data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, calculate his age in 2013."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 15200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that YQZC is a rectangle with YC = 8 and CZ = 15, and equilateral triangles ABC and PQR each with a side length of 9, where R and B are on sides YQ and CZ, respectively, determine the length of AP."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 15201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( 0<a<b<c<d<300 \\) and the equations:\n\\[ a + d = b + c \\]\n\\[ bc - ad = 91 \\]\n\nFind the number of ordered quadruples of positive integers \\((a, b, c, d)\\) that satisfy the above conditions."
}
] | math | {
"style": "rule",
"ground_truth": "486"
} | {
"index": 15202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \\), find the value of \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 15203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( 2^{a} \\times 3^{b} \\times 5^{c} \\times 7^{d} = 252000 \\), what is the probability that a three-digit number formed by any 3 of the natural numbers \\( a, b, c, d \\) is divisible by 3 and less than 250?"
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 15204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( 5^{\\log 30} \\times \\left(\\frac{1}{3}\\right)^{\\log 0.5} = d \\), find the value of \\( d \\)."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 15205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( 990 \\times 991 \\times 992 \\times 993 = \\overline{966428 A 91 B 40} \\), find the values of \\( \\overline{A B} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 15206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( A \\) and \\( B \\) are two distinct points on the parabola \\( y = 3 - x^2 \\) that are symmetric with respect to the line \\( x + y = 0 \\), calculate the distance |AB|."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 15207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( A \\) and \\( B \\) are two points on the surface of a sphere with a radius of 5, and \\( AB = 8 \\). Planes \\( O_1AB \\) and \\( O_2AB \\) are perpendicular to each other and pass through \\( AB \\). The intersections of these planes with the sphere create cros... | math | {
"style": "rule",
"ground_truth": "41 \\pi"
} | {
"index": 15208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( A \\), \\( B \\), and \\( C \\) are any three non-collinear points on a plane, and point \\( O \\) is inside \\( \\triangle ABC \\) such that:\n\\[\n\\angle AOB = \\angle BOC = \\angle COA = 120^\\circ.\n\\]\nFind the maximum value of \\( \\frac{OA + OB + OC}{AB ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 15209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( ABC - A_1B_1C_1 \\) is a right prism with \\(\\angle BAC = 90^\\circ\\), points \\( D_1 \\) and \\( F_1 \\) are the midpoints of \\( A_1B_1 \\) and \\( B_1C_1 \\), respectively. If \\( AB = CA = AA_1 \\), find the cosine of the angle between \\( BD_1 \\) and \\( ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30}}{10}"
} | {
"index": 15210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( AE \\) and \\( BD \\) are straight lines that intersect at \\( C \\), \\( BD = 16 \\), \\( AB = 9 \\), \\( CE = 5 \\), and \\( DE = 3 \\), calculate the length of \\( AC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 15211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( F \\) is the right focus of the hyperbola \\( x^{2} - y^{2} = 1 \\), \\( l \\) is the right directrix of the hyperbola, and \\( A \\) and \\( B \\) are two moving points on the right branch of the hyperbola such that \\( A F \\perp B F \\). The projection of the ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 15212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( F_1 \\) and \\( F_2 \\) are the left and right foci of the ellipse \\( C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) where \\( a > b > 0 \\), and \\( P \\) is a point on the ellipse \\( C \\). The incenter of triangle \\( \\triangle F_{1}PF_{2} \\) is \\( I ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( I \\) is the incenter of \\( \\triangle ABC \\) and \\( 5 \\overrightarrow{IA} = 4(\\overrightarrow{BI} + \\overrightarrow{CI}) \\). Let \\( R \\) and \\( r \\) be the radii of the circumcircle and the incircle of \\( \\triangle ABC \\) respectively. If \\( r = 1... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 15214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( I \\) is the incenter of \\( \\triangle ABC \\), and\n\\[ 9 \\overrightarrow{CI} = 4 \\overrightarrow{CA} + 3 \\overrightarrow{CB}. \\]\n\nLet \\( R \\) and \\( r \\) be the circumradius and inradius of \\( \\triangle ABC \\), respectively. Find \\(\\frac{r}{R} =... | math | {
"style": "rule",
"ground_truth": "5/16"
} | {
"index": 15215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( M \\) is a subset of \\(\\{1, 2, 3, \\cdots, 15\\}\\) such that the product of any 3 distinct elements of \\( M \\) is not a perfect square, determine the maximum possible number of elements in \\( M \\)."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 15216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( M \\) is the midpoint of the height \\( D D_{1} \\) of a regular tetrahedron \\( ABCD \\), find the dihedral angle \\( A-M B-C \\) in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 15217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( O \\) is the circumcenter of \\( \\triangle ABC \\), and the equation\n\\[\n\\overrightarrow{A O} \\cdot \\overrightarrow{B C} + 2 \\overrightarrow{B O} \\cdot \\overrightarrow{C A} + 3 \\overrightarrow{C O} \\cdot \\overrightarrow{A B} = 0,\n\\]\nfind the minimu... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 15218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( O \\) is the circumcenter of \\(\\triangle ABC\\), and \\( 3 \\overrightarrow{OA} + 4 \\overrightarrow{OB} + 5 \\overrightarrow{OC} = \\overrightarrow{0} \\), find the value of \\( \\cos \\angle BAC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{10}"
} | {
"index": 15219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( O \\) is the circumcenter of \\(\\triangle ABC\\), where \\(|AB|=2\\), \\(|AC|=1\\), and \\(\\angle BAC = \\frac{2}{3} \\pi\\). Let \\(\\overrightarrow{AB} = \\mathbf{a}\\) and \\(\\overrightarrow{AC} = \\mathbf{b}\\). If \\(\\overrightarrow{AO} = \\lambda_1 \\ma... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{6}"
} | {
"index": 15220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( P Q R S \\) is a square, and point \\( O \\) is on the line \\( R Q \\) such that the distance from \\( O \\) to point \\( P \\) is 1, what is the maximum possible distance from \\( O \\) to point \\( S \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"index": 15221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( P \\) is a point on the hyperbola \\( C: \\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1 \\), and \\( F_{1} \\) and \\( F_{2} \\) are the left and right foci of the hyperbola \\( C \\), and \\( M \\) and \\( I \\) are the centroid and incenter of the triangle \\( \\tria... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 15222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( P \\) is a point on the hyperbola \\( C: \\frac{x^{2}}{4}-\\frac{y^{2}}{12}=1 \\), \\( F_{1} \\) and \\( F_{2} \\) are the left and right foci of \\( C \\), and \\( M \\) and \\( I \\) are the centroid and incenter of \\(\\triangle P F_{1} F_{2}\\) respectively, ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 15223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( P \\) is any point on the arc \\( \\overparen{AD} \\) of the circumscribed circle of square \\( ABCD \\), find the value of \\( \\frac{PA + PC}{PB} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 15224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( P \\) is the vertex of a right circular cone with an isosceles right triangle as its axial cross-section, \\( PA \\) is a generatrix of the cone, and \\( B \\) is a point on the base of the cone. Find the maximum value of \\(\\frac{PA + AB}{PB}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{4 + 2\\sqrt{2}}"
} | {
"index": 15225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( PQ = 4 \\), \\( QR = 8 \\), \\( RS = 8 \\), and \\( ST = 3 \\), if \\( PQ \\) is perpendicular to \\( QR \\), \\( QR \\) is perpendicular to \\( RS \\), and \\( RS \\) is perpendicular to \\( ST \\), calculate the distance from \\( P \\) to \\( T \\)."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 15226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( \\cos A + \\cos B + \\cos C = \\sin A + \\sin B + \\sin C = 0 \\), find the value of \\( \\cos^4 A + \\cos^4 B + \\cos^4 C \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"index": 15227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( \\mathrm{a}, \\mathrm{b}, \\mathrm{c} \\) are three natural numbers, and the least common multiple (LCM) of \\( \\mathrm{a} \\) and \\( \\mathrm{b} \\) is 60, and the LCM of \\( \\mathrm{a} \\) and \\( \\mathrm{c} \\) is 270, find the LCM of \\( \\mathrm{b} \\) a... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 15228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( \\text{rem} \\left(\\frac{5}{7}, \\frac{3}{4}\\right) \\) must be calculated, determine the value of the remainder."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 15229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a > 0 \\), if \\( f(g(h(a))) = 17 \\), where \\( f(x) = x^2 + 5 \\), \\( g(x) = x^2 - 3 \\), and \\( h(x) = 2x + 1 \\), what is the value of \\( a \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{3 + 2\\sqrt{3}}}{2}"
} | {
"index": 15230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a \\) and \\( b \\) are positive integers, and \\( b - a = 2013 \\). If the equation \\( x^{2} - a x + b = 0 \\) has a positive integer solution, what is the smallest value of \\( a \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 15231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a \\) and \\( b \\) are real numbers, and the following system of inequalities in terms of \\( x \\):\n\\[\n\\left\\{\\begin{array}{l}\n20x + a > 0, \\\\\n15x - b \\leq 0\n\\end{array}\\right.\n\\]\nhas integer solutions of only 2, 3, and 4, find the maximum valu... | math | {
"style": "rule",
"ground_truth": "-1200"
} | {
"index": 15232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a \\) is a positive real number and \\( b \\) is an integer between \\( 2 \\) and \\( 500 \\), inclusive, find the number of ordered pairs \\( (a,b) \\) that satisfy the equation \\( (\\log_b a)^{1001}=\\log_b(a^{1001}) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1497"
} | {
"index": 15233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a \\) is a real number, and for any \\( k \\in [-1,1] \\), when \\( x \\in (0,6] \\), the following inequality is always satisfied:\n\\[ 6 \\ln x + x^2 - 8 x + a \\leq k x. \\]\n\nFind the maximum value of \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 6 \\ln 6"
} | {
"index": 15234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a \\) is an integer, if \\( 50! \\) is divisible by \\( 2^a \\), find the largest possible value of \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 15235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a, b, \\) and \\( c \\) are all positive integers, and the parabola \\( y = ax^2 + bx + c \\) intersects the x-axis at two distinct points \\( A \\) and \\( B \\). If both \\( A \\) and \\( B \\) are less than 1 unit away from the origin, what is the minimum valu... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 15236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a, b, c \\) are positive integers, and the parabola \\( y = ax^2 + bx + c \\) intersects the x-axis at two distinct points \\( A \\) and \\( B \\). If the distances from \\( A \\) and \\( B \\) to the origin are both less than 1, find the minimum value of \\( a +... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 15237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a, b, c, d \\) are prime numbers (they can be the same), and \\( abcd \\) is the sum of 35 consecutive positive integers, find the minimum value of \\( a + b + c + d \\)."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 15238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( a_{k} \\) is the number of integer terms in \\( \\log_{2} k, \\log_{3} k, \\cdots, \\log_{2018} k \\). Calculate \\( \\sum_{k=1}^{2018} a_{k} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4102"
} | {
"index": 15239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( b \\) is an even number between 1 and 11 (inclusive), and \\( c \\) is any natural number, determine the number of quadratic equations \\( x^{2} + b x + c = 0 \\) that have two distinct real roots."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 15240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( b\\left[\\frac{1}{1 \\times 3}+\\frac{1}{3 \\times 5}+\\cdots+\\frac{1}{1999 \\times 2001}\\right]=2 \\times\\left[\\frac{1^{2}}{1 \\times 3}+\\frac{2^{2}}{3 \\times 5}+\\cdots+\\frac{1000^{2}}{1999 \\times 2001}\\right] \\), find the value of \\( b \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"index": 15241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f \\) is a mapping from the set \\( M = \\{a, b, c\\} \\) to the set \\( N = \\{-3, -2, \\cdots, 3\\} \\). Determine the number of mappings \\( f \\) that satisfy\n$$\nf(a) + f(b) + f(c) = 0\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 15242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f(x) \\) and \\( g(x) \\) are two quadratic functions both with a leading coefficient of 1, where \\( g(6) = 35 \\) and \\( \\frac{f(-1)}{g(-1)} = \\frac{f(1)}{g(1)} = \\frac{21}{20} \\), what is \\( f(6) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 15243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f(x) \\) is a polynomial of degree \\( n \\) with non-negative integer coefficients, and that \\( f(1)=6 \\) and \\( f(7)=3438 \\), find \\( f(2) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 15244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f(x) \\) is an odd function defined on \\( \\mathbf{R} \\), and for any \\( x \\in \\mathbf{R} \\), the following holds:\n$$\nf(2+x) + f(2-x) = 0.\n$$\n\nWhen \\( x \\in [-1, 0) \\), it is given that\n$$\nf(x) = \\log_{2}(1-x).\n$$\n\nFind \\( f(1) + f(2) + \\cdo... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f(x) \\) is an odd function with a period of 4, and for \\( x \\in (0,2) \\), \\( f(x) = x^2 - 16x + 60 \\). Find the value of \\( f(2 \\sqrt{10}) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"index": 15246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( f(x-1)=|x|-|x-2| \\) and \\( f(f(m))=f(2002)-\\frac{7}{2} \\), find the value of the real number \\( m \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 15247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( m \\) and \\( n \\) are two distinct positive integers and the last four digits of \\( 2019^{m} \\) and \\( 2019^{n} \\) are the same, find the minimum value of \\( m+n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"index": 15248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( n \\) is a positive integer and \\( n \\leq 144 \\), ten questions of the type \"Is \\( n \\) smaller than \\( a \\)?\" are allowed. The answers are provided with a delay such that the answer to the \\( i \\)-th question is given only after the \\((i+1)\\)-st que... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( n \\) is a positive integer, \\( P \\) is a prime number, and \\( pn \\) has exactly 8 positive divisors, arrange them in ascending order as \\( 1=d_{1}<d_{2}< \\cdots <d_{8}=pn \\). Additionally, let \\( d_{17p-d_{3}}=\\left(d_{1}+d_{2}+d_{3}\\right)\\left(d_{3}... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 15250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( n! \\) is evenly divisible by \\( 1 + 2 + \\cdots + n \\), find the number of positive integers \\( n \\) less than or equal to 50."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 15251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( n! \\), in decimal notation, has exactly 57 ending zeros, find the sum of all possible values of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1185"
} | {
"index": 15252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( p \\) and \\( q \\) are positive integers such that \\( p + q > 2017 \\), \\( 0 < p < q \\leq 2017 \\), and \\((p, q) = 1\\), find the sum of all fractions of the form \\(\\frac{1}{pq}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 15253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( p \\) is a prime number and \\( r \\) is the remainder when \\( p \\) is divided by 210, if \\( r \\) is a composite number that can be expressed as the sum of two perfect squares, find \\( r \\)."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 15254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( p \\) is a prime number, the decimal part of \\( \\sqrt{p} \\) is \\( x \\). The decimal part of \\( \\frac{1}{x} \\) is \\( \\frac{\\sqrt{p} - 31}{75} \\). Find all prime numbers \\( p \\) that satisfy these conditions."
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 15255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( r, s, t \\) are integers, and the set \\( \\{a \\mid a = 2^r + 2^s + 2^t, 0 \\leq t < s < r\\} \\) forms a sequence \\(\\{a_n\\} \\) from smallest to largest as \\(7, 11, 13, 14, \\cdots\\), find \\( a_{36} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "131"
} | {
"index": 15256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( t = \\frac{1}{1 - \\sqrt[4]{2}} \\), simplify the expression for \\( t \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-(1 + \\sqrt[4]{2})(1 + \\sqrt{2})"
} | {
"index": 15257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x + y + z = xy + yz + zx \\), find the minimum value of \\( \\frac{x}{x^2 + 1} + \\frac{y}{y^2 + 1} + \\frac{z}{z^2 + 1} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1/2"
} | {
"index": 15258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x - \\frac{1}{x} = 5 \\), find the value of \\( x^4 - \\frac{1}{x^4} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "727"
} | {
"index": 15259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x - \\frac{1}{x} = \\sqrt{3} \\), find \\( x^{2048} - \\frac{1}{x^{2048}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "277526"
} | {
"index": 15260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are non-zero real numbers and they satisfy \\(\\frac{x \\sin \\frac{\\pi}{5} + y \\cos \\frac{\\pi}{5}}{x \\cos \\frac{\\pi}{5} - y \\sin \\frac{\\pi}{5}} = \\tan \\frac{9 \\pi}{20}\\),\n1. Find the value of \\(\\frac{y}{x}\\).\n2. In \\(\\tri... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 15261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are positive integers such that \\( 56 \\leq x + y \\leq 59 \\) and \\( 0.9 < \\frac{x}{y} < 0.91 \\), find the value of \\( y^2 - x^2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"index": 15262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are positive numbers, determine the minimum value of \\(\\left(x+\\frac{1}{y}\\right)^{2}+\\left(y+\\frac{1}{2x}\\right)^{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2 \\sqrt{2}"
} | {
"index": 15263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are real numbers greater than 10, the leading digit of \\( \\lg x \\) is \\( a \\) and the trailing digit is \\( b \\); the leading digit of \\( \\lg y \\) is \\( c \\) and the trailing digit is \\( d \\). Additionally, it is known that \\( |1... | math | {
"style": "rule",
"ground_truth": "10^7"
} | {
"index": 15264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are real numbers satisfying the following equations:\n\\[\nx + xy + y = 2 + 3 \\sqrt{2} \\quad \\text{and} \\quad x^2 + y^2 = 6,\n\\]\nfind the value of \\( |x + y + 1| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{2}"
} | {
"index": 15265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) is a four-digit number and the sum of its digits is \\( y \\). When the value of \\( \\frac{x}{y} \\) is minimized, \\( x = \\) _______"
}
] | math | {
"style": "rule",
"ground_truth": "1099"
} | {
"index": 15266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) is a multiple of \\( 7200 \\), what is the greatest common divisor of \\( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \\) and \\( x \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 15267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x \\) is a real number and \\( y = \\sqrt{x^2 - 2x + 2} + \\sqrt{x^2 - 10x + 34} \\). Find the minimum value of \\( y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 15268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x, y, z \\in \\mathbf{R}_{+} \\) and \\( x^{2} + y^{2} + z^{2} = 1 \\), find the value of \\( z \\) when \\(\\frac{(z+1)^{2}}{x y z} \\) reaches its minimum."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 15269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x_{1}, x_{2}, \\cdots, x_{n} \\) are real numbers, find the minimum value of\n\\[ \nE(x_{1}, x_{2}, \\cdots, x_{n}) = \\sum_{i=1}^{n} x_{i}^{2} + \\sum_{i=1}^{n-1} x_{i} x_{i+1} + \\sum_{i=1}^{n} x_{i} \n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-1/2"
} | {
"index": 15270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x_{1}, x_{2}, x_{3}, x_{4} \\) are all positive numbers and \\( x_{1} + x_{2} + x_{3} + x_{4} = \\pi \\), find the minimum value of the expression \\(\\left(2 \\sin^2 x_{1}+\\frac{1}{\\sin^2 x_{1}}\\right)\\left(2 \\sin^2 x_{2}+\\frac{1}{\\sin^2 x_{2}}\\right)\\l... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 15271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( x_{i}=\\frac{i}{101} \\), find the value of \\( S=\\sum_{i=0}^{101} \\frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 15272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( z_{1} \\) and \\( z_{2} \\) are complex numbers with \\( \\left|z_{1}\\right|=3 \\), \\( \\left|z_{2}\\right|=5 \\), and \\( \\left|z_{1}+z_{2}\\right|=7 \\), find the value of \\( \\arg \\left(\\left(\\frac{z_{2}}{z_{1}}\\right)^{3}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 15273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\( |z| = 2 \\) and \\( u = \\left|z^{2} - z + 1\\right| \\), find the minimum value of \\( u \\) (where \\( z \\in \\mathbf{C} \\))."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} \\sqrt{3}"
} | {
"index": 15274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(0 \\leq a_{k} \\leq 1 \\) for \\(k=1,2, \\ldots, 2020\\), and defining \\(a_{2021} = a_{1}\\), \\(a_{2022} = a_{2}\\), find the maximum value of \\(\\sum_{k=1}^{2020}\\left(a_{k} - a_{k+1}a_{k+2}\\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 15275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(1 \\leqslant a_{1} \\leqslant a_{2} \\leqslant a_{3} \\leqslant a_{4} \\leqslant a_{5} \\leqslant a_{6} \\leqslant 64\\), find the minimum value of \\(Q = \\frac{a_{1}}{a_{2}} + \\frac{a_{3}}{a_{4}} + \\frac{a_{5}}{a_{6}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"index": 15276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(11 \\cdot 14 n\\) is a non-negative integer and \\(f\\) is defined by \\(f(0)=0\\), \\(f(1)=1\\), and \\(f(n)=f\\left(\\left\\lfloor \\frac{n}{2} \\right\\rfloor \\right)+n-2\\left\\lfloor \\frac{n}{2} \\right\\rfloor\\), find the maximum value of \\(f(n)\\) for \... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 15277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(15^{-1} \\equiv 31 \\pmod{53}\\), find \\(38^{-1} \\pmod{53}\\), as a residue modulo 53."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 15278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(2 \\cdot 50N\\) is an integer and its representation in base \\(b\\) is 777, find the smallest positive integer \\(b\\) such that \\(N\\) is a fourth power of an integer."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 15279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(7^{-1} \\equiv 55 \\pmod{101}\\), find \\(49^{-1} \\pmod{101}\\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 15280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(9^{-1} \\equiv 90 \\pmod{101}\\), find \\(81^{-1} \\pmod{101}\\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 15281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(A, B, C,\\) and \\(D\\) are points on a circle with radius 1, \\(\\overrightarrow{AB} + 2 \\overrightarrow{AC} = \\overrightarrow{AD}\\), and \\(|AC| = 1\\). Find the area of the quadrilateral \\(ABDC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{4}"
} | {
"index": 15282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(ABCD-A_{1}B_{1}C_{1}D_{1}\\) is a cube and \\(P-A_{1}B_{1}C_{1}D_{1}\\) is a regular tetrahedron, find the cosine of the angle between the skew lines \\(A_{1}P\\) and \\(BC_{1}\\), given that the distance from point \\(P\\) to plane \\(ABC\\) is \\(\\frac{3}{2}AB\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 15283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(ABCD\\) is a square, points \\(E\\) and \\(F\\) lie on the side \\(BC\\) and \\(CD\\) respectively, such that \\(BE = CF = \\frac{1}{3} AB\\). \\(G\\) is the intersection of \\(BF\\) and \\(DE\\). If\n\\[\n\\frac{\\text{Area of } ABGD}{\\text{Area of } ABCD} = \\f... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 15284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(AD\\), \\(BE\\), and \\(CF\\) are the altitudes of the acute triangle \\(\\triangle ABC\\). If \\(AB = 26\\) and \\(\\frac{EF}{BC} = \\frac{5}{13}\\), what is the length of \\(BE\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 15285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\([x]\\) represents the largest integer not exceeding \\( x \\), if \\([x+0.1] + [x+0.2] + \\ldots + [x+0.9] = 104\\), what is the minimum value of \\( x \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "11.5"
} | {
"index": 15286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\) and \\(\\sin \\beta = 2 \\cos (\\alpha + \\beta) \\cdot \\sin \\alpha \\left(\\alpha + \\beta \\neq \\frac{\\pi}{2}\\right)\\), find the maximum value of \\(\\tan \\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 15287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta \\in \\left(\\frac{3\\pi}{4}, \\pi \\right)\\), \\(\\cos (\\alpha + \\beta) = \\frac{4}{5}\\), and \\(\\sin \\left(\\alpha - \\frac{\\pi}{4}\\right) = \\frac{12}{13}\\), find \\(\\cos \\left(\\beta + \\frac{\\pi}{4}\\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{56}{65}"
} | {
"index": 15288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta, \\gamma\\) satisfy \\(0<\\alpha<\\beta<\\gamma<2 \\pi\\), and for any \\(x \\in \\mathbf{R}\\), \\(\\cos (x+\\alpha) + \\cos (x+\\beta) + \\cos (x+\\gamma) = 0\\), determine the value of \\(\\gamma - \\alpha\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"index": 15289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha\\) and \\(\\beta\\) are acute angles, and the following equations hold:\n$$\n\\left\\{\\begin{array}{l}\n3 \\sin ^{2} \\alpha + 2 \\sin ^{2} \\beta = 1, \\\\\n3 \\sin 2 \\alpha - 2 \\sin 2 \\beta = 0.\n\\end{array}\\right.\n$$\n\nDetermine \\(\\alpha + 2\\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 15290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha\\) and \\(\\beta\\) are both acute angles, and\n$$(1+\\tan \\alpha)(1+\\tan \\beta)=2.$$\nFind \\(\\alpha + \\beta =\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 15291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha\\) is an acute angle and \\(\\beta\\) is an obtuse angle, and \\(\\sec (\\alpha - 2\\beta)\\), \\(\\sec \\alpha\\), and \\(\\sec (\\alpha + 2\\beta)\\) form an arithmetic sequence, find the value of \\(\\frac{\\cos \\alpha}{\\cos \\beta}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 15292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\alpha\\), \\(\\beta\\), and \\(\\gamma\\) are acute angles such that \\(\\cos^2 \\alpha + \\cos^2 \\beta + \\cos^2 \\gamma = 1\\), find the minimum value of \\(\\tan \\alpha \\cdot \\tan \\beta \\cdot \\tan \\gamma\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\frac{810 \\times 811 \\times 812 \\times \\cdots \\times 2010}{810^{n}}\\) is an integer, find the maximum value of \\(n\\)."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 15294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\frac{x+y}{x-y}+\\frac{x-y}{x+y}=3\\). Find the value of the expression \\(\\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "13/6"
} | {
"index": 15295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\log_8 2 = 0.2525\\) in base 8 (to 4 decimal places), find \\(\\log_8 4\\) in base 8 (to 4 decimal places)."
}
] | math | {
"style": "rule",
"ground_truth": "0.5050"
} | {
"index": 15296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\sin (x + \\sin x) = \\cos (x - \\cos x)\\), where \\(x \\in [0, \\pi]\\). Find \\(x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 15297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\tan \\frac{\\alpha+\\beta}{2}=\\frac{\\sqrt{6}}{2}\\) and \\(\\cot \\alpha \\cdot \\cot \\beta=\\frac{7}{13}\\), find the value of \\(\\cos (\\alpha-\\beta)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 15298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(\\theta\\) is an angle in the third quadrant, and \\(\\sin^{4}\\theta + \\cos^{4}\\theta = \\frac{5}{9}\\), determine the value of \\(\\sin 2\\theta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{2}}{3}"
} | {
"index": 15299,
"solution": ""
} |
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