data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The set $S=\\{1,2,3, \\ldots, 49,50\\}$ contains the first 50 positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34200,
"solution": "The set $S$ contains 25 multiples of 2 (that is, even numbers). When these are removed, the set $S$ is left with only the odd integers from 1 to 49. At this point, there are $50-25=25$ integers in $S$. We still need to remove the multiples of 3 from $S$. Since $S$ only contains odd in... |
DeepScaleR | [
{
"role": "user",
"content": "The set $T = \\{1, 2, 3, \\ldots, 59, 60\\}$ contains the first 60 positive integers. After the multiples of 2, the multiples of 3, and multiples of 5 are removed, how many integers remain in the set $T$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set $\\{3,6,9,10\\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 34202,
"solution": "We are given the set $\\{3,6,9,10\\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal.\n\n#### Case 1: Median is $6$\nFor $6$ to be the medi... |
DeepScaleR | [
{
"role": "user",
"content": "The set $\\{[x] + [2x] + [3x] \\mid x \\in \\mathbb{R}\\} \\mid \\{x \\mid 1 \\leq x \\leq 100, x \\in \\mathbb{Z}\\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 34203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set $\\{[x]+[2x]+[3x] \\mid x \\in \\mathbf{R}\\} \\bigcap \\{1, 2, \\cdots, 100\\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 34204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set \\( A = \\left\\{ x \\left\\lvert\\, x = \\left\\lfloor \\frac{5k}{6} \\right\\rfloor \\right., k \\in \\mathbb{Z}, 100 \\leqslant k \\leqslant 999 \\right\\} \\), where \\(\\left\\lfloor x \\right\\rfloor\\) denotes the greatest integer less than or equal to \\( x \\).... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 34205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set \\( M = \\left\\{(x, y) \\mid \\log_{4} x + \\log_{4} y \\leq 1, x, y \\in \\mathbf{N}^{*}\\right\\} \\) has how many subsets?"
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 34206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set \\( S \\) is given by \\( S = \\{1, 2, 3, 4, 5, 6\\} \\). A non-empty subset \\( T \\) of \\( S \\) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \\( T \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set containing three real numbers can be represented as $\\{a, \\frac{b}{a}, 1\\}$, and also as $\\{a^2, a+b, 0\\}$. Find the value of $a^{2009} + b^{2009}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 34208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of $x$-values satisfying the equation $x^{\\log_{10} x} = \\frac{x^3}{100}$ consists of:"
}
] | math | {
"style": "rule",
"ground_truth": "10 \\text{ or } 100"
} | {
"index": 34209,
"solution": "1. **Rewrite the given equation**: We start with the equation \\(x^{\\log_{10} x} = \\frac{x^3}{100}\\).\n\n2. **Take logarithm on both sides**: To simplify, we take the logarithm with base \\(x\\) on both sides:\n \\[\n \\log_x \\left(x^{\\log_{10} x}\\right) = \\log_x \\left(\\f... |
DeepScaleR | [
{
"role": "user",
"content": "The set of $x$-values satisfying the inequality $2 \\leq |x-1| \\leq 5$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "-4\\leq x\\leq-1\\text{ or }3\\leq x\\leq 6"
} | {
"index": 34210,
"solution": "The inequality given is $2 \\leq |x-1| \\leq 5$. We can interpret this as the absolute value of $x-1$ being at least 2 and at most 5. To solve this, we break it into two separate inequalities based on the definition of absolute value:\n\n1. **Case 1: $x-1 \\geq 0$**\n - Here, $|x-1|... |
DeepScaleR | [
{
"role": "user",
"content": "The set of all solutions of the system $$\n\\begin{cases}\nx+y\\leq 3 \\\\\n2x+y\\geq 2 \\\\\nx\\geq 0 \\\\\ny\\geq 0\n\\end{cases}\n$$ is a quadrilateral region. Find the number of units in the length of the longest side. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 34211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of points $(x,y)$ such that $|x - 3| \\le y \\le 4 - |x - 1|$ defines a region in the $xy$-plane. Compute the area of this region."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of points $(x,y,z)$ that are equidistant to $(1,2,-5)$ and point $P$ satisfy an equation of the form\n\\[10x - 4y + 24z = 55.\\]Find the point $P.$"
}
] | math | {
"style": "rule",
"ground_truth": "(6,0,7)"
} | {
"index": 34213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of points $\\left(x_{1}, x_{2}, x_{3}, x_{4}\\right)$ in $\\mathbf{R}^{4}$ such that $x_{1} \\geq x_{2} \\geq x_{3} \\geq x_{4}$ is a cone (or hypercone, if you insist). Into how many regions is this cone sliced by the hyperplanes $x_{i}-x_{j}=1$ for $1 \\leq i<j \\leq ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34214,
"solution": "$C(4)=14$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:"
}
] | math | {
"style": "rule",
"ground_truth": "I and II"
} | {
"index": 34215,
"solution": "1. **Graph the inequalities**:\n - The inequality $y > 2x$ represents the region above the line $y = 2x$. This line passes through the origin and has a positive slope, dividing the plane into two regions. The region of interest is above this line.\n - The inequality $y > 4 - x$ re... |
DeepScaleR | [
{
"role": "user",
"content": "The set of points with spherical coordinates of the form\n\\[(\\rho, \\theta, \\phi) = \\left( 1, \\theta, \\frac{\\pi}{6} \\right)\\]forms a circle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of positive even numbers $\\{2, 4, 6, \\cdots\\}$ is grouped in increasing order such that the $n$-th group has $3n-2$ numbers:\n\\[\n\\{2\\}, \\{4, 6, 8, 10\\}, \\{12, 14, 16, 18, 20, 22, 24\\}, \\cdots\n\\]\n\nDetermine which group contains the number 2018."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of positive odd numbers $\\{1, 3, 5, \\cdots\\}$ is arranged in ascending order and grouped by the $n$th group having $(2n-1)$ odd numbers as follows:\n$$\n\\begin{array}{l}\n\\{1\\}, \\quad \\{3,5,7\\}, \\quad \\{9,11,13,15,17\\}, \\cdots \\\\\n\\text{ (First group) (S... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 34218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely:"
}
] | math | {
"style": "rule",
"ground_truth": "0, 12"
} | {
"index": 34219,
"solution": "1. **Testing $m = 0$:** \n Substitute $m = 0$ into the expression $x^2 + 3xy + x + my - m$:\n \\[\n x^2 + 3xy + x\n \\]\n Factoring out $x$:\n \\[\n x(x + 3y + 1)\n \\]\n This expression has two factors, $(x)$ and $(x + 3y + 1)$, both linear in $x$ and $y$ with inte... |
DeepScaleR | [
{
"role": "user",
"content": "The set of vectors $\\left\\{ \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}, \\begin{pmatrix} 3 \\\\ k \\end{pmatrix} \\right\\}$ is linearly dependent. Find all possible values of $k.$ Enter all the possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of vectors $\\mathbf{u}$ such that\n\\[\\mathbf{u} \\cdot \\mathbf{u} = \\mathbf{u} \\cdot \\begin{pmatrix} 6 \\\\ -28 \\\\ 12 \\end{pmatrix}\\] forms a solid in space. Find the volume of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi \\cdot 241^{3/2}"
} | {
"index": 34221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of vectors $\\mathbf{u}$ such that\n\\[\\mathbf{u} \\cdot \\mathbf{u} = \\mathbf{u} \\cdot \\begin{pmatrix} 8 \\\\ -28 \\\\ 12 \\end{pmatrix}\\]forms a solid in space. Find the volume of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi \\cdot 248^{3/2}"
} | {
"index": 34222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of vectors $\\mathbf{v}$ such that\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} 10 \\\\ -40 \\\\ 8 \\end{pmatrix}\\]forms a solid in space. Find the volume of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "12348 \\pi"
} | {
"index": 34223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set of vectors $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\begin{pmatrix} 5 \\\\ 2 \\end{pmatrix}} \\mathbf{v} = \\begin{pmatrix} -\\frac{5}{2} \\\\ -1 \\end{pmatrix}\\]lie on a line. Enter the equation of this line in the form \"$y = mx + b$\"."
}
] | math | {
"style": "rule",
"ground_truth": "y = -\\frac{5}{2} x - \\frac{29}{4}"
} | {
"index": 34224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sets $A = \\{z : z^{18} = 1\\}$ and $B = \\{w : w^{48} = 1\\}$ are both sets of complex roots of unity. The set $C = \\{zw : z \\in A ~ \\mbox{and} ~ w \\in B\\}$ is also a set of complex roots of unity. How many distinct elements are in $C_{}^{}$?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 34225,
"solution": "The values in polar form will be $(1, 20x)$ and $(1, 7.5x)$. Multiplying these gives $(1, 27.5x)$. Then, we get $27.5$, $55$, $82.5$, $110$, $...$ up to $3960$ $(\\text{lcm}(55,360)) \\implies \\frac{3960 \\cdot 2}{55}=\\boxed{144}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The seven digits in Sam's phone number and the four digits in his house number have the same sum. The four digits in his house number are distinct, and his phone number is 271-3147. What is the largest possible value of Sam's house number?"
}
] | math | {
"style": "rule",
"ground_truth": "9871"
} | {
"index": 34226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The seventh and tenth terms of a geometric sequence are $7$ and $21$, respectively. What is the $13$th term of this progression?"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 34227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shaded design shown in the diagram is made by drawing eight circular arcs, all with the same radius. The centers of four arcs are the vertices of the square; the centers of the four touching arcs are the midpoints of the sides of the square. The diagonals of the square have... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} \\pi"
} | {
"index": 34228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\frac{3}{2}$ and center $(0,\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{8}"
} | {
"index": 34229,
"solution": "1. **Identify the areas involved**:\n - The larger quarter-circle has a radius of $3$ and is centered at $(0,0)$. It occupies the first quadrant.\n - The smaller semicircle has a radius of $\\frac{3}{2}$ and is centered at $(0,\\frac{3}{2})$. It also lies in the first quadrant.\n\... |
DeepScaleR | [
{
"role": "user",
"content": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is"
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 34230,
"solution": "1. **Identify the areas of the individual rectangles**: \n - The first rectangle has dimensions $2 \\times 10$. Therefore, its area is:\n \\[\n 2 \\cdot 10 = 20 \\text{ square units}\n \\]\n - The second rectangle has dimensions $3 \\times 8$. Therefore, its area is:\n... |
DeepScaleR | [
{
"role": "user",
"content": "The shaded region shown consists of 11 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\\pi$... | math | {
"style": "rule",
"ground_truth": "37\\pi"
} | {
"index": 34231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shape of a bridge arch is a parabola. It is known that the width of the parabolic arch is 8 meters, and the area of the parabolic arch is 160 square meters. Then, the height of the parabolic arch is"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shelf life $y$ (in hours) of a certain food product and its storage temperature $x$ (in °C) satisfy the function relationship $y=e^{kx+b}$ (where $e=2.718\\ldots$ is the base of the natural logarithm, and $k$, $b$ are constants). It is known that the shelf life of this food... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 34234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean."
}
] | math | {
"style": "rule",
"ground_truth": "6000"
} | {
"index": 34235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distance for an ant to crawl along the surface of a rectangular box with length and width both being $6 \\mathrm{~cm}$ from vertex $A$ to vertex $B$ is $20 \\mathrm{~cm}$. What is the volume of this rectangular box in $\\mathrm{cm}^{3}$?"
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 34236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distance from a moving point P on circle C: $\\rho = -4\\sin\\theta$ to the line $l: \\rho\\sin(\\theta + \\frac{\\pi}{4}) = \\sqrt{2}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"index": 34237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distance from a point on the curve $f(x) = \\ln(2x-1)$ to the line $2x - y + 3 = 0$ is what?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 34238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distance from a point on the curve $y=\\ln x$ to the line $y=x+2$ is what value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 34239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distance from a point on the parabola $x^2=y$ to the line $y=2x+m$ is $\\sqrt{5}$. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 34240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The shortest distances between an interior diagonal of a rectangular parallelepiped, $P$, and the edges it does not meet are $2\\sqrt{5}$, $\\frac{30}{\\sqrt{13}}$, and $\\frac{15}{\\sqrt{10}}$. Determine the volume of $P$."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 34241,
"solution": "In the above diagram, we focus on the line that appears closest and is parallel to $BC$. All the blue lines are perpendicular lines to $BC$ and their other points are on $AB$, the main diagonal. The green lines are projections of the blue lines onto the bottom face; all of the green l... |
DeepScaleR | [
{
"role": "user",
"content": "The side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \\ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \\ldots, C_{n+1} = C$). The triangles $C_... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side \\( AB \\) of a regular hexagon \\( ABCDEF \\) is equal to \\( \\sqrt{3} \\) and serves as a chord of a certain circle, while the other sides of the hexagon lie outside this circle. The length of the tangent \\( CM \\), drawn to the same circle from vertex \\( C \\), i... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 34243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side \\( AB \\) of triangle \\( ABC \\) is longer than side \\( AC \\), and \\(\\angle A = 40^\\circ\\). Point \\( D \\) lies on side \\( AB \\) such that \\( BD = AC \\). Points \\( M \\) and \\( N \\) are the midpoints of segments \\( BC \\) and \\( AD \\) respectively. F... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side \\( AD \\) of the rectangle \\( ABCD \\) is equal to 2. On the extension of the side \\( AD \\) beyond point \\( A \\), a point \\( E \\) is taken such that \\( EA = 1 \\), and \\( \\angle BEC = 30^\\circ \\). Find \\( BE \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 34245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side edge of a regular tetrahedron \\( S-ABC \\) is 2, and the base is an equilateral triangle with side length 1. A section passing through \\( AB \\) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{15}}"
} | {
"index": 34246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side length of an equilateral triangle ABC is 2. Calculate the area of the orthographic (isometric) projection of triangle ABC."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"index": 34247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side length of square $A$ is 36 cm. The side length of square $B$ is 42 cm. What is the ratio of the area of square $A$ to the area of square $B$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{36}{49}"
} | {
"index": 34248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side length of square \\(ABCD\\) is 4. Point \\(E\\) is the midpoint of \\(AB\\), and point \\(F\\) is a moving point on side \\(BC\\). Triangles \\(\\triangle ADE\\) and \\(\\triangle DCF\\) are folded up along \\(DE\\) and \\(DF\\) respectively, making points \\(A\\) and ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{5}}{5}"
} | {
"index": 34249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side length of the regular hexagon is 10 cm. What is the number of square centimeters in the area of the shaded region? Express your answer in simplest radical form.\n\n[asy]\nsize(100);\n\npair A,B,C,D,E,F;\nA = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240)... | math | {
"style": "rule",
"ground_truth": "100\\sqrt{3}"
} | {
"index": 34250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths \\(a, b, c\\) of triangle \\(\\triangle ABC\\) satisfy the conditions:\n1. \\(a, b, c\\) are all integers;\n2. \\(a, b, c\\) form a geometric sequence;\n3. At least one of \\(a\\) or \\(c\\) is equal to 100. \n\nFind all possible sets of the triplet \\((a, b, c... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Calculate the diameter of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 34252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. What is the diameter of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 34253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths of a triangle are 14 cm, 48 cm and 50 cm. How many square centimeters are in the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 34254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42, and another is a multiple of 72. What is the minimum possible length of the third side?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34255,
"solution": "Suppose that two of the side lengths are $42 a$ and $72 b$, for some positive integers $a$ and $b$. Let $c$ be the third side length. We know that $42 a$ is not equal to $72 b$, since the side lengths are distinct. Also, $6 \\mid 42 a-72 b$. Therefore, by the triangle inequality, we g... |
DeepScaleR | [
{
"role": "user",
"content": "The side lengths of both triangles to the right are given in centimeters. What is the length of segment $AB$?\n\n[asy]\npair A,B,C,D,E,F,G;\n\nA=(0,0);\nB=12*dir(0);\nC=20*dir(120);\nD=8+B;\nE=D+6*dir(0);\nF=D+10*dir(120);\n\ndraw(A--B--C--cycle);\ndraw(D--E--F--cycle);\n\nlabe... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of a square has the length $(x-2)$, while a rectangle has a length of $(x-3)$ and a width of $(x+4)$. If the area of the rectangle is twice the area of the square, what is the sum of the possible values of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of a square is increased by $20\\%$. To keep the area of the square unchanged, what percentage must the other side be reduced?"
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"index": 34258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of a triangle are 2, $\\sqrt{6},$ and $1 + \\sqrt{3}.$ Enter the angles of the triangle in degrees, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"index": 34259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of a triangle are 2, 2, and $\\sqrt{6} - \\sqrt{2}.$ Enter the angles of the triangle in degrees, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ, 75^\\circ"
} | {
"index": 34260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of rhombus \\(ABCD\\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.\n\nFind the distance between the points where this circle touches the sides \\(AB\\) and \\(BC\\), if the diagonal \\(AC\\) is less than the diagonal \\(BD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3.84"
} | {
"index": 34261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side of the base of a regular quadrilateral pyramid \\( \\operatorname{ABCDP} \\) (with \\( P \\) as the apex) is \\( 4 \\sqrt{2} \\), and the angle between adjacent lateral faces is \\( 120^{\\circ} \\). Find the area of the cross-section of the pyramid by a plane passing ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{6}"
} | {
"index": 34262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The side surface of a cylinder unfolds into a rectangle with side lengths of $6\\pi$ and $4\\pi$. The surface area of the cylinder is ______."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi^2 + 8\\pi"
} | {
"index": 34263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides $x$ and $y$ of a scalene triangle satisfy $x + \\frac{2\\Delta }{x}=y+ \\frac{2\\Delta }{y}$ , where $\\Delta$ is the area of the triangle. If $x = 60, y = 63$ , what is the length of the largest side of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 34264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\\frac{1}{2}$ inches. Then $QR$, in inches, is:"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34265,
"solution": "1. **Using the Median Formula**: The formula for the length of a median in a triangle, which connects a vertex to the midpoint of the opposite side, is given by:\n \\[\n PM = \\frac{1}{2}\\sqrt{2PQ^2 + 2PR^2 - QR^2}\n \\]\n where $PQ$, $PR$, and $QR$ are the lengths of the sid... |
DeepScaleR | [
{
"role": "user",
"content": "The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of $\\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a non-degenerate isosceles triangle are \\(x\\), \\(x\\), and \\(24\\) units. How many integer values of \\(x\\) are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a regular hexagon are trisected, resulting in 18 points, including vertices. These points, starting with a vertex, are numbered clockwise as $A_{1}, A_{2}, \\ldots, A_{18}$. The line segment $A_{k} A_{k+4}$ is drawn for $k=1,4,7,10,13,16$, where indices are taken m... | math | {
"style": "rule",
"ground_truth": "9/13"
} | {
"index": 34269,
"solution": "Let us assume all sides are of side length 3. Consider the triangle $A_{1} A_{4} A_{5}$. Let $P$ be the point of intersection of $A_{1} A_{5}$ with $A_{4} A_{8}$. This is a vertex of the inner hexagon. Then $\\angle A_{4} A_{1} A_{5}=\\angle A_{5} A_{4} P$, by symmetry. It follows tha... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(n-2)180}{n}"
} | {
"index": 34270,
"solution": "1. **Identify the internal angle of the regular polygon**: \n The internal angle of a regular polygon with $n$ sides can be calculated using the formula:\n \\[\n \\text{Internal angle} = \\frac{(n-2) \\times 180^\\circ}{n}\n \\]\n\n2. **Calculate the external angle of the poly... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "1 : 9"
} | {
"index": 34271,
"solution": "Given a right triangle with sides $a$, $b$, and hypotenuse $c$, where $a:b = 1:3$. Let's denote $a = k$ and $b = 3k$ for some positive constant $k$. By the Pythagorean theorem, we have:\n\\[ c^2 = a^2 + b^2 = k^2 + (3k)^2 = k^2 + 9k^2 = 10k^2 \\]\nThus, the hypotenuse $c = \\sqrt{10k^... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is:"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 34272,
"solution": "1. **Assign Variables:**\n Let the shorter segment of the side of length $80$ be $x$, and let the altitude dropped to this side be $y$. The larger segment is then $80 - x$.\n\n2. **Apply the Pythagorean Theorem:**\n Since the altitude divides the triangle into two right triangles,... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30 \\sqrt{6}}{7}"
} | {
"index": 34273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle are all integers, and the longest side is 11. Calculate the number of such triangles."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths $11, 15,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34277,
"solution": "To find the smallest possible value of $s$ such that the sides $6.5$, $10$, and $s$ can form a triangle, we must apply the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of ... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths \\( 13, 17, \\) and \\( k, \\) where \\( k \\) is a positive integer. For how many values of \\( k \\) is the triangle obtuse?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude."
}
] | math | {
"style": "rule",
"ground_truth": "12.8470588235"
} | {
"index": 34279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of a triangle have lengths of $15$, $20$, and $25$. Find the length of the shortest altitude."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of an isosceles triangle are $\\cos x,$ $\\cos x,$ and $\\cos 7x,$ and its vertex angle is $2x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "10^\\circ, 50^\\circ, 54^\\circ"
} | {
"index": 34281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ are $AB=3$ and $BC=2$. Point $P$ is on side $AB$ such that line $PD$ touches the circle with diameter $BC$ at point $E$. The line passing through the center of the circle and point $E$ intersects side $AB$ at point $Q$. What is the area of triangle... | math | {
"style": "rule",
"ground_truth": "1/24"
} | {
"index": 34282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle."
}
] | math | {
"style": "rule",
"ground_truth": "221 \\sqrt{3} - 330"
} | {
"index": 34283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers... | math | {
"style": "rule",
"ground_truth": "554"
} | {
"index": 34284,
"solution": "Since $\\angle{BAD}=90$ and $\\angle{EAF}=60$, it follows that $\\angle{DAF}+\\angle{BAE}=90-60=30$. Rotate triangle $ADF$ $60$ degrees clockwise. Note that the image of $AF$ is $AE$. Let the image of $D$ be $D'$. Since angles are preserved under rotation, $\\angle{DAF}=\\angle{D'AE}$... |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $12$ (height) and $15$ (width). An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle."
}
] | math | {
"style": "rule",
"ground_truth": "369 \\sqrt{3} - 540"
} | {
"index": 34285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle."
}
] | math | {
"style": "rule",
"ground_truth": "36\\sqrt{3}"
} | {
"index": 34286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $12$ and $5$. A right triangle is drawn so that no point of the triangle lies outside $ABCD$, and one of its angles is $30^\\circ$. The maximum possible area of such a triangle can be written in the form $p \\sqrt{q} - r$, where $p$, $... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of rhombus \\( EFGH \\) are the hypotenuses of the isosceles right triangles \\( EAF, FDG, GCH, \\) and \\( HBE \\), and all these triangles have common interior points with the rhombus \\( EFGH \\). The sum of the areas of quadrilateral \\( ABCD \\) and rhombus \\( E... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 34288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of the base of a brick are 28 cm and 9 cm, and its height is 6 cm. A snail crawls rectilinearly along the faces of the brick from one vertex of the lower base to the opposite vertex of the upper base. The horizontal and vertical components of its speed $v_{x}$ and $v_... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 34289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of this parallelogram measure 7,9, $8y-1$ and $2x+3$ units, consecutively. What is the value of $x+y$?\n\n[asy]draw((0,0)--(21,0)--(30,25)--(9,25)--cycle);\nlabel(\"$8y-1$\",(10,0),S);\nlabel(\"9\",(25.5,12.5),E);\nlabel(\"7\",(19.5,25),N);\nlabel(\"$2x+3$\",(4.5,12.5... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $CAB$ are in the ratio of $2:3:4$. Segment $BD$ is the angle bisector drawn to the shortest side, dividing it into segments $AD$ and $DC$. What is the length, in inches, of the longer subsegment of side $AC$ if the length of side $AC$ is $10$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac {40}7"
} | {
"index": 34291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $DEF$ are in the ratio $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. If the length of side $DE$ (the base) is 12 inches, what is the length, in inches, of the longer segment of side $EF$ on... | math | {
"style": "rule",
"ground_truth": "\\frac{80}{7}"
} | {
"index": 34292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac{48}{7}"
} | {
"index": 34293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{7}"
} | {
"index": 34294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $QS$ is the angle bisector drawn to the longest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the shorter subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express ... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{7}"
} | {
"index": 34295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{7}"
} | {
"index": 34296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 63^\\circ$ and $\\angle QPR = 59^\\circ$, find $\\angle QRC$, in degrees.\n\n[asy]\n\nunitsize(1.0 cm);\n\npair Q, P, R, C;\n\nQ = (2.43,3.46);\n\nP = (0,0);\n\nR = (4.43,0);... | math | {
"style": "rule",
"ground_truth": "29^\\circ"
} | {
"index": 34297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 65^\\circ$ and $\\angle QRC = 30^\\circ$, find $\\angle QPR$, in degrees.\n\n[asy]\nunitsize(1.0 cm);\n\npair Q, P, R, C;\n\nQ = (2.43,3.46);\nP = (0,0);\nR = (4.43,0);\nC = ... | math | {
"style": "rule",
"ground_truth": "55^\\circ"
} | {
"index": 34298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides of triangle \\(ABC\\) are divided by points \\(M, N\\), and \\(P\\) such that \\(AM : MB = BN : NC = CP : PA = 1 : 4\\). Find the ratio of the area of the triangle bounded by lines \\(AN, BP\\), and \\(CM\\) to the area of triangle \\(ABC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"index": 34299,
"solution": ""
} |
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