data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $X\\sim N(2, \\sigma ^{2})$, $P(X\\leqslant 0)=0.15$, calculate $P(2\\leqslant X\\leqslant 4)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"index": 13500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $\\eta = 8 - \\xi$, if $\\xi \\sim B(10, 0.6)$, then calculate $E\\eta$ and $D\\eta$."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 13501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $\\xi \\sim N(1, \\sigma ^{2})$, $a \\gt 0$, $b \\gt 0$, if $P(\\xi \\leq a) = P(\\xi \\geq b)$, then the minimum value of $\\frac{{4a+b}}{{ab}}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 13502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $\\xi \\sim N(1,4)$, and $P(\\xi < 3)=0.84$, then $P(-1 < \\xi < 1)=$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "0.34"
} | {
"index": 13503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $\\xi$ follows the normal distribution $N(0, \\sigma^2)$. If $P(\\xi > 2) = 0.023$, calculate $P(-2 \\leq \\xi \\leq 2)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.954"
} | {
"index": 13504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $\\xi$ that follows the normal distribution $N(0, \\sigma^2)$, if $P(\\xi > 2) = 0.023$, calculate $P(-2 \\leq \\xi \\leq 2)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.954"
} | {
"index": 13505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \\leqslant 4-a) = P(ξ \\geqslant 2+3a)$, solve for the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a randomly selected number $x$ in the interval $[0,\\pi]$, determine the probability of the event \"$-1 \\leqslant \\tan x \\leqslant \\sqrt {3}$\"."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{12}"
} | {
"index": 13507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between $0$ and $1$ will $20_{}^{}!$ be the resulting product?"
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 13508,
"solution": "If the fraction is in the form $\\frac{a}{b}$, then $a < b$ and $\\gcd(a,b) = 1$. There are 8 prime numbers less than 20 ($2, 3, 5, 7, 11, 13, 17, 19$), and each can only be a factor of one of $a$ or $b$. There are $2^8$ ways of selecting some combination of numbers for $a$; however, ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a real coefficient fourth-degree polynomial with a leading coefficient of 1 that has four imaginary roots, where the product of two of the roots is \\(32+\\mathrm{i}\\) and the sum of the other two roots is \\(7+\\mathrm{i}\\), determine the coefficient of the quadratic t... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 13509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a real number $a$ satisfying ${a}^{\\frac{1}{2}}\\leqslant 3$ and $\\log _{a}3\\leqslant \\frac{1}{2}$. \n$(1)$ Find the range of real number $a$; \n$(2)$ If $a \\gt 1$, $f\\left(x\\right)=mx^{a}+\\ln \\left(1+x\\right)^{a}-a\\ln \\left(1-x\\right)-2\\left(m\\in R\\righ... | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"index": 13510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}... | math | {
"style": "rule",
"ground_truth": "553"
} | {
"index": 13511,
"solution": "Obviously $k$ is positive. Then, we can let $n_1$ equal $k^3$ and similarly let $n_i$ equal $k^3 + (i - 1)k$.\n The wording of this problem (which uses \"exactly\") tells us that $k^3+69k<(k+1)^3 = k^3 + 3k^2 + 3k+1 \\leq k^3+70k$. Taking away $k^3$ from our inequality results in $69k... |
DeepScaleR | [
{
"role": "user",
"content": "Given a real number \\(a\\), and for any \\(k \\in [-1, 1]\\), when \\(x \\in (0, 6]\\), the inequality \\(6 \\ln x + x^2 - 8x + a \\leq kx\\) always holds. Determine the maximum value of \\(a\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 6 \\ln 6"
} | {
"index": 13512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutiv... | math | {
"style": "rule",
"ground_truth": "3^{n}-(n+1) \\cdot 2^{n}+n(n+1) / 2"
} | {
"index": 13513,
"solution": "Notice that each such permutation consists of 3 disjoint subsets of $\\{1, \\ldots, n\\}$ whose union is $\\{1, \\ldots, n\\}$, each arranged in decreasing order. For instance, if $n=6$, in the permutation 415326 (which has the two increasing pairs 15 and 26), the three sets are $\\{4... |
DeepScaleR | [
{
"role": "user",
"content": "Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by ma... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 13514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle $A B C D$, let $X$ and $Y$ be points on $A B$ and $B C$, respectively. Suppose the areas of the triangles $\\triangle A X D$, $\\triangle B X Y$, and $\\triangle D Y C$ are 5, 4, and 3, respectively. Find the area of $\\triangle D X Y$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{21}"
} | {
"index": 13515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle $ABCD$ with all vertices on a sphere centered at $O$, where $AB = \\sqrt{3}$, $BC = 3$, and the volume of the pyramid $O-ABCD$ is $4\\sqrt{3}$, find the surface area of the sphere $O$."
}
] | math | {
"style": "rule",
"ground_truth": "76\\pi"
} | {
"index": 13516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle \\(ABCD\\), a circle intersects the side \\(AB\\) at points \\(K\\) and \\(L\\), and the side \\(CD\\) at points \\(M\\) and \\(N\\). Find the length of segment \\(MN\\) if \\(AK = 10\\), \\(KL = 17\\), and \\(DN = 7\\)."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 13517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle \\(ABCD\\). On two sides of the rectangle, different points are chosen: six points on \\(AB\\) and seven points on \\(BC\\). How many different triangles can be formed with vertices at the chosen points?"
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"index": 13518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 13519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangle with dimensions \\(100 \\times 101\\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 13520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort."
}
] | math | {
"style": "rule",
"ground_truth": "594"
} | {
"index": 13521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular coordinate system with origin point $O$, vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ satisfy $|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$ and $\\overrightarrow {a}\\cdot \\overrightarrow {b}=\\frac {1}{2}$. Let $\\overrightarrow {c} = (m, ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}"
} | {
"index": 13522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid."
}
] | math | {
"style": "rule",
"ground_truth": "387"
} | {
"index": 13523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected fro... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{47}"
} | {
"index": 13524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 13525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular painting that measures $12$ inches high and $16$ inches wide, and is placed in a frame that forms a border three inches wide on all sides, find the area of the border, in square inches."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 13526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular pan of brownies measuring 15 inches by 24 inches, cut into triangular pieces with a base of 3 inches and a height of 4 inches, determine the number of triangular pieces that can be cut from the pan."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular parallelepiped with edge lengths AB = 5, BC = 3, and CG = 4, and point M as the midpoint of diagonal AG, determine the volume of the rectangular pyramid with base BCHE and apex M."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{34}"
} | {
"index": 13528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction."
}
] | math | {
"style": "rule",
"ground_truth": "1592"
} | {
"index": 13529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rectangular yard containing two congruent isosceles right triangles in the form of flower beds and a trapezoidal remainder, with the parallel sides of the trapezoid having lengths $15$ and $25$ meters."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 13530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a region bounded by a larger quarter-circle with a radius of $5$ units, centered at the origin $(0,0)$ in the first quadrant, a smaller circle with radius $2$ units, centered at $(0,4)$ that lies entirely in the first quadrant, and the line segment from $(0,0)$ to $(5,0)$... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{4}"
} | {
"index": 13531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular 14-gon, where each vertex is connected to every other vertex by a segment, three distinct segments are chosen at random. What is the probability that the lengths of these three segments refer to a triangle with positive area?\nA) $\\frac{73}{91}$\nB) $\\frac{74}... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{91}"
} | {
"index": 13532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon."
}
] | math | {
"style": "rule",
"ground_truth": "1506"
} | {
"index": 13533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular 2017-sided polygon \\( A_{1} A_{2} \\cdots A_{2017} \\) inscribed in a unit circle \\(\\odot O\\), choose any two distinct vertices \\( A_{i} \\) and \\( A_{j} \\). What is the probability that \\( \\overrightarrow{O A_{i}} \\cdot \\overrightarrow{O A_{j}} > \\f... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 13534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular 2017-sided polygon \\(A_{1} A_{2} \\cdots A_{2017}\\) inscribed in a unit circle \\(\\odot O\\), choose any two different vertices \\(A_{i}, A_{j}\\). Find the probability that \\(\\overrightarrow{O A_{i}} \\cdot \\overrightarrow{O A_{j}} > \\frac{1}{2}\\)."
}... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 13535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular decagon $ABCDEFGHIJ$ with area $n$, calculate the area $m$ of pentagon $ACEGI$, which is defined using every second vertex of the decagon, and then determine the value of $\\frac{m}{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular decagon, calculate the number of distinct points in the interior of the decagon where two or more diagonals intersect."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 13537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular decagon, the probability that exactly one of the sides of the triangle formed by connecting three randomly chosen vertices of the decagon is also a side of the decagon."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular hexagon \\( A B C D E F \\) with a side length of 1, calculate \\((\\overrightarrow{A B}+\\overrightarrow{D C}) \\cdot(\\overrightarrow{A D}+\\overrightarrow{B E})\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 13539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular hexagonal pyramid \\( M A B C D E F \\). Point \\( K \\) bisects edge \\( B M \\). Find the ratio in which the plane \\( F E K \\) divides edge \\( A M \\) (at point \\( X \\))."
}
] | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"index": 13541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular pentagon \\(ABCDE\\). Point \\(K\\) is marked on side \\(AE\\), and point \\(L\\) is marked on side \\(CD\\). It is known that \\(\\angle LAE + \\angle KCD = 108^\\circ\\) and \\(AK: KE = 3:7\\). Find \\(CL: AB\\).\n\nA regular pentagon is a pentagon where all s... | math | {
"style": "rule",
"ground_truth": "0.7"
} | {
"index": 13542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon's vertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. A pivot point is a point inside the pentagon with only finitely many non-pivo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}(7-3 \\sqrt{5})"
} | {
"index": 13543,
"solution": "Let the pentagon be labeled $ABCDE$. First, no pivot point can be on the same side of $AC$ as vertex $B$. Any such point $P$ has the infinite set of non-pivot lines within the hourglass shape formed by the acute angles between lines $PA$ and $PC$. Similar logic can be applied to point... |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a **right triangle**. What is the value of $n$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 13544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular pyramid V-ABCD with a base edge length of 4 and a lateral edge length of $\\sqrt{13}$, its surface area is ______."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 13545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular quadrangular pyramid \\( S-ABCD \\) with a side edge length of 4, and \\( \\angle ASB = 30^\\circ \\). Points \\( E \\), \\( F \\), and \\( G \\) are taken on side edges \\( SB \\), \\( SC \\), and \\( SD \\), respectively. Find the minimum value of \\( AE + EF ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 13546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\\angle ASB = 30^\\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$."
}... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 13547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular quadrilateral pyramid $S-ABCD$ with side edges of length $4$ and $\\angle ASB = 30^\\circ$, a plane passing through point $A$ intersects the side edges $SB$, $SC$, and $SD$ at points $E$, $F$, and $G$ respectively. Find the minimum perimeter of the cross-section... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 13548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular quadrilateral pyramid $S-ABCD$, with a base side length of $2$ and a volume of $\\frac{{4\\sqrt{3}}}{3}$, the length of the lateral edge of this quadrilateral pyramid is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 13549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular square pyramid \\( P-ABCD \\) with a base side length \\( AB=2 \\) and height \\( PO=3 \\). \\( O' \\) is a point on the segment \\( PO \\). Through \\( O' \\), a plane parallel to the base of the pyramid is drawn, intersecting the edges \\( PA, PB, PC, \\) and ... | math | {
"style": "rule",
"ground_truth": "16/27"
} | {
"index": 13550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron $ABCD$ with edge length of 1 cm, an ant starts at point $A$ and follows these rules: at each vertex, it randomly chooses one of the three edges connected to that vertex and crawls along this edge to the next vertex. What is the probability that, afte... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"index": 13551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron $P-ABC$, where points $P$, $A$, $B$, and $C$ are all on the surface of a sphere with radius $\\sqrt{3}$. If $PA$, $PB$, and $PC$ are mutually perpendicular, calculate the distance from the center of the sphere to the plane $ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{3}"
} | {
"index": 13552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron $S-ABC$ with a base that is an equilateral triangle of side length 1 and side edges of length 2. If a plane passing through line $AB$ divides the tetrahedron's volume into two equal parts, the cosine of the dihedral angle between the plane and the ba... | math | {
"style": "rule",
"ground_truth": "$\\frac{2 \\sqrt{15}}{15}$"
} | {
"index": 13553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron A-BCD with an edge length of 1, and $\\overrightarrow{AE} = 2\\overrightarrow{EB}$, $\\overrightarrow{AF} = 2\\overrightarrow{FD}$, calculate $\\overrightarrow{EF} \\cdot \\overrightarrow{DC}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 13554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron \\(P-ABC\\) with edge length 1, let \\(E\\), \\(F\\), \\(G\\), and \\(H\\) be points on edges \\(PA\\), \\(PB\\), \\(CA\\), and \\(CB\\) respectively. If \\(\\overrightarrow{EF}+\\overrightarrow{GH}=\\overrightarrow{AB}\\) and \\(\\overrightarrow{EH}... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 13555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron of volume 1, obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 13556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron with an edge length of \\(2 \\sqrt{6}\\), a sphere is centered at the centroid \\(O\\) of the tetrahedron. The total length of the curves where the sphere intersects with the four faces of the tetrahedron is \\(4 \\pi\\). Find the radius of the spher... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 13557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron with four vertices \\(A, B, C, D\\) and each edge of length 1 meter, a bug starts at point \\(A\\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability a... | math | {
"style": "rule",
"ground_truth": "\\frac{182}{729}"
} | {
"index": 13558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular tetrahedron with vertices $A, B, C$, and $D$ and edge length 1, where point $P$ lies on edge $AC$ one third of the way from vertex $A$ to vertex $C$, and point $Q$ lies on edge $BD$ one third of the way from vertex $B$ to vertex $D$, calculate the least possible... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 13559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular triangular prism $ABC-A_1B_1C_1$ with side edges equal to the edges of the base, the sine of the angle formed by $AB_1$ and the lateral face $ACCA_1$ is equal to _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"index": 13560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular triangular prism \\(ABC-A_1B_1C_1\\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \\(A_1ABC\\), \\(B_1ABC\\), and \\(C_1ABC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{36}"
} | {
"index": 13561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral trian... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}\\pi}{2}"
} | {
"index": 13562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular triangular pyramid \\( S A B C \\). Point \\( S \\) is the apex of the pyramid, \\( AB = 1 \\), \\( AS = 2 \\), \\( BM \\) is the median of triangle \\( ABC \\), and \\( AD \\) is the angle bisector of triangle \\( SAB \\). Find the length of segment \\( DM \\).... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{31}}{6}"
} | {
"index": 13563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a regular triangular pyramid \\(P-ABC\\), where points \\(P\\), \\(A\\), \\(B\\), and \\(C\\) all lie on the surface of a sphere with radius \\(\\sqrt{3}\\), and \\(PA\\), \\(PB\\), and \\(PC\\) are mutually perpendicular, find the distance from the center of the sphere t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:\n- Less than 1 year: 3 marks\n- 1 to less than 2 years: 6 marks\n- 2 to less than 3 years: 5 marks\n- 3 to less than 4 years: 4 marks\n- 4 to les... | math | {
"style": "rule",
"ground_truth": "24.14\\%"
} | {
"index": 13565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rhombus \\(ABCD\\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \\(B\\), draw the altitudes \\(BE\\) and \\(BF\\). Calculate the area of the quadrilateral \\(BFDE\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4.32"
} | {
"index": 13566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{90\\sqrt{261}}{261}"
} | {
"index": 13567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \\sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 13568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right circular cone with a base radius of \\(1 \\, \\text{cm}\\) and a slant height of \\(3 \\, \\text{cm}\\), point \\(P\\) is on the circumference of the base. Determine the shortest distance from the vertex \\(V\\) of the cone to the shortest path from \\(P\\) back t... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 13569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right circular cone with three mutually perpendicular side edges, each with a length of $\\sqrt{3}$, determine the surface area of the circumscribed sphere."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 13570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 13571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \\perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism."
}
] | math | {
"style": "rule",
"ground_truth": "33\\pi"
} | {
"index": 13572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______."
}
] | math | {
"style": "rule",
"ground_truth": "32\\pi"
} | {
"index": 13573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right quadrilateral pyramid \\( V-ABCD \\) with the height equal to half the length of \\( AB \\), \\( M \\) is the midpoint of the lateral edge \\( VB \\), and \\( N \\) is a point on the lateral edge \\( VD \\) such that \\( DN=2VN \\). Find the cosine of the angle be... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{11}}{11}"
} | {
"index": 13574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right rectangular prism $B$ with edge lengths $1,$ $3,$ and $4$, and the set $S(r)$ of points within a distance $r$ of some point in $B$, express the volume of $S(r)$ as $ar^{3} + br^{2} + cr + d$ and determine the ratio $\\frac{bc}{ad}$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right square prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a base edge length of $1$, and $AB_{1}$ forms a $60^{\\circ}$ angle with the base $ABCD$, find the distance from $A_{1}C_{1}$ to the base $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 13577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle $PQR$ with $\\angle PQR = 90^\\circ$, suppose $\\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 13578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with \\(\\angle A = 60^\\circ\\) and hypotenuse \\( AB = 2 + 2\\sqrt{3} \\), a line \\( p \\) is drawn through vertex \\( B \\) parallel to \\( AC \\). Points \\( D \\) and \\( E \\) are placed on line \\( p \\) such that \\( AB = BD \\) and \... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3} + \\sqrt{6}"
} | {
"index": 13579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with hypotenuse \\( AB \\). One leg \\( AC = 15 \\) and the altitude from \\( C \\) to \\( AB \\) divides \\( AB \\) into segments \\( AH \\) and \\( HB \\) with \\( HB = 16 \\). What is the area of triangle \\( ABC \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 13580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with leg lengths equal to 1, a point \\( P \\) is chosen on one of the sides of the triangle. Find the maximum value of \\( PA \\cdot PB \\cdot PC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 13581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with legs \\( AC = 3 \\) and \\( BC = 4 \\). Construct triangle \\( A_1 B_1 C_1 \\) by successively translating point \\( A \\) a certain distance parallel to segment \\( BC \\) to get point \\( A_1 \\), then translating point \\( B \\) parall... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD \\) to form a right dihedral angle \\( A-CD-B \\). When the length of \\( AB \\) is minimized... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\) with a right angle at \\(A\\). On the leg \\(AC\\), a point \\(D\\) is marked such that \\(AD:DC = 1:3\\). Circles \\(\\Gamma_1\\) and \\(\\Gamma_2\\) are then drawn with centers at \\(A\\) and \\(C\\) respectively, both passing through point \\... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\) with a right angle at \\(C\\), a circle is drawn with diameter \\(BC\\) of length 26. A tangent \\(AP\\) from point \\(A\\) to this circle (distinct from \\(AC\\)) is drawn. The perpendicular \\(PH\\) dropped onto segment \\(BC\\) intersects seg... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 13585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\) with legs \\(BC = 30\\) and \\(AC = 40\\). Points \\(C_1\\), \\(A_1\\), and \\(B_1\\) are chosen on the sides \\(AB\\), \\(BC\\), and \\(CA\\), respectively, such that \\(AC_1 = BA_1 = CB_1 = 1\\). Find the area of triangle \\(A_1 B_1 C_1\\)."
... | math | {
"style": "rule",
"ground_truth": "554.2"
} | {
"index": 13586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\), point \\(D\\) is located on the extension of hypotenuse \\(BC\\) such that line \\(AD\\) is tangent to the circumcircle \\(\\omega\\) of triangle \\(ABC\\). Line \\(AC\\) intersects the circumcircle of triangle \\(ABD\\) at point \\(E\\). It tu... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 13587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\). On the extension of the hypotenuse \\(BC\\), a point \\(D\\) is chosen such that the line \\(AD\\) is tangent to the circumscribed circle \\(\\omega\\) of triangle \\(ABC\\). The line \\(AC\\) intersects the circumscribed circle of triangle \\(... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 13588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle where one leg has length $4x + 2$ feet and the other leg has length $(x-3)^2$ feet, find the value of $x$ if the hypotenuse is $5x + 1$ feet."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}}"
} | {
"index": 13589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right tria... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{51}"
} | {
"index": 13591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side face $ACC_{1}A_{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"index": 13592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $\\angle BAC=90^{\\circ}$, the area of the side face $BCC_{1}B_{1}$ is $16$. Find the minimum value of the radius of the circumscribed sphere of the right triangular prism $ABC-A_{1}B_{1}C_{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"index": 13593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a right-angled triangle, one of whose acute angles is $\\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\\alpha$ this ratio will be the smallest."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 13594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 13595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"index": 13597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a school library with four types of books: A, B, C, and D, and a student limit of borrowing at most 3 books, determine the minimum number of students $m$ such that there must be at least two students who have borrowed the same type and number of books."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sector with a central angle of $\\alpha$ and a radius of $r$. \n$(1)$ If $\\alpha = 60^{\\circ}$ and $r = 3$, find the arc length of the sector. \n$(2)$ If the perimeter of the sector is $16$, at what angle $\\alpha$ will the area of the sector be maximized? Also, fin... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13599,
"solution": ""
} |
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