[["Problem: The function $f(x)$ satisfies\n\\[f(x) + f(x + 2y) = 6x + 6y - 8\\]for all real numbers $x$ and $y.$ Find the value of $x$ such that $f(x) = 0.$\nAnswer:",["\n"]],["Problem: Suppose $x \\in [-5,-3]$ and $y \\in [2,4]$. What is the largest possible value of $\\frac{x+y}{x-y}$?\nAnswer:",["\n"]],["Problem: Find all the solutions to\n\\[\\sqrt{3x^2 - 8x + 1} + \\sqrt{9x^2 - 24x - 8} = 3.\\]Enter all the solutions, separated by commas.\nAnswer:",["\n"]],["Problem: The ellipse shown below is defined by the equation\n\\[PF_1 + PF_2 = d.\\]Find $d.$\n\n[asy]\nunitsize(0.3 cm);\n\nint i, n = 10;\n\nfor (i = -n; i <= n; ++i) {\n draw((i,-n)--(i,n),gray(0.7));\n draw((-n,i)--(n,i),gray(0.7));\n}\n\ndraw((0,-n)--(0,n));\ndraw((-n,0)--(n,0));\n\ndraw(shift((-1,0))*xscale(7)*yscale(5)*Circle((0,0),1),red);\n\ndot((-1,0));\n[/asy]\nAnswer:",["\n"]],["Problem: If $a,$ $b,$ and $c$ are real numbers such that $a + b + c = 4$ and $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = 5,$ find the value of\n\\[\\frac{a}{b} + \\frac{b}{a} + \\frac{a}{c} + \\frac{c}{a} + \\frac{b}{c} + \\frac{c}{b}.\\]\nAnswer:",["\n"]],["Problem: Find the number of solutions to the equation\n\\[\\frac{1}{(7 - x)(1 + x)(1 - x)} + \\frac{3x^2 - 18x - 22}{(x^2 - 1)(x - 7)} + \\frac{3}{x - 2} = \\frac{3}{(x - 1)(x - 2)}.\\]\nAnswer:",["\n"]],["Problem: Which type of conic section is described by the equation \\[(x-y)(x+y) = -2y^2 + 1?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above.\nAnswer:",["\n"]],["Problem: Compute the number of ordered pairs $(a,b)$ of integers such that the polynomials $x^2 - ax + 24$ and $x^2 - bx + 36$ have one root in common.\nAnswer:",["\n"]],["Problem: There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that\n$$\\frac {z}{z + n} = 4i.$$Find $n$.\nAnswer:",["\n"]],["Problem: Find the sum of all complex values of $a,$ such that the polynomial $x^4 + (a^2 - 1) x^2 + a^3$ has exactly two distinct complex roots.\nAnswer:",["\n"]]]
