data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{q(x)}{2x^5+x^4-7x^2+1}$ has a horizontal asymptote. What is the largest possible degree of $q(x)$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 32700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest?"
}
] | math | {
"style": "rule",
"ground_truth": "second (1-2)"
} | {
"index": 32701,
"solution": "To determine during which hour the average speed of the airplane was the largest, we need to analyze the slope of the graph of distance versus time. The average speed for any given hour is calculated as the change in distance divided by the change in time (which is 1 hour in this case... |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the birth month of 100 famous Americans. What percent of these people have March as their birth month?\n\n[asy]\ndraw((0,0)--(24,0)--(24,16)--(0,16)--(0,0)--cycle,linewidth(1));\n\nfor(int i = 1; i < 16; ++i)\n{\n\ndraw((-.5,i)--(24,i),gray);\n}\n\nfilldraw((.5,... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32703,
"solution": "1. **Interpreting the Graph**: The problem states that Suzanna rides her bike at a constant rate, and the graph shows that every 5 minutes, her distance increases by 1 mile.\n\n2. **Total Riding Time**: Suzanna rides for a total of half an hour. Half an hour is equivalent to 30 minute... |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32704,
"solution": "To find the median number of children in the families of the students in Ms. Jordan's English class, we follow these steps:\n\n1. **Identify the Total Number of Data Points**: The total number of families (data points) is given as thirteen.\n\n2. **Determine the Position of the Median... |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32705,
"solution": "To find how many more minutes per day on average Sasha studied than Asha, we need to calculate the daily differences in their study times and then find the average of these differences.\n\n1. **Calculate the daily differences**: From the problem, the differences between Sasha's and As... |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 32706,
"solution": "1. **Identify the highest and lowest prices:** From the problem, we know that the highest price of gasoline was $17 in Month 1, and the lowest price was $10 in Month 3.\n\n2. **Calculate the percentage increase from the lowest to the highest price:** To find by what percent the highes... |
DeepScaleR | [
{
"role": "user",
"content": "The graph shows the total distance Sam drove from 6 a.m to 11 a.m. How many miles per hour is the car's average speed for the period from 6 a.m. to 11 a.m.?\n\n[asy]\nunitsize(0.2inch);\ndraw((0,0)--(5.5,0));\ndraw((0,0)--(0,8.5));\ndraw((1,0)--(1,8.5));\ndraw((2,0)--(2... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 32707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$?"
}
] | math | {
"style": "rule",
"ground_truth": "10^{-x}"
} | {
"index": 32708,
"solution": "1. **Understanding the rotation**: Rotating a point $(x, y)$ $90^\\circ$ counterclockwise about the origin results in the point $(-y, x)$. This can be verified using rotation matrices:\n \\[\n \\begin{bmatrix}\n \\cos(90^\\circ) & -\\sin(90^\\circ) \\\\\n \\sin(90^\\circ) & \\... |
DeepScaleR | [
{
"role": "user",
"content": "The graphs $y = 3(x-h)^2 + j$ and $y = 2(x-h)^2 + k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 32709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have y-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer x-intercepts. Find $h$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 32710,
"solution": "Begin by setting $x$ to 0, then set both equations to $h^2=\\frac{2013-j}{3}$ and $h^2=\\frac{2014-k}{2}$, respectively. Notice that because the two parabolas have to have positive x-intercepts, $h\\ge32$.\n We see that $h^2=\\frac{2014-k}{2}$, so we now need to find a positive intege... |
DeepScaleR | [
{
"role": "user",
"content": "The graphs \\( y = 2 \\cos 3x + 1 \\) and \\( y = - \\cos 2x \\) intersect at many points. Two of these points, \\( P \\) and \\( Q \\), have \\( x \\)-coordinates between \\(\\frac{17 \\pi}{4}\\) and \\(\\frac{21 \\pi}{4}\\). The line through \\( P \\) and \\( Q \\) intersects... | math | {
"style": "rule",
"ground_truth": "\\frac{361\\pi}{8}"
} | {
"index": 32711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 32712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $y = -|x-a| + b$ and $y = |x-c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find $a+c$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32713,
"solution": "1. **Identify the vertex of each graph:**\n - The graph of $y = -|x-a| + b$ has a vertex at $(a, b)$, where the graph reaches its maximum value $b$.\n - The graph of $y = |x-c| + d$ has a vertex at $(c, d)$, where the graph reaches its minimum value $d$.\n\n2. **Understand the int... |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $y = x^3 - 3x + 2$ and $x + 4y = 4$ intersect in the points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$ If $x_1 + x_2 + x_3 = A$ and $y_1 + y_2 + y_3 = B,$ compute the ordered pair $(A,B).$"
}
] | math | {
"style": "rule",
"ground_truth": "(0,3)"
} | {
"index": 32714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $y=\\log_3 x$, $y=\\log_x 3$, $y=\\log_{\\frac{1}{3}} x$, and $y=\\log_x \\dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32716,
"solution": "To solve this problem, we need to analyze the intersections of the given logarithmic functions. We start by rewriting each function in terms of a common logarithmic base or expression for easier comparison.\n\n1. **Rewriting the functions:**\n - $y = \\log_3 x$\n - $y = \\log_x 3$... |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of a function $f(x)=3x+b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-3,a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 32718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:\n 1. The domain of function (2) is from $$\\{-6,-5,-4,-3,-2,-1,0,1,2,3\\}.$$ It is graphed as a set of discrete points.\n 2. Function (4) is defined by the equation... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of four functions, labelled (2) through (5), are shown below. Note that the domain of function (3) is $$\\{-5,-4,-3,-2,-1,0,1,2\\}.$$ Find the product of the labels of the functions which are invertible. [asy]\nsize(8cm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 32720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\n are drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\\tfrac{2}{\\sqrt{3}}.\\,$ How many such... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"index": 32721,
"solution": "Picturing the diagram in your head should give you an illustration similar to the one above. The distance from parallel sides of the center hexagon is 20 units, and by extending horizontal lines to the sides of the hexagon. This shows that for every side of the hexagon there are 10 sp... |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\nare drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side $2/\\sqrt{3}.\\,$ How many such triangles are for... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"index": 32722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of the lines $y = \\frac{3}{2}x$ and $y=2x - 1$ intersect and form an acute angle. Find the slope $k$ of the angle bisector of this acute angle that is oriented positively with respect to the x-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7 - \\sqrt{29}}{4}"
} | {
"index": 32723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickd... | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 32724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 400 Newtons. What is the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 32725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Bill is on the surface of Earth, 4,000 miles from the center, the gravitational force is 600 Newtons. What is the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 32726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values?"
}
] | math | {
"style": "rule",
"ground_truth": "15,450"
} | {
"index": 32727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor (GCD) of 17 and 51 is , and the least common multiple (LCM) is . The GCD of 6 and 8 is , and the LCM of 8 and 9 is ."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 32728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 32729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of 21 and some number between 50 and 60 is 7. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 32730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number, if the least common multiple of these two numbers is also between 200 and 300?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 32731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 32732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of natural numbers \\( m \\) and \\( n \\) is 1. What is the greatest possible value of \\(\\text{GCD}(m + 2000n, n + 2000m) ?\\)"
}
] | math | {
"style": "rule",
"ground_truth": "3999999"
} | {
"index": 32733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two integers is $(x+2)$ and their least common multiple is $x(x+2)$, where $x$ is a positive integer. If one of the integers is 24, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 32735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 32736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)$, where $x$ is a positive integer. If one of the integers is 50, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two positive integers is $(x+6)$ and their least common multiple is $x(x+6)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest common divisor of two positive integers is $(x+7)$ and their least common multiple is $x(x+7)$, where $x$ is a positive integer. If one of the integers is 56, what is the smallest possible value of the other one?"
}
] | math | {
"style": "rule",
"ground_truth": "294"
} | {
"index": 32739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest integer function of $x$ is expressed as $[x]$ and is defined to be the greatest integer less than or equal to $x$. Find $[\\pi - 4]$"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest integer function, $\\lfloor x\\rfloor$, denotes the largest integer less than or equal to $x$. For example, $\\lfloor3.5\\rfloor=3$, $\\lfloor\\pi\\rfloor=3$ and $\\lfloor -\\pi\\rfloor=-4$. Find the sum of the three smallest positive solutions to $x-\\lfloor x\\rf... | math | {
"style": "rule",
"ground_truth": "10\\frac{1}{12}"
} | {
"index": 32741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 32742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32743,
"solution": "1. **Identify the number to factorize**: We start with the number $16{,}383$. We know that $16{,}384 = 2^{14}$, so $16{,}383 = 2^{14} - 1$.\n\n2. **Factorize $16{,}383$ using difference of squares**:\n \\[\n 16{,}383 = 2^{14} - 1 = (2^7)^2 - 1^2 = (2^7 + 1)(2^7 - 1) = 129 \\cdot 1... |
DeepScaleR | [
{
"role": "user",
"content": "The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 32744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"index": 32746,
"solution": "1. **Calculate the reciprocals of the numbers**: \n Given numbers are 1, 2, and 4. Their reciprocals are:\n \\[\n \\frac{1}{1}, \\frac{1}{2}, \\text{ and } \\frac{1}{4}\n \\]\n\n2. **Sum the reciprocals**:\n \\[\n \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{4} = 1 + 0.5 + 0.25... |
DeepScaleR | [
{
"role": "user",
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x \\neq y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 32747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?"
}
] | math | {
"style": "rule",
"ground_truth": "799"
} | {
"index": 32748,
"solution": "The harmonic mean of $x$ and $y$ is equal to $\\frac{1}{\\frac{\\frac{1}{x}+\\frac{1}{y}}2} = \\frac{2xy}{x+y}$, so we have $xy=(x+y)(3^{20}\\cdot2^{19})$, and by SFFT, $(x-3^{20}\\cdot2^{19})(y-3^{20}\\cdot2^{19})=3^{40}\\cdot2^{38}$. Now, $3^{40}\\cdot2^{38}$ has $41\\cdot39=1599$ f... |
DeepScaleR | [
{
"role": "user",
"content": "The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 32749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height $BL$ of the rhombus $ABCD$, dropped perpendicular to the side $AD$, intersects the diagonal $AC$ at point $E$. Find $AE$ if $BL = 8$ and $AL:LD = 3:2$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"index": 32750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height \\( PO \\) of the regular quadrilateral pyramid \\( PABC D \\) is 4, and the side of the base \\( ABCD \\) is 6. Points \\( M \\) and \\( K \\) are the midpoints of segments \\( BC \\) and \\( CD \\). Find the radius of the sphere inscribed in the pyramid \\( PMKC \\... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13+\\sqrt{41}}"
} | {
"index": 32751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height \\(CH\\), dropped from the vertex of the right angle of the triangle \\(ABC\\), bisects the angle bisector \\(BL\\) of this triangle. Find the angle \\(BAC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 32752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a ball when it is thrown off a cliff can be represented by the equation $h=45-7t-6t^2$, where $t$ is time in seconds. In how many seconds will the ball reach a height of 25 feet?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac43"
} | {
"index": 32753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1152}{125} \\pi"
} | {
"index": 32754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{15}}{16}"
} | {
"index": 32755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{15}}{16}"
} | {
"index": 32756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a right triangle $ABC$ dropped to the hypotenuse is 9.6. From the vertex $C$ of the right angle, a perpendicular $CM$ is raised to the plane of the triangle $ABC$, with $CM = 28$. Find the distance from point $M$ to the hypotenuse $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "29.6"
} | {
"index": 32757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a right truncated quadrilateral pyramid is 3 cm, its volume is 38 cm³, and the areas of its bases are in the ratio 4:9. Determine the lateral surface area of the truncated pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\sqrt{19}"
} | {
"index": 32758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 32759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"index": 32760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "11/3"
} | {
"index": 32761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of an isosceles trapezoid is \\( h \\). The upper base of the trapezoid is viewed from the midpoint of the lower base at an angle of \\( 2\\alpha \\), and the lower base is viewed from the midpoint of the upper base at an angle of \\( 2\\beta \\). Find the area of th... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 32762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of the pyramid $P-ABCD$ with a square base of side length $2\\sqrt{2}$ is $1$. If the radius of the circumscribed sphere of the pyramid is $2\\sqrt{2}$, then the distance between the center of the square $ABCD$ and the point $P$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 32763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The height of trapezoid $ABCD$ is 5, and the bases $BC$ and $AD$ are 3 and 5 respectively. Point $E$ is on side $BC$ such that $BE=2$. Point $F$ is the midpoint of side $CD$, and $M$ is the intersection point of segments $AE$ and $BF$. Find the area of quadrilateral $AMFD$."
... | math | {
"style": "rule",
"ground_truth": "12.25"
} | {
"index": 32764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hexagon with the R is colored red. Each hexagon is colored either red, yellow or green, such that no two hexagons with a common side are colored the same color. In how many different ways can the figure be colored?\n\n[asy]\npath a=(0,0)--(10,0)--(15,8.7)--(10,17.3)--(0... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The high-speed train \"Sapsan,\" approaching a railway station at a speed of \\( v = 216 \\) km/h, emits a warning sound signal lasting \\( \\Delta t = 5 \\) seconds when it is half a kilometer away from the station. What will be the duration of the signal \\( \\Delta t_{1} \\)... | math | {
"style": "rule",
"ground_truth": "4.12"
} | {
"index": 32766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The highest power of 2 that is a factor of \\(15.13^{4} - 11^{4}\\) needs to be determined."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 32767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The highest temperatures from April 1st to April 6th in a certain region were 28℃, 21℃, 22℃, 26℃, 28℃, and 25℃, respectively. Calculate the variance of the highest temperature data for these six days."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{22}{3}"
} | {
"index": 32768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32769,
"solution": "To solve this problem, we need to first understand the positions of points $A$, $B$, $C$, and $D$ on the grid. However, the problem statement does not provide specific coordinates for these points, and the solution provided seems to assume a specific configuration without describing i... |
DeepScaleR | [
{
"role": "user",
"content": "The horse walks a distance that is half of the previous day's distance each day, and after walking for 7 days, it has covered a total distance of 700 li. Calculate the total distance it will cover from the 8th day to the 14th day."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {175}{32}"
} | {
"index": 32770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the m... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 32771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.\n\n*Pro... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 32772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \\ldots$), and that the sum of the house numbers from the beginning of the street t... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 32773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 32774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street? \n\nA) 41 \nB... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 32775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32776,
"solution": "1. **Define the digits of the number**: Let the hundreds, tens, and units digits of the original three-digit number be $a$, $b$, and $c$, respectively.\n\n2. **Use the given relationship**: We know that the hundreds digit $a$ is $2$ more than the units digit $c$. Therefore, we can exp... |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola $C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ has an asymptote that is perpendicular to the line $x + 2y + 1 = 0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola such that $|F_1A| = 2|F_2A|$. Find $\\cos \\angle AF... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 32777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\\triangle A P F$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2} + \\sqrt{10}"
} | {
"index": 32778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola $C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ has an asymptote perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola. If $|F_1A|=2|F_2A|$, then $\\cos \\angle AF_2F_1=$ __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{5}"
} | {
"index": 32779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola $M$: $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has left and right foci $F_l$ and $F_2$. The parabola $N$: $y^{2} = 2px (p > 0)$ has a focus at $F_2$. Point $P$ is an intersection point of hyperbola $M$ and parabola $N$. If the midpoint of $P... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 32780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 32781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola C: $\\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) has one focus that is also the focus of the parabola $y^{2} = 4x$. Line $l$ is an asymptote of $C$ and intersects the circle $(x-1)^{2} + y^{2} = a^{2}$ at points $A$ and $B$. If $|AB| = b$, d... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{5}}{5}"
} | {
"index": 32782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola \\[-x^2+2y^2-10x-16y+1=0\\]has two foci. Find the coordinates of either of them. (Enter your answer as an ordered pair. Enter only one of the foci, not both.)"
}
] | math | {
"style": "rule",
"ground_truth": "(-5, 1)"
} | {
"index": 32783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola \\[\\frac{(x-3)^2}{5^2} - \\frac{(y-17)^2}{12^2} = 1\\]has two foci, which have different $x-$coordinates. Find the coordinates of the one with the larger $x-$coordinate."
}
] | math | {
"style": "rule",
"ground_truth": "(16,17)"
} | {
"index": 32784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola given by the equation \\[\\frac{x^2}{100} - \\frac{y^2}{64} = 1\\]has asymptotes $y = \\pm mx,$ where $m$ is positive. Find $m.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac45"
} | {
"index": 32785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbola given by the equation \\[\\frac{y^2}{9}-\\frac{x^2}{4} = 1\\]has asymptotes $y = \\pm mx,$ where $m$ is positive. Find $m.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac32"
} | {
"index": 32786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hyperbolas \\[\\frac{x^2}{4} - \\frac{y^2}{9} = 1\\]and \\[\\frac{y^2}{18} - \\frac{x^2}{N} = 1\\]have the same asymptotes. Find $N.$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is:"
}
] | math | {
"style": "rule",
"ground_truth": "c+a"
} | {
"index": 32788,
"solution": "1. Let's denote the sides of the right triangle as $a$, $b$, and $c$, where $c$ is the hypotenuse and $a$ and $b$ are the legs. According to the Pythagorean theorem, we have:\n \\[\n a^2 + b^2 = c^2\n \\]\n\n2. Given that $c$ and $a$ are consecutive integers, we can express $c$ ... |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is:"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle measures $6\\sqrt{2}$ inches and one angle is $45^{\\circ}$. What is the number of square inches in the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 32790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle measures $8\\sqrt{2}$ inches and one angle is $45^{\\circ}$. Calculate both the area and the perimeter of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "16 + 8\\sqrt{2}"
} | {
"index": 32791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle measures $9$ inches, and one angle is $30^{\\circ}$. What is the number of square inches in the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "10.125\\sqrt{3}"
} | {
"index": 32792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle measures 10 inches and one angle is $45^{\\circ}$. What is the number of square inches in the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 32793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs?"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 32794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 32795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle whose legs are consecutive whole numbers is 29 units. What is the sum of the lengths of the two legs?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 32796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs?"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 32797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The hypotenuse of an isosceles right triangle is $4\\sqrt{2}$ units. How many square units are in the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the nu... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 32799,
"solution": ""
} |
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