answer:We need to prove that if ( frac{p}{q} ) is a rational number and (cosleft(frac{p}{q} cdot 180^circright) neq 0, pmfrac{1}{2}, pm 1), then (cosleft(frac{p}{q} cdot 180^circright)) is an irrational number. 1. **Express ( frac{p}{q} ) in radians**: Let ( x = frac{p}{q} cdot pi ). 2. **Chebyshev polynomials**: The Chebyshev polynomials ( T_n(x) ) are defined such that ( T_n(cos theta) = cos(ntheta) ). Notice that if ( x = cosleft(frac{p}{q} piright) ), then ( T_q(x) = cos(p pi) ). 3. **Roots of Chebyshev polynomials**: The roots of Chebyshev polynomials are precisely the values of (cos(p pi)) for ( p in mathbb{Z} ). According to these roots, for any rational number ( frac{p}{q} ), we have the following: If ( x = cosleft(frac{p}{q} piright) ), then it satisfies ( T_q(x) = cos(p pi) ). 4. **Irrationality from rational roots theorem**: The Rational Root Theorem states that any rational solution ( frac{p}{q} ) of a polynomial equation with integer coefficients is a fraction where ( p ) is a factor of the constant term and ( q ) is a factor of the leading coefficient. Chebyshev polynomials have integer coefficients. 5. **Specific cases**: If (cosleft(frac{p}{q} piright) = 0, pmfrac{1}{2}, pm 1), these correspond to rational roots. In other words, if ( cos(p pi) ) were rational, then the arguments of the cosine function would either be (0, frac{pi}{3}, frac{pi}{2}, pi, dots). Given the specific condition ( x = cosleft(frac{ppi}{q}right) neq 0, pmfrac{1}{2}, pm 1 ), it implies that ( x ) cannot be a rational number since we’ve removed these rational ( cos(p pi) ). 6. **Conclusion**: If ( x = cos left( frac{p}{q} pi right) ) for rational ( frac{p}{q} ) and ( x neq 0, pm frac{1}{2}, pm 1 ), then ( x ) must be irrational by the properties and roots of Chebyshev polynomials and the rational roots theorem. Therefore, (cosleft(frac{p}{q} cdot 180^circright)) is irrational. (blacksquare)

answer:The direction vector of the first line is begin{pmatrix} b -3 2 end{pmatrix}. The direction vector of the second line is begin{pmatrix} 2 3 4 end{pmatrix}. For the lines to be perpendicular, the dot product of their direction vectors must be zero: [ (b)(2) + (-3)(3) + (2)(4) = 0. ] Solving the equation: [ 2b - 9 + 8 = 0 implies 2b - 1 = 0 implies 2b = 1 implies b = frac{1}{2}. ] Thus, b = boxed{frac{1}{2}}.

answer:First, we complete the square on the parabola's equation to simplify it and determine its vertex form. The given equation is y = 2x^2 - 4x - 1. We factor out the coefficient of x^2, and complete the square for the x-terms: [ y = 2(x^2 - 2x) - 1 = 2((x - 1)^2 - 1) - 1 = 2(x - 1)^2 - 3. ] Now the equation is in the vertex form: y = 2(x - 1)^2 - 3. The vertex form of a parabola is y - k = a(x - h)^2, where (h, k) is the vertex, and the standard form to identify the focus distance from the vertex for a parabola is y = ax^2 originates 1/(4a) where a = 2 here. So, the distance from vertex for our parabola becomes: [ frac{1}{4a} = frac{1}{4 cdot 2} = frac{1}{8}. ] The parabola opens upward, and therefore the focus is frac{1}{8} units above the vertex. The vertex is at (1, -3). Adding frac{1}{8} to the y-coordinate of the vertex: [ y = -3 + frac{1}{8} = -frac{23}{8}. ] Therefore, the focus of the parabola y = 2x^2 - 4x - 1 is: [ boxed{left(1, -frac{23}{8}right)}. ]

answer:1. **Calculate the total expenses**: The total amount spent by Olivia, LeRoy, and Bernardo is the sum of the amounts each paid. The total expenses, (T), are: [ T = O + L + B ] 2. **Determine the amount each should pay**: Since they agreed to share the costs equally among three people, each should pay one-third of the total expenses. Thus, the amount each should pay, (E), is: [ E = frac{T}{3} = frac{O + L + B}{3} ] 3. **Calculate the difference Olivia will receive**: Olivia has already paid (O) dollars. To find out how much she needs to receive to make her total contribution equal to (E), we subtract the amount she has paid from (E): [ text{Amount Olivia needs to receive} = E - O = frac{O + L + B}{3} - O ] Simplifying this expression, we find: [ frac{O + L + B}{3} - O= frac{O + L + B - 3O}{3} = frac{L + B - 2O}{3} ] 4. **Split the above amount between LeRoy and Bernardo**: They should equally share what must be given to Olivia: [ text{Each pays to Olivia } = frac{L + B - 2O}{6} ] Conclusion: Olivia should each receive frac{L + B - 2O{6}} dollars from both LeRoy and Bernardo so that the expenses are equally shared among everyone. The final answer is The correct choice is boxed{text{B)} frac{L + B - 2O}{6}}