does this ensure that all products are distinct?Let me check. Suppose ( (ik + 1)(jm + 1) equiv (sk + 1)(tm + 1) pmod{mk} ). Then, expanding both sides:( ijk m + ik + jm + 1 equiv sktm + sk + tm + 1 pmod{mk} ).Subtracting the right side from the left:( ijk m - sktm + ik - sk + jm - tm equiv 0 pmod{mk} ).Factor out terms:( km(ij - st) + k(i - s) + m(j - t) equiv 0 pmod{mk} ).Since ( m ) and ( k ) are coprime, if ( km(ij - st) + k(i - s) + m(j - t) ) is divisible by ( mk ), then each term must be divisible by ( m ) and ( k ) respectively.Looking at the term ( km(ij - st) ), it's divisible by ( mk ), so it doesn't affect the congruence. Then, ( k(i - s) ) must be divisible by ( m ), and ( m(j - t) ) must be divisible by ( k ).Since ( gcd(m, k) = 1 ), ( k(i - s) equiv 0 pmod{m} ) implies ( i equiv s pmod{m} ). But ( i ) and ( s ) are both between 1 and ( m ), so ( i = s ). Similarly, ( m(j - t) equiv 0 pmod{k} ) implies ( j equiv t pmod{k} ), so ( j = t ).Therefore, ( (i, j) = (s, t) ), meaning all products ( a_i b_j ) are distinct modulo ( mk ). So, this construction works. Therefore, part (a) is proven.Moving on to part (b): If ( (m, k) > 1 ), then for any integers ( a_1, a_2, ldots, a_m ) and ( b_1, b_2, ldots, b_k ), there must be two products ( a_i b_j ) and ( a_x b_t ) with ( (i, j) neq (x, t) ) that give the same residue modulo ( mk ).So, this is the opposite of part (a). When ( m ) and ( k ) are not coprime, it's impossible to have all products distinct modulo ( mk ). I need to show that no matter how you choose the sequences ( a ) and ( b ), there will always be some collision in the residues.Let me think about the pigeonhole principle. The number of possible residues modulo ( mk ) is ( mk ). The number of products ( a_i b_j ) is also ( mk ). So, if all residues were distinct, they would cover all residues exactly once.But wait, that's only if the sequences are constructed in a way that their products cover all residues. However, when ( m ) and ( k ) are not coprime, maybe there's some overlap forced by the common divisors.Let me consider the common divisor ( d = gcd(m, k) ). So, ( d > 1 ). Then, ( m = d cdot m' ) and ( k = d cdot k' ) where ( gcd(m', k') = 1 ).Now, if I think about the residues modulo ( d ), since ( d ) divides both ( m ) and ( k ), the products ( a_i b_j ) modulo ( d ) might not be unique. But how does this affect the residues modulo ( mk )?Alternatively, maybe I can use the fact that the number of distinct residues modulo ( mk ) is less than ( mk ) when ( m ) and ( k ) are not coprime. Wait, no, the number of residues modulo ( mk ) is always ( mk ), regardless of whether ( m ) and ( k ) are coprime.Hmm, maybe I need to think about the structure of the residues. Since ( d > 1 ), there are multiple residues that are congruent modulo ( d ). Perhaps, the products ( a_i b_j ) must share some common factors, leading to overlaps.Wait, another approach: Suppose that all products ( a_i b_j ) are distinct modulo ( mk ). Then, in particular, they must be distinct modulo ( d ). But since ( d ) divides both ( m ) and ( k ), the number of distinct residues modulo ( d ) is ( d ). However, the number of products is ( mk = d^2 m' k' ), which is much larger than ( d ). Therefore, by the pigeonhole principle, there must be multiple products that are congruent modulo ( d ).But wait, that's not directly leading to a contradiction because the residues modulo ( mk ) could still be distinct even if some residues modulo ( d ) are the same. I need a better approach.Let me think about the Chinese Remainder Theorem again. Since ( m ) and ( k ) are not coprime, the ring ( mathbb{Z}/mkmathbb{Z} ) is not isomorphic to ( mathbb{Z}/mmathbb{Z} times mathbb{Z}/kmathbb{Z} ). Therefore, the mapping from ( a_i b_j ) to their residues modulo ( m ) and ( k ) might not be injective.Alternatively, consider that if ( d = gcd(m, k) ), then ( mathbb{Z}/mkmathbb{Z} ) has a non-trivial annihilator, specifically multiples of ( d ). This might cause some products to coincide.Wait, maybe I can use the fact that the number of invertible elements modulo ( mk ) is less when ( m ) and ( k ) are not coprime. But I'm not sure if that's directly applicable.Another idea: If ( d > 1 ), then ( m ) and ( k ) share a common factor. Suppose ( p ) is a prime dividing both ( m ) and ( k ). Then, consider the residues modulo ( p ). Since ( p ) divides both ( m ) and ( k ), the products ( a_i b_j ) modulo ( p ) must be considered.But how does this lead to a contradiction? Maybe if I fix ( p ), then the number of distinct residues modulo ( p ) is limited, and with ( mk ) products, some must coincide modulo ( p ), hence modulo ( mk ).Wait, but residues modulo ( p ) don't directly translate to residues modulo ( mk ). Maybe I need a different approach.Let me think about the total number of possible products. If ( m ) and ( k ) are not coprime, the number of distinct products modulo ( mk ) is less than ( mk ). Therefore, by the pigeonhole principle, some products must coincide.But I need to formalize this. Let me consider the mapping from the set ( {1, 2, ldots, m} times {1, 2, ldots, k} ) to ( mathbb{Z}/mkmathbb{Z} ) given by ( (i, j) mapsto a_i b_j ). If this mapping is injective, then all residues are distinct. But I need to show that it's impossible when ( gcd(m, k) > 1 ).Suppose, for contradiction, that all products ( a_i b_j ) are distinct modulo ( mk ). Then, the mapping is injective, implying that the number of distinct residues is exactly ( mk ). However, since ( gcd(m, k) = d > 1 ), the ring ( mathbb{Z}/mkmathbb{Z} ) has zero divisors, meaning there exist non-zero elements whose product is zero. This might cause some products to coincide.Wait, but how does that lead to a contradiction? Maybe I need to consider the structure of the sequences ( a_i ) and ( b_j ). If ( d ) divides both ( m ) and ( k ), then ( d ) divides ( mk ). Therefore, any multiple of ( d ) modulo ( mk ) will be a zero divisor.Suppose that one of the ( a_i ) is divisible by ( d ). Then, all products ( a_i b_j ) will be divisible by ( d ), meaning they are congruent to 0 modulo ( d ). But since ( d > 1 ), there are multiple residues modulo ( mk ) that are congruent to 0 modulo ( d ). Therefore, multiple products could be congruent to the same residue modulo ( mk ).Wait, but I'm assuming that all products are distinct modulo ( mk ). So, if one ( a_i ) is divisible by ( d ), then all products ( a_i b_j ) are divisible by ( d ), but since ( d ) divides ( mk ), these products are 0 modulo ( d ), but not necessarily 0 modulo ( mk ). However, there are ( k ) such products, each congruent to 0 modulo ( d ), but possibly different modulo ( mk ).Hmm, maybe this isn't the right path. Let me try another approach.Consider the sequences ( a_i ) and ( b_j ). Since ( m ) and ( k ) are not coprime, let ( d = gcd(m, k) ). Then, ( m = d cdot m' ) and ( k = d cdot k' ) with ( gcd(m', k') = 1 ).Now, consider the residues of ( a_i ) modulo ( m' ) and ( b_j ) modulo ( k' ). Since ( m' ) and ( k' ) are coprime, by part (a), we can have sequences ( a_i ) and ( b_j ) such that their products modulo ( m' k' ) are distinct. But since ( mk = d^2 m' k' ), the residues modulo ( mk ) are more refined.Wait, maybe I can use the fact that the number of distinct residues modulo ( mk ) is larger than the number of distinct residues modulo ( m' k' ). But I'm not sure.Alternatively, think about the fact that since ( d > 1 ), the number of distinct residues modulo ( mk ) that are multiples of ( d ) is ( mk / d ). But the number of products ( a_i b_j ) is ( mk ), so if all products are distinct modulo ( mk ), then exactly ( mk / d ) of them must be multiples of ( d ). But this might not necessarily lead to a contradiction.Wait, maybe I need to consider the fact that if ( d > 1 ), then the sequences ( a_i ) and ( b_j ) cannot be chosen such that their products cover all residues modulo ( mk ). Because the presence of a common divisor introduces dependencies that prevent the products from being unique.Alternatively, let's use the concept of the least common multiple. Since ( gcd(m, k) = d ), the least common multiple ( text{lcm}(m, k) = frac{mk}{d} ). Therefore, the number of distinct residues modulo ( text{lcm}(m, k) ) is ( frac{mk}{d} ), which is less than ( mk ) when ( d > 1 ). Therefore, by the pigeonhole principle, some residues must coincide.But wait, we're considering residues modulo ( mk ), not modulo ( text{lcm}(m, k) ). So, maybe this isn't directly applicable.Another idea: If ( d > 1 ), then ( m ) and ( k ) share a common factor, say ( p ). Then, consider the residues modulo ( p ). Since ( p ) divides both ( m ) and ( k ), the products ( a_i b_j ) modulo ( p ) must be considered. There are ( mk ) products, but only ( p ) residues modulo ( p ). Therefore, by the pigeonhole principle, at least ( mk / p ) products must be congruent modulo ( p ).But how does this affect residues modulo ( mk )? If two products are congruent modulo ( p ), they might not necessarily be congruent modulo ( mk ). However, if they are congruent modulo ( p ), and also congruent modulo ( mk / p ), then by the Chinese Remainder Theorem, they would be congruent modulo ( mk ).Wait, maybe I can use the fact that if two products are congruent modulo both ( m ) and ( k ), then they are congruent modulo ( mk ). But I need to show that two products are congruent modulo ( mk ).Alternatively, consider that since ( d > 1 ), the ring ( mathbb{Z}/mkmathbb{Z} ) has non-trivial idempotents, which might cause some products to coincide.I'm getting a bit stuck here. Maybe I need to think about the problem differently. Let's consider that if ( gcd(m, k) = d > 1 ), then ( m = d cdot m' ) and ( k = d cdot k' ) with ( gcd(m', k') = 1 ).Now, consider the residues modulo ( m' ) and ( k' ). Since ( m' ) and ( k' ) are coprime, by part (a), we can have sequences ( a_i ) and ( b_j ) such that their products modulo ( m' k' ) are distinct. However, since ( mk = d^2 m' k' ), the residues modulo ( mk ) are more refined.But I'm not sure how this helps. Maybe I need to consider that the number of distinct residues modulo ( mk ) is ( mk ), but the number of distinct residues modulo ( m' k' ) is ( m' k' ). Since ( mk = d^2 m' k' ), the number of residues modulo ( mk ) is larger, but the number of distinct products modulo ( m' k' ) is limited.Wait, perhaps if I fix the residues modulo ( m' ) and ( k' ), the products modulo ( mk ) can still be distinct. But I'm not sure.Another approach: Suppose that all products ( a_i b_j ) are distinct modulo ( mk ). Then, in particular, they must be distinct modulo ( m ) and modulo ( k ). But since ( gcd(m, k) = d > 1 ), the number of distinct residues modulo ( m ) is ( m ), and modulo ( k ) is ( k ). However, the number of products is ( mk ), which is larger than both ( m ) and ( k ). Therefore, by the pigeonhole principle, some products must coincide modulo ( m ) or modulo ( k ).But how does this lead to a contradiction? If two products are congruent modulo ( m ) and modulo ( k ), then by the Chinese Remainder Theorem, they are congruent modulo ( mk ). Therefore, if two products are congruent modulo both ( m ) and ( k ), they must be congruent modulo ( mk ), which contradicts the assumption that all products are distinct modulo ( mk ).Therefore, there must be two products ( a_i b_j ) and ( a_x b_t ) such that ( a_i b_j equiv a_x b_t pmod{m} ) and ( a_i b_j equiv a_x b_t pmod{k} ), hence ( a_i b_j equiv a_x b_t pmod{mk} ).This shows that when ( gcd(m, k) > 1 ), it's impossible for all products ( a_i b_j ) to be distinct modulo ( mk ). Therefore, part (b) is proven.So, in summary, for part (a), by constructing specific sequences ( a_i ) and ( b_j ) that are designed to cover all residues modulo ( mk ) when multiplied, we ensure that all products are distinct. For part (b), the presence of a common divisor greater than 1 forces some products to coincide modulo ( mk ) due to the pigeonhole principle and the structure of the ring ( mathbb{Z}/mkmathbb{Z} ).I think I've got a good grasp on both parts now. The key idea is that when ( m ) and ( k ) are coprime, we can construct sequences that avoid overlaps in residues, but when they share a common divisor, overlaps become inevitable.