[["Problem: All solutions of the equation $\\cos 4x = -\\frac{1}{2}$ can be expressed in the form $\\frac{(kn \\pm 1) \\pi}{6},$ where $n$ is an integer. Find the positive value of $k.$\nAnswer:",["\n"]],["Problem: Simplify\n\\[\\frac{\\cos x}{1 - \\sin x} - \\frac{\\cos x}{1 + \\sin x}.\\]\nAnswer:",["\n"]],["Problem: Compute $\\cos 180^\\circ$.\nAnswer:",["\n"]],["Problem: If $e^{i \\alpha} = \\frac{3}{5} +\\frac{4}{5} i$ and $e^{i \\beta} = -\\frac{12}{13} + \\frac{5}{13} i,$ then find $\\cos (\\alpha - \\beta).$\nAnswer:",["\n"]],["Problem: Let $0, a, b, c$ be the vertices of a square in counterclockwise order. Compute\n\\[\\frac{ac + b^2}{ab}.\\]Enter your answer in rectangular form.\nAnswer:",["\n"]],["Problem: What is the value of $ \\sum_{n=1}^\\infty (\\tan^{-1}\\sqrt{n}-\\tan^{-1}\\sqrt{n+1})$?\n\nYour answer should be in radians.\nAnswer:",["\n"]],["Problem: Let $L$ be the line in space that passes through the origin and the point $(2,1,-2).$ Find the reflection of the point $(3,6,15)$ across $L.$\nAnswer:",["\n"]],["Problem: Let $S$ be the set of all points $(x,y,z)$ such that $x^2 + y^2 + z^2 \\le 25$ and $z \\ge 0.$ Compute the side length of the largest cube contained in $S.$\nAnswer:",["\n"]],["Problem: Vectors $\\mathbf{a}$ and $\\mathbf{b}$ satisfy $\\|\\mathbf{a}\\| = 5$ and $\\|\\mathbf{b}\\| = 4.$ Also, the angle between vectors $\\mathbf{a}$ and $\\mathbf{b}$ is $60^\\circ.$ Find $\\|\\mathbf{a} - \\mathbf{b}\\|.$\nAnswer:",["\n"]],["Problem: Let $\\mathbf{a}$ and $\\mathbf{b}$ be two nonzero vectors such that $\\mathbf{a} + \\mathbf{b}$ and $\\mathbf{b}$ are orthogonal, and $\\mathbf{a} + 2 \\mathbf{b}$ and $\\mathbf{a}$ are orthogonal. Find $\\frac{\\|\\mathbf{a}\\|}{\\|\\mathbf{b}\\|}.$\nAnswer:",["\n"]]]
