data_source
stringclasses 1
value | prompt
stringlengths 80
5.22k
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict |
|---|---|---|---|---|
DeepScaleR-Qwen-base
|
Find the ordered pair $(a,b)$ of integers such that
\[\sqrt{9 - 8 \sin 50^\circ} = a + b \csc 50^\circ.\] Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "(3,-1)",
"style": "rule"
}
|
{
"index": "DeepScaleR-40300"
}
|
DeepScaleR-Qwen-base
|
Evaluate
\[\begin{vmatrix} 1 & x & y \\ 1 & x + y & y \\ 1 & x & x + y \end{vmatrix}.\] Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "xy",
"style": "rule"
}
|
{
"index": "DeepScaleR-40301"
}
|
DeepScaleR-Qwen-base
|
Compute $\cos 72^\circ.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "\\frac{-1 + \\sqrt{5}}{4}",
"style": "rule"
}
|
{
"index": "DeepScaleR-40302"
}
|
DeepScaleR-Qwen-base
|
The matrix $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 5 & d \end{pmatrix}$ satisfies
\[\mathbf{A}^{-1} = k \mathbf{A}\]for some constant $k.$ Enter the ordered pair $(d,k).$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "\\left( -2, \\frac{1}{19} \\right)",
"style": "rule"
}
|
{
"index": "DeepScaleR-40303"
}
|
DeepScaleR-Qwen-base
|
The projection of $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$ onto a certain vector $\mathbf{w}$ is $\begin{pmatrix} -9/10 \\ 3/10 \end{pmatrix}.$ Find the projection of $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ onto $\mathbf{w}.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "\\begin{pmatrix} 33/10 \\\\ -11/10 \\end{pmatrix}",
"style": "rule"
}
|
{
"index": "DeepScaleR-40304"
}
|
DeepScaleR-Qwen-base
|
A plane is expressed parametrically by
\[\mathbf{v} = \begin{pmatrix} 1 + s - t \\ 2 - s \\ 3 - 2s + 2t \end{pmatrix}.\]Find the equation of the plane. Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "2x + z - 5 = 0",
"style": "rule"
}
|
{
"index": "DeepScaleR-40305"
}
|
DeepScaleR-Qwen-base
|
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "84",
"style": "rule"
}
|
{
"index": "DeepScaleR-40306"
}
|
DeepScaleR-Qwen-base
|
If
\[\sin x + \cos x + \tan x + \cot x + \sec x + \csc x = 7,\]then find $\sin 2x.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "22 - 8 \\sqrt{7}",
"style": "rule"
}
|
{
"index": "DeepScaleR-40307"
}
|
DeepScaleR-Qwen-base
|
Find the phase shift of the graph of $y = \sin (3x - \pi).$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "-\\frac{\\pi}{3}",
"style": "rule"
}
|
{
"index": "DeepScaleR-40308"
}
|
DeepScaleR-Qwen-base
|
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 $a_n < 0$. Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "628",
"style": "rule"
}
|
{
"index": "DeepScaleR-40309"
}
|
DeepScaleR-Qwen-base
|
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\] Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "63",
"style": "rule"
}
|
{
"index": "DeepScaleR-40310"
}
|
DeepScaleR-Qwen-base
|
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\] Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "0",
"style": "rule"
}
|
{
"index": "DeepScaleR-40311"
}
|
DeepScaleR-Qwen-base
|
Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that
\[PA^2 + PB^2 + PC^2 = k \cdot PG^2 + GA^2 + GB^2 + GC^2.\]Find $k.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": "DeepScaleR-40312"
}
|
DeepScaleR-Qwen-base
|
If angle $A$ lies in the second quadrant and $\sin A = \frac{3}{4},$ find $\cos A.$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "-\\frac{\\sqrt{7}}{4}",
"style": "rule"
}
|
{
"index": "DeepScaleR-40313"
}
|
DeepScaleR-Qwen-base
|
The real numbers $a$ and $b$ satisfy
\[\begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix} \times \begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = \mathbf{0}.\]Enter the ordered pair $(a,b).$ Let's think step by step and output the final answer within \boxed{}.
|
math
|
{
"ground_truth": "\\left( \\frac{8}{5}, -\\frac{35}{2} \\right)",
"style": "rule"
}
|
{
"index": "DeepScaleR-40314"
}
|
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