messages
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|---|---|---|
[
{
"content": "Compute $(\\cos 185^\\circ + i \\sin 185^\\circ)^{54}.$",
"role": "user"
}
] | 19,494
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"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{pmatrix} \\frac{1}{a} & \\frac{1}{b} \\\\ \\frac{1}{c} & \\frac{1}{d} \\end{pmatrix} \\renewcommand{\\arraystretch}{1}.\\]",
"role": "user"
}
] | 19,495
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "If $\\sin x,$ $\\cos x,$ $\\tan x$ form a geometric sequence, in this order, then find $\\cot^6 x - \\cot^2 x.$",
"role": "user"
}
] | 19,496
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "The matrices\n\\[\\begin{pmatrix} 3 & -8 \\\\ a & 11 \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} 11 & b \\\\ 4 & 3 \\end{pmatrix}\\]are inverses. Enter the ordered pair $(a,b).$",
"role": "user"
}
] | 19,497
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"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 coordinates of vertex $A.$",
"role": "user"
}
] | 19,498
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "For a certain value of $k,$ the system\n\\begin{align*}\nx + ky + 3z &= 0, \\\\\n3x + ky - 2z &= 0, \\\\\n2x + 4y - 3z &= 0\n\\end{align*}has a solution where $x,$ $y,$ and $z$ are all nonzero. Find $\\frac{xz}{y^2}.$",
"role": "user"
}
] | 19,499
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Find $\\tan \\left( -\\frac{3 \\pi}{4} \\right).$",
"role": "user"
}
] | 19,500
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "If $\\det \\mathbf{M} = -2,$ then find $ \\det (\\mathbf{M}^4).$",
"role": "user"
}
] | 19,501
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Find the ordered pair $(a,b)$ of integers such that\n\\[\\sqrt{9 - 8 \\sin 50^\\circ} = a + b \\csc 50^\\circ.\\]",
"role": "user"
}
] | 19,502
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Evaluate\n\\[\\begin{vmatrix} 1 & x & y \\\\ 1 & x + y & y \\\\ 1 & x & x + y \\end{vmatrix}.\\]",
"role": "user"
}
] | 19,503
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"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 answer in the form\n\\[Ax + By + Cz + D = 0,\\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\\gcd(|A|,|B|,|C|,|D|) = 1.$",
"role": "user"
}
] | 19,504
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Find the smallest positive integer $k$ such that $\nz^{10} + z^9 + z^6+z^5+z^4+z+1\n$ divides $z^k-1$.",
"role": "user"
}
] | 19,505
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"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 $a_n < 0$.",
"role": "user"
}
] | 19,506
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Find the number of real solutions of the equation\n\\[\\frac{x}{100} = \\sin x.\\]",
"role": "user"
}
] | 19,507
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate\n\\[\\begin{vmatrix} \\sin^2 A & \\cot A & 1 \\\\ \\sin^2 B & \\cot B & 1 \\\\ \\sin^2 C & \\cot C & 1 \\end{vmatrix}.\\]",
"role": "user"
}
] | 19,508
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
[
{
"content": "Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that\n\\[PA^2 + PB^2 + PC^2 = k \\cdot PG^2 + GA^2 + GB^2 + GC^2.\\]Find $k.$",
"role": "user"
}
] | 19,509
|
mnoukhov/deepscaler_20k_medhard_nolatex_rlvr
|
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