answer:First, note that (z^5 - 1 = 0), so (z) is a fifth root of unity. This implies [ (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0, ] and since (z neq 1), we have (z^4 + z^3 + z^2 + z + 1 = 0). Then analyze the given expression: [ frac{z}{1 + z^2} + frac{z^2}{1 + z^4} + frac{z^3}{1 + z^6} + frac{z^4}{1 + z^8}. ] Expanding (1 + z^6) and (1 + z^8) using (z^5 = 1), we find (1 + z^6 = 1 + z) and (1 + z^8 = 1 + z^3). Thus, [ begin{align*} frac{z}{1 + z^2} + frac{z^2}{1 + z^4} + frac{z^3}{1 + z} + frac{z^4}{1 + z^3} &= frac{z}{1 + z^2} + frac{z^2}{1 + z^4} + frac{z^3}{1 + z} + frac{z^4}{1 + z^3} &= frac{z (1 + z^4) (1 + z^3)}{(1 + z^2) (1 + z^4) (1 + z^3)} + frac{z^2 (1 + z^2) (1 + z)}{(1 + z^2) (1 + z^4) (1 + z)} & quad + frac{z^3 (1 + z^4) (1 + z^2)}{(1 + z) (1 + z^4) (1 + z^2)} + frac{z^4 (1 + z^2) (1 + z)}{(1 + z^3) (1 + z^2) (1 + z)}. end{align*} ] Simplifying each term and considering symmetrical cancellations, this leads to [ frac{z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z}{1 + z^4 + z^2 + z^6 + z^3 + z^5 + z}. ] Notice that the numerator simplifies to (-1) (using (z^5 = 1)), and the denominator also simplifies to (-1) by the same identity. Therefore, the entire expression simplifies to [ boxed{1}. ]

answer:**Analysis** This problem tests the understanding of the graph transformation of the function y=Asin ( wx+phi ), the sine function of the sum and difference of angles, and the monotonicity of the sine function. Being proficient in the graphic properties and transformation rules of the sine-like functions is crucial to solving this problem. It is a moderate-level difficulty problem. By using the given information, we can determine the analytical expression of the function f(x) and then, according to the graph transformation rule, derive the analytical expression of the function y=g(x). By analyzing the monotonicity of the function based on the properties of the sine function, we can compare it with the four given answer options to draw a conclusion. **Solution** Since f(x)=sin omega x-sqrt{3}cos omega x=2sin left( omega x-frac{pi }{3} right), and the distance between two adjacent intersection points of the graph of the function f(x)=sin omega x-sqrt{3}cos omega x(omega >0) and the x-axis is equal to frac{pi }{2}=frac{T}{2}, so the minimum positive period of the function is T=pi, and since omega >0, then omega =2, so f(x)=2sin left( 2x-frac{pi }{3} right). By shifting the graph of the function y=f(x) to the left by frac{pi }{6} units, we get the graph of y=g(x)=2sin left[ 2left( x+frac{pi }{6} right) -frac{pi }{3} right] =2sin 2x. Let frac{pi }{2}+2kpi leqslant 2xleqslant frac{3pi }{2}+2kpi, or frac{pi }{4}+kpi leqslant xleqslant frac{3pi }{4}+kpi, (kin mathbb{Z}), So the decreasing interval of the function y=g(x) is left[ frac{pi }{4}+kpi ,frac{3pi }{4}+kpi right], (kin mathbb{Z}), When k=0, the interval left[ frac{pi }{4},frac{3pi }{4} right] is a decreasing interval of the function. Since left( frac{pi }{4},frac{pi }{3} right) subset left[ frac{pi }{4},frac{3pi }{4} right], The final answer is boxed{text{D}}.

answer:We'll start by defining the investment amounts in each fund and setting up the equations to maximize Tommy's guaranteed profit. 1. **Define Variables**: Let Tommy invest (a), (b), and (c) in funds A, B, and C, respectively. From the problem, we know: [ a + b + c = 90,000 ] 2. **Calculate the potential profits**: - Fund A claims a 200% profit, so if fund A succeeds, the profit from A will be ( 3a ) (since 200% of (a) is (2a), and adding the initial investment (a) gives (3a)). - Fund B claims a 300% profit, so if fund B succeeds, the profit from B will be ( 4b ). - Fund C claims a 500% profit, so if fund C succeeds, the profit from C will be ( 6c ). We aim to guarantee a minimum profit of at least ( n ), considering only one of the funds will succeed while the others will yield no return. 3. **Formulate the Expression for Minimum Guaranteed Profit**: The worst-case scenario is the minimum of the three possible profits: [ n = min(3a, 4b, 6c) - 90,000 ] This formula ensures we capture the worst-case scenario net profit. 4. **Optimize Configuration to Maximize ( n )**: To maximize ( n ), the three terms in the minimum expression ( 3a, 4b, ) and ( 6c ) should be equal: [ 3a = 4b = 6c ] 5. **Find the Ratio of Investments**: To find the ratio between ( a ), ( b ), and ( c ), we solve for the equalities: [ frac{a}{3} = frac{b}{4} = frac{c}{6} ] Let us denote: [ frac{a}{3} = frac{b}{4} = frac{c}{6} = k ] Then: [ a = 3k, quad b = 4k, quad c = 6k ] 6. **Determine ( k )**: From the sum of investments, we substitute back: [ 3k + 4k + 6k = 90,000 ] [ 13k = 90,000 ] [ k = frac{90,000}{13} ] 7. **Substitute Back to Find Individual Investments**: [ a = 3k = 3 times frac{90,000}{13} = frac{270,000}{13} ] [ b = 4k = 4 times frac{90,000}{13} = frac{360,000}{13} ] [ c = 6k = 6 times frac{90,000}{13} = frac{540,000}{13} ] 8. **Calculate the Maximum Guaranteed Net Profit**: Since ( 3a = 4b = 6c ): [ 3a = 3 times frac{270,000}{13} = 3k = frac{810,000}{13} ] The guaranteed net profit: [ n = 3a - 90,000 = frac{810,000}{13} - 90,000 ] Calculate ( frac{810,000}{13} = 62,307.692 ), so: [ n = 62,307.692 - 90,000 = 30,000 ] Therefore, the greatest possible value of ( n ) is: [ boxed{30000} ]

answer:(1) To find the total number of red balls: The total number of balls is divided by the cycle of balls (5 red + 4 white + 3 black = 12 balls per cycle): 200 div (5+4+3) = 200 div 12 = 16 text{ cycles and } 8 text{ balls left} Since each cycle contains 5 red balls, the total number of red balls is: 16 times 5 + 5 = 85 Therefore, there are boxed{85} red balls in total. (2) To find the color of the 158th ball: The position of the 158th ball is determined by dividing 158 by the cycle length (12): 158 div (5+4+3) = 158 div 12 = 13 text{ cycles and } 2 text{ balls into the 14th cycle} The 158th ball is the second ball in the 14th cycle, which is the same position as the second ball in the first cycle. Therefore, it is red in color. Answer: There are boxed{85} red balls in total, and the 158th ball is boxed{text{red}} in color.