messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the equation of the plane passing through $(-1,1,1)$ and $(1,-1,1),$ and which is perpendicular to the plane $x + 2y + 3z = 5.$ Enter your answer in the form\n\\[Ax + By + C... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If\n\\[\\frac{\\sin^4 \\theta}{a} + \\frac{\\cos^4 \\theta}{b} = \\frac{1}{a + b},\\]then find the value of\n\\[\\frac{\\sin^8 \\theta}{a^3} + \\frac{\\cos^8 \\theta}{b^3}\\]in te... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^2 = \\begin{pmatrix} 7 & 0 \\\\ 0 & 7 \\end{pmatrix}.\\]Find the smallest... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\\begin{pmatrix} 1 \\\\ 3 \\\\ 8 \\end{pmatrix},$ and the vector on the line at $t = 2$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute $\\begin{pmatrix} \\sqrt{3} & -1 \\\\ 1 & \\sqrt{3} \\end{pmatrix}^6.$",
"role": "user"
},
{
"content": "To compute the matrix \\(\\begin{pmatrix} \\sqrt{3} & ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the lea... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If $\\mathbf{A} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix},$ then its transpose is given by\n\\[\\mathbf{A}^T = \\begin{pmatrix} a & c \\\\ b & d \\end{pmatrix}.\\]Given t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that\n\\[(\\mathbf{a} \\times \\mathbf{b}) \\times \\mathbf{c} = \\frac{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute\n\\[\\cos^2 0^\\circ + \\cos^2 1^\\circ + \\cos^2 2^\\circ + \\dots + \\cos^2 90^\\circ.\\]",
"role": "user"
},
{
"content": "To find the value of \\(\\cos^2 0... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ are unit vectors, then find the largest possible value of\n\\[\\|\\mathbf{a} - \\mathbf{b}\\|^2 + \\|\\mathbf{a} - \\mathbf{c}\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{c}\\| = 2,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a}... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $G$ be the centroid of triangle $ABC.$ If $GA^2 + GB^2 + GC^2 = 58,$ then find $AB^2 + AC^2 + BC^2.$",
"role": "user"
},
{
"content": "To solve for \\( AB^2 + AC^... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For real numbers $t,$ the point\n\\[(x,y) = \\left( e^t + e^{-t}, 3 (e^t - e^{-t}) \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circl... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If $x + \\frac{1}{x} = \\sqrt{3}$, then find $x^{18}$.",
"role": "user"
},
{
"content": "To find \\( x^{18} \\) given that \\( x + \\frac{1}{x} = \\sqrt{3} \\), we wil... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{A} = \\begin{pmatrix} 2 & 3 \\\\ 0 & 1 \\end{pmatrix}.$ Find $\\mathbf{A}^{20} - 2 \\mathbf{A}^{19}.$",
"role": "user"
},
{
"content": "To find \\(\\mat... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute $\\arccos (\\cos 7).$ All functions are in radians.",
"role": "user"
},
{
"content": "To solve \\(\\arccos (\\cos 7)\\), we need to understand the properties ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The quantity $\\tan 7.5^\\circ$ can be expressed in the form\n\\[\\tan 7.5^\\circ = \\sqrt{a} - \\sqrt{b} + \\sqrt{c} - d,\\]where $a \\ge b \\ge c \\ge d$ are positive integers. ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the solutions to\n\\[\\frac{1}{\\sin x} + \\frac{1}{\\cos x} = 2 \\sqrt{2}\\]in the interval $0 \\le x \\le 2 \\pi.$",
"role": "user"
},
{
"content": "... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The domain of the function $f(x) = \\arcsin(\\log_{m}(nx))$ is a closed interval of length $\\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smal... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of complex numbers $z$ satisfying $|z| = 1$ and\n\\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1.\\]",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Line segment $\\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some consta... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "There exists a scalar $k$ such that for any vectors $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ such that $\\mathbf{a} + \\mathbf{b} + \\mathbf{c} = \\mathbf{0},$ the equation... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{u}$ and $\\mathbf{v}$ be unit vectors, and let $\\mathbf{w}$ be a vector such that $\\mathbf{u} \\times \\mathbf{v} + \\mathbf{u} = \\mathbf{w}$ and $\\mathbf{w} \\t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Below is the graph of $y = a \\sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $b.$\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal f(real x)\n{\n\tre... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 2:7.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC,$ $AB = 20$ and $BC = 15.$ Find the largest possible value of $\\tan A.$",
"role": "user"
},
{
"content": "To find the largest possible value of \\(\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given vectors $\\mathbf{a}$ and $\\mathbf{b},$ let $\\mathbf{p}$ be a vector such that\n\\[\\|\\mathbf{p} - \\mathbf{b}\\| = 2 \\|\\mathbf{p} - \\mathbf{a}\\|.\\]Among all such ve... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $x=\\frac{\\sum\\limits_{n=1}^{44} \\cos n^\\circ}{\\sum\\limits_{n=1}^{44} \\sin n^\\circ}$. What is the greatest integer that does not exceed $100x$?\n",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The sum\\[\\sum_{x=2}^{44} 2\\sin{x}\\sin{1}[1 + \\sec (x-1) \\sec (x+1)]\\]can be written in the form $\\sum_{n=1}^{4} (-1)^n \\frac{\\Phi(\\theta_n)}{\\Psi(\\theta_n)}$, where $... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A projection takes $\\begin{pmatrix} 1 \\\\ -2 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{3}{2} \\\\ -\\frac{3}{2} \\end{pmatrix}.$ Which vector does the projection take $\\begi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{p}$ be the projection of $\\mathbf{v}$ onto $\\mathbf{w},$ and let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{v}.$ If $\\frac{\\|\\mathbf{p}\\|... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\\[(a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)\\]f... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Solve $\\arcsin x + \\arcsin (1 - x) = \\arccos x.$",
"role": "user"
},
{
"content": "To solve the equation \\(\\arcsin x + \\arcsin (1 - x) = \\arccos x\\), we need t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find $k$ if\n\\[(\\sin \\alpha + \\csc \\alpha)^2 + (\\cos \\alpha + \\sec \\alpha)^2 = k + \\tan^2 \\alpha + \\cot^2 \\alpha.\\]",
"role": "user"
},
{
"content": "To ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The line $y = \\frac{3x - 5}{4}$ is parameterized in the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\mathbf{v} + t \\mathbf{d},\\]so that for $x \\ge 3,$ the distance be... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the phase shift of the graph of $y = 2 \\sin \\left( 2x + \\frac{\\pi}{3} \\right).$",
"role": "user"
},
{
"content": "To find the phase shift of the graph of the... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{w} = \\begin{pmatrix} 2 \\\\ -1 \\\\ 2 \\end{pmatrix}.$ The set of vectors $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmat... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^{-1} = \\renewcommand{\\arraystretch}{1.5} \\begin... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$",
"role": "user"
},
{
"content": "To find the mi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For a constant $c,$ in cylindrical coordinates $(r,\\theta,z),$ find the shape described by the equation\n\\[\\theta = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylind... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The vectors $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ 1 \\\\ -2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\\ 2 \\\\ -1 \\end{pmatrix}.$ There exist scalars $p,$ $q,$ a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In spherical coordinates, the point $\\left( 3, \\frac{2 \\pi}{7}, \\frac{8 \\pi}{5} \\right)$ is equivalent to what other point, in the standard spherical coordinate representati... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC,$ the midpoint of $\\overline{BC}$ is $(1,5,-1),$ the midpoint of $\\overline{AC}$ is $(0,4,-2),$ and the midpoint of $\\overline{AB}$ is $(2,3,4).$ Find the coo... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\cos 170^\\circ & -\\sin 170^\\circ \\\\ \\sin 170^\\circ & \\cos 170^\\circ \\end{pmatrix}^n = \\mathbf{I}.... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find $\\tan \\left( -\\frac{3 \\pi}{4} \\right).$",
"role": "user"
},
{
"content": "To find \\(\\tan \\left( -\\frac{3 \\pi}{4} \\right)\\), we will follow these steps... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute $\\cos 72^\\circ.$",
"role": "user"
},
{
"content": "To compute \\(\\cos 72^\\circ\\), we can use the properties of a regular pentagon and some trigonometric i... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A plane is expressed parametrically by\n\\[\\mathbf{v} = \\begin{pmatrix} 1 + s - t \\\\ 2 - s \\\\ 3 - 2s + 2t \\end{pmatrix}.\\]Find the equation of the plane. Enter your answe... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$."... |
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