answer：First, I need to determine the points scored by both teams in the first half. Team A scored 8 points, and Team B scored half of that, which is 4 points.Next, in the second half, Team B scored the same number of points as Team A did in the first half, which is 8 points. Team A scored 2 goals less than Team B in the second half, so Team A scored 6 points in the second half.Finally, I add up all the points scored by both teams in both halves to find the total points scored during the entire match.

answer：Okay, so I have this problem about an arithmetic sequence. The sum of the first n terms is denoted as S_n. I know that S_{10} is 0 and S_{15} is 25. I need to find the minimum value of nS_n. Hmm, okay, let's break this down step by step.First, I remember that the sum of the first n terms of an arithmetic sequence can be calculated using the formula:S_n = (n/2) * [2a_1 + (n - 1)d]where a_1 is the first term and d is the common difference. Given that S_{10} = 0, I can plug n = 10 into the formula:0 = (10/2) * [2a_1 + (10 - 1)d]Simplifying that, 0 = 5 * [2a_1 + 9d]. So, 2a_1 + 9d = 0. Let me write that down as equation (1):2a_1 + 9d = 0Similarly, for S_{15} = 25, plugging n = 15 into the formula:25 = (15/2) * [2a_1 + (15 - 1)d]Simplifying, 25 = (15/2) * [2a_1 + 14d]. Multiply both sides by 2 to eliminate the denominator:50 = 15 * [2a_1 + 14d]Divide both sides by 15:50/15 = 2a_1 + 14dSimplify 50/15 to 10/3:10/3 = 2a_1 + 14dLet me write that as equation (2):2a_1 + 14d = 10/3Now, I have two equations:1) 2a_1 + 9d = 02) 2a_1 + 14d = 10/3I can subtract equation (1) from equation (2) to eliminate a_1:(2a_1 + 14d) - (2a_1 + 9d) = (10/3) - 0Simplify:5d = 10/3So, d = (10/3) / 5 = (10/3) * (1/5) = 2/3Wait, hold on, 10 divided by 3 is approximately 3.333, divided by 5 is approximately 0.666, which is 2/3. So, d = 2/3.Now, plug d = 2/3 back into equation (1):2a_1 + 9*(2/3) = 0Simplify 9*(2/3) = 6So, 2a_1 + 6 = 0Thus, 2a_1 = -6Therefore, a_1 = -3Alright, so now I know that the first term a_1 is -3 and the common difference d is 2/3.Now, I need to find the minimum value of nS_n. So, first, let's express S_n in terms of n.Using the sum formula again:S_n = (n/2) * [2a_1 + (n - 1)d]Plugging in a_1 = -3 and d = 2/3:S_n = (n/2) * [2*(-3) + (n - 1)*(2/3)]Simplify inside the brackets:2*(-3) = -6(n - 1)*(2/3) = (2/3)n - 2/3So, combining these:-6 + (2/3)n - 2/3 = (2/3)n - 6 - 2/3Convert 6 to thirds: 6 = 18/3, so:(2/3)n - 18/3 - 2/3 = (2/3)n - 20/3Therefore, S_n = (n/2) * [(2/3)n - 20/3]Simplify this expression:First, factor out 2/3 from the bracket:(2/3)(n - 10)So, S_n = (n/2) * (2/3)(n - 10) = (n/2)*(2/3)*(n - 10)The 2s cancel out:S_n = (n/3)*(n - 10)So, S_n = (n^2 - 10n)/3Therefore, nS_n = n * (n^2 - 10n)/3 = (n^3 - 10n^2)/3So, nS_n = (n^3 - 10n^2)/3Now, I need to find the minimum value of this expression. Since n is a positive integer (as it's the number of terms), but to find the minimum, I can treat n as a continuous variable and use calculus to find the critical points, then check the integer values around it.Let me denote f(n) = (n^3 - 10n^2)/3To find the critical points, take the derivative f'(n):f'(n) = (3n^2 - 20n)/3 = n^2 - (20/3)nSet f'(n) = 0:n^2 - (20/3)n = 0Factor out n:n(n - 20/3) = 0So, n = 0 or n = 20/3 ≈ 6.666...Since n = 0 doesn't make sense in this context (we can't have 0 terms), the critical point is at n = 20/3 ≈ 6.666...Since this is a minimum, we can check the second derivative to confirm:f''(n) = 2n - 20/3At n = 20/3, f''(20/3) = 2*(20/3) - 20/3 = (40/3 - 20/3) = 20/3 > 0, so it's a minimum.Therefore, the minimum occurs around n ≈ 6.666. Since n must be an integer, we need to check n = 6 and n = 7.But wait, hold on. The problem didn't specify that n has to be an integer? Wait, no, in the context of the problem, n is the number of terms, so it must be a positive integer. So, n must be an integer. Therefore, we need to evaluate nS_n at n = 6 and n = 7 to see which gives the minimum.But wait, let's compute f(n) at n = 6 and n = 7.First, f(n) = (n^3 - 10n^2)/3At n = 6:f(6) = (216 - 360)/3 = (-144)/3 = -48At n = 7:f(7) = (343 - 490)/3 = (-147)/3 = -49So, f(7) is -49, which is less than f(6) = -48. Therefore, the minimum occurs at n = 7, and the minimum value is -49.Wait, but hold on, earlier I thought the critical point was at n ≈ 6.666, so n = 7 is the closest integer, and indeed, it gives a lower value. So, the minimum value is -49.But let me double-check my calculations.First, a_1 = -3, d = 2/3.So, S_n = (n/2)[2a_1 + (n - 1)d] = (n/2)[-6 + (2/3)(n - 1)]Simplify:(n/2)[-6 + (2n/3 - 2/3)] = (n/2)[(2n/3 - 2/3 - 6)]Convert 6 to thirds: 6 = 18/3So, (2n/3 - 2/3 - 18/3) = (2n/3 - 20/3)Therefore, S_n = (n/2)*(2n/3 - 20/3) = (n/2)*(2/3)(n - 10) = (n/3)(n - 10)So, S_n = (n^2 - 10n)/3Therefore, nS_n = n*(n^2 - 10n)/3 = (n^3 - 10n^2)/3Yes, that's correct.Taking derivative: f'(n) = (3n^2 - 20n)/3 = n^2 - (20/3)nSet to zero: n(n - 20/3) = 0, so n = 0 or n = 20/3 ≈ 6.666...So, yes, the critical point is at n ≈ 6.666...Therefore, checking n = 6 and n = 7.At n = 6:f(6) = (216 - 360)/3 = (-144)/3 = -48At n = 7:f(7) = (343 - 490)/3 = (-147)/3 = -49So, indeed, n = 7 gives a lower value.But wait, let me check n = 5 and n = 8 just to be thorough.At n = 5:f(5) = (125 - 250)/3 = (-125)/3 ≈ -41.666...At n = 8:f(8) = (512 - 640)/3 = (-128)/3 ≈ -42.666...So, n = 7 gives the lowest value so far. Let's check n = 9:f(9) = (729 - 810)/3 = (-81)/3 = -27Wait, that's higher than -49. So, n = 7 is the minimum.Wait, but hold on, n = 7 gives -49, which is less than n = 6 (-48), n = 5 (-41.666), n = 8 (-42.666), and n = 9 (-27). So, yes, n = 7 is indeed the minimum.But wait, let me also check n = 10:f(10) = (1000 - 1000)/3 = 0/3 = 0And n = 11:f(11) = (1331 - 1210)/3 = 121/3 ≈ 40.333...So, it's increasing after n = 10.Therefore, the minimum value is at n = 7, which is -49.But wait, hold on, in the initial problem statement, the user wrote:"For an arithmetic sequence {a_n}, the sum of the first n terms is denoted as S_n. Given that S_{10} = 0 and S_{15} = 25, the minimum value of nS_n is ______."But in my calculation, I got a_1 = -3 and d = 2/3, leading to S_n = (n^2 - 10n)/3, so nS_n = (n^3 - 10n^2)/3.But wait, when I computed f(7) = -49, that's correct. But let me check if the arithmetic sequence is correct.Given a_1 = -3, d = 2/3.So, the sequence is: -3, -3 + 2/3 = -7/3, -7/3 + 2/3 = -5/3, -5/3 + 2/3 = -1, -1 + 2/3 = -1/3, -1/3 + 2/3 = 1/3, 1/3 + 2/3 = 1, 1 + 2/3 = 5/3, 5/3 + 2/3 = 7/3, 7/3 + 2/3 = 3, etc.Let me compute S_10:Sum of first 10 terms: S_10 = (10/2)[2*(-3) + 9*(2/3)] = 5*(-6 + 6) = 5*0 = 0. Correct.S_15 = (15/2)[2*(-3) + 14*(2/3)] = (15/2)*(-6 + 28/3) = (15/2)*(-18/3 + 28/3) = (15/2)*(10/3) = (15*10)/(2*3) = 150/6 = 25. Correct.So, the arithmetic sequence is correctly determined.Therefore, the expression for nS_n is correct, and the minimum occurs at n = 7 with value -49.But wait, in the initial problem, the user wrote:"For an arithmetic sequence {a_n}, the sum of the first n terms is denoted as S_n. Given that S_{10} = 0 and S_{15} = 25, the minimum value of nS_n is ______."So, the answer should be -49.But wait, in the initial assistant's answer, it was -49 as well, but the user's initial problem had a different setup? Wait, no, the user's initial problem is the same as the one I'm solving.Wait, but in the initial problem, the user wrote:"For an arithmetic sequence {a_n}, the sum of the first n terms is denoted as S_n. Given that S_{10} = 0 and S_{15} = 25, the minimum value of nS_n is ______."So, my conclusion is that the minimum value is -49.But let me just double-check my calculations once more.Given a_1 = -3, d = 2/3.So, S_n = (n/2)[2a_1 + (n - 1)d] = (n/2)[-6 + (2/3)(n - 1)] = (n/2)[-6 + (2n/3 - 2/3)] = (n/2)[(2n/3 - 20/3)] = (n/3)(n - 10)Thus, nS_n = n*(n^2 - 10n)/3 = (n^3 - 10n^2)/3Taking derivative: f'(n) = (3n^2 - 20n)/3 = n^2 - (20/3)nSet to zero: n(n - 20/3) = 0, so n = 0 or n = 20/3 ≈ 6.666...Thus, the minimum occurs at n ≈ 6.666, so checking n = 6 and n = 7.At n = 6: f(6) = (216 - 360)/3 = -144/3 = -48At n = 7: f(7) = (343 - 490)/3 = -147/3 = -49Thus, the minimum value is -49 at n = 7.Therefore, the answer is -49.

answer：First, calculate the total amount spent by the 6 people who each spent 11:Total spent by 6 people = 6 * 11 = 66Let A be the average expenditure of all 7 people. The seventh person spent 6 more than this average, so their expenditure is A + 6.The total amount spent by all 7 people is the sum of the expenditures of the first 6 people and the seventh person:Total spent by 7 people = 66 + (A + 6)Since the average expenditure A is the total amount spent divided by the number of people (7), we can write:A = Total spent by 7 people / 7Substitute A back into the equation for the total amount spent by 7 people:Total spent by 7 people = 66 + (Total spent by 7 people / 7 + 6)Let T represent the total amount spent by 7 people:T = 66 + (T / 7 + 6)Multiply both sides by 7 to eliminate the fraction:7T = 7 * 66 + T + 7 * 67T = 462 + T + 427T - T = 462 + 426T = 504Divide both sides by 6 to solve for T:T = 504 / 6T = 84Therefore, the total amount of money that the 7 people spent is 84.

answer：Okay, so I have this problem where there's a geometric sequence {a_n} and an arithmetic sequence {b_n} with b₁ = 0. Then they define c_n as a_n + b_n. The sequence {c_n} starts with 1, 1, 2, and I need to find the sum of the first 10 terms of {c_n}. The options are A: 979, B: 557, C: 467, D: 978.Alright, let's break this down. First, let's recall what a geometric sequence is. A geometric sequence has a common ratio between consecutive terms. So, if the first term is a₁, then the next terms are a₁*r, a₁*r², and so on. Similarly, an arithmetic sequence has a common difference between consecutive terms. Since b₁ = 0, the arithmetic sequence starts at 0, and each term increases by a common difference, say d. So, b₂ = d, b₃ = 2d, etc.Given that c_n = a_n + b_n, and the first three terms of {c_n} are 1, 1, 2. So, let's write down what we know:c₁ = a₁ + b₁ = 1c₂ = a₂ + b₂ = 1c₃ = a₃ + b₃ = 2Since b₁ = 0, c₁ = a₁ = 1. So, a₁ = 1.Now, let's denote the common ratio of the geometric sequence as r. So, a₂ = a₁*r = r, and a₃ = a₁*r² = r².For the arithmetic sequence, since b₁ = 0, the common difference is d, so b₂ = d, b₃ = 2d.So, plugging into c₂ and c₃:c₂ = a₂ + b₂ = r + d = 1c₃ = a₃ + b₃ = r² + 2d = 2So now we have a system of equations:1) r + d = 12) r² + 2d = 2We can solve this system for r and d. Let's solve equation 1) for d: d = 1 - r.Then substitute into equation 2):r² + 2*(1 - r) = 2Simplify:r² + 2 - 2r = 2Subtract 2 from both sides:r² - 2r = 0Factor:r(r - 2) = 0So, r = 0 or r = 2. But if r = 0, then the geometric sequence would be 1, 0, 0, 0,... which would make c₂ = 0 + d = 1, so d = 1, but then c₃ = 0 + 2d = 2, which is consistent. However, if r = 0, the geometric sequence becomes trivial after the first term. But let's check if that's acceptable.Wait, if r = 0, then a₁ = 1, a₂ = 0, a₃ = 0, etc. Then b₁ = 0, b₂ = d, b₃ = 2d. So, c₁ = 1 + 0 = 1, c₂ = 0 + d = 1, so d = 1. Then c₃ = 0 + 2d = 2, which is correct. So, that works.But if r = 2, then a₁ = 1, a₂ = 2, a₃ = 4. Then from equation 1), d = 1 - r = 1 - 2 = -1. So, b₁ = 0, b₂ = -1, b₃ = -2. Then c₁ = 1 + 0 = 1, c₂ = 2 + (-1) = 1, c₃ = 4 + (-2) = 2. That also works.So, we have two possibilities: either r = 0 and d = 1, or r = 2 and d = -1.But let's think about whether r = 0 is a valid geometric sequence. Typically, a geometric sequence is defined with a non-zero common ratio, otherwise, after the first term, all terms are zero. So, maybe r = 2 is the intended solution here.So, let's proceed with r = 2 and d = -1.Therefore, the sequences are:a_n = 1 * 2^{n-1} = 2^{n-1}b_n = 0 + (n - 1)*d = (n - 1)*(-1) = 1 - nThus, c_n = a_n + b_n = 2^{n-1} + (1 - n)Now, we need to find the sum of the first 10 terms of {c_n}, which is S₁₀ = c₁ + c₂ + ... + c₁₀.Since c_n = a_n + b_n, the sum S₁₀ = Σ_{n=1}^{10} c_n = Σ_{n=1}^{10} a_n + Σ_{n=1}^{10} b_n.So, we can compute the sum of the geometric sequence {a_n} and the sum of the arithmetic sequence {b_n} separately.First, let's compute Σ_{n=1}^{10} a_n. Since {a_n} is a geometric sequence with a₁ = 1 and r = 2, the sum of the first n terms is S_n = a₁*(r^n - 1)/(r - 1). So, S₁₀ = (2^{10} - 1)/(2 - 1) = (1024 - 1)/1 = 1023.Next, let's compute Σ_{n=1}^{10} b_n. Since {b_n} is an arithmetic sequence with b₁ = 0 and common difference d = -1, the sum of the first n terms is S_n = n/2*(2b₁ + (n - 1)d). So, S₁₀ = 10/2*(2*0 + 9*(-1)) = 5*(0 - 9) = 5*(-9) = -45.Therefore, the total sum S₁₀ = 1023 + (-45) = 1023 - 45 = 978.Wait, but let me double-check the arithmetic. 1023 - 45 is indeed 978. So, the sum is 978, which corresponds to option D.But just to make sure, let's verify the first few terms:c₁ = 2^{0} + (1 - 1) = 1 + 0 = 1c₂ = 2^{1} + (1 - 2) = 2 - 1 = 1c₃ = 2^{2} + (1 - 3) = 4 - 2 = 2c₄ = 2^{3} + (1 - 4) = 8 - 3 = 5c₅ = 16 + (1 - 5) = 16 - 4 = 12c₆ = 32 + (1 - 6) = 32 - 5 = 27c₇ = 64 + (1 - 7) = 64 - 6 = 58c₈ = 128 + (1 - 8) = 128 - 7 = 121c₉ = 256 + (1 - 9) = 256 - 8 = 248c₁₀ = 512 + (1 - 10) = 512 - 9 = 503Now, let's add these up:1 + 1 = 22 + 2 = 44 + 5 = 99 + 12 = 2121 + 27 = 4848 + 58 = 106106 + 121 = 227227 + 248 = 475475 + 503 = 978Yes, that adds up to 978. So, the sum of the first 10 terms is indeed 978.Therefore, the correct answer is D: 978.

answer：Okay, so I have this problem where two ordered positive integers add up to 915, and their greatest common divisor (gcd) is 61. I need to find how many possible pairs of these two numbers there are. The options given are A: 4, B: 6, C: 8, D: 14.Hmm, let me think about how to approach this. I remember that when dealing with problems involving gcd, it's often useful to express the numbers in terms of their gcd. So, if the gcd of the two numbers is 61, I can write the numbers as multiples of 61. Let me denote the two numbers as ( a ) and ( b ). So, ( a = 61x ) and ( b = 61y ), where ( x ) and ( y ) are positive integers.Since the gcd of ( a ) and ( b ) is 61, it means that ( x ) and ( y ) must be coprime, right? Because if they had a common divisor greater than 1, then the gcd of ( a ) and ( b ) would be higher than 61. So, ( gcd(x, y) = 1 ).Also, the sum of ( a ) and ( b ) is 915. So, substituting the expressions for ( a ) and ( b ), we get:[61x + 61y = 915]I can factor out 61:[61(x + y) = 915]Now, to find ( x + y ), I can divide both sides by 61:[x + y = frac{915}{61}]Let me calculate that. 61 times 15 is 915 because 60 times 15 is 900, and 1 times 15 is 15, so 900 + 15 = 915. So, ( x + y = 15 ).So, now I have two conditions:1. ( x + y = 15 )2. ( gcd(x, y) = 1 )I need to find all pairs of positive integers ( (x, y) ) such that their sum is 15 and they are coprime. Since the problem mentions ordered pairs, the order matters, meaning ( (x, y) ) and ( (y, x) ) are considered different if ( x neq y ).Let me list all pairs of positive integers ( (x, y) ) where ( x + y = 15 ). Starting from ( x = 1 ) up to ( x = 14 ) (since ( y ) has to be at least 1 as well):1. ( (1, 14) )2. ( (2, 13) )3. ( (3, 12) )4. ( (4, 11) )5. ( (5, 10) )6. ( (6, 9) )7. ( (7, 8) )8. ( (8, 7) )9. ( (9, 6) )10. ( (10, 5) )11. ( (11, 4) )12. ( (12, 3) )13. ( (13, 2) )14. ( (14, 1) )Now, I need to check which of these pairs have ( gcd(x, y) = 1 ). Let's go through them one by one.1. ( (1, 14) ): The gcd of 1 and 14 is 1. So, this pair is valid.2. ( (2, 13) ): The gcd of 2 and 13 is 1. Valid.3. ( (3, 12) ): The gcd of 3 and 12 is 3. Not valid.4. ( (4, 11) ): The gcd of 4 and 11 is 1. Valid.5. ( (5, 10) ): The gcd of 5 and 10 is 5. Not valid.6. ( (6, 9) ): The gcd of 6 and 9 is 3. Not valid.7. ( (7, 8) ): The gcd of 7 and 8 is 1. Valid.8. ( (8, 7) ): Same as above, gcd is 1. Valid.9. ( (9, 6) ): Same as pair 6, gcd is 3. Not valid.10. ( (10, 5) ): Same as pair 5, gcd is 5. Not valid.11. ( (11, 4) ): Same as pair 4, gcd is 1. Valid.12. ( (12, 3) ): Same as pair 3, gcd is 3. Not valid.13. ( (13, 2) ): Same as pair 2, gcd is 1. Valid.14. ( (14, 1) ): Same as pair 1, gcd is 1. Valid.So, the valid pairs are:1. ( (1, 14) )2. ( (2, 13) )3. ( (4, 11) )4. ( (7, 8) )5. ( (8, 7) )6. ( (11, 4) )7. ( (13, 2) )8. ( (14, 1) )That's 8 pairs in total. Each of these pairs corresponds to a pair of numbers ( (a, b) ) where ( a = 61x ) and ( b = 61y ). Since the problem asks for ordered pairs, both ( (x, y) ) and ( (y, x) ) are counted separately if they are different.Let me double-check to make sure I didn't miss any pairs or count any incorrectly. Starting from 1 to 14, I considered all possible pairs adding up to 15, checked their gcd, and only kept those with gcd 1. It seems correct. There are 8 such pairs.So, the number of possible pairs is 8, which corresponds to option C.**Final Answer**boxed{C}

answer：Alright, so I have this problem here about a sine function, and I need to find the phase shift φ. Let me try to understand what's given and what I need to do step by step.First, the function is given as f(x) = sin(ωx + φ), where ω is greater than 0, and φ is between 0 and π/2. Okay, so it's a sine wave with some frequency ω and a phase shift φ. The first condition is that f(0) = -f(π/2). Hmm, that means when I plug in x = 0 into the function, it should be the negative of when I plug in x = π/2. Let me write that down:f(0) = sin(ω*0 + φ) = sin(φ)f(π/2) = sin(ω*(π/2) + φ)So according to the condition, sin(φ) = -sin(ω*(π/2) + φ). That equation should help me find a relationship between ω and φ.I remember that sin(A) = -sin(B) implies that A = -B + 2πk or A = π + B + 2πk for some integer k. Let me apply that here.Case 1: φ = - (ω*(π/2) + φ) + 2πkSimplify this:φ = -ω*(π/2) - φ + 2πkBring φ to the left side:2φ = -ω*(π/2) + 2πkDivide both sides by 2:φ = -ω*(π/4) + πkBut φ is between 0 and π/2, so let's see what this gives us. Let me plug in k = 1:φ = -ω*(π/4) + πBut since ω is positive, this would make φ negative unless ω is very small. But ω is positive, so maybe k=0?If k=0:φ = -ω*(π/4)But φ is supposed to be positive, so this would require ω to be negative, which contradicts ω > 0. So this case doesn't seem to work.Case 2: φ = π + (ω*(π/2) + φ) + 2πkSimplify:φ = π + ω*(π/2) + φ + 2πkSubtract φ from both sides:0 = π + ω*(π/2) + 2πkSo:ω*(π/2) = -π - 2πkDivide both sides by π/2:ω = (-π - 2πk) / (π/2) = (-2 - 4k)But ω has to be positive, so (-2 - 4k) > 0Which implies -2 -4k > 0 => -4k > 2 => k < -0.5Since k is an integer, k ≤ -1Let me try k = -1:ω = (-2 -4*(-1)) = (-2 +4) = 2So ω = 2Let me check k = -2:ω = (-2 -4*(-2)) = (-2 +8) = 6So ω = 6Wait, so ω can be 2, 6, 10, etc., increasing by 4 each time. So ω = 4n + 2 where n is a non-negative integer.But I don't know which one it is yet. Maybe the next condition will help.The next part says that if we shift the graph of f(x) π/12 units to the left, the resulting function is symmetric about the origin. So shifting left by π/12 means replacing x with x + π/12 in the function.So the new function g(x) = sin(ω*(x + π/12) + φ) = sin(ωx + ω*(π/12) + φ)And this function g(x) is symmetric about the origin, meaning it's an odd function. So g(-x) = -g(x) for all x.Let me write that condition:sin(ω*(-x) + ω*(π/12) + φ) = -sin(ωx + ω*(π/12) + φ)Simplify the left side:sin(-ωx + ω*(π/12) + φ) = sin(-ωx + (ωπ/12 + φ))Using the identity sin(-A + B) = sin(B - A) = sin(B - A)So sin(ωπ/12 + φ - ωx) = -sin(ωx + ωπ/12 + φ)Hmm, so we have sin(B - A) = -sin(A + B), where A = ωx and B = ωπ/12 + φIs there a relationship between A and B that would make this identity hold for all x?Let me think about the sine function properties. For sin(B - A) = -sin(A + B), let's see:sin(B - A) = sin(-A + B) = -sin(A - B)So, -sin(A - B) = -sin(A + B)Multiply both sides by -1:sin(A - B) = sin(A + B)When does sin(A - B) = sin(A + B)?This occurs when either:1. A - B = A + B + 2πk, which simplifies to -B = B + 2πk => -2B = 2πk => B = -πkBut B is ωπ/12 + φ, which is positive since ω > 0 and φ > 0, so B = -πk would require k to be negative. Let's see:B = -πk => ωπ/12 + φ = -πkBut ωπ/12 + φ is positive, so -πk must be positive => k < 0Let me set k = -m where m is a positive integer:Then B = ωπ/12 + φ = πmSo ωπ/12 + φ = πmWhich gives φ = πm - ωπ/12Since φ is between 0 and π/2, let's see possible m:For m = 1:φ = π - ωπ/12But φ must be less than π/2, so π - ωπ/12 < π/2 => ωπ/12 > π/2 => ω > 6But earlier, ω could be 2, 6, 10, etc. So if ω > 6, possible ω is 10, 14, etc.But let's check m = 1 and ω = 10:φ = π - 10π/12 = π - 5π/6 = π/6Which is within 0 and π/2. Okay, that's possible.For m = 2:φ = 2π - ωπ/12But φ must be less than π/2, so 2π - ωπ/12 < π/2 => ωπ/12 > 2π - π/2 = 3π/2 => ω > (3π/2)*(12/π) = 18So ω would have to be 22, 26, etc., but earlier ω is 4n + 2, so 22 is 4*5 + 2, which is okay, but let's see if that's necessary.Alternatively, maybe another case.Case 2: sin(A - B) = sin(A + B) implies that A - B = π - (A + B) + 2πkSo:A - B = π - A - B + 2πkBring A to the left and constants to the right:2A = π + 2πkSo A = π/2 + πkBut A = ωx, so ωx = π/2 + πkBut this has to hold for all x, which is not possible unless ω = 0, which contradicts ω > 0. So this case doesn't work.Therefore, the only valid case is the first one where B = πm, leading to φ = πm - ωπ/12.From earlier, when m=1, φ = π - ωπ/12. And we found that for ω=10, φ=π/6, which is within the required range.Wait, but earlier when solving the first condition, we found ω=4n+2. So ω=2,6,10,14,...For m=1, φ=π - ωπ/12.If ω=2:φ=π - 2π/12= π - π/6=5π/6, which is greater than π/2, so invalid.If ω=6:φ=π -6π/12=π - π/2=π/2, which is the boundary, but φ must be less than π/2, so invalid.If ω=10:φ=π -10π/12=π -5π/6=π/6, which is valid.If ω=14:φ=π -14π/12=π -7π/6= -π/6, which is negative, invalid.So the only valid solution is ω=10 and φ=π/6.Wait, but earlier when solving the first condition, I thought ω=4n+2, so n=0 gives ω=2, n=1 gives ω=6, n=2 gives ω=10, etc.But in the first condition, when I set f(0)=-f(π/2), I got ω=4k+2, which is similar.So combining both conditions, the only possible ω is 10, leading to φ=π/6.Therefore, the answer should be B: π/6.But let me double-check.Given ω=10 and φ=π/6, let's verify the conditions.First, f(0)=sin(0 + π/6)=sin(π/6)=1/2f(π/2)=sin(10*(π/2) + π/6)=sin(5π + π/6)=sin(π/6 + π)=sin(π/6 + π)= -sin(π/6)= -1/2So f(0)=1/2, f(π/2)=-1/2, so f(0)=-f(π/2). That checks out.Now, shifting f(x) left by π/12 gives g(x)=sin(10*(x + π/12) + π/6)=sin(10x +10π/12 + π/6)=sin(10x +5π/6 + π/6)=sin(10x + π)But sin(10x + π)= -sin(10x), because sin(A + π)= -sin A.So g(x)= -sin(10x), which is an odd function because -sin(10*(-x))= -(-sin(10x))=sin(10x)= -g(x). Wait, no, let me check:g(-x)=sin(10*(-x) + π)=sin(-10x + π)=sin(π -10x)=sin(10x) because sin(π - θ)=sin θ.But g(x)= -sin(10x), so g(-x)=sin(10x)= -g(x). Therefore, g(-x)= -g(x), which means g(x) is odd. So it's symmetric about the origin. Perfect.So everything checks out. Therefore, φ=π/6.