data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The radian measure of 300° is $$\\frac {5π}{3}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{3}"
} | {
"index": 33900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radian measure of a 60° angle is $$\\frac {\\pi}{6}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 33901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radii of the excircles of a triangle are 3, 10, and 15 respectively. What is the perimeter of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a$, $b$ and $c$ is given by\n\\[\\frac{1}{r} = \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + 2 \\sqrt{\\frac{1}{ab} + \\frac{1}{ac} + \\frac{1}{bc}}.\\]What is the value of $r$ when $... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{7}"
} | {
"index": 33903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the numb... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 33904,
"solution": "1. **Calculate the circumference of the Earth at the equator:** \n The formula for the circumference, $C$, of a circle is given by:\n \\[\n C = 2\\pi r\n \\]\n where $r$ is the radius of the circle. Given that the radius of the Earth at the equator is approximately 4000 mil... |
DeepScaleR | [
{
"role": "user",
"content": "The radius of a cylinder is doubled and its height is tripled. If its original volume was 10 cubic feet, what is its volume now, in cubic feet?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 33905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 33906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of a sphere is \\( r = 10 \\text{ cm} \\). Determine the volume of the spherical segment whose surface area is in the ratio 10:7 compared to the area of its base."
}
] | math | {
"style": "rule",
"ground_truth": "288 \\pi"
} | {
"index": 33907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of a sphere that touches all the edges of a regular tetrahedron is 1. Find the edge length of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"index": 33908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the Earth measures approximately $6378 \\mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.\n\nNext, suppose that the length of the wire is increased by $1 \\mathrm{~m}$, so that the wire and the Equator are concentric circles a... | math | {
"style": "rule",
"ground_truth": "0.159"
} | {
"index": 33909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the circle inscribed in triangle \\(ABC\\) is 4, with \\(AC = BC\\). On the line \\(AB\\), point \\(D\\) is chosen such that the distances from \\(D\\) to the lines \\(AC\\) and \\(BC\\) are 11 and 3 respectively. Find the cosine of the angle \\(DBC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 33910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the circumcircle of an acute triangle \\( ABC \\) is 1. It is known that the center of the circle passing through the vertices \\( A \\), \\( C \\), and the orthocenter of triangle \\( ABC \\) lies on this circumcircle. Find the length of side \\( AC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 33911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the circumcircle of the acute-angled triangle \\(ABC\\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \\(A\\), \\(C\\), and the orthocenter of triangle \\(ABC\\). Find \\(AC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 33912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the first circle is $1$ inch, that of the second $\\frac{1}{2}$ inch, that of the third $\\frac{1}{4}$ inch and so on indefinitely. The sum of the areas of the circles is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"index": 33913,
"solution": "1. **Identify the sequence of radii**: The radii of the circles are given as $1$ inch, $\\frac{1}{2}$ inch, $\\frac{1}{4}$ inch, and so on. This forms a geometric sequence where each term is half of the previous term. The general term for the radius of the $n$-th circle can be express... |
DeepScaleR | [
{
"role": "user",
"content": "The radius of the inscribed circle is 6 cm. What is the number of centimeters in the length of $\\overline{AB}$? Express your answer in simplest radical form. [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\ndraw((sqrt(3),0)--origin--(0,1)--cycle... | math | {
"style": "rule",
"ground_truth": "AB=12+12\\sqrt{3}"
} | {
"index": 33914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. \nThe speed of the train in miles per hour is approximately the number of clicks heard in:"
}
] | math | {
"style": "rule",
"ground_truth": "20 seconds"
} | {
"index": 33915,
"solution": "1. **Convert the train's speed from miles per hour to feet per minute:**\n Given the speed of the train is $x$ miles per hour, we first convert this speed to feet per minute. We know:\n \\[\n 1 \\text{ mile} = 5280 \\text{ feet} \\quad \\text{and} \\quad 1 \\text{ hour} = 60 \\t... |
DeepScaleR | [
{
"role": "user",
"content": "The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two ... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 33916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\\mu = 1$ and the variance $\\sigma^2 = 4$. Given that $P(X \\geq 2) = 0.2$, calculate the probability $P(0 \\leq X \\leq 1)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 33917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \\geqslant 2) = 0.2$, calculate the probability that $0 \\leqslant X \\leqslant 1$."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 33918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The range of the function $g(x) = \\frac{2}{2+4x^2}$ can be written as an interval $(a,b]$. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rank of a rational number $q$ is the unique $k$ for which $q=\\frac{1}{a_{1}}+\\cdots+\\frac{1}{a_{k}}$, where each $a_{i}$ is the smallest positive integer such that $q \\geq \\frac{1}{a_{1}}+\\cdots+\\frac{1}{a_{i}}$. Let $q$ be the largest rational number less than \\fra... | math | {
"style": "rule",
"ground_truth": "(5,21,421)"
} | {
"index": 33920,
"solution": "Suppose that $A$ and $B$ were rational numbers of rank 3 less than $\\frac{1}{4}$, and let $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ be positive integers so that $A=\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\frac{1}{a_{3}}$ and $B=\\frac{1}{b_{1}}+\\frac{1}{b_{2}}+\\frac{1}{b_{3}}$ are the ... |
DeepScaleR | [
{
"role": "user",
"content": "The rapid growth of the Wuhan Economic and Technological Development Zone's economy cannot be separated from the support of the industrial industry. In 2022, the total industrial output value of the entire area reached 3462.23 billion yuan, ranking first in the city. Express th... | math | {
"style": "rule",
"ground_truth": "3.46223 \\times 10^{3}"
} | {
"index": 33921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio $AC:CB$ is $3:4$, in $\\triangle ABC$. The external angle bisector of $\\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$."
}
] | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"index": 33922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio $\\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which whole number?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio $\\frac{2^{2001} \\cdot 3^{2003}}{6^{2002}}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 33924,
"solution": "1. **Rewrite the given expression using the properties of exponents:**\n \\[\n \\frac{2^{2001} \\cdot 3^{2003}}{6^{2002}}\n \\]\n\n2. **Express $6^{2002}$ in terms of $2$ and $3$:**\n \\[\n 6^{2002} = (2 \\cdot 3)^{2002} = 2^{2002} \\cdot 3^{2002}\n \\]\n\n3. **Substitute ... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "16:3"
} | {
"index": 33925,
"solution": "1. **Convert the given ratios to equations:**\n - The ratio of $w$ to $x$ is given as $4:3$. This can be written as:\n \\[\n \\frac{w}{x} = \\frac{4}{3}\n \\]\n - The ratio of $y$ to $z$ is given as $3:2$. This can be written as:\n \\[\n \\frac{y}{z} = \\frac{3... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$"
}
] | math | {
"style": "rule",
"ground_truth": "16:3"
} | {
"index": 33926,
"solution": "1. **Set up the given ratios**:\n - The ratio of $w$ to $x$ is given as $\\frac{w}{x} = \\frac{4}{3}$.\n - The ratio of $y$ to $z$ is given as $\\frac{y}{z} = \\frac{3}{2}$.\n - The ratio of $z$ to $x$ is given as $\\frac{z}{x} = \\frac{1}{6}$.\n\n2. **Express the equations in t... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 33927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of a geometric sequence is an integer. We know that there is a term in the sequence which is equal to the sum of some other terms of the sequence. What can the ratio of the sequence be?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of apples to bananas in a box is $3: 2$. What total number of apples and bananas in the box cannot be equal to?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33929,
"solution": "Since the ratio of apples to bananas is $3: 2$, then we can let the numbers of apples and bananas equal $3n$ and $2n$, respectively, for some positive integer $n$. Therefore, the total number of apples and bananas is $3n + 2n = 5n$, which is a multiple of 5. Of the given choices, only... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33930,
"solution": "1. **Understanding the Ratio**: The problem states that the ratio of boys to girls in Mr. Brown's math class is $2:3$. This means for every 2 boys, there are 3 girls.\n\n2. **Setting Up Variables**: Let the number of boys be $2x$ and the number of girls be $3x$. Here, $x$ is a common ... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys."
}
] | math | {
"style": "rule",
"ground_truth": "42.86\\%"
} | {
"index": 33931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of butter:flour:sugar in a recipe is 1:6:4. When using 8 cups of sugar in this recipe, how many total cups of these three ingredients will be used?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 33932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of girls to boys in Ms. Snow's math class is 3:2. If there is a total of 45 students, how many girls are in Ms. Snow's math class?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of irises to roses in Nikki's garden is 2:5. She currently has 25 roses. She is planning to add 20 more roses and enough irises to keep the same ratio. How many irises will she have in total after this addition?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\\%$. By what percent must the larger measure be decreased so that the two angles remain complementary?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\%"
} | {
"index": 33935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in a full circle is:"
}
] | math | {
"style": "rule",
"ground_truth": "2: 5"
} | {
"index": 33936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 33937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of a square and a circle is $\\frac{250}{196}$. After rationalizing the denominator, the ratio of the side length of the square to the radius of the circle can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integer... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 33938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the radii is best approximated by:"
}
] | math | {
"style": "rule",
"ground_truth": "0.73r"
} | {
"index": 33939,
"solution": "1. **Identify the given information**: The ratio of the areas of two concentric circles is given as $1:3$. Let the radius of the smaller circle be $r$ and the radius of the larger circle be $R$.\n\n2. **Express the areas of the circles**: The area of a circle is given by $\\pi r^2$. T... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{192}{80}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 33940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the si... | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"index": 33941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{32}{63}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 33942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{50}{98}$. After rationalizing the denominator, express the simplified form of the ratio of their side lengths in the form $\\frac{a \\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Find the sum $a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 33943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the dividend to the divisor is 9:2, and the ratio of the divisor to the quotient is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 33944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33945,
"solution": "1. **Identify the formula for the interior angle of a regular polygon**: The measure of an interior angle of a regular polygon with $n$ sides is given by the formula:\n \\[\n 180^\\circ - \\frac{360^\\circ}{n}\n \\]\n This formula arises from the fact that the sum of the exter... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere."
}
] | math | {
"style": "rule",
"ground_truth": "171:113"
} | {
"index": 33946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"index": 33947,
"solution": "1. **Assign Variables to Dimensions:**\n Let the length of the rectangle be $4x$ and the width be $3x$. This assignment is based on the given ratio of length to width, which is $4:3$.\n\n2. **Use the Pythagorean Theorem:**\n The diagonal $d$ of the rectangle forms a right triangle... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"index": 33948,
"solution": "1. **Assign Variables to Dimensions**:\n Let the length of the rectangle be $4x$ and the width be $3x$. This assignment is based on the given ratio of length to width, which is $4:3$.\n\n2. **Use the Pythagorean Theorem**:\n The diagonal $d$ of the rectangle forms a right triangle... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{29}"
} | {
"index": 33949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{7}}{4}"
} | {
"index": 33950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the measures of the acute angles of a right triangle is $8:1$. In degrees, what is the measure of the largest angle of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"index": 33951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the measures of two angles of a triangle is 2, and the difference in lengths of the sides opposite to these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "3.75\\sqrt{7}"
} | {
"index": 33952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33953,
"solution": "1. **Understanding the Ratio**: The problem states that the ratio of the number of games won to the number of games lost by the Middle School Middies is $\\frac{11}{4}$. This implies that for every 11 games won, there are 4 games lost.\n\n2. **Setting Up the Expression**: Let $x$ be a... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent."
}
] | math | {
"style": "rule",
"ground_truth": "24\\%"
} | {
"index": 33954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the radii of two concentric circles is $1:3$. If $\\overline{AC}$ is a diameter of the larger circle, $\\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33955,
"solution": "1. **Identify the Geometry and Setup the Problem:**\n - We have two concentric circles with radii ratio $1:3$. Let the radius of the smaller circle be $r$ and the radius of the larger circle be $3r$.\n - $\\overline{AC}$ is a diameter of the larger circle, so $AC = 2 \\times 3r = ... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"index": 33956,
"solution": "1. **Define the sides and diagonal of the rectangle**: Let the short side of the rectangle be \\(a\\) and the long side be \\(b\\). The diagonal \\(d\\) of the rectangle can be expressed using the Pythagorean Theorem as \\(d = \\sqrt{a^2 + b^2}\\).\n\n2. **Set up the given ratio condi... |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the sum of the first $n$ terms of two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ is $\\frac{S_n}{T_n} = \\frac{7n+1}{4n+2}$. Calculate the value of $\\frac{a_{11}}{b_{11}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{74}{43}"
} | {
"index": 33957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the sums of the first \\( n \\) terms of two arithmetic sequences is \\(\\frac{9n+2}{n+7}\\). Find the ratio of their 5th terms."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{16}"
} | {
"index": 33958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ratio of the sums of the first n terms of the arithmetic sequences {a_n} and {b_n} is given by S_n / T_n = (2n) / (3n + 1). Find the ratio of the fifth terms of {a_n} and {b_n}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{14}"
} | {
"index": 33959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 33960,
"solution": "Write $x+y=a / 60+b / 70=(7 a+6 b) / 420$. Since $a$ is relatively prime to 60 and $b$ is relatively prime to 70 , it follows that none of the primes $2,3,7$ can divide $7 a+6 b$, so we won't be able to cancel any of these factors in the denominator. Thus, after reducing to lowest ter... |
DeepScaleR | [
{
"role": "user",
"content": "The real function $f$ has the property that, whenever $a, b, n$ are positive integers such that $a+b=2^{n}$, the equation $f(a)+f(b)=n^{2}$ holds. What is $f(2002)$?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 33961,
"solution": "We know $f(a)=n^{2}-f\\left(2^{n}-a\\right)$ for any $a$, $n$ with $2^{n}>a$; repeated application gives $$f(2002)=11^{2}-f(46)=11^{2}-\\left(6^{2}-f(18)\\right)=11^{2}-\\left(6^{2}-\\left(5^{2}-f(14)\\right)\\right) =11^{2}-\\left(6^{2}-\\left(5^{2}-\\left(4^{2}-f(2)\\right)\\right)\... |
DeepScaleR | [
{
"role": "user",
"content": "The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation\n\\[f(a) + f(b) = n^2\\]holds. What is $f(2002)$?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 33962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation\n\\[g(x) + g(y) = 2m^2\\]holds. What is $g(2187)$?"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 33963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real number $x$ satisfies $x^2 - 5x + 6 < 0.$ Find all possible values of $x^2 + 5x + 6.$"
}
] | math | {
"style": "rule",
"ground_truth": "(20,30)"
} | {
"index": 33964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real number $x$ satisfies the equation $x+\\frac{1}{x} = \\sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 33965,
"solution": "1. **Start with the given equation and manipulate it:**\n Given that \\( x + \\frac{1}{x} = \\sqrt{5} \\), we multiply both sides by \\( x \\) to eliminate the fraction:\n \\[\n x^2 + 1 = \\sqrt{5}x\n \\]\n Rearranging gives:\n \\[\n x^2 - \\sqrt{5}x + 1 = 0\n \\]\n\n2... |
DeepScaleR | [
{
"role": "user",
"content": "The real number \\( a \\) is such that \\( 2a - \\frac{1}{a} = 3 \\). What is \\( 16a^{4} + \\frac{1}{a^{4}} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"index": 33966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real number \\( a \\) makes the equation \\( 4^{x} - 4^{-x} = 2 \\cos(ax) \\) have exactly 2015 solutions. For this \\( a \\), how many solutions does the equation \\( 4^{x} + 4^{-x} = 2 \\cos(ax) + 4 \\) have?"
}
] | math | {
"style": "rule",
"ground_truth": "4030"
} | {
"index": 33967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real number \\(x\\) satisfies the equation \\(x + \\frac{1}{x} = \\sqrt{3}\\). Evaluate the expression \\(x^{7} - 5x^{5} + x^{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $a$ and $b$ satisfy\n\\[\\begin{pmatrix} 2 \\\\ a \\\\ -7 \\end{pmatrix} \\times \\begin{pmatrix} 5 \\\\ 4 \\\\ b \\end{pmatrix} = \\mathbf{0}.\\]Enter the ordered pair $(a,b).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{8}{5}, -\\frac{35}{2} \\right)"
} | {
"index": 33969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\\frac{{b}^{2}}{4}+\\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"index": 33970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $a,$ $b,$ $c,$ and $d$ satisfy\n\\[a^2 + b^2 + c^2 + 1 = d + \\sqrt{a + b + c - d}.\\]Find $d.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 33971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $c, b, a$ form an arithmetic sequence with $a \\geq b \\geq c \\geq 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?"
}
] | math | {
"style": "rule",
"ground_truth": "-2-\\sqrt{3}"
} | {
"index": 33972,
"solution": "\nGiven that $a, b, c$ form an arithmetic sequence and $a \\geq b \\geq c \\geq 0$, we can express $b$ and $c$ in terms of $a$ and a common difference $d$:\n\\[ b = a - d, \\quad c = a - 2d. \\]\n\nThe quadratic equation is $ax^2 + bx + c = 0$. Since it has exactly one root, the discr... |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $x$ and $y$ satisfy the equation $2\\cos ^{2}(x+y-1)= \\frac {(x+1)^{2}+(y-1)^{2}-2xy}{x-y+1}$. Find the minimum value of $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 33973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $x, y$ and $z$ satisfy the three equations $x+y=7$, $xz=-180$, and $(x+y+z)^{2}=4$. If $S$ is the sum of the two possible values of $y$, what is $-S$?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 33974,
"solution": "Since $x+y=7$, then $x+y+z=7+z$. Thus, the equation $(x+y+z)^{2}=4$ becomes $(7+z)^{2}=4$. Since the square of $7+z$ equals 4, then $7+z=2$ or $7+z=-2$. If $7+z=2$, then $z=-5$. In this case, since $xz=-180$, we get $x=\\frac{-180}{-5}=36$ which gives $y=7-x=-29$. If $7+z=-2$, then $z... |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $x, y, z$ satisfy $0 \\leq x \\leq y \\leq z \\leq 4$. If their squares form an arithmetic progression with common difference 2, determine the minimum possible value of $|x-y|+|y-z|$."
}
] | math | {
"style": "rule",
"ground_truth": "4-2\\sqrt{3}"
} | {
"index": 33975,
"solution": "Clearly $|x-y|+|y-z|=z-x=\\frac{z^{2}-x^{2}}{z+x}=\\frac{4}{z+x}$, which is minimized when $z=4$ and $x=\\sqrt{12}$. Thus, our answer is $4-\\sqrt{12}=4-2 \\sqrt{3}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers $x, y, z, w$ satisfy $$\\begin{aligned} & 2 x+y+z+w=1 \\\\ & x+3 y+z+w=2 \\\\ & x+y+4 z+w=3 \\\\ & x+y+z+5 w=25 \\end{aligned}$$ Find the value of $w$."
}
] | math | {
"style": "rule",
"ground_truth": "11/2"
} | {
"index": 33976,
"solution": "Multiplying the four equations by $12,6,4,3$ respectively, we get $$\\begin{aligned} 24 x+12 y+12 z+12 w & =12 \\\\ 6 x+18 y+6 z+6 w & =12 \\\\ 4 x+4 y+16 z+4 w & =12 \\\\ 3 x+3 y+3 z+15 w & =75 \\end{aligned}$$ Adding these yields $37 x+37 y+37 z+37 w=111$, or $x+y+z+w=3$. Subtract t... |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers \\( x, y, z \\) satisfy \\( x + y + z = 2 \\) and \\( xy + yz + zx = 1 \\). Find the maximum possible value of \\( x - y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"index": 33977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers \\( x, y, z \\) satisfy the equations \\( x + y + z = 2 \\) and \\( xy + yz + zx = 1 \\). Find the maximum possible value of \\( x - y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"index": 33978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers \\( x_{1}, x_{2}, \\cdots, x_{2001} \\) satisfy \\( \\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right| = 2001 \\). Let \\( y_{k} = \\frac{1}{k} \\left( x_{1} + x_{2} + \\cdots + x_{k} \\right) \\) for \\( k = 1, 2, \\cdots, 2001 \\). Find the maximum possible value... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 33979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers \\(a, b, c\\) satisfy the following system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\n\\frac{a b}{a+b}=4 \\\\\n\\frac{b c}{b+c}=5 \\\\\n\\frac{c a}{c+a}=7\n\\end{array}\\right.\n$$\n\nFind the value of the expression \\(\\frac{a b c}{a b + b c + c a}\\)."... | math | {
"style": "rule",
"ground_truth": "280/83"
} | {
"index": 33980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real numbers \\(x_{1}, x_{2}, \\cdots, x_{2001}\\) satisfy \\(\\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right|=2001\\). Let \\(y_{k}=\\frac{1}{k}\\left(x_{1}+ x_{2} + \\cdots + x_{k}\\right)\\) for \\(k=1, 2, \\cdots, 2001\\). Find the maximum possible value of \\(\\sum_{k=1}... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 33981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\\frac{\\sqrt[3]a + \\sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 33982,
"solution": "We note that $8x^3 - 3x^2 - 3x - 1 = 9x^3 - x^3 - 3x^2 - 3x - 1 = 9x^3 - (x + 1)^3$. Therefore, we have that $9x^3 = (x+1)^3$, so it follows that $x\\sqrt[3]{9} = x+1$. Solving for $x$ yields $\\frac{1}{\\sqrt[3]{9}-1} = \\frac{\\sqrt[3]{81}+\\sqrt[3]{9}+1}{8}$, so the answer is $\\bo... |
DeepScaleR | [
{
"role": "user",
"content": "The real roots of the equations \\( x^{5} + x + 1 = 0 \\) and \\( x + \\sqrt[5]{x} + 1 = 0 \\) are \\(\\alpha\\) and \\(\\beta\\), respectively. What is the value of \\(\\alpha + \\beta\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real sequence \\( x_0, x_1, x_2, \\ldots \\) is defined by \\( x_0 = 1 \\), \\( x_1 = 2 \\), and \\( n(n+1) x_{n+1} = n(n-1) x_n - (n-2) x_{n-1} \\). Find \\( \\frac{x_0}{x_1} + x_1 x_2 + \\cdots + \\frac{x_{50}}{x_{51}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1326"
} | {
"index": 33984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 33985,
"solution": "To solve the equation $\\frac{64^{x-1}}{4^{x-1}} = 256^{2x}$, we start by expressing all terms with base 2:\n\n1. **Expressing each term with base 2:**\n - $64 = 2^6$, so $64^{x-1} = (2^6)^{x-1} = 2^{6(x-1)}$\n - $4 = 2^2$, so $4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)}$\n - $256 = 2^8$... |
DeepScaleR | [
{
"role": "user",
"content": "The real value of $x$ such that $\\frac{81^{x-2}}{9^{x-2}} = 27^{3x+2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{10}{7}"
} | {
"index": 33986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the pr... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"index": 33987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reciprocal of $-2$ is equal to $\\frac{1}{-2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 33988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reciprocal of $-2024$ is ______; $7.8963\\approx$ ______ (rounded to the nearest hundredth)."
}
] | math | {
"style": "rule",
"ground_truth": "7.90"
} | {
"index": 33989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reciprocal of $\\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 33990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reciprocal of $\\left( \\frac{1}{2}+\\frac{1}{3}\\right)$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 33991,
"solution": "1. **Identify the Expression to Simplify**:\n The problem asks for the reciprocal of the expression $\\left( \\frac{1}{2}+\\frac{1}{3}\\right)$.\n\n2. **Simplify the Expression**:\n To simplify $\\frac{1}{2} + \\frac{1}{3}$, find a common denominator. The least common multiple of ... |
DeepScaleR | [
{
"role": "user",
"content": "The reciprocal of the opposite number of \\(-(-3)\\) is \\(\\frac{1}{3}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 33992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. Ahead of Peter, there were 50 people; ahead of Nicholas, there were 100; ahead of Denis, there were 170. When the command \"About-face!\"... | math | {
"style": "rule",
"ground_truth": "211"
} | {
"index": 33993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The recruits were standing in a row, one behind the other, facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. There were 50 people ahead of Peter, 100 ahead of Nicholas, and 170 ahead of Denis. Upon the command \"About face!\", everyone turne... | math | {
"style": "rule",
"ground_truth": "211"
} | {
"index": 33994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rectangle $ABCD$ below has dimensions $AB = 12 \\sqrt{3}$ and $BC = 13 \\sqrt{3}$. Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\\overline{AP}$ and $\\overline{BP}$ are joined, and the figure is then crea... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"index": 33995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \\sqrt{3}$ and $BC^{}_{} = 13 \\sqrt{3}$. Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P^{}_{}$. If triangle $ABP^{}_{}$ is cut out and removed, edges $\\overline{AP}$ and $\\overline{BP}$ are joine... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"index": 33996,
"solution": "Let $X$ be the apex of the pyramid and $M$ be the midpoint of $\\overline{CD}$. We find the side lengths of $\\triangle XMP$.\n$MP = \\frac{13\\sqrt3}{2}$. $PX$ is half of $AC$, which is $\\frac{\\sqrt{3\\cdot13^2+3\\cdot12^2}}{2} = \\frac{\\sqrt{939}}{2}$. To find $MX$, consider righ... |
DeepScaleR | [
{
"role": "user",
"content": "The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is"
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"index": 33997,
"solution": "1. **Identify the dimensions of the rectangle and the midpoints:**\n - The rectangle $ACDE$ has length $AC = 32$ and width $AE = 20$.\n - $B$ is the midpoint of $\\overline{AC}$, so $AB = BC = \\frac{32}{2} = 16$.\n - $F$ is the midpoint of $\\overline{AE}$, so $AF = FE = \\frac... |
DeepScaleR | [
{
"role": "user",
"content": "The rectangle with vertices $(-1, y), (7, y), (-1, 3)$, and $(7, 3)$ has an area of 72 square units, and $y$ is positive. What is the value of $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rectangular flag shown is divided into seven stripes of equal height. The height of the flag is $h$ and the length of the flag is twice its height. The total area of the four shaded regions is $1400 \\mathrm{~cm}^{2}$. What is the height of the flag?"
}
] | math | {
"style": "rule",
"ground_truth": "35 \\mathrm{~cm}"
} | {
"index": 33999,
"solution": "Since the flag shown is rectangular, then its total area is its height multiplied by its width, or $h \\times 2h=2h^{2}$. Since the flag is divided into seven stripes of equal height and each stripe has equal width, then the area of each stripe is the same. Since the four shaded strip... |
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