problem stringlengths 8 2.17k | answer stringlengths 1 1.88k | data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
Given an arithmetic sequence $\{a_n\}$ where the common difference is nonzero, and it satisfies that $a_1 \neq 6$, $a_2$, $a_6$, $a_{14}$ form a geometric sequence.
(1) Determine the general term formula of the sequence $\{a_n\}$;
(2) Let $b_n = \frac{2}{(n+1)a_n}$, calculate the sum of the first $n$ terms of the seque... | S_n = \frac{n}{2(n+2)} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ where the common difference is nonzero, and it satisfies that $a_1 \\neq 6$, $a_2$, $a_6$, $a_{14}$ form a geo... | math | {
"ground_truth": "S_n = \\frac{n}{2(n+2)}",
"style": "rule"
} | {
"index": 0,
"split": "train/test"
} |
Circle $C$ with radius 5 has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is four times the radius of circle $E$. Find the radius of circle $D$ i... | 5 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Circle $C$ with radius 5 has diameter $\\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 1,
"split": "train/test"
} |
How many three-digit numbers can be formed using the digits 0, 1, and 2 without repeating any digit? | 4 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many three-digit numbers can be formed using the digits 0, 1, and 2 without repeating any digit? Let's think step by step and output the final answ... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 2,
"split": "train/test"
} |
If
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0,\]
find \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}.\) | 16 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If\n\\[\\frac{x}{a} + \\frac{y}{b} + \\frac{z}{c} = 4 \\quad \\text{and} \\quad \\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 0,\\]\nfind \\(\\frac{x^2}... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 3,
"split": "train/test"
} |
If the line $x+2ay-1=0$ is parallel to the line $(a-1)x-ay-1=0$, then the value of $a$ is \_\_\_\_\_\_. | \frac{1}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the line $x+2ay-1=0$ is parallel to the line $(a-1)x-ay-1=0$, then the value of $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the f... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 4,
"split": "train/test"
} |
The fifth term of a geometric sequence of positive numbers is $16$ and the ninth term is $4$. What is the seventh term of the sequence? | 8 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The fifth term of a geometric sequence of positive numbers is $16$ and the ninth term is $4$. What is the seventh term of the sequence? Let's think ste... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 5,
"split": "train/test"
} |
Given $cosα+\sqrt{3}sinα=\frac{3}{5}$, find the value of $cos({2α+\frac{π}{3}})$. | \frac{41}{50} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $cosα+\\sqrt{3}sinα=\\frac{3}{5}$, find the value of $cos({2α+\\frac{π}{3}})$. Let's think step by step and output the final answer within \\boxe... | math | {
"ground_truth": "\\frac{41}{50}",
"style": "rule"
} | {
"index": 6,
"split": "train/test"
} |
Determine the positive value of \( m \) for which the quadratic \( x^2 + 6mx + m = 0 \) has exactly one real root. | \frac{1}{9} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the positive value of \\( m \\) for which the quadratic \\( x^2 + 6mx + m = 0 \\) has exactly one real root. Let's think step by step and out... | math | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": 7,
"split": "train/test"
} |
Given the function $f(x) = |x-2| + |x+3|$, the minimum value is $m$.
(Ⅰ) Find $m$;
(Ⅱ) When $a+2b+c=m$, find the minimum value of $a^2+2b^2+3c^2$. | \frac{15}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x) = |x-2| + |x+3|$, the minimum value is $m$.\n(Ⅰ) Find $m$;\n(Ⅱ) When $a+2b+c=m$, find the minimum value of $a^2+2b^2+3c^2$. Le... | math | {
"ground_truth": "\\frac{15}{2}",
"style": "rule"
} | {
"index": 8,
"split": "train/test"
} |
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircas... | 1512 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 c... | math | {
"ground_truth": "1512",
"style": "rule"
} | {
"index": 9,
"split": "train/test"
} |
In the triangular pyramid \( S-ABC \), given \( SA=4 \), \( SB \geq 7 \), \( SC \geq 9 \), \( AB=5 \), \( BC \leq 6 \), \( AC \leq 8 \), find the maximum volume of the pyramid. | 8\sqrt{6} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the triangular pyramid \\( S-ABC \\), given \\( SA=4 \\), \\( SB \\geq 7 \\), \\( SC \\geq 9 \\), \\( AB=5 \\), \\( BC \\leq 6 \\), \\( AC \\leq 8 \... | math | {
"ground_truth": "8\\sqrt{6}",
"style": "rule"
} | {
"index": 10,
"split": "train/test"
} |
Given the function $f(x)=\dfrac{a(x-1)+b{e}^{x}}{{e}^{x}}(a\ne 0)$.
$(1)$ When $a=-1$, $b=0$, find the extremum of the function $f(x)$;
$(2)$ When $b=1$, if the function $f(x)$ has no zeros, find the range of the real number $a$. | (-e^{2},0) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x)=\\dfrac{a(x-1)+b{e}^{x}}{{e}^{x}}(a\\ne 0)$.\n\n$(1)$ When $a=-1$, $b=0$, find the extremum of the function $f(x)$;\n\n$(2)$ W... | math | {
"ground_truth": "(-e^{2},0)",
"style": "rule"
} | {
"index": 11,
"split": "train/test"
} |
Let $n$ be an integer. Express $A_{n}=a^{n}+a^{-n}$ in terms of $A_{1}=A$ and $n$. | A_{n} = \frac{1}{2^{n}} \left\{\left( A + \sqrt{A^2 - 4} \right)^n + \left( A - \sqrt{A^2 - 4} \right)^n \right\} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be an integer. Express $A_{n}=a^{n}+a^{-n}$ in terms of $A_{1}=A$ and $n$. Let's think step by step and output the final answer within \\boxed{... | math | {
"ground_truth": "A_{n} = \\frac{1}{2^{n}} \\left\\{\\left( A + \\sqrt{A^2 - 4} \\right)^n + \\left( A - \\sqrt{A^2 - 4} \\right)^n \\right\\}",
"style": "rule"
} | {
"index": 12,
"split": "train/test"
} |
Simplify: \\((x-5+ \dfrac{16}{x+3} )÷ \dfrac{x-1}{{x}^{2}-9} \\). | x^{2}-4x+3 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Simplify: \\\\((x-5+ \\dfrac{16}{x+3} )÷ \\dfrac{x-1}{{x}^{2}-9} \\\\). Let's think step by step and output the final answer within \\boxed{}.",
"r... | math | {
"ground_truth": "x^{2}-4x+3",
"style": "rule"
} | {
"index": 13,
"split": "train/test"
} |
When a car's brakes are applied, it travels 9 feet less in each second than the previous second until it comes to a complete stop. A car goes 36 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops? | 90 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "When a car's brakes are applied, it travels 9 feet less in each second than the previous second until it comes to a complete stop. A car goes 36 feet i... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 14,
"split": "train/test"
} |
If $|a|=4$, $\sqrt{{b}^{2}}=3$, and $a+b > 0$, determine the possible value of $a+b$. | 1 \text{ or } 7 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $|a|=4$, $\\sqrt{{b}^{2}}=3$, and $a+b > 0$, determine the possible value of $a+b$. Let's think step by step and output the final answer within \\bo... | math | {
"ground_truth": "1 \\text{ or } 7",
"style": "rule"
} | {
"index": 15,
"split": "train/test"
} |
If the equation \(\cot (\arcsin x) = \sqrt{a^{2} - x^{2}}\) has a real solution for \(x\), then the range of the real number \(a\) is \(\qquad\). | (-\infty, -1] \cup [1, +\infty) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the equation \\(\\cot (\\arcsin x) = \\sqrt{a^{2} - x^{2}}\\) has a real solution for \\(x\\), then the range of the real number \\(a\\) is \\(\\qqu... | math | {
"ground_truth": "(-\\infty, -1] \\cup [1, +\\infty)",
"style": "rule"
} | {
"index": 16,
"split": "train/test"
} |
Given that the line $3x+(3^a-3)y=0$ is perpendicular to the line $2x-y-3=0$, find the value of $a$. | 2 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the line $3x+(3^a-3)y=0$ is perpendicular to the line $2x-y-3=0$, find the value of $a$. Let's think step by step and output the final answe... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 17,
"split": "train/test"
} |
Given the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then shift all points on the graph to the right by $\frac{\pi}{6}$ units, and find the equation of one of the axes of symmetry for the resulting function $g(x)$. | \frac{\pi}{3} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x) = \\sin 2x + \\sqrt{3}\\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then... | math | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": 18,
"split": "train/test"
} |
Points $A_{1}, \ldots, A_{12}$ are the vertices of a regular 12-gon. How many distinct 11-segment open polygons without self-intersections are there with vertices at these points? Polygons that can be transformed into each other by rotation are considered the same. | 1024 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A_{1}, \\ldots, A_{12}$ are the vertices of a regular 12-gon. How many distinct 11-segment open polygons without self-intersections are there w... | math | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": 19,
"split": "train/test"
} |
How many perfect squares are there between 50 and 500? | 15 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many perfect squares are there between 50 and 500? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 20,
"split": "train/test"
} |
On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1? | 45 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \\( k \\) cards were taken. What is the smallest \\( k \\) such... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 21,
"split": "train/test"
} |
Grandfather distributed a bag of candies to Donald, Henry, and John, who got 70%, 25% and 5% respectively. Later Donald gave John 20 candies, and after that Henry and John shared their candies equally. By then Donald had three times as many candies as Henry. The next day Grandfather gave \( x \) candies to each of the ... | 40 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Grandfather distributed a bag of candies to Donald, Henry, and John, who got 70%, 25% and 5% respectively. Later Donald gave John 20 candies, and after... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 22,
"split": "train/test"
} |
If $|a|=4$, $|b|=7$, and $a \lt b$, find the value of $a+b$. | 3 \text{ or } 11 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $|a|=4$, $|b|=7$, and $a \\lt b$, find the value of $a+b$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "use... | math | {
"ground_truth": "3 \\text{ or } 11",
"style": "rule"
} | {
"index": 23,
"split": "train/test"
} |
Given that $x + \cos y = 3012$ and $x + 3012 \sin y = 3010$, where $0 \leq y \leq \pi$, find the value of $x + y$. | 3012 + \frac{\pi}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $x + \\cos y = 3012$ and $x + 3012 \\sin y = 3010$, where $0 \\leq y \\leq \\pi$, find the value of $x + y$. Let's think step by step and ou... | math | {
"ground_truth": "3012 + \\frac{\\pi}{2}",
"style": "rule"
} | {
"index": 24,
"split": "train/test"
} |
Inside a square with side length 20, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square, but this time they are positioned in opposite corners of the square. What is the side length of the largest square that can be inscribed in the space inside... | 10 - \frac{10}{1 + 2\sqrt{3}} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Inside a square with side length 20, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of... | math | {
"ground_truth": "10 - \\frac{10}{1 + 2\\sqrt{3}}",
"style": "rule"
} | {
"index": 25,
"split": "train/test"
} |
Find the greatest integer not exceeding \( 1 + \frac{1}{2^k} + \frac{1}{3^k} + \ldots + \frac{1}{N^k} \), where \( k = \frac{1982}{1983} \) and \( N = 2^{1983} \). | 1983 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the greatest integer not exceeding \\( 1 + \\frac{1}{2^k} + \\frac{1}{3^k} + \\ldots + \\frac{1}{N^k} \\), where \\( k = \\frac{1982}{1983} \\) an... | math | {
"ground_truth": "1983",
"style": "rule"
} | {
"index": 26,
"split": "train/test"
} |
Given the line y = (1/2)x + 3 and a line K that is parallel to the given line but 5 units away from it, find the equation of line K. | y = \dfrac{1}{2}x + \left(3 - \dfrac{5\sqrt{5}}{2}\right) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the line y = (1/2)x + 3 and a line K that is parallel to the given line but 5 units away from it, find the equation of line K. Let's think step b... | math | {
"ground_truth": "y = \\dfrac{1}{2}x + \\left(3 - \\dfrac{5\\sqrt{5}}{2}\\right)",
"style": "rule"
} | {
"index": 27,
"split": "train/test"
} |
Given $f(x) = \ln9 \cdot \log_{3}x$, then $[f(2)]' + f'(2) = \_\_\_\_\_\_$. | 1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $f(x) = \\ln9 \\cdot \\log_{3}x$, then $[f(2)]' + f'(2) = \\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxe... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 28,
"split": "train/test"
} |
Given that the zero point of the function $f(x)=2^{x}+2x-6$ is $x\_0$, and the smallest integer solution of the inequality $x-4 > x\_0$ is $k$, find the value of $k$. | 6 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the zero point of the function $f(x)=2^{x}+2x-6$ is $x\\_0$, and the smallest integer solution of the inequality $x-4 > x\\_0$ is $k$, find ... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 29,
"split": "train/test"
} |
Given a sphere is intersected by plane $\alpha$ to form circle M. A plane $\beta$ that passes through the center of circle M and forms a dihedral angle of $60^\circ$ with plane $\alpha$ intersects the sphere to form circle N. If the radius of the sphere is 4, and the area of circle M is $4\pi$, calculate the area of ci... | 13\pi | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a sphere is intersected by plane $\\alpha$ to form circle M. A plane $\\beta$ that passes through the center of circle M and forms a dihedral ang... | math | {
"ground_truth": "13\\pi",
"style": "rule"
} | {
"index": 30,
"split": "train/test"
} |
Find the sum of the digits of the following product (as a function of \( n \)):
$$
9 \cdot 99 \cdot 9999 \cdot \cdots \cdot \left(10^{2^{n}} - 1\right).
$$
In this product, the number of digits of each factor is twice the number of digits of the previous factor.
(21st United States of America Mathematical Olympiad, 19... | 9 \cdot 2^n | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the digits of the following product (as a function of \\( n \\)):\n$$\n9 \\cdot 99 \\cdot 9999 \\cdot \\cdots \\cdot \\left(10^{2^{n}} ... | math | {
"ground_truth": "9 \\cdot 2^n",
"style": "rule"
} | {
"index": 31,
"split": "train/test"
} |
What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse? | 60 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse? Let's think step by step and output the final answ... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 32,
"split": "train/test"
} |
Find the number of real solutions of the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -\frac{3}{2}.\] | 0 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of real solutions of the equation\n\\[\\frac{6x}{x^2 + 2x + 5} + \\frac{7x}{x^2 - 7x + 5} = -\\frac{3}{2}.\\] Let's think step by step ... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 33,
"split": "train/test"
} |
What is the smallest prime divisor of $3^{25} + 11^{19}$? | 2 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest prime divisor of $3^{25} + 11^{19}$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 34,
"split": "train/test"
} |
Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$ . Find the area of $S$ . | \sqrt{3} + \frac{\pi}{6} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with... | math | {
"ground_truth": "\\sqrt{3} + \\frac{\\pi}{6}",
"style": "rule"
} | {
"index": 35,
"split": "train/test"
} |
Three balls are randomly and independently tossed into bins numbered with the positive integers such that the probability of ball ending up in bin \(i\) is \(3^{-i}\) for \(i=1,2,3,...\). More than one ball is allowed in each bin. The probability that the balls end up in an arithmetic progression in distinct bins (exam... | 341 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three balls are randomly and independently tossed into bins numbered with the positive integers such that the probability of ball ending up in bin \\(i... | math | {
"ground_truth": "341",
"style": "rule"
} | {
"index": 36,
"split": "train/test"
} |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, satisfying $c\cos B+(b-2a)\cos C=0.$ Also, $c=2\sqrt{3}.$
$(1)$ Find the size of angle $C$;
$(2)$ Find the maximum area of $\Delta ABC$ and determine the shape of $\Delta ABC$ at this time. | 3\sqrt {3} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In $\\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, satisfying $c\\cos B+(b-2a)\\cos C=0.$ Also, $c=2\... | math | {
"ground_truth": "3\\sqrt {3}",
"style": "rule"
} | {
"index": 37,
"split": "train/test"
} |
On a Monday, all prices in Isla's shop are 10% more than normal. On Friday, all prices in Isla’s shop are 10% less than normal. James bought a book on Monday for £5.50. What would be the price of another copy of this book on Friday?
A) £5.50
B) £5.00
C) £4.95
D) £4.50
E) £4.40 | 4.50 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a Monday, all prices in Isla's shop are 10% more than normal. On Friday, all prices in Isla’s shop are 10% less than normal. James bought a book on ... | math | {
"ground_truth": "4.50",
"style": "rule"
} | {
"index": 38,
"split": "train/test"
} |
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference? | 672 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is ... | math | {
"ground_truth": "672",
"style": "rule"
} | {
"index": 39,
"split": "train/test"
} |
Given the Fibonacci sequence defined as follows: \( F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n} \) (for \( n \geq 1 \)), find \( \left(F_{2017}, F_{99} F_{101}+1\right) \). | 1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the Fibonacci sequence defined as follows: \\( F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n} \\) (for \\( n \\geq 1 \\)), find \\( \\left(F_{2017}, F_{... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 40,
"split": "train/test"
} |
Four fair six-sided dice are rolled. Calculate the probability that at least three of the four dice show the same value. | \frac{7}{72} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Four fair six-sided dice are rolled. Calculate the probability that at least three of the four dice show the same value. Let's think step by step and o... | math | {
"ground_truth": "\\frac{7}{72}",
"style": "rule"
} | {
"index": 41,
"split": "train/test"
} |
Consider an arithmetic sequence $\{a\_n\}$. It is given that $a\_2+a\_3+a\_4=15$ and $a\_5=9$.
(I) Find the general term formula of the sequence $\{a\_n\}$.
(II) Let $b\_n=3^{\frac{a\_n+1}{2}}$. Find the sum $S\_n$ of the first $n$ terms of the sequence $\{\frac{a\_n+1}{2}\cdot b\_n\}$. | \frac{3+(2n-1)\cdot 3^{n+1}}{4} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider an arithmetic sequence $\\{a\\_n\\}$. It is given that $a\\_2+a\\_3+a\\_4=15$ and $a\\_5=9$.\n(I) Find the general term formula of the sequenc... | math | {
"ground_truth": "\\frac{3+(2n-1)\\cdot 3^{n+1}}{4}",
"style": "rule"
} | {
"index": 42,
"split": "train/test"
} |
In $\triangle ABC$, if $AB= \sqrt {13}$, $BC=3$, $\angle C=60^{\circ}$, then $AC=$ \_\_\_\_\_\_. | 4 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, if $AB= \\sqrt {13}$, $BC=3$, $\\angle C=60^{\\circ}$, then $AC=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and output the fina... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 43,
"split": "train/test"
} |
What is the value of $330^2 - 270^2$? | 36000 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $330^2 - 270^2$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "36000",
"style": "rule"
} | {
"index": 44,
"split": "train/test"
} |
A shoe factory produced 20% of its monthly plan in the first week. In the second week, it produced 120% of the output of the first week. In the third week, it produced 60% of the output produced in the first two weeks combined. What is the monthly production plan if it is known that to complete the plan, 1480 pairs of ... | 5000 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A shoe factory produced 20% of its monthly plan in the first week. In the second week, it produced 120% of the output of the first week. In the third w... | math | {
"ground_truth": "5000",
"style": "rule"
} | {
"index": 45,
"split": "train/test"
} |
Given that the terminal side of angle $\alpha$ passes through $P(-3, 4)$, compute $\cos 2\alpha+\sin 2\alpha$. | -\frac{31}{25} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the terminal side of angle $\\alpha$ passes through $P(-3, 4)$, compute $\\cos 2\\alpha+\\sin 2\\alpha$. Let's think step by step and output... | math | {
"ground_truth": "-\\frac{31}{25}",
"style": "rule"
} | {
"index": 46,
"split": "train/test"
} |
A group wants to spend exactly 60 dollars on flowers consisting of roses and carnations for a community event. Each rose costs 4 dollars and each carnation costs 5 dollars. How many different combinations of roses and carnations can be purchased for exactly 60 dollars? | 4 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A group wants to spend exactly 60 dollars on flowers consisting of roses and carnations for a community event. Each rose costs 4 dollars and each carna... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 47,
"split": "train/test"
} |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1.$ Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}.$ | [8, \infty) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1.$ Find the set of all possible values of $\\frac{1}{x} + \\frac{1}{y}.$ Let's think s... | math | {
"ground_truth": "[8, \\infty)",
"style": "rule"
} | {
"index": 48,
"split": "train/test"
} |
What is the volume, and the length of a diagonal, of a cube whose surface area is 864 square units? | 12\sqrt{3} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the volume, and the length of a diagonal, of a cube whose surface area is 864 square units? Let's think step by step and output the final answe... | math | {
"ground_truth": "12\\sqrt{3}",
"style": "rule"
} | {
"index": 49,
"split": "train/test"
} |
Find the coordinates of the point in the plane at which the sum of the distances from it to the three points $(0, 0)$ , $(2, 0)$ , $(0, \sqrt{3})$ is minimal. | \left(1, \frac{5.5\sqrt{3} - 3}{13}\right) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the coordinates of the point in the plane at which the sum of the distances from it to the three points $(0, 0)$ , $(2, 0)$ , $(0, \\sqrt{3})$ ... | math | {
"ground_truth": "\\left(1, \\frac{5.5\\sqrt{3} - 3}{13}\\right)",
"style": "rule"
} | {
"index": 50,
"split": "train/test"
} |
Uncle Zhang delivered a truckload of vegetables to the market. It took him 3 hours to get there at a speed of 40 km/h. It only took him 2 hours to return. What was his speed on the return trip? | 60 \, \text{km/h} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Uncle Zhang delivered a truckload of vegetables to the market. It took him 3 hours to get there at a speed of 40 km/h. It only took him 2 hours to retu... | math | {
"ground_truth": "60 \\, \\text{km/h}",
"style": "rule"
} | {
"index": 51,
"split": "train/test"
} |
For a cone and a cylinder with the same base diameter and height, what is the ratio of their lateral surface areas? | \sqrt{5} : 4 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a cone and a cylinder with the same base diameter and height, what is the ratio of their lateral surface areas? Let's think step by step and output... | math | {
"ground_truth": "\\sqrt{5} : 4",
"style": "rule"
} | {
"index": 52,
"split": "train/test"
} |
Consider a regular hexagon where each side is 12 units. Find the length of the diagonal $AC$, where $A$ and $C$ are opposite vertices of the hexagon. Express your answer in simplest radical form. | AC = 12 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a regular hexagon where each side is 12 units. Find the length of the diagonal $AC$, where $A$ and $C$ are opposite vertices of the hexagon. E... | math | {
"ground_truth": "AC = 12",
"style": "rule"
} | {
"index": 53,
"split": "train/test"
} |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = x - 3$.
1. Find the analytical expression of $f(x)$.
2. Find the solution set of the inequality $f(x) \geq 1$. | [-2, 0) \cup [4, +\infty) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x > 0$, $f(x) = x - 3$.\n1. Find the analytical expression of $f(... | math | {
"ground_truth": "[-2, 0) \\cup [4, +\\infty)",
"style": "rule"
} | {
"index": 54,
"split": "train/test"
} |
Given the sets $M=\{x||x-1| > 1\}$ and $N=\{x|x^{2}-3x\leqslant 0\}$, find the intersection $M\cap N$. | \{x|2 < x\leqslant 3\} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the sets $M=\\{x||x-1| > 1\\}$ and $N=\\{x|x^{2}-3x\\leqslant 0\\}$, find the intersection $M\\cap N$. Let's think step by step and output the fi... | math | {
"ground_truth": "\\{x|2 < x\\leqslant 3\\}",
"style": "rule"
} | {
"index": 55,
"split": "train/test"
} |
An integer is between $0$ and $999999$ (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be $19$ ? | \frac{7623}{250000} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An integer is between $0$ and $999999$ (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that... | math | {
"ground_truth": "\\frac{7623}{250000}",
"style": "rule"
} | {
"index": 56,
"split": "train/test"
} |
The equation $x^2 - (2A)x + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist? (Since $A$ represents a digit, $A0$ represents the integer 10A.) | A = 5 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The equation $x^2 - (2A)x + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist? (Since $A$ represent... | math | {
"ground_truth": "A = 5",
"style": "rule"
} | {
"index": 57,
"split": "train/test"
} |
Calculate: $|-\sqrt{3}|+\sqrt{12}+\sqrt{3}(\sqrt{3}-3)$. | 3 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate: $|-\\sqrt{3}|+\\sqrt{12}+\\sqrt{3}(\\sqrt{3}-3)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 58,
"split": "train/test"
} |
Find the derivative.
$$
y=\sin \sqrt[3]{\operatorname{tg} 2}-\frac{\cos ^{2} 28 x}{56 \sin 56 x}
$$ | \frac{1}{4 \sin^2 (28x)} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the derivative.\n\n$$\ny=\\sin \\sqrt[3]{\\operatorname{tg} 2}-\\frac{\\cos ^{2} 28 x}{56 \\sin 56 x}\n$$ Let's think step by step and output the ... | math | {
"ground_truth": "\\frac{1}{4 \\sin^2 (28x)}",
"style": "rule"
} | {
"index": 59,
"split": "train/test"
} |
Given that the function $f(x)$ is an odd function defined on the interval $[-a, a]$ (where $a > 0$), and it has maximum and minimum values, determine the sum of the maximum and minimum values of the function $g(x) = f(x) + 2$. | 4 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the function $f(x)$ is an odd function defined on the interval $[-a, a]$ (where $a > 0$), and it has maximum and minimum values, determine t... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 60,
"split": "train/test"
} |
Let \( f(x) = 3x^2 + 3 \) and \( g(x) = 4x + 1 \). Calculate \( f(g(f(1))) \). | 1878 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( f(x) = 3x^2 + 3 \\) and \\( g(x) = 4x + 1 \\). Calculate \\( f(g(f(1))) \\). Let's think step by step and output the final answer within \\boxe... | math | {
"ground_truth": "1878",
"style": "rule"
} | {
"index": 61,
"split": "train/test"
} |
What is $(3^4)^2$? | 6561 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is $(3^4)^2$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6561",
"style": "rule"
} | {
"index": 62,
"split": "train/test"
} |
Given the operation $\oplus$ on $\mathbb{R}$: $a \oplus b = ab + 2a + b$. Find the range of real numbers $x$ such that $x \oplus (x - 2) < 0$. | (-2,1) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the operation $\\oplus$ on $\\mathbb{R}$: $a \\oplus b = ab + 2a + b$. Find the range of real numbers $x$ such that $x \\oplus (x - 2) < 0$. Let'... | math | {
"ground_truth": "(-2,1)",
"style": "rule"
} | {
"index": 63,
"split": "train/test"
} |
Given that the function $f(x)$ is an odd function on $\mathbb{R}$, and for $x > 0$, $f(x)= \sqrt{x}+1$, find the expression for $f(x)$ when $x < 0$. | -\sqrt{-x}-1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the function $f(x)$ is an odd function on $\\mathbb{R}$, and for $x > 0$, $f(x)= \\sqrt{x}+1$, find the expression for $f(x)$ when $x < 0$. ... | math | {
"ground_truth": "-\\sqrt{-x}-1",
"style": "rule"
} | {
"index": 64,
"split": "train/test"
} |
Find the smallest solution to the equation \[\frac{3x}{x-3} + \frac{3x^2 - 27x}{x} = 14.\] | \frac{-41 - \sqrt{4633}}{12} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest solution to the equation \\[\\frac{3x}{x-3} + \\frac{3x^2 - 27x}{x} = 14.\\] Let's think step by step and output the final answer wit... | math | {
"ground_truth": "\\frac{-41 - \\sqrt{4633}}{12}",
"style": "rule"
} | {
"index": 65,
"split": "train/test"
} |
A magician thought of two natural numbers and told their sum to Sima, while Prow knew their product was 2280. Knowing that the product is 2280, Prow was able to guess the numbers only after Sima revealed that their sum is an odd two-digit number. What are the numbers the magician thought of? | 40 \; \text{and} \; 57 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A magician thought of two natural numbers and told their sum to Sima, while Prow knew their product was 2280. Knowing that the product is 2280, Prow wa... | math | {
"ground_truth": "40 \\; \\text{and} \\; 57",
"style": "rule"
} | {
"index": 66,
"split": "train/test"
} |
Consider two sequences: an arithmetic sequence ${a_{n}}$ with sum $S_{n}$ of its first $n$ terms, and a geometric sequence ${b_{n}}$. Given that $a_{1}=3$, $b_{1}=1$, $b_{2}+S_{2}=10$, and $a_{5}-2b_{2}=a_{3}$.
(I) Find the general term formulas for both sequences ${a_{n}}$ and ${b_{n}}$.
(II) Define a new sequence ${... | 2^{2n}-\frac{5}{8}-\frac{1}{4}\left(\frac{1}{2n+1}+\frac{1}{2n+2}\right) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider two sequences: an arithmetic sequence ${a_{n}}$ with sum $S_{n}$ of its first $n$ terms, and a geometric sequence ${b_{n}}$. Given that $a_{1}... | math | {
"ground_truth": "2^{2n}-\\frac{5}{8}-\\frac{1}{4}\\left(\\frac{1}{2n+1}+\\frac{1}{2n+2}\\right)",
"style": "rule"
} | {
"index": 67,
"split": "train/test"
} |
Determine the coefficient of $x^8$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9$. | 0 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the coefficient of $x^8$ in the expansion of $\\left(\\frac{x^3}{3} - \\frac{3}{x^2}\\right)^9$. Let's think step by step and output the fina... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 68,
"split": "train/test"
} |
Given $$|\overrightarrow{a}|=1, |\overrightarrow{b}|= \sqrt{2}$$, and $$\overrightarrow{a} \perp (\overrightarrow{a} - \overrightarrow{b})$$, determine the angle between vector $$\overrightarrow{a}$$ and vector $$\overrightarrow{b}$$. | \frac{\pi}{4} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $$|\\overrightarrow{a}|=1, |\\overrightarrow{b}|= \\sqrt{2}$$, and $$\\overrightarrow{a} \\perp (\\overrightarrow{a} - \\overrightarrow{b})$$, de... | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": 69,
"split": "train/test"
} |
According to tax laws, if a one-time labor income is less than 800 yuan, no income tax is required. If it exceeds 800 yuan, income tax must be paid. The specific standard is: for the part of income between 800 and 2000 yuan, the income tax is calculated at 20%. Someone paid 130 yuan in taxes for a one-time labor servic... | 1320 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "According to tax laws, if a one-time labor income is less than 800 yuan, no income tax is required. If it exceeds 800 yuan, income tax must be paid. Th... | math | {
"ground_truth": "1320",
"style": "rule"
} | {
"index": 70,
"split": "train/test"
} |
The coefficient of $x^2$ in the expansion of $(x^2+x+1)(1-x)^6$ is __________. | 10 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The coefficient of $x^2$ in the expansion of $(x^2+x+1)(1-x)^6$ is __________. Let's think step by step and output the final answer within \\boxed{}.",... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 71,
"split": "train/test"
} |
Calculate: $(3.14-\pi)^{0}+|\sqrt{2}-1|+(\frac{1}{2})^{-1}-\sqrt{8}$. | 2-\sqrt{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate: $(3.14-\\pi)^{0}+|\\sqrt{2}-1|+(\\frac{1}{2})^{-1}-\\sqrt{8}$. Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "2-\\sqrt{2}",
"style": "rule"
} | {
"index": 72,
"split": "train/test"
} |
Compute $\cos 240^\circ$. | -\frac{1}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute $\\cos 240^\\circ$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": 73,
"split": "train/test"
} |
In Oregon, 600 people were asked their preferred term for soft drinks. The survey results are displayed in a pie chart, where the central angle for the "Soda" sector is $108^\circ$. Calculate the number of people who referred to soft drinks as "Soda". | 180 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In Oregon, 600 people were asked their preferred term for soft drinks. The survey results are displayed in a pie chart, where the central angle for the... | math | {
"ground_truth": "180",
"style": "rule"
} | {
"index": 74,
"split": "train/test"
} |
Given the sequence (1, 1, 1), (2, 4, 8), (3, 9, 27), ..., what is the sum of the three numbers in the 100th group? | 1010100 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the sequence (1, 1, 1), (2, 4, 8), (3, 9, 27), ..., what is the sum of the three numbers in the 100th group? Let's think step by step and output ... | math | {
"ground_truth": "1010100",
"style": "rule"
} | {
"index": 75,
"split": "train/test"
} |
Four friends were initially sharing the cost of renting a boat. Two additional friends joined later, which resulted in each of the original four paying $\$15$ less than initially planned. If the total cost of the boat rental was $\$180$, determine how much each of the six friends ultimately paid. | 30 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Four friends were initially sharing the cost of renting a boat. Two additional friends joined later, which resulted in each of the original four paying... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 76,
"split": "train/test"
} |
In a plane, $n$ ($n\geqslant 3$) lines can divide the plane into a maximum of $f(n)$ regions. Derive an expression for $f(n)$. | \dfrac {n^{2}+n+2}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a plane, $n$ ($n\\geqslant 3$) lines can divide the plane into a maximum of $f(n)$ regions. Derive an expression for $f(n)$. Let's think step by ste... | math | {
"ground_truth": "\\dfrac {n^{2}+n+2}{2}",
"style": "rule"
} | {
"index": 77,
"split": "train/test"
} |
Given five differently colored balls, we are to select four of them and place them into three distinct boxes in such a way that no box remains empty. The total number of different ways to accomplish this task is ______. | 180 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given five differently colored balls, we are to select four of them and place them into three distinct boxes in such a way that no box remains empty. T... | math | {
"ground_truth": "180",
"style": "rule"
} | {
"index": 78,
"split": "train/test"
} |
Given the set A={-1, 0, 1} and set B={y | y=cos(πx), x∈A}, find A∩B. | \{-1,1\} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the set A={-1, 0, 1} and set B={y | y=cos(πx), x∈A}, find A∩B. Let's think step by step and output the final answer within \\boxed{}.",
"role... | math | {
"ground_truth": "\\{-1,1\\}",
"style": "rule"
} | {
"index": 79,
"split": "train/test"
} |
A coin is flipped ten times, and the sequence of heads and tails occurring is recorded. However, it is found out that exactly five of these flips resulted in heads. How many such distinct sequences are possible? | 252 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A coin is flipped ten times, and the sequence of heads and tails occurring is recorded. However, it is found out that exactly five of these flips resul... | math | {
"ground_truth": "252",
"style": "rule"
} | {
"index": 80,
"split": "train/test"
} |
In $\Delta ABC$, $\overline{FG} \parallel \overline{AB}, CF = 5$ cm, $FA = 15$ cm, and $CG = 9$ cm. What is the number of centimeters in the length of $\overline{CB}$? | 36 \text{ cm} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In $\\Delta ABC$, $\\overline{FG} \\parallel \\overline{AB}, CF = 5$ cm, $FA = 15$ cm, and $CG = 9$ cm. What is the number of centimeters in the length... | math | {
"ground_truth": "36 \\text{ cm}",
"style": "rule"
} | {
"index": 81,
"split": "train/test"
} |
Suppose the random variable $\xi$ follows a Binomial distribution $B(6, \frac{1}{2})$. Find the value of $P(\xi=3)$. | \frac{5}{16} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose the random variable $\\xi$ follows a Binomial distribution $B(6, \\frac{1}{2})$. Find the value of $P(\\xi=3)$. Let's think step by step and ou... | math | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": 82,
"split": "train/test"
} |
Let $M=\{1,3,5,7,9\}$, $N=\{x|2x \gt 7\}$, find the intersection of the sets M and N. | \{5, 7, 9\} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $M=\\{1,3,5,7,9\\}$, $N=\\{x|2x \\gt 7\\}$, find the intersection of the sets M and N. Let's think step by step and output the final answer within ... | math | {
"ground_truth": "\\{5, 7, 9\\}",
"style": "rule"
} | {
"index": 83,
"split": "train/test"
} |
Given that the foci and the endpoints of the real axis of hyperbola $C$ are exactly the endpoints of the major axis and the foci, respectively, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, determine the equation of hyperbola $C$. | \frac{x^2}{9} - \frac{y^2}{16} = 1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the foci and the endpoints of the real axis of hyperbola $C$ are exactly the endpoints of the major axis and the foci, respectively, of the ... | math | {
"ground_truth": "\\frac{x^2}{9} - \\frac{y^2}{16} = 1",
"style": "rule"
} | {
"index": 84,
"split": "train/test"
} |
Given the sides of $\triangle ABC$, where $a$, $b$, and $c$ satisfy the equation $(2b)^2 = 4(c+a)(c-a)$, and $5a-3c=0$, find the value of $\sin A + \sin B + \sin C$. | \frac{12}{5} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the sides of $\\triangle ABC$, where $a$, $b$, and $c$ satisfy the equation $(2b)^2 = 4(c+a)(c-a)$, and $5a-3c=0$, find the value of $\\sin A + \... | math | {
"ground_truth": "\\frac{12}{5}",
"style": "rule"
} | {
"index": 85,
"split": "train/test"
} |
Given the function \( f(x) = \begin{cases} 2^{-x} - 1, & x \leq 0 \\ \log_{2}x, & x > 0 \end{cases} \) satisfies \( f(x) < 1 \), determine the range of values for \( x \). | (-1, 2) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function \\( f(x) = \\begin{cases} 2^{-x} - 1, & x \\leq 0 \\\\ \\log_{2}x, & x > 0 \\end{cases} \\) satisfies \\( f(x) < 1 \\), determine th... | math | {
"ground_truth": "(-1, 2)",
"style": "rule"
} | {
"index": 86,
"split": "train/test"
} |
Given a square piece of paper with $n$ distinct points inside it, let $M$ represent the set of $n+4$ points consisting of the four vertices of the square and the $n$ points. The square is to be cut into triangles according to the following rules:
(1) Each triangle must have its 3 vertices as elements of the set $M$.
(2... | 3n + 1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a square piece of paper with $n$ distinct points inside it, let $M$ represent the set of $n+4$ points consisting of the four vertices of the squa... | math | {
"ground_truth": "3n + 1",
"style": "rule"
} | {
"index": 87,
"split": "train/test"
} |
Proposition $p$: The equation $x^{2}+mx+1=0$ has two distinct positive real roots; Proposition $q$: The equation $4x^{2}+4(m+2)x+1=0$ has no real roots. If "Proposition $p$ or Proposition $q$" is true, find the range of values for $m$. | (-\infty, -1) | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Proposition $p$: The equation $x^{2}+mx+1=0$ has two distinct positive real roots; Proposition $q$: The equation $4x^{2}+4(m+2)x+1=0$ has no real roots... | math | {
"ground_truth": "(-\\infty, -1)",
"style": "rule"
} | {
"index": 88,
"split": "train/test"
} |
A round cake is $\frac{1}{2}$ of an inch thick and has a diameter of 16 inches. It is cut into 8 equal pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\pi$. | 4\pi | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A round cake is $\\frac{1}{2}$ of an inch thick and has a diameter of 16 inches. It is cut into 8 equal pieces. What is the number of cubic inches in t... | math | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": 89,
"split": "train/test"
} |
Two circles \(C_1\) and \(C_2\) are defined by the equations \(x^2 + y^2 = 4\) and \((x - 3)^2 + y^2 = 25\), respectively. Find the locus of the centers \((a,b)\) of all circles externally tangent to \(C_1\) and internally tangent to \(C_2\). Express your answer in the form
\[ Pa^2 + Qb^2 + Ra + Sb + T = 0, \]
where al... | 13a^2 + 49b^2 - 12a - 1 = 0 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two circles \\(C_1\\) and \\(C_2\\) are defined by the equations \\(x^2 + y^2 = 4\\) and \\((x - 3)^2 + y^2 = 25\\), respectively. Find the locus of th... | math | {
"ground_truth": "13a^2 + 49b^2 - 12a - 1 = 0",
"style": "rule"
} | {
"index": 90,
"split": "train/test"
} |
If $A B C$ is a triangle, $O$ is the center of the circumscribed circle with radius $R$, and $H$ is the orthocenter, show the following:
1. $\overrightarrow{O A} \cdot \overrightarrow{O B}=R^{2}-\frac{c^{2}}{2}$;
2. $\overrightarrow{O H}=\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}$;
3. $O H^{2}=9 R^... | O H^2 = 9 R^2 - a^2 - b^2 - c^2 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $A B C$ is a triangle, $O$ is the center of the circumscribed circle with radius $R$, and $H$ is the orthocenter, show the following:\n\n1. $\\overr... | math | {
"ground_truth": "O H^2 = 9 R^2 - a^2 - b^2 - c^2",
"style": "rule"
} | {
"index": 91,
"split": "train/test"
} |
Let $x$ and $y$ be real numbers such that
\[
3 < \frac{x - y}{x + y} < 4.
\]
If $\frac{x}{y}$ is a rational number and $x+y=10$, what is the value of $\frac{x}{y}$? | \frac{a}{b} = -2 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x$ and $y$ be real numbers such that\n\\[\n 3 < \\frac{x - y}{x + y} < 4.\n\\]\nIf $\\frac{x}{y}$ is a rational number and $x+y=10$, what is the ... | math | {
"ground_truth": "\\frac{a}{b} = -2",
"style": "rule"
} | {
"index": 92,
"split": "train/test"
} |
The sum of the interior angles of a hexagon equals to degrees. | 720^\circ | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sum of the interior angles of a hexagon equals to degrees. Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | {
"ground_truth": "720^\\circ",
"style": "rule"
} | {
"index": 93,
"split": "train/test"
} |
There are 2 male students and 2 female students left in the classroom. If they leave the classroom one after another, determine the probability that the second person to leave is a male student. | \frac{1}{2} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are 2 male students and 2 female students left in the classroom. If they leave the classroom one after another, determine the probability that th... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 94,
"split": "train/test"
} |
Given that $m$ and $n$ are two non-coincident lines, and $\alpha$ and $\beta$ are two non-coincident planes, consider the following propositions:
① If $m \subset \beta$ and $\alpha \parallel \beta$, then $m \parallel \alpha$;
② If $m \parallel \beta$ and $\alpha \parallel \beta$, then $m \parallel \alpha$;
③ If $... | ①④ | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $m$ and $n$ are two non-coincident lines, and $\\alpha$ and $\\beta$ are two non-coincident planes, consider the following propositions: \n... | math | {
"ground_truth": "①④",
"style": "rule"
} | {
"index": 95,
"split": "train/test"
} |
Given the estimated attendance of 50,000 fans in Atlanta and 60,000 fans in Boston, and the actual attendance being within 10% of Anita's estimate and Bob's estimate being within 10% of the actual attendance in Boston, calculate the largest possible difference between the numbers attending the two games to the nearest ... | 22000 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the estimated attendance of 50,000 fans in Atlanta and 60,000 fans in Boston, and the actual attendance being within 10% of Anita's estimate and ... | math | {
"ground_truth": "22000",
"style": "rule"
} | {
"index": 96,
"split": "train/test"
} |
If $z-\text{i}= \frac{4-2\text{i}}{1+2\text{i}}$, then the imaginary part of the complex number $z$ is __________. | -1 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $z-\\text{i}= \\frac{4-2\\text{i}}{1+2\\text{i}}$, then the imaginary part of the complex number $z$ is __________. Let's think step by step and out... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 97,
"split": "train/test"
} |
Find the limit as \( n \) approaches infinity of the sum given by:
\[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{n}{N + i^2} \]
where \( N = n^2 \). | \frac{\pi}{4} | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the limit as \\( n \\) approaches infinity of the sum given by:\n\n\\[ \\lim_{n \\to \\infty} \\sum_{i=1}^{n} \\frac{n}{N + i^2} \\]\n\nwhere \\( ... | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": 98,
"split": "train/test"
} |
Given the function $f(x)= \begin{cases}2^{x}-1,(x\leqslant 0) \\ f(x-2)+1,(x > 0)\end{cases}$, arrange the even zero points of the function $g(x)=f(x)- \dfrac {1}{2}x$ in ascending order to form a sequence and find the sum of the first 10 terms of the sequence, $S_{10}$. | 90 | my_numinamath | [
{
"content": "You are given a problem. Think about it step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x)= \\begin{cases}2^{x}-1,(x\\leqslant 0) \\\\ f(x-2)+1,(x > 0)\\end{cases}$, arrange the even zero points of the function $g(x)=... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 99,
"split": "train/test"
} |
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