answer:Given that log_a left( frac{2}{3} right) > 1, we have the inequality: log_a left( frac{2}{3} right) > log_a (a) = 1, since by the definition of logarithm, log_a (a) = 1. Now, considering the properties of logarithmic functions, we have two cases to analyze based on the value of a: Case 1: If a>1, the logarithmic function is increasing. However, if frac{2}{3} > a, this means log_a left( frac{2}{3} right) would be less than log_a (a), which is impossible because we are given log_a left( frac{2}{3} right) > 1. Thus, there are no solutions in this range as it would contradict the given condition. Case 2: If 0 < a < 1, the logarithmic function is decreasing. This means if log_a left( frac{2}{3} right) > log_a (a), then frac{2}{3} < a. In this case, the value of a lies in the interval frac{2}{3} < a < 1. Combining both cases, the range of values for a is given by the interval (frac{2}{3}, 1). The final answer is: boxed{(frac{2}{3}, 1)}

answer:1. **Identify the total goal and the amount already raised:** - Total fundraising goal: 2400 dollars. - Amount already raised: 600 dollars. 2. **Calculate the remaining amount to be raised:** - Remaining amount = Total goal - Amount already raised = 2400 - 600 = 1800. 3. **Determine the number of people involved:** - Total number of people: 8 (you and seven friends). 4. **Calculate the amount each person needs to raise:** - To find out how much each person needs to raise, divide the remaining amount by the number of people: [ text{Amount per person} = frac{text{Remaining amount}}{text{Number of people}} = frac{1800}{8} ] 5. **Perform the division:** - Calculate the division: [ frac{1800}{8} = 225 ] - Each person needs to raise 225 dollars. 6. **Conclusion:** - To meet the remaining goal of 1800 dollars, each of the eight people needs to raise 225 dollars. [ 225 ] The final answer is boxed{mathrm{(B)} 225}

answer:To determine the range of m for which f(x) is a decreasing function on (1, +infty), we need to compute its derivative and analyze its sign over the given interval. Differentiating f(x) with respect to x, we get f'(x) = -x + frac{m}{x} = frac{-x^2 + m}{x}. Let's analyze this derivative based on the values of m: 1. When m leq 0, we have f'(x) < 0 for all x > 0, including x in the interval (1, +infty). In this case, f(x) is decreasing as required. 2. When m > 0, we need the condition -x^2 + m leq 0 to hold for all x in (1, +infty). This inequality holds when 0 < m leq 1, because if m > 1, then there would exist x > 1 such that -x^2 + m > 0, which means f(x) would not be decreasing for all values in the interval. Combining both conditions, the value of m for which f(x) is decreasing on (1, +infty) is m leq 1. Therefore, the range of m is boxed{(-infty, 1]}, which corresponds to option C.

answer:# Solution: Part 1: Completing the Contingency Table and Conducting an Independence Test Given data allows us to fill the contingency table as follows: - There are 60 male students in total. - The proportion of female students who enjoy watching football matches is frac{1}{4} of the total number of female students. Since there are 100 students in total and 60 are male, there are 40 female students. Thus, 40 times frac{1}{4} = 10 female students enjoy watching football. - 10 male students do not like watching football matches, implying 60 - 10 = 50 male students enjoy watching football matches. Filling in the table: [ begin{array}{|c|c|c|c|} hline & text{Male} & text{Female} & text{Total} hline text{Enjoy watching football} & 50 & 10 & 60 hline text{Do not enjoy watching football} & 10 & 30 & 40 hline text{Total} & 60 & 40 & 100 hline end{array} ] To conduct the independence test, we use the given formula for {chi}^{2}: [ {chi}^{2} = frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)} ] Substituting a=50, b=10, c=10, d=30, and n=100: [ {chi}^{2} = frac{100 times (50 times 30 - 10 times 10)^{2}}{60 times 40 times 60 times 40} = frac{100 times (1500 - 100)^{2}}{1440000} = frac{100 times 1400^{2}}{1440000} = frac{1960000}{14400} = frac{1225}{36} approx 34.028 ] Comparing with the critical value x_{0.001} = 10.828: [ 34.028 > 10.828 ] Therefore, we conclude that the preference for watching football matches is related to gender. boxed{text{The preference for watching football matches is related to gender.}} Part 2: Probability Distribution and Expectation of X Given that 8 people are selected, with 2 males and 6 females, and 2 are randomly selected to participate in the interview program. The possible values of X (number of male students selected) are 0, 1, 2. Calculating probabilities: - P(X=0) = frac{{C}_{6}^{2}}{{C}_{8}^{2}} = frac{15}{28} - P(X=1) = frac{{C}_{6}^{1}{C}_{2}^{1}}{{C}_{8}^{2}} = frac{12}{28} = frac{3}{7} - P(X=2) = frac{{C}_{2}^{2}}{{C}_{8}^{2}} = frac{1}{28} The probability distribution of X is: [ begin{array}{|c|c|c|c|} hline X & 0 & 1 & 2 hline P & frac{15}{28} & frac{3}{7} & frac{1}{28} hline end{array} ] Calculating the expectation E(X): [ E(X) = 0 times frac{15}{28} + 1 times frac{3}{7} + 2 times frac{1}{28} = 0 + frac{3}{7} + frac{2}{28} = frac{3}{7} + frac{1}{14} = frac{6}{14} + frac{1}{14} = frac{7}{14} = frac{1}{2} ] boxed{E(X) = frac{1}{2}}