[["Problem: Find $(\\log_2 x)^2$ if $\\log_2 (\\log_8 x) = \\log_8 (\\log_2 x).$\nAnswer:",["\n"]],["Problem: Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} + 2018 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} + 2018 \\right).\\]\nAnswer:",["\n"]],["Problem: The complex numbers $z$ and $w$ satisfy the system\n\\begin{align*}\nz + \\frac{20i}w &= 5+i, \\\\\nw+\\frac{12i}z &= -4+10i.\n\\end{align*}Find the smallest possible value of $\\vert zw\\vert^2$.\nAnswer:",["\n"]],["Problem: Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?\nAnswer:",["\n"]],["Problem: Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty.\n\n$(x-3)^2 + y^2 = 10$\nAnswer:",["\n"]],["Problem: Find the sum of the roots, real and non-real, of the equation $x^{2001}+\\left(\\frac 12-x\\right)^{2001}=0$, given that there are no multiple roots.\nAnswer:",["\n"]],["Problem: Find all real numbers $k$ such that $x^4+kx^3+x^2+4kx+16=0$ is true for exactly one real number $x = r$. Enter all the possible values of $k,$ separated by commas.\nAnswer:",["\n"]],["Problem: Find the greatest integer less than $(\\sqrt{7} + \\sqrt{5})^6.$ (Do not use a calculator!)\nAnswer:",["\n"]],["Problem: Let $S$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are integers. We say that $z \\in S$ is a unit if there exists a $w \\in S$ such that $zw = 1.$ Find the number of units in $S.$\nAnswer:",["\n"]],["Problem: Let $\\omega$ be a complex number such that\n\\[\\omega + \\frac{1}{\\omega} = 1.\\]Find all possible values of\n\\[\\omega^n + \\frac{1}{\\omega^n},\\]where $n$ is a positive integer.\n\nEnter all possible values, separated by commas.\nAnswer:",["\n"]]]
