answer:1. **Identify the position and relationship of the six numbers**: According to the problem, we have a series of numbers given to six students: A sim F. Each student has different numbers, and there are specific interrelations and the way students respond to these numbers. 2. **Extract Information from Responses**: - A and B say they know the number after their turn. - After hearing A and B, C and D also claim they know the number. - E mentions that they have a strictly larger number than F. - F says their number is between the numbers C and D. 3. **Analyzing the Constraints**: - A and B's statements indicate they are able to identify their positions uniquely in the sequence. This implies A and B have larger or distinct numbers than the following cases. This can also be assumed for C and D. Thus: - Assume A > B > C > D > E > F. - From E and F's statement: - E's number is larger than F. - F's number is between C and D. 4. **Determine a suitable number set:**: - Since each student has a distinct value and the statements are valid, consider typical unique values known (common divisors or harmonic numbers). Following the simplest arithmetic progression or set of simple common factors yields: - Numbers could be derived from distinct, smaller factorial components or primes. 5. **Summing up the proposed values**: - Consider a plausible set of numbers, where these values fulfill all the preceding conditions regarding order, uniqueness, and logical deduction of numerical sequence: - Suppose: [ A Rightarrow 70 text{ (largest common numbers factor)} ] [ B Rightarrow 35 ] [ C Rightarrow 14 ] [ D Rightarrow 10 text{ (sequence between (C) and (D))} ] [ E Rightarrow 7 text{ (greater than (F))} ] [ F Rightarrow 5 text{ (smallest factor)} ] 6. **Sum these values**: [ 70 + 35 + 14 + 10 + 7 + 5 = 141 ] # Conclusion: (boxed{A})

answer:1. **Calculate the total area of the rectangle**: The rectangle has dimensions of 15 by 24 units. Therefore, its total area is: [ text{Area} = 15 times 24 = 360 text{ square units} ] 2. **Determine the area of the shaded region**: One-third of the rectangle is shaded. Thus, the area of this shaded region is: [ text{Area of shaded region} = frac{1}{3} times 360 = 120 text{ square units} ] 3. **Calculate the area of the distinctly marked region within the shaded area**: Half of this shaded area is distinctly marked. Therefore, the area of the distinctly marked region is: [ text{Area of distinctly marked region} = frac{1}{2} times 120 = 60 text{ square units} ] 4. **Find the fraction of the large rectangle that is distinctly marked**: The fraction of the large rectangle that is distinctly marked is calculated as follows: [ text{Fraction of distinctly marked area} = frac{text{Area of distinctly marked region}}{text{Area of large rectangle}} = frac{60}{360} = frac{1}{6} ] 5. **Conclusion**: The fraction of the large rectangular region that is distinctly marked is frac{1}{6}. Therefore, the correct answer is frac{1}{6}, which will be shown as: [ frac{1{6}} ] The correct answer is B) boxed{frac{1}{6}}.

answer:First, we find the derivative of f(x): f'(x) = frac{x^2 - 2x}{(x-1)^2} = frac{x(x-2)}{(x-1)^2}. Solving f'(x) leq 0, we get 0 leq x < 1 or 1 < x leq 2. Therefore, the monotonic decreasing intervals of the original function are [0, 1) and (1, 2]. Hence, the answer is: boxed{[0, 1), (1, 2]}. By deriving the function to get f'(x) = frac{x(x-2)}{(x-1)^2}, we only need to solve f'(x) leq 0 to find the monotonic decreasing intervals of f(x). This problem examines the method of finding the monotonic intervals of a function based on its derivative, the formula for the derivative of a quotient, and the method of solving fractional inequalities.

answer:From the given condition 3sinalpha - cosalpha = 0, we find: [ tanalpha = frac{1}{3}. ] Using the double-angle formula for tangent, we get: [ tan(2alpha) = frac{2tanalpha}{1-tan^2alpha} = frac{2 times frac{1}{3}}{1 - left(frac{1}{3}right)^2} = frac{3}{4}. ] From the given condition 7sinbeta + cosbeta = 0, we find: [ tanbeta = -frac{1}{7}. ] Given the intervals for alpha and beta, 2alpha falls in the interval (0, pi) and 2alpha - beta falls in the interval (-pi, 0). Using the formula for the tangent of the difference of two angles, we get: [ tan(2alpha - beta) = frac{tan(2alpha) - tan(beta)}{1 + tan(2alpha) tan(beta)} = frac{frac{3}{4} + frac{1}{7}}{1 + left(frac{3}{4}right) times left(-frac{1}{7}right)} = 1. ] Since tan(2alpha - beta) = 1 and 2alpha - beta lies within the interval (-pi, 0), the value of 2alpha - beta that corresponds to tan(2alpha - beta) = 1 is: [ boxed{- frac{3pi}{4}}. ] Therefore, the correct answer is option D.