data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Convert -630° to radians."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\pi}{2}"
} | {
"index": 7200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert 2 meters plus 3 centimeters plus 5 millimeters into meters."
}
] | math | {
"style": "rule",
"ground_truth": "2.035 \\text{ m}"
} | {
"index": 7201,
"solution": "Since there are 100 cm in 1 m, then 1 cm is 0.01 m. Thus, 3 cm equals 0.03 m. Since there are 1000 mm in 1 m, then 1 mm is 0.001 m. Thus, 5 mm equals 0.005 m. Therefore, 2 m plus 3 cm plus 5 mm equals $2+0.03+0.005=2.035 \\mathrm{~m}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert 89 into a binary number."
}
] | math | {
"style": "rule",
"ground_truth": "1011001_{(2)}"
} | {
"index": 7202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert \\(531_8\\) to base 7."
}
] | math | {
"style": "rule",
"ground_truth": "1002_7"
} | {
"index": 7203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert \\\\(101 111 011_{(2)}\\\\) to its decimal value."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"index": 7204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.\n\nSuppose we want to find out how many miles are in 30 kilometers. For this, we represent the n... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the base 2 number \\(1011111010_2\\) to its base 4 representation."
}
] | math | {
"style": "rule",
"ground_truth": "23322_4"
} | {
"index": 7206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $1000001$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 7207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $100111$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 7208,
"solution": "$100111_2 = 39_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $1010$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7209,
"solution": "$1010_2 = 10_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $10101010$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"index": 7210,
"solution": "$10101010_2 = 170_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $101101$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 7211,
"solution": "$101101_2 = 45_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $10110100$ to a decimal number."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 7212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $101101110_{(2)}$ to a decimal number and then to an octal number $({\\ }\\_{(8)})$."
}
] | math | {
"style": "rule",
"ground_truth": "556_{(8)}"
} | {
"index": 7213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $101101_2$ to an octal number. The result is"
}
] | math | {
"style": "rule",
"ground_truth": "55_8"
} | {
"index": 7214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $101110_2$ to an octal number."
}
] | math | {
"style": "rule",
"ground_truth": "56_8"
} | {
"index": 7215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $101111011_{(2)}$ to its decimal equivalent."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"index": 7216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $1101$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $110101_{(2)}$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 7218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $11011001_2$ to base 4."
}
] | math | {
"style": "rule",
"ground_truth": "3121_4"
} | {
"index": 7219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $1101100_{(2)}$ to a decimal number."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 7220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $110110100_2$ to base 4."
}
] | math | {
"style": "rule",
"ground_truth": "31220_4"
} | {
"index": 7221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $111000$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 7222,
"solution": "$111000_2 = 56_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $1110011_2$ to its decimal equivalent."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 7223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number."
}
] | math | {
"style": "rule",
"ground_truth": "3785"
} | {
"index": 7224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $111011_{(2)}$ to a decimal number."
}
] | math | {
"style": "rule",
"ground_truth": "1 \\times 2^{5} + 1 \\times 2^{4} + 1 \\times 2^{3} + 0 \\times 2^{2} + 1 \\times 2^{1} + 1"
} | {
"index": 7225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number $11111111$ to decimal."
}
] | math | {
"style": "rule",
"ground_truth": "255"
} | {
"index": 7226,
"solution": "$11111111_2 = 255_{10}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the binary number \\(11111011111_2\\) to its decimal representation."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"index": 7227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the complex number \\(1 + i \\sqrt{3}\\) into its exponential form \\(re^{i \\theta}\\) and find \\(\\theta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 7228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the decimal number 2011 to a base-7 number."
}
] | math | {
"style": "rule",
"ground_truth": "5602_7"
} | {
"index": 7229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the decimal number 89 to binary."
}
] | math | {
"style": "rule",
"ground_truth": "1011001"
} | {
"index": 7230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the following expressions between different bases: $110011_{(2)} = \\_{(10)} = \\_{(5)}$"
}
] | math | {
"style": "rule",
"ground_truth": "51_{(10)} = 201_{(5)}"
} | {
"index": 7231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the following radians to degrees and degrees to radians: \n(1) $$\\frac {\\pi}{12}$$ = \\_\\_\\_\\_\\_\\_ ; (2) $$\\frac {13\\pi}{6}$$ = \\_\\_\\_\\_\\_\\_ ; (3) -$$\\frac {5}{12}$$π = \\_\\_\\_\\_\\_\\_ . \n(4) 36° = \\_\\_\\_\\_\\_\\_ rad; (5) -105° = \\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "- \\frac {7}{12}\\pi"
} | {
"index": 7232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the following radians to degrees: convert degrees to radians:\n(1) $\\frac{\\pi}{12} =$ \\_\\_\\_\\_\\_\\_ ; (2) $\\frac{13\\pi}{6} =$ \\_\\_\\_\\_\\_\\_ ; (3) $- \\frac{5}{12}\\pi =$ \\_\\_\\_\\_\\_\\_ .\n(4) $36^{\\circ} =$ \\_\\_\\_\\_\\_\\_ $rad$ ; (5) $-105^{\\circ... | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\pi}{12}"
} | {
"index": 7233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the fraction $\\frac{123456789}{2^{26} \\times 5^{4}}$ to a decimal. How many decimal places does the resulting number have?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 7234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $( -2, -2 \\sqrt{3}, -1)$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r,\\theta,z),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 4, \\frac{4 \\pi}{3}, -1 \\right)"
} | {
"index": 7235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $( -5, 0, -8 )$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r,\\theta,z),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "(5,\\pi,-8)"
} | {
"index": 7236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(-2,-2)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 2 \\sqrt{2}, \\frac{5 \\pi}{4} \\right)"
} | {
"index": 7237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(0, -3 \\sqrt{3}, 3)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \\pi,$ and $0 \\le \\phi \\le \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 6, \\frac{3 \\pi}{2}, \\frac{\\pi}{3} \\right)"
} | {
"index": 7238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(1,-\\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 2, \\frac{5 \\pi}{3} \\right)"
} | {
"index": 7239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(2 \\sqrt{3}, 6, -4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \\pi,$ and $0 \\le \\phi \\le \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 8, \\frac{\\pi}{3}, \\frac{2 \\pi}{3} \\right)"
} | {
"index": 7240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(6,2 \\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 4 \\sqrt{3}, \\frac{\\pi}{6} \\right)"
} | {
"index": 7241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 12, \\frac{7 \\pi}{6}, \\frac{\\pi}{3} \\right)$ in spherical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(-9, -3 \\sqrt{3}, 6)"
} | {
"index": 7242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 2, \\pi, \\frac{\\pi}{4} \\right)$ in spherical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(-\\sqrt{2}, 0, \\sqrt{2})"
} | {
"index": 7243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 3, \\frac{5 \\pi}{12}, 0 \\right)$ in spherical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0,3)"
} | {
"index": 7244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 4, \\frac{5 \\pi}{3}, \\frac{\\pi}{2} \\right)$ in spherical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(2, -2 \\sqrt{3}, 0)"
} | {
"index": 7245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 2, \\frac{7 \\pi}{4} \\right)"
} | {
"index": 7246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $\\left( 2 \\sqrt{3}, \\frac{2 \\pi}{3} \\right)$ in polar coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(-\\sqrt{3}, 3)"
} | {
"index": 7247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $\\left( 5, \\frac{3 \\pi}{2} \\right)$ in polar coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(0,-5)"
} | {
"index": 7248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $\\left( 5, \\frac{3 \\pi}{2}, 4 \\right)$ in cylindrical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(0, -5, 4)"
} | {
"index": 7249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $\\left( 8, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(-4 \\sqrt{3},-4)"
} | {
"index": 7250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the point $\\left( 8, \\frac{\\pi}{4}, \\sqrt{3} \\right)$ in cylindrical coordinates to rectangular coordinates."
}
] | math | {
"style": "rule",
"ground_truth": "(4 \\sqrt{2}, 4 \\sqrt{2}, \\sqrt{3})"
} | {
"index": 7251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 7252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert the quinary (base-5) number $234_{(5)}$ to a decimal (base-10) number, then convert it to a binary (base-2) number."
}
] | math | {
"style": "rule",
"ground_truth": "1000101_{(2)}"
} | {
"index": 7253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convex hexagon $ABCDEF$ has exactly two distinct side lengths. Side $AB$ measures 5 units, and side $BC$ measures 6 units. The perimeter of hexagon $ABCDEF$ is 34 units. How many sides of hexagon $ABCDEF$ have measure 6 units?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7255,
"solution": "This pentagon is very close to a regular pentagon with side lengths $6$. The area of a regular pentagon with side lengths $s$ is $\\frac{5s^2}{4\\sqrt{5-2\\sqrt{5}}}$. $5-2\\sqrt{5}$ is slightly greater than $\\frac{1}{2}$ given that $2\\sqrt{5}$ is slightly less than $\\frac{9}{2}$. $... |
DeepScaleR | [
{
"role": "user",
"content": "Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "n_1n_2"
} | {
"index": 7256,
"solution": "1. **Understanding the Problem**: We are given two convex polygons $P_1$ and $P_2$ with $n_1$ and $n_2$ sides respectively, where $n_1 \\leq n_2$. The polygons do not share any line segment, and we need to find the maximum number of intersections between these two polygons.\n\n2. **Pro... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \\angle{A}\\equal{}60^o$, $ BC\\equal{}CD\\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle ... | math | {
"style": "rule",
"ground_truth": "\\frac{38}{7}"
} | {
"index": 7257,
"solution": "\nGiven a convex quadrilateral \\(ABCD\\) inscribed in a circle with \\(\\angle A = 60^\\circ\\), \\(BC = CD = 1\\), and the intersections of rays \\(AB\\) and \\(DC\\) at point \\(E\\), and rays \\(BC\\) and \\(AD\\) at point \\(F\\), we aim to find the perimeter of quadrilateral \\(A... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $A B C D$ has right angles $\\angle A$ and $\\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7258,
"solution": "Note that $m \\angle Q P B=m \\angle M P B=m \\angle M A B=m \\angle C A B=\\angle B C A=\\angle C D B$. Thus, $M P \\cdot M Q=M B \\cdot M D$. On the other hand, segment $C M$ is an altitude of right triangle $B C D$, so $M B \\cdot M D=M C^{2}=36$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. Diagonals $AC$ and $BD$ intersect at $E$, where $AC = 18$, and $\\triangle AED$ and $\\triangle BEC$ have equal perimeters. Calculate the length of $AE$.\n**A)** $6$ \n**B)** $7.2$ \n**C)** $7.5$ \n**D)** $9$ \n**E)**... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"index": 7259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 18$, $\\angle A = 60^\\circ$, and $\\overline{AB} \\parallel \\overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\\overline{AB}$ is a side of maximum length. The length of another side is ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 7260,
"solution": "1. **Identify the properties of the quadrilateral**: Given that $ABCD$ is a convex quadrilateral with $AB = 18$, $\\angle A = 60^\\circ$, and $\\overline{AB} \\parallel \\overline{CD}$. The lengths of the sides form an arithmetic progression, and $AB$ is the longest side.\n\n2. **Intro... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\\triangle AED$ and $\\triangle BEC$ have equal areas. What is $AE$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 7261,
"solution": "1. **Given Information and Assumptions**:\n - Convex quadrilateral $ABCD$ with $AB = 9$, $CD = 12$, and diagonals $AC = 14$.\n - Diagonals $AC$ and $BD$ intersect at $E$.\n - $\\triangle AED$ and $\\triangle BEC$ have equal areas.\n\n2. **Using Equal Areas to Infer Similarity**:\... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\\angle ABC=90^{\\circ}$, as shown. What is the area of the quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7262,
"solution": "1. **Identify the right triangle**: Given $\\angle ABC = 90^\\circ$ and the sides $AB = 3$ and $BC = 4$, we can apply the Pythagorean theorem to find $AC$:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5.\n \\]\n\n2. **Check if $\\triangle C... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral $B C D E$ lies in the plane. Lines $E B$ and $D C$ intersect at $A$, with $A B=2$, $A C=5, A D=200, A E=500$, and $\\cos \\angle B A C=\\frac{7}{9}$. What is the largest number of nonoverlapping circles that can lie in quadrilateral $B C D E$ such that all ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7263,
"solution": "Let $\\theta=\\angle B A C$, and $\\cos \\theta=\\frac{7}{9}$ implies $\\cos \\frac{\\theta}{2}=\\sqrt{\\frac{1+\\frac{7}{9}}{2}}=\\frac{2 \\sqrt{2}}{3} ; \\sin \\frac{\\theta}{2}=\\frac{1}{3} ; B C=$ $\\sqrt{4+25-2(2)(5) \\frac{7}{9}}=\\frac{11}{3}$. Let $O_{1}$ be the excircle of $\\... |
DeepScaleR | [
{
"role": "user",
"content": "Convex quadrilateral \\(ABCD\\) is such that \\(\\angle BAC = \\angle BDA\\) and \\(\\angle BAD = \\angle ADC = 60^\\circ\\). Find the length of \\(AD\\) given that \\(AB = 14\\) and \\(CD = 6\\)."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert."
}
] | math | {
"style": "rule",
"ground_truth": "13\\pi"
} | {
"index": 7265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Coordinate System and Parametric Equation\nGiven the ellipse $(C)$: $\\frac{x^{2}}{16} + \\frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\\tri... | math | {
"style": "rule",
"ground_truth": "6(\\sqrt{2} + 1)"
} | {
"index": 7266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7267,
"solution": "\nTo construct a unit segment using only a ruler and compass, given the graph of the quadratic trinomial \\( y = x^2 + ax + b \\) without any coordinate marks, we can use geometric properties of the parabola.\n\n### Steps:\n\n1. **Identify the Vertex and Axis of Symmetry**\n\n The pa... |
DeepScaleR | [
{
"role": "user",
"content": "Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10-7\\sqrt{2}}{3}"
} | {
"index": 7268,
"solution": "1. **Identify the shape of the faces after slicing**: After slicing off the corners of the unit cube, each face becomes a regular octagon. This implies that each edge of the cube is divided into three segments: two segments forming the slanted edges of the octagon and one segment formi... |
DeepScaleR | [
{
"role": "user",
"content": "Corners are sliced off from a cube of side length 2 so that all its six faces each become regular octagons. Find the total volume of the removed tetrahedra.\nA) $\\frac{80 - 56\\sqrt{2}}{3}$\nB) $\\frac{80 - 48\\sqrt{2}}{3}$\nC) $\\frac{72 - 48\\sqrt{2}}{3}$\nD) $\\frac{60 - 42... | math | {
"style": "rule",
"ground_truth": "\\frac{80 - 56\\sqrt{2}}{3}"
} | {
"index": 7269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cory has $3$ apples, $2$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for a week and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit? One such order is $AAAOOBB.$"
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 7270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 7271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cory made a complete list of the prime numbers between 1 and 25. What is the sum of the smallest prime number and the largest prime number on his list?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 7272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Count all the distinct anagrams of the word \"YOANN\"."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Count how many 8-digit numbers there are that contain exactly four nines as digits."
}
] | math | {
"style": "rule",
"ground_truth": "433755"
} | {
"index": 7274,
"solution": "There are $\\binom{8}{4} \\cdot 9^{4}$ sequences of 8 numbers with exactly four nines. A sequence of digits of length 8 is not an 8-digit number, however, if and only if the first digit is zero. There are $\\binom{7}{4} 9^{3}$ 8-digit sequences that are not 8-digit numbers. The answer ... |
DeepScaleR | [
{
"role": "user",
"content": "Count the number of functions $f: \\mathbb{Z} \\rightarrow\\{$ 'green', 'blue' $\\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'."
}
] | math | {
"style": "rule",
"ground_truth": "39601"
} | {
"index": 7275,
"solution": "It is clear that $f$ is determined by $f(0), \\ldots, f(21)$. The colors of the 11 even integers are independent of those of the odd integers because evens and odds are never exactly 2 apart. First, we count the number of ways to 'color' the even integers. $f(0)$ can either be 'green' ... |
DeepScaleR | [
{
"role": "user",
"content": "Count the number of permutations $a_{1} a_{2} \\ldots a_{7}$ of 1234567 with longest decreasing subsequence of length at most two (i.e. there does not exist $i<j<k$ such that $a_{i}>a_{j}>a_{k}$ )."
}
] | math | {
"style": "rule",
"ground_truth": "429"
} | {
"index": 7276,
"solution": "$C(7)=429$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Count the number of sequences $1 \\leq a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{5}$ of integers with $a_{i} \\leq i$ for all $i$."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 7277,
"solution": "$C$ (number of terms) $=C(5)=42$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Count the number of sequences $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of integers such that $a_{i} \\leq 1$ for all $i$ and all partial sums $\\left(a_{1}, a_{1}+a_{2}\\right.$, etc.) are non-negative."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 7278,
"solution": "$C$ (length +1$)=C(6)=132$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Country $X$ has $40\\%$ of the world's population and $60\\%$ of the world's wealth. Country $Y$ has $20\\%$ of the world's population but $30\\%$ of its wealth. Country $X$'s top $50\\%$ of the population owns $80\\%$ of the wealth, and the wealth in Country $Y$ is equally sha... | math | {
"style": "rule",
"ground_truth": "1.6"
} | {
"index": 7280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Country music songs are added to a playlist so that now $40\\%$ of the songs are Country. If the ratio of Hip Hop songs to Pop songs remains the same, what percentage of the total number of songs are now Hip Hop?"
}
] | math | {
"style": "rule",
"ground_truth": "39\\%"
} | {
"index": 7281,
"solution": "Since $40\\%$ of the songs on the updated playlist are Country, then the remaining $100\\%-40\\%$ or $60\\%$ must be Hip Hop or Pop songs. Since the ratio of Hip Hop songs to Pop songs does not change, then $65\\%$ of this remaining $60\\%$ must be Hip Hop songs. Overall, this is $65\\... |
DeepScaleR | [
{
"role": "user",
"content": "Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes fiv... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7282,
"solution": "To solve this problem, we need to find the number of steps in the staircase such that Dash takes 19 fewer jumps than Cozy. Let $n$ be the total number of steps in the staircase. Cozy jumps 2 steps at a time, and Dash jumps 5 steps at a time. We can express the number of jumps each take... |
DeepScaleR | [
{
"role": "user",
"content": "Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five s... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7283,
"solution": "1. **Understanding the Problem:**\n Cozy the Cat jumps two steps at a time, and Dash the Dog jumps five steps at a time. Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. We need to find the sum of all possible numbers of steps in the staircase and then find the ... |
DeepScaleR | [
{
"role": "user",
"content": "Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface area... | math | {
"style": "rule",
"ground_truth": "\\frac{54+9 \\sqrt{3}}{8}"
} | {
"index": 7284,
"solution": "The lengths of the second cube are one-third of the lengths of the first cube, so the surface area decreases by a factor of one-ninth. Since the first cube has surface area 6 and the first octahedron has surface area $\\sqrt{3}$, the total area is $(6+\\sqrt{3}) \\cdot\\left(1+\\frac{1... |
DeepScaleR | [
{
"role": "user",
"content": "Create a five-digit number without repeated digits using the numbers 0, 3, 4, 5, and 6, with the condition that even numbers must be adjacent to each other, as well as odd numbers. How many such five-digit numbers exist?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Create three-digit numbers without repeating digits using the numbers 0, 1, 2, 3, 4, 5: \n(1) How many of them have a ones digit smaller than the tens digit? \n(2) How many of them are divisible by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Crestview's school colors are purple and gold. The students are designing a flag using three solid-colored horizontal stripes, as shown. Using one or both of the school colors, how many different flags are possible if adjacent stripes may be the same color?\n\n[asy]\nsize(75);\... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 7287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Crestview's school has expanded its official colors to include blue along with the original purple and gold. The students need to design a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 7288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Crestwood Elementary School has an active four-square league, consisting of ten players, including Justin and Tim. Each day at recess, the ten players split into two four-square games, each with five players in no relevant order. Over the course of a semester, each possible mat... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 7289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Crisp All, a basketball player, is dropping dimes and nickels on a number line. Crisp drops a dime on every positive multiple of 10 , and a nickel on every multiple of 5 that is not a multiple of 10. Crisp then starts at 0 . Every second, he has a $\\frac{2}{3}$ chance of jumpi... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{31}"
} | {
"index": 7290,
"solution": "Let \"a 3\" mean a move in which Crisp moves from $x$ to $x+3$, and \"a 7 \" mean a move in which Crisp moves from $x$ to $x+7$. Note that Crisp stops precisely the first time his number of 3's and number of 7 's differs by a multiple of 5 , and that he'll stop on a dime if they differ... |
DeepScaleR | [
{
"role": "user",
"content": "Crystal decides to alter her running routine slightly. She heads due north for 2 miles, then goes northwest for 2 miles, followed by a southwest direction for 2 miles, and she finishes with a final segment directly back to her starting point. Calculate the distance of this fina... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 7291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in m... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 7292,
"solution": "1. **Understanding the Path**: Crystal's path consists of four segments:\n - North for 1 mile.\n - Northeast for 1 mile.\n - Southeast for 1 mile.\n - A straight line back to the starting point.\n\n2. **Breaking Down the Northeast and Southeast Runs**:\n - **Northeast Run**: ... |
DeepScaleR | [
{
"role": "user",
"content": "Ctibor marked a square land plot on a map with a scale of 1:50000 and calculated that its side corresponds to $1 \\mathrm{~km}$ in reality. He then resized the map on a copier such that the marked square had an area $1.44 \\mathrm{~cm}^{2}$ smaller than the original.\n\nWhat wa... | math | {
"style": "rule",
"ground_truth": "1:62500"
} | {
"index": 7293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cube $A B C D E F G H$ has edge length 100. Point $P$ is on $A B$, point $Q$ is on $A D$, and point $R$ is on $A F$, as shown, so that $A P=x, A Q=x+1$ and $A R=\\frac{x+1}{2 x}$ for some integer $x$. For how many integers $x$ is the volume of triangular-based pyramid $A P Q R$... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 7294,
"solution": "Triangular-based pyramid $A P Q R$ can be thought of as having triangular base $\\triangle A P Q$ and height $A R$. Since this pyramid is built at a vertex of the cube, then $\\triangle A P Q$ is right-angled at $A$ and $A R$ is perpendicular to the base. The area of $\\triangle A P Q$... |
DeepScaleR | [
{
"role": "user",
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the larger of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{48}"
} | {
"index": 7295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two sol... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 7296,
"solution": "Define a coordinate system with $D = (0,0,0)$, $M = (1, \\frac{1}{2}, 0)$, $N = (0,1,\\frac{1}{2})$. The plane formed by $D$, $M$, and $N$ is $z = \\frac{y}{2} - \\frac{x}{4}$. It intersects the base of the unit cube at $y = \\frac{x}{2}$. The z-coordinate of the plane never exceeds th... |
DeepScaleR | [
{
"role": "user",
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $2$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the smaller of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 7297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cubes. As is known, the whole space can be filled with equal cubes. At each vertex, eight cubes will converge. Therefore, by appropriately truncating the vertices of the cubes and joining the adjacent truncated parts into a single body, it is possible to fill the space with reg... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 7298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Currently, 7 students are to be assigned to participate in 5 sports events, with the conditions that students A and B cannot participate in the same event, each event must have at least one participant, and each student can only participate in one event. How many different ways... | math | {
"style": "rule",
"ground_truth": "15000"
} | {
"index": 7299,
"solution": ""
} |
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