data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The letter T is formed by placing two $3\\:\\text{inch}\\!\\times\\!5\\:\\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches?... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letters $A$, $B$, $C$ and $D$ represent digits. If $\\begin{array}{ccc}&A&B\\\\ +&C&A\\\\ \\hline &D&A\\end{array}$and $\\begin{array}{ccc}&A&B\\\\ -&C&A\\\\ \\hline &&A\\end{array}$,what digit does $D$ represent?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33001,
"solution": "1. **Analyze the first equation:**\n Given the addition problem:\n \\[\n \\begin{array}{cc}\n & A\\ B \\\\\n + & C\\ A \\\\\n \\hline\n & D\\ A \\\\\n \\end{array}\n \\]\n From the units column, we have $B + A = A$ or $B + A = A + 10$ (considering a possible ca... |
DeepScaleR | [
{
"role": "user",
"content": "The letters $\\text{A}$, $\\text{J}$, $\\text{H}$, $\\text{S}$, $\\text{M}$, $\\text{E}$ and the digits $1$, $9$, $8$, $9$ are \"cycled\" separately as follows and put together in a numbered list:\n\\[\\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 989... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33002,
"solution": "1. **Identify the cycling pattern for letters and digits:**\n - The letters $\\text{AJHSME}$ are cycled by moving the first letter to the end. This cycle repeats every $6$ positions because there are $6$ letters.\n - The digits $1989$ are cycled by moving the first digit to the en... |
DeepScaleR | [
{
"role": "user",
"content": "The letters L, K, R, F, O and the digits 1, 7, 8, 9 are \"cycled\" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letters \\( A, J, H, S, M, E \\) and the numbers \\( 1, 9, 8, 9 \\) are \"rotated\" as follows:\n\\begin{tabular}{rrr} \nAJHSME & 1989 & \\\\\n1. JHSMEA & 9891 & (1st rotation) \\\\\n2. HSMEAJ & 8919 & (2nd rotation) \\\\\n3. SMEAJH & 9198 & (3rd rotation) \\\\\n..... & &\n... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letters of the alphabet are each assigned a random integer value, and $H=10$. The value of a word comes from the sum of its letters' values. If $MATH$ is 35 points, $TEAM$ is 42 points and $MEET$ is 38 points, what is the value of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 33005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letters of the alphabet are given numeric values based on the two conditions below.\n\n$\\bullet$ Only the numeric values of $-2,$ $-1,$ $0,$ $1$ and $2$ are used.\n\n$\\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to t... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letters of the word 'GAUSS' and the digits in the number '1998' are each cycled separately. If the pattern continues in this way, what number will appear in front of 'GAUSS 1998'?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lighting power increased by \n\\[ \\Delta N = N_{\\text {after}} - N_{\\text {before}} = 300\\, \\text{BT} - 240\\, \\text{BT} = 60\\, \\text{BT} \\]"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 33008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The limit of $\\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33009,
"solution": "To find the limit of the function $\\frac{x^2-1}{x-1}$ as $x$ approaches 1, we start by simplifying the expression.\n\n1. **Factorize the Numerator**: \n The numerator $x^2 - 1$ is a difference of squares, which can be factored as:\n \\[\n x^2 - 1 = (x + 1)(x - 1)\n \\]\n\n2. ... |
DeepScaleR | [
{
"role": "user",
"content": "The limit of the sum of an infinite number of terms in a geometric progression is $\\frac {a}{1 - r}$ where $a$ denotes the first term and $- 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {a^2}{1 - r^2}"
} | {
"index": 33010,
"solution": "1. **Identify the Original Series**: \n The original geometric series is given by the terms $a, ar, ar^2, ar^3, \\ldots$. Here, $a$ is the first term and $r$ is the common ratio.\n\n2. **Form the Series of Squares**: \n Squaring each term of the original series, we get the seque... |
DeepScaleR | [
{
"role": "user",
"content": "The limiting sum of the infinite series, $\\frac{1}{10} + \\frac{2}{10^2} + \\frac{3}{10^3} + \\dots$ whose $n$th term is $\\frac{n}{10^n}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{81}"
} | {
"index": 33011,
"solution": "To find the sum of the series $\\frac{1}{10} + \\frac{2}{10^2} + \\frac{3}{10^3} + \\dots$, where the $n$th term is $\\frac{n}{10^n}$, we can use a technique that involves rewriting the series in a more manageable form.\n\n1. **Rewrite the series**: Notice that each term in the series... |
DeepScaleR | [
{
"role": "user",
"content": "The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\nA) $\\frac{18\\sqrt{41} + 40}{\\sqrt{41}}$\nB) $\\frac{360}{17}$\nC) $\\frac{107}{5}$\nD) $\\frac{43}{2}$\nE) $\\frac{281}{13}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18\\sqrt{41} + 40}{\\sqrt{41}}"
} | {
"index": 33012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{281}{13}"
} | {
"index": 33013,
"solution": "1. **Identify the triangle formed by the line and the axes**: The line $12x + 5y = 60$ intersects the x-axis and y-axis, forming a right triangle with the axes. To find the intercepts:\n - **x-intercept**: Set $y = 0$ in the equation $12x + 5y = 60$:\n \\[\n 12x = 60 \\impli... |
DeepScaleR | [
{
"role": "user",
"content": "The line $2x+3y-6=0$ intersects the $x$-axis and $y$-axis at points A and B, respectively. Point P is on the line $y=-x-1$. The minimum value of $|PA|+|PB|$ is ________."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{37}"
} | {
"index": 33014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $2x+ay-2=0$ is parallel to the line $ax+(a+4)y-4=0$. Find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 33015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $4kx-4y-k=0$ intersects the parabola $y^{2}=x$ at points $A$ and $B$. If the distance between $A$ and $B$ is $|AB|=4$, determine the distance from the midpoint of chord $AB$ to the line $x+\\frac{1}{2}=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 33016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 33017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $\\sqrt{2}ax+by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\\triangle AOB$ is a right-angled triangle (where $O$ is the origin). The maximum distance between point $P(a,b)$ and point $(0,1)$ is ______."
}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 33018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $ax+(a+1)y=a+2$ passes through the point $(4,-8)$. Find $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 33019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 33020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $l: x - 2y + 2 = 0$ passes through the left focus F<sub>1</sub> and a vertex B of an ellipse. Find the eccentricity of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 33021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $l_{1}$: $x+my+6=0$ is parallel to the line $l_{2}$: $(m-2)x+3y+2m=0$. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $l_{1}: x+a^{2}y+6=0$ and the line $l_{2}: (a-2)x+3ay+2a=0$ are parallel, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $x = k$ intersects the graph of the parabola $x = -2y^2 - 3y + 5$ at exactly one point. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{8}"
} | {
"index": 33025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $x+2ay-1=0$ is parallel to $\\left(a-1\\right)x+ay+1=0$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 33026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y - x \\sqrt{3} + 3 = 0$ intersects the parabola $2y^2 = 2x + 3$ at points $A$ and $B.$ Let $P = (\\sqrt{3},0).$ Find $|AP - BP|.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 33027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = -\\frac{2}{3}x + 6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\\triangle POQ$ is four times the area of $\\triangle TOP$, then what is the value of $r+s$?"
}
] | math | {
"style": "rule",
"ground_truth": "8.25"
} | {
"index": 33028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\\frac{\\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{22}}{3}"
} | {
"index": 33029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 2x + 7$ is to be parameterized using vectors. Which of the following options are valid parameterizations?\n\n(A) $\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}$\n\n(B) $\\begin{pma... | math | {
"style": "rule",
"ground_truth": "\\text{B,E}"
} | {
"index": 33030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 2x + c$ is tangent to the parabola $y^2 = 8x.$ Find $c.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\\frac{\\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$."
... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 33032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 3x - 11$ is parameterized by the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} r \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 4 \\\\ k \\end{pmatrix}.\\]Enter the ordered pair $(r,k).$"
}
] | math | {
"style": "rule",
"ground_truth": "(4,12)"
} | {
"index": 33033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = 5$ intersects the graph of $y = 3x^2 + 2x - 2$ at points $C$ and $D$. Find the distance between $C$ and $D$ and express it in the form $\\frac{\\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. Find the value of $p - q$."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 33034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = \\frac{1}{2} x + 4$ is paramterized by the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -7 \\\\ s \\end{pmatrix} + t \\begin{pmatrix} l \\\\ -5 \\end{pmatrix}.\\]Enter the ordered pair $(s,l).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{2}, -10 \\right)"
} | {
"index": 33035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"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 between $\\begin{pmatrix} x \\\\ y \\end{pmatrix}$ and $\\begin{pmatrix} 3 \\\\ 1 \\end{pmatrix}$ is ... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4/5 \\\\ 3/5 \\end{pmatrix}"
} | {
"index": 33036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = \\frac{3}{2} x - 25$ is parameterized by $(x,y) = (f(t),15t - 7).$ Enter the function $f(t).$"
}
] | math | {
"style": "rule",
"ground_truth": "10t + 12"
} | {
"index": 33037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = \\frac{5}{3} x - \\frac{17}{3}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?\n\n(A) $\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} -3 \\\\ -5 \\end{pma... | math | {
"style": "rule",
"ground_truth": "\\text{A,C}"
} | {
"index": 33038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = a$ intersects the curves $y = 2(x + 1)$ and $y = x + \\ln x$ at points $A$ and $B$, respectively. Find the minimum value of $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 33039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,-3)-... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 33040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y=-\\frac{3}{4}x+9$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\\triangle POQ$ is three times the area of $\\triangle TOP$, then what is the value of $r+s$? [asy]\n\n// draw axes\ndraw((-3, 0)--(15, 0... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 33041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y=-\\frac{5}{3}x+15$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on the line segment $PQ$. If the area of $\\triangle POQ$ is twice the area of $\\triangle TOP$, what is the value of $r+s$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y=2b$ intersects the left and right branches of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ at points $B$ and $C$ respectively, with $A$ being the right vertex and $O$ the origin. If $\\angle AOC = \\angle BOC$, then calculate the eccentr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{19}}{2}"
} | {
"index": 33043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y=kx$ intersects the graph of the function $y=\\tan x$ ($-\\frac{π}{2}＜x＜\\frac{π}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\\frac{π}{2},0)$. Find $(\\overrightarrow{AM}+\\overrightarrow{AN})\\cdot\\overright... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi^2}{2}"
} | {
"index": 33044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line $y=kx+b$ is tangent to the curve $y=x^2+\\frac{a}{x}+1$ at the point $(2,3)$. Find the value of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 33045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line RS does not intersect with △ABC. Perpendiculars are drawn from the vertices A, B, and C of △ABC to line RS, and the corresponding feet of the perpendiculars are D, E, and F respectively. It is given that AD = 10, BE = 6, and CF = 24. Let H be the foot of the perpendicu... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"index": 33046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line \\( L \\) crosses the \\( x \\)-axis at \\((-8,0)\\). The area of the shaded region is 16. Find the slope of the line \\( L \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 33047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line \\( l: (2m+1)x + (m+1)y - 7m - 4 = 0 \\) intersects the circle \\( C: (x-1)^{2} + (y-2)^{2} = 25 \\) to form the shortest chord length of \\(\\qquad \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{5}"
} | {
"index": 33048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line \\(y = -\\frac{1}{2}x + 8\\) crosses the \\(x\\)-axis at \\(P\\) and the \\(y\\)-axis at \\(Q\\). Point \\(T(r, s)\\) is on line segment \\(PQ\\). If the area of \\(\\triangle POQ\\) is twice the area of \\(\\triangle TOP\\), then what is the value of \\(r+s\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line \\(y = -\\frac{2}{3}x + 10\\) crosses the \\(x\\)-axis at point \\(P\\) and the \\(y\\)-axis at point \\(Q\\). Point \\(T(r, s)\\) is on the segment \\(PQ\\). If the area of \\(\\triangle POQ\\) is four times the area of \\(\\triangle TOP\\), find the value of \\(r + s... | math | {
"style": "rule",
"ground_truth": "13.75"
} | {
"index": 33050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line \\(y = M\\) intersects the graph of the function \\(y = x^{3} - 84x\\) at points with abscissas \\(a\\), \\(b\\), and \\(c\\) (\\(a < b < c\\)). It is given that the distance between \\(a\\) and \\(b\\) is half the distance between \\(b\\) and \\(c\\). Find \\(M\\)."
... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 33051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 33052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.\n\n[asy]\ndraw((-2,0)--(7,0),linewidth(1),Arrows);\ndraw((0,-1)--(0,4),linewidth(1),Arrows);\n\ndr... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 33053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line joining $(4,3)$ and $(7,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Assume the square has vertices at $(4,0)$, $(7,0)$, $(7,3)$, and $(4,3)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 33054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97,$ then the shorter base is:"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 33055,
"solution": "1. **Identify the properties of the trapezoid**: In a trapezoid, the line segment joining the midpoints of the diagonals is parallel to the bases and its length is equal to half the difference of the lengths of the bases.\n\n2. **Set up the equation**: Let $x$ be the length of the sho... |
DeepScaleR | [
{
"role": "user",
"content": "The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\\triangle OAB$, where $O$ is the origin."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 33056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line passing through the points (3, 9) and (-1, 1) intersects the x-axis at a point whose x-coordinate is $\\frac{9-1}{3-(-1)}$"
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{3}{2}"
} | {
"index": 33057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line passing through the points (3,9) and (-1,1) has an x-intercept of ( )."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 33058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line segment connecting $(-2,7)$ and $(3,11)$ can be parameterized by the equations\n\\begin{align*}\nx &= at + b, \\\\\ny &= ct + d,\n\\end{align*}where $0 \\le t \\le 1,$ and $t = 0$ corresponds to the point $(-2,7).$ Find $a^2 + b^2 + c^2 + d^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 33059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line segment connecting the focus F of the parabola $y^2=4x$ and the point M(0,1) intersects the parabola at point A. Let O be the origin, then the area of △OAM is _____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2} - \\sqrt {2}"
} | {
"index": 33060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line with equation $y = 3x + 5$ is translated 2 units to the right. What is the equation of the resulting line?"
}
] | math | {
"style": "rule",
"ground_truth": "y = 3x - 1"
} | {
"index": 33061,
"solution": "The line with equation $y = 3x + 5$ has slope 3 and $y$-intercept 5. Since the line has $y$-intercept 5, it passes through $(0, 5)$. When the line is translated 2 units to the right, its slope does not change and the new line passes through $(2, 5)$. A line with slope $m$ that passes ... |
DeepScaleR | [
{
"role": "user",
"content": "The line with equation $y = x$ is an axis of symmetry of the curve with equation\n\\[y = \\frac{px + q}{rx + s},\\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?\n\n(A) $p + q = 0$\n\n(B) $p + r = 0$\n\n(C) $p + s = 0$\n\n(D) $q + r = 0$... | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"index": 33062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The line with equation $y = x$ is translated 3 units to the right and 2 units down. What is the $y$-intercept of the resulting line?"
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 33063,
"solution": "The line with equation $y = x$ has slope 1 and passes through $(0,0)$. When this line is translated, its slope does not change. When this line is translated 3 units to the right and 2 units down, every point on the line is translated 3 units to the right and 2 units down. Thus, the po... |
DeepScaleR | [
{
"role": "user",
"content": "The line with equation $y=2x-6$ is translated upwards by 4 units. What is the $x$-intercept of the resulting line?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33064,
"solution": "The line with equation $y=2 x-6$ has slope 2. When this line is translated, the slope does not change. The line with equation $y=2 x-6$ has $y$-intercept -6. When this line is translated upwards by 4 units, its $y$-intercept is translated upwards by 4 units and so becomes -2. This mea... |
DeepScaleR | [
{
"role": "user",
"content": "The line with equation $y=3x+6$ is reflected in the $y$-axis. What is the $x$-intercept of the new line?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33065,
"solution": "When a line is reflected in the $y$-axis, its $y$-intercept does not change (since it is on the line of reflection) and its slope is multiplied by -1 . Therefore, the new line has slope -3 and $y$-intercept 6 , which means that its equation is $y=-3 x+6$. The $x$-intercept of this new... |
DeepScaleR | [
{
"role": "user",
"content": "The lines $-2x + y = k$ and $0.5x + y = 14$ intersect when $x = -8.4$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 33066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 33067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines $x = \\frac{1}{4}y + a$ and $y = \\frac{1}{4}x + b$ intersect at the point $(1,2)$. What is $a + b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 33068,
"solution": "1. **Given Equations and Intersection Point**: We are given the equations of two lines:\n - $x = \\frac{1}{4}y + a$\n - $y = \\frac{1}{4}x + b$\n These lines intersect at the point $(1,2)$.\n\n2. **Substitute the Intersection Point into Each Equation**:\n - Substituting $x = 1... |
DeepScaleR | [
{
"role": "user",
"content": "The lines $x=\\frac{1}{4}y+a$ and $y=\\frac{1}{4}x+b$ intersect at the point $(1,2)$. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 33069,
"solution": "1. **Substitute $(1,2)$ into the first equation:** \n Given the equation $x = \\frac{1}{4}y + a$, substitute $x = 1$ and $y = 2$:\n \\[\n 1 = \\frac{1}{4} \\cdot 2 + a\n \\]\n Simplify the equation:\n \\[\n 1 = \\frac{1}{2} + a\n \\]\n Solve for $a$:\n \\[\n a =... |
DeepScaleR | [
{
"role": "user",
"content": "The lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square. Find the product of the possible values for $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 33070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines $y=5x+3$, $y=-2x-25$ and $y=3x+k$ intersect at the same point. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 33071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines $y=x, y=2 x$, and $y=3 x$ are the three medians of a triangle with perimeter 1. Find the length of the longest side of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{\\sqrt{58}}{2+\\sqrt{34}+\\sqrt{58}}}"
} | {
"index": 33072,
"solution": "The three medians of a triangle contain its vertices, so the three vertices of the triangle are $(a, a),(b, 2 b)$ and $(c, 3 c)$ for some $a, b$, and $c$. Then, the midpoint of $(a, a)$ and $(b, 2 b)$, which is $\\left(\\frac{a+b}{2}, \\frac{a+2 b}{2}\\right)$, must lie along the line... |
DeepScaleR | [
{
"role": "user",
"content": "The lines containing the altitudes of the scalene triangle \\( ABC \\) intersect at point \\( H \\). Let \\( I \\) be the incenter of triangle \\( ABC \\), and \\( O \\) be the circumcenter of triangle \\( BHC \\). It is known that point \\( I \\) lies on the segment \\( OA \\)... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 33073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 33075,
"solution": "1. **Convert to slope-intercept form**: \n The given equations are $ax-2y=c$ and $2x+by=-c$. We can rearrange these to solve for $y$:\n \\[\n y = \\frac{a}{2}x - \\frac{c}{2}\n \\]\n \\[\n y = -\\frac{2}{b}x - \\frac{c}{b}\n \\]\n The slopes of the lines are $\\frac{a}... |
DeepScaleR | [
{
"role": "user",
"content": "The list $p, q, r, s$ consists of four consecutive integers listed in increasing order. If $p + s = 109$, what is the value of $q + r$?"
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 33076,
"solution": "Since $p, q, r, s$ is a list of consecutive integers in increasing order, then $q$ is 1 more than $p$ and $r$ is 1 less than $s$. This means that $q + r = (p + 1) + (s - 1) = p + s = 109$. Therefore, $q + r = 109$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The local cinema has two ticket windows opening simultaneously. In how many ways can eight people line up to buy a ticket if they can choose any of the two windows?"
}
] | math | {
"style": "rule",
"ground_truth": "10321920"
} | {
"index": 33077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{15}"
} | {
"index": 33078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The local library has two service windows. In how many ways can eight people line up to be served if there are two lines, one for each window?"
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"index": 33079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The local theater has one ticket window. In how many ways can six people line up to buy a ticket?"
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 33080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one?"
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"index": 33081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The logarithm of $27\\sqrt[4]{9}\\sqrt[3]{9}$ to the base $3$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$4\\frac{1}{6}$"
} | {
"index": 33082,
"solution": "1. **Rewrite the expression using properties of exponents and logarithms:**\n The given expression is $27\\sqrt[4]{9}\\sqrt[3]{9}$. We know that $27 = 3^3$, $\\sqrt[4]{9} = 9^{1/4}$, and $\\sqrt[3]{9} = 9^{1/3}$. Since $9 = 3^2$, we can rewrite these as:\n \\[\n 27 = 3^3, \\quad... |
DeepScaleR | [
{
"role": "user",
"content": "The longer leg of a right triangle is $1$ foot shorter than twice the length of the shorter leg. The area of the triangle is $60$ square feet. What is the length of the hypotenuse, in feet?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 33083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?"
}
] | math | {
"style": "rule",
"ground_truth": "665"
} | {
"index": 33084,
"solution": "To find the total duration of the tennis match in minutes, we need to convert the hours into minutes and then add the remaining minutes.\n\n1. **Convert hours to minutes**: \n - There are 60 minutes in one hour.\n - Therefore, for 11 hours, the total minutes are:\n \\[\n 1... |
DeepScaleR | [
{
"role": "user",
"content": "The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes."
}
] | math | {
"style": "rule",
"ground_truth": "428"
} | {
"index": 33085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle? Also, determine if the triangle has any acute angles."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The longest side of a right triangle is 5 meters and the shortest side is 3 meters. What is the area of the triangle in square meters?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lottery in our state consists of two drawings. First, a MegaBall is picked from among 27 numbered balls. Second, five WinnerBalls are picked from among 44 numbered balls. To win the lottery, you must pick the MegaBall number correctly and also pick the numbers on the five... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{29,\\!322,\\!216}"
} | {
"index": 33088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lottery now consists of two drawings. First, a PowerBall is picked from among 30 numbered balls. Second, six LuckyBalls are picked from among 49 numbered balls. To win the lottery, you must pick the PowerBall number correctly and also correctly pick the numbers on all six L... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{419,512,480}"
} | {
"index": 33089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The magician has a set of \\(15^{2}\\) different cards. Each card has one red side and one blue side; on each card, there is a natural number from 1 to 15 on both sides. A card is called a \"duplicate\" if the numbers on both sides of the card are the same. The magician wants t... | math | {
"style": "rule",
"ground_truth": "2835"
} | {
"index": 33090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The magnitude of the vector $\\overset{→}{a} +2 \\overset{→}{b}$, where $\\overset{→}{a} =(2,0)$, $\\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 33091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The main structure of the Chinese space station includes the Tianhe core module, the Wentian experiment module, and the Mengtian experiment module. Assuming that the space station needs to arrange 6 astronauts, including astronauts A and B, to conduct experiments, with each mod... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 33092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. The amount of money in th... | math | {
"style": "rule",
"ground_truth": "995"
} | {
"index": 33093,
"solution": "Let $n$ be the number of employees in the company. According to the problem, the manager initially planned to give each employee a $50$ bonus, but the fund was $5$ short. Therefore, the total amount required for the $50$ bonus per employee would be $50n$, and the amount in the fund wa... |
DeepScaleR | [
{
"role": "user",
"content": "The marble statue of Abraham Lincoln in the Lincoln Memorial has a height of 60 feet. A scale model of the statue has a height of 4 inches. How many feet of the statue does one inch of the model represent?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The marching band has more than 100 members but fewer than 200 members. When they line up in rows of 4 there is one extra person; when they line up in rows of 5 there are two extra people; and when they line up in rows of 7 there are three extra people. How many members are in ... | math | {
"style": "rule",
"ground_truth": "157"
} | {
"index": 33095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?"
}
] | math | {
"style": "rule",
"ground_truth": "35\\%"
} | {
"index": 33096,
"solution": "1. **Assume the suggested retail price (SRP) of the book**: Let the suggested retail price of the book be $P$. For simplicity, we can assume $P = 100$ USD.\n\n2. **Calculate the marked price (MP)**: The problem states that the marked price is 30% less than the suggested retail price. ... |
DeepScaleR | [
{
"role": "user",
"content": "The market demand for a certain product over the next four years forms a sequence $\\left\\{a_{n}\\right\\}(n=1,2,3,4)$. It is predicted that the percentage increase in annual demand from the first year to the second year is $p_{1}$, from the second year to the third year is $p... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{3}$"
} | {
"index": 33097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The mass of the first cast iron ball is $1462.5\\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\\frac{4}{3} \\pi R^{3}$, and the surface area of ... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 33098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The math scores of a high school's college entrance examination, denoted as $\\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\\xi < 110) = 0.98$, find the value of $P(90 < \\xi < 100)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.48"
} | {
"index": 33099,
"solution": ""
} |
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