answer:Let's denote the time Martha spent on hold with Comcast as ( x ) minutes. According to the problem, she spent half as much time yelling at a customer service representative, which would be ( frac{x}{2} ) minutes. We know she spent 10 minutes turning the router off and on again. So, the total time spent on all activities is the sum of these three times: [ 10 + x + frac{x}{2} = 100 ] To solve for ( x ), we first combine like terms: [ frac{3x}{2} = 90 ] Now, we multiply both sides by ( frac{2}{3} ) to isolate ( x ): [ x = 90 times frac{2}{3} ] [ x = 60 ] So, Martha spent 60 minutes on hold with Comcast. The ratio of the time she spent on hold with Comcast to the time she spent turning the router off and on again is: [ frac{60}{10} = 6 ] Therefore, the ratio is boxed{6:1} .

answer:Let's denote Bill's current years of experience as ( B ) and Joan's current years of experience as ( J ). According to the information given, 5 years ago, Joan had 3 times as much experience as Bill. This can be written as: [ J - 5 = 3(B - 5) ] Now, Joan has twice as much experience as Bill, which can be written as: [ J = 2B ] We now have a system of two equations with two variables: 1. ( J - 5 = 3B - 15 ) 2. ( J = 2B ) We can substitute the second equation into the first to solve for ( B ): [ 2B - 5 = 3B - 15 ] Now, we'll solve for ( B ): [ 2B - 3B = -15 + 5 ] [ -B = -10 ] [ B = 10 ] So, Bill currently has boxed{10} years of experience.

answer:To find the maximum value of the given expression, we start with the given condition |beta| = 1. This implies that the magnitude of the conjugate of beta, denoted as |overline{beta}|, is also 1. We use this fact to manipulate the given expression: [ left| frac{beta - alpha}{1 - overline{alpha} beta} right| = frac{1}{|overline{beta}|} cdot left| frac{beta - alpha}{1 - overline{alpha} beta} right| ] Since |overline{beta}| = 1, we can rewrite the expression without changing its value: [ = left| frac{beta - alpha}{overline{beta} - overline{alpha} beta overline{beta}} right| ] Next, we use the fact that |beta|^2 = 1 (since |beta| = 1) to simplify the denominator: [ = left| frac{beta - alpha}{overline{beta} - overline{alpha} |beta|^2} right| ] This simplifies further to: [ = left| frac{beta - alpha}{overline{beta} - overline{alpha}} right| ] Given that the numerator and the denominator have the same form, just conjugated, the magnitude of this fraction is 1. Therefore, the maximum value of the given expression is: [ boxed{1}. ]

answer:We consider a cubic polynomial ax^3 + bx^2 + cx + d = 0 with real coefficients, where the set of roots equals the set of coefficients {a, b, c, d}. We will analyze different cases based on possible equalities among a, b, c, and d. Case 1: a = b = r, c = s neq r, d = t neq r, s The polynomial becomes ax^3 + ax^2 + cx + d = 0. By Vieta's formulas: - r + s + t = -b/a = -1 - rs + rt + st = c/a - rst = -d/a If a = b = r, then r + s + t = -1 and rst = -d/r. Solving for specific values: - Let r = 1, then s + t = -2. - Suppose s = 0, then t = -2, giving us the polynomial x^3 + x^2 = 0, which has roots 0, -2. This does not meet our set condition. Case 2: a = c = r, b = s neq r, d = t neq r, s The polynomial becomes ax^3 + bx^2 + ax + d = 0. By Vieta's: - r + s + t = -b/a = -s/r - rs + rt + st = a/a = 1 - rst = -d/a Solving for specific values: - Let r = 1, then s + t = -s, s = -1/2, t = -1/2, resulting in the polynomial x^3 - frac{1}{2}x^2 + x = 0, which has roots 1, -1/2. This case does not meet our set condition. Case 3: a = d = r, b = s neq r, c = t neq r, s The polynomial becomes ax^3 + bx^2 + cx + a = 0. By Vieta's: - r + s + t = -b/a = -s/r - rs + rt + st = c/a = t/r - rst = -a/a = -1 Solving for specific values: - If r = 1, s + t = -s, s = -1/2, t = -1/2, results in a contradiction with non-distinct values of s and t. From the analysis, all cases do not meet the condition of having distinct coefficients matching distinct roots. Conclusion: This problem results in 0 valid cubic polynomials meeting the condition. The final answer is boxed{textbf{(A) } 0}