answer：Okay, so I need to find the last three digits of (7^{123}). Hmm, that means I need to compute (7^{123} mod 1000). I remember that modular arithmetic can help with finding the last few digits of a number because the last three digits of a number are equivalent to the number modulo 1000.First, I should recall if there's a theorem or method that can simplify calculating such a large exponent. Euler's theorem comes to mind. It states that if two numbers are coprime, then (a^{phi(n)} equiv 1 mod n), where (phi(n)) is Euler's totient function. Since 7 and 1000 are coprime (their greatest common divisor is 1), this theorem should apply here.Let me calculate (phi(1000)). Since 1000 is (2^3 times 5^3), the totient function is multiplicative for coprime factors. So,[phi(1000) = phi(2^3) times phi(5^3)]I know that (phi(p^k) = p^k - p^{k-1}) for a prime (p). Therefore,[phi(2^3) = 8 - 4 = 4][phi(5^3) = 125 - 25 = 100]Multiplying these together:[phi(1000) = 4 times 100 = 400]So, Euler's theorem tells us that:[7^{400} equiv 1 mod 1000]But we need (7^{123}), which is less than 400. Maybe I can express 123 in terms of 400? Hmm, 123 is just 123, so perhaps I need another approach. Maybe breaking down the exponent into smaller parts.Alternatively, I can use the method of successive squaring to compute (7^{123} mod 1000). Let me try that.First, let's compute powers of 7 modulo 1000 step by step:1. (7^1 = 7)2. (7^2 = 49)3. (7^3 = 343)4. (7^4 = 343 times 7 = 2401). But (2401 mod 1000 = 401)5. (7^5 = 401 times 7 = 2807). (2807 mod 1000 = 807)6. (7^6 = 807 times 7 = 5649). (5649 mod 1000 = 649)7. (7^7 = 649 times 7 = 4543). (4543 mod 1000 = 543)8. (7^8 = 543 times 7 = 3801). (3801 mod 1000 = 801)9. (7^9 = 801 times 7 = 5607). (5607 mod 1000 = 607)10. (7^{10} = 607 times 7 = 4249). (4249 mod 1000 = 249)Hmm, this is getting tedious. Maybe I should look for a pattern or a cycle in the powers of 7 modulo 1000. If I can find the period, I can reduce the exponent 123 modulo the period length.Wait, earlier I found that (phi(1000) = 400), so the multiplicative order of 7 modulo 1000 must divide 400. Let me check if the order is less than 400.To find the order, I need the smallest positive integer (k) such that (7^k equiv 1 mod 1000). Let's see:We know (7^{400} equiv 1 mod 1000), but maybe a smaller exponent works. Let's check the divisors of 400.The prime factors of 400 are (2^4 times 5^2), so the divisors are numerous. Maybe I can test some exponents.Alternatively, perhaps using the Chinese Remainder Theorem (CRT) would be more efficient. Since 1000 factors into 8 and 125, which are coprime, I can compute (7^{123} mod 8) and (7^{123} mod 125), then combine the results using CRT.Let me try that approach.First, compute (7^{123} mod 8):Since 7 is congruent to -1 modulo 8, so:[7^{123} equiv (-1)^{123} equiv -1 equiv 7 mod 8]So, (7^{123} equiv 7 mod 8).Next, compute (7^{123} mod 125):This is more involved. Let's compute (phi(125)). Since 125 is (5^3), (phi(125) = 125 - 25 = 100). So, Euler's theorem tells us that (7^{100} equiv 1 mod 125).Therefore, (7^{123} = 7^{100 + 23} = (7^{100})(7^{23}) equiv 1 times 7^{23} equiv 7^{23} mod 125).So, I need to compute (7^{23} mod 125).Let me compute powers of 7 modulo 125 step by step:1. (7^1 = 7)2. (7^2 = 49)3. (7^3 = 343). (343 mod 125 = 343 - 2*125 = 343 - 250 = 93)4. (7^4 = 93 times 7 = 651). (651 mod 125 = 651 - 5*125 = 651 - 625 = 26)5. (7^5 = 26 times 7 = 182). (182 mod 125 = 182 - 125 = 57)6. (7^6 = 57 times 7 = 399). (399 mod 125 = 399 - 3*125 = 399 - 375 = 24)7. (7^7 = 24 times 7 = 168). (168 mod 125 = 168 - 125 = 43)8. (7^8 = 43 times 7 = 301). (301 mod 125 = 301 - 2*125 = 301 - 250 = 51)9. (7^9 = 51 times 7 = 357). (357 mod 125 = 357 - 2*125 = 357 - 250 = 107)10. (7^{10} = 107 times 7 = 749). (749 mod 125 = 749 - 5*125 = 749 - 625 = 124)11. (7^{11} = 124 times 7 = 868). (868 mod 125 = 868 - 6*125 = 868 - 750 = 118)12. (7^{12} = 118 times 7 = 826). (826 mod 125 = 826 - 6*125 = 826 - 750 = 76)13. (7^{13} = 76 times 7 = 532). (532 mod 125 = 532 - 4*125 = 532 - 500 = 32)14. (7^{14} = 32 times 7 = 224). (224 mod 125 = 224 - 125 = 99)15. (7^{15} = 99 times 7 = 693). (693 mod 125 = 693 - 5*125 = 693 - 625 = 68)16. (7^{16} = 68 times 7 = 476). (476 mod 125 = 476 - 3*125 = 476 - 375 = 101)17. (7^{17} = 101 times 7 = 707). (707 mod 125 = 707 - 5*125 = 707 - 625 = 82)18. (7^{18} = 82 times 7 = 574). (574 mod 125 = 574 - 4*125 = 574 - 500 = 74)19. (7^{19} = 74 times 7 = 518). (518 mod 125 = 518 - 4*125 = 518 - 500 = 18)20. (7^{20} = 18 times 7 = 126). (126 mod 125 = 1)21. (7^{21} = 1 times 7 = 7)22. (7^{22} = 7 times 7 = 49)23. (7^{23} = 49 times 7 = 343). (343 mod 125 = 343 - 2*125 = 343 - 250 = 93)Wait, so (7^{23} equiv 93 mod 125). Therefore, (7^{123} equiv 93 mod 125).Now, we have:[7^{123} equiv 7 mod 8][7^{123} equiv 93 mod 125]We need to find a number (x) such that:[x equiv 7 mod 8][x equiv 93 mod 125]We can express (x) as (x = 125k + 93) for some integer (k). Now, substitute into the first congruence:[125k + 93 equiv 7 mod 8]Compute (125 mod 8) and (93 mod 8):- (125 div 8 = 15 times 8 = 120), so (125 equiv 5 mod 8)- (93 div 8 = 11 times 8 = 88), so (93 equiv 5 mod 8)So,[5k + 5 equiv 7 mod 8][5k equiv 2 mod 8]To solve for (k), we need the inverse of 5 modulo 8. Since (5 times 5 = 25 equiv 1 mod 8), the inverse of 5 is 5.Multiply both sides by 5:[k equiv 2 times 5 mod 8][k equiv 10 mod 8][k equiv 2 mod 8]So, (k = 8m + 2) for some integer (m). Substitute back into (x):[x = 125(8m + 2) + 93 = 1000m + 250 + 93 = 1000m + 343]Therefore, the smallest positive solution is (x = 343). So, (7^{123} equiv 343 mod 1000).Wait, but earlier when I tried using the binomial theorem, I got 717. Hmm, that's different. Did I make a mistake somewhere?Let me double-check my calculations. Maybe I messed up in the CRT step.So, (x = 125k + 93). Then, (x equiv 7 mod 8).Compute (125k + 93 mod 8):125 mod 8 is 5, and 93 mod 8 is 5, so:5k + 5 ≡ 7 mod 8So, 5k ≡ 2 mod 8Inverse of 5 mod 8 is 5, since 5*5=25≡1 mod8.So, k ≡ 2*5=10≡2 mod8.Thus, k=8m+2.Thus, x=125*(8m+2)+93=1000m +250 +93=1000m +343.So, x≡343 mod1000.But wait, earlier using the binomial theorem, I got 717. Which one is correct?Let me compute 7^123 mod1000 using another method to verify.Alternatively, perhaps I made a mistake in the binomial theorem approach.Wait, in the initial approach, the user expanded (10 - 3)^123, but 7 is not equal to 10 - 3. 10 - 3 is 7, so that part is correct.But when expanding, they considered only the last three terms, which correspond to k=121,122,123.But let's see:(10 - 3)^123 = sum_{k=0}^{123} C(123,k) 10^{123 -k} (-3)^kTo compute mod1000, we need to consider terms where 10^{123 -k} is at least 10^3, but since 10^3 is 1000, which is 0 mod1000, so only terms where 123 -k <=2, i.e., k >=121.Thus, the relevant terms are k=121,122,123.Compute each term:For k=121:C(123,121)=C(123,2)= (123*122)/2= (123*61)=7503Multiply by 10^{2}=100 and (-3)^{121}= (-3)^{121}= -3^{121}Wait, 3^{121} mod something? Wait, but in the initial approach, they just multiplied 7503 *100 *9, but that seems incorrect.Wait, hold on, (-3)^{121}= (-1)^{121}*3^{121}= -3^{121}But 3^{121} is a huge number, but in the initial approach, they just wrote 9, which is 3^2.Wait, that seems wrong. Because (-3)^2=9, but for k=121, it's (-3)^{121}= -3^{121}, not 9.So, the initial approach incorrectly took (-3)^k as 9 when k=121, which is wrong.Similarly, for k=122, (-3)^{122}=3^{122}, but they took it as (-3)^{122}= (-1)^{122}*3^{122}=3^{122}, but in their calculation, they wrote (-3)^{122}= (-1), which is incorrect.Wait, let me see:In the initial solution, they wrote:= C(123,2)*10^2*9 + C(123,1)*10*(-3)^{122} + (-3)^{123}But (-3)^{122}= (3)^{122}, which is a huge number, but they approximated it as (-1), which is wrong.Similarly, (-3)^{123}= -3^{123}, which they approximated as -3, which is also wrong.So, the initial approach is flawed because they incorrectly simplified (-3)^k for large exponents, which isn't valid modulo 1000.Therefore, the correct approach is the one using Euler's theorem and CRT, which gave me 343. But wait, when I computed 7^123 mod1000, I got 343, but the initial approach got 717. So, which one is correct?Wait, let me compute 7^123 mod1000 using another method.Alternatively, perhaps using exponentiation by squaring.Let me compute 7^123 mod1000.First, note that 123 in binary is 64 + 32 + 16 + 8 + 2 + 1, so 64+32=96, +16=112, +8=120, +2=122, +1=123.So, compute 7^1, 7^2, 7^4, 7^8, 7^16, 7^32, 7^64 mod1000, then multiply the necessary ones.Compute step by step:1. (7^1 = 7)2. (7^2 = 49)3. (7^4 = (7^2)^2 = 49^2 = 2401 mod 1000 = 401)4. (7^8 = (7^4)^2 = 401^2). Compute 401^2:400^2 = 160000, 2*400*1=800, 1^2=1. So, 160000 + 800 + 1 = 160801. 160801 mod1000=801.So, (7^8 equiv 801 mod 1000).5. (7^{16} = (7^8)^2 = 801^2). Compute 801^2:800^2=640000, 2*800*1=1600, 1^2=1. So, 640000 + 1600 +1=641601. 641601 mod1000=601.So, (7^{16} equiv 601 mod 1000).6. (7^{32} = (7^{16})^2 = 601^2). Compute 601^2:600^2=360000, 2*600*1=1200, 1^2=1. So, 360000 + 1200 +1=361201. 361201 mod1000=201.So, (7^{32} equiv 201 mod 1000).7. (7^{64} = (7^{32})^2 = 201^2). Compute 201^2:200^2=40000, 2*200*1=400, 1^2=1. So, 40000 + 400 +1=40401. 40401 mod1000=401.So, (7^{64} equiv 401 mod 1000).Now, we need to compute (7^{123} = 7^{64} times 7^{32} times 7^{16} times 7^8 times 7^2 times 7^1).So, multiply all these together modulo 1000:First, multiply (7^{64} times 7^{32} = 401 times 201).Compute 401*201:400*200=80000, 400*1=400, 1*200=200, 1*1=1. So, 80000 + 400 + 200 +1=80601. 80601 mod1000=601.So, 401*201=80601≡601 mod1000.Next, multiply by (7^{16}=601):601*601. Compute 600*600=360000, 600*1=600, 1*600=600, 1*1=1. So, 360000 + 600 + 600 +1=361201. 361201 mod1000=201.So, 601*601=361201≡201 mod1000.Next, multiply by (7^8=801):201*801. Compute 200*800=160000, 200*1=200, 1*800=800, 1*1=1. So, 160000 + 200 + 800 +1=161001. 161001 mod1000=1.So, 201*801=161001≡1 mod1000.Next, multiply by (7^2=49):1*49=49.Finally, multiply by (7^1=7):49*7=343.So, (7^{123} equiv 343 mod 1000).Therefore, the last three digits are 343.Wait, but earlier using CRT, I also got 343, so that seems consistent. So, the initial approach using binomial theorem was incorrect because they incorrectly simplified the exponents.Therefore, the correct last three digits are 343.But wait, in the initial problem, the user wrote that the last three digits are 717, but according to my calculations, it's 343. So, which one is correct?Wait, perhaps I made a mistake in the CRT step. Let me double-check.We had:(7^{123} equiv 7 mod 8)(7^{123} equiv 93 mod 125)We set (x = 125k + 93), then (x equiv 7 mod 8).Compute (125k + 93 mod 8):125 mod8=5, 93 mod8=5, so:5k +5 ≡7 mod85k ≡2 mod8Multiply both sides by inverse of 5 mod8, which is 5:k ≡10 mod8 => k≡2 mod8Thus, k=8m+2.Thus, x=125*(8m+2)+93=1000m +250 +93=1000m +343.So, x≡343 mod1000.So, that's correct.But when I computed using exponentiation by squaring, I also got 343.So, the initial approach was wrong, and the correct answer is 343.Therefore, the last three digits of (7^{123}) are 343.But wait, in the initial problem, the user wrote that the last three digits are 717, but that's incorrect. So, the correct answer is 343.Wait, but let me check with another method.Alternatively, perhaps using the fact that 7^4=2401≡401 mod1000.Then, 7^4=4017^8=(7^4)^2=401^2=160801≡801 mod10007^16=(7^8)^2=801^2=641601≡601 mod10007^32=(7^16)^2=601^2=361201≡201 mod10007^64=(7^32)^2=201^2=40401≡401 mod1000So, 7^64=401, 7^32=201, 7^16=601, 7^8=801, 7^4=401, 7^2=49, 7^1=7.Now, 123=64+32+16+8+2+1.So, 7^123=7^64 *7^32 *7^16 *7^8 *7^2 *7^1.Multiply step by step:Start with 1.1 *7^64=401401 *7^32=401*201=80601≡601601 *7^16=601*601=361201≡201201 *7^8=201*801=161001≡11 *7^2=1*49=4949 *7^1=49*7=343So, 343 mod1000.Thus, 343 is correct.Therefore, the initial approach was wrong, and the correct last three digits are 343.But wait, in the initial problem, the user wrote that the last three digits are 717, but that's incorrect. So, the correct answer is 343.Wait, but let me check with another method.Alternatively, perhaps using the fact that 7^100≡1 mod125, so 7^123=7^(100+23)=7^100 *7^23≡1*7^23≡93 mod125.And 7^123≡7 mod8.Thus, solving x≡93 mod125 and x≡7 mod8, we get x=343 mod1000.Thus, 343 is correct.Therefore, the last three digits of (7^{123}) are 343.

answer：First, I need to express the number 30,067 in scientific notation. Scientific notation requires the number to be written in the form ( a times 10^n ), where ( 1 leq |a| < 10 ) and ( n ) is an integer.To achieve this, I will move the decimal point in 30,067 so that there is only one non-zero digit to the left of the decimal point. Moving the decimal point four places to the left will give me 3.0067.Next, I need to determine the exponent ( n ). Since I moved the decimal point four places to the left, the exponent will be positive 4.Therefore, the number 30,067 in scientific notation is ( 3.0067 times 10^4 ).

answer：Okay, so I have this algebra problem here: I need to rewrite the quadratic expression (8j^2 - 6j + 20) in the form (c(j + p)^2 + q), where (c), (p), and (q) are constants. Then, I have to find the value of (frac{q}{p}). Hmm, this looks like it involves completing the square. I remember that completing the square is a method used to convert a quadratic expression from standard form to vertex form, which is similar to what's being asked here.First, let me recall how completing the square works. For a quadratic expression (ax^2 + bx + c), the process involves factoring out the coefficient of (x^2) from the first two terms, then adding and subtracting a specific value to complete the square. The specific value is ((frac{b}{2a})^2), which allows the expression to be written as a perfect square trinomial plus some constant.In this problem, the expression is (8j^2 - 6j + 20). So, the coefficient of (j^2) is 8, the coefficient of (j) is -6, and the constant term is 20. Let me write this down:(8j^2 - 6j + 20)I need to express this in the form (c(j + p)^2 + q). Let me see. The first step is to factor out the coefficient of (j^2) from the first two terms. So, factoring out 8 from (8j^2 - 6j), I get:(8(j^2 - frac{6}{8}j) + 20)Simplify (frac{6}{8}) to (frac{3}{4}), so:(8(j^2 - frac{3}{4}j) + 20)Now, inside the parentheses, I have (j^2 - frac{3}{4}j). To complete the square, I need to add and subtract the square of half the coefficient of (j). The coefficient here is (-frac{3}{4}), so half of that is (-frac{3}{8}), and squaring that gives (frac{9}{64}). So, I add and subtract (frac{9}{64}) inside the parentheses:(8left(j^2 - frac{3}{4}j + frac{9}{64} - frac{9}{64}right) + 20)This can be rewritten as:(8left(left(j - frac{3}{8}right)^2 - frac{9}{64}right) + 20)Now, distribute the 8 across the terms inside the parentheses:(8left(j - frac{3}{8}right)^2 - 8 times frac{9}{64} + 20)Calculate (8 times frac{9}{64}):(8 times frac{9}{64} = frac{72}{64} = frac{9}{8})So, the expression becomes:(8left(j - frac{3}{8}right)^2 - frac{9}{8} + 20)Now, combine the constant terms (-frac{9}{8}) and (20). To do this, I need a common denominator. 20 can be written as (frac{160}{8}), so:(-frac{9}{8} + frac{160}{8} = frac{151}{8})Putting it all together, the expression is now:(8left(j - frac{3}{8}right)^2 + frac{151}{8})So, comparing this to the desired form (c(j + p)^2 + q), I can see that:- (c = 8)- The term inside the square is (j - frac{3}{8}), which can be written as (j + (-frac{3}{8})), so (p = -frac{3}{8})- (q = frac{151}{8})Now, I need to find (frac{q}{p}). Let me compute that:(frac{q}{p} = frac{frac{151}{8}}{-frac{3}{8}})Dividing these fractions, I can multiply by the reciprocal:(frac{151}{8} times frac{-8}{3} = frac{151 times (-8)}{8 times 3})The 8s cancel out:(frac{151 times (-1)}{3} = -frac{151}{3})So, (frac{q}{p} = -frac{151}{3}).Let me double-check my steps to make sure I didn't make any mistakes. Starting from the original expression, I factored out 8 correctly, completed the square by adding and subtracting (frac{9}{64}), distributed the 8, and combined the constants properly. The final form matches the desired structure, and the calculation of (frac{q}{p}) seems correct. I think this is the right answer.

answer：Okay, so I'm trying to solve this problem about triangle ABC. The sides opposite angles A, B, and C are a, b, and c respectively. The given equation is:[frac{2sin A - sin C}{sin C} = frac{a^2 + b^2 - c^2}{a^2 + c^2 - b^2}]And I need to find the maximum value of (sin^2 A + sin^2 C).First, I remember that in a triangle, the Law of Cosines relates the sides and the angles. Specifically, for any triangle:[a^2 = b^2 + c^2 - 2bc cos A][b^2 = a^2 + c^2 - 2ac cos B][c^2 = a^2 + b^2 - 2ab cos C]Looking at the given equation, the numerator on the right side is (a^2 + b^2 - c^2), which from the Law of Cosines is equal to (2ab cos C). Similarly, the denominator (a^2 + c^2 - b^2) is equal to (2ac cos B). So, substituting these into the equation, we get:[frac{2sin A - sin C}{sin C} = frac{2ab cos C}{2ac cos B}]Simplifying the right side, the 2's and a's cancel out, so it becomes:[frac{b cos C}{c cos B}]Now, from the Law of Sines, we know that:[frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R]Where R is the radius of the circumscribed circle. So, we can express b and c in terms of sines:[b = 2R sin B][c = 2R sin C]Substituting these into the fraction:[frac{b cos C}{c cos B} = frac{2R sin B cos C}{2R sin C cos B} = frac{sin B cos C}{sin C cos B}]So, the equation now is:[frac{2sin A - sin C}{sin C} = frac{sin B cos C}{sin C cos B}]Wait, both sides have (sin C) in the denominator, so we can multiply both sides by (sin C) to eliminate the denominators:[2sin A - sin C = frac{sin B cos C}{cos B}]Simplify the right side:[frac{sin B}{cos B} = tan B]So, we have:[2sin A - sin C = tan B cos C]Hmm, not sure if that helps directly. Maybe I should express everything in terms of angles. Since in a triangle, the sum of angles is (pi), so (A + B + C = pi). Therefore, (B = pi - A - C).So, maybe I can express (tan B) in terms of A and C. Let me try that.[tan B = tan(pi - A - C) = -tan(A + C)]And,[tan(A + C) = frac{tan A + tan C}{1 - tan A tan C}]So,[tan B = -frac{tan A + tan C}{1 - tan A tan C}]But this seems complicated. Maybe there's a better approach.Looking back at the equation:[2sin A - sin C = frac{sin B cos C}{cos B}]Let me express (sin B) and (cos B) in terms of A and C.Since (B = pi - A - C), we have:[sin B = sin(pi - A - C) = sin(A + C)][cos B = cos(pi - A - C) = -cos(A + C)]So, substituting back into the equation:[2sin A - sin C = frac{sin(A + C) cos C}{-cos(A + C)}]Simplify the right side:[frac{sin(A + C) cos C}{-cos(A + C)} = -tan(A + C) cos C]But (A + C = pi - B), so (tan(A + C) = tan(pi - B) = -tan B). Wait, this seems circular. Maybe I should use angle addition formulas.Let me expand (sin(A + C)):[sin(A + C) = sin A cos C + cos A sin C]Similarly, (cos(A + C) = cos A cos C - sin A sin C)So, substituting back into the equation:[2sin A - sin C = frac{(sin A cos C + cos A sin C) cos C}{-(cos A cos C - sin A sin C)}]Simplify numerator and denominator:Numerator:[(sin A cos C + cos A sin C) cos C = sin A cos^2 C + cos A sin C cos C]Denominator:[-(cos A cos C - sin A sin C) = -cos A cos C + sin A sin C]So, the equation becomes:[2sin A - sin C = frac{sin A cos^2 C + cos A sin C cos C}{-cos A cos C + sin A sin C}]This seems messy. Maybe there's a better way. Let me think.Alternatively, perhaps I can express everything in terms of sides using the Law of Sines.From the Law of Sines, we have:[frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R]So, (a = 2R sin A), (b = 2R sin B), (c = 2R sin C).Let me substitute these into the original equation:[frac{2sin A - sin C}{sin C} = frac{(2R sin A)^2 + (2R sin B)^2 - (2R sin C)^2}{(2R sin A)^2 + (2R sin C)^2 - (2R sin B)^2}]Simplify numerator and denominator:Numerator:[(4R^2 sin^2 A) + (4R^2 sin^2 B) - (4R^2 sin^2 C) = 4R^2 (sin^2 A + sin^2 B - sin^2 C)]Denominator:[(4R^2 sin^2 A) + (4R^2 sin^2 C) - (4R^2 sin^2 B) = 4R^2 (sin^2 A + sin^2 C - sin^2 B)]So, the equation becomes:[frac{2sin A - sin C}{sin C} = frac{sin^2 A + sin^2 B - sin^2 C}{sin^2 A + sin^2 C - sin^2 B}]Hmm, this might not be helpful. Maybe I should consider another approach.Wait, earlier I had:[2sin A - sin C = frac{sin B cos C}{cos B}]Let me write this as:[2sin A - sin C = sin B cdot frac{cos C}{cos B}]But from the Law of Sines, (frac{sin B}{sin C} = frac{b}{c}). Maybe not helpful.Alternatively, perhaps I can use the fact that in a triangle, angles sum to (pi), so (A + B + C = pi), which might allow me to express one angle in terms of the others.Let me assume that (A) and (C) are variables, and (B = pi - A - C). Then, I can express everything in terms of (A) and (C).But before that, maybe I can consider specific cases or look for symmetries.Wait, the problem asks for the maximum value of (sin^2 A + sin^2 C). So, perhaps after finding a relationship between A and C, I can express (sin^2 A + sin^2 C) as a function of a single variable and then find its maximum.So, let's try to find a relationship between A and C.From the given equation:[frac{2sin A - sin C}{sin C} = frac{a^2 + b^2 - c^2}{a^2 + c^2 - b^2}]We had earlier simplified the right side to:[frac{2ab cos C}{2ac cos B} = frac{b cos C}{c cos B}]Which, using the Law of Sines, is:[frac{sin B cos C}{sin C cos B}]So, the equation is:[frac{2sin A - sin C}{sin C} = frac{sin B cos C}{sin C cos B}]Multiplying both sides by (sin C):[2sin A - sin C = frac{sin B cos C}{cos B}]So,[2sin A - sin C = sin B cdot frac{cos C}{cos B}]But (B = pi - A - C), so (sin B = sin(A + C)) and (cos B = -cos(A + C)).So,[2sin A - sin C = sin(A + C) cdot frac{cos C}{-cos(A + C)}]Simplify:[2sin A - sin C = -sin(A + C) cdot frac{cos C}{cos(A + C)}]Hmm, this is getting complicated. Maybe I can express (sin(A + C)) and (cos(A + C)) using angle addition formulas.[sin(A + C) = sin A cos C + cos A sin C][cos(A + C) = cos A cos C - sin A sin C]So, substituting back:[2sin A - sin C = -frac{(sin A cos C + cos A sin C) cos C}{cos A cos C - sin A sin C}]Simplify numerator:[(sin A cos C + cos A sin C) cos C = sin A cos^2 C + cos A sin C cos C]Denominator:[cos A cos C - sin A sin C]So, the equation becomes:[2sin A - sin C = -frac{sin A cos^2 C + cos A sin C cos C}{cos A cos C - sin A sin C}]Let me factor out (cos C) from the numerator:[2sin A - sin C = -frac{cos C (sin A cos C + cos A sin C)}{cos A cos C - sin A sin C}]Notice that the numerator inside the parentheses is (sin(A + C)):[sin A cos C + cos A sin C = sin(A + C)]So,[2sin A - sin C = -frac{cos C sin(A + C)}{cos(A + C)}]But (A + C = pi - B), so (sin(A + C) = sin(pi - B) = sin B), and (cos(A + C) = cos(pi - B) = -cos B).So,[2sin A - sin C = -frac{cos C sin B}{-cos B} = frac{cos C sin B}{cos B}]Which brings us back to:[2sin A - sin C = frac{sin B cos C}{cos B}]This seems like a loop. Maybe I need a different approach.Let me consider using the Law of Cosines again. From the given equation:[frac{2sin A - sin C}{sin C} = frac{a^2 + b^2 - c^2}{a^2 + c^2 - b^2}]We know that:[a^2 + b^2 - c^2 = 2ab cos C][a^2 + c^2 - b^2 = 2ac cos B]So, substituting these in:[frac{2sin A - sin C}{sin C} = frac{2ab cos C}{2ac cos B} = frac{b cos C}{c cos B}]From the Law of Sines, (frac{b}{c} = frac{sin B}{sin C}). So,[frac{b cos C}{c cos B} = frac{sin B cos C}{sin C cos B}]So, the equation becomes:[frac{2sin A - sin C}{sin C} = frac{sin B cos C}{sin C cos B}]Multiplying both sides by (sin C):[2sin A - sin C = frac{sin B cos C}{cos B}]So,[2sin A - sin C = sin B cdot frac{cos C}{cos B}]But (B = pi - A - C), so (sin B = sin(A + C)) and (cos B = -cos(A + C)).So,[2sin A - sin C = sin(A + C) cdot frac{cos C}{-cos(A + C)}]Simplify:[2sin A - sin C = -sin(A + C) cdot frac{cos C}{cos(A + C)}]Again, this seems complicated. Maybe I can express (sin(A + C)) and (cos(A + C)) in terms of A and C.[sin(A + C) = sin A cos C + cos A sin C][cos(A + C) = cos A cos C - sin A sin C]So,[2sin A - sin C = -frac{(sin A cos C + cos A sin C) cos C}{cos A cos C - sin A sin C}]Simplify numerator:[(sin A cos C + cos A sin C) cos C = sin A cos^2 C + cos A sin C cos C]Denominator:[cos A cos C - sin A sin C]So,[2sin A - sin C = -frac{sin A cos^2 C + cos A sin C cos C}{cos A cos C - sin A sin C}]Let me factor out (cos C) from the numerator:[2sin A - sin C = -frac{cos C (sin A cos C + cos A sin C)}{cos A cos C - sin A sin C}]Notice that (sin A cos C + cos A sin C = sin(A + C)), so:[2sin A - sin C = -frac{cos C sin(A + C)}{cos(A + C)}]But (A + C = pi - B), so (sin(A + C) = sin B) and (cos(A + C) = -cos B). Therefore:[2sin A - sin C = -frac{cos C sin B}{-cos B} = frac{cos C sin B}{cos B}]Which brings us back to:[2sin A - sin C = frac{sin B cos C}{cos B}]This seems like a loop. Maybe I need to consider specific values or make an assumption.Alternatively, perhaps I can express everything in terms of one variable. Let me assume that (A = C). Then, since (A + B + C = pi), (2A + B = pi), so (B = pi - 2A).Let me see if this assumption satisfies the given equation.If (A = C), then:Left side:[frac{2sin A - sin A}{sin A} = frac{sin A}{sin A} = 1]Right side:[frac{a^2 + b^2 - c^2}{a^2 + c^2 - b^2}]But if (A = C), then sides a and c are equal, so (a = c). Therefore, the numerator becomes:[a^2 + b^2 - a^2 = b^2]Denominator:[a^2 + a^2 - b^2 = 2a^2 - b^2]So, the right side is:[frac{b^2}{2a^2 - b^2}]But from the Law of Sines, since (A = C), (a = c), so (b = 2a sin B). Wait, no, from the Law of Sines:[frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C}]Since (A = C), (sin A = sin C), so (a = c). Therefore, (b = 2a sin B / sin A).But (B = pi - 2A), so (sin B = sin(2A)).Therefore,[b = 2a cdot frac{sin(2A)}{sin A} = 2a cdot 2 cos A = 4a cos A]So, (b = 4a cos A).Now, substituting back into the right side:[frac{b^2}{2a^2 - b^2} = frac{(4a cos A)^2}{2a^2 - (4a cos A)^2} = frac{16a^2 cos^2 A}{2a^2 - 16a^2 cos^2 A} = frac{16 cos^2 A}{2 - 16 cos^2 A}]Simplify numerator and denominator:Divide numerator and denominator by 2:[frac{8 cos^2 A}{1 - 8 cos^2 A}]But from the left side, we have 1 = right side, so:[1 = frac{8 cos^2 A}{1 - 8 cos^2 A}]Solving for (cos^2 A):Multiply both sides by denominator:[1 - 8 cos^2 A = 8 cos^2 A]Bring terms together:[1 = 16 cos^2 A]So,[cos^2 A = frac{1}{16}][cos A = pm frac{1}{4}]But since A is an angle in a triangle, (0 < A < pi), so (cos A) can be positive or negative, but in this case, since (A = C), and (B = pi - 2A), we need (A < pi/2) to have (B > 0). So, (cos A) is positive.Thus,[cos A = frac{1}{4}][A = arccosleft(frac{1}{4}right)]So, in this case, (A = C = arccos(1/4)), and (B = pi - 2arccos(1/4)).Now, let's compute (sin^2 A + sin^2 C). Since (A = C), this is:[2 sin^2 A]We know that (sin^2 A = 1 - cos^2 A = 1 - frac{1}{16} = frac{15}{16}). So,[2 cdot frac{15}{16} = frac{30}{16} = frac{15}{8} = 1.875]Hmm, but the problem asks for the maximum value. Is this the maximum? Maybe not. Perhaps there's a higher value when (A neq C).Alternatively, maybe I can express (sin^2 A + sin^2 C) in terms of a single variable and find its maximum.Let me denote (A) as a variable, then (C = pi - B - A). But since (B) is also related, maybe I can find a relationship between A and C.Wait, earlier I had:[2sin A - sin C = frac{sin B cos C}{cos B}]But (B = pi - A - C), so maybe I can write everything in terms of A and C.Alternatively, perhaps I can use the fact that in a triangle, the sides are proportional to the sines of the opposite angles. So, (a = 2R sin A), (b = 2R sin B), (c = 2R sin C).Given that, the original equation:[frac{2sin A - sin C}{sin C} = frac{a^2 + b^2 - c^2}{a^2 + c^2 - b^2}]Substituting (a = 2R sin A), etc., we get:[frac{2sin A - sin C}{sin C} = frac{(2R sin A)^2 + (2R sin B)^2 - (2R sin C)^2}{(2R sin A)^2 + (2R sin C)^2 - (2R sin B)^2}]Simplify numerator and denominator:Numerator:[4R^2 (sin^2 A + sin^2 B - sin^2 C)]Denominator:[4R^2 (sin^2 A + sin^2 C - sin^2 B)]So, the equation becomes:[frac{2sin A - sin C}{sin C} = frac{sin^2 A + sin^2 B - sin^2 C}{sin^2 A + sin^2 C - sin^2 B}]Let me denote (x = sin A), (y = sin B), (z = sin C). Then, the equation is:[frac{2x - z}{z} = frac{x^2 + y^2 - z^2}{x^2 + z^2 - y^2}]Cross-multiplying:[(2x - z)(x^2 + z^2 - y^2) = z(x^2 + y^2 - z^2)]Expanding the left side:[2x(x^2 + z^2 - y^2) - z(x^2 + z^2 - y^2) = 2x^3 + 2x z^2 - 2x y^2 - x^2 z - z^3 + z y^2]Right side:[z x^2 + z y^2 - z^3]So, bringing all terms to the left:[2x^3 + 2x z^2 - 2x y^2 - x^2 z - z^3 + z y^2 - z x^2 - z y^2 + z^3 = 0]Simplify:- (2x^3)- (2x z^2)- (-2x y^2)- (-x^2 z)- (-z^3 + z^3 = 0)- (z y^2 - z y^2 = 0)- (-z x^2)So, combining like terms:[2x^3 + 2x z^2 - 2x y^2 - x^2 z - z x^2 = 0]Factor terms:- (2x^3 - 2x y^2 = 2x(x^2 - y^2))- (2x z^2 - x^2 z - z x^2 = 2x z^2 - 2x^2 z = 2x z(z - x))So,[2x(x^2 - y^2) + 2x z(z - x) = 0]Factor out 2x:[2x[(x^2 - y^2) + z(z - x)] = 0]Since x ≠ 0 (as A is an angle in a triangle), we have:[(x^2 - y^2) + z(z - x) = 0]Expand:[x^2 - y^2 + z^2 - x z = 0]Rearrange:[x^2 + z^2 - x z = y^2]But from the Law of Sines, (y = sin B = sin(pi - A - C) = sin(A + C)). Also, (x = sin A), (z = sin C).So,[sin^2 A + sin^2 C - sin A sin C = sin^2(A + C)]Let me compute (sin^2(A + C)):[sin^2(A + C) = (sin A cos C + cos A sin C)^2 = sin^2 A cos^2 C + 2 sin A cos A sin C cos C + cos^2 A sin^2 C]So, the equation becomes:[sin^2 A + sin^2 C - sin A sin C = sin^2 A cos^2 C + 2 sin A cos A sin C cos C + cos^2 A sin^2 C]Let me bring all terms to one side:[sin^2 A + sin^2 C - sin A sin C - sin^2 A cos^2 C - 2 sin A cos A sin C cos C - cos^2 A sin^2 C = 0]Factor terms:- (sin^2 A (1 - cos^2 C) = sin^2 A sin^2 C)- (sin^2 C (1 - cos^2 A) = sin^2 C sin^2 A)- (- sin A sin C - 2 sin A cos A sin C cos C)So,[sin^2 A sin^2 C + sin^2 C sin^2 A - sin A sin C - 2 sin A cos A sin C cos C = 0]Combine like terms:[2 sin^2 A sin^2 C - sin A sin C - 2 sin A cos A sin C cos C = 0]Factor out (sin A sin C):[sin A sin C (2 sin A sin C - 1 - 2 cos A cos C) = 0]Since (sin A sin C neq 0) (as A and C are angles in a triangle), we have:[2 sin A sin C - 1 - 2 cos A cos C = 0]Simplify:[2 (sin A sin C - cos A cos C) = 1]But (sin A sin C - cos A cos C = -cos(A + C)). So,[2 (-cos(A + C)) = 1][-2 cos(A + C) = 1][cos(A + C) = -frac{1}{2}]Since (A + C = pi - B), we have:[cos(pi - B) = -cos B = -frac{1}{2}][cos B = frac{1}{2}]So, (B = frac{pi}{3}) or (B = frac{5pi}{3}). But since B is an angle in a triangle, (0 < B < pi), so (B = frac{pi}{3}).Therefore, (A + C = pi - frac{pi}{3} = frac{2pi}{3}).Now, we need to find the maximum value of (sin^2 A + sin^2 C) given that (A + C = frac{2pi}{3}).Let me denote (A = t), then (C = frac{2pi}{3} - t), where (0 < t < frac{2pi}{3}).So,[sin^2 A + sin^2 C = sin^2 t + sin^2left(frac{2pi}{3} - tright)]Let me compute (sinleft(frac{2pi}{3} - tright)):[sinleft(frac{2pi}{3} - tright) = sinleft(pi - frac{pi}{3} - tright) = sinleft(frac{pi}{3} + tright)]Wait, no:Actually,[sinleft(frac{2pi}{3} - tright) = sinleft(pi - frac{pi}{3} - tright) = sinleft(frac{pi}{3} + tright)]Wait, that's not correct. Let me compute it correctly.Using the sine of a difference:[sinleft(frac{2pi}{3} - tright) = sinleft(frac{2pi}{3}right)cos t - cosleft(frac{2pi}{3}right)sin t]We know that:[sinleft(frac{2pi}{3}right) = frac{sqrt{3}}{2}, quad cosleft(frac{2pi}{3}right) = -frac{1}{2}]So,[sinleft(frac{2pi}{3} - tright) = frac{sqrt{3}}{2} cos t - left(-frac{1}{2}right) sin t = frac{sqrt{3}}{2} cos t + frac{1}{2} sin t]Therefore,[sin^2left(frac{2pi}{3} - tright) = left(frac{sqrt{3}}{2} cos t + frac{1}{2} sin tright)^2]Expanding this:[= left(frac{sqrt{3}}{2} cos tright)^2 + 2 cdot frac{sqrt{3}}{2} cos t cdot frac{1}{2} sin t + left(frac{1}{2} sin tright)^2][= frac{3}{4} cos^2 t + frac{sqrt{3}}{2} sin t cos t + frac{1}{4} sin^2 t]So, the expression for (sin^2 A + sin^2 C) becomes:[sin^2 t + frac{3}{4} cos^2 t + frac{sqrt{3}}{2} sin t cos t + frac{1}{4} sin^2 t]Combine like terms:[left(1 + frac{1}{4}right) sin^2 t + frac{3}{4} cos^2 t + frac{sqrt{3}}{2} sin t cos t][= frac{5}{4} sin^2 t + frac{3}{4} cos^2 t + frac{sqrt{3}}{2} sin t cos t]Now, let me express this in terms of double angles to simplify.Recall that:[sin^2 t = frac{1 - cos 2t}{2}][cos^2 t = frac{1 + cos 2t}{2}][sin t cos t = frac{sin 2t}{2}]Substituting these into the expression:[frac{5}{4} cdot frac{1 - cos 2t}{2} + frac{3}{4} cdot frac{1 + cos 2t}{2} + frac{sqrt{3}}{2} cdot frac{sin 2t}{2}]Simplify each term:First term:[frac{5}{4} cdot frac{1 - cos 2t}{2} = frac{5}{8} (1 - cos 2t)]Second term:[frac{3}{4} cdot frac{1 + cos 2t}{2} = frac{3}{8} (1 + cos 2t)]Third term:[frac{sqrt{3}}{2} cdot frac{sin 2t}{2} = frac{sqrt{3}}{4} sin 2t]Combine all terms:[frac{5}{8} (1 - cos 2t) + frac{3}{8} (1 + cos 2t) + frac{sqrt{3}}{4} sin 2t]Expand:[frac{5}{8} - frac{5}{8} cos 2t + frac{3}{8} + frac{3}{8} cos 2t + frac{sqrt{3}}{4} sin 2t]Combine like terms:- Constants: (frac{5}{8} + frac{3}{8} = 1)- (cos 2t) terms: (-frac{5}{8} cos 2t + frac{3}{8} cos 2t = -frac{2}{8} cos 2t = -frac{1}{4} cos 2t)- (sin 2t) term: (frac{sqrt{3}}{4} sin 2t)So, the expression becomes:[1 - frac{1}{4} cos 2t + frac{sqrt{3}}{4} sin 2t]Now, let me combine the sine and cosine terms into a single sine function using the identity:[a sin x + b cos x = R sin(x + phi)]Where (R = sqrt{a^2 + b^2}) and (phi = arctanleft(frac{b}{a}right)).Here, (a = frac{sqrt{3}}{4}) and (b = -frac{1}{4}).So,[R = sqrt{left(frac{sqrt{3}}{4}right)^2 + left(-frac{1}{4}right)^2} = sqrt{frac{3}{16} + frac{1}{16}} = sqrt{frac{4}{16}} = sqrt{frac{1}{4}} = frac{1}{2}]And,[phi = arctanleft(frac{-1/4}{sqrt{3}/4}right) = arctanleft(-frac{1}{sqrt{3}}right) = -frac{pi}{6}]So,[-frac{1}{4} cos 2t + frac{sqrt{3}}{4} sin 2t = frac{1}{2} sinleft(2t - frac{pi}{6}right)]Therefore, the expression for (sin^2 A + sin^2 C) becomes:[1 + frac{1}{2} sinleft(2t - frac{pi}{6}right)]Now, to find the maximum value, we need to find the maximum of this expression. Since the sine function oscillates between -1 and 1, the maximum value of (sinleft(2t - frac{pi}{6}right)) is 1. Therefore, the maximum value of the expression is:[1 + frac{1}{2} cdot 1 = frac{3}{2}]So, the maximum value of (sin^2 A + sin^2 C) is (frac{3}{2}).**Final Answer**The maximum value of (sin^2 A + sin^2 C) is boxed{dfrac{3}{2}}.

answer：Okay, so I have this problem about an ideal gas where pressure and volume are inversely proportional when temperature is held constant. That means if one goes up, the other goes down in such a way that their product remains the same. I remember from my classes that this is related to Boyle's Law, which states that for a given mass of gas at constant temperature, the pressure is inversely proportional to the volume.The question is asking, if the pressure increases by 20%, how does the volume change? The options are A) 15%, B) 16.67%, C) 20%, and D) 25%. I need to figure out which one is correct.First, let me recall what inversely proportional means. If two quantities are inversely proportional, their product is a constant. So, if P (pressure) increases, V (volume) must decrease to keep the product PV constant.Let me denote the initial pressure as P and the initial volume as V. According to the problem, the pressure increases by 20%. So, the new pressure P' is P plus 20% of P. Mathematically, that would be:P' = P + 0.20P = 1.20PNow, since PV = constant, let's call this constant k. So, initially, we have:PV = kAfter the pressure increases, the new pressure P' times the new volume V' should also equal k:P'V' = kSubstituting P' with 1.20P, we get:1.20P * V' = kBut we know that PV = k, so we can set them equal:1.20P * V' = PVNow, I can solve for V' by dividing both sides by 1.20P:V' = PV / (1.20P)The P cancels out:V' = V / 1.20Calculating V / 1.20, that's the same as multiplying V by (1/1.20). Let me compute 1/1.20:1 / 1.20 = 0.8333...So, V' = 0.8333...VThis means the new volume is approximately 83.33% of the original volume. To find the percentage decrease, I subtract this from 100%:100% - 83.33% = 16.67%So, the volume decreases by approximately 16.67%. Looking back at the options, that's option B.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting from PV = k, and P increases by 20%, so P' = 1.20P. Then, V' = k / P' = (PV) / (1.20P) = V / 1.20 = 0.8333V. Yes, that seems correct. The percentage decrease is indeed 16.67%.I think I got it right. The volume decreases by 16.67% when the pressure increases by 20%.

answer：Okay, so I need to find the equation of a linear function that passes through the point (0,2) and where y decreases as x increases. Hmm, let's break this down step by step.First, I remember that a linear function can be written in the form y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis, which is given as (0,2). So, that means when x is 0, y is 2. Therefore, b should be 2. So, the equation so far is y = mx + 2.Now, the problem says that y decreases as x increases. I think this has to do with the slope. If y decreases when x increases, that means the slope should be negative. Because if the slope is positive, y would increase as x increases, right? So, m needs to be a negative number.But how do I choose the specific value for m? The problem doesn't give me another point to determine the slope, so maybe I can just pick any negative number. Let me think... If I choose m = -1, then the equation becomes y = -x + 2. That seems simple enough. Let me check if this works.If x increases by 1, then y decreases by 1 because of the -1 slope. So, starting at (0,2), if x is 1, y would be 1; if x is 2, y would be 0; and so on. That does show y decreasing as x increases. So, that seems to fit the criteria.Wait, but could I have chosen a different negative slope? Like m = -2 or m = -0.5? Yeah, technically, any negative slope would work because the key point is that y decreases as x increases. So, maybe the simplest answer is with m = -1, but other slopes are possible too.But since the problem doesn't specify any other conditions, like another point the line must pass through, I think it's safe to go with the simplest form, which is y = -x + 2. That way, the equation is straightforward and meets all the given requirements.Let me just visualize this line to make sure. Starting at (0,2), it goes downward as x increases, which makes sense because the slope is negative. So, it's a line that's decreasing from left to right. Yeah, that looks correct.I guess another way to think about it is that the y-intercept is fixed at (0,2), and since the slope is negative, the line tilts downward. So, any negative slope would work, but without more information, the simplest choice is m = -1.Alright, I think I've got it. The equation is y = -x + 2.