🧠:[<think>Okay, so I have this problem here with 1999 non-negative real numbers: a₁, a₂, ..., a₁₉₉₉. They have two conditions. The first one is that their sum is 2, and the second one is that the sum of their cyclic products (like a₁a₂, a₂a₃, ..., a₁₉₉₉a₁) is 1. I need to find the maximum and minimum values of S, which is the sum of their squares. Let me start by recalling some algebraic identities that might help here. I know that the sum of squares can be related to the square of the sum and the sum of products. Specifically, the formula is:S = (Σa_i)² - 2Σa_i a_j (where i < j)But here, the sum of the cyclic products isn't exactly the same as the sum over all i < j. Wait, the cyclic products are consecutive terms multiplied together. So, in the given problem, condition (2) is Σa_i a_{i+1} (with a₁₉₉₉a₁ at the end) equals 1. So maybe I can use the identity for the square of the sum. Let's write that out:(Σa_i)² = Σa_i² + 2Σa_i a_j (for all i < j)But we have Σa_i a_{i+1} = 1. However, Σa_i a_j over all i < j is a much larger sum. So perhaps this identity isn't directly helpful. Hmm.Wait, the problem gives Σa_i = 2 and Σa_i a_{i+1} = 1, and we need to find Σa_i². Let's denote the sum of squares as S. Then from the square of the sum:(Σa_i)² = S + 2Σa_i a_j (over all i < j)But we know Σa_i = 2, so 4 = S + 2Σ_{i < j} a_i a_j.But how does this relate to the given cyclic sum? The cyclic sum Σa_i a_{i+1} = 1. But the total sum of all products a_i a_j (i ≠ j) is Σa_i a_j = (Σa_i)^2 - Σa_i² = 4 - S. But Σa_i a_j includes all pairs, not just consecutive ones. So the cyclic sum is part of this total. Therefore, Σ_{i=1 to 1999} a_i a_{i+1} = 1 is a part of the total Σa_i a_j, which is (4 - S)/2. Wait, because:Σ_{i < j} a_i a_j = (Σa_i)^2 - Σa_i² / 2 = (4 - S)/2.But the cyclic sum is Σ_{i=1 to 1999} a_i a_{i+1} = 1. However, each term in the cyclic sum is a pair of consecutive indices, considering the sequence as cyclic (so a₁₉₉₉a₁ is included). So the cyclic sum is a subset of the total sum of products. Therefore, 1 ≤ Σ_{i < j} a_i a_j = (4 - S)/2.But wait, no. Actually, the cyclic sum is a specific part of all possible products. So if we have 1999 terms, each a_i is multiplied by its next neighbor. So there are 1999 terms in the cyclic sum. However, the total number of terms in Σ_{i < j} a_i a_j is C(1999, 2) which is way larger. Therefore, Σcyclic a_i a_{i+1} = 1 is just a small part of the total sum. So maybe we can't directly relate these two, unless there is some structure.Alternatively, maybe we can use Cauchy-Schwarz inequality or some other inequalities to bound S. Let me think.First, to find the maximum and minimum of S given Σa_i = 2 and Σa_i a_{i+1} = 1. Since all a_i are non-negative, this might help.For the maximum of S: The sum of squares is maximized when the variables are as unequal as possible. Since we have non-negative numbers, the maximum occurs when one variable is as large as possible, and the others are as small as possible. However, we also have the constraint that the sum of consecutive products is 1. So maybe distributing the values such that two adjacent variables are large and the rest are zero. Let's check.Suppose two consecutive variables are non-zero. Let's say a₁ = x and a₂ = y, and the rest are zero. Then the sum would be x + y = 2. The cyclic product sum would be a₁a₂ + a₂a₃ + ... + a₁₉₉₉a₁. But since a₃ to a₁₉₉₉ are zero, the only non-zero terms are a₁a₂ and a₁₉₉₉a₁. But a₁₉₉₉ is zero, so the cyclic sum is just a₁a₂ = xy. So in this case, we have x + y = 2 and xy = 1. Solving these equations: x + y = 2, xy = 1. This is a quadratic equation: t² - 2t + 1 = 0, which factors as (t - 1)² = 0, so x = y = 1. Then the sum of squares S = 1² + 1² + 0 + ... + 0 = 2. But is this the maximum? Wait, if we have three consecutive variables non-zero, maybe we can get a higher sum of squares. Let's test that. Suppose a₁ = x, a₂ = y, a₃ = z, and the rest are zero. Then the sum is x + y + z = 2. The cyclic product sum is a₁a₂ + a₂a₃ + a₃a₄ + ... + a₁₉₉₉a₁. Since a₄ to a₁₉₉₉ are zero, the only non-zero terms are a₁a₂ = xy, a₂a₃ = yz, and a₃a₁ (since a₃a₄ is zero, etc., and a₁₉₉₉a₁ is zero because a₁₉₉₉ is zero). Wait, actually, in the cyclic product sum, after a₃a₄ (which is zero), all subsequent terms are zero until a₁₉₉₉a₁, but since a₁₉₉₉ is zero, that term is also zero. So the cyclic sum is xy + yz. So we have:x + y + z = 2,xy + yz = 1.We need to maximize S = x² + y² + z².Express z = 2 - x - y. Substitute into the second equation:xy + y(2 - x - y) = 1,xy + 2y - xy - y² = 1,2y - y² = 1,y² - 2y + 1 = 0,(y - 1)² = 0,so y = 1. Then z = 2 - x - 1 = 1 - x. Since all variables are non-negative, x ≤ 1. Then S = x² + 1² + (1 - x)² = x² + 1 + 1 - 2x + x² = 2x² - 2x + 2. To maximize this, take derivative: 4x - 2 = 0 ⇒ x = 0.5. But x must be between 0 and 1. At x = 0.5, S = 2*(0.25) - 2*(0.5) + 2 = 0.5 - 1 + 2 = 1.5. At x=0, S=0 +1 +1=2. At x=1, S=1 +1 +0=2. So maximum S here is 2, same as before. So having three variables doesn't give a higher S.What if we have four variables? Let's say a₁, a₂, a₃, a₄ non-zero. Then the cyclic sum would include a₁a₂, a₂a₃, a₃a₄, a₄a₅, ..., but since a₅ onwards are zero, the cyclic sum is a₁a₂ + a₂a₃ + a₃a₄. Then the sum a₁ + a₂ + a₃ + a₄ = 2. Let me suppose that a₁, a₂, a₃, a₄ are non-zero. Then we have:a₁a₂ + a₂a₃ + a₃a₄ = 1,and a₁ + a₂ + a₃ + a₄ = 2.Trying to maximize S = a₁² + a₂² + a₃² + a₄².This might complicate. Maybe a better approach is to think that regardless of how many variables we take, the maximum S is 2. But let's check if there's a configuration where S is larger.Alternatively, consider distributing more mass into a single variable. Suppose a₁ = 2, and all others are zero. Then S = 4. But then the cyclic product sum would be a₁a₂ + a₂a₃ + ... + a₁₉₉₉a₁. Since a₁=2, a₂ to a₁₉₉₉=0, so the only non-zero term is a₁₉₉₉a₁ = 0*2 = 0. So the cyclic sum is 0, which doesn't satisfy the second condition. Therefore, such a case is invalid.Therefore, to satisfy the cyclic sum being 1, we need at least two adjacent variables to be non-zero. If we set a₁ = x, a₂ = y, rest zero, then cyclic sum is xy = 1, and x + y = 2. Then x and y must both be 1, giving S = 2. If we set three variables, as before, we still get S maximum 2. So maybe 2 is the maximum? But let's verify with another configuration.Suppose we have two pairs of adjacent variables. For example, a₁ = x, a₂ = y, a₃ = z, a₄ = w, with the rest zero. Then the cyclic sum would be a₁a₂ + a₂a₃ + a₃a₄ + a₄a₅ + ... + a₁₉₉₉a₁. Since a₅ onwards are zero, the cyclic sum is xy + yz + zw. If we set x + y + z + w = 2 and xy + yz + zw = 1. Let's see if we can get a higher S here.Let me try setting x = w = a and y = z = b. Then sum is 2a + 2b = 2 ⇒ a + b = 1. The cyclic sum is xy + yz + zw = ab + b² + ab = 2ab + b². So 2ab + b² = 1. Since a = 1 - b, substitute:2(1 - b)b + b² = 1,2b - 2b² + b² = 1,2b - b² = 1,b² - 2b + 1 = 0,(b - 1)² = 0 ⇒ b = 1. Then a = 0. So this reduces to y = z = 1, x = w = 0. Then the variables are a₂ = 1, a₃ = 1, rest zero. Then the cyclic sum is a₁a₂ + a₂a₃ + a₃a₄ + ... = 0*1 + 1*1 + 1*0 + ... = 1. The sum S = 0² + 1² + 1² + 0² + ... = 2. Same as before. So no improvement.Alternatively, perhaps setting non-adjacent variables? Wait, but non-adjacent variables would not contribute to the cyclic product sum. For example, if a₁ and a₃ are non-zero, but a₂ is zero. Then the cyclic sum would have a₁a₂ = 0, a₂a₃ = 0, so those terms are zero. Unless other adjacent pairs are non-zero. But if you have multiple non-adjacent variables, the cyclic product sum would only include the products of consecutive ones. So maybe distributing the mass in such a way that more products contribute. But since the variables are non-negative, you can't have overlapping products unless variables are adjacent.Alternatively, maybe spreading out the variables more. Suppose we have every other variable set to some value. For example, a₁ = x, a₃ = x, a₅ = x, etc. But given that 1999 is odd, this might not cycle evenly. However, if we spread out the variables, their products would be zero since they are not adjacent. Then the cyclic sum would be zero, which doesn't meet the requirement. Therefore, such a configuration is invalid.Alternatively, suppose all variables are equal. Let's test that. If all a_i = 2/1999. Then the cyclic sum would be 1999*(2/1999)^2 = (4/1999). Which is way less than 1. So that's not going to work. Therefore, equality is not the case here.So perhaps the maximum S is 2. Now, for the minimum S. The sum of squares is minimized when the variables are as equal as possible, but given the constraints. However, the second condition complicates things. Let me think.We need to minimize S = Σa_i², given Σa_i = 2 and Σa_i a_{i+1} = 1, with all a_i ≥ 0.To minimize the sum of squares, given the sum is fixed, we usually use the Cauchy-Schwarz inequality, which tells us that the sum of squares is minimized when the variables are equal. But here, the constraint Σa_i a_{i+1} = 1 complicates things.Alternatively, maybe using Lagrange multipliers. Let's consider the problem:Minimize S = Σa_i²Subject to:1. Σa_i = 22. Σa_i a_{i+1} = 1With a_i ≥ 0.This is a constrained optimization problem. The Lagrangian would be:L = Σa_i² - λ(Σa_i - 2) - μ(Σa_i a_{i+1} - 1)Taking partial derivatives with respect to each a_i:For each i, dL/da_i = 2a_i - λ - μ(a_{i-1} + a_{i+1}) = 0Here, indices are cyclic, so a₀ = a₁₉₉₉ and a₂₀₀₀ = a₁.So we get the system of equations:2a_i - λ - μ(a_{i-1} + a_{i+1}) = 0 for all i.This seems complex for 1999 variables, but maybe there's a symmetric solution where all a_i are equal. Let's check if that's possible.Suppose all a_i = c. Then Σa_i = 1999c = 2 ⇒ c = 2/1999.Then Σa_i a_{i+1} = 1999c² = 1999*(4/1999²) = 4/1999 ≈ 0.002, which is much less than 1. So this doesn't satisfy the second condition. Therefore, the variables cannot all be equal.Alternatively, maybe a periodic pattern, like alternating values. Let's suppose that the variables alternate between two values: a, b, a, b, etc. Since 1999 is odd, the last term will be a, making the cycle a, b, a, b, ..., a.Then the sum Σa_i = (number of a's)*a + (number of b's)*b. Since 1999 is odd, there are 1000 a's and 999 b's. So 1000a + 999b = 2.The cyclic product sum Σa_i a_{i+1} = sum over i of a_i a_{i+1}. In the alternating case, this becomes a*b + b*a + a*b + ... + a*a (since the last term is a₁₉₉₉ a₁ = a*a). Wait, because the sequence is a, b, a, b, ..., a, so the last pair is a (from a₁₉₉₉) and a (from a₁). So the cyclic product sum is (a*b + b*a)*999 + a*a. Because each a is followed by b, except the last a is followed by a.So Σa_i a_{i+1} = 999*(2ab) + a² = 1998ab + a² = 1.So we have two equations:1000a + 999b = 2,1998ab + a² = 1.Let me denote equation 1: 1000a + 999b = 2,equation 2: a² + 1998ab = 1.Let me solve equation 1 for b: b = (2 - 1000a)/999.Substitute into equation 2:a² + 1998a*(2 - 1000a)/999 = 1Simplify:a² + (1998/999)*a*(2 - 1000a) = 11998/999 = 2, so:a² + 2a*(2 - 1000a) = 1Expand:a² + 4a - 2000a² = 1Combine like terms:-1999a² + 4a - 1 = 0Multiply both sides by -1:1999a² - 4a + 1 = 0Discriminant D = 16 - 4*1999*1 = 16 - 7996 = -7980 < 0Negative discriminant, so no real solutions. Therefore, this alternating pattern is impossible.Hmm, so maybe another pattern. What if all non-zero a_i are equal and spaced in some way? For example, having every third variable non-zero? But then the products would be zero, so cyclic sum would be zero, which doesn't work. Alternatively, having blocks of two consecutive variables.Alternatively, suppose that the sequence is composed of k pairs of consecutive variables, each pair contributing some value, and the rest zero. For example, if we have k pairs like (a, a), separated by zeros. But given the cyclic nature, the number of pairs would need to divide 1999? Since 1999 is prime, this complicates.Alternatively, perhaps just two pairs of consecutive variables. Wait, but 1999 is prime, so arranging pairs might not fit.Alternatively, suppose that all the non-zero variables are consecutive. For example, a block of m consecutive variables each equal to c, and the rest zero. Then the sum would be mc = 2. The cyclic product sum would be sum_{i=1}^{m} c*c + 0 + ... + 0 = (m - 1)c². Because in the block of m consecutive variables, there are m-1 products of consecutive terms. Also, the cyclic product would include the term between the last variable of the block and the first variable, which is zero. So total cyclic sum is (m - 1)c². We need this to be 1. So:mc = 2,(m - 1)c² = 1.From the first equation, c = 2/m. Substitute into the second:(m - 1)(4/m²) = 1,4(m - 1)/m² = 1,4m - 4 = m²,m² - 4m + 4 = 0,(m - 2)² = 0,m = 2. Then c = 1. So two consecutive variables each equal to 1, rest zero. Then cyclic sum is (2 - 1)*1² = 1*1 = 1. Which satisfies the second condition. The sum of squares S = 1² + 1² = 2. Wait, this is the same as previous cases. So this configuration gives S = 2, which we already have. But we are looking for the minimum S.Wait, so in this case, the sum of squares is 2. But maybe there are configurations with more variables contributing, leading to a lower S. Let's see.Suppose we have three consecutive variables each equal to c, so m = 3. Then sum 3c = 2 ⇒ c = 2/3. The cyclic sum would be (3 - 1)c² = 2*(4/9) = 8/9 ≈ 0.888, which is less than 1. So we need to adjust.But if we have three consecutive variables with different values. Let's say a₁ = x, a₂ = y, a₃ = z, rest zero. Then sum x + y + z = 2, cyclic sum xy + yz = 1. Let's see if we can get a lower S.Express z = 2 - x - y. Then cyclic sum:xy + y(2 - x - y) = 1 ⇒ 2y - y² = 1 ⇒ y² - 2y + 1 = 0 ⇒ y = 1. Then z = 1 - x.Sum of squares S = x² + 1² + (1 - x)² = 2x² - 2x + 2. As before, this is minimized when x = 0.5, giving S = 1.5. But x must be between 0 and 1. So at x = 0.5, S = 1.5. This is lower than 2. So maybe 1.5 is a candidate for the minimum. Wait, but does this configuration satisfy the cyclic sum? Let's check.If x = 0.5, y = 1, z = 0.5. Then cyclic sum is (0.5)(1) + (1)(0.5) = 0.5 + 0.5 = 1. Correct. Then sum of squares is 0.25 + 1 + 0.25 = 1.5. So S = 1.5. That's lower. So this gives a lower S. Therefore, the minimum might be 1.5. But wait, is this the minimal possible?Wait, let's see if we can spread more variables to get even a lower S. For example, if we have four consecutive variables: a, b, c, d. Let's assume they are all non-zero. Then the sum a + b + c + d = 2. The cyclic sum includes ab + bc + cd + da (since after d, the next term is a, but if the rest are zero, then da is part of the cyclic sum). Wait, no. If there are four consecutive non-zero variables a, b, c, d, followed by zeros, then the cyclic sum would be ab + bc + cd + d*0 + ... + 0*a = ab + bc + cd. So three terms. Then the sum would need ab + bc + cd = 1.Let me attempt to set variables symmetrically. Suppose a = d and b = c. Then sum is a + b + b + a = 2a + 2b = 2 ⇒ a + b = 1. The cyclic sum is ab + bc + cd = ab + b² + ba = 2ab + b². So 2ab + b² = 1. Since a = 1 - b, substitute:2(1 - b)b + b² = 1,2b - 2b² + b² = 1,2b - b² = 1,b² - 2b + 1 = 0,(b - 1)² = 0 ⇒ b = 1. Then a = 0. So variables are 0, 1, 1, 0. Then cyclic sum is 0*1 + 1*1 + 1*0 = 1. Sum of squares S = 0 + 1 + 1 + 0 = 2. Again, same as before. So this doesn't help.Alternatively, with four variables but not symmetric. Let's say a, b, c, d. Sum a + b + c + d = 2. Cyclic sum ab + bc + cd = 1. Let's set a = x, b = y, c = z, d = w. Then x + y + z + w = 2, and xy + yz + zw = 1. To minimize S = x² + y² + z² + w².Express w = 2 - x - y - z. Then substitute into cyclic sum:xy + yz + z(2 - x - y - z) = 1.Expand:xy + yz + 2z - xz - y z - z² = 1.Simplify:xy + yz + 2z - xz - y z - z² = xy - xz + 2z - z² = 1.Factor x(y - z) + z(2 - z) = 1.This seems complicated. Maybe assume some relations between variables. For example, set x = z. Then:x(y - x) + x(2 - x) = 1,x(y - x + 2 - x) = 1,x(y + 2 - 2x) = 1.But also, from the sum, x + y + x + w = 2 ⇒ w = 2 - 2x - y.But this might not lead anywhere. Alternatively, set y = z. Then:x(y - y) + y(2 - y) = 0 + 2y - y² = 1,So 2y - y² = 1,Which gives y² - 2y + 1 = 0 ⇒ y = 1. Then x(y - y) + y(2 - y) = 1 ⇒ 1*1 = 1. Then w = 2 - x - 2y = 2 - x - 2. So w = -x. But variables are non-negative, so x must be zero. Then w = 0. Then variables are x=0, y=1, z=1, w=0. Sum of squares S = 0 + 1 + 1 + 0 = 2. Again, same result.Alternatively, maybe three variables. Wait, earlier with three variables, we got S = 1.5. Let me check again. If we set a₁ = 0.5, a₂ = 1, a₃ = 0.5, rest zero. Then S = 0.25 + 1 + 0.25 = 1.5. Cyclic sum is 0.5*1 + 1*0.5 = 1. That works. So that's a valid configuration with S = 1.5. Is that the minimum?Wait, maybe we can have four variables with more spreading. Let's suppose a₁ = a, a₂ = b, a₃ = c, a₄ = d, rest zero. Then sum a + b + c + d = 2. Cyclic sum ab + bc + cd = 1. Let's try to set a = d and b = c for symmetry.Then sum 2a + 2b = 2 ⇒ a + b = 1.Cyclic sum ab + bc + cd = ab + b² + a b = 2ab + b² = 1.From a + b = 1, a = 1 - b. Substitute:2(1 - b)b + b² = 1,2b - 2b² + b² = 1,2b - b² = 1,b² - 2b + 1 = 0,(b - 1)^2 = 0 ⇒ b = 1, a = 0. So variables are 0, 1, 1, 0, rest zero. Then S = 0 + 1 + 1 + 0 = 2.Alternatively, if we don't assume symmetry. Let's take three variables: a, b, c, d with a, b, c non-zero and d = 0. Wait, but maybe this complicates. Alternatively, let's think of a different approach.Let me consider the Cauchy-Schwarz inequality. For the sum of products, Σa_i a_{i+1} = 1. By Cauchy-Schwarz, (Σa_i a_{i+1}) ≤ sqrt(Σa_i²) * sqrt(Σa_{i+1}²). But since Σa_i² = Σa_{i+1}² = S, then 1 ≤ S. So S ≥ 1. But this is a lower bound. But our previous example gave S = 1.5, so maybe the actual minimum is higher. Wait, but Cauchy-Schwarz gives 1 ≤ S, but in reality, can we reach S =1? Let's check.Suppose S =1. Then all a_i² sum to 1. Since all a_i are non-negative, each a_i ≤1. The sum of a_i is 2. But by Cauchy-Schwarz, (Σa_i)^2 ≤ 1999 * Σa_i². Here, 4 ≤ 1999 * 1 ⇒ 4 ≤ 1999, which is true. So it's possible. But how to arrange the variables?Wait, but if S = 1, then Σa_i² =1, but Σa_i =2. This would require that the variables have a certain distribution. For instance, if we have two variables as 1 and the rest zero, then Σa_i = 2, Σa_i² = 2, which is more than 1. So that's not it. If we have more variables, but spread out. For example, if all a_i = 2/1999, then Σa_i² = 1999*(4/1999²) = 4/1999 ≈ 0.002, which is much less than 1, but then Σa_i a_{i+1} = 1999*(2/1999)^2 = 4/1999 ≈ 0.002, which is not 1. So this is invalid. Therefore, maybe there's a middle ground.But how to achieve Σa_i a_{i+1} =1 with minimal Σa_i². Perhaps distribute the products such that the products are as large as possible given the sum constraints. Wait, but products are maximized when variables are equal, so maybe spreading the products equally.Alternatively, think of the problem as optimizing S given the constraints. Let's consider variables in pairs. Since the cyclic sum involves consecutive terms, perhaps modeling this as a system where each pair contributes to the sum.Alternatively, use Lagrange multipliers again. The minimal S occurs when the derivative conditions are met. Earlier, we saw that with three variables, we can get S = 1.5. Is that the minimal?Wait, let's consider the three-variable case. If we have three consecutive variables: a, b, c, with a + b + c = 2, ab + bc = 1. Then, as before, c = 2 - a - b. Then ab + b(2 - a - b) =1 ⇒ 2b - b² =1 ⇒ b=1. Then a + c =1. The sum of squares is a² +1 +c². To minimize a² + c², given a + c =1. The minimum occurs at a = c =0.5, giving 0.25 +0.25 =0.5, so total S=0.5 +1=1.5. This is the minimal for this configuration.Is there a configuration with more variables that gives a lower S?Suppose four consecutive variables: a, b, c, d. With a + b + c + d =2 and ab + bc + cd =1. Let's try to set a = d and b = c for symmetry.Then 2a + 2b =2 ⇒ a + b=1.Cyclic sum ab + bc + cd= ab + b² +ab= 2ab + b²=1.Let’s solve 2ab + b²=1 with a + b=1 ⇒ a=1 - b.Substitute:2(1 - b)b + b²=12b - 2b² + b²=1 ⇒ 2b - b²=1 ⇒ b² -2b +1=0 ⇒ (b -1)^2=0 ⇒ b=1, a=0.So variables are 0,1,1,0. Sum of squares=0+1+1+0=2. No improvement.Alternatively, without symmetry. Let’s set a, b, c, d with a + b + c + d=2 and ab + bc + cd=1.Let’s set c =0. Then the cyclic sum becomes ab +0 +0=ab=1. Then a + b +0 +d=2. Since ab=1, and a, b ≥0. Then possible only if a and b are positive. Let’s set d=2 -a -b. Then S= a² +b² +0 +d²= a² +b² + (2 -a -b)^2.We need to minimize this given ab=1. Let’s use Lagrange multipliers here.Let f(a,b)=a² +b² + (2 -a -b)^2,subject to g(a,b)=ab -1=0.The Lagrangian is L = f - λg.Partial derivatives:dL/da = 2a + 2(2 -a -b)(-1) - λb =0,dL/db = 2b + 2(2 -a -b)(-1) - λa =0,dL/dλ = -(ab -1)=0.So equations:2a - 2(2 -a -b) - λb =0,2b - 2(2 -a -b) - λa =0,ab=1.Let’s denote 2 -a -b = c. Then:2a - 2c - λb =0,2b - 2c - λa =0.From the first equation: 2a -2c = λb,From the second: 2b -2c = λa.Divide the first equation by the second:(2a - 2c)/(2b - 2c) = (λb)/(λa) ⇒ (a - c)/(b - c) = b/a.Cross-multiplying:a(a - c) = b(b - c),a² - a c = b² - b c,a² - b² = c(a - b),(a - b)(a + b) = c(a - b).If a ≠ b, then divide both sides by (a - b):a + b = c.But c = 2 -a -b, so:a + b = 2 -a -b,2a + 2b =2,a + b=1.But ab=1 and a + b=1. Then the solutions for a and b are roots of t² - t +1=0, which has discriminant 1 -4= -3 <0. No real solutions. Therefore, a = b.If a = b, then from ab=1, a²=1 ⇒a=1. So a =b=1, then c=2 -1 -1=0. Then S=1 +1 +0 +0=2. Which is the same as before. So no improvement.Therefore, the minimal S in this configuration is 2. But previously, with three variables, we had S=1.5. So perhaps 1.5 is the minimal. But let's test another configuration with more variables.Suppose five consecutive variables: a, b, c, d, e. Sum a + b + c + d + e=2. Cyclic sum ab + bc + cd + de + ea=1. Wait, but if they are consecutive and followed by zeros, then the cyclic sum would be ab + bc + cd + de + e*0 + ... +0*a=ab + bc + cd + de. So four terms. If I set a, b, c, d, e with some values.Alternatively, this is getting too complicated. Let's think back to the three-variable case, which gives us a lower S. Maybe that's the minimal. But how to confirm?Alternatively, consider using the Cauchy-Schwarz inequality on the cyclic sum. We have:(Σa_i a_{i+1}) ≤ sqrt(Σa_i²) * sqrt(Σa_{i+1}²) = S.But given that Σa_i a_{i+1}=1, so 1 ≤ S. But in our three-variable case, S=1.5 >1, so this inequality is satisfied. But this only gives a lower bound of 1, but we have a configuration achieving 1.5, which is higher. So maybe the actual minimum is higher than 1.Alternatively, consider the following approach. Let's model this as a quadratic form. Let’s arrange the variables in a circle (since it's cyclic). Then the problem is to minimize S = Σa_i² given Σa_i =2 and Σa_i a_{i+1}=1.This can be considered as optimizing a quadratic function with linear constraints. The minimum of such a function is attained at a point where the gradient is a linear combination of the gradients of the constraints. So using Lagrange multipliers.But with 1999 variables, solving this directly is not feasible. However, maybe the minimal occurs when only a few variables are non-zero. For instance, in the three-variable case, S=1.5. Is there a way to have more variables contributing to lower S?Suppose we have four variables with two separated pairs. For example, a₁ = x, a₂ = y, a₄ = z, a₅ = w, with the rest zero. But then the cyclic sum would be a₁a₂ + a₂a₃ + a₃a₄ + a₄a₅ + ... Since a₃, a₆,... are zero, the cyclic sum is a₁a₂ + a₄a₅. But if a₁ is adjacent to a₂, and a₄ is adjacent to a₅ (if the indices are cyclic, but in this case, a₅ is adjacent to a₆, which is zero. So unless the pairs are adjacent cyclically. Wait, if a₁₉₉₉ is also non-zero, but this complicates. Maybe this approach isn't useful.Alternatively, let's consider using the method of Lagrange multipliers for the three-variable case and see if that gives us the minimal S.In the three-variable case, we have a + b + c =2, ab + bc=1, and S= a² + b² + c². We found that S=1.5 when a=c=0.5 and b=1. Is this the minimal?Suppose we try to set more variables. Let's say four variables: a, b, c, d, where a + b + c + d=2 and ab + bc + cd=1. If we set a=0.5, b=1, c=0.5, d=0, then the cyclic sum is 0.5*1 +1*0.5 +0.5*0=1, and the sum of squares is 0.25 +1 +0.25 +0=1.5. Wait, same as the three-variable case. So by setting d=0, we effectively have three variables. So this is the same as before.Similarly, if we set five variables with a similar pattern: a=0.25, b=1, c=0.25, d=0, e=0.5, etc., but need to check if the cyclic sum condition holds. However, this might not necessarily give a lower S.Alternatively, consider distributing the values over more adjacent variables. Suppose we have a sequence where multiple consecutive variables are non-zero, forming a block. Let’s say we have a block of k variables, each equal to c, followed by zeros. Then:Sum: k*c =2 ⇒ c=2/k.Cyclic sum: The number of adjacent products is k (since it's cyclic), but if followed by zeros, actually the products are (k-1)*c² + c*0 = (k-1)c². So (k-1)c² =1. Substitute c=2/k:(k-1)*(4/k²)=1 ⇒ 4(k-1)/k²=1 ⇒ 4k -4 =k² ⇒k² -4k +4=0 ⇒(k-2)²=0 ⇒k=2. Which brings us back to the case with two variables, S=2. So no improvement.Therefore, the minimal S seems to be 1.5, achieved by three consecutive variables: 0.5, 1, 0.5, rest zero. But is there a way to get lower?Wait, let's think differently. Suppose we have two separate pairs of consecutive variables. For example, a₁ = x, a₂ = y, a₃ = 0, a₄ = z, a₅ = w, rest zero. Then the sum x + y + z + w=2. The cyclic sum is a₁a₂ + a₂a₃ + a₃a₄ + a₄a₅ + ... + a₁₉₉₉a₁. Since a₃=0, a₅=w, followed by zeros, then cyclic sum is xy + 0 + 0 + zw + 0 + ... +0. So we need xy + zw =1. Then S =x² + y² + z² + w². To minimize S given x + y + z + w=2 and xy + zw=1.Let’s set x=y and z=w for symmetry. Then 2x + 2z=2 ⇒x + z=1. Also, x² + z²=1. Then the sum of squares S=2x² + 2z²=2(x² + z²). Given x + z=1, x² + z²= (x + z)² - 2xz=1 - 2xz. So S=2(1 - 2xz). But we also have the cyclic sum condition: xy + zw= x² + z²=1 - 2xz=1. Wait, but x² + z²=1 - 2xz. So 1 - 2xz=1 ⇒xz=0. Which implies either x=0 or z=0. If x=0, then z=1, so variables are 0,0,1,1, rest zero. Then cyclic sum=0*0 +0*0 +1*1 +1*0 +...=1. Sum of squares=0 +0 +1 +1=2. Similarly for z=0. So this gives S=2. No improvement.Alternatively, set x ≠ y and z ≠w. Let’s set x= a, y= b, z= c, w= d. Then a + b + c + d=2, ab + cd=1. To minimize S= a² + b² + c² + d².Express d=2 -a -b -c.Then S= a² + b² + c² + (2 -a -b -c)².But with the constraint ab + cd=1.This seems complicated. Maybe use Lagrange multipliers again. But this might take a while. Alternatively, assume that a = c and b = d. Then symmetry gives:a + b + a + b=2 ⇒2a + 2b=2 ⇒a + b=1.ab + ab=2ab=1 ⇒ab=0.5.Then S=2a² + 2b²=2(a² + b²).Given a + b=1, then a² + b²=1 - 2ab=1 -1=0. So S=0, which is impossible. Wait, contradiction. So such symmetry isn't possible.Alternatively, maybe set a = d and b = c. Then:a + b + b + a=2 ⇒2a +2b=2 ⇒a + b=1.Cyclic sum ab + ba=2ab=1 ⇒ab=0.5.Then S= a² + b² + b² + a²=2a² +2b²=2(a² + b²).Again, since a + b=1, a² + b²=1 -2ab=1 -1=0 ⇒S=0. Contradiction. Impossible.Therefore, this approach doesn't work. Hence, the minimal S seems to be 1.5, achieved by three consecutive variables: 0.5,1,0.5, rest zero. Let's check if this is indeed the minimum.Alternatively, consider four variables with three non-zero: a, b, c, 0. Then sum a + b + c=2. Cyclic sum ab + bc +0=ab + bc=1. Let's set a = x, b= y, c= z. Then x + y + z=2, xy + y z=1. Let's express z=2 -x - y. Substitute:xy + y(2 -x - y)=1 ⇒2y - y²=1 ⇒y=1. Then z=1 -x. Sum of squares S=x² +1 + (1 -x)^2=2x² -2x +2. Minimum at x=0.5, giving S=1.5. Same as before. So even with four variables, the minimal S is 1.5. Therefore, regardless of how many variables we consider, the minimal S seems to be 1.5.Therefore, the minimal value of S is 1.5, and the maximal value is 2.Wait, but 1.5 is 3/2. To confirm, let me check if there's a configuration with S lower than 1.5.Suppose we have four variables a, b, c, d with a + b + c + d=2 and cyclic sum ab + bc + cd=1. Let’s set a=0.25, b=1, c=0.25, d=0.5. Then sum=0.25 +1 +0.25 +0.5=2. Cyclic sum=0.25*1 +1*0.25 +0.25*0.5 +0.5*0=0.25 +0.25 +0.125=0.625≠1. Doesn't work.Alternatively, set a=0.5, b=1, c=0.5, d=0. Then sum=0.5+1+0.5+0=2. Cyclic sum=0.5*1 +1*0.5 +0.5*0 +0*0=0.5 +0.5=1. Sum of squares=0.25 +1 +0.25 +0=1.5. So yes, same as before. If we spread variables more, we might get higher S, but not lower. Therefore, it seems that 1.5 is the minimal.But to ensure there are no other configurations, let's consider using the method of Lagrange multipliers for the three-variable case.Variables a, b, c, with a + b + c=2, ab + bc=1, and S= a² + b² + c².Express c=2 -a -b. Then S= a² + b² + (2 -a -b)².Expand S= a² + b² +4 -4a -4b +a² + 2ab + b²=2a² +2b² +2ab -4a -4b +4.Constraint: ab + b(2 -a -b)=1 ⇒2b -b²=1 ⇒b=1.So when b=1, then a + c=1. Then S= a² +1 +c²= a² +c² +1. Since a + c=1, the minimum of a² +c² is when a=c=0.5, giving 0.25 +0.25=0.5. Thus, S=0.5 +1=1.5.Therefore, this is indeed the minimal for the three-variable case, and since adding more variables either keeps S the same or increases it, this must be the global minimum.For the maximum S, as previously seen, the maximum occurs when two adjacent variables are 1 and the rest zero, giving S=2. Any other distribution with more variables would result in a lower sum of squares due to the nature of the square function (convexity).Therefore, the maximum value of S is 2, and the minimum value is 1.5, which is 3/2.Final AnswerThe maximum value of ( S ) is (boxed{2}) and the minimum value of ( S ) is (boxed{dfrac{3}{2}}).</think>Given 1999 non-negative real numbers (a_1, a_2, ldots, a_{1999}) satisfying the conditions:1. (a_1 + a_2 + cdots + a_{1999} = 2)2. (a_1a_2 + a_2a_3 + cdots + a_{1998}a_{1999} + a_{1999}a_1 = 1)We need to find the maximum and minimum values of (S = a_1^2 + a_2^2 + cdots + a_{1999}^2). Maximum Value of (S):To maximize (S), we consider the configuration where the sum of squares is maximized. This occurs when the values are as unequal as possible. By setting two adjacent variables to 1 and the rest to 0, we satisfy both conditions:- The sum (1 + 1 + 0 + cdots + 0 = 2)- The cyclic product sum (1 cdot 1 + 0 + cdots + 0 = 1)Thus, the maximum value of (S) is:[S_{text{max}} = 1^2 + 1^2 + 0^2 + cdots + 0^2 = 2] Minimum Value of (S):To minimize (S), we consider configurations where the values are spread out more evenly. By setting three consecutive variables to (0.5), (1), and (0.5) (with the rest zero), we satisfy both conditions:- The sum (0.5 + 1 + 0.5 + 0 + cdots + 0 = 2)- The cyclic product sum (0.5 cdot 1 + 1 cdot 0.5 + 0.5 cdot 0 + cdots + 0 cdot 0.5 = 1)Thus, the minimum value of (S) is:[S_{text{min}} = 0.5^2 + 1^2 + 0.5^2 + 0^2 + cdots + 0^2 = 0.25 + 1 + 0.25 = 1.5 = frac{3}{2}] Final AnswerThe maximum value of (S) is (boxed{2}) and the minimum value of (S) is (boxed{dfrac{3}{2}}).]

🔑:Solution: For a_{1}+a_{2}+cdots+a_{1999}=2, the following calculation is made:begin{array}{l}fleft(a_{1}, a_{2}, cdots, a_{1999}right) =a_{1} a_{2}+a_{2} a_{3}+cdots+a_{1998} a_{1999}+a_{1999} a_{1}end{array}Find the maximum value.Lemma: There exists an i in{1,2, cdots, 1999}, such that a_{i}>a_{i+4} (with a_{1999+i}=a_{i}).Proof: Otherwise, we havea_{1} leqslant a_{5} leqslant a_{9} leqslant cdots leqslant a_{1997} leqslant a_{2} leqslant a_{6} leqslant cdots leqslant a_{1998}leqslant a_{3} leqslant a_{7} leqslant cdots leqslant a_{1999} leqslant a_{4} leqslant a_{8} leqslant cdots leqslant a_{1996} leqslant a_{1},Thus, a_{1}=a_{2}=cdots=a_{1999}=frac{2}{1999}, which does not meet the condition.The lemma is proved.Without loss of generality, assume a_{1996}>a_{1}.For given a_{1}, a_{2}, cdots, a_{1996}, a_{1998}, letbegin{array}{c}a_{1}^{prime}=a_{1}, a_{2}^{prime}=a_{2}, cdots, a_{1996}^{prime}=a_{1996}, a_{1998}^{prime}=a_{1998}, a_{1999}^{prime}=0, a_{1997}^{prime}=a_{1997}+a_{1999},end{array}Then sum_{i} a_{i}^{prime}=2.begin{array}{l} fleft(a_{1}^{prime}, a_{2}^{prime}, cdots, a_{1999}^{prime}right) = a_{1} a_{2}+a_{2} a_{3}+cdots+a_{1995} a_{1996} +a_{1996}left(a_{1997}+a_{1999}right)+left(a_{1997}+a_{1999}right) a_{1998} = a_{1} a_{2}+cdots+a_{1996} a_{1997}+a_{1996} a_{1999}+a_{1997} a_{1998} quad+a_{1998} a_{1999} > a_{1} a_{2}+cdots+a_{1998} a_{1999}+a_{1999} a_{1} = fleft(a_{1}, a_{2}, cdots, a_{1999}right) .end{array}Next, let a_{1}^{prime prime}=a_{1}^{prime}, cdots, a_{1996}^{prime prime}=a_{1996}^{prime}+a_{1998}^{prime}, a_{1997}^{prime prime}=a_{1997}^{prime}, a_{1998}^{prime prime}=0, thenbegin{aligned}f & left(a_{1}^{prime prime}, a_{2}^{prime prime}, cdots, a_{1998}^{prime prime}, 0right) = & fleft(a_{1}^{prime}, a_{2}^{prime}, cdots, a_{1995}^{prime},left(a_{1996}^{prime}+a_{1998}^{prime}right), a_{1997}^{prime}, 0,0right) = & a_{1} a_{2}+cdots+a_{1995} a_{1996}+a_{1995} a_{1998}+a_{1996} a_{1997}^{prime} & +a_{1997}^{prime} a_{1998} > & fleft(a_{1}^{prime}, a_{2}^{prime}, cdots, a_{1998}^{prime}, 0right) .end{aligned}That is, each time the last non-zero number a_{i} is set to zero, and the sum of a_{i} and a_{i-2} is set as the new a_{i-2}, the value of f increases.Until only three numbers remain:begin{array}{l}a_{1}+o_{2}+u_{3}=2 . jleft(a_{1}, a_{2}, a_{3}, 0, cdots, hat{v}right)=a_{1} a_{2}+a_{2} a_{3} =a_{2}left(2-a_{2}right)=2 a_{2}-a_{2}^{2} leqslant 1 .end{array}Thus, the maximum value of fleft(a_{1}, a_{2}, cdots, a_{1099}right) is 1.It is achieved when a_{4}=a_{5}=cdots=a_{1999}=0, a_{2}=1. Therefore,begin{aligned}S & =a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=1+a_{1}^{2}+a_{3}^{2} & geqslant 1+frac{left(a_{1}+a_{3}right)^{2}}{2}=1 frac{1}{2} .end{aligned}Hence, S_{min }=1 frac{1}{2}.text { Also, } begin{aligned}S & =1+a_{1}^{2}+a_{3}^{2} & =2left(a_{1}-frac{1}{2}right)^{2}+frac{3}{2}, 0 leqslant a_{1} leqslant 1,end{aligned}Therefore, S_{max }=2.

🧠:[<think>Okay, so I have this problem about two circles, C₁ and C₂, and I need to figure out their positional relationship. The options are externally tangent, separate, intersecting, or internally tangent. Hmm, I remember that to determine how two circles are positioned relative to each other, I need to find their centers and radii, then calculate the distance between the centers and compare it to the sum and difference of the radii. Let me try to recall the steps.First, I need to rewrite the equations of the circles in standard form, which is (x - h)² + (y - k)² = r². The given equations are in general form, so I have to complete the squares for both x and y terms. Let me start with circle C₁: x² + y² + 4x + 3y + 2 = 0.For C₁, let's group the x terms and the y terms:(x² + 4x) + (y² + 3y) = -2.Now, completing the square for x: take the coefficient of x, which is 4, divide by 2 to get 2, square it to get 4. So, add and subtract 4. Similarly for y: coefficient of y is 3, half of that is 1.5, square is 2.25. So add and subtract 2.25.So, the equation becomes:(x² + 4x + 4) - 4 + (y² + 3y + 2.25) - 2.25 = -2Simplify:(x + 2)² + (y + 1.5)² - 6.25 = -2Then, move the constant to the right side:(x + 2)² + (y + 1.5)² = -2 + 6.25 = 4.25So, the center of C₁ is (-2, -1.5) and the radius is sqrt(4.25). Let me compute sqrt(4.25). Since 4.25 is 17/4, sqrt(17/4) is (√17)/2 ≈ 2.0616.Now, moving on to circle C₂: x² + y² + 2x + 3y + 1 = 0.Again, group x and y terms:(x² + 2x) + (y² + 3y) = -1.Completing the square for x: coefficient of x is 2, half is 1, square is 1. Add and subtract 1. For y terms, same as before: coefficient 3, half 1.5, square 2.25. Add and subtract 2.25.Equation becomes:(x² + 2x + 1) - 1 + (y² + 3y + 2.25) - 2.25 = -1Simplify:(x + 1)² + (y + 1.5)² - 3.25 = -1Move the constant:(x + 1)² + (y + 1.5)² = -1 + 3.25 = 2.25So, center of C₂ is (-1, -1.5) and radius is sqrt(2.25) which is 1.5.Alright, now I have both centers and radii. Let me write them down:C₁: Center (-2, -1.5), radius ≈ 2.0616C₂: Center (-1, -1.5), radius 1.5Next step is to compute the distance between the centers. The centers are (-2, -1.5) and (-1, -1.5). Let's see, the x-coordinates differ by (-1) - (-2) = 1, and the y-coordinates are the same, both -1.5. So the distance between centers is sqrt[(1)^2 + (0)^2] = sqrt(1) = 1.Wait, that's interesting. The distance between centers is 1 unit. Now, the radii of C₁ and C₂ are approximately 2.0616 and 1.5 respectively. Let me compute the sum and the difference of the radii.Sum of radii: 2.0616 + 1.5 ≈ 3.5616Difference of radii: |2.0616 - 1.5| ≈ 0.5616Now, the distance between centers is 1. So, we need to see where 1 falls in relation to the sum and difference.If the distance is equal to the sum, they are externally tangent. If equal to the difference, internally tangent. If between sum and difference, they intersect. If greater than the sum, they are separate. If less than the difference, one is inside the other.So here, 1 is compared to approximately 3.5616 (sum) and 0.5616 (difference). Since 0.5616 < 1 < 3.5616, that means the distance is between the difference and the sum. Wait, but actually, the logic is: if the distance is greater than the sum, they're separate. If equal to sum, externally tangent. If between the difference and sum, they intersect. If equal to the difference, internally tangent. If less than the difference, one is inside the other without touching.But wait, in our case, the distance is 1. Let's use exact values instead of approximate decimals to be precise. Let me compute the exact radii.For C₁, the radius squared is 4.25, which is 17/4. So the radius is sqrt(17)/2. For C₂, radius squared is 2.25, which is 9/4, so radius is 3/2.So the sum of radii is sqrt(17)/2 + 3/2 = (sqrt(17) + 3)/2. The difference is |sqrt(17)/2 - 3/2| = |sqrt(17) - 3|/2.Compute sqrt(17) first: sqrt(16) is 4, sqrt(17) ≈ 4.1231. Therefore, sum of radii ≈ (4.1231 + 3)/2 ≈ 7.1231/2 ≈ 3.5616, which matches the approximate value. The difference is |4.1231 - 3|/2 ≈ 1.1231/2 ≈ 0.5616.The distance between centers is exactly 1. So, comparing 1 with sum ≈3.5616 and difference≈0.5616.So 1 is greater than the difference (0.5616) and less than the sum (3.5616). Therefore, according to the criteria:- If distance > sum: separate- If distance = sum: externally tangent- If difference < distance < sum: intersecting- If distance = difference: internally tangent- If distance < difference: one inside the other (no intersection)But wait, actually, I think the difference is |r₁ - r₂|. If the distance is less than |r₁ - r₂|, then one circle is entirely inside the other. If it's equal, they are internally tangent. If it's between |r₁ - r₂| and r₁ + r₂, they intersect at two points. If equal to sum, externally tangent. If greater than sum, separate.Yes, so in this case, since 0.5616 < 1 < 3.5616, so the distance is between the difference and the sum. Therefore, the circles intersect at two points. Hence, the answer should be C. Intersecting.Wait, but let me confirm once again with exact values.Let me calculate |r₁ - r₂| = |sqrt(17)/2 - 3/2| = |sqrt(17) - 3|/2.Compute sqrt(17) exactly? Well, sqrt(17) is irrational, but sqrt(17) ≈4.123, so sqrt(17) - 3 ≈1.123, divided by 2 is≈0.5616.Distance between centers is exactly 1. So 1 is greater than |r₁ - r₂| (≈0.5616) and less than r₁ + r₂ (≈3.5616). Therefore, the circles intersect at two distinct points. So the answer is C. Intersecting.But let me make sure there isn't a miscalculation. Let's check centers again.C₁: x² + y² +4x +3y +2=0. Completing squares:x² +4x + y² +3y = -2.x² +4x +4 + y² +3y + (9/4) = -2 +4 +9/4.Which is (x+2)^2 + (y + 1.5)^2 = 4.25. Correct.C₂: x² + y² +2x +3y +1=0.x² +2x + y² +3y = -1.x² +2x +1 + y² +3y + (9/4) = -1 +1 +9/4.Which is (x +1)^2 + (y +1.5)^2 = 9/4. That is 2.25. Correct.Centers: C₁ at (-2, -1.5), C₂ at (-1, -1.5). So both centers are on the same horizontal line y = -1.5. The x-coordinates are -2 and -1. So the horizontal distance is |-1 - (-2)| = |1| = 1. So indeed, distance between centers is 1. Correct.So sum of radii is sqrt(17)/2 + 3/2. Let me compute sqrt(17) ≈4.123, so sum is (4.123 + 3)/2 ≈7.123/2≈3.5615. Difference is (4.123 -3)/2≈1.123/2≈0.5615. Therefore, distance between centers is 1, which is between 0.5615 and 3.5615. So, they must intersect. So answer is C.But wait, the options are A. Externally tangent, B. Separate, C. Intersecting, D. Internally tangent.So yes, intersecting is C. Therefore, the answer should be C.But wait, just to be thorough, let's check if 1 is actually between |r1 - r2| and r1 + r2.r1 + r2 ≈2.0616 + 1.5 =3.5616|r1 - r2|≈2.0616 -1.5≈0.5616Distance is 1, which is between 0.5616 and 3.5616, so yes, intersecting. So correct.Therefore, the answer is C.Final Answerboxed{C}</think>To determine the positional relationship between the circles ( C_1 ) and ( C_2 ), we first convert their equations to standard form by completing the square.For ( C_1: x^2 + y^2 + 4x + 3y + 2 = 0 ):1. Group and complete the square for ( x ) and ( y ): [ (x^2 + 4x) + (y^2 + 3y) = -2 ] [ (x^2 + 4x + 4) + (y^2 + 3y + left(frac{9}{4}right)) = -2 + 4 + frac{9}{4} ] [ (x + 2)^2 + left(y + frac{3}{2}right)^2 = frac{17}{4} ] The center is ((-2, -1.5)) and the radius is (frac{sqrt{17}}{2} approx 2.0616).For ( C_2: x^2 + y^2 + 2x + 3y + 1 = 0 ):1. Group and complete the square for ( x ) and ( y ): [ (x^2 + 2x) + (y^2 + 3y) = -1 ] [ (x^2 + 2x + 1) + (y^2 + 3y + left(frac{9}{4}right)) = -1 + 1 + frac{9}{4} ] [ (x + 1)^2 + left(y + frac{3}{2}right)^2 = frac{9}{4} ] The center is ((-1, -1.5)) and the radius is (frac{3}{2} = 1.5).Next, we calculate the distance between the centers ((-2, -1.5)) and ((-1, -1.5)):[text{Distance} = sqrt{(-1 - (-2))^2 + (-1.5 - (-1.5))^2} = sqrt{1^2 + 0^2} = 1]We compare this distance with the sum and difference of the radii:- Sum of radii: (frac{sqrt{17}}{2} + frac{3}{2} approx 3.5616)- Difference of radii: (left| frac{sqrt{17}}{2} - frac{3}{2} right| approx 0.5616)Since (0.5616 < 1 < 3.5616), the distance between the centers is between the difference and the sum of the radii. Therefore, the circles intersect at two points.[boxed{C}],<think>Okay, so I need to figure out the positional relationship between the two circles C₁ and C₂. The options are externally tangent, separate, intersecting, or internally tangent. Hmm. Let me recall what each of these terms means.First, I think the key here is to find the centers and radii of both circles. Once I have those, I can calculate the distance between the centers and compare it to the sum and difference of the radii. That should tell me their relative positions.Starting with circle C₁: x² + y² + 4x + 3y + 2 = 0. To find the center and radius, I need to rewrite this equation in the standard form (x - h)² + (y - k)² = r². To do that, I'll complete the square for both x and y terms.For the x terms: x² + 4x. Completing the square: take half of 4, which is 2, square it to get 4. So, x² + 4x = (x + 2)² - 4.For the y terms: y² + 3y. Completing the square: take half of 3, which is 1.5, square it to get 2.25. So, y² + 3y = (y + 1.5)² - 2.25.Now substitute these back into the original equation:(x + 2)² - 4 + (y + 1.5)² - 2.25 + 2 = 0Combine the constants: -4 -2.25 + 2 = -4.25So, (x + 2)² + (y + 1.5)² = 4.25Therefore, the center of C₁ is (-2, -1.5) and the radius is √4.25. Let me compute that. √4.25 is the same as √(17/4) which is (√17)/2 ≈ 2.0616.Okay, moving on to circle C₂: x² + y² + 2x + 3y + 1 = 0. Again, I need to complete the square for x and y.For the x terms: x² + 2x. Completing the square: half of 2 is 1, square is 1. So, x² + 2x = (x + 1)² - 1.For the y terms: y² + 3y. Wait, this is the same as in C₁. Completing the square: same as before, (y + 1.5)² - 2.25.Substitute back into the equation:(x + 1)² - 1 + (y + 1.5)² - 2.25 + 1 = 0Combine the constants: -1 -2.25 + 1 = -2.25So, (x + 1)² + (y + 1.5)² = 2.25Therefore, the center of C₂ is (-1, -1.5) and the radius is √2.25 = 1.5.Alright, so now I have both centers and radii. Let me note them down:C₁: Center (-2, -1.5), radius ≈ 2.0616C₂: Center (-1, -1.5), radius 1.5Next step is to find the distance between the centers. The centers are (-2, -1.5) and (-1, -1.5). Let me calculate the distance.The formula for distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²].Here, x₁ = -2, y₁ = -1.5; x₂ = -1, y₂ = -1.5.So, the x difference is (-1) - (-2) = 1, the y difference is 0.Therefore, distance = √(1² + 0²) = √1 = 1.So the distance between the centers is 1 unit.Now, let's find the sum and difference of the radii. The radii are approximately 2.0616 (C₁) and 1.5 (C₂).Sum of radii: 2.0616 + 1.5 ≈ 3.5616Difference of radii: 2.0616 - 1.5 ≈ 0.5616Compare the distance between centers (1) with these two numbers.If the distance is equal to the sum, they are externally tangent.If the distance is equal to the difference, they are internally tangent.If the distance is greater than the sum, they are separate.If the distance is less than the sum but greater than the difference, they intersect.So here, distance = 1.Sum ≈ 3.5616, difference ≈ 0.5616.Since 0.5616 < 1 < 3.5616, that would mean the circles intersect.Wait, but let me confirm with exact values instead of approximate decimals, to be precise.Original radii:C₁'s radius squared is 4.25, so radius is √(17)/2.C₂'s radius is 3/2.Sum of radii: √17/2 + 3/2 = (√17 + 3)/2Difference of radii: √17/2 - 3/2 = (√17 - 3)/2Distance between centers is 1.So we need to compare 1 with (√17 + 3)/2 and (√17 - 3)/2.First, compute (√17 - 3)/2:√17 is approximately 4.123, so √17 - 3 ≈ 1.123, then divided by 2 ≈ 0.5615.So (√17 - 3)/2 ≈ 0.5615, which is the difference.Sum is (4.123 + 3)/2 ≈ 7.123/2 ≈ 3.5615.So, 0.5615 ≈ difference < 1 < 3.5615 ≈ sum.Therefore, the distance between centers is between the difference and sum of radii. Therefore, the circles intersect.But wait, let me check with exact values.Let me square the distance between centers and compare with sum and difference squared? Wait, no. Wait, actually, another approach is to compute the distance squared and compare with (r1 + r2)^2 and (r1 - r2)^2.Wait, but in this case, since we already have the distance as 1, and the sum and difference, maybe it's straightforward.Alternatively, maybe I can compute 1 and compare to (√17 + 3)/2 and (√17 - 3)/2.But perhaps there's a miscalculation. Wait, (√17 + 3)/2: √17 is about 4.123, so 4.123 + 3 = 7.123, divided by 2 is 3.5615. Similarly, √17 - 3 ≈ 1.123, divided by 2 ≈ 0.5615.So yes, 1 is between these two values. Therefore, the circles intersect. So the answer should be C. Intersecting.Wait, but let me double-check. Maybe I made a mistake in calculating the centers or the radii.For C₁: x² + y² +4x +3y +2 =0.Completing the square:x² +4x = (x+2)^2 -4y² +3y = (y + 1.5)^2 - 2.25So equation becomes (x+2)^2 -4 + (y+1.5)^2 -2.25 +2 =0So (x+2)^2 + (y+1.5)^2 -4 -2.25 +2 =0-4 -2.25 is -6.25 +2 = -4.25Hence, (x+2)^2 + (y +1.5)^2 = 4.25, so radius is sqrt(4.25) = sqrt(17)/2 ≈ 4.123/2 ≈2.0616. That seems right.C₂: x² + y² +2x +3y +1=0x² +2x = (x+1)^2 -1y² +3y = (y +1.5)^2 - 2.25So equation becomes (x+1)^2 -1 + (y +1.5)^2 -2.25 +1=0Thus, (x+1)^2 + (y +1.5)^2 -1 -2.25 +1 =0-1 -2.25 +1 = -2.25Therefore, (x+1)^2 + (y +1.5)^2 = 2.25, so radius is sqrt(2.25)=1.5. Correct.Centers: C₁ at (-2, -1.5), C₂ at (-1, -1.5). So the y-coordinates are the same, both at -1.5. The x-coordinates are -2 and -1. So the distance between them is |-1 - (-2)| = |1| =1. Correct, since they are aligned horizontally.So centers are 1 unit apart. C₁ has radius ≈2.0616, C₂ has radius 1.5.So, if the distance between centers (1) is less than the sum of radii (≈3.5616) and greater than the difference (≈0.5616), then they intersect.Yes, so the answer should be C. Intersecting.Wait, but let me check again. Maybe I need to verify using exact values without approximation.Let me compute (r1 + r2) and |r1 - r2| exactly.r1 = sqrt(17)/2, r2 = 3/2.Sum: sqrt(17)/2 + 3/2 = (sqrt(17) + 3)/2Difference: sqrt(17)/2 - 3/2 = (sqrt(17) - 3)/2Distance between centers: 1.So we need to compare 1 with (sqrt(17) + 3)/2 and (sqrt(17) - 3)/2.Compute (sqrt(17) - 3)/2:sqrt(17) ≈4.123, 4.123 -3=1.123, divided by 2≈0.5615. So approximately 0.5615.Similarly, (sqrt(17) +3)/2≈(4.123 +3)/2≈7.123/2≈3.5615.So, 0.5615 < 1 < 3.5615. Therefore, 1 is between the difference and the sum.Thus, the circles intersect. Therefore, the answer is C.But let me just confirm with another method. Let's see if the circles actually intersect.Alternatively, we can solve the two equations simultaneously and see if there are real solutions.But solving them might be time-consuming, but maybe feasible.Given:C₁: x² + y² +4x +3y +2 =0C₂: x² + y² +2x +3y +1 =0Subtract C₂ from C₁: (x² + y² +4x +3y +2) - (x² + y² +2x +3y +1) =0 -0Simplifies to: 2x +1 =0 => 2x = -1 => x= -0.5.So the radical axis is the line x = -0.5. Therefore, the points of intersection lie on x = -0.5.Now substitute x = -0.5 into one of the circle equations to find y.Let's use C₂: x² + y² +2x +3y +1 =0Plug x = -0.5:(-0.5)^2 + y² + 2*(-0.5) +3y +1 =00.25 + y² -1 +3y +1 =0Simplify: 0.25 -1 +1 + y² +3y =0 => 0.25 + y² +3y =0Multiply through by 4 to eliminate decimal:1 +4y² +12y =0 =>4y² +12y +1=0Quadratic equation: 4y² +12y +1 =0Solutions: y = [-12 ± sqrt(144 - 16)] /8 = [-12 ± sqrt(128)] /8 = [-12 ± 8*sqrt(2)] /8 = [-3 ± 2*sqrt(2)] /2Therefore, real solutions exist because the discriminant is positive (144 -16=128>0). So there are two points of intersection: at x=-0.5, y=(-3 + 2√2)/2 and y=(-3 -2√2)/2.Therefore, the circles intersect at two distinct points, confirming that the answer is C. Intersecting.Hence, the correct answer is C.Final Answerboxed{C}</think>To determine the positional relationship between the circles ( C_1 ) and ( C_2 ), we first find their centers and radii by rewriting the given equations in standard form.For ( C_1: x^2 + y^2 + 4x + 3y + 2 = 0 ):- Completing the square for ( x ) terms: ( x^2 + 4x = (x + 2)^2 - 4 )- Completing the square for ( y ) terms: ( y^2 + 3y = (y + 1.5)^2 - 2.25 )- Substituting back, we get: ( (x + 2)^2 + (y + 1.5)^2 - 4 - 2.25 + 2 = 0 )- Simplifying constants: ( -4.25 )- Standard form: ( (x + 2)^2 + (y + 1.5)^2 = 4.25 )- Center: ( (-2, -1.5) ), Radius: ( sqrt{4.25} = frac{sqrt{17}}{2} approx 2.0616 )For ( C_2: x^2 + y^2 + 2x + 3y + 1 = 0 ):- Completing the square for ( x ) terms: ( x^2 + 2x = (x + 1)^2 - 1 )- Completing the square for ( y ) terms: ( y^2 + 3y = (y + 1.5)^2 - 2.25 )- Substituting back, we get: ( (x + 1)^2 + (y + 1.5)^2 - 1 - 2.25 + 1 = 0 )- Simplifying constants: ( -2.25 )- Standard form: ( (x + 1)^2 + (y + 1.5)^2 = 2.25 )- Center: ( (-1, -1.5) ), Radius: ( sqrt{2.25} = 1.5 )Next, we calculate the distance between the centers:- Centers: ( (-2, -1.5) ) and ( (-1, -1.5) )- Distance: ( sqrt{(-1 - (-2))^2 + (-1.5 - (-1.5))^2} = sqrt{1^2 + 0^2} = 1 )Comparing the distance with the sum and difference of the radii:- Sum of radii: ( frac{sqrt{17}}{2} + 1.5 approx 3.5616 )- Difference of radii: ( frac{sqrt{17}}{2} - 1.5 approx 0.5616 )- Since ( 0.5616 < 1 < 3.5616 ), the distance between the centers is between the difference and sum of the radii.To confirm, we solve the equations simultaneously:- Subtracting ( C_2 ) from ( C_1 ) gives ( 2x + 1 = 0 ) leading to ( x = -0.5 )- Substituting ( x = -0.5 ) into ( C_2 ) and solving the quadratic equation yields real solutions for ( y ), confirming two points of intersection.Thus, the circles intersect. The answer is (boxed{C}).]

🔑:To solve for the center and radius of circle C1, we complete the square for both x and y: C_{1} text{ center } = (-2, -frac{3}{2}), text{ with radius } r_{1}= frac{1}{2} sqrt{16+9-8} = frac{sqrt{17}}{2}. Similarly for circle C2: C_{2} text{ center } = (-1, -frac{3}{2}), text{ with radius } r_{2}= frac{1}{2} sqrt{4+9-4} = frac{3}{2}. Next, we calculate the distance between the centers of the two circles, which is given by: d = sqrt{(-2 - (-1))^2 + (-frac{3}{2} - (-frac{3}{2}))^2} = sqrt{1+0} = 1. Now let's establish the relationship between the distance d and the radii r1 and r2: because r_{1} - r_{2} < d < r_{1} + r_{2}, therefore text{circle } C_{1} text{ and } C_{2} text{ are intersecting.} Thus the correct answer is:[boxed{C}]

🧠:[<think>Okay, so I need to find the general term formula for the sequence {a_n} where all terms are positive numbers. The given condition is that for any positive integer n, 4S_n = a_n² + 2a_n, where S_n is the sum of the first n terms. Hmm, let's start by recalling that S_n is the sum of the first n terms, so S_n = a_1 + a_2 + ... + a_n. Since the problem relates S_n and a_n, maybe I can express S_n in terms of a_n and then find a recurrence relation? Let me try to write down the given equation again:4S_n = a_n² + 2a_nSo, S_n = (a_n² + 2a_n)/4.But S_n is also equal to S_{n-1} + a_n. So, substituting that into the equation, we have:S_{n-1} + a_n = (a_n² + 2a_n)/4But S_{n-1} can also be expressed using the given formula for n-1. Let's see:For n-1, the equation would be 4S_{n-1} = a_{n-1}² + 2a_{n-1} => S_{n-1} = (a_{n-1}² + 2a_{n-1}) /4Substituting this into the previous equation:(a_{n-1}² + 2a_{n-1}) /4 + a_n = (a_n² + 2a_n)/4Let me multiply both sides by 4 to eliminate the denominators:a_{n-1}² + 2a_{n-1} + 4a_n = a_n² + 2a_nThen, bringing all terms to one side:a_{n-1}² + 2a_{n-1} + 4a_n - a_n² - 2a_n = 0Simplify the terms:a_{n-1}² + 2a_{n-1} + (4a_n - a_n² - 2a_n) = 0Which simplifies to:a_{n-1}² + 2a_{n-1} + (2a_n - a_n²) = 0Rearranged:-a_n² + 2a_n + a_{n-1}² + 2a_{n-1} = 0Hmm, this looks a bit complicated. Maybe I should rearrange the terms differently. Let's move the a_n² terms to the other side:a_{n-1}² + 2a_{n-1} = a_n² - 2a_nWait, so:a_{n-1}² + 2a_{n-1} = a_n² - 2a_nHmm, this seems like a recursive relationship. Maybe there's a pattern here. Let's consider the left-hand side (LHS) and the right-hand side (RHS). The LHS is in terms of a_{n-1} and the RHS is in terms of a_n. Maybe if I can express both sides as something squared or factor them?Let me try to complete the square on both sides.For the left-hand side: a_{n-1}² + 2a_{n-1}Adding and subtracting 1: (a_{n-1}² + 2a_{n-1} +1) -1 = (a_{n-1} +1)^2 -1Similarly, the right-hand side: a_n² - 2a_nAdding and subtracting 1: (a_n² - 2a_n +1) -1 = (a_n -1)^2 -1So substituting back into the equation:(a_{n-1} +1)^2 -1 = (a_n -1)^2 -1Then, adding 1 to both sides:(a_{n-1} +1)^2 = (a_n -1)^2Taking square roots on both sides, since all terms are positive:a_{n-1} +1 = ±(a_n -1)But since all terms a_n are positive, let's check the possible signs.If we take the positive square root:a_{n-1} +1 = a_n -1Then, rearranged:a_n = a_{n-1} +2Alternatively, if we take the negative square root:a_{n-1} +1 = - (a_n -1) = -a_n +1Then, rearranged:a_{n-1} +1 = -a_n +1 => a_{n-1} = -a_nBut since all terms are positive, a_{n-1} and a_n are positive, so this would imply a negative a_n, which is impossible. Therefore, the only valid possibility is the positive square root:a_n = a_{n-1} +2So this suggests that the sequence {a_n} is an arithmetic sequence with common difference 2. Wait, but let me check if that's the case. If a_n = a_{n-1} +2, then the sequence is arithmetic with d=2. Let's verify this with the initial terms.First, we need to find a_1. For n=1, S_1 = a_1. According to the given equation:4S_1 = a_1² + 2a_1 => 4a_1 = a_1² + 2a_1 => a_1² -2a_1 =0 => a_1(a_1 -2)=0Since a_1 is positive, a_1=2.Therefore, the first term is 2. If the sequence is arithmetic with d=2, then a_n = 2 + (n-1)*2 = 2n. Let's check if this satisfies the given condition.Suppose a_n=2n. Then S_n = sum_{k=1}^n 2k = 2*(n(n+1)/2) = n(n+1)So 4S_n =4n(n+1)On the other hand, a_n² +2a_n = (2n)^2 +2*(2n)=4n² +4n=4n(n +1)Hence, 4S_n = a_n² +2a_n. Which matches the given condition. So the general term is a_n=2n. But wait, let me check this for n=2 as well. If a_1=2 and a_2=4, then S_2=2+4=6. Then 4S_2=24. a_2² +2a_2=16+8=24. So yes, it works. Similarly, for n=3, a_3=6, S_3=2+4+6=12. 4*12=48, a_3² +2a_3=36+12=48. Correct. So it seems to hold.But wait, earlier when we derived the recurrence relation, we found that a_n = a_{n-1} +2. Which is an arithmetic progression with common difference 2. So starting from a_1=2, this gives a_n=2n. So the general term is 2n.But let me think again. Is there another possible solution? Because when we took the square root, we considered only the positive case, but perhaps there's a different sequence that satisfies the original equation? Let's check.Suppose that instead of a_n = a_{n-1} +2, there's another relation. Wait, but when we derived the equation (a_{n-1} +1)^2 = (a_n -1)^2, we considered the square roots. Since both sides are squares, the only possibilities are a_{n-1} +1 = a_n -1 or a_{n-1} +1 = - (a_n -1). But as I saw before, the second case leads to a_{n-1} +1 = -a_n +1 => a_{n-1} = -a_n, which is impossible because all terms are positive. So only the first case is valid, leading to a_n = a_{n-1} +2. Therefore, the sequence must be arithmetic with d=2. And since a_1=2, then indeed a_n=2n.Alternatively, let's try to approach this problem using mathematical induction. Suppose that a_n=2n for all n. Then S_n = n(n+1). Let's check if 4S_n = a_n² + 2a_n.Left side: 4n(n+1) =4n² +4nRight side: (2n)^2 +2*(2n)=4n² +4nThey are equal. So the formula holds. Therefore, the general term is a_n=2n.Alternatively, suppose that there's another sequence that satisfies the given condition. Let me assume that there is a different sequence. But given the recurrence relation and the initial condition a_1=2, which is uniquely determined, the recurrence a_n =a_{n-1} +2 must generate the entire sequence. Therefore, the sequence is uniquely determined as a_n=2n.Hence, the answer should be a_n=2n.Final AnswerThe general term formula for the sequence is boxed{2n}.</think>Given that all terms of the sequence ({a_n}) are positive numbers, and for any (n in mathbb{N}^*), (4S_n = a_n^2 + 2a_n) holds, where (S_n) is the sum of the first (n) terms of the sequence ({a_n}), we need to find the general term formula for the sequence ({a_n}).First, we know that (S_n = frac{a_n^2 + 2a_n}{4}). Also, (S_n = S_{n-1} + a_n). Substituting (S_{n-1}) from the given formula for (n-1), we have:[S_{n-1} = frac{a_{n-1}^2 + 2a_{n-1}}{4}]Substituting this into the equation for (S_n), we get:[frac{a_{n-1}^2 + 2a_{n-1}}{4} + a_n = frac{a_n^2 + 2a_n}{4}]Multiplying through by 4 to clear the denominators:[a_{n-1}^2 + 2a_{n-1} + 4a_n = a_n^2 + 2a_n]Rearranging terms:[a_{n-1}^2 + 2a_{n-1} = a_n^2 - 2a_n]Completing the square on both sides:[(a_{n-1} + 1)^2 - 1 = (a_n - 1)^2 - 1]Simplifying, we get:[(a_{n-1} + 1)^2 = (a_n - 1)^2]Taking the positive square root (since all terms are positive):[a_{n-1} + 1 = a_n - 1]Solving for (a_n):[a_n = a_{n-1} + 2]This indicates the sequence is an arithmetic progression with common difference 2. To find the first term (a_1), we use (n = 1) in the given formula:[4S_1 = a_1^2 + 2a_1 implies 4a_1 = a_1^2 + 2a_1 implies a_1(a_1 - 2) = 0]Since (a_1) is positive, (a_1 = 2). Thus, the general term for the arithmetic sequence is:[a_n = 2 + (n - 1) cdot 2 = 2n]Verifying, if (a_n = 2n), then (S_n = sum_{k=1}^n 2k = n(n + 1)). Substituting into the given equation:[4S_n = 4n(n + 1) = (2n)^2 + 2(2n) = 4n^2 + 4n]This confirms the solution is correct.The general term formula for the sequence is (boxed{2n}).,<think>Okay, let's see. The problem says that for all natural numbers n, 4S_n = a_n² + 2a_n, where S_n is the sum of the first n terms of the sequence {a_n}. And all the terms are positive numbers. We need to find the general term formula for a_n.First, I remember that S_n is the sum of the first n terms, so S_n = a_1 + a_2 + ... + a_n. Also, S_n is related to a_n through the equation 4S_n = a_n² + 2a_n. Hmm, interesting. Since this is true for any n, maybe we can express S_n in terms of a_n and then relate it to S_{n-1} somehow?Let me try writing the given equation for n and for n-1. For n, we have 4S_n = a_n² + 2a_n. For n-1, substituting n-1 into the equation, we get 4S_{n-1} = a_{n-1}² + 2a_{n-1}. But since S_n = S_{n-1} + a_n, maybe I can substitute that into the first equation. Let's try that. So, 4(S_{n-1} + a_n) = a_n² + 2a_n.But from the equation for n-1, S_{n-1} = (a_{n-1}² + 2a_{n-1}) / 4. Substitute that into the equation above:4[(a_{n-1}² + 2a_{n-1}) / 4 + a_n] = a_n² + 2a_nSimplifying the left side:(a_{n-1}² + 2a_{n-1}) + 4a_n = a_n² + 2a_nBring all terms to one side:a_{n-1}² + 2a_{n-1} + 4a_n - a_n² - 2a_n = 0Simplify:a_{n-1}² + 2a_{n-1} + 2a_n - a_n² = 0Hmm, this looks a bit complicated. Let me rearrange the terms:a_n² - 2a_n = a_{n-1}² + 2a_{n-1}So, a_n² - 2a_n - (a_{n-1}² + 2a_{n-1}) = 0Wait, that's:a_n² - 2a_n = a_{n-1}² + 2a_{n-1}So, maybe this can be expressed as a quadratic in terms of a_n? Let me see.Alternatively, perhaps factor both sides. Let me see:Left side: a_n² - 2a_n = a_n(a_n - 2)Right side: a_{n-1}² + 2a_{n-1} = a_{n-1}(a_{n-1} + 2)Hmm, not sure if that helps. Maybe another approach.Wait, since the equation relates a_n and a_{n-1}, maybe this is a recurrence relation. If I can express a_n in terms of a_{n-1}, maybe I can solve it.Let me rearrange the equation:a_n² - 2a_n - a_{n-1}² - 2a_{n-1} = 0Let me group terms:(a_n² - a_{n-1}²) - 2(a_n + a_{n-1}) = 0Factor the difference of squares:(a_n - a_{n-1})(a_n + a_{n-1}) - 2(a_n + a_{n-1}) = 0Factor out (a_n + a_{n-1}):(a_n + a_{n-1})(a_n - a_{n-1} - 2) = 0Since all terms are positive, a_n + a_{n-1} can't be zero. Therefore, the other factor must be zero:a_n - a_{n-1} - 2 = 0So, a_n = a_{n-1} + 2Ah! That's a simple recurrence relation. So the sequence is arithmetic with common difference 2. So if I can find the first term a_1, then the general term is a_n = a_1 + 2(n - 1).But wait, let's confirm this. If the recurrence is a_n = a_{n-1} + 2, then the sequence is indeed arithmetic with difference 2. Let's check if this satisfies the original equation.First, let's find a_1. For n = 1, S_1 = a_1. So the equation becomes 4a_1 = a_1² + 2a_1. Let's solve for a_1:4a_1 = a_1² + 2a_1Subtract 4a_1:0 = a_1² - 2a_1Factor:a_1(a_1 - 2) = 0Since all terms are positive, a_1 = 2.So the first term is 2, then each subsequent term is 2 more than the previous. So the sequence is 2, 4, 6, 8, ..., which is an arithmetic sequence with a_1 = 2 and common difference 2.Therefore, the general term is a_n = 2 + 2(n - 1) = 2n.Wait, let's verify this with n=2. If a_1=2 and a_2=4, then S_2 = 2 + 4 = 6. Let's check the equation 4S_2 = a_2² + 2a_2. 4*6 = 24, and a_2² + 2a_2 = 16 + 8 = 24. So that works.Similarly for n=3: a_3=6, S_3=2+4+6=12. 4*12=48, and 6² + 2*6=36 +12=48. Correct.So seems like it works. Therefore, the general term is a_n=2n.Wait, but let me check the recurrence again. The key step was when we derived that (a_n + a_{n-1})(a_n - a_{n-1} -2)=0, so since the sum is positive, the other factor must be zero. So a_n = a_{n-1} + 2. Therefore, arithmetic sequence with difference 2. Then a_1 is 2, so a_n = 2n. That seems correct.But just to be thorough, maybe check one more term. For n=4, a_4=8, S_4=2+4+6+8=20. 4*20=80. a_4² + 2a_4 =64 +16=80. Correct.Okay, seems consistent. Therefore, the general term is a_n=2n.Wait, but let me think again. The problem says "for any n∈N*", so the equation holds for all n. So if the recurrence relation holds for all n≥2, and the base case n=1 gives a_1=2, then the formula a_n=2n is valid.Alternatively, since we derived that the sequence is arithmetic with a_1=2 and d=2, so yes, a_n=2n.But maybe we can also find it through another way. Let's try using the formula for S_n.Since S_n is the sum of an arithmetic sequence. The sum of the first n terms of an arithmetic sequence is (n/2)(2a_1 + (n-1)d). Here, a_1=2, d=2. So S_n=(n/2)(4 + 2(n-1)) = (n/2)(2n + 2) = n(n +1). Then 4S_n=4n(n+1). On the other hand, a_n² +2a_n= (2n)^2 +2*(2n)=4n² +4n=4n(n +1). So indeed, 4S_n = a_n² + 2a_n. Therefore, this formula works.Therefore, the general term is a_n=2n.Final AnswerThe general term formula for the sequence is boxed{2n}.</think>Given that all terms of the sequence ({a_n}) are positive numbers, and for any (n in mathbb{N}^{*}), the equation (4S_n = a_n^2 + 2a_n) holds, where (S_n) is the sum of the first (n) terms of the sequence ({a_n}), we need to find the general term formula for the sequence ({a_n}).First, we note that (S_n = a_1 + a_2 + cdots + a_n). For (n) and (n-1), we have the equations:[4S_n = a_n^2 + 2a_n][4S_{n-1} = a_{n-1}^2 + 2a_{n-1}]Since (S_n = S_{n-1} + a_n), substituting this into the first equation gives:[4(S_{n-1} + a_n) = a_n^2 + 2a_n]Substituting (S_{n-1} = frac{a_{n-1}^2 + 2a_{n-1}}{4}) into the equation above, we get:[4left(frac{a_{n-1}^2 + 2a_{n-1}}{4} + a_nright) = a_n^2 + 2a_n]Simplifying this, we have:[a_{n-1}^2 + 2a_{n-1} + 4a_n = a_n^2 + 2a_n]Rearranging terms, we get:[a_n^2 - 2a_n - a_{n-1}^2 - 2a_{n-1} = 0]Factoring the equation, we find:[(a_n - a_{n-1} - 2)(a_n + a_{n-1}) = 0]Since all terms are positive, (a_n + a_{n-1} neq 0), so we must have:[a_n - a_{n-1} - 2 = 0 implies a_n = a_{n-1} + 2]This indicates the sequence is arithmetic with a common difference of 2. To find the first term (a_1), we use (n = 1):[4S_1 = a_1^2 + 2a_1 implies 4a_1 = a_1^2 + 2a_1 implies a_1^2 - 2a_1 = 0 implies a_1(a_1 - 2) = 0]Since (a_1) is positive, we get (a_1 = 2). Thus, the general term of the arithmetic sequence is:[a_n = 2 + 2(n - 1) = 2n]Verifying this, the sum (S_n) of the first (n) terms of the arithmetic sequence is (S_n = n(n + 1)). Substituting into the given equation:[4S_n = 4n(n + 1) quad text{and} quad a_n^2 + 2a_n = (2n)^2 + 2(2n) = 4n^2 + 4n = 4n(n + 1)]Both sides are equal, confirming the solution. Therefore, the general term formula for the sequence is (boxed{2n}).]

🔑:When n=1, from 4S_1=a_1^2+2a_1 and a_1 > 0, we get a_1=2.When ngeqslant 2, from 4a_n=4S_n-4S_{n-1}=(a_n^2+2a_n)-(a_{n-1}^2+2a_{n-1}), we get (a_n+a_{n-1})(a_n-a_{n-1}-2)=0.Since a_n+a_{n-1} > 0, we have a_n-a_{n-1}=2.Thus, a_n=2+(n-1)×2=2n.Therefore, the answer is: boxed{2n}.When n=1, we get a_1=2; when ngeqslant 2, from 4a_n=4S_n-4S_{n-1}, we get a_n-a_{n-1}=2, and hence the conclusion can be drawn.This problem tests the knowledge of the general term formula and the sum of the first n terms of a sequence. Pay attention to the accumulation of problem-solving methods, and it is a moderately difficult problem.

🧠:[<think>Okay, let's see. I need to find the measure of angle AMB in a parallelogram ABCD where M is the midpoint of side DC, and M lies on the angle bisector of angle DAB. Hmm, first, let me visualize the parallelogram. In a parallelogram, opposite sides are equal and parallel. So, AB is parallel to DC, and AD is parallel to BC. Also, opposite angles are equal, and consecutive angles are supplementary. Since M is the midpoint of DC, that means DM = MC. And M is on the angle bisector of angle DAB. So, angle DAB is one of the angles of the parallelogram. Let me denote angle DAB as angle A. Since it's a parallelogram, angle ABC would be equal to angle ADC, and so on. Wait, maybe drawing a diagram would help. Let me sketch parallelogram ABCD with AB and DC as the base sides, and AD and BC as the other sides. Let's label the vertices in order: A at the bottom left, B at the bottom right, C at the top right, D at the top left. Then, DC is the top side, so M is the midpoint of DC. So, DM = MC. Now, the angle bisector of angle DAB. Angle DAB is at vertex A, between sides AD and AB. The bisector would split angle DAB into two equal parts. So, if angle DAB is, say, 2θ, then each bisected angle would be θ. Since M is on this bisector, point M must lie somewhere such that the bisector passes through it. But M is the midpoint of DC. Hmm, interesting. In a parallelogram, the coordinates might help. Maybe assigning coordinates to the points will make this easier. Let's place point A at the origin (0,0). Since it's a parallelogram, let me assign coordinates based on vectors. Let's assume vector AB is along the x-axis. Let me denote the length of AB as 2a, so that point B is at (2a, 0). Then, since it's a parallelogram, point D can be at some point (0, b), so that vector AD is (0, b). Therefore, point C, which is AB + AD, would be at (2a, b). Wait, but in a parallelogram, opposite sides are equal. So DC would be from D(0, b) to C(2a, b), which is length 2a. Then, midpoint M of DC would be at (a, b). Now, angle bisector of angle DAB. Angle DAB is the angle at A between AD and AB. Since AD is vertical (from (0,0) to (0, b)) and AB is horizontal (from (0,0) to (2a,0)), angle DAB is 90 degrees? Wait, no, that's only if AD is vertical. Wait, but in a general parallelogram, angle DAB can be any angle, not necessarily 90 degrees. So, maybe my coordinate system is too restrictive. Because if I set AD as vertical, then the angle DAB would be 90 degrees, but in a general parallelogram, angles can be different. Hmm, perhaps a better coordinate system is needed. Let's try to generalize. Let me place point A at (0,0). Let me let vector AB be along the x-axis with length 2a, so point B is at (2a, 0). Then, vector AD can be some vector making an angle θ with the x-axis, so point D is at (c, d). Then, since ABCD is a parallelogram, point C would be at (2a + c, d). Then, the midpoint M of DC would be the midpoint between D(c, d) and C(2a + c, d). So, coordinates of M would be ((c + 2a + c)/2, (d + d)/2) = ( (2a + 2c)/2, d ) = (a + c, d). But M is supposed to lie on the angle bisector of angle DAB. Let's find the angle bisector of angle DAB. Angle DAB is the angle at point A between vectors AD and AB. So, vector AD is (c, d) - (0,0) = (c, d), and vector AB is (2a, 0) - (0,0) = (2a, 0). The angle bisector of angle DAB would be a line starting at A(0,0) and going in the direction that divides angle DAB into two equal parts. The angle bisector theorem might help here. The angle bisector theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. But in this case, the angle bisector is in angle DAB, so it would intersect side DC (opposite to angle DAB) at some point, say M. Wait, according to the problem, M is the midpoint of DC and lies on the angle bisector. Therefore, by the angle bisector theorem, the ratio of the segments DM to MC should be equal to the ratio of AD to AB. But since M is the midpoint, DM = MC, so the ratio is 1:1. Therefore, AD/AB = 1, so AD = AB. Wait, that seems important. So if the angle bisector cuts DC at its midpoint M, then according to the angle bisector theorem, AD/AB = DM/MC = 1. Therefore, AD = AB. So in this parallelogram, sides AD and AB are equal. Since it's a parallelogram, AB = DC and AD = BC. Therefore, if AD = AB, then all sides are equal? Wait, in a parallelogram, if adjacent sides are equal, then it's a rhombus. So, if AD = AB, then ABCD is a rhombus. But in a rhombus, all sides are equal. However, in our case, if AD = AB, then since AB is the base of the parallelogram and AD is the side, making them equal would make it a rhombus. Wait, but in the problem, it's stated as a parallelogram, not necessarily a rhombus. But according to the angle bisector theorem, since the bisector meets DC at its midpoint, which divides DC into DM and MC with a 1:1 ratio, therefore AD must equal AB. Therefore, ABCD must be a rhombus. Therefore, the given parallelogram is actually a rhombus with all sides equal. Therefore, all sides are equal, and the angles are determined by that. But wait, angle DAB is being bisected by the bisector that goes through M. If it's a rhombus, then the diagonals bisect the angles. But in a rhombus, the diagonals are the angle bisectors. However, point M is the midpoint of DC, not the intersection point of the diagonals. Wait, but in a rhombus, the midpoint of DC would be the same as the intersection point of the diagonals? No, the diagonals of a rhombus intersect at their midpoints. So, in a rhombus, the diagonals bisect each other. So, the midpoint of DC is point M, but in a rhombus, the diagonals intersect at the midpoint of both diagonals. So, if the angle bisector of angle DAB is a diagonal? Wait, in a rhombus, the diagonals bisect the angles. So, the diagonal from A would bisect angle DAB. Therefore, if the angle bisector of angle DAB is the diagonal AC, then point M, the midpoint of DC, lies on diagonal AC. But in a rhombus, diagonals are not necessarily along the midpoints of the sides unless it's a square. Wait, in a rhombus, the diagonals bisect each other but not necessarily the sides. Wait, in a rhombus, the diagonals intersect at 90 degrees and bisect each other. Wait, perhaps I need to verify. Let me think again. If ABCD is a rhombus, then all sides are equal: AB = BC = CD = DA. The diagonals AC and BD intersect at point O, which is the midpoint of both diagonals. The diagonals in a rhombus bisect the angles. So, diagonal AC bisects angle DAB into two equal angles. Therefore, if point M is the midpoint of DC, then in the rhombus, is point M lying on diagonal AC? Let me consider coordinates again. Let’s take a rhombus with side length 2a. Let’s place point A at (0,0), point B at (2a,0). Since it's a rhombus, point D would be at (a, b), so that AD = AB = 2a. Wait, if AB is 2a, then AD must also be 2a. So, the distance from A(0,0) to D(a, b) must be 2a. So sqrt(a² + b²) = 2a ⇒ a² + b² = 4a² ⇒ b² = 3a² ⇒ b = a√3. Therefore, point D is at (a, a√3). Then, point C would be at (2a + a, a√3) = (3a, a√3). Wait, no. In a rhombus, the coordinates should be such that vectors AB and AD define the rhombus. If AB is (2a, 0) and AD is (a, a√3), then point C is AB + AD = (2a + a, 0 + a√3) = (3a, a√3). Then, side DC is from D(a, a√3) to C(3a, a√3), so midpoint M is at ( (a + 3a)/2, (a√3 + a√3)/2 ) = (2a, a√3). Now, the angle bisector of angle DAB. In a rhombus, diagonal AC is the angle bisector. Let's see, point A(0,0) to point C(3a, a√3). The diagonal AC goes from (0,0) to (3a, a√3). But point M is at (2a, a√3). Is this point on diagonal AC? Let's check. The line AC can be parametrized as x = 3a*t, y = a√3*t, for t from 0 to 1. So, when x = 2a, then t = 2a / 3a = 2/3. Then y = a√3*(2/3) = (2a√3)/3. But point M is at (2a, a√3), which has y-coordinate a√3, which is greater than (2a√3)/3. Therefore, point M is not on diagonal AC. Hmm, that contradicts the previous thought. Therefore, maybe in this case, the angle bisector isn't the diagonal? Wait, maybe I made a mistake in assuming the rhombus. Let me go back. The angle bisector theorem says that if a bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides. But here, we're dealing with a parallelogram, not a triangle. However, perhaps considering triangle DAB. Wait, in the parallelogram ABCD, angle DAB is at point A, and the bisector of angle DAB meets DC at point M. Then, applying the angle bisector theorem to triangle DAB, but wait, DC isn't a side of triangle DAB. Maybe need to extend lines or something. Alternatively, think of ABCD as a trapezoid? Wait, no, it's a parallelogram. Maybe using vectors would help. Let me consider vectors. Let me place point A at the origin. Let vector AB = u, and vector AD = v. Then, point M, being the midpoint of DC, which is the same as the midpoint of AB + v and v, wait, no. Wait, in the parallelogram, point D is at v, point C is at u + v, so DC is from v to u + v. Therefore, midpoint M is at (v + u + v)/2 = (u + 2v)/2. But angle bisector of angle DAB. The angle at A between vectors u and v. The bisector direction can be found by normalizing the vectors u and v and then adding them. So, the angle bisector direction would be u/|u| + v/|v|. Since M lies on this bisector, the vector from A to M should be parallel to u/|u| + v/|v|. But vector AM is M - A = M - 0 = M = (u + 2v)/2. So, (u + 2v)/2 should be parallel to u/|u| + v/|v|. Let me denote |u| = AB = length of AB, and |v| = AD = length of AD. Let’s denote |u| = m and |v| = n. Then, the direction vector of the bisector is u/m + v/n. The vector AM is (u + 2v)/2. So, these two vectors must be scalar multiples of each other. Therefore, there exists a scalar k such that:(u + 2v)/2 = k(u/m + v/n)Multiplying both sides by 2:u + 2v = 2k(u/m + v/n)This gives us two equations for the coefficients of u and v:1 = 2k/m (coefficient of u)2 = 2k/n (coefficient of v)From the first equation: k = m/2From the second equation: 2 = 2*(m/2)/n ⇒ 2 = m/n ⇒ m = 2nTherefore, the length of AB (which is |u| = m) is twice the length of AD (|v| = n). So, AB = 2 AD. So, in the parallelogram, side AB is twice side AD. Therefore, it's not a rhombus, but a parallelogram with sides AB = 2 AD. Okay, that makes sense. So, the conclusion from the angle bisector theorem is that AB = 2 AD. Therefore, the sides are in a 2:1 ratio. Now, with that ratio, we can assign coordinates accordingly. Let's set point A at (0,0). Let’s take AD as a vector along the y-axis for simplicity, since the actual angles might complicate things. Wait, but if we take AD along the y-axis, then angle DAB would be 90 degrees, but we found that AB = 2 AD, which might not hold. Maybe better to assign coordinates such that AD is along the y-axis. Let’s try. Let me set AD along the y-axis. Let’s let AD = n, so point D is at (0, n). Then, AB is twice AD, so AB = 2n. Let’s take AB along the x-axis, so point B is at (2n, 0). Then, since it's a parallelogram, point C is at (2n, n). Then, midpoint M of DC is between D(0, n) and C(2n, n), so M is at (n, n). Now, angle bisector of angle DAB. Angle DAB is the angle at A(0,0) between AD (from A(0,0) to D(0,n)) and AB (from A(0,0) to B(2n,0)). So, AD is vertical, AB is horizontal. Therefore, angle DAB is 90 degrees. The angle bisector of a right angle would be a line that makes 45 degrees with both axes. So, the bisector would be the line y = x. But in our coordinate system, point M is at (n, n). So, the line y = x passes through (n, n). Therefore, yes, M lies on the angle bisector. So in this coordinate system, angle DAB is 90 degrees, and M is at (n, n). Now, we need to find angle AMB. Points A(0,0), M(n, n), and B(2n, 0). Let's compute the measure of angle AMB. To find angle AMB, which is the angle at point M between points A and B. So, angle AMB is the angle between vectors MA and MB. First, find the coordinates:Point A: (0, 0)Point M: (n, n)Point B: (2n, 0)Vector MA: A - M = (0 - n, 0 - n) = (-n, -n)Vector MB: B - M = (2n - n, 0 - n) = (n, -n)Now, to find the angle between vectors MA and MB, we can use the dot product formula:cosθ = (MA • MB) / (|MA| |MB|)Compute the dot product:MA • MB = (-n)(n) + (-n)(-n) = -n² + n² = 0Wait, the dot product is zero, which means the vectors are perpendicular. Therefore, angle AMB is 90 degrees. So, in this coordinate system, angle AMB is 90 degrees. Therefore, the measure of angle AMB is 90 degrees. But let me verify this with another approach to ensure that this isn't a coincidence due to the coordinate system chosen. Alternative approach: Since in the coordinate system we found angle AMB is 90 degrees, is this always the case regardless of the actual lengths? Let's see. Given AB = 2 AD, and angle DAB is 90 degrees, which might be a result of the coordinate system. Wait, but in reality, angle DAB doesn't have to be 90 degrees. Wait, in the previous coordinate system, we set AD along the y-axis and AB along the x-axis, making angle DAB 90 degrees. However, in a general case, if AB = 2 AD, angle DAB could be different. Wait, but according to the angle bisector theorem, we found that AB = 2 AD, but how does that affect the angle? Wait, maybe in the general case, even if angle DAB isn't 90 degrees, angle AMB is still 90 degrees. Let's check with another coordinate system. Let me assume that angle DAB is θ, and AB = 2 AD. Let me assign coordinates accordingly. Let’s place point A at (0,0). Let’s take AD as a vector making an angle θ/2 with the angle bisector. Wait, this might get complicated. Alternatively, use trigonometry. Suppose angle DAB is θ. Then, since M lies on the angle bisector, the angle between AD and AM is θ/2, and the angle between AB and AM is also θ/2. Given that AB = 2 AD, let’s denote AD = x, so AB = 2x. Then, in triangle ABM and ADM... Wait, perhaps using the Law of Sines or Cosines. Wait, but since ABCD is a parallelogram, we know that AD = BC and AB = DC. Also, M is the midpoint of DC, so DM = MC = x (since DC = AB = 2x). But I think the coordinate approach was more straightforward. Let me try another coordinate system where angle DAB is not 90 degrees. Let's say point A is at (0,0), point B is at (2a,0), and point D is at (b,c), such that AD = sqrt(b² + c²) = x, and AB = 2a = 2x, so a = x. Therefore, AB = 2x. Then, DC is from D(b,c) to C(b + 2x, c). Midpoint M is at (b + x, c). Now, the angle bisector of angle DAB. The angle bisector will divide angle DAB into two equal parts. The direction of the angle bisector can be found using the angle bisector formula in coordinates. The angle bisector from point A(0,0) between vectors AB(2x,0) and AD(b,c). The angle bisector direction vector is proportional to (2x/|AB| + b/|AD|, 0/|AB| + c/|AD|). Since |AB| = 2x and |AD| = sqrt(b² + c²) = x (since AD = x). Therefore, direction vector is (2x/(2x) + b/x, 0/(2x) + c/x) = (1 + b/x, c/x). But point M(b + x, c) lies on this angle bisector. Therefore, the line from A(0,0) in the direction (1 + b/x, c/x) should pass through M(b + x, c). So, parametric equations of the angle bisector are:x(t) = t*(1 + b/x)y(t) = t*(c/x)We need this line to pass through (b + x, c). Therefore,b + x = t*(1 + b/x)c = t*(c/x)From the second equation: c = t*(c/x) ⇒ t = x (assuming c ≠ 0). Substitute t = x into the first equation:b + x = x*(1 + b/x) = x + bWhich holds true. Therefore, the angle bisector passes through M(b + x, c). Therefore, regardless of the coordinates, as long as AB = 2 AD, the midpoint M lies on the angle bisector. Now, to compute angle AMB. Let's compute vectors MA and MB. Point A: (0,0)Point M: (b + x, c)Point B: (2x, 0)Vector MA: A - M = (-b - x, -c)Vector MB: B - M = (2x - b - x, -c) = (x - b, -c)Dot product MA • MB = (-b - x)(x - b) + (-c)(-c) = (-bx + b² - x² + bx) + c² = (b² - x²) + c²But since |AD| = sqrt(b² + c²) = x ⇒ b² + c² = x². Therefore, substituting:MA • MB = (b² - x²) + c² = (b² + c²) - x² = x² - x² = 0Therefore, the dot product is zero, which means vectors MA and MB are perpendicular, so angle AMB is 90 degrees. Therefore, regardless of the angle θ = angle DAB, as long as AB = 2 AD (which is forced by the angle bisector condition and M being the midpoint), angle AMB is always 90 degrees. Therefore, the measure of angle AMB is 90 degrees. Final AnswerThe measure of angle AMB is boxed{90} degrees.</think>Given a parallelogram (ABCD) with (M) as the midpoint of side (overline{DC}) and (M) lying on the angle bisector of (angle DAB), we need to determine the measure of (angle AMB).1. Parallelogram Properties: - Opposite sides are equal and parallel. - (AB = DC) and (AD = BC).2. Midpoint and Angle Bisector: - (M) is the midpoint of (DC), so (DM = MC). - (M) lies on the angle bisector of (angle DAB).3. Angle Bisector Theorem: - The angle bisector of (angle DAB) divides (DC) into segments (DM) and (MC) such that the ratio (DM/MC = AD/AB). - Since (M) is the midpoint, (DM = MC), implying (AD/AB = 1). Therefore, (AD = AB).4. Coordinate System: - Place (A) at the origin ((0,0)), (B) at ((2a, 0)), (D) at ((0, b)), and (C) at ((2a, b)). - Midpoint (M) of (DC) is ((a, b)).5. Vectors and Dot Product: - Vectors (overrightarrow{MA} = (-a, -b)) and (overrightarrow{MB} = (a, -b)). - Dot product (overrightarrow{MA} cdot overrightarrow{MB} = (-a)(a) + (-b)(-b) = -a^2 + b^2).6. Using Parallelogram Properties: - Since (AD = AB), the coordinates simplify such that (b = asqrt{3}) (for a rhombus), but the key insight is the dot product result.7. Conclusion: - The dot product (overrightarrow{MA} cdot overrightarrow{MB} = 0) implies vectors (overrightarrow{MA}) and (overrightarrow{MB}) are perpendicular. - Therefore, (angle AMB = 90^circ).The measure of (angle AMB) is (boxed{90}) degrees.,<think>Okay, so I need to find the measure of angle AMB in a parallelogram ABCD where M is the midpoint of DC, and M lies on the angle bisector of angle DAB. Hmm, let me start by drawing a diagram to visualize the problem.First, let's recall that in a parallelogram, opposite sides are equal and parallel, and opposite angles are equal. Also, the diagonals bisect each other. But here, M is the midpoint of DC, so that's specific. Also, M is on the angle bisector of angle DAB. So angle DAB is one of the angles of the parallelogram, let's say at vertex A.Let me label the parallelogram: A is at the bottom left, B at the bottom right, C at the top right, D at the top left. So sides AB and DC are parallel and equal, sides AD and BC are parallel and equal. Then M is the midpoint of DC, so if DC is the top side from D to C, then M is halfway between D and C.Now, the angle bisector of angle DAB. Angle DAB is the angle at vertex A between sides AD and AB. The bisector of this angle would split it into two equal angles. Since M is on this bisector, that gives us some relationship between the sides or angles.I need to find angle AMB. So points A, M, and B. Let me note that in the parallelogram, AB is the base, DC is the top side. Point M is the midpoint of DC. So coordinates might help here. Maybe assigning coordinates to the points will make this easier. Let's try that.Let me place point A at the origin (0,0). Since it's a parallelogram, let's let vector AB be along the x-axis. Let’s denote AB as length 2a for simplicity, so point B is at (2a, 0). Then, since AD is another side, let's denote AD as length 2b, but since in a parallelogram opposite sides are equal, BC will also be 2b. Wait, but if we use coordinates, maybe it's better to use vectors. Let's think.Alternatively, set coordinates such that point A is at (0,0), point B is at (2,0), so that the midpoint calculations might be simpler. Wait, since M is the midpoint of DC, maybe scaling the figure so that DC has length 2 units, making M at (1, something). But perhaps that's complicating.Alternatively, let's assign coordinates:Let’s set point A at (0,0).Let’s let AB be along the x-axis. Let’s let the length of AB be 2a (so point B is at (2a, 0)), and the length of AD be 2b (so point D is at (0, 2b)). Then, since ABCD is a parallelogram, point C would be at (2a, 2b). Then, midpoint M of DC: since D is (0, 2b) and C is (2a, 2b), the midpoint M is ((0 + 2a)/2, (2b + 2b)/2) = (a, 2b).Now, angle bisector of angle DAB. Angle DAB is the angle at A between AD and AB. Since AD is from A(0,0) to D(0, 2b), which is along the y-axis, and AB is from A(0,0) to B(2a,0), along the x-axis. So angle DAB is 90 degrees? Wait, no. Wait, if AD is vertical and AB is horizontal, then angle DAB is 90 degrees. But in a general parallelogram, angle DAB can be any angle. Wait, no. If AD is vertical and AB is horizontal, then it's a rectangle, which is a type of parallelogram, but in a general parallelogram, the sides are not necessarily aligned with the axes. Wait, I think I made a mistake here by fixing the coordinates that way.Wait, perhaps I need to assign coordinates more carefully. Let me think again.In a general parallelogram, adjacent angles are supplementary. So angle DAB + angle ABC = 180 degrees. But if we assign coordinates such that point A is at (0,0), and side AB is along the x-axis with length 'a', and point D is at (0, b), but then ABCD would be a rectangle. Wait, no, in a general parallelogram, the coordinates can be set with vectors.Alternatively, let's define the coordinates as follows:Let’s place point A at (0,0).Let’s let vector AB be (2a, 0), so point B is (2a, 0).Let’s let vector AD be (p, q), so point D is (p, q). Then, since ABCD is a parallelogram, point C is B + AD = (2a + p, q).Then, midpoint M of DC: points D(p, q) and C(2a + p, q). The midpoint M is ((p + 2a + p)/2, (q + q)/2) = ((2p + 2a)/2, q) = (p + a, q).Now, the angle bisector of angle DAB. Angle DAB is the angle at A between AD and AB. The angle bisector would be a line starting at A(0,0) and going in the direction that bisects the angle between vectors AB(2a,0) and AD(p,q). Since M(p + a, q) lies on this angle bisector, we can use the angle bisector theorem.The angle bisector theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides. Wait, but here M is not on side BC or anything; it's on DC. Hmm, maybe not directly applicable. Alternatively, in terms of coordinates, the angle bisector can be parametrized.Alternatively, since M lies on the angle bisector, the direction vector from A to M should satisfy the angle bisector condition. Let me recall that the angle bisector between two vectors can be found by normalizing the vectors and adding them. So if we have two vectors u and v, the angle bisector direction is u/|u| + v/|v|.So, vector AB is (2a, 0), so its length is 2a. Vector AD is (p, q), so its length is sqrt(p² + q²). Then, the direction of the angle bisector would be ( (2a,0)/(2a) ) + ( (p, q)/sqrt(p² + q²) ) = (1, 0) + (p, q)/sqrt(p² + q²). So the direction vector is (1 + p/sqrt(p² + q²), q/sqrt(p² + q²)).But point M(p + a, q) lies on this angle bisector. So the line from A(0,0) through M(p + a, q) must be in the direction of the angle bisector. Therefore, the vector AM = (p + a, q) must be a scalar multiple of the angle bisector direction vector.Therefore, (p + a, q) = k*(1 + p/sqrt(p² + q²), q/sqrt(p² + q²)) for some scalar k.So we have two equations:1. p + a = k*(1 + p/sqrt(p² + q²))2. q = k*( q/sqrt(p² + q²) )From equation 2:q = k*q / sqrt(p² + q²)Assuming q ≠ 0 (otherwise, the parallelogram would be degenerate), we can divide both sides by q:1 = k / sqrt(p² + q²)So k = sqrt(p² + q²)Plugging into equation 1:p + a = sqrt(p² + q²) * (1 + p / sqrt(p² + q²))Simplify the right-hand side:sqrt(p² + q²) * 1 + sqrt(p² + q²)*(p / sqrt(p² + q²)) = sqrt(p² + q²) + pTherefore, equation 1 becomes:p + a = sqrt(p² + q²) + pSubtract p from both sides:a = sqrt(p² + q²)So sqrt(p² + q²) = aBut sqrt(p² + q²) is the length of vector AD, which is side AD. In a parallelogram, AD and BC are equal, and AB and DC are equal. So here, AB is length 2a, and AD is length a. Wait, but in our coordinate setup, AB is vector (2a, 0), so its length is 2a. But from the equation above, length AD is a. So sides AB and DC are length 2a, and sides AD and BC are length a. Therefore, the sides are in a 2:1 ratio.So that gives us a relationship between the sides. So if AB is twice AD, then the parallelogram has sides of length 2a and a. Interesting. So in this case, the angle at A (angle DAB) can be found using the cosine law if needed, but maybe we can now find coordinates.Since sqrt(p² + q²) = a, and vector AD is (p, q), so p² + q² = a².Also, in our coordinate system, vector AB is (2a, 0), and vector AD is (p, q) with p² + q² = a².We need to find angle AMB. Points A(0,0), M(p + a, q), B(2a, 0). So we need to find angle at M between points A, M, B. Wait, no, angle AMB is the angle at M between points A, M, B. Wait, no, angle AMB is the angle at M between points A, M, B? Wait, no. Wait, angle AMB is the angle at M formed by points A, M, B. Wait, no, in standard notation, angle AMB is the angle at M between points A and B. Wait, actually, no. Wait, in angle AMB, the vertex is M, so it's the angle at M between points A and B. Wait, but in the problem statement, it says "the measure of angle AMB". So vertex at M, with segments MA and MB. So angle AMB is ∠AMB, which is at point M. Wait, but sometimes notation can be ambiguous. Wait, in standard notation, angle AMB would be the angle at M. But in the problem statement, it's written as angle AMB, which is the angle at M. Wait, but the user says "determine the measure of angle AMB". Let me confirm.In standard geometry notation, angle AMB is the angle at M between points A and B. So vertex at M, with sides MA and MB. So yes, angle at M. But in the problem, the points are A, M, B. So angle at M between A and B.But to compute angle at M, we need coordinates of points A, M, B. Let's get back to the coordinates.We have points:A: (0,0)B: (2a, 0)M: (p + a, q)But we also know that p² + q² = a², from earlier.So we can express M's coordinates as (p + a, q), where p² + q² = a². Hmm.To find angle at M between points A, M, B, which is angle AMB (at M). To find this angle, we can use vectors. The vectors MA and MB. Then, the angle between vectors MA and MB will give us the angle at M.First, compute vectors:Vector MA = A - M = (0 - (p + a), 0 - q) = (-p - a, -q)Vector MB = B - M = (2a - (p + a), 0 - q) = (a - p, -q)Then, the angle between vectors MA and MB can be found using the dot product formula:cosθ = (MA • MB) / (|MA| |MB|)Compute MA • MB:(-p - a)(a - p) + (-q)(-q)First term: (-p - a)(a - p) = (-p)(a - p) - a(a - p) = -pa + p² - a² + ap = p² - a²Second term: (-q)(-q) = q²Therefore, MA • MB = (p² - a²) + q²But we know from earlier that p² + q² = a². So substituting:MA • MB = (a² - q² - a²) + q² = (-q²) + q² = 0Wait, that's interesting. Wait, hold on. Wait, p² + q² = a², so p² = a² - q². Then, p² - a² = (a² - q²) - a² = -q². Then, adding q² gives -q² + q² = 0. Therefore, MA • MB = 0.Therefore, the cosine of the angle between vectors MA and MB is 0/( |MA| |MB| ) = 0. Therefore, the angle between them is 90 degrees. Therefore, angle AMB is 90 degrees.Wait, so is the answer 90 degrees? That seems straightforward, but let me verify the steps again to make sure I didn't make a mistake.First, assigned coordinates with A at (0,0), B at (2a,0), D at (p,q), so C is (2a + p, q). Then, M is the midpoint of DC, which is ((p + 2a + p)/2, (q + q)/2) = (p + a, q). Then, since M lies on the angle bisector of angle DAB, we found that sqrt(p² + q²) = a. Then, vectors MA and MB are computed, their dot product is zero, hence angle AMB is 90 degrees.Therefore, the measure of angle AMB is 90 degrees, so the answer is 90°, which is π/2 radians. Therefore, boxed, it's boxed{90^circ}.But let me check with a specific case. Suppose a = 1, so AB is length 2, AD is length 1. Let’s take AD as along the y-axis for simplicity, so p = 0, q = 1. Then point D is (0,1), C is (2,1). Midpoint M of DC is ((0 + 2)/2, (1 + 1)/2) = (1,1). Then, angle bisector of angle DAB. Since angle DAB is 90 degrees (since AD is vertical and AB is horizontal), the angle bisector would be the line y = x, since it bisects the right angle. But point M is (1,1), which lies on y = x. Then, angle AMB: points A(0,0), M(1,1), B(2,0). So vectors MA = (-1,-1), MB = (1,-1). The dot product is (-1)(1) + (-1)(-1) = -1 + 1 = 0. Therefore, angle AMB is 90 degrees. So in this specific case, it works. So the answer is indeed 90 degrees.Therefore, the measure of angle AMB is boxed{90^circ}.Final Answerboxed{90^circ}</think>Given a parallelogram (ABCD) with (M) as the midpoint of side (overline{DC}) and (M) lying on the angle bisector of (angle DAB), we need to determine the measure of (angle AMB).1. Coordinate Setup: - Place point (A) at the origin ((0,0)). - Let (AB) be along the x-axis with (B) at ((2a, 0)). - Let (AD) be a vector ((p, q)) such that point (D) is ((p, q)) and point (C) is ((2a + p, q)).2. Midpoint Calculation: - Midpoint (M) of (DC) is (left(frac{p + (2a + p)}{2}, frac{q + q}{2}right) = (p + a, q)).3. Angle Bisector Condition: - Since (M) lies on the angle bisector of (angle DAB), the vector (AM) must be a scalar multiple of the angle bisector direction. - Using the angle bisector theorem and vector analysis, we find that the length of (AD) is (a), leading to (p^2 + q^2 = a^2).4. Vectors Calculation: - Vector (MA = (0 - (p + a), 0 - q) = (-p - a, -q)). - Vector (MB = (2a - (p + a), 0 - q) = (a - p, -q)).5. Dot Product and Angle Calculation: - The dot product of vectors (MA) and (MB) is calculated as: [ (-p - a)(a - p) + (-q)(-q) = (p^2 - a^2) + q^2 = 0 ] - Since the dot product is zero, the vectors are perpendicular, implying (angle AMB = 90^circ).Thus, the measure of (angle AMB) is (boxed{90^circ}).]

🔑:## Solution.Let varangle B A D=alpha.Since D C | A B andvarangle B A M=varangle A M D=varangle D A M=frac{alpha}{2},triangle D A M is isosceles:|D M|=|D A| quad text{and} quad varangle D M A=frac{alpha}{2} .Since triangle M C B is also isosceles (because |M C|=|M D|=|D A|=|B C|), it follows thatvarangle B C M=varangle B C D=varangle D A B=alphaandvarangle C M B=varangle C B M=frac{180^{circ}-varangle B C M}{2}=frac{180^{circ}-alpha}{2}=90^{circ}-frac{alpha}{2} .Now it is easy to calculate that varangle A M B=180^{circ}-varangle A M D-varangle C M B=90^{circ}.

🧠:[<think>Okay, so I have this problem where there's a piecewise function f(x). Let me first write that down properly to make sure I understand it. The function f(x) is defined as:f(x) = { lg x, when x > 0 x + ∫ from 0 to a of 3t² dt, when x ≤ 0 }And we're told that f(f(1)) = 27. The question is asking for the real number a, which is supposed to be 3 according to the answer provided. But I need to work through it step by step.First, let me recall that lg x typically stands for logarithm base 10, right? So lg x = log₁₀ x. Just want to confirm that notation since sometimes "lg" can be log base 2, but in many contexts, especially in higher mathematics, "lg" is base 10. Anyway, let's proceed with that assumption.So, f(f(1)) = 27. Let's break this down. First, compute f(1), then plug that result into f again, and set it equal to 27. Then solve for a.Starting with f(1). Since 1 is greater than 0, we use the first part of the piecewise function, which is lg x. Therefore, f(1) = lg(1). What's lg(1)? Well, log base 10 of 1 is 0 because 10^0 = 1. So f(1) = 0.Now, we need to compute f(f(1)) = f(0). Because f(1) is 0, so next we evaluate f at 0. Since 0 is less than or equal to 0, we use the second part of the piecewise function. That part is x + the integral from 0 to a of 3t² dt. So plugging x = 0 into that, we get:f(0) = 0 + ∫₀ᵃ 3t² dtSo now I need to compute that integral. The integral of 3t² with respect to t is t³ + C, because the antiderivative of t² is (1/3)t³, so multiplying by 3 gives t³. Therefore, evaluating from 0 to a:∫₀ᵃ 3t² dt = [t³]₀ᵃ = a³ - 0³ = a³So f(0) = 0 + a³ = a³But we know that f(f(1)) = f(0) = a³, and according to the problem, this equals 27. Therefore:a³ = 27Solving for a, we take the cube root of both sides:a = ∛27 = 3So that gives a = 3. That seems straightforward, but let me double-check each step to make sure I haven't made any mistakes.First, checking f(1). Since x=1 >0, use lg x. log₁₀(1) is indeed 0. Correct.Then f(0). Since x=0 ≤0, use the second part. The integral ∫₀ᵃ 3t² dt. The integral of 3t² is t³, evaluated at a gives a³, subtract at 0 gives 0, so integral is a³. So f(0) = 0 + a³ = a³. Then set that equal to 27, so a³=27, so a=3. All steps check out.Wait, but just to make sure there's no trick here. Let me check if the integral was computed correctly. The integrand is 3t², so integrating from 0 to a. The antiderivative is 3*(t³/3) = t³, right? So yes, evaluated from 0 to a, that's a³ - 0 = a³. So that part is correct.Is there any possibility that the problem uses lg as natural logarithm or base 2? Hmm. If lg were natural logarithm, then f(1) would be ln(1) which is also 0, so same result. If it were base 2, log₂(1) is also 0. So regardless of the base, as long as it's a logarithm, log_b(1) is 0 because any number to the power of 0 is 1. So the first step is safe.Therefore, f(f(1)) is f(0) which is a³=27, leading to a=3. All steps seem correct. Therefore, the answer is indeed 3. I don't see any errors here.Final AnswerThe real number ( a = boxed{3} ).</think>Given the piecewise function ( f(x) ):[f(x) = begin{cases} lg x, & x > 0 x + int_{0}^{a} 3t^2 , dt, & x leq 0 end{cases}]we need to find the real number ( a ) such that ( f(f(1)) = 27 ).1. Compute ( f(1) ): - Since ( 1 > 0 ), we use the first part of the function: ( f(1) = lg 1 ). - ( lg 1 = 0 ) because ( 10^0 = 1 ).2. Compute ( f(f(1)) = f(0) ): - Since ( 0 leq 0 ), we use the second part of the function: ( f(0) = 0 + int_{0}^{a} 3t^2 , dt ). - Evaluate the integral: [ int_{0}^{a} 3t^2 , dt = left[ t^3 right]_{0}^{a} = a^3 - 0^3 = a^3 ] - Therefore, ( f(0) = a^3 ).3. Set ( f(f(1)) = 27 ): - ( a^3 = 27 ) - Solving for ( a ), we get ( a = sqrt[3]{27} = 3 ).Thus, the real number ( a = boxed{3} ).,<think>Okay, let's try to figure out how to solve this problem. The function f(x) is defined piecewise: for x > 0, it's the logarithm base 10 of x, and for x ≤ 0, it's x plus the integral from 0 to a of 3t² dt. We need to find the real number a such that f(f(1)) equals 27. Hmm, alright. Let's break this down step by step.First, let's understand the function f(x). For positive x, it's straightforward: just log base 10 of x. For non-positive x, it's x plus some integral. The integral part is from 0 to a of 3t² dt. Maybe I should compute that integral first so I know what exactly the function looks like for x ≤ 0.Let me compute the integral ∫₀ᵃ 3t² dt. The integral of 3t² is t³, right? Because the antiderivative of t² is (1/3)t³, so multiplying by 3 gives t³. So evaluating from 0 to a, it's a³ - 0³ = a³. So the integral simplifies to a³. Therefore, for x ≤ 0, f(x) = x + a³.So now the function is:f(x) = { lg x, if x > 0 x + a³, if x ≤ 0}Got it. So now, we need to compute f(f(1)) and set it equal to 27, then solve for a.Starting with f(1): since 1 is greater than 0, we use the first part of the function, which is lg 1. Log base 10 of 1 is 0, because 10^0 = 1. So f(1) = 0.Now, compute f(f(1)) which is f(0). Since 0 is less than or equal to 0, we use the second part of the function: f(0) = 0 + a³ = a³.But the problem states that f(f(1)) = 27. Therefore, a³ = 27. Solving for a, we take the cube root of both sides: a = ∛27 = 3. So a should be 3. That seems straightforward. Wait, let me double-check to make sure I didn't skip any steps or make a mistake.First, confirm f(1): lg(1) is indeed 0. Then f(0) is 0 + a³. Set that equal to 27: a³ = 27, so a = 3. Seems correct. Let me check the integral again. ∫₀ᵃ 3t² dt. The integral of 3t² is t³ evaluated from 0 to a, which is a³ - 0 = a³. Yes, that's correct. So substituting back into f(x) for x ≤ 0, it's x + a³. Then f(0) = 0 + a³. Therefore, yes, a³ = 27 gives a = 3. That checks out.Is there any chance I messed up the integral? Let me think. The integrand is 3t². The antiderivative would be 3*(t³/3) = t³. So yes, the integral from 0 to a is a³. Correct. So, no mistakes there. Then f(0) is indeed a³, which is 27. Therefore, a is 3. Alright, that seems right.I guess the key steps here were recognizing how the piecewise function works, computing the integral correctly, and then applying the functions step by step. Starting with f(1), which is straightforward, then using that result (0) as the input into f again, which leads us to the second part of the piecewise function. Then solving for a is simple once we have the equation a³ = 27. So, the answer should be 3. I don't see any errors in this reasoning.Final AnswerThe real number ( a ) is (boxed{3}).</think>To solve for the real number ( a ) such that ( f(f(1)) = 27 ), we start by analyzing the given piecewise function ( f(x) ):[f(x) = begin{cases} lg x, & x > 0 x + int_{0}^{a} 3t^2 , dt, & x leq 0 end{cases}]First, we compute the integral in the definition of ( f(x) ) for ( x leq 0 ):[int_{0}^{a} 3t^2 , dt = left[ t^3 right]_{0}^{a} = a^3 - 0 = a^3]Thus, the function ( f(x) ) for ( x leq 0 ) simplifies to:[f(x) = x + a^3]Next, we evaluate ( f(1) ). Since ( 1 > 0 ), we use the logarithmic part of the function:[f(1) = lg 1 = 0]Now, we need to evaluate ( f(f(1)) = f(0) ). Since ( 0 leq 0 ), we use the second part of the function:[f(0) = 0 + a^3 = a^3]We are given that ( f(f(1)) = 27 ), so:[a^3 = 27]Solving for ( a ), we take the cube root of both sides:[a = sqrt[3]{27} = 3]Thus, the real number ( a ) is (boxed{3}).]

🔑:From the piecewise function, we know that f(1) = lg 1 = 0,Therefore, f(f(1)) = f(0) = int_{0}^{a} 3t^{2}dt = t^{3} bigg|_{0}^{a} = a^{3},Since f(f(1)) = 27 = a^{3},We solve to get a = 3.Hence, the answer is boxed{3}.The problem involves first finding the value of f(1), and then solving for the value of a based on the equation.This question mainly tests the application of piecewise functions and the key to solving it lies in using the integral formula.

🧠:[<think>Okay, so I need to figure out when Peter finishes the triathlon. Let's break down the problem step by step. First, the triathlon has three parts: swimming, cycling, and running. The distances are 1 km for swimming, 24 km for cycling, and 6 km for running. Peter started at 9:30. He completed the swimming in 20 minutes. Then, his cycling speed is ten times his swimming speed, and his running speed is four times his swimming speed. The answer choices are from 10:56 to 11:12, so I need to calculate the total time he took and add it to the start time.Let me start with the swimming part. He swam 1 km in 20 minutes. So, his swimming speed is distance divided by time. But let's convert the time to hours because the other distances are in km and speeds will be in km/h. 20 minutes is 1/3 of an hour. So, speed = 1 km / (1/3 hour) = 3 km/h. Wait, that's his swimming speed. Then, his cycling speed is ten times that, which would be 3 * 10 = 30 km/h. And his running speed is four times his swimming speed, so 3 * 4 = 12 km/h. Okay, so now we know his speeds for each activity. Let's compute the time he took for each part.Swimming time is already given as 20 minutes. So that's straightforward.For cycling: distance is 24 km, speed is 30 km/h. Time = distance / speed = 24 / 30 hours. Let's compute that. 24 divided by 30 is 0.8 hours. To convert that to minutes, multiply by 60: 0.8 * 60 = 48 minutes. So cycling takes him 48 minutes.For running: distance is 6 km, speed is 12 km/h. Time = 6 / 12 = 0.5 hours, which is 30 minutes.So total time is swimming (20) + cycling (48) + running (30) = 20 + 48 is 68, plus 30 is 98 minutes. Now, convert 98 minutes to hours and minutes. 98 divided by 60 is 1 hour and 38 minutes. He started at 9:30. Adding 1 hour brings us to 10:30. Then add 38 minutes: 10:30 + 30 minutes is 11:00, plus 8 minutes is 11:08. Wait, but wait, let me check again. 9:30 plus 1 hour is 10:30. Then 10:30 plus 38 minutes. 30 minutes would make it 11:00, then 8 more minutes is 11:08. Hmm, but the answer choices are (A) 10:56, (B) 11:00, (C)11:04, (D)11:08, (E)11:12. So 11:08 is option D.But wait, let me double-check my calculations because the answer might be different. Maybe I made a mistake in converting the swimming speed or the time.Swimming: 1 km in 20 minutes. So 20 minutes is 1/3 hour. Speed = 1 / (1/3) = 3 km/h. That's correct. Then cycling speed is 10 times that, so 30 km/h. Cycling distance is 24 km. Time = 24 / 30 = 0.8 hours. 0.8 *60 = 48 minutes. Correct. Running: 6 km at 12 km/h. 6/12 = 0.5 hours = 30 minutes. Correct. Total time 20 + 48 +30=98 minutes. 98 minutes is 1 hour (60 minutes) and 38 minutes. So 9:30 + 1 hour = 10:30, plus 38 minutes: 10:30 + 30 minutes is 11:00, plus 8 minutes is 11:08. So D.Wait, but maybe I made a mistake in the problem statement. Let me check again. The triathlon consists of 1 km swim, 24 km cycling, 6 km running. Peter completed the swimming in 20 minutes. Then he cycled at ten times the speed he swam. So swimming speed is 3 km/h, so cycling speed is 30 km/h. Then running speed is four times the swimming speed, so 12 km/h. Then time for cycling is 24 /30 = 0.8 hours = 48 minutes. Running time is 6 /12 = 0.5 hours =30 minutes. Total time: 20 +48 +30=98 minutes. 98 minutes is 1 hour 38 minutes. 9:30 +1 hour is 10:30, plus 38 minutes is 11:08. So answer D.But let me check the answer options again. The options are (A)10:56, (B)11:00, (C)11:04, (D)11:08, (E)11:12. So D is 11:08. That's the answer.Wait, but just to be thorough, maybe I miscalculated the start time. He started at 9:30. If he took 98 minutes, adding 98 minutes to 9:30. Let's do it step by step. 9:30 plus 60 minutes is 10:30. Then 38 minutes more: 10:30 + 30 minutes is 11:00. Then 8 minutes is 11:08. Yes, that's correct. So answer D. But wait, let me check if the problem said he started at 9:30, so adding 1h38m to that. Alternatively, sometimes time problems can be tricky with 12-hour clock, but here we are only dealing with morning times so 9:30 to 11:08 is straightforward. Alternatively, maybe I messed up the swimming time. Wait, he completed swimming in 20 minutes. So starting at 9:30, he finishes swimming at 9:50. Then he starts cycling. Then cycling takes 48 minutes. 9:50 +48 minutes: 9:50 +10 minutes is 10:00, then 38 minutes more is 10:38. Then he starts running. Running takes 30 minutes. 10:38 +30 minutes is 11:08. Yes, that's another way to compute it. So that's correct. So he finishes at 11:08, which is option D. So the answer is D.Final Answerboxed{D}</think>Peter started the triathlon at 9:30. The swimming segment took him 20 minutes, so he finished swimming at 9:50. Calculating speeds:- Swimming speed: ( frac{1 text{ km}}{frac{20}{60} text{ hours}} = 3 text{ km/h} )- Cycling speed: ( 10 times 3 text{ km/h} = 30 text{ km/h} )- Running speed: ( 4 times 3 text{ km/h} = 12 text{ km/h} )Calculating times for each segment:- Cycling time: ( frac{24 text{ km}}{30 text{ km/h}} = 0.8 text{ hours} = 48 text{ minutes} )- Running time: ( frac{6 text{ km}}{12 text{ km/h}} = 0.5 text{ hours} = 30 text{ minutes} )Total time for each segment:- Swimming: 20 minutes- Cycling: 48 minutes- Running: 30 minutesTotal time spent:[ 20 + 48 + 30 = 98 text{ minutes} = 1 text{ hour and 38 minutes} ]Calculating finish time:- Start time: 9:30- Adding 1 hour: 10:30- Adding 38 minutes: 10:30 + 38 minutes = 11:08Thus, Peter finished the competition at (boxed{D}) (11:08).,<think>Alright, let's try to figure out when Peter finishes the triathlon. Okay, so the triathlon has three parts: swimming, cycling, and running. The distances are 1 km for swimming, 24 km for cycling, and 6 km for running. Peter started at 9:30 AM. First, he did the swimming part in 20 minutes. Then he cycled at ten times his swimming speed, and ran at four times his swimming speed. We need to calculate the total time he took for all three events and add that to the start time to find the finish time. The options are from 10:56 to 11:12, so probably around an hour and a half or so?Let me break it down step by step. First, let's find Peter's swimming speed because the cycling and running speeds are given relative to that. He swam 1 km in 20 minutes. Speed is distance divided by time. But the units here are km and minutes, so maybe convert minutes to hours to have km per hour?Yes, because the other parts might be easier in km/h. So 20 minutes is 1/3 of an hour, right? 20 divided by 60 is 1/3. So his swimming speed is 1 km divided by 1/3 hour. That equals 3 km/h. Let me check that: if he swims 3 km in one hour, then in 20 minutes (which is a third of an hour), he swims 1 km. Yep, that makes sense.Okay, so his swimming speed is 3 km/h. Then, his cycling speed is ten times that, which would be 3 * 10 = 30 km/h. And his running speed is four times his swimming speed, so 3 * 4 = 12 km/h. Got that.Now, we need to find the time he took for each activity. We already know the swimming time is 20 minutes. Let's calculate the time for cycling and running.Starting with cycling: he cycled 24 km at 30 km/h. Time is distance divided by speed. So 24 km divided by 30 km/h. 24 divided by 30 is 0.8 hours. Hmm, 0.8 hours. To convert that to minutes, multiply by 60. 0.8 * 60 = 48 minutes. So cycling took 48 minutes.Then running: he ran 6 km at 12 km/h. Again, time is distance over speed. 6 divided by 12 is 0.5 hours. 0.5 hours is 30 minutes. So running took 30 minutes.Now, total time is swimming + cycling + running. That's 20 + 48 + 30 minutes. Let's add them up. 20 + 48 is 68, plus 30 is 98 minutes. So total time is 98 minutes.Now, he started at 9:30 AM. Adding 98 minutes to 9:30 AM. Let's convert 98 minutes into hours and minutes. 60 minutes is 1 hour, so 98 - 60 = 38 minutes. So that's 1 hour and 38 minutes.Adding 1 hour to 9:30 AM gets us to 10:30 AM. Then adding 38 minutes: 10:30 + 38 minutes. 10:30 + 30 minutes is 11:00 AM, then +8 minutes is 11:08 AM. Wait, but 38 minutes after 10:30 is 11:08? Let me check that again. 10:30 + 30 minutes is 11:00, then +8 minutes is 11:08. Yes. So total finish time is 11:08 AM.Looking at the options: (A) 10:56, (B) 11:00, (C)11:04, (D)11:08, (E)11:12. So the answer should be D) 11:08.Wait, let me double-check all the steps to make sure I didn't make a mistake. First, swimming speed: 1 km in 20 minutes. 20 minutes is 1/3 hour. Speed = 1 / (1/3) = 3 km/h. Correct. Cycling speed is 10 times that, so 30 km/h. Time cycling: 24 km / 30 km/h = 0.8 hours = 48 minutes. Correct. Running speed is 4 times swimming speed: 12 km/h. Time running: 6 / 12 = 0.5 hours = 30 minutes. Correct. Total time: 20 + 48 + 30 = 98 minutes. 98 minutes is 1 hour 38 minutes. Starting at 9:30, adding 1 hour is 10:30, then 38 minutes: 10:30 + 30 = 11:00, then +8 is 11:08. Yep, seems right. So answer is D.Final Answerboxed{D}</think>Peter completed the triathlon which consists of 1 km swimming, 24 km cycling, and 6 km running. He started at 9:30 AM. 1. Swimming Speed and Time: - Peter swam 1 km in 20 minutes. - Convert 20 minutes to hours: ( frac{20}{60} = frac{1}{3} ) hours. - Swimming speed: ( frac{1 text{ km}}{frac{1}{3} text{ hours}} = 3 text{ km/h} ). - Swimming time: 20 minutes.2. Cycling Speed and Time: - Cycling speed is 10 times swimming speed: ( 3 text{ km/h} times 10 = 30 text{ km/h} ). - Cycling time: ( frac{24 text{ km}}{30 text{ km/h}} = 0.8 text{ hours} ). - Convert 0.8 hours to minutes: ( 0.8 times 60 = 48 text{ minutes} ).3. Running Speed and Time: - Running speed is 4 times swimming speed: ( 3 text{ km/h} times 4 = 12 text{ km/h} ). - Running time: ( frac{6 text{ km}}{12 text{ km/h}} = 0.5 text{ hours} ). - Convert 0.5 hours to minutes: ( 0.5 times 60 = 30 text{ minutes} ).4. Total Time: - Total time: ( 20 text{ minutes} + 48 text{ minutes} + 30 text{ minutes} = 98 text{ minutes} ). - Convert 98 minutes to hours and minutes: ( 98 text{ minutes} = 1 text{ hour} 38 text{ minutes} ).5. Finish Time: - Start time: 9:30 AM. - Add 1 hour 38 minutes: 9:30 AM + 1 hour = 10:30 AM, then add 38 minutes: 10:30 AM + 38 minutes = 11:08 AM.Thus, Peter finished the competition at (boxed{D}) 11:08.]

🔑:A5. Peter swims 1 mathrm{~km} in 20 mathrm{~min}, cycles 1 mathrm{~km} in 2 min, and runs 1 mathrm{~km} in 5 mathrm{~min}. In the competition, he spent 20 min for 1 mathrm{~km} of swimming, 48 min for 24 mathrm{~km} of cycling, and 30 min for 6 mathrm{~km} of running, totaling 98 min. If the competition started at 9:30, he arrived at the finish line at 11:08.