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 triangle \\( \\triangle ABC \\) with interior angles \\( \\angle A, \\angle B, \\angle C \\) and opposite sides \\( a, b, c \\) respectively, where \\( \\angle A - \\angle C = \\frac{\\pi}{2} \\) and \\( a, b, c \\) are in arithmetic progression, find the value of \\( \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( \\triangle ABC \\) with sides \\( a, b, c \\) and corresponding medians \\( m_a, m_b, m_c \\), and angle bisectors \\( w_a, w_b, w_c \\). Let \\( w_a \\cap m_b = P \\), \\( w_b \\cap m_c = Q \\), and \\( w_c \\cap m_a = R \\). Denote the area of \\( \\trian... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 13801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( \\triangle ABC \\) with sides \\( a, b, c \\) opposite to angles \\( A, B, C \\) respectively, and \\( a^{2} + b^{2} = c^{2} + \\frac{2}{3}ab \\). If the circumradius of \\( \\triangle ABC \\) is \\( \\frac{3\\sqrt{2}}{2} \\), what is the maximum possible a... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 13802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(A B C\\) with \\(A B = A C\\) and \\(\\angle A = 110^{\\circ}\\). Inside the triangle, a point \\(M\\) is chosen such that \\(\\angle M B C = 30^{\\circ}\\) and \\(\\angle M C B = 25^{\\circ}\\). Find \\(\\angle A M C\\)."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 13803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) where \\(AB = AC\\) and \\(\\angle A = 80^\\circ\\). Inside triangle \\(ABC\\) is a point \\(M\\) such that \\(\\angle MBC = 30^\\circ\\) and \\(\\angle MCB = 10^\\circ\\). Find \\(\\angle AMC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 13804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with an area of 1. Points \\(P\\), \\(Q\\), and \\(R\\) are taken on the medians \\(AK\\), \\(BL\\), and \\(CN\\) respectively such that \\(AP = PK\\), \\(BQ : QL = 1 : 2\\), and \\(CR : RN = 5 : 4\\). Find the area of triangle \\(PQR\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"index": 13805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with an area of 1. The first player chooses a point \\(X\\) on side \\(AB\\), the second player chooses a point \\(Y\\) on side \\(BC\\), and then the first player chooses a point \\(Z\\) on side \\(AC\\). The first player's goal is to maximize the ar... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with an area of 2. Points \\(P\\), \\(Q\\), and \\(R\\) are taken on the medians \\(AK\\), \\(BL\\), and \\(CN\\) of the triangle \\(ABC\\) respectively, such that \\(AP : PK = 1\\), \\(BQ : QL = 1:2\\), and \\(CR : RN = 5:4\\). Find the area of the t... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"index": 13807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with side lengths \\(AB = BC = 80\\) and \\(AC = 96\\).\n\nCircle \\(Q_1\\) is inscribed in the triangle \\(ABC\\). Circle \\(Q_2\\) is tangent to \\(Q_1\\) and to the sides \\(AB\\) and \\(BC\\). Circle \\(Q_3\\) is tangent to \\(Q_2\\) and also to t... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 13808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with sides \\(AB = 13\\), \\(BC = 14\\), and \\(AC = 15\\). Point \\(K\\) is marked on side \\(AB\\), point \\(L\\) is marked on side \\(BC\\), and point \\(N\\) is marked on side \\(AC\\). It is known that \\(BK = \\frac{14}{13}\\), \\(AN = 10\\), an... | math | {
"style": "rule",
"ground_truth": "\\frac{36503}{1183}"
} | {
"index": 13809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with sides opposite to the angles \\(A\\), \\(B\\), and \\(C\\) being \\(a\\), \\(b\\), and \\(c\\) respectively, and it is known that \\( \\sqrt {3}\\sin A-\\cos (B+C)=1\\) and \\( \\sin B+\\sin C= \\dfrac {8}{7}\\sin A\\) with \\(a=7\\):\n(Ⅰ) Find t... | math | {
"style": "rule",
"ground_truth": "\\dfrac {15 \\sqrt {3}}{4}"
} | {
"index": 13810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with the midpoints of sides \\(BC\\), \\(AC\\), and \\(AB\\) denoted as \\(D\\), \\(E\\), and \\(F\\) respectively, it is known that the medians \\(AD\\) and \\(BE\\) are perpendicular to each other, with lengths \\(\\overline{AD} = 18\\) and \\(\\ove... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 13811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\(PQR\\). Point \\(T\\) is the center of the inscribed circle.\n\nThe rays \\(PT\\) and \\(QT\\) intersect side \\(PQ\\) at points \\(E\\) and \\(F\\) respectively. It is known that the areas of triangles \\(PQR\\) and \\(TFE\\) are equal. What part of side \\... | math | {
"style": "rule",
"ground_truth": "\\frac{3 - \\sqrt{5}}{2}"
} | {
"index": 13812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is ."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 13813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle whose three sides are all positive integers, with only one side length equal to 5 and not the shortest side, find the number of such triangles."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle with integral sides and an isosceles perimeter of 11, calculate the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{2.75}}{2}"
} | {
"index": 13815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle with side lengths \\( l, m, n \\), where \\( l > m > n \\) are integers, and it is known that:\n\\[\n\\left\\{\\frac{3^{l}}{10^{4}}\\right\\} = \\left\\{\\frac{3^{m}}{10^{4}}\\right\\} = \\left\\{\\frac{3^{n}}{10^{4}}\\right\\}\n\\]\nwhere \\( \\{x\\} = x - \\l... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"index": 13816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 13817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$, replace the midpoint triangle of every face o... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 13818,
"solution": "On the first construction, $P_1$, four new tetrahedra will be constructed with side lengths $\\frac 12$ of the original one. Since the ratio of the volume of similar polygons is the cube of the ratio of their corresponding lengths, it follows that each of these new tetrahedra will hav... |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular prism $S-ABC$, where the base is an isosceles right triangle with $AB$ as the hypotenuse, $SA = SB = SC = 2$, and $AB = 2$, and points $S$, $A$, $B$, and $C$ all lie on a sphere centered at point $O$, find the distance from point $O$ to the plane $ABC$."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular prism \\( S-ABC \\) with a base that is an isosceles right triangle with \\( AB \\) as the hypotenuse, and \\( SA = SB = SC = AB = 2 \\). If the points \\( S, A, B, C \\) all lie on the surface of a sphere centered at \\( O \\), what is the surface area of th... | math | {
"style": "rule",
"ground_truth": "\\frac{16 \\pi}{3}"
} | {
"index": 13820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\\bot $ plane $ABC$, $\\angle ACB=60^{\\circ}$, $AB=3\\sqrt{2}$, and $DC=2\\sqrt{3}$, calculate the surface area of sphere $O$."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 13821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid $S-ABC$ whose base is an equilateral triangle, the projection of point $A$ on the face $SBC$ is the orthocenter $H$ of triangle $\\triangle SBC$. The dihedral angle between the planes $H-AB-C$ is $30^{\\circ}$, and $SA = 2\\sqrt{3}$. Find the volume o... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"index": 13822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, and $SA = SB = SC = 2$, $AB = 2$, let points $S$, $A$, $B$, and $C$ all lie on the surface of a sphere centered at $O$. What is the distance from point $O$ to the pla... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid \\( P-ABC \\) with a base that is an equilateral triangle of side length \\( 4 \\sqrt{3} \\), and with \\( PA=3 \\), \\( PB=4 \\), and \\( PC=5 \\). If \\( O \\) is the center of the triangle \\( ABC \\), find the length of \\( PO \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 13824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid \\( S-ABC \\) whose base \\( ABC \\) is an isosceles right triangle with \\( AB \\) as the hypotenuse, and satisfying \\( SA = SB = SC = 2 \\) and \\( AB = 2 \\), assume that the four points \\( S, A, B, C \\) are all on the surface of a sphere center... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid \\( S-ABC \\) with a base that is an isosceles right triangle with \\( AB \\) as the hypotenuse, \\( SA = SB = SC = 2 \\), and \\( AB = 2 \\), suppose points \\( S, A, B, C \\) are all on a sphere with center \\( Q \\). Find the distance from point \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid \\( S-ABC \\) with vertex \\( S \\). The projection of \\( S \\) onto the base \\( \\triangle ABC \\) is the orthocenter \\( H \\) of \\( \\triangle ABC \\). Additionally, \\( BC = 2 \\), \\( SB = SC \\), and the dihedral angle between the face \\( SB... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangular pyramid where two of the three lateral faces are isosceles right triangles and the third face is an equilateral triangle with a side length of 1, calculate the volume of this triangular pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"index": 13828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a unit square region $R$ and an integer $n \\geq 4$, determine how many points are $80$-ray partitional but not $50$-ray partitional."
}
] | math | {
"style": "rule",
"ground_truth": "7062"
} | {
"index": 13829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a wire of length \\(150 \\mathrm{~cm}\\) that needs to be cut into \\(n (n>2)\\) smaller pieces, with each piece being an integer length of at least \\(1 \\mathrm{~cm}\\). If any 3 pieces cannot form a triangle, what is the maximum value of \\(n\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a woman was x years old in the year $x^2$, determine her birth year."
}
] | math | {
"style": "rule",
"ground_truth": "1980"
} | {
"index": 13832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given acute angles $ \\alpha $ and $ \\beta $ satisfy $ \\sin \\alpha =\\frac{\\sqrt{5}}{5},\\sin (\\alpha -\\beta )=-\\frac{\\sqrt{10}}{10} $, then $ \\beta $ equals \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 13833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given acute angles $\\alpha$ and $\\beta$ such that $\\sin \\alpha= \\frac { \\sqrt {5}}{5}$ and $\\sin(\\alpha-\\beta)=- \\frac { \\sqrt {10}}{10}$, determine the value of $\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 13834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given acute angles $α$ and $β$ that satisfy: $\\cos α= \\frac {1}{3}$ and $\\cos (α+β)=- \\frac {1}{3}$, find $\\cos (α-β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"index": 13835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given acute angles \\\\(\\alpha\\\\) and \\\\(\\beta\\\\) satisfy \\\\((\\tan \\alpha-1)(\\tan \\beta-1)=2\\\\), then the value of \\\\(\\alpha+\\beta\\\\) is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {3\\pi}{4}"
} | {
"index": 13836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given algebraic expressions $A=2m^{2}+3my+2y-1$ and $B=m^{2}-my$. Find:<br/>\n$(1)$ Simplify $3A-2\\left(A+B\\right)$.<br/>\n$(2)$ If $\\left(m-1\\right)^{2}+|y+2|=0$, find the value of $3A-2\\left(A+B\\right)$.<br/>\n$(3)$ If the value of $3A-2\\left(A+B\\right)$ is independen... | math | {
"style": "rule",
"ground_truth": "-0.4"
} | {
"index": 13837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an $8 \\times 6$ grid, consider a triangle with vertices at $D=(2,1)$, $E=(7,1)$, and $F=(5,5)$. Determine the fraction of the grid covered by this triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{24}"
} | {
"index": 13838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an $8 \\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?"
}
] | math | {
"style": "rule",
"ground_truth": "20736"
} | {
"index": 13839,
"solution": "Number both the rows and the columns from 1 to 8, and say that black squares are the ones where the rows and columns have the same parity. We will use, e.g. 'even rows' to refer to rows 2, 4, 6,8. Choosing 8 squares all in different rows and columns is equivalent to matching rows to c... |
DeepScaleR | [
{
"role": "user",
"content": "Given an acute angle \\( \\theta \\), the equation \\( x^{2} + 4x \\cos \\theta + \\cot \\theta = 0 \\) has a double root. Find the radian measure of \\( \\theta \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"index": 13840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an acute triangle $\\triangle ABC$ where the internal angles $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, and $a=3$, $b=\\sqrt{13}$. If $a\\sin 2B=b\\sin A$. Find:<br/>\n$(1)$ the value of angle $B$;<br/>\n$(2)$ if point $D$ satisfies $\\overrig... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{19}}{3}"
} | {
"index": 13841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an angle $a$ and a point $P(-4,3)$ on its terminal side, find the value of $\\dfrac{\\cos \\left( \\dfrac{\\pi}{2}+a\\right)\\sin \\left(-\\pi-a\\right)}{\\cos \\left( \\dfrac{11\\pi}{2}-a\\right)\\sin \\left( \\dfrac{9\\pi}{2}+a\\right)}$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac{3}{4}"
} | {
"index": 13842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an angle $α$ whose terminal side passes through the point $(-1, \\sqrt{2})$, find the values of $\\tan α$ and $\\cos 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 13843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an angle \\(\\theta\\), consider the polynomial \\(P(x)=\\sin(\\theta)x^{2}+\\cos(\\theta)x+\\tan(\\theta)x+1\\). Given that \\(P\\) only has one real root, find all possible values of \\(\\sin(\\theta)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "0, \\frac{\\sqrt{5}-1}{2}"
} | {
"index": 13844,
"solution": "Note that if \\(\\sin(\\theta)=0\\), then the polynomial has 1 root. Now assume this is not the case then the polynomial is a quadratic in \\(x\\). Factor the polynomial as \\((\\tan(\\theta)x+1)(x+\\sec(\\theta))\\). Then the condition is equivalent to \\(\\sec(\\theta)=\\frac{1}{\\t... |
DeepScaleR | [
{
"role": "user",
"content": "Given an angle measuring $54^{\\circ}$, use only a compass to divide it into three equal parts (that is, find such points that rays passing through the vertex of the given angle and these points divide the angle into three equal parts)."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an angle α with its vertex at the origin of coordinates, its initial side coinciding with the non-negative half-axis of the x-axis, and two points on its terminal side A(1,a), B(2,b), and cos(2α) = 2/3, determine the value of |a-b|."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{5}"
} | {
"index": 13846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$."
}
] | math | {
"style": "rule",
"ground_truth": "1- \\dfrac { \\sqrt {3}\\pi}{24}"
} | {
"index": 13847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arc length of 50cm and the central angle corresponding to the arc is 200°, (1) find the radius of the circle containing this arc, (2) find the area of the sector formed by this arc and the radius."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1125}{\\pi}"
} | {
"index": 13848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\\\{a_{n}\\\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:\n\n$(1)$ Find the general term formula for the sequence;\n$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "-90"
} | {
"index": 13849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\left\\{a_{n}\\right\\}$ with the first term $a_{1}>0$, and the following conditions:\n$$\na_{2013} + a_{2014} > 0, \\quad a_{2013} a_{2014} < 0,\n$$\n\nfind the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds... | math | {
"style": "rule",
"ground_truth": "4026"
} | {
"index": 13850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\left\\{a_{n}\\right\\}$ with the sum of the first 12 terms being 60, find the minimum value of $\\left|a_{1}\\right| + \\left|a_{2}\\right| + \\cdots + \\left|a_{12}\\right|$."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ where all terms are positive, the sum of the first $n$ terms is $S\\_n$. If $S\\_n=2$ and $S\\_3n=14$, find $S\\_6n$."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 13852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with a common difference $d > 0$, and $a\\_2$, $a\\_5-1$, $a\\_{10}$ form a geometric sequence. If $a\\_1=5$, and $S\\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\\frac{2S\\_n+n+32}{a\\_n+1... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"index": 13853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with a common ratio $q > 1$, and it satisfies: $a\\_2 + a\\_3 + a\\_4 = 28$, and $a\\_3 + 2$ is the arithmetic mean of $a\\_2$ and $a\\_4$.\n(1) Find the general term formula of the sequence $\\{a\\_n\\}$;\n(2) If $b\\_n = a\\_n \\log\... | math | {
"style": "rule",
"ground_truth": "n = 6"
} | {
"index": 13854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with first term $a\\_1$ and common difference $d$, let $S\\_n$ denote the sum of its first $n$ terms. If the two intersection points of the line $y=a\\_1x+m$ and the circle $x^2+(y-1)^2=1$ are symmetric about the line $x+y-d=0$, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{200}{101}"
} | {
"index": 13855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, the sum of its first $n$ terms, $S\\_n$, satisfies $S\\_3=0$ and $S\\_5=-5$. The sum of the first 2016 terms of the sequence $\\{ \\frac{1}{a_{2n-1}a_{2n+1}} \\}$ is $\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2016}{4031}"
} | {
"index": 13856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_1+a\\_2=3$, $a\\_4+a\\_5=5$.\n\n(I) Find the general term formula of the sequence.\n(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\\_n=[a\\_1]+[a\\_2]+…+[a\\_n]$. Find... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 13857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_1=\\tan 225^{\\circ}$ and $a\\_5=13a\\_1$, let $S\\_n$ denote the sum of the first $n$ terms of the sequence $\\{(-1)^na\\_n\\}$. Determine the value of $S\\_{2015}$."
}
] | math | {
"style": "rule",
"ground_truth": "-3022"
} | {
"index": 13858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_n \\in \\mathbb{N}^*$, and $S\\_n = \\frac{1}{8}(a\\_n + 2)^2$. If $b\\_n = \\frac{1}{2}a\\_n - 30$, find the minimum value of the sum of the first $\\_\\_\\_\\_\\_\\_$ terms of the sequence $\\{b\\_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ that satisfies $a_3 + a_5 = 14$ and $a_2a_6 = 33$, find the value of $a_1a_7$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ where $a_1=1$ and $a_n=70$ (for $n\\geq3$), find all possible values of $n$ if the common difference is a natural number."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 13861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ where the sum of the first $n$ terms is $S_n = (a+1)n^2 + a$, if the sides of a certain triangle are $a_2$, $a_3$, and $a_4$, then the area of this triangle is ________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{4} \\sqrt{3}"
} | {
"index": 13862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \\ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \\ldots + a_{99}$."
}
] | math | {
"style": "rule",
"ground_truth": "-66"
} | {
"index": 13863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\\{a_n^2\\}$ is $S_n$. A geometric sequence $\\{b_n\\}$ has a common ratio $q$ less than 1 and consists of rational sine v... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$, and $S_n$ as the sum of its first $n$ terms, if $a_2$, $a_3$, and $a_6$ form a geometric sequence and $a_{10} = -17$, determine the minimum value of $\\frac{S_n}{2^n}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\\frac{2S_n+8}{a_n+3}$ for all positive integers $n$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 13866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference of $2$ and an even number of terms, the sum of all odd terms is $15$, and the sum of all even terms is $25$, calculate the number of terms in this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a non-zero common difference, the sum of the first $n$ terms is $S_n$, satisfying $S_3 = a_4 + 4$, and $a_2$, $a_6$, $a_{18}$ form a geometric sequence.\n1. Find the general formula for the sequence $\\{a_n\\}$.\n2. Let $b_n = \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$, determine the minimum value of $$\\frac{2S_{n} + ... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"index": 13869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with common difference $d \\neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\\{a_n^2\\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\\{b_n\\}$ with a common ratio $q$ that is a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\\in \\mathbb{N}^*$ \n$(1)$ Find $k$ and $a_n$ \n$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\\{a_n\\}$ is $S... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{13}-1}{2}"
} | {
"index": 13871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term $a$ ($a \\in \\mathbb{R}, a \\neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\\frac {a_{2n}}{a_{n}}= \\frac {4n-1}{2n-1}$$.\n(1) Find the genera... | math | {
"style": "rule",
"ground_truth": "k=3"
} | {
"index": 13872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term $a_1=11$ and common difference $d=2$, and $a_n=2009$, find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 13873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for whi... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 13874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_1 > 0$, $a_3 + a_{10} > 0$, $a_6a_7 < 0$, determine the maximum natural number $n$ that satisfies $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, if $\\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms $S_n$ has a maximum value, find the maximum value of $n$ for which $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, if $d < 0$, $T_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $T_3=15$, also $a_1+1, a_2+3, a_3+9$ form a geometric sequence with common ratio $q$, calculate the value of $q$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, it is known that $\\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, it is known that $\\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\\{b_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {175}{264}"
} | {
"index": 13880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \\leq 6$ and $S_{11} \\geq 27$. Determine the minimum value of $S_{19}$."
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"index": 13881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_2=1$, $S_4=8$. Find $a_5$ and $S_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 13882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$. If $\\frac{S_6}{S_3} = 3$, then calculate $\\frac{S_{12}}{S_{9}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 13883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}} \\right\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100}{101}"
} | {
"index": 13884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \\leqslant S_5$. The sum of the first $9$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}} \\right\\}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{9}"
} | {
"index": 13885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, where $a_n \\in \\mathbb{N}^*$, and $S_n = \\frac{1}{8}(a_n+2)^2$. If $b_n = \\frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \\_\\_\\_\\_\\_\\_ terms of the sequence $\\{b_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ and $\\{b_{n}\\}$ with the sums of the first $n$ terms being $S_{n}$ and $T_{n}$, respectively, if $\\frac{S_n}{T_n}=\\frac{3n+4}{n+2}$, find $\\frac{a_3+a_7+a_8}{b_2+b_{10}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{111}{26}"
} | {
"index": 13887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ and $\\{b_{n}\\}$, where the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively, and $\\left(2n+3\\right)S_{n}=nT_{n}$, calculate the value of $\\frac{{{a_5}}}{{{b_6}}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{25}"
} | {
"index": 13888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with a common difference of $\\frac{{2π}}{3}$, let $S=\\{\\cos a_{n}|n\\in N^{*}\\}$. If $S=\\{a,b\\}$, find the value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with a non-zero common difference, $S_{n}$ represents the sum of its first $n$ terms. If $S_{5}=0$, then the number of distinct values in $S_{i}$ for $i=1,2,\\ldots,100$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 13890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\\{b_{n}\\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\\{a... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"index": 13891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\\{b_{n}\\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$. <br/>$(1)$ If $b_{2}=-2a_{4}$, find the ge... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms as $S_{n}$, if $a_{2}+a_{4}+3a_{7}+a_{9}=24$, calculate the value of $S_{11}$."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 13893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms as $S_{n}$, where the common difference $d\\neq 0$, and $S_{3}+S_{5}=50$, $a_{1}$, $a_{4}$, $a_{13}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{27}"
} | {
"index": 13894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$, $\\{b_{n}\\}$, with their sums of the first $n$ terms being $S_{n}$ and $T_{n}$ respectively, and satisfying $\\frac{S_n}{T_n}=\\frac{n+3}{2n-1}$, find $\\frac{a_5}{T_9}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{51}"
} | {
"index": 13895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$, where $a_{1}+a_{8}=2a_{5}-2$ and $a_{3}+a_{11}=26$, calculate the sum of the first 2022 terms of the sequence $\\{a_{n} \\cdot \\cos n\\pi\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 13896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence ${\\_{a\\_n}}$ with a non-zero common difference $d$, and $a\\_7$, $a\\_3$, $a\\_1$ are three consecutive terms of a geometric sequence ${\\_{b\\_n}}$.\n\n(1) If $a\\_1=4$, find the sum of the first 10 terms of the sequence ${\\_{a\\_n}}$, denoted a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 13897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 13898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an arithmetic sequence ${a_{n}}$, let $S_{n}$ denote the sum of its first $n$ terms. The first term $a_{1}$ is given as $-20$. The common difference is a real number in the interval $(3,5)$. Determine the probability that the minimum value of $S_{n}$ is only $S_{6}$."
}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"index": 13899,
"solution": ""
} |
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