💡First, I need to understand the problem. I have a set of numbers ({0, 1, 2, 3, 4, 5}), and I need to select two distinct numbers (a) and (b) to form a complex number (a + bi). The goal is to determine how many such complex numbers can be formed.Next, I recall that a complex number (a + bi) consists of a real part (a) and an imaginary part (b). Importantly, for the number to be considered a complex number, the imaginary part (b) must not be zero. This means (b) can be any of the numbers (1, 2, 3, 4,) or (5), giving me 5 possible choices for (b).For each choice of (b), the real part (a) can be any of the remaining numbers in the set, including zero. Since (a) must be distinct from (b), there are 5 possible choices for (a) for each choice of (b).Finally, to find the total number of distinct complex numbers, I multiply the number of choices for (b) by the number of choices for (a). This gives me (5 times 5 = 25) distinct complex numbers that can be formed.

💡Okay, so I have this problem here about a complex number being purely imaginary. Let me try to figure it out step by step. The complex number given is ((m^2 - 3m + 2) + (m^2 - 2m)i). I remember that a complex number is purely imaginary if its real part is zero and the imaginary part is not zero. So, I need to set the real part equal to zero and make sure the imaginary part isn't zero. First, let's identify the real and imaginary parts. The real part is (m^2 - 3m + 2) and the imaginary part is (m^2 - 2m). So, according to the definition, for the complex number to be purely imaginary, the real part must be zero. That gives me the equation:[ m^2 - 3m + 2 = 0 ]I need to solve this quadratic equation for (m). Let me factor it. Looking at (m^2 - 3m + 2), I can factor it as:[ (m - 1)(m - 2) = 0 ]So, the solutions are (m = 1) and (m = 2). But wait, I also need to make sure that the imaginary part isn't zero because if the imaginary part is zero, then the complex number would just be zero, which isn't purely imaginary. So, I need to check when (m^2 - 2m neq 0). Let's solve (m^2 - 2m = 0) to find the values of (m) that make the imaginary part zero:[ m(m - 2) = 0 ]So, (m = 0) or (m = 2). This means that for the imaginary part to be non-zero, (m) cannot be 0 or 2. Going back to the solutions we found earlier, (m = 1) and (m = 2). But (m = 2) makes the imaginary part zero, which we don't want. So, (m = 2) is not a valid solution. That leaves us with (m = 1). Let me check this. If (m = 1), then the real part is:[ (1)^2 - 3(1) + 2 = 1 - 3 + 2 = 0 ]And the imaginary part is:[ (1)^2 - 2(1) = 1 - 2 = -1 ]Which is not zero, so that works. So, the only valid solution is (m = 1). Looking back at the options, A is 2, B is 1, C is 1 or 2, and D is 0. So, the correct answer should be B: 1.Wait, just to make sure I didn't make any mistakes. Let me go through it again. 1. Real part: (m^2 - 3m + 2 = 0) leads to (m = 1) or (m = 2).2. Imaginary part: (m^2 - 2m neq 0) leads to (m neq 0) and (m neq 2).3. So, from the real part solutions, (m = 2) is excluded because it makes the imaginary part zero.4. Therefore, only (m = 1) is valid.Yes, that seems correct. So, I think I'm confident that the answer is B: 1.**Final Answer**boxed{B}

💡Okay, so I have this problem about an ellipse centered at the origin with foci on the x-axis. The point P(2, √3) lies on the ellipse, and the distances |PF₁|, |F₁F₂|, and |PF₂| form an arithmetic sequence. I need to find the equation of the ellipse from the given options.First, let me recall some properties of an ellipse. The standard equation of an ellipse centered at the origin with major axis along the x-axis is (x²/a²) + (y²/b²) = 1, where a is the semi-major axis, b is the semi-minor axis, and the distance from the center to each focus is c, with c² = a² - b².Given that the foci F₁ and F₂ are on the x-axis, their coordinates should be (-c, 0) and (c, 0) respectively.Now, the point P(2, √3) lies on the ellipse, so plugging this into the ellipse equation gives:(2²)/a² + (√3²)/b² = 1=> 4/a² + 3/b² = 1That's one equation relating a² and b².Next, the problem states that |PF₁|, |F₁F₂|, and |PF₂| form an arithmetic sequence. Let me unpack that.An arithmetic sequence means that the difference between consecutive terms is constant. So, if we denote the three terms as t₁, t₂, t₃, then t₂ - t₁ = t₃ - t₂, which implies 2t₂ = t₁ + t₃.In this case, the terms are |PF₁|, |F₁F₂|, and |PF₂|. So, 2|F₁F₂| = |PF₁| + |PF₂|.Let me compute each of these distances.First, |F₁F₂| is the distance between the two foci. Since F₁ is (-c, 0) and F₂ is (c, 0), the distance between them is 2c.Next, |PF₁| is the distance from P(2, √3) to F₁(-c, 0). Using the distance formula:|PF₁| = sqrt[(2 - (-c))² + (√3 - 0)²] = sqrt[(2 + c)² + (√3)²] = sqrt[(4 + 4c + c²) + 3] = sqrt(c² + 4c + 7)Similarly, |PF₂| is the distance from P(2, √3) to F₂(c, 0):|PF₂| = sqrt[(2 - c)² + (√3 - 0)²] = sqrt[(4 - 4c + c²) + 3] = sqrt(c² - 4c + 7)So, we have:2|F₁F₂| = |PF₁| + |PF₂|=> 2*(2c) = sqrt(c² + 4c + 7) + sqrt(c² - 4c + 7)=> 4c = sqrt(c² + 4c + 7) + sqrt(c² - 4c + 7)Hmm, this looks a bit complicated. Maybe I can square both sides to eliminate the square roots. Let's try that.Let me denote S = sqrt(c² + 4c + 7) + sqrt(c² - 4c + 7). Then, 4c = S.Squaring both sides:(4c)² = [sqrt(c² + 4c + 7) + sqrt(c² - 4c + 7)]²16c² = (c² + 4c + 7) + (c² - 4c + 7) + 2*sqrt[(c² + 4c + 7)(c² - 4c + 7)]16c² = 2c² + 14 + 2*sqrt[(c² + 4c + 7)(c² - 4c + 7)]Simplify the right side:16c² = 2c² + 14 + 2*sqrt[(c² + 4c + 7)(c² - 4c + 7)]Subtract 2c² + 14 from both sides:16c² - 2c² - 14 = 2*sqrt[(c² + 4c + 7)(c² - 4c + 7)]14c² - 14 = 2*sqrt[(c² + 4c + 7)(c² - 4c + 7)]Divide both sides by 2:7c² - 7 = sqrt[(c² + 4c + 7)(c² - 4c + 7)]Now, square both sides again to eliminate the square root:(7c² - 7)² = (c² + 4c + 7)(c² - 4c + 7)Let me compute both sides.Left side: (7c² - 7)² = [7(c² - 1)]² = 49(c² - 1)² = 49(c⁴ - 2c² + 1)Right side: (c² + 4c + 7)(c² - 4c + 7)Let me multiply these two quadratics. Notice that it's of the form (A + B)(A - B) where A = c² + 7 and B = 4c.So, (A + B)(A - B) = A² - B² = (c² + 7)² - (4c)²Compute A²: (c² + 7)² = c⁴ + 14c² + 49Compute B²: (4c)² = 16c²So, right side = c⁴ + 14c² + 49 - 16c² = c⁴ - 2c² + 49Therefore, the equation becomes:49(c⁴ - 2c² + 1) = c⁴ - 2c² + 49Let me expand the left side:49c⁴ - 98c² + 49 = c⁴ - 2c² + 49Bring all terms to the left side:49c⁴ - 98c² + 49 - c⁴ + 2c² - 49 = 0(49c⁴ - c⁴) + (-98c² + 2c²) + (49 - 49) = 048c⁴ - 96c² = 0Factor out 48c²:48c²(c² - 2) = 0So, either 48c² = 0 => c² = 0 => c = 0, which is not possible because then the foci would coincide at the center, making it a circle, but the problem specifies an ellipse with foci on the x-axis, so c ≠ 0.Or, c² - 2 = 0 => c² = 2 => c = √2 or c = -√2, but since c is a distance, c = √2.So, c = √2.Now, since c² = a² - b², we have a² = b² + c² = b² + 2.Also, from the point P(2, √3) lying on the ellipse, we have:4/a² + 3/b² = 1But since a² = b² + 2, let's substitute that into the equation:4/(b² + 2) + 3/b² = 1Let me denote b² = k for simplicity. Then:4/(k + 2) + 3/k = 1Multiply both sides by k(k + 2) to eliminate denominators:4k + 3(k + 2) = k(k + 2)4k + 3k + 6 = k² + 2k7k + 6 = k² + 2kBring all terms to one side:k² + 2k - 7k - 6 = 0k² - 5k - 6 = 0Factor the quadratic:(k - 6)(k + 1) = 0So, k = 6 or k = -1Since k = b², which cannot be negative, so k = 6. Therefore, b² = 6.Then, a² = b² + 2 = 6 + 2 = 8.So, the equation of the ellipse is:x²/8 + y²/6 = 1Looking back at the options, that's option A.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I found c = √2 correctly by solving the arithmetic sequence condition.2. Then, using the point P(2, √3), I set up the equation 4/a² + 3/b² = 1.3. Since a² = b² + c² = b² + 2, I substituted and solved for b², getting b² = 6.4. Then, a² = 8.So, the equation is x²/8 + y²/6 = 1, which is option A.Yes, that seems correct. I think I didn't make any mistakes in the calculations.**Final Answer**boxed{A}

💡Alright, let's tackle this problem step by step. It's about points on a line segment and some relationships between their lengths. I need to find the length of segment BC. Here's what I know:We have points A, B, D, C, and E arranged in that order on a straight line. So, the order is A---B---D---C---E. The total length of AE is 20 units. The problem gives me two specific relationships:1. The length of AB is 3 units plus three times the length of BD.2. The length of CE is 2 units plus twice the length of CD.I need to find the length of BC.First, let's assign some variables to make this easier. Let me denote:- Let BD = x. Then, AB = 3 + 3x.- Let CD = y. Then, CE = 2 + 2y.Now, let's express the lengths of AD and DE in terms of x and y.AD is the segment from A to D, which includes AB and BD. So, AD = AB + BD = (3 + 3x) + x = 3 + 4x.Similarly, DE is the segment from D to E, which includes CD and CE. So, DE = CD + CE = y + (2 + 2y) = 2 + 3y.Since the entire segment AE is 20 units, we can write:AE = AD + DE = (3 + 4x) + (2 + 3y) = 20.Simplifying this equation:3 + 4x + 2 + 3y = 20 5 + 4x + 3y = 20 4x + 3y = 15.So, we have the equation 4x + 3y = 15.Now, I need another equation to solve for x and y. Let's think about BC. BC is the segment between B and C. Since B is on AD and C is on DE, BC is essentially the distance from D to C minus the distance from D to B. Wait, no, that's not quite right.Actually, BC is the segment from B to C, which spans from B to D to C. So, BC = BD + DC. But BD is x and DC is y, so BC = x + y.Wait, that doesn't seem right because B is on AD and C is on DE, so BC is actually the distance from B to D plus the distance from D to C. So, yes, BC = BD + DC = x + y.But I need to find BC, which is x + y, so if I can find x and y, I can find BC.But I only have one equation: 4x + 3y = 15. I need another equation to solve for x and y.Let me think again. Maybe I can express BC in terms of x and y and then see if I can find another relationship.Wait, perhaps I can express BC in terms of the total length.Let me try to visualize the entire segment AE:A---B---D---C---ESo, AE = AB + BD + DC + CE.But wait, that's not correct because AE is just A to E, which is AD + DE.But AD is AB + BD, and DE is DC + CE.So, AE = (AB + BD) + (DC + CE) = (3 + 3x + x) + (y + 2 + 2y) = 3 + 4x + 3y + 2 = 5 + 4x + 3y = 20.Which is the same as before: 4x + 3y = 15.Hmm, so I still only have one equation. Maybe I need to find another relationship.Wait, let's think about BC. BC is the segment from B to C. Since B is on AD and C is on DE, BC is equal to AD - AB + DE - CE. Wait, that might not be correct.Alternatively, BC can be thought of as the distance from B to D plus the distance from D to C, which is x + y.But I need another equation to relate x and y.Wait, maybe I can express BC in terms of x and y and then see if I can find another equation.But I only have one equation: 4x + 3y = 15.Wait, maybe I can express BC as x + y and then find x and y in terms of BC.But that might not help directly.Alternatively, maybe I can express BC in terms of the total length.Wait, let's think differently. Let's denote BC = k. Then, since BC = x + y, we have k = x + y.So, we have:4x + 3y = 15 k = x + y.We can solve this system of equations.From the second equation, y = k - x.Substitute into the first equation:4x + 3(k - x) = 15 4x + 3k - 3x = 15 x + 3k = 15 x = 15 - 3k.Now, substitute back into y = k - x:y = k - (15 - 3k) = k - 15 + 3k = 4k - 15.Now, since lengths can't be negative, we have:x = 15 - 3k > 0 15 - 3k > 0 3k < 15 k < 5.Similarly, y = 4k - 15 > 0 4k - 15 > 0 4k > 15 k > 15/4 k > 3.75.So, k must be greater than 3.75 and less than 5.Looking at the options, the possible values are 2, 3, 4, 5. So, k must be 4 units.Wait, that seems a bit quick. Let me verify.If k = 4, then:x = 15 - 3*4 = 15 - 12 = 3 y = 4*4 - 15 = 16 - 15 = 1.So, x = 3 and y = 1.Now, let's check if these values satisfy the original relationships.AB = 3 + 3x = 3 + 3*3 = 3 + 9 = 12 BD = x = 3 So, AD = AB + BD = 12 + 3 = 15.CE = 2 + 2y = 2 + 2*1 = 4 CD = y = 1 So, DE = CD + CE = 1 + 4 = 5.Now, AE = AD + DE = 15 + 5 = 20, which matches the given total length.Also, BC = x + y = 3 + 1 = 4.So, BC is 4 units.Therefore, the answer is 4 units.

💡Okay, so I have this problem about a geometric sequence. Let me try to understand what it's asking. First, part (I) says that the sequence {a_n} is a geometric sequence and satisfies two conditions: a₂ + a₅ = 36 and a₃ · a₄ = 128. I need to find the general term formula of the sequence {a_n}. Alright, let's recall what a geometric sequence is. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as q. So, the general term of a geometric sequence can be written as a_n = a₁ * q^(n-1), where a₁ is the first term.Given that, let's write down the terms mentioned in the problem:- a₂ = a₁ * q^(2-1) = a₁ * q- a₅ = a₁ * q^(5-1) = a₁ * q⁴- a₃ = a₁ * q^(3-1) = a₁ * q²- a₄ = a₁ * q^(4-1) = a₁ * q³So, the two given conditions become:1. a₂ + a₅ = a₁*q + a₁*q⁴ = 362. a₃ * a₄ = (a₁*q²) * (a₁*q³) = a₁² * q⁵ = 128Hmm, okay. So, I have two equations:1. a₁*q + a₁*q⁴ = 362. a₁² * q⁵ = 128I need to solve for a₁ and q. Let me see how I can do this.Looking at the first equation, I can factor out a₁*q:a₁*q*(1 + q³) = 36And the second equation is:a₁² * q⁵ = 128Hmm, maybe I can express a₁ from the first equation and substitute it into the second equation.From the first equation:a₁ = 36 / (q*(1 + q³))Plugging this into the second equation:(36 / (q*(1 + q³)))² * q⁵ = 128Let me compute this step by step.First, square the numerator and denominator:(36²) / (q²*(1 + q³)²) * q⁵ = 128Simplify the exponents:36² * q⁵ / (q²*(1 + q³)²) = 128Simplify q⁵ / q² = q³:36² * q³ / (1 + q³)² = 128Compute 36²: 36*36 is 1296.So, 1296 * q³ / (1 + q³)² = 128Let me write this as:(1296 / 128) * (q³ / (1 + q³)²) = 1Compute 1296 / 128:Divide numerator and denominator by 16: 1296 ÷ 16 = 81, 128 ÷ 16 = 8. So, 81/8.So, (81/8) * (q³ / (1 + q³)²) = 1Let me denote t = q³ to simplify the equation.Then, the equation becomes:(81/8) * (t / (1 + t)²) = 1Multiply both sides by 8/81:t / (1 + t)² = 8/81So, t / (1 + t)² = 8/81Let me cross-multiply:81t = 8(1 + t)²Expand the right side:81t = 8(1 + 2t + t²)81t = 8 + 16t + 8t²Bring all terms to one side:8t² + 16t + 8 - 81t = 0Simplify:8t² - 65t + 8 = 0So, quadratic equation in t: 8t² - 65t + 8 = 0Let me try to solve this quadratic equation.Using the quadratic formula, t = [65 ± sqrt(65² - 4*8*8)] / (2*8)Compute discriminant:65² = 42254*8*8 = 256So, sqrt(4225 - 256) = sqrt(3969) = 63So, t = [65 ± 63] / 16Compute both possibilities:First, t = (65 + 63)/16 = 128/16 = 8Second, t = (65 - 63)/16 = 2/16 = 1/8So, t = 8 or t = 1/8But t = q³, so q³ = 8 or q³ = 1/8Therefore, q = 2 or q = 1/2So, the common ratio q can be 2 or 1/2.Now, let's find a₁ for each case.Case 1: q = 2From the first equation: a₁*q*(1 + q³) = 36Compute 1 + q³: 1 + 8 = 9So, a₁*2*9 = 36 => a₁*18 = 36 => a₁ = 2Case 2: q = 1/2Again, from the first equation: a₁*q*(1 + q³) = 36Compute 1 + q³: 1 + (1/8) = 9/8So, a₁*(1/2)*(9/8) = 36Compute (1/2)*(9/8) = 9/16So, a₁*(9/16) = 36 => a₁ = 36*(16/9) = 4*16 = 64So, we have two possible geometric sequences:Either a₁ = 2 and q = 2, so the general term is a_n = 2 * 2^(n-1) = 2^nOr a₁ = 64 and q = 1/2, so the general term is a_n = 64 * (1/2)^(n-1) = 64 / 2^(n-1) = 2^(6) / 2^(n-1) = 2^(7 - n)So, the general term is either a_n = 2^n or a_n = 2^(7 - n)Wait, let me check if both satisfy the original conditions.First, for a_n = 2^n:a₂ = 4, a₅ = 32, so a₂ + a₅ = 4 + 32 = 36, which matches.a₃ = 8, a₄ = 16, so a₃ * a₄ = 8 * 16 = 128, which also matches.Now, for a_n = 2^(7 - n):a₂ = 2^(5) = 32, a₅ = 2^(2) = 4, so a₂ + a₅ = 32 + 4 = 36, which is correct.a₃ = 2^(4) = 16, a₄ = 2^(3) = 8, so a₃ * a₄ = 16 * 8 = 128, which is also correct.So, both sequences satisfy the given conditions.Therefore, part (I) is solved, and the general term is either a_n = 2^n or a_n = 2^(7 - n).Moving on to part (II). It says that if the sequence {a_n} is an increasing sequence, and b_n = a_n + log₂ a_n for n ∈ N*, find the sum of the first n terms of the sequence {b_n}, denoted as S_n.First, let's recall that a geometric sequence is increasing if the common ratio q > 1. So, in our case, we have two possibilities for q: 2 and 1/2. Since 2 > 1 and 1/2 < 1, the sequence with q = 2 is increasing, while the one with q = 1/2 is decreasing.Therefore, since the sequence is increasing, we must take the general term a_n = 2^n.So, now, b_n = a_n + log₂ a_n = 2^n + log₂(2^n)Simplify log₂(2^n): that's just n.Therefore, b_n = 2^n + nSo, the sequence {b_n} is 2^1 + 1, 2^2 + 2, 2^3 + 3, ..., 2^n + nWe need to find the sum S_n = b₁ + b₂ + ... + b_nWhich is equal to (2^1 + 2^2 + ... + 2^n) + (1 + 2 + 3 + ... + n)So, S_n is the sum of two separate series: a geometric series and an arithmetic series.Let me compute each part separately.First, the geometric series: sum_{k=1}^n 2^kThe formula for the sum of a geometric series is S = a₁*(r^n - 1)/(r - 1), where a₁ is the first term, r is the common ratio.Here, a₁ = 2, r = 2, so sum = 2*(2^n - 1)/(2 - 1) = 2*(2^n - 1)/1 = 2^(n+1) - 2Second, the arithmetic series: sum_{k=1}^n kThe formula for the sum of the first n natural numbers is S = n(n + 1)/2Therefore, putting it all together:S_n = (2^(n+1) - 2) + (n(n + 1)/2)So, S_n = 2^(n+1) - 2 + (n(n + 1))/2I think that's the final expression. Let me just write it neatly:S_n = 2^{n+1} - 2 + frac{n(n + 1)}{2}Let me check if this makes sense. For example, let's compute S₁:b₁ = 2^1 + 1 = 2 + 1 = 3Using the formula: 2^{2} - 2 + (1*2)/2 = 4 - 2 + 1 = 3. Correct.Compute S₂:b₁ + b₂ = (2 + 1) + (4 + 2) = 3 + 6 = 9Using the formula: 2^{3} - 2 + (2*3)/2 = 8 - 2 + 3 = 9. Correct.Compute S₃:b₁ + b₂ + b₃ = 3 + 6 + (8 + 3) = 3 + 6 + 11 = 20Using the formula: 2^{4} - 2 + (3*4)/2 = 16 - 2 + 6 = 20. Correct.So, the formula seems to work for small n. Therefore, I think it's correct.**Final Answer**(I) The general term is boxed{a_n = 2^n} or boxed{a_n = 2^{7 - n}}.(II) The sum of the first ( n ) terms is boxed{2^{n+1} - 2 + frac{n(n + 1)}{2}}.

💡Okay, so I have this problem about knights and liars on an island with 2021 people. Knights always tell the truth, and liars always lie. Every islander knows who is who, and they all line up in a row. Then, each one says, "The number of liars standing to my right is greater than the number of knights standing to my left." I need to figure out how many knights are on the island, considering all possible scenarios and proving there are no other options.Hmm, let me start by understanding the problem step by step. First, there are 2021 people, each either a knight or a liar. Knights tell the truth, so whatever they say must be true. Liars always lie, so whatever they say must be false. Each person makes a statement about the number of liars to their right and the number of knights to their left. Specifically, they say that the number of liars to their right is greater than the number of knights to their left.I think it would help to consider the positions of the people in the line. Let's imagine them standing from left to right as positions 1 to 2021. Each person at position i makes a statement about the number of liars to their right (from i+1 to 2021) and the number of knights to their left (from 1 to i-1). So, for each person, we can denote:- Let L(i) be the number of liars to the right of person i.- Let K(i) be the number of knights to the left of person i.Each person i says: L(i) > K(i).Now, if person i is a knight, then L(i) > K(i) must be true. If person i is a liar, then L(i) > K(i) must be false, meaning L(i) ≤ K(i).I need to figure out how many knights there are in total. Let me think about the implications of these statements.First, consider the rightmost person, person 2021. To their right, there are no people, so L(2021) = 0. To their left, there are 2020 people. So, person 2021 is saying that 0 > K(2021). But K(2021) is the number of knights to the left of person 2021, which is at least 0 and at most 2020. So, 0 > K(2021) is impossible because K(2021) is non-negative. Therefore, person 2021 must be lying, which means person 2021 is a liar.Similarly, consider the leftmost person, person 1. To their left, there are no people, so K(1) = 0. To their right, there are 2020 people. Person 1 says that L(1) > 0. Since L(1) is the number of liars to the right of person 1, which is at least 0 and at most 2020. If person 1 is a knight, then L(1) > 0 must be true, which means there is at least one liar to the right of person 1. If person 1 is a liar, then L(1) > 0 is false, meaning L(1) ≤ 0, which would imply L(1) = 0, meaning there are no liars to the right of person 1. But we already established that person 2021 is a liar, so L(1) must be at least 1. Therefore, person 1 cannot be a liar because that would lead to a contradiction. Hence, person 1 must be a knight.So, from this, we know that person 1 is a knight and person 2021 is a liar.Now, let's consider person 2. To their left, there is only person 1, who is a knight, so K(2) = 1. To their right, there are 2019 people. Person 2 says that L(2) > 1. If person 2 is a knight, then L(2) > 1 must be true. If person 2 is a liar, then L(2) ≤ 1 must be true.But we know that person 2021 is a liar, so L(2) is at least 1 (since person 2021 is a liar). However, if person 2 is a knight, then L(2) must be greater than 1, meaning there must be at least two liars to the right of person 2. If person 2 is a liar, then L(2) ≤ 1, meaning there is at most one liar to the right of person 2.But wait, we already know that person 2021 is a liar, so L(2) is at least 1. If person 2 is a liar, then L(2) must be exactly 1. That would mean that apart from person 2021, there are no other liars to the right of person 2. But if that's the case, then person 3 to person 2020 must all be knights.But let's check person 2020. To their right, there is only person 2021, who is a liar, so L(2020) = 1. To their left, there are 2019 people. Person 2020 says that L(2020) > K(2020). Since L(2020) = 1, this means person 2020 is saying that 1 > K(2020). If person 2020 is a knight, then K(2020) must be less than 1, which is impossible because K(2020) is the number of knights to the left of person 2020, which is at least 1 (person 1 is a knight). Therefore, person 2020 must be a liar, meaning their statement is false, so L(2020) ≤ K(2020). Since L(2020) = 1, this means K(2020) ≥ 1, which is true because person 1 is a knight.Wait, but if person 2020 is a liar, that means L(2020) ≤ K(2020). Since L(2020) = 1, K(2020) must be at least 1, which is true. So, person 2020 is a liar.But if person 2020 is a liar, then L(2) is at least 2 (person 2020 and person 2021). Therefore, if person 2 is a knight, L(2) > 1 is true, which is consistent. If person 2 is a liar, then L(2) ≤ 1, but we have at least two liars to the right of person 2, so person 2 cannot be a liar. Therefore, person 2 must be a knight.So, person 2 is a knight. Therefore, K(3) = 2 (persons 1 and 2 are knights). Person 3 says that L(3) > 2. If person 3 is a knight, then L(3) > 2 must be true. If person 3 is a liar, then L(3) ≤ 2 must be true.But we know that person 2021 and person 2020 are liars, so L(3) is at least 2. If person 3 is a knight, then L(3) must be greater than 2, meaning there must be at least three liars to the right of person 3. If person 3 is a liar, then L(3) ≤ 2, meaning there are at most two liars to the right of person 3.But we already have person 2020 and 2021 as liars, so L(3) is at least 2. If person 3 is a liar, then L(3) must be exactly 2, meaning there are no other liars to the right of person 3. But let's check person 2019.Person 2019 has to their right persons 2020 and 2021, both liars, so L(2019) = 2. Person 2019 says that L(2019) > K(2019). Since L(2019) = 2, this means person 2019 is saying that 2 > K(2019). If person 2019 is a knight, then K(2019) must be less than 2, which is impossible because K(2019) is the number of knights to the left of person 2019, which is at least 2 (persons 1 and 2). Therefore, person 2019 must be a liar, meaning their statement is false, so L(2019) ≤ K(2019). Since L(2019) = 2, this means K(2019) ≥ 2, which is true because persons 1 and 2 are knights.So, person 2019 is a liar. Therefore, L(3) is at least 3 (persons 2019, 2020, 2021). Therefore, if person 3 is a knight, L(3) > 2 is true. If person 3 is a liar, then L(3) ≤ 2, but we have at least three liars to the right of person 3, so person 3 cannot be a liar. Therefore, person 3 must be a knight.Continuing this pattern, it seems that each time we check a person from the left, they must be a knight because the number of liars to their right is increasing, and each time we check a person from the right, they must be a liar because the number of knights to their left is increasing.This suggests that the number of knights is equal to the number of people minus the number of liars. But we need to find the exact number.Wait, let's think about the total number of people, which is 2021. If we keep removing the rightmost liar and the leftmost knight, we can see how many knights there are.Starting with 2021 people, we know person 1 is a knight and person 2021 is a liar. Removing them, we have 2019 people left. Now, the new leftmost person (person 2) is a knight, and the new rightmost person (person 2020) is a liar. Removing them, we have 2017 people left. Continuing this process, each time we remove a knight from the left and a liar from the right.Since 2021 is an odd number, this process will continue until we have one person left. That last person must be a liar because we've been removing knights from the left and liars from the right alternately. Therefore, the number of knights is equal to the number of times we removed a knight from the left, which is (2021 - 1)/2 = 1010.So, there are 1010 knights and 1011 liars on the island.