data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A line containing the points $(9, 1)$ and $(5, 5)$ intersects the $x$ -axis at what point?"
}
] | math | {
"style": "rule",
"ground_truth": "(10, 0)"
} | {
"index": 2100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line contains the points $(6,8)$, $(-2, k)$ and $(-10, 4)$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi - 2}{3\\pi + 2}"
} | {
"index": 2103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line has equation $y=mx-50$ for some positive integer $m$. The line passes through the point $(a, 0)$ for some positive integer $a$. What is the sum of all possible values of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 2105,
"solution": "Since the line with equation $y=mx-50$ passes through the point $(a, 0)$, then $0=ma-50$ or $ma=50$. Since $m$ and $a$ are positive integers whose product is 50, then $m$ and $a$ are divisor pair of 50. Therefore, the possible values of $m$ are the positive divisors of 50, which are $1... |
DeepScaleR | [
{
"role": "user",
"content": "A line in the plane is called strange if it passes through \\((a, 0)\\) and \\((0, 10-a)\\) for some \\(a\\) in the interval \\([0,10]\\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set... | math | {
"style": "rule",
"ground_truth": "50/3"
} | {
"index": 2106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line initially 1 inch long grows according to a new law, where the first term is the initial length and an additional linear term is introduced.\n\\[1+\\frac{1}{3}\\sqrt{3}+2\\cdot\\frac{1}{9}+3\\cdot\\frac{1}{27}\\sqrt{3}+4\\cdot\\frac{1}{81}+\\cdots\\]\nDetermine the limit ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(4+\\sqrt{3})"
} | {
"index": 2107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line initially 1 inch long grows according to the following law, where the first term is the initial length.\n\\[1+\\frac{1}{4}\\sqrt{2}+\\frac{1}{4}+\\frac{1}{16}\\sqrt{2}+\\frac{1}{16}+\\frac{1}{64}\\sqrt{2}+\\frac{1}{64}+\\cdots\\]\nIf the growth process continues forever,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(4+\\sqrt{2})"
} | {
"index": 2108,
"solution": "1. **Identify the pattern and separate the terms**:\n The given series is:\n \\[\n 1 + \\frac{1}{4}\\sqrt{2} + \\frac{1}{4} + \\frac{1}{16}\\sqrt{2} + \\frac{1}{16} + \\frac{1}{64}\\sqrt{2} + \\frac{1}{64} + \\cdots\n \\]\n We can observe that this series can be split into tw... |
DeepScaleR | [
{
"role": "user",
"content": "A line is described by the equation $y-4=4(x-8)$. What is the sum of its $x$-intercept and $y$-intercept?"
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"index": 2109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola."
}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 2110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is expressed in the form\n\\[\\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} 1 \\\\ 11 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the orde... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{2}{5}, \\frac{57}{5} \\right)"
} | {
"index": 2111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is expressed in the form\n\\[\\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} -2 \\\\ 8 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordere... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{1}{3}, \\frac{22}{3} \\right)"
} | {
"index": 2112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ -2 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -8 \\\\ 12 \\end... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt{10}}"
} | {
"index": 2113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 5 \\\\ -9 \\end{... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ -8 \\end{pmatrix}"
} | {
"index": 2114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix} + s \\begin{pmatrix} 4 \\\\ -1 \\end{pmatrix}.\\]\nA second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 7 \\end... | math | {
"style": "rule",
"ground_truth": "\\frac{-13}{\\sqrt{493}}"
} | {
"index": 2115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} + t \\begin{pmatrix} -1 \\\\ 5 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{p... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6 \\\\ -17 \\end{pmatrix}"
} | {
"index": 2116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"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 $t = 2$ is $\\begin{pmatrix} 0 \\\\ -2 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = 3.$"
... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -1/3 \\\\ -11/3 \\\\ -8 \\end{pmatrix}"
} | {
"index": 2117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\\begin{pmatrix} 1 \\\\ 4 \\end{pmatrix},$ and the vector on the line at $t = 3$ is $\\begin{pmatrix} 3 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = -7.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -17 \\\\ 76 \\end{pmatrix}"
} | {
"index": 2118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2119,
"solution": "30."
} |
DeepScaleR | [
{
"role": "user",
"content": "A line parallel to leg \\(AC\\) of right triangle \\(ABC\\) intersects leg \\(BC\\) at point \\(K\\) and the hypotenuse \\(AB\\) at point \\(N\\). On leg \\(AC\\), a point \\(M\\) is chosen such that \\(MK = MN\\). Find the ratio \\(\\frac{AM}{MC}\\) if \\(\\frac{BK}{BC} = 14\\... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 2120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line parallel to side \\( AB \\) of triangle \\( ABC \\) intersects side \\( BC \\) at point \\( M \\) and side \\( AC \\) at point \\( N \\). The area of triangle \\( MCN \\) is twice the area of trapezoid \\( ABMN \\). Find the ratio \\( CM:MB \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{6}"
} | {
"index": 2121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "(\\sqrt{6} + 2) : 1"
} | {
"index": 2122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line parallel to the side $AC$ of a triangle $ABC$ with $\\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 2123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through $(2,2,1)$ and $(5,1,-2).$ A point on this line has an $x$-coordinate of 4. Find the $z$-coordinate of the point."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 2124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through $A\\ (1,1)$ and $B\\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2125,
"solution": "1. **Identify the slope of the line passing through points $A$ and $B$:**\n Given points $A(1,1)$ and $B(100,1000)$, the slope of the line can be calculated using the formula:\n \\[\n \\text{slope} = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{1000 - 1}{100 - 1} = \\frac{999}{99}\n \... |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through point $Q(\\frac{1}{3}, \\frac{4}{3})$ and intersects the hyperbola $x^{2}- \\frac{y^{2}}{4}=1$ at points $A$ and $B$. Point $Q$ is the midpoint of chord $AB$.\n\n1. Find the equation of the line containing $AB$.\n2. Find the length of $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{3}"
} | {
"index": 2126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through the distinct vectors $\\mathbf{a}$ and $\\mathbf{b}.$ Then for a certain value of $k,$ the vector\n\\[k \\mathbf{a} + \\frac{3}{4} \\mathbf{b}\\]must also lie on the line. Find $k.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through the distinct vectors $\\mathbf{u}$ and $\\mathbf{v}.$ For some value of $k$, the vector\n\\[k \\mathbf{u} + \\frac{5}{8} \\mathbf{v}\\] must also lie on the line. Find $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 2128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through the point $(-2, \\sqrt{3})$ at an angle of inclination of $\\frac{\\pi}{3}$. Find the parametric equation of the line, and if it intersects the curve $y^2 = -x - 1$ at points A and B, find the length $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 2129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passes through the vectors $\\mathbf{a}$ and $\\mathbf{b}$. For a certain value of $k$, the vector\n\\[ k \\mathbf{a} + \\frac{5}{8} \\mathbf{b} \\]\nmust also lie on the line. Find $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 2130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them."
}
] | math | {
"style": "rule",
"ground_truth": "174"
} | {
"index": 2131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through one focus F_1 of the ellipse 4x^{2}+2y^{2}=1 intersects the ellipse at points A and B. Then, points A, B, and the other focus F_2 of the ellipse form ∆ABF_2. Calculate the perimeter of ∆ABF_2."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"index": 2132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{30}"
} | {
"index": 2133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through point P(1, 2) is tangent to the circle $x^2+y^2=4$ and perpendicular to the line $ax-y+1=0$. Find the value of the real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 2134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=12$, then $x_1+x_2=$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=7$, find the distance from the midpoint $M$ of $AB$ to the directrix of the parabola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 2136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the intersection point of the medians of triangle \\(ABC\\) intersects the sides \\(BA\\) and \\(BC\\) at points \\(A^{\\prime}\\) and \\(C_1\\) respectively. Given that:\n\n\\(BA^{\\prime} < BA = 3\\), \\(BC = 2\\), and \\(BA^{\\prime} \\cdot BC_1 = 3\\)... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 2137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyp... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 2138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 2139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line passing through the vertex \\( A \\) of triangle \\( ABC \\) perpendicular to its median \\( BD \\) bisects this median.\n\nFind the ratio of the sides \\( AB \\) and \\( AC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment $AB$ with a fixed length of $4$ has its endpoints moving along the positive $x$-axis and the positive $y$-axis, respectively, and $P(x,y)$ is a point on the circumcircle of triangle $OAB$. Find the maximum value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 2141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment begins at $(1, 3)$. It is 5 units long and ends at the point $(4, x)$ and $x > 0$. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment begins at $(2, 5)$. It is 10 units long and ends at the point $(-6, y)$ where $y > 0$. What is the value of $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 2144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment is divided into three parts, $x$, $y$, and $z$, such that, $x < y < z$ and $x$ to $y$ is as $y$ to $z$. If all three parts combined form a segment of length $s$, and $x + y = z$, determine the value of the ratio $\\frac{x}{y}$.\nA) $\\frac{-1 - \\sqrt{5}}{2}$\nB)... | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"index": 2145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of\n\\[R^{\\left(R^{(R^2+R^{-1})}+R^{-1}\\right)}+R^{-1}\\]is"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2146,
"solution": "1. **Identify the Relationship**: Given that a line segment is divided into two parts, $w$ (lesser part) and $l$ (greater part), such that the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole segment. This can be expressed as:\n \... |
DeepScaleR | [
{
"role": "user",
"content": "A line segment starts at $(2, 4)$ and ends at the point $(7, y)$ with $y > 0$. The segment is 6 units long. Find the value of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "4 + \\sqrt{11}"
} | {
"index": 2147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line that always passes through a fixed point is given by the equation $mx - ny - m = 0$, and it intersects with the parabola $y^2 = 4x$ at points $A$ and $B$. Find the number of different selections of distinct elements $m$ and $n$ from the set ${-3, -2, -1, 0, 1, 2, 3}$ suc... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 2148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \\frac{\\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{3}"
} | {
"index": 2149,
"solution": "1. **Identify the slopes of the lines**: \n - The line $y = 1 + \\frac{\\sqrt{3}}{3}x$ has a slope of $\\frac{\\sqrt{3}}{3}$.\n - Since the triangle formed is equilateral, and one side is vertical ($x=1$), the line passing through the origin that forms another side of the triangle ... |
DeepScaleR | [
{
"role": "user",
"content": "A line that passes through the two foci of the hyperbola $$\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 2150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$?"
}
] | math | {
"style": "rule",
"ground_truth": "-37"
} | {
"index": 2151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line through the points $(5, -12)$ and $(k, 23)$ is parallel to the line $4x + 6y = 12$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "-47.5"
} | {
"index": 2152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with a slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 2153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with a slope of $2$ passing through the right focus of the ellipse $\\frac{x^2}{5} + \\frac{y^2}{4} = 1$ intersects the ellipse at points $A$ and $B$. If $O$ is the origin, then the area of $\\triangle OAB$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 2154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with a slope of 2 and a line with a slope of -4 each have a $y$-intercept of 6. What is the distance between the $x$-intercepts of these lines?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 2155,
"solution": "The line with a slope of 2 and $y$-intercept 6 has equation $y=2x+6$. To find its $x$-intercept, we set $y=0$ to obtain $0=2x+6$ or $2x=-6$, which gives $x=-3$. The line with a slope of -4 and $y$-intercept 6 has equation $y=-4x+6$. To find its $x$-intercept, we set $y=0$ to obtain $0=... |
DeepScaleR | [
{
"role": "user",
"content": "A line with equation \\( y = 2x + b \\) passes through the point \\((-4, 0)\\). What is the value of \\(b\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2156,
"solution": "Since the line with equation \\( y = 2x + b \\) passes through the point \\((-4, 0)\\), the coordinates of the point must satisfy the equation of the line. Substituting \\(x = -4\\) and \\(y = 0\\) gives \\(0 = 2(-4) + b\\) and so \\(0 = -8 + b\\) which gives \\(b = 8\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$ , respectively. What is the smallest possible value of $a+b$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 2157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\\triangle BCD$ is $6\\sqrt{5}$, th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 2158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 2159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope 3 intersects a line with slope 5 at the point $(10,15)$. What is the distance between the $x$-intercepts of these two lines?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope equal to $-1$ and a line with slope equal to $-2$ intersect at the point $P(2,5)$. What is the area of $\\triangle PQR$?"
}
] | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"index": 2161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope equal to $1$ and a line with slope equal to $2$ intersect at the point $P(1,6),$ as shown. [asy]\nunitsize(0.5 cm);\ndraw((0,-1)--(0,10),EndArrow);\ndraw((-10,0)--(5,0),EndArrow);\ndraw((-6,-1)--(5,10),linewidth(0.8));\ndraw((-2.5,-1)--(3,10),linewidth(0.8));\... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 2162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line with slope of $-2$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(8,0)$ and the $y$-axis at $D$. The lines intersect at $E(4,4)$. What is the area of the shaded quadrilateral $OBEC$? [asy]\ndraw((0,-1... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 2163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A linear function \\( f(x) \\) is given. It is known that the distance between the points of intersection of the graphs \\( y = x^{2} \\) and \\( y = f(x) \\) is \\( 2 \\sqrt{3} \\), and the distance between the points of intersection of the graphs \\( y = x^{2}-2 \\) and \\( y... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"index": 2164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A liquid $Y$ which does not mix with water spreads out on the surface to form a circular film $0.15$ cm thick. If liquid $Y$ is poured from a rectangular holder measuring $10$ cm by $4$ cm by $8$ cm onto a large water surface, what will be the radius in centimeters of the forne... | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{2133.33}{\\pi}}"
} | {
"index": 2165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 2166,
"solution": "To find the least number of distinct values in the list, we need to consider how the integers are distributed among these values. Given that the mode occurs exactly 10 times, we want to minimize the number of distinct values while ensuring that no other number appears 10 or more times,... |
DeepScaleR | [
{
"role": "user",
"content": "A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. Determine the least number of distinct values that can occur in the list."
}
] | math | {
"style": "rule",
"ground_truth": "203"
} | {
"index": 2167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list."
}
] | math | {
"style": "rule",
"ground_truth": "218"
} | {
"index": 2168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 2169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element?"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 2170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of five positive integers has all of the following properties:\n\n$\\bullet$ The only integer in the list that occurs more than once is $8,$\n\n$\\bullet$ its median is $9,$ and\n\n$\\bullet$ its average (mean) is $10.$\n\nWhat is the largest possible integer that cou... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2172,
"solution": "1. **Define Variables:**\n Let the integers in the list be $a_1, a_2, a_3, a_4, a_5$ such that $a_1 \\leq a_2 \\leq a_3 \\leq a_4 \\leq a_5$. Given that the mode and median are both $8$, we know $a_3 = 8$. Since the mode is $8$, at least one other number must be $8$. Without loss of ... |
DeepScaleR | [
{
"role": "user",
"content": "A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replac... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2173,
"solution": "1. **Understanding the problem and setting up equations:**\n Let the number of integers in the list be $n$. The list has a mean of $22$, so the sum of all integers in the list is $22n$.\n\n2. **Analyzing the effect of replacing $m$ with $m+10$:**\n When $m$ is replaced by $m+10$, t... |
DeepScaleR | [
{
"role": "user",
"content": "A list of seven positive integers has a median of 4 and a mean of 10. What is the maximum possible value of the list's largest element?"
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 2174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?"
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 2175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 2176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all thre... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 2177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. ... | math | {
"style": "rule",
"ground_truth": "4100"
} | {
"index": 2178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.\nA) $960\nB) $984\nC) $1008\nD) $1032"
}
] | math | {
"style": "rule",
"ground_truth": "984"
} | {
"index": 2179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A local government intends to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between 500,000 and 5,000,000 RMB. The reward scheme follows these principles: The bonus amount $y$ (in ten thousand RMB) increases wi... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 2180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the c... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 2181,
"solution": "We want to count the number of four-digit codes $abcd$ that satisfy the given rules. From the first rule, at least one of the digits must be 4, but $b \\neq 4$ and $d \\neq 4$. Therefore, either $a=4$ or $c=4$. The fourth rule tells us that we could have both $a=4$ and $c=4$. Suppose t... |
DeepScaleR | [
{
"role": "user",
"content": "A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles w... | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 2182,
"solution": "1. **Understanding the Problem:**\n The paper is wrapped around a cardboard tube, forming concentric circles. Each wrap increases the diameter of the roll by the thickness of the paper, which is $5$ cm. The total number of wraps is $600$, and the final diameter of the roll is $10$ cm... |
DeepScaleR | [
{
"role": "user",
"content": "A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is... | math | {
"style": "rule",
"ground_truth": "593"
} | {
"index": 2183,
"solution": "We can keep track of the position of the square labeled 942 in each step. We use an $(x,y)$ coordinate system, so originally the 942 square is in the position $(942,1)$. In general, suppose that we've folded the strip into an array $r=2^k$ squares wide and $c=1024/r=2^{10-k}$ squares t... |
DeepScaleR | [
{
"role": "user",
"content": "A loonie is a $\\$ 1$ coin and a dime is a $\\$ 0.10$ coin. One loonie has the same mass as 4 dimes. A bag of dimes has the same mass as a bag of loonies. The coins in the bag of loonies are worth $\\$ 400$ in total. How much are the coins in the bag of dimes worth?"
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 2184,
"solution": "Since the coins in the bag of loonies are worth $\\$ 400$, then there are 400 coins in the bag. Since 1 loonie has the same mass as 4 dimes, then 400 loonies have the same mass as $4(400)$ or 1600 dimes. Therefore, the bag of dimes contains 1600 dimes, and so the coins in this bag are ... |
DeepScaleR | [
{
"role": "user",
"content": "A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of t... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 6 that is not a lucky integer?"
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 2186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 2187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\\log 101$ and $\\log 2018$ . The last side of his triangle has side length $\\log n$ , where $n$ is an integer. How many possible values are there for $n$ ?\n\n*2018 ... | math | {
"style": "rule",
"ground_truth": "203797"
} | {
"index": 2188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of de... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?\n(a) 35\n(b) 36\n(c) ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 2190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the cent... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 2191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the cent... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 2192,
"solution": "We use coordinates. Let the circle have center $(0,0)$ and radius $\\sqrt{50}$; this circle has equation $x^2 + y^2 = 50$. Let the coordinates of $B$ be $(a,b)$. We want to find $a^2 + b^2$. $A$ and $C$ with coordinates $(a,b+6)$ and $(a+2,b)$, respectively, both lie on the circle. Fro... |
DeepScaleR | [
{
"role": "user",
"content": "A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 2193,
"solution": "To solve this problem, we need to determine the probability of correctly guessing the match between each celebrity and their corresponding baby picture.\n\n1. **Total Possible Matches**:\n There are three celebrities and each has one corresponding baby picture. The task is to match e... |
DeepScaleR | [
{
"role": "user",
"content": "A magic square is an array of numbers in which the sum of the numbers in each row, in each column, and along the two main diagonals are equal. The numbers in the magic square shown are not written in base 10. For what base will this be a magic square?\n\n[asy]\nunitsize(0.75cm)... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A magician and their assistant are planning to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. Their task is to guess both of the covered digi... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 2195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in wh... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 2196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A magician has $5432_{9}$ tricks for his magical performances. How many tricks are there in base 10?"
}
] | math | {
"style": "rule",
"ground_truth": "3998"
} | {
"index": 2197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those car... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2198,
"solution": "\nGiven the problem, let's denote the three boxes as \\( R \\) (red), \\( W \\) (white), and \\( B \\) (blue). Each box must contain at least one card, and the numbers on the cards range from 1 to 100. The magician must be able to determine the box from which no card has been drawn usi... |
DeepScaleR | [
{
"role": "user",
"content": "A magician writes the numbers 1 to 16 on 16 positions of a spinning wheel. Four audience members, A, B, C, and D, participate in the magic show. The magician closes his eyes, and then A selects a number from the wheel. B, C, and D, in that order, each choose the next number in ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 2199,
"solution": ""
} |
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