answer：First, calculate the total cost of the pizzas before the discount. Each pizza costs 11, so for 3 pizzas, the cost would be:3 pizzas * 11/pizza = 33Next, calculate the cost of the toppings for all three pizzas. Since there are two different toppings on each pizza, the first topping costs 1.50 per pizza and the second topping costs 2 per pizza, the total cost for toppings would be:(3 pizzas * 1.50/topping) + (3 pizzas * 2/topping) = 4.50 + 6 = 10.50Adding the cost of the pizzas and the toppings together, we get:33 (pizzas) + 10.50 (toppings) = 43.50However, there is a 10% discount for purchasing 3 pizzas. The discount would be 10% of the cost of the pizzas, which is:10% of 33 = 0.10 * 33 = 3.30Now subtract the discount from the total cost before the discount:43.50 (total cost before discount) - 3.30 (discount) = 40.20Frank and Bill had 42 to start with, and the final cost after the discount is 40.20. So, Frank would have:42 - 40.20 = 1.80 leftFrank gives the remaining 1.80 to Bill. Since Bill already had 30, the total amount of money Bill has now is:30 + 1.80 = 31.80Therefore, Bill now has boxed{31.80}.

answer：Okay, so I have this problem where I start with a triangle T₁ with side lengths 1001, 1002, and 1003. Then, for each n ≥ 1, if Tₙ is a triangle ABC, I need to find the points where the incircle touches the sides AB, BC, and AC, which are points D, E, and F respectively. Then, Tₙ₊₁ is a triangle with side lengths AD, BE, and CF, if it exists. I need to find the perimeter of the last triangle in this sequence.Hmm, okay. So, first, let me recall that in a triangle, the lengths from the vertices to the points of tangency can be calculated using the formula:AD = AF = (b + c - a)/2BE = BD = (a + c - b)/2CF = CE = (a + b - c)/2Where a, b, c are the side lengths of the triangle. So, for each triangle Tₙ, I can compute these lengths and then form the next triangle Tₙ₊₁.Let me try to compute the first few terms to see if I can spot a pattern.Starting with T₁: sides 1001, 1002, 1003.Let me label them as a = 1002, b = 1003, c = 1001 for consistency.So, AD = (b + c - a)/2 = (1003 + 1001 - 1002)/2 = (1002)/2 = 501Wait, that's not right. Wait, 1003 + 1001 is 2004, minus 1002 is 1002, divided by 2 is 501. So AD = 501.Similarly, BE = (a + c - b)/2 = (1002 + 1001 - 1003)/2 = (1000)/2 = 500And CF = (a + b - c)/2 = (1002 + 1003 - 1001)/2 = (1004)/2 = 502So, T₂ has sides 501, 500, 502.Wait, let me check that again.Wait, AD = AF = (b + c - a)/2 = (1003 + 1001 - 1002)/2 = (1002)/2 = 501BE = BD = (a + c - b)/2 = (1002 + 1001 - 1003)/2 = (1000)/2 = 500CF = CE = (a + b - c)/2 = (1002 + 1003 - 1001)/2 = (1004)/2 = 502So, yes, T₂ has sides 501, 500, 502.Wait, so the perimeter of T₁ is 1001 + 1002 + 1003 = 3006.The perimeter of T₂ is 501 + 500 + 502 = 1503.Hmm, that's exactly half of 3006. Interesting.Let me compute T₃.So, for T₂: sides 501, 500, 502.Let me label them as a = 500, b = 502, c = 501.So, AD = (b + c - a)/2 = (502 + 501 - 500)/2 = (503)/2 = 251.5BE = (a + c - b)/2 = (500 + 501 - 502)/2 = (499)/2 = 249.5CF = (a + b - c)/2 = (500 + 502 - 501)/2 = (501)/2 = 250.5So, T₃ has sides 251.5, 249.5, 250.5.Wait, the perimeter is 251.5 + 249.5 + 250.5 = 751.5Which is half of 1503, so 751.5.Hmm, so each time, the perimeter is halved.Wait, so T₁: 3006T₂: 1503T₃: 751.5T₄: 375.75T₅: 187.875T₆: 93.9375T₇: 46.96875T₈: 23.484375T₉: 11.7421875T₁₀: 5.87109375Wait, but at some point, the side lengths might not form a valid triangle anymore.Wait, but in the problem statement, it says "if it exists". So, we need to stop when the next triangle doesn't satisfy the triangle inequality.So, let me check when the triangle inequality fails.So, for each triangle, we need to ensure that the sum of any two sides is greater than the third.So, let's see:Starting from T₁: 1001, 1002, 1003. Obviously, it's a valid triangle.T₂: 501, 500, 502. Let's check:501 + 500 > 502? 1001 > 502, yes.501 + 502 > 500? 1003 > 500, yes.500 + 502 > 501? 1002 > 501, yes.So, valid.T₃: 251.5, 249.5, 250.5Check:251.5 + 249.5 > 250.5? 501 > 250.5, yes.251.5 + 250.5 > 249.5? 502 > 249.5, yes.249.5 + 250.5 > 251.5? 500 > 251.5, yes.Valid.T₄: 125.75, 124.75, 125.25Wait, hold on, let me compute T₄ properly.Wait, T₃ sides: 251.5, 249.5, 250.5So, a = 249.5, b = 250.5, c = 251.5AD = (b + c - a)/2 = (250.5 + 251.5 - 249.5)/2 = (252.5)/2 = 126.25BE = (a + c - b)/2 = (249.5 + 251.5 - 250.5)/2 = (250.5)/2 = 125.25CF = (a + b - c)/2 = (249.5 + 250.5 - 251.5)/2 = (248.5)/2 = 124.25So, T₄ has sides 126.25, 125.25, 124.25Perimeter: 126.25 + 125.25 + 124.25 = 375.75Which is half of 751.5, so correct.Check triangle inequality:126.25 + 125.25 > 124.25? 251.5 > 124.25, yes.126.25 + 124.25 > 125.25? 250.5 > 125.25, yes.125.25 + 124.25 > 126.25? 249.5 > 126.25, yes.Valid.T₅: Let's compute T₅.T₄ sides: 126.25, 125.25, 124.25Label a = 124.25, b = 125.25, c = 126.25AD = (b + c - a)/2 = (125.25 + 126.25 - 124.25)/2 = (127.25)/2 = 63.625BE = (a + c - b)/2 = (124.25 + 126.25 - 125.25)/2 = (125.25)/2 = 62.625CF = (a + b - c)/2 = (124.25 + 125.25 - 126.25)/2 = (123.25)/2 = 61.625So, T₅ has sides 63.625, 62.625, 61.625Perimeter: 63.625 + 62.625 + 61.625 = 187.875Which is half of 375.75, correct.Check triangle inequality:63.625 + 62.625 > 61.625? 126.25 > 61.625, yes.63.625 + 61.625 > 62.625? 125.25 > 62.625, yes.62.625 + 61.625 > 63.625? 124.25 > 63.625, yes.Valid.T₆: Compute T₆.T₅ sides: 63.625, 62.625, 61.625Label a = 61.625, b = 62.625, c = 63.625AD = (b + c - a)/2 = (62.625 + 63.625 - 61.625)/2 = (64.625)/2 = 32.3125BE = (a + c - b)/2 = (61.625 + 63.625 - 62.625)/2 = (62.625)/2 = 31.3125CF = (a + b - c)/2 = (61.625 + 62.625 - 63.625)/2 = (60.625)/2 = 30.3125So, T₆ has sides 32.3125, 31.3125, 30.3125Perimeter: 32.3125 + 31.3125 + 30.3125 = 93.9375Which is half of 187.875, correct.Check triangle inequality:32.3125 + 31.3125 > 30.3125? 63.625 > 30.3125, yes.32.3125 + 30.3125 > 31.3125? 62.625 > 31.3125, yes.31.3125 + 30.3125 > 32.3125? 61.625 > 32.3125, yes.Valid.T₇: Compute T₇.T₆ sides: 32.3125, 31.3125, 30.3125Label a = 30.3125, b = 31.3125, c = 32.3125AD = (b + c - a)/2 = (31.3125 + 32.3125 - 30.3125)/2 = (33.3125)/2 = 16.65625BE = (a + c - b)/2 = (30.3125 + 32.3125 - 31.3125)/2 = (31.3125)/2 = 15.65625CF = (a + b - c)/2 = (30.3125 + 31.3125 - 32.3125)/2 = (29.3125)/2 = 14.65625So, T₇ has sides 16.65625, 15.65625, 14.65625Perimeter: 16.65625 + 15.65625 + 14.65625 = 46.96875Which is half of 93.9375, correct.Check triangle inequality:16.65625 + 15.65625 > 14.65625? 32.3125 > 14.65625, yes.16.65625 + 14.65625 > 15.65625? 31.3125 > 15.65625, yes.15.65625 + 14.65625 > 16.65625? 30.3125 > 16.65625, yes.Valid.T₈: Compute T₈.T₇ sides: 16.65625, 15.65625, 14.65625Label a = 14.65625, b = 15.65625, c = 16.65625AD = (b + c - a)/2 = (15.65625 + 16.65625 - 14.65625)/2 = (17.65625)/2 = 8.828125BE = (a + c - b)/2 = (14.65625 + 16.65625 - 15.65625)/2 = (15.65625)/2 = 7.828125CF = (a + b - c)/2 = (14.65625 + 15.65625 - 16.65625)/2 = (13.65625)/2 = 6.828125So, T₈ has sides 8.828125, 7.828125, 6.828125Perimeter: 8.828125 + 7.828125 + 6.828125 = 23.484375Which is half of 46.96875, correct.Check triangle inequality:8.828125 + 7.828125 > 6.828125? 16.65625 > 6.828125, yes.8.828125 + 6.828125 > 7.828125? 15.65625 > 7.828125, yes.7.828125 + 6.828125 > 8.828125? 14.65625 > 8.828125, yes.Valid.T₉: Compute T₉.T₈ sides: 8.828125, 7.828125, 6.828125Label a = 6.828125, b = 7.828125, c = 8.828125AD = (b + c - a)/2 = (7.828125 + 8.828125 - 6.828125)/2 = (9.828125)/2 = 4.9140625BE = (a + c - b)/2 = (6.828125 + 8.828125 - 7.828125)/2 = (7.828125)/2 = 3.9140625CF = (a + b - c)/2 = (6.828125 + 7.828125 - 8.828125)/2 = (5.828125)/2 = 2.9140625So, T₉ has sides 4.9140625, 3.9140625, 2.9140625Perimeter: 4.9140625 + 3.9140625 + 2.9140625 = 11.7421875Which is half of 23.484375, correct.Check triangle inequality:4.9140625 + 3.9140625 > 2.9140625? 8.828125 > 2.9140625, yes.4.9140625 + 2.9140625 > 3.9140625? 7.828125 > 3.9140625, yes.3.9140625 + 2.9140625 > 4.9140625? 6.828125 > 4.9140625, yes.Valid.T₁₀: Compute T₁₀.T₉ sides: 4.9140625, 3.9140625, 2.9140625Label a = 2.9140625, b = 3.9140625, c = 4.9140625AD = (b + c - a)/2 = (3.9140625 + 4.9140625 - 2.9140625)/2 = (5.9140625)/2 = 2.95703125BE = (a + c - b)/2 = (2.9140625 + 4.9140625 - 3.9140625)/2 = (3.9140625)/2 = 1.95703125CF = (a + b - c)/2 = (2.9140625 + 3.9140625 - 4.9140625)/2 = (1.9140625)/2 = 0.95703125So, T₁₀ has sides 2.95703125, 1.95703125, 0.95703125Perimeter: 2.95703125 + 1.95703125 + 0.95703125 = 5.87109375Which is half of 11.7421875, correct.Check triangle inequality:2.95703125 + 1.95703125 > 0.95703125? 4.9140625 > 0.95703125, yes.2.95703125 + 0.95703125 > 1.95703125? 3.9140625 > 1.95703125, yes.1.95703125 + 0.95703125 > 2.95703125? 2.9140625 > 2.95703125? Wait, 2.9140625 is less than 2.95703125.So, 2.9140625 < 2.95703125, which means 1.95703125 + 0.95703125 < 2.95703125Therefore, triangle inequality fails here.So, T₁₀ is not a valid triangle because the sum of the two smaller sides is not greater than the largest side.Therefore, the last valid triangle is T₉ with perimeter 11.7421875.Wait, but looking at the answer choices, they are in fractions with denominators as powers of 2: 64, 128, 256, 512, 1024.11.7421875 is equal to 11 + 0.7421875.0.7421875 is 47/64, because 47/64 = 0.734375, which is close but not exact.Wait, let me compute 11.7421875 as a fraction.11.7421875 = 11 + 0.74218750.7421875 * 65536 = ?Wait, perhaps better to note that 0.7421875 = 7421875/10000000But that's messy.Alternatively, since each time the perimeter is halved, starting from 3006.So, T₁: 3006T₂: 3006 / 2 = 1503T₃: 1503 / 2 = 751.5T₄: 751.5 / 2 = 375.75T₅: 375.75 / 2 = 187.875T₆: 187.875 / 2 = 93.9375T₇: 93.9375 / 2 = 46.96875T₈: 46.96875 / 2 = 23.484375T₉: 23.484375 / 2 = 11.7421875T₁₀: 11.7421875 / 2 = 5.87109375So, T₉ is the last valid triangle with perimeter 11.7421875.Expressed as a fraction, 11.7421875 = 11 + 0.74218750.7421875 = 7421875/10000000But let's see:0.7421875 * 1048576 = ?Wait, perhaps better to note that 0.7421875 = 47/64Wait, 47/64 = 0.734375Hmm, not exact.Wait, 0.7421875 = 7421875/10000000Simplify numerator and denominator by dividing numerator and denominator by 15625:7421875 ÷ 15625 = 47510000000 ÷ 15625 = 640So, 475/640 = 95/128Wait, 95/128 = 0.7421875Yes, because 95 ÷ 128 = 0.7421875So, 11.7421875 = 11 + 95/128 = (11*128 + 95)/128 = (1408 + 95)/128 = 1503/128Wait, 11*128 = 1408, 1408 + 95 = 1503So, 1503/128 = 11.7421875Therefore, the perimeter of T₉ is 1503/128.Wait, but looking back, T₉ is the last valid triangle, so the perimeter is 1503/128.But wait, in the answer choices, option B is 1503/128, which is 11.7421875.But earlier, I thought T₉ was the last one, but let me check.Wait, T₁₀ is invalid, so T₉ is the last one.But wait, in the problem statement, it says "if it exists". So, T₁₀ does not exist, so the last triangle is T₉.But in the answer choices, 1503/128 is option B, but in my earlier calculation, I thought the perimeter was 1503/256, which is option C.Wait, perhaps I made a mistake in the fraction conversion.Wait, 11.7421875 is equal to 1503/128.But let me verify:1503 ÷ 128 = 11.7421875Yes, because 128*11 = 1408, 1503 - 1408 = 95, so 95/128 = 0.7421875So, 1503/128 = 11.7421875But in my earlier thought process, I thought the perimeter was 1503/256, which is half of that.Wait, perhaps I made a mistake in the number of divisions.Wait, starting from T₁: 3006Each time, perimeter halves:T₁: 3006T₂: 1503T₃: 751.5T₄: 375.75T₅: 187.875T₆: 93.9375T₇: 46.96875T₈: 23.484375T₉: 11.7421875T₁₀: 5.87109375So, T₉ is 11.7421875, which is 1503/128.But in the answer choices, option C is 1503/256, which is half of that.Wait, perhaps I miscounted the number of times we halved.Wait, starting from T₁: 3006After 1 halving: T₂: 1503After 2 halvings: T₃: 751.5After 3 halvings: T₄: 375.75After 4 halvings: T₅: 187.875After 5 halvings: T₆: 93.9375After 6 halvings: T₇: 46.96875After 7 halvings: T₈: 23.484375After 8 halvings: T₉: 11.7421875After 9 halvings: T₁₀: 5.87109375So, T₉ is after 8 halvings, so 3006 / (2^8) = 3006 / 256 = 11.7421875Wait, 3006 / 256 = 11.7421875Yes, because 256*11 = 2816, 3006 - 2816 = 190, 190/256 = 95/128 ≈ 0.7421875So, 3006 / 256 = 11.7421875 = 1503/128Wait, because 3006 / 2 = 1503, so 3006 / 256 = 1503 / 128Yes, so 1503/128 is equal to 3006/256.So, the perimeter of T₉ is 1503/128.But in the answer choices, option B is 1503/128, which is 11.7421875.But earlier, I thought the answer was 1503/256, which is option C.Wait, perhaps I made a mistake in the number of halvings.Wait, T₁: 3006T₂: 1503 (1 halving)T₃: 751.5 (2 halvings)T₄: 375.75 (3 halvings)T₅: 187.875 (4 halvings)T₆: 93.9375 (5 halvings)T₇: 46.96875 (6 halvings)T₈: 23.484375 (7 halvings)T₉: 11.7421875 (8 halvings)T₁₀: 5.87109375 (9 halvings)So, T₉ is after 8 halvings, so perimeter is 3006 / 2^8 = 3006 / 256 = 11.7421875 = 1503/128Therefore, the perimeter is 1503/128, which is option B.But in my initial thought process, I thought it was 1503/256, which is option C.Wait, perhaps I confused the number of halvings.Wait, 2^8 = 256, so 3006 / 256 = 11.7421875But 1503/128 is the same as 3006/256.Yes, because 1503*2 = 3006, and 128*2 = 256.So, 1503/128 = 3006/256Therefore, the perimeter is 1503/128, which is option B.But wait, the answer choices have option C as 1503/256.Wait, perhaps I made a mistake in the number of halvings.Wait, starting from T₁, each Tₙ is formed from Tₙ₋₁, so T₂ is after 1 halving, T₃ after 2 halvings, ..., T₉ after 8 halvings.So, perimeter of T₉ is 3006 / 2^8 = 3006 / 256 = 11.7421875 = 1503/128Therefore, the correct answer is 1503/128, which is option B.But in the initial problem statement, the answer choices are:(A) 1503/64(B) 1503/128(C) 1503/256(D) 1503/512(E) 1503/1024So, the correct answer is B.But in my earlier thought process, I thought it was C, but that was a mistake.Wait, let me double-check.T₁: 3006T₂: 1503 (1 halving)T₃: 751.5 (2 halvings)T₄: 375.75 (3 halvings)T₅: 187.875 (4 halvings)T₆: 93.9375 (5 halvings)T₇: 46.96875 (6 halvings)T₈: 23.484375 (7 halvings)T₉: 11.7421875 (8 halvings)T₁₀: 5.87109375 (9 halvings)So, T₉ is after 8 halvings, so perimeter is 3006 / 2^8 = 3006 / 256 = 11.7421875Which is 1503/128, because 3006 / 256 = (1503*2)/(128*2) = 1503/128Yes, so the perimeter is 1503/128, which is option B.Therefore, the correct answer is B.But wait, in the initial problem, the answer choices are:(A) 1503/64(B) 1503/128(C) 1503/256(D) 1503/512(E) 1503/1024So, 1503/128 is option B.But in my initial calculation, I thought it was 1503/256, which is option C.Wait, perhaps I made a mistake in the number of halvings.Wait, starting from T₁, each Tₙ is formed from Tₙ₋₁, so T₂ is after 1 halving, T₃ after 2 halvings, ..., T₉ after 8 halvings.So, perimeter of T₉ is 3006 / 2^8 = 3006 / 256 = 11.7421875 = 1503/128Therefore, the correct answer is B.But in the initial problem, the answer choices are:(A) 1503/64(B) 1503/128(C) 1503/256(D) 1503/512(E) 1503/1024So, the correct answer is B.But wait, in the initial problem, the answer choices are:(A) 1503/64(B) 1503/128(C) 1503/256(D) 1503/512(E) 1503/1024So, the correct answer is B.But in the initial thought process, I thought it was C, but that was a mistake.Therefore, the correct answer is B.

answer：First, I need to calculate ( 3 star 8 ) using the definition of the operation ( star ).[3 star 8 = frac{3 + 8}{4} = frac{11}{4}]Next, I use the result from the first step to calculate ( left(3 star 8right) star 6 ).[left(frac{11}{4}right) star 6 = frac{frac{11}{4} + 6}{4} = frac{frac{11}{4} + frac{24}{4}}{4} = frac{frac{35}{4}}{4} = frac{35}{16}]Therefore, the final answer is ( frac{35}{16} ).

answer：Alright, so I have this problem where I need to find the standard equation of a circle. The circle has its center at point C, which lies on the line given by the equation x - y + 1 = 0. Additionally, the circle passes through two specific points, A (1, -5) and B (2, -2). First, I remember that the standard equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Since the center C is on the line x - y + 1 = 0, that means the coordinates of C, which I'll call (h, k), must satisfy this equation. So, h - k + 1 = 0. That gives me one equation relating h and k.Next, since the circle passes through points A and B, the distance from the center C to each of these points must be equal to the radius r. So, I can set up two equations based on the distance formula. For point A (1, -5), the distance to C (h, k) is sqrt[(1 - h)^2 + (-5 - k)^2] = r. Similarly, for point B (2, -2), the distance to C is sqrt[(2 - h)^2 + (-2 - k)^2] = r. Since both distances equal r, I can set them equal to each other:sqrt[(1 - h)^2 + (-5 - k)^2] = sqrt[(2 - h)^2 + (-2 - k)^2]To eliminate the square roots, I'll square both sides:(1 - h)^2 + (-5 - k)^2 = (2 - h)^2 + (-2 - k)^2Now, I'll expand both sides:Left side: (1 - 2h + h^2) + (25 + 10k + k^2) = 1 - 2h + h^2 + 25 + 10k + k^2 = h^2 + k^2 - 2h + 10k + 26Right side: (4 - 4h + h^2) + (4 + 4k + k^2) = 4 - 4h + h^2 + 4 + 4k + k^2 = h^2 + k^2 - 4h + 4k + 8Now, subtract the right side from the left side:(h^2 + k^2 - 2h + 10k + 26) - (h^2 + k^2 - 4h + 4k + 8) = 0Simplify:(-2h + 10k + 26) - (-4h + 4k + 8) = 0Which becomes:(-2h + 10k + 26) + 4h - 4k - 8 = 0Combine like terms:(2h) + (6k) + 18 = 0So, 2h + 6k + 18 = 0Divide the entire equation by 2 to simplify:h + 3k + 9 = 0Now, I have two equations:1. h - k + 1 = 0 (from the line)2. h + 3k + 9 = 0 (from equating distances)I can solve this system of equations to find h and k.From the first equation: h = k - 1Substitute h = k - 1 into the second equation:(k - 1) + 3k + 9 = 0Combine like terms:4k + 8 = 04k = -8k = -2Now, substitute k = -2 back into h = k - 1:h = -2 - 1 = -3So, the center C is at (-3, -2).Now, to find the radius r, I can use the distance from C to either point A or B. Let's use point A (1, -5):r^2 = (1 - (-3))^2 + (-5 - (-2))^2 = (4)^2 + (-3)^2 = 16 + 9 = 25So, r = 5.Therefore, the standard equation of the circle is:(x - (-3))^2 + (y - (-2))^2 = 25Simplifying:(x + 3)^2 + (y + 2)^2 = 25I think that's the answer. Let me just double-check my steps to make sure I didn't make any mistakes.First, I set up the equations correctly based on the distance formula and the line equation. Then, I expanded both sides correctly and simplified to get 2h + 6k + 18 = 0, which I simplified to h + 3k + 9 = 0. Solving the system gave me h = -3 and k = -2, which seems correct. Calculating the radius using point A gave me r = 5, which makes sense. Plugging everything back into the standard equation looks right.Yeah, I think that's correct.

answer：Alright, so I have this problem here: In triangle ABC, angle A is an internal angle, and it's given that sin A + cos A equals 7/12. I need to figure out what type of triangle ABC is—whether it's obtuse, acute, right, or equilateral. Hmm, okay, let's break this down step by step.First off, I remember that for any angle, sin and cos values are related, and there are some identities that connect them. Maybe I can use one of those identities to simplify the equation sin A + cos A = 7/12. Let me think... Oh, right! If I square both sides of the equation, I can use the Pythagorean identity sin²A + cos²A = 1.So, let's try that. Squaring both sides:(sin A + cos A)² = (7/12)²Expanding the left side:sin²A + 2 sin A cos A + cos²A = 49/144Now, I know that sin²A + cos²A equals 1, so substituting that in:1 + 2 sin A cos A = 49/144Hmm, okay, so now I can solve for sin A cos A. Let's subtract 1 from both sides:2 sin A cos A = 49/144 - 1Calculating the right side:49/144 - 144/144 = (49 - 144)/144 = (-95)/144So, 2 sin A cos A = -95/144Wait a minute, 2 sin A cos A is equal to sin 2A, right? So, sin 2A = -95/144.Now, sin 2A is negative. That means that 2A must be in a quadrant where sine is negative. Sine is negative in the third and fourth quadrants. But since A is an internal angle of a triangle, it must be between 0 and 180 degrees. So, 2A would be between 0 and 360 degrees.If sin 2A is negative, then 2A must be between 180 and 360 degrees. So, 2A is in the third or fourth quadrant. Therefore, A must be between 90 and 180 degrees. Wait, but if A is between 90 and 180, that would make triangle ABC an obtuse triangle because one of its angles is greater than 90 degrees.But hold on, let me double-check. If A is between 90 and 180, then 2A is between 180 and 360, which is correct. But does that necessarily mean that A itself is obtuse? Yes, because if 2A is greater than 180, then A is greater than 90. So, angle A is obtuse.But wait, the answer choices don't have "obtuse" as the only option. Let me see: (A) Obtuse triangle, (B) Acute triangle, (C) Right triangle, (D) Equilateral triangle.So, if angle A is obtuse, then triangle ABC is an obtuse triangle. So, the answer should be (A).But let me think again—maybe I made a mistake. Because sin A + cos A = 7/12 is a positive value. If A is obtuse, then sin A is positive, but cos A is negative. So, sin A + cos A would be positive only if sin A is greater in magnitude than cos A. Is that possible?Let me check. If A is 120 degrees, for example, sin 120 is √3/2 ≈ 0.866, and cos 120 is -1/2 = -0.5. So, sin A + cos A would be approximately 0.866 - 0.5 = 0.366, which is roughly 7/12 (which is about 0.583). Wait, 0.366 is less than 0.583, so maybe 120 degrees is too big.Let me try A = 135 degrees. Sin 135 is √2/2 ≈ 0.707, cos 135 is -√2/2 ≈ -0.707. So, sin A + cos A would be approximately 0.707 - 0.707 = 0, which is less than 7/12. Hmm, so maybe A is less than 135 degrees.Wait, but earlier, we found that 2A is between 180 and 360, so A is between 90 and 180. But when I tried 120, the sum was 0.366, and at 135, it was 0. So, maybe the angle is somewhere between 90 and 120 degrees.But regardless, if A is greater than 90 degrees, the triangle is obtuse. So, even if the sum sin A + cos A is positive, as long as A is greater than 90, the triangle is obtuse.But wait, let me think about the value of sin A + cos A. If A is 90 degrees, sin A is 1, cos A is 0, so sin A + cos A is 1. If A is 60 degrees, sin A is √3/2 ≈ 0.866, cos A is 0.5, so sin A + cos A ≈ 1.366. If A is 45 degrees, sin A and cos A are both √2/2 ≈ 0.707, so their sum is about 1.414.Wait, so as A decreases from 90 to 0 degrees, sin A + cos A increases from 1 to √2 ≈ 1.414. But in our case, sin A + cos A is 7/12 ≈ 0.583, which is much less than 1. So, that suggests that A is greater than 90 degrees because as A increases beyond 90, sin A starts decreasing, and cos A becomes negative, making their sum smaller.Wait, but when A is 90 degrees, sin A + cos A is 1. When A is 180 degrees, sin A is 0, cos A is -1, so sin A + cos A is -1. So, as A increases from 90 to 180, sin A + cos A decreases from 1 to -1. So, 7/12 is between 0 and 1, so A must be between 90 and 180 degrees, but closer to 90 than to 180 because 7/12 is closer to 1 than to -1.Wait, but 7/12 is approximately 0.583, which is less than 1, so A is greater than 90 degrees. So, the triangle is obtuse.But wait, let me think again. Maybe I can find the exact value of A to confirm.We have sin A + cos A = 7/12.Let me write this as:sin A + cos A = 7/12I can write this as:√2 sin(A + 45°) = 7/12Because sin A + cos A can be written as √2 sin(A + 45°). Let me verify that:sin A + cos A = √2 sin(A + 45°)Yes, because sin(A + 45°) = sin A cos 45° + cos A sin 45° = (sin A + cos A)/√2. So, multiplying both sides by √2 gives sin A + cos A = √2 sin(A + 45°).So, √2 sin(A + 45°) = 7/12Therefore, sin(A + 45°) = 7/(12√2) ≈ 7/(16.97) ≈ 0.412So, A + 45° = arcsin(0.412) ≈ 24.3 degrees or 180 - 24.3 = 155.7 degrees.But since A is between 0 and 180, A + 45° is between 45 and 225 degrees.So, the two possible solutions are:A + 45° ≈ 24.3°, which would make A ≈ -20.7°, which is not possible because angles can't be negative.Or, A + 45° ≈ 155.7°, so A ≈ 155.7 - 45 = 110.7°So, A is approximately 110.7 degrees, which is obtuse.Therefore, triangle ABC is an obtuse triangle.Wait, but earlier I thought that sin A + cos A = 7/12 is less than 1, so A must be greater than 90 degrees. And now, calculating, A is approximately 110.7 degrees, which confirms it's obtuse.So, the answer should be (A) Obtuse triangle.But wait, let me check if there's any other possibility. Could A be in a different quadrant? No, because A is an internal angle of a triangle, so it's between 0 and 180 degrees.Alternatively, maybe I can use another identity or method to confirm.Let me consider that sin A + cos A = 7/12. Let me denote sin A = x, cos A = y.So, x + y = 7/12.Also, we know that x² + y² = 1.We can solve these two equations to find x and y.From x + y = 7/12, we can write y = 7/12 - x.Substitute into x² + y² = 1:x² + (7/12 - x)² = 1Expanding:x² + (49/144 - (14/12)x + x²) = 1Combine like terms:2x² - (14/12)x + 49/144 - 1 = 0Simplify:2x² - (7/6)x + (49/144 - 144/144) = 0Which is:2x² - (7/6)x - 95/144 = 0Multiply both sides by 144 to eliminate denominators:288x² - 168x - 95 = 0Now, solve for x using quadratic formula:x = [168 ± √(168² - 4*288*(-95))]/(2*288)Calculate discriminant:168² = 282244*288*95 = 4*288*95 = 4*27360 = 109440So, discriminant = 28224 + 109440 = 137664√137664 = 371.0 (since 371² = 137641, which is close, but let me check:371² = (370 + 1)² = 370² + 2*370*1 + 1 = 136900 + 740 + 1 = 137641So, √137664 is slightly more than 371, maybe 371.002.But for simplicity, let's approximate it as 371.So, x = [168 ± 371]/576First solution:x = (168 + 371)/576 = 539/576 ≈ 0.935Second solution:x = (168 - 371)/576 = (-203)/576 ≈ -0.352So, sin A is either approximately 0.935 or -0.352.But since A is an internal angle of a triangle, it's between 0 and 180 degrees, so sin A is non-negative. Therefore, sin A ≈ 0.935.So, A ≈ arcsin(0.935) ≈ 69.5 degrees or 180 - 69.5 = 110.5 degrees.Wait, so A could be approximately 69.5 degrees or 110.5 degrees.But earlier, we found that A is approximately 110.7 degrees, which is consistent with this.So, A is approximately 110.5 degrees, which is obtuse.Therefore, triangle ABC is an obtuse triangle.Wait, but if A is 69.5 degrees, that's acute. So, why do we have two solutions?Because when we squared the equation, we introduced an extraneous solution. So, we need to check which one is valid.If A is 69.5 degrees, then sin A ≈ 0.935, cos A ≈ 0.352, so sin A + cos A ≈ 1.287, which is greater than 7/12 ≈ 0.583. So, that's not possible.Therefore, the valid solution is A ≈ 110.5 degrees, which is obtuse.So, the triangle is obtuse.Therefore, the answer is (A) Obtuse triangle.

answer：Alright, so I have this problem here with complex numbers a, b, c, x, y, and z. The equations given are:a = (b + c)/(x - 2),b = (c + a)/(y - 2),c = (a + b)/(z - 2).And I know that x + y + z = 2010 and xy + yz + zx = 67. I need to find the value of xyz.Hmm, okay. Let me see how I can approach this. It seems like these equations are interconnected, so maybe I can find a relationship between x, y, z by manipulating these equations.First, let me try to express x, y, z in terms of a, b, c. From the first equation, a = (b + c)/(x - 2), so if I rearrange that, I get x - 2 = (b + c)/a. Similarly, from the second equation, y - 2 = (c + a)/b, and from the third, z - 2 = (a + b)/c.So, x = 2 + (b + c)/a,y = 2 + (c + a)/b,z = 2 + (a + b)/c.Hmm, that's interesting. So, each of x, y, z is 2 plus some ratio involving the other two variables. Maybe I can find a way to relate these expressions.Let me denote S = a + b + c. Then, from the expressions above, x - 2 = (S - a)/a = (S/a) - 1,Similarly, y - 2 = (S - b)/b = (S/b) - 1,and z - 2 = (S - c)/c = (S/c) - 1.So, x = 2 + (S/a) - 1 = 1 + S/a,y = 1 + S/b,z = 1 + S/c.Okay, so x = 1 + S/a, y = 1 + S/b, z = 1 + S/c.Now, if I take reciprocals, 1/(x - 1) = a/S,1/(y - 1) = b/S,1/(z - 1) = c/S.So, adding these up: 1/(x - 1) + 1/(y - 1) + 1/(z - 1) = (a + b + c)/S = S/S = 1.So, 1/(x - 1) + 1/(y - 1) + 1/(z - 1) = 1.That's a nice equation. Maybe I can use this to find a relationship between x, y, z.Let me denote u = x - 1, v = y - 1, w = z - 1. Then, the equation becomes:1/u + 1/v + 1/w = 1.Also, from the definitions, x = u + 1, y = v + 1, z = w + 1.Given that x + y + z = 2010, so (u + 1) + (v + 1) + (w + 1) = 2010,which simplifies to u + v + w + 3 = 2010,so u + v + w = 2007.Similarly, xy + yz + zx = 67.Let me compute xy + yz + zx in terms of u, v, w.xy = (u + 1)(v + 1) = uv + u + v + 1,yz = (v + 1)(w + 1) = vw + v + w + 1,zx = (w + 1)(u + 1) = wu + w + u + 1.Adding them up: xy + yz + zx = (uv + vw + wu) + 2(u + v + w) + 3.We know that xy + yz + zx = 67, so:uv + vw + wu + 2(u + v + w) + 3 = 67.We already know that u + v + w = 2007, so substituting:uv + vw + wu + 2*2007 + 3 = 67,uv + vw + wu + 4014 + 3 = 67,uv + vw + wu + 4017 = 67,uv + vw + wu = 67 - 4017 = -3950.So, uv + vw + wu = -3950.Now, we also have 1/u + 1/v + 1/w = 1.Let me write that as (vw + uw + uv)/(uvw) = 1.So, (uv + vw + wu) = uvw.But we know that uv + vw + wu = -3950, so:-3950 = uvw.Therefore, uvw = -3950.But uvw = (x - 1)(y - 1)(z - 1).So, (x - 1)(y - 1)(z - 1) = -3950.Now, I need to find xyz. Let me expand (x - 1)(y - 1)(z - 1):(x - 1)(y - 1)(z - 1) = xyz - xy - yz - zx + x + y + z - 1.We know that:(x - 1)(y - 1)(z - 1) = -3950,xy + yz + zx = 67,x + y + z = 2010.So, substituting:xyz - 67 + 2010 - 1 = -3950,xyz + (2010 - 67 - 1) = -3950,xyz + 1942 = -3950,xyz = -3950 - 1942,xyz = -5892.So, the value of xyz is -5892.**Final Answer**The value of ( xyz ) is boxed{-5892}.