data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP(\"C\",(0,0),SW);MP(\"A\",(0,3),NW);MP(\"B\",(4,0),S);MP(\"E\",(7,0),SE);MP(\"D\",(7,10),NE); [/asy]\nTriangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 39100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy] draw((0,0)--(2,2)--(5/2,1/2)--(2,0)--cycle,dot); MP(\"A\",(0,0),W);MP(\"B\",(2,2),N);MP(\"C\",(5/2,1/2),SE);MP(\"D\",(2,0),S); MP(\"a\",(1,0),N);MP(\"b\",(17/8,1/8),N); [/asy]\nIn the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is tw... | math | {
"style": "rule",
"ground_truth": "a+b"
} | {
"index": 39101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP(\"B\",(-8/3,16*sqrt(2)/3),W);MP(\"B'\",(8/3,16*sqrt(2)/3),E);... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 39102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP(\"A\",(0,0),SW);MP(\"B\",(8,0),SE);MP(\"C\",(8,6),NE);MP(\"P\",(4,1),NW); MP(\"8\",(4,0),S);MP(\"6\",(8,3),E);MP(\"10\",(4,3),NW); MP(\"->\",(5,1),E); dot((4,1)); [/asy]T... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 39103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((7,0),1),white); fill(circle((3,0),3),white); draw((0,0)--(8,0),black+linewidth(1)); draw((6,0)--(6,sqrt(12)),black+linewidth(1)); MP(\"A\", (0,0), W); MP(\"B\", (8,0), E); MP(\"C\"... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"index": 39104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(... | math | {
"style": "rule",
"ground_truth": "4.2"
} | {
"index": 39105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[asy]size(8cm);\n\nreal w = 2.718; // width of block\nreal W = 13.37; // width of the floor\nreal h = 1.414; // height of block\nreal H = 7; // height of block + string\nreal t = 60; // measure of theta\n\npair apex = (w/2, H); // point where the strings meet\n\npath block = (0... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 39106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[b]Problem Section #1\na) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .\n\n<span style=\"color:red\">NOTE: There is a high chance ... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"index": 39107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n?"
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"index": 39108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,\n\\[\n\\sqrt[N]{\\frac{x^{2 N}+1}{2}} \\leqslant a_{n}(x-1)^{2}+x .\n\\]\n[i]Version 2[/i]. For every positive integer $N$, determine the ... | math | {
"style": "rule",
"ground_truth": "{a_n = 2^{n-1}}"
} | {
"index": 39109,
"solution": "\nWe are tasked with finding the smallest real number \\( a_n \\) for a given positive integer \\( n \\) and \\( N = 2^n \\), such that the inequality\n\n\\[\n\\sqrt[N]{\\frac{x^{2N} + 1}{2}} \\leq a_{n}(x-1)^{2} + x\n\\]\n\nholds for all real \\( x \\).\n\n### Step-by-Step Analysis:\... |
DeepScaleR | [
{
"role": "user",
"content": "\\( 427 \\div 2.68 \\times 16 \\times 26.8 \\div 42.7 \\times 16 \\)"
}
] | math | {
"style": "rule",
"ground_truth": "25600"
} | {
"index": 39110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( A, B, C \\) are positive integers. It is known that \\( A \\) has 7 divisors, \\( B \\) has 6 divisors, \\( C \\) has 3 divisors, \\( A \\times B \\) has 24 divisors, and \\( B \\times C \\) has 10 divisors. What is the minimum value of \\( A + B + C \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 39111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( AB \\) and \\( AC \\) are tangents to a circle with center \\( O \\), \\( M \\) is the point of intersection of the line \\( AO \\) with the circle; \\( DE \\) is the segment of the tangent drawn through the point \\( M \\) between \\( AB \\) and \\( AC \\). Find \\( DE \\)... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 39112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( AB \\) and \\( AC \\) are two chords forming an angle \\( BAC \\) equal to \\( 70^\\circ \\). Tangents are drawn through points \\( B \\) and \\( C \\) until they intersect at point \\( M \\). Find \\(\\angle BMC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 39113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( BL \\) is the angle bisector of triangle \\( ABC \\). Find its area if it is known that \\( |AL| = 3 \\), \\( |BL| = 6\\sqrt{5} \\), and \\( |CL| = 4 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21 \\sqrt{55}}{4}"
} | {
"index": 39114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( N \\) is an integer whose representation in base \\( b \\) is 777. Find the smallest positive integer \\( b \\) such that \\( N \\) is the fourth power of a decimal integer."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 39115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( P \\) is a moving point on the ellipse \\(\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\\). The tangent line to the ellipse at point \\( P \\) intersects the circle \\(\\odot O\\): \\(x^{2}+y^{2}=12\\) at points \\( M \\) and \\( N \\). The tangents to \\(\\odot O\\) at \\( M \\) and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 39116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( P \\) is the midpoint of the height \\( V O \\) of the regular quadrilateral pyramid \\( V-ABCD \\). If the distance from point \\( P \\) to the side faces is 3, and the distance to the base is 5, what is the volume of the regular quadrilateral pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 39117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( S \\) is a set of 5 coplanar points, no 3 of which are collinear. \\( M(S) \\) is the largest area of a triangle with vertices in \\( S \\). Similarly, \\( m(S) \\) is the smallest area of such a triangle. What is the smallest possible value of \\( \\frac{M(S)}{m(S)} \\) as... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"index": 39118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( S \\) is the set of all ordered tuples \\((a, b, c, d, e, f)\\) where \\(a, b, c, d, e, f\\) are integers and \\(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\\). Find the largest \\( k \\) such that \\( k \\) divides \\( a b c d e f \\) for all elements in \\( S \\)."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 39119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( \\Delta ABC \\) is an isosceles triangle with \\( AB = 2 \\) and \\( \\angle ABC = 90^{\\circ} \\). Point \\( D \\) is the midpoint of \\( BC \\) and point \\( E \\) is on \\( AC \\) such that the area of quadrilateral \\( AEDB \\) is twice the area of triangle \\( ECD \\).... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{3}"
} | {
"index": 39120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( \\mathrm{n} \\) is a positive integer not greater than 100 and not less than 10, and \\( \\mathrm{n} \\) is a multiple of the sum of its digits. How many such \\( \\mathrm{n} \\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 39121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( n \\) is a positive integer that is not greater than 100 and not less than 10, and \\( n \\) is a multiple of the sum of its digits. How many such \\( n \\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 39122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( p(x, y, z) \\) is a polynomial with real coefficients such that:\n1. \\( p(tx, ty, tz) = t^2 f(y - x, z - x) \\) for all real \\( x, y, z, t \\) (and some function \\( f \\));\n2. \\( p(1, 0, 0) = 4 \\), \\( p(0, 1, 0) = 5 \\), and \\( p(0, 0, 1) = 6 \\);\n3. \\( p(\\alpha,... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{30}}{3}"
} | {
"index": 39123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( x_{1} = 2001 \\). When \\( n > 1, x_{n} = \\frac{n}{x_{n-1}} \\). Given that \\( x_{1} x_{2} x_{3} \\ldots x_{10} = a \\), find the value of \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3840"
} | {
"index": 39124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\( z_{1}, z_{2}, z_{3} \\) are the three roots of the polynomial\n\\[ P(z) = z^{3} + a z + b \\]\nand satisfy the condition\n\\[ \\left|z_{1}\\right|^{2} + \\left|z_{2}\\right|^{2} + \\left|z_{3}\\right|^{2} = 250 \\]\nMoreover, the three points \\( z_{1}, z_{2}, z_{3} \\) in ... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{15}"
} | {
"index": 39125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(1.25 \\times 67.875 + 125 \\times 6.7875 + 1250 \\times 0.053375\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 39126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(5, 6, 7\\)"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 39127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(5^{-2 \\log_{0.04}\\left(3-4x^{2}\\right)} + 1.5 \\log_{\\frac{1}{8}} 4^{x} = 0\\)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 39128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(A, B, C, D\\) are consecutive vertices of a parallelogram. Points \\(E, F, P, H\\) lie on sides \\(AB\\), \\(BC\\), \\(CD\\), and \\(AD\\) respectively. Segment \\(AE\\) is \\(\\frac{1}{3}\\) of side \\(AB\\), segment \\(BF\\) is \\(\\frac{1}{3}\\) of side \\(BC\\), and poin... | math | {
"style": "rule",
"ground_truth": "37/72"
} | {
"index": 39129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABCD\\) is a convex quadrilateral where \\(AB = 7\\), \\(BC = 4\\), and \\(AD = DC\\). Also, \\(\\angle ABD = \\angle DBC\\). Point \\(E\\) is on segment \\(AB\\) such that \\(\\angle DEB = 90^\\circ\\). Find the length of segment \\(AE\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 39130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABCD\\) is a parallelogram with \\(AB = 7\\), \\(BC = 2\\), and \\(\\angle DAB = 120^\\circ\\). Parallelogram \\(ECFA\\) is contained within \\(ABCD\\) and is similar to it. Find the ratio of the area of \\(ECFA\\) to the area of \\(ABCD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "39/67"
} | {
"index": 39131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABCD\\) is a square and \\(X\\) is a point on the side \\(DA\\) such that the semicircle with diameter \\(CX\\) touches the side \\(AB\\). Find the ratio \\(AX: XD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1 : 3"
} | {
"index": 39132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABCD\\) is a square with sides \\(8 \\, \\text{cm}\\). \\(M\\) is a point on \\(CB\\) such that \\(CM = 2 \\, \\text{cm}\\). If \\(N\\) is a variable point on the diagonal \\(DB\\), find the least value of \\(CN + MN\\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 39133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABCD\\) is a square-based pyramid with base \\(ABCD\\) and apex \\(E\\). Point \\(E\\) is directly above point \\(A\\), with \\(AE = 1024\\) units and \\(AB = 640\\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \\(ABCD\\), at a heigh... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 39134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(ABC\\) is a triangle with \\(AB = 15\\), \\(BC = 14\\), and \\(CA = 13\\). The altitude from \\(A\\) to \\(BC\\) is extended to meet the circumcircle of \\(ABC\\) at \\(D\\). Find \\(AD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{4}"
} | {
"index": 39135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(AB\\) is the diameter of a circle; \\(BC\\) is a tangent; \\(D\\) is the point where line \\(AC\\) intersects the circle. It is known that \\(AD = 32\\) and \\(DC = 18\\). Find the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 39136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(PQRS\\) is a square. The points \\(T\\) and \\(U\\) are the midpoints of \\(QR\\) and \\(RS\\) respectively. The line \\(QS\\) cuts \\(PT\\) and \\(PU\\) at \\(W\\) and \\(V\\) respectively. What fraction of the total area of the square \\(PQRS\\) is the area of the pentagon... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 39137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\cos \\frac{\\pi}{11} - \\cos \\frac{2 \\pi}{11} + \\cos \\frac{3 \\pi}{11} - \\cos \\frac{4 \\pi}{11} + \\cos \\frac{5 \\pi}{11} = \\) (Answer with a number)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 39138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\cos \\frac{\\pi}{15} - \\cos \\frac{2\\pi}{15} - \\cos \\frac{4\\pi}{15} + \\cos \\frac{7\\pi}{15} =\\)"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 39139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\log _{\\sqrt{3}} x+\\log _{\\sqrt{3}} x+\\log _{\\sqrt[6]{3}} x+\\ldots+\\log _{\\sqrt{3}} x=36\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 39140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\log_{5} x + \\log_{25} x = \\log_{1/5} \\sqrt{3}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt[3]{3}}"
} | {
"index": 39141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\triangle ABC\\) is equilateral with side length 4. \\(D\\) is a point on \\(BC\\) such that \\(BD = 1\\). If \\(r\\) and \\(s\\) are the radii of the inscribed circles of \\(\\triangle ADB\\) and \\(\\triangle ADC\\) respectively, find \\(rs\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4 - \\sqrt{13}"
} | {
"index": 39142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(\\triangle ABC\\) is isosceles with base \\(AC\\). Points \\(P\\) and \\(Q\\) are respectively in \\(CB\\) and \\(AB\\) and such that \\(AC=AP=PQ=QB\\). \nThe number of degrees in \\(\\angle B\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "25\\frac{5}{7}"
} | {
"index": 39143,
"solution": "1. **Identify the properties of the triangles**: Given that $\\triangle ABC$ is isosceles with base $AC$, and $AC = AP = PQ = QB$, we can infer that $\\triangle APQ$ and $\\triangle BQP$ are also isosceles. Additionally, $\\triangle APC$ and $\\triangle BQC$ are isosceles because $AP ... |
DeepScaleR | [
{
"role": "user",
"content": "\\(a, b, c\\) are distinct positive integers such that \\(\\{a+b, b+c, c+a\\} = \\left\\{n^2, (n+1)^2, (n+2)^2\\right\\}\\), where \\(n\\) is a positive integer. What is the minimum value of \\(a^2 + b^2 + c^2\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1297"
} | {
"index": 39144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(a_n\\) is the last digit of \\(1 + 2 + \\ldots + n\\). Find \\(a_1 + a_2 + \\ldots + a_{1992}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6984"
} | {
"index": 39145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(a_{1}, a_{2}, a_{3}, \\ldots\\) is an increasing sequence of natural numbers. It is known that \\(a_{a_{k}} = 3k\\) for any \\(k\\).\n\nFind\na) \\(a_{100}\\)\n\nb) \\(a_{1983}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3762"
} | {
"index": 39146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(f(x)\\) is a linear function, and the equation \\(f(f(x)) = x + 1\\) has no solutions. Find all possible values of \\(f(f(f(f(f(2022)))))-f(f(f(2022)))-f(f(2022))\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-2022"
} | {
"index": 39147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\(x, y\\) are real numbers, \\(z_{1}=x+\\sqrt{11}+yi\\), \\(z_{6}=x-\\sqrt{11}+yi\\) (where \\(i\\) is the imaginary unit). Find \\(|z_{1}| + |z_{6}|\\)."
}
] | math | {
"style": "rule",
"ground_truth": "30(\\sqrt{2} + 1)"
} | {
"index": 39148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\n1.047. \\left(\\frac{\\sqrt{561^{2} - 459^{2}}}{4 \\frac{2}{7} \\cdot 0.15 + 4 \\frac{2}{7} : \\frac{20}{3}} + 4 \\sqrt{10}\\right) : \\frac{1}{3} \\sqrt{40}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 39149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\n\\frac{\\left(\\left(4.625 - \\frac{13}{18} \\cdot \\frac{9}{26}\\right) : \\frac{9}{4} + 2.5 : 1.25 : 6.75\\right) : 1 \\frac{53}{68}}{\\left(\\frac{1}{2} - 0.375\\right) : 0.125 + \\left(\\frac{5}{6} - \\frac{7}{12}\\right) : (0.358 - 1.4796 : 13.7)}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{27}"
} | {
"index": 39150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\n\\frac{\\sin ^{2}\\left(135^{\\circ}-\\alpha\\right)-\\sin ^{2}\\left(210^{\\circ}-\\alpha\\right)-\\sin 195^{\\circ} \\cos \\left(165^{\\circ}-2 \\alpha\\right)}{\\cos ^{2}\\left(225^{\\circ}+\\alpha\\right)-\\cos ^{2}\\left(210^{\\circ}-\\alpha\\right)+\\sin 15^{\\circ} ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 39151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\n\\sum_{k=1}^{70} \\frac{k}{x-k} \\geq \\frac{5}{4}\n\\]\nis a union of disjoint intervals the sum of whose lengths is 1988."
}
] | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"index": 39152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[ y = x + \\cos(2x) \\] in the interval \\((0, \\pi / 4)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12} + \\frac{\\sqrt{3}}{2}"
} | {
"index": 39153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\\frac{\\tan 96^{\\circ} - \\tan 12^{\\circ} \\left( 1 + \\frac{1}{\\sin 6^{\\circ}} \\right)}{1 + \\tan 96^{\\circ} \\tan 12^{\\circ} \\left( 1 + \\frac{1}{\\sin 6^{\\circ}} \\right)} =\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 39154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\\left(\\frac{a}{b} + \\frac{b}{a} + 2\\right) \\cdot \\left(\\frac{a+b}{2a} - \\frac{b}{a+b}\\right) \\div \\left(\\left(a + 2b + \\frac{b^2}{a}\\right) \\cdot \\left(\\frac{a}{a+b} + \\frac{b}{a-b}\\right)\\right);\\ a = 0.75,\\ b = \\frac{4}{3}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{24}"
} | {
"index": 39155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\[\\log_{10} x + \\log_{\\sqrt{10}} x + \\log_{\\sqrt[3]{10}} x + \\ldots + \\log_{\\sqrt[1]{10}} x = 5.5\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[10]{10}"
} | {
"index": 39156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\angle 1 + \\angle 2 = 180^\\circ\n\\angle 3 = \\angle 4\nFind \\angle 4."
}
] | math | {
"style": "rule",
"ground_truth": "35^\\circ"
} | {
"index": 39157,
"solution": "1. **Using the left triangle's angle sum property**: The sum of angles in any triangle is $180^\\circ$. Given $\\angle 1 + 70^\\circ + 40^\\circ = 180^\\circ$, we can solve for $\\angle 1$:\n \\[\n \\angle 1 + 110^\\circ = 180^\\circ\n \\]\n \\[\n \\angle 1 = 180^\\circ - 11... |
DeepScaleR | [
{
"role": "user",
"content": "\\begin{align*}\n2a + 3b + 5c + 7d &= 34 \\\\\n3(d+c) &= b \\\\\n3b + c &= a \\\\\nc - 1 &= d \\\\\n\\end{align*} Given the above system of equations, find $a \\cdot b \\cdot c \\cdot d$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 39158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\begin{align*}\n4a + 2b + 5c + 8d &= 67 \\\\\n4(d+c) &= b \\\\\n2b + 3c &= a \\\\\nc + 1 &= d \\\\\n\\end{align*}\nGiven the above system of equations, find \\(a \\cdot b \\cdot c \\cdot d\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1201 \\times 572 \\times 19 \\times 124}{105^4}"
} | {
"index": 39159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\\dfrac{1}{10} + \\dfrac{9}{100} + \\dfrac{9}{1000} + \\dfrac{7}{10000} ="
}
] | math | {
"style": "rule",
"ground_truth": "0.1997"
} | {
"index": 39160,
"solution": "1. **Convert each fraction to its decimal form**:\n - $\\dfrac{1}{10} = 0.1$\n - $\\dfrac{9}{100} = 0.09$\n - $\\dfrac{9}{1000} = 0.009$\n - $\\dfrac{7}{10000} = 0.0007$\n\n2. **Add the decimals**:\n - Start by adding the largest decimal places first:\n \\[\n 0.1 + 0... |
DeepScaleR | [
{
"role": "user",
"content": "\\frac{1}{10} + \\frac{2}{10} + \\frac{3}{10} + \\frac{4}{10} + \\frac{5}{10} + \\frac{6}{10} + \\frac{7}{10} + \\frac{8}{10} + \\frac{9}{10} + \\frac{55}{10}="
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 39161,
"solution": "1. **Convert fractions to a common denominator**: All the fractions given in the problem have the same denominator, which is 10. Therefore, we can focus on adding the numerators directly.\n\n2. **Add the numerators**: The numerators are $1, 2, 3, 4, 5, 6, 7, 8, 9, 55$. We add these nu... |
DeepScaleR | [
{
"role": "user",
"content": "\\frac{2\\sqrt{6}}{\\sqrt{2}+\\sqrt{3}+\\sqrt{5}} equals"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+\\sqrt{3}-\\sqrt{5}"
} | {
"index": 39162,
"solution": "To simplify the expression $\\frac{2\\sqrt{6}}{\\sqrt{2}+\\sqrt{3}+\\sqrt{5}}$, we will use the technique of rationalizing the denominator.\n\n1. **Rationalize the Denominator**:\n Multiply the numerator and the denominator by the conjugate of the denominator, which is $\\sqrt{2} - ... |
DeepScaleR | [
{
"role": "user",
"content": "\\frac{3 \\times 5}{9 \\times 11} \\times \\frac{7 \\times 9 \\times 11}{3 \\times 5 \\times 7}="
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 39163,
"solution": "1. **Identify the expression to simplify**: \n \\[\n \\frac{3 \\times 5}{9 \\times 11} \\times \\frac{7 \\times 9 \\times 11}{3 \\times 5 \\times 7}\n \\]\n\n2. **Rewrite the expression using the associative property**:\n The associative property allows us to regroup the facto... |
DeepScaleR | [
{
"role": "user",
"content": "\\frac{3}{2} + \\frac{5}{4} + \\frac{9}{8} + \\frac{17}{16} + \\frac{33}{32} + \\frac{65}{64} - 7 ="
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{64}"
} | {
"index": 39164,
"solution": "1. **Convert each fraction to have a common denominator of 64:**\n - $\\frac{3}{2} = \\frac{3 \\times 32}{2 \\times 32} = \\frac{96}{64}$\n - $\\frac{5}{4} = \\frac{5 \\times 16}{4 \\times 16} = \\frac{80}{64}$\n - $\\frac{9}{8} = \\frac{9 \\times 8}{8 \\times 8} = \\frac{72}{64... |
DeepScaleR | [
{
"role": "user",
"content": "a and b are real numbers for which the equation \\(x^4 + ax^3 + bx^2 + ax + 1 = 0\\) has at least one real solution. Find the least possible value of \\(a^2 + b^2\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 39165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) Calculate the number of triangles whose three vertices are vertices of the cube.\n\nb) How many of these triangles are not contained in a face of the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 39166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) Find all the divisors of the number 572 based on its prime factorization.\n\nb) How many divisors does the number \\(572 a^{3} b c\\) have if:\n\n I. \\(a, b\\), and \\(c\\) are prime numbers greater than 20 and different from each other? \n\n II. \\(a = 31\\), \\(b = 32... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"index": 39167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) In Mexico City, there are regulations that restrict private cars from driving on city streets on two specified days of the week. A family needs to have at least ten cars available every day. What is the minimum number of cars the family needs if they can choose the restricte... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 39168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table? \nb) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate ... | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"index": 39169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) In how many ways can a rectangle $8 \\times 2$ be divided into $1 \\times 2$ rectangles? \n\nb) Imagine and describe a shape that can be divided into $1 \\times 2$ rectangles in exactly 555 ways."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 39170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) Is the equation \\( A = B \\) true if\n\n\\[ \nA = \\frac{1 + \\frac{1 + \\frac{1 + \\frac{1}{2}}{4}}{2}}{2}, \\quad B = \\frac{1}{1 + \\frac{1}{2 + \\frac{1}{1 + \\frac{1}{2 + \\frac{1}{4}}}}} \n\\]\n\nb) Which is greater between the numbers \\( \\frac{23}{31} \\) and \\( \... | math | {
"style": "rule",
"ground_truth": "0.7433"
} | {
"index": 39171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?\n\nb) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 39172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "a) What is the maximum number of squares on an $8 \\times 8$ board that can be painted black such that in each corner of three squares, there is at least one unpainted square?\n\nb) What is the minimum number of squares on an $8 \\times 8$ board that need to be painted black su... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 39173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P."
}
] | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 39174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \\{f^n(x)-x| x \\in \\mathbb{R} \\}=\\{a_1+n,a_2+n,...,a_k+n\\} $$ ."
}
] | math | {
"style": "rule",
"ground_truth": "{0}"
} | {
"index": 39175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "in a right-angled triangle $ABC$ with $\\angle C=90$ , $a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$ ; $a,c$ respectively,with radii $r,t$ .find the greatest real number $p$ such that the inequality $\\frac{1... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 39176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the\n $$ \\max( [x]y + [y]z + [z]x ) $$ \n( $[a]$ is the biggest integer not exceeding $a$ )"
}
] | math | {
"style": "rule",
"ground_truth": "652400"
} | {
"index": 39177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "【Information Extraction】 \nIn some cases, it is not necessary to calculate the result to remove the absolute value symbol. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7$. \n【Initial Experience】 \n$(1)$ According to the above rules, write the following expre... | math | {
"style": "rule",
"ground_truth": "\\frac{505}{1011}"
} | {
"index": 39178,
"solution": ""
} |
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