idx int64 0 5k | prompt stringlengths 228 4.87k | answers listlengths 8 8 | gt dict |
|---|---|---|---|
3,574 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(x)$ be an odd function defined on $\mathbb{R}$, and for $x \geq 0$, $f(x) = 3^x - 2x + c$ (where $c$ is a constant). Find $f(-1)$. Let's think step by step and output the final answer within \boxe... | [
"To solve this problem, let's first understand what it means for a function to be odd. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$. Given that $f(x) = 3^x - 2x + c$ for $x \\geq 0$, we need to find the value of $c$ using the property of the odd function.\n\nSince $f(x)$ is odd, we k... | {
"ground_truth": "0",
"style": "rule"
} |
3,386 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a Cartesian coordinate system, the parametric equation of a circle C is given by (with α as the parameter). If we establish a polar coordinate system with the origin O as the pole and the positive x-a... | [
"To find the polar equation of the circle given by the parametric equations \\(x = \\cos\\alpha\\) and \\(y = \\sin\\alpha\\), we can follow these steps:\n\n1. **Convert Parametric Equations to Cartesian Form:**\n The parametric equations \\(x = \\cos\\alpha\\) and \\(y = \\sin\\alpha\\) represent a circle center... | {
"ground_truth": "\\rho = 4\\sin(\\theta)",
"style": "rule"
} |
3,106 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all the solutions of $x = \left|2x-|100-2x|\right|$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"What is the sum of all the solutions of $x = \\left|2x-|100-2x|\\right|$? Let's think step by step and output the final answer within \\boxed{}",
"To solve the equation \\( x = \\left|2x - |100 - 2x|\\right| \\), we need to consider different cases for the expression inside the absolute value functions. Let's br... | {
"ground_truth": "\\frac{460}{3}",
"style": "rule"
} |
2,827 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The domain of the function $$y= \sqrt {(x-1)(x-2)}+ \sqrt {x-1}$$ is \_\_\_\_\_\_. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The domain of the function is determined by the constraints on the expressions inside the square roots. Specifically, the expressions under the square roots must be non-negative.\n\n1. For the term \\(\\sqrt{(x-1)(x-2)}\\), we need \\((x-1)(x-2) \\geq 0\\).\n2. For the term \\(\\sqrt{x-1}\\), we need \\(x-1 \\geq ... | {
"ground_truth": "\\{x | x=1 \\text{ or } x\\geq2\\}",
"style": "rule"
} |
4,572 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a geometric sequence $\{a_n\}$ with the first term being 3 and the common ratio $\frac{2}{5}$, find the value of $n$ that maximizes the product of the first $n$ terms. Let's think step by step and ... | [
"### Problem Analysis\n\n1. **Geometric Sequence**: The $n$-th term of a geometric sequence is given by $a_n = a_1 \\cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.\n2. **First Term and Common Ratio**: Given $a_1 = 3$ and $r = \\frac{2}{5}$, the $n$-th term of the sequence is $a_n = 3 \\cd... | {
"ground_truth": "n = 3",
"style": "rule"
} |
1,837 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the value of the integer \( n \) for the following pair of simultaneous equations to have no solution.
\[
\begin{array}{l}
2x = 1 + ny, \\
nx = 1 + 2y.
\end{array}
\] Let's think step by step and ... | [
"To determine the value of \\( n \\) for which the system of equations has no solution, we need to analyze the system:\n\n\\[\n\\begin{cases}\n2x = 1 + ny, \\\\\nnx = 1 + 2y.\n\\end{cases}\n\\]\n\nFirst, we can rearrange the equations to make them in the standard form \\( Ax + By = C \\):\n\n\\[\n\\begin{cases}\n2x... | {
"ground_truth": "-2",
"style": "rule"
} |
4,620 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $f(x)$ is an even function, when $x \geqslant 0$, $f(x) = x + 1$. Find $f(-1)$ = \_\_\_\_\_\_. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assist... | [
"Since \\(f(x)\\) is an even function, we have \\(f(-x) = f(x)\\) for all \\(x\\). In particular, \\(f(-1) = f(1)\\). Given that \\(f(x) = x + 1\\) for \\(x \\geqslant 0\\), we can find \\(f(1)\\) by substituting \\(x = 1\\) into the function:\n\\[ f(1) = 1 + 1 = 2. \\]\nSince \\(f(-1) = f(1)\\), we have:\n\\[ f(-1... | {
"ground_truth": "2",
"style": "rule"
} |
4,642 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The height of a cone is $h$. The development of the lateral surface of this cone is a sector with a central angle of $120^\circ$. Calculate the volume of the cone. Let's think step by step and output the... | [
"Given the central angle of the sector is $120^\\circ$, we can find the ratio of the arc length to the circumference of the base of the cone. The arc length of the sector is the circumference of the base of the cone, and the central angle is $120^\\circ$ out of a full $360^\\circ$ circle. Therefore, the ratio is $\... | {
"ground_truth": "\\frac{\\pi h^3}{24}",
"style": "rule"
} |
4,421 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The purchase price of a certain product is 40 yuan. If the selling price is set at 50 yuan, 50 units can be sold. However, for every 1 yuan increase in the selling price, the sales volume decreases by 1 ... | [
"To find the optimal selling price that maximizes profit, let's define the variables and relationships:\n\n1. Let \\( p \\) be the selling price in yuan.\n2. The cost price is 40 yuan.\n3. The number of units sold at price \\( p \\) is \\( 50 - (p - 50) = 100 - p \\).\n\nThe profit per unit is \\( p - 40 \\). The t... | {
"ground_truth": "70",
"style": "rule"
} |
4,022 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given circle $C: (x-1)^2 + (y-2)^2 = 2$, if one side $AB$ of the equilateral triangle $\triangle PAB$ is a chord of circle $C$, then the maximum value of $|PC|$ is _______. Let's think step by step and o... | [
"Given the circle \\(C: (x-1)^2 + (y-2)^2 = 2\\), we need to find the maximum value of \\(|PC|\\) where \\(P\\) is the center of an equilateral triangle \\(\\triangle PAB\\) and \\(AB\\) is a chord of the circle.\n\n1. First, let's find the radius of the circle. The equation of the circle is \\((x-1)^2 + (y-2)^2 = ... | {
"ground_truth": "|PC|_{\\text{max}} = 2r = 2\\sqrt{2}",
"style": "rule"
} |
1,435 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $1 < (x-2)^2 < 25$, find the sum of all integer solutions for x. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the problem, we need to find the integer values of \\(x\\) that satisfy the inequality \\(1 < (x-2)^2 < 25\\). We can break this down into two separate inequalities:\n\n1. \\(1 < (x-2)^2\\)\n2. \\((x-2)^2 < 25\\)\n\nFirst, let's solve the inequality \\(1 < (x-2)^2\\). This inequality is true when \\(x-2 >... | {
"ground_truth": "12",
"style": "rule"
} |
2,828 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain battery has a voltage of $48V$. When using this battery, the current $I$ (unit: $A$) is related to the resistance $R$ (unit: $\Omega$) by the function $I=\frac{48}{R}$. When $R=12\Omega$, the v... | [
"Given the function \\(I = \\frac{48}{R}\\), we need to find the value of \\(I\\) when \\(R = 12 \\Omega\\). We can substitute \\(R = 12\\) into the equation to find \\(I\\):\n\n\\[I = \\frac{48}{12}\\]\n\nWe can perform this calculation using Python to ensure accuracy.\n```python\n# Given function I = 48 / R\n# We... | {
"ground_truth": "4\\,A",
"style": "rule"
} |
3,102 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the original price of a backpack is $120.00 and the sales tax rate is 7%, if Alex calculates the final price by adding 7% sales tax on the original price first and then applying a 15% discount to t... | [
"Let's break down the calculations step by step for both Alex and Sam.\n\n### Alex's Calculation:\n1. First, calculate the sales tax on the original price: \\( 120.00 \\times 0.07 = 8.40 \\).\n2. Add the sales tax to the original price to get the total before the discount: \\( 120.00 + 8.40 = 128.40 \\).\n3. Apply ... | {
"ground_truth": "-\\$2.50",
"style": "rule"
} |
353 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the interior angles of a polygon is $1800^{\circ}$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The sum of the interior angles of a polygon is given by the formula $(n-2) \\cdot 180^{\\circ}$, where $n$ is the number of sides of the polygon. Given that the sum of the interior angles is $1800^{\\circ}$, we can set up the equation:\n\n\\[\n(n-2) \\cdot 180 = 1800\n\\]\n\nNow we can solve for $n$:\n\n\\[\nn-2 =... | {
"ground_truth": "9",
"style": "rule"
} |
3,797 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Factorization: $a^{2}-9=$____. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To factorize \\(a^2 - 9\\), we recognize that this is a difference of squares. The difference of squares formula is \\(a^2 - b^2 = (a - b)(a + b)\\). In this case, \\(a^2 - 9\\) can be written as \\(a^2 - 3^2\\).\n\nSo, \\(a^2 - 9 = (a - 3)(a + 3)\\).\n\nLet's confirm this using Python code.\n```python\nfrom sympy... | {
"ground_truth": "(a + 3)(a - 3)",
"style": "rule"
} |
605 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate the integral $$∫_{ -1 }^{ 1 }(\sqrt {1-x^{2}} - 1)dx$$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To evaluate the integral $$∫_{ -1 }^{ 1 }(\\sqrt {1-x^{2}} - 1)dx$$, we can break it into two separate integrals:\n\n$$∫_{ -1 }^{ 1 }(\\sqrt {1-x^{2}} - 1)dx = ∫_{ -1 }^{ 1 } \\sqrt {1-x^{2}} \\, dx - ∫_{ -1 }^{ 1 } 1 \\, dx.$$\n\nFirst, let's consider the integral $$∫_{ -1 }^{ 1 } \\sqrt {1-x^{2}} \\, dx$$. The f... | {
"ground_truth": "\\frac {π}{2} - 2",
"style": "rule"
} |
2,735 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first three terms of an increasing arithmetic progression is equal to 21. If 1 is subtracted from each of the first two terms of this progression and 2 is added to the third term, the resu... | [
"The sum of the first three terms of an increasing arithmetic progression is equal to 21. If 1 is subtracted from each of the first two terms of this progression and 2 is added to the third term, the resulting three numbers form a geometric progression. Find the sum of the first eight terms of the geometric progres... | {
"ground_truth": "765",
"style": "rule"
} |
3,316 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the universal set is the set of real numbers, and the sets A and B are defined as $A=\{x|1<x<4\}$, $B=\{x|3x-1<x+5\}$.
1. Find the set B and the complement of A with respect to the real numbers, $\... | [
"```python\nimport sympy as sp\n\n# Define the symbol\nx = sp.symbols('x')\n\n# Define the sets A and B\nA = sp.Interval(1, 4, left_open=True, right_open=True)\nB = sp.solve(3*x - 1 < x + 5, x)\n\n# The complement of A with respect to the real numbers\ncomplement_A = sp.S.Reals - A\n\n# The intersection of the comp... | {
"ground_truth": "a\\leq 1",
"style": "rule"
} |
266 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculat... | [
"Given the functions $f(x)$ and $g(x)$ with the following properties:\n1. $f(x) + g(2-x) = 5$\n2. $g(x) - f(x-4) = 7$\n3. The graph of $y = g(x)$ is symmetric about the line $x = 2$\n4. $g(2) = 4$\n\nFirst, we use the symmetry property of $g(x)$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, then ... | {
"ground_truth": "-24",
"style": "rule"
} |
3,061 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the greatest root of the function $g(x) = 20x^4 - 21x^2 + 5$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The function g(x) is a polynomial of degree 4, so it has at most 4 real roots. We can find these roots by solving the equation $20x^4 - 21x^2 + 5 = 0$. This can be done by making a substitution to reduce it to a quadratic equation. Let's use the substitution $y = x^2$. Then the equation becomes $20y^2 - 21y + 5 = ... | {
"ground_truth": "1",
"style": "rule"
} |
4,622 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that all terms in the geometric sequence {a_n} are positive numbers, $a_2a_4a_6=6$ and $a_8a_{10}a_{12}=24$. Find the value of $a_5a_7a_9$. Let's think step by step and output the final answer with... | [
"To solve the problem, let's break it down step by step:\n\n1. Let the first term of the geometric sequence be \\(a\\) and the common ratio be \\(r\\). Then, the terms of the sequence can be written as:\n \\[\n a_1 = a, \\quad a_2 = ar, \\quad a_3 = ar^2, \\quad a_4 = ar^3, \\quad a_5 = ar^4, \\quad a_6 = ar^5,... | {
"ground_truth": "12",
"style": "rule"
} |
435 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a magical grid paper of dimensions \(2000 \times 70\), initially all cells are grey. A painter starts at a certain cell and paints it red. Every second, the painter takes two steps: one cell to ... | [
"The initial dimensions of the grid are \\(2000 \\times 70\\). The painter starts at a certain cell and moves one cell to the left and one cell downwards every second. If the painter stands in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row. If he stands in the bottom ... | {
"ground_truth": "14000",
"style": "rule"
} |
4,252 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the least integer value of $x$ for which $3|x| + 6 < 24$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Think about the problem, and output your answer within \\boxed{}. Shayra\nThe given inequality is $3|x| + 6 < 24$. Let's solve this inequality step by step.\n\nFirst, we can isolate the term with the absolute value by subtracting 6 from both sides of the inequality:\n\\[ 3|x| < 18. \\]\n\nNext, we can divide both ... | {
"ground_truth": "-5",
"style": "rule"
} |
2,770 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) = $$\begin{cases} \overset{3^{x-4},x<0}{log_{2}x,x>0}\end{cases}$$, if the solution set of the inequality f(x) > a is (a², +∞), then the sum of all possible values of the real num... | [
"Given the piecewise function:\n\\[ f(x) = \\begin{cases} \n3^{x-4} & \\text{if } x < 0 \\\\ \n\\log_{2}x & \\text{if } x > 0 \n\\end{cases} \\]\n\nWe need to determine the values of \\(a\\) such that the solution set of the inequality \\(f(x) > a\\) is \\((a^2, +\\infty)\\).\n\n1. First, we need to analyze the fun... | {
"ground_truth": "6",
"style": "rule"
} |
4,749 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$ . Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_sta... | [
"To find the smallest integer \\( k > 1 \\) such that \\( n^k - n \\) is a multiple of \\( 2010 \\) for every integer positive \\( n \\), we need to ensure that \\( n^k \\equiv n \\pmod{2010} \\).\n\nFirst, we factorize \\( 2010 \\):\n\\[ 2010 = 2 \\times 3 \\times 5 \\times 67. \\]\n\nFor \\( n^k \\equiv n \\pmod{... | {
"ground_truth": "133",
"style": "rule"
} |
802 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sides of the bases of a truncated square pyramid are in the ratio \( m:n \) (\( m > n \)). The height of the pyramid is \( H \). The lateral edge forms an angle \( \alpha \) with the plane of the bas... | [
"To find the lateral surface area of the truncated square pyramid, we need to follow these steps:\n\n1. Determine the slant height (slant edge) of the truncated pyramid.\n2. Use the slant height to find the lateral surface area.\n\nGiven:\n- The ratio of the sides of the bases is \\( m:n \\).\n- The height of the p... | {
"ground_truth": "\\frac{2(m+n)H^2 \\cot \\alpha}{m-n} \\sqrt{2 + \\cot^2 \\alpha}",
"style": "rule"
} |
1,759 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer $n$ such that $4n$ is a perfect square and $5n$ is a perfect fifth power? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>... | [
"To find the smallest positive integer \\( n \\) such that \\( 4n \\) is a perfect square and \\( 5n \\) is a perfect fifth power, we need to consider the prime factorization of \\( n \\).\n\n1. **Prime Factorization of \\( n \\):**\n Let \\( n = 2^a \\cdot 5^b \\cdot k \\), where \\( k \\) is an integer that is ... | {
"ground_truth": "625",
"style": "rule"
} |
4,448 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $a$, $b$, $c \in \{1, 2, 3, 4, 5, 6\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \_\_\_\_\_\_ such triangles. Let's think step by step... | [
"To solve this problem, we need to count the number of isosceles triangles that can be formed with sides chosen from the set \\(\\{1, 2, 3, 4, 5, 6\\}\\). An isosceles triangle has at least two sides of equal length. We also need to ensure that the triangle inequality holds, i.e., the sum of the lengths of any two ... | {
"ground_truth": "27",
"style": "rule"
} |
3,134 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The degree of an interior angle of a regular octagon is ____. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
" A regular octagon has 8 sides. The sum of the interior angles of a polygon with \\( n \\) sides is given by \\((n-2) \\cdot 180\\) degrees. For an octagon (\\( n = 8 \\)), the sum of the interior angles is \\((8-2) \\cdot 180 = 6 \\cdot 180 = 1080\\) degrees. Since it is a regular octagon, all interior angles are... | {
"ground_truth": "135^{\\circ}",
"style": "rule"
} |
1,180 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer $n$ for which $15^n$ evenly divides $30!$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the largest integer \\( n \\) for which \\( 15^n \\) evenly divides \\( 30! \\), we need to determine how many times the prime factors of 15 (which are 3 and 5) appear in the prime factorization of \\( 30! \\). The number of times a prime \\( p \\) appears in the factorization of \\( n! \\) is given by the... | {
"ground_truth": "7",
"style": "rule"
} |
1,658 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the equation of the plane containing the points $(2, -3, 1)$, $(6, -3, 3)$, and $(4, -5, 2)$. Enter your answer in the form
\[Ax + By + Cz + D = 0,\]
where $A$, $B$, $C$, $D$ are integers such that ... | [
"To find the equation of the plane containing the points \\(P_1 = (2, -3, 1)\\), \\(P_2 = (6, -3, 3)\\), and \\(P_3 = (4, -5, 2)\\), we can follow these steps:\n\n1. **Find two vectors that lie on the plane**: We can use the vectors \\(\\vec{v_1} = P_2 - P_1\\) and \\(\\vec{v_2} = P_3 - P_1\\).\n2. **Compute the no... | {
"ground_truth": "2x + 3y - 4z + 9 = 0",
"style": "rule"
} |
1,447 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest four-digit palindromic prime? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The smallest four-digit palindromic number is 1001, and we need to check if it is a prime number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. We can check the divisibility of 1001 to determine if it is a prime number. If it is not, we will move on to the next pa... | {
"ground_truth": "1101",
"style": "rule"
} |
4,255 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all prime numbers \( p \) and all integers \( x \) such that \( 1 \leq x \leq 2p \) and \( x^{p-1} \) divides \((p-1)^{x} + 1 \). Let's think step by step and output the final answer within \boxed{}... | [
"We need to find all prime numbers \\( p \\) and all integers \\( x \\) such that \\( 1 \\leq x \\leq 2p \\) and \\( x^{p-1} \\) divides \\((p-1)^{x} + 1 \\).\n\nLet's break down the problem:\n\n1. **Understand the Divisibility Condition:**\n We need \\( x^{p-1} \\) to divide \\((p-1)^x + 1\\). This means that \\... | {
"ground_truth": "(2, 1), (2, 2), (3, 1), (3, 3), (p, 1) \\text{ for any prime } p",
"style": "rule"
} |
353 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the least positive integer \( n \) for which there exists a set \(\{s_{1}, s_{2}, \ldots, s_{n}\}\) consisting of \( n \) distinct positive integers such that
\[
\left(1-\frac{1}{s_{1}}\right)\left(... | [
"To solve the problem, we need to determine the smallest positive integer \\( n \\) for which there exists a set \\(\\{s_1, s_2, \\ldots, s_n\\}\\) of distinct positive integers such that the product \\(\\left(1 - \\frac{1}{s_1}\\right)\\left(1 - \\frac{1}{s_2}\\right) \\cdots \\left(1 - \\frac{1}{s_n}\\right) = \\... | {
"ground_truth": "39",
"style": "rule"
} |
295 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiao Ming was solving a math problem: "Find the value of the algebraic expression $10x^9 + 9x^8 + 8x^7 + 7x^6 + 6x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1$ when $x = -1$." Due to mistaking a "+" sign for a "-" s... | [
"To solve the problem, we need to determine which term Xiao Ming misread. The original expression is:\n\n\\[10x^9 + 9x^8 + 8x^7 + 7x^6 + 6x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1\\]\n\nWhen \\(x = -1\\), the expression becomes:\n\n\\[10(-1)^9 + 9(-1)^8 + 8(-1)^7 + 7(-1)^6 + 6(-1)^5 + 5(-1)^4 + 4(-1)^3 + 3(-1)^2 + 2(-1) + ... | {
"ground_truth": "6",
"style": "rule"
} |
750 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate $\lfloor\sqrt{30}\rfloor^2$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the problem, we need to find the largest integer less than or equal to the square root of 30, then square that integer. Let's break it down:\n\n1. Calculate the square root of 30.\n2. Take the floor of the square root to get the largest integer less than or equal to the square root.\n3. Square that intege... | {
"ground_truth": "25",
"style": "rule"
} |
603 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the curve $y=f(x)= \frac{4}{x}$
$(1)$ Find the equation of the tangent line to the curve $y=f(x)$ at point $A(2,2)$;
$(2)$ Find the equation of the line that is tangent to the curve $y=f(x)$ an... | [
"1. **Find the equation of the tangent line to the curve \\( y = f(x) = \\frac{4}{x} \\) at the point \\( A(2,2) \\):**\n\n The equation of a tangent line at a point \\((x_0, y_0)\\) on a curve \\( y = f(x) \\) is given by:\n \\[\n y - y_0 = f'(x_0)(x - x_0)\n \\]\n where \\( f'(x) \\) is the derivative o... | {
"ground_truth": "4x+y-8=0",
"style": "rule"
} |
3,543 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular $2n$-gon ($n \geqslant 2$) inscribed in a circle, there is a frog on each vertex. At a certain moment, all the frogs simultaneously jump to an adjacent vertex (it is allowed for multiple fro... | [
"To determine the values of $n$ such that for a regular $2n$-gon inscribed in a circle, there exists a jumping method where no line segment connecting any two different vertices with frogs passes through the center of the circle after the jump, we need to analyze the geometric properties of the polygon and the effe... | {
"ground_truth": "n \\equiv 2 \\ (\\bmod \\ 4)",
"style": "rule"
} |
1,675 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number of zeros of the function $f(x)=x- \sqrt {x}-2$ must be calculated. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the number of zeros of the function \\(f(x) = x - \\sqrt{x} - 2\\), we need to find the solutions to the equation \\(x - \\sqrt{x} - 2 = 0\\).\n\nLet's start by solving the equation step by step.\n\n1. Let \\(y = \\sqrt{x}\\). Then \\(x = y^2\\).\n2. Substituting \\(y\\) into the equation, we get \\(y^2 - ... | {
"ground_truth": "1",
"style": "rule"
} |
3,348 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four boys and three girls stand in a row. (Points are awarded only if an analytical method is provided before calculating the result)
(1) If the three girls must stand together, how many different arrang... | [
"First, let's solve each part of the problem one by one.\n\n### (1) If the three girls must stand together, how many different arrangements are possible?\n\nIf the three girls must stand together, we can treat the three girls as a single \"block\". This means we have 5 \"blocks\" to arrange (4 boys + 1 block of 3 g... | {
"ground_truth": "960",
"style": "rule"
} |
612 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = x^2 - 6x - 9$, determine the range of $f(x)$ when $x \in (1, 4)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the range of the function \\( f(x) = x^2 - 6x - 9 \\) for \\( x \\in (1, 4) \\), we need to follow these steps:\n\n1. Determine the vertex of the parabola, as the function \\( f(x) = x^2 - 6x - 9 \\) is a quadratic function that opens upwards (since the coefficient of \\( x^2 \\) is positive).\n2. Check th... | {
"ground_truth": "[-18, -14)",
"style": "rule"
} |
4,392 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are eight ways to evaluate the expression " $\pm 2^{11} \pm 2^{5} \pm 2$ ". When these eight values are listed in decreasing order, what is the third value in the list? Let's think step by step and... | [
"We start by considering the eight ways to evaluate the expression \" $\\pm 2^{11} \\pm 2^{5} \\pm 2$ \". Each term can independently be either added or subtracted. Let's list all the possible combinations and then sort them in decreasing order. Here are the steps:\n\n1. List all possible combinations of the expres... | {
"ground_truth": "2018",
"style": "rule"
} |
3,909 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the solution set of the inequality $ax ^{2}+5x-2 \gt 0$ with respect to $x$ is $ \left\{x\left|\right.\dfrac{1}{2} \lt x \lt 2\right\}$, find:
1. The value of $a$.
2. The solution set of the ... | [
"### Step 1: Determine the value of $a$\nGiven that the solution set of the inequality $ax^2 + 5x - 2 > 0$ is $\\left\\{x\\left|\\right.\\dfrac{1}{2} < x < 2\\right\\}$, we know that the roots of the equation $ax^2 + 5x - 2 = 0$ must be $\\frac{1}{2}$ and $2$. We can use Vieta's formulas to find the value of $a$.\n... | {
"ground_truth": "\\left\\{x\\left|\\right.-3 < x < \\dfrac{1}{2}\\right\\}",
"style": "rule"
} |
1,385 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on bot... | [
"First, let's understand the problem. The problem involves four two-inch squares, and the second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. We need to determine the distance of point P, the top vertex of the rotated squ... | {
"ground_truth": "1 + \\sqrt{2}",
"style": "rule"
} |
3,425 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Isosceles trapezoid $ ABCD$ , with $ AB \parallel CD$ , is such that there exists a circle $ \Gamma$ tangent to its four sides. Let $ T \equal{} \Gamma \cap BC$ , and $ P \equal{} \Gamma \cap AT$ ... | [
"Given that \\( \\frac{AP}{AT} = \\frac{2}{5} \\), we need to find the ratio \\( \\frac{AB}{CD} \\). \n\nTo solve this problem, we will use some properties of the trapezoid and the tangency points of the circle. \n\n1. Since \\(AB \\parallel CD\\), and the circle is tangent to all four sides of the trapezoid, the t... | {
"ground_truth": "\\frac{13}{3}",
"style": "rule"
} |
210 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $n$ that is not less than 9, satisfying: For any given $n$ integers (which can be the same) $a_{1}, a_{2}, \cdots, a_{n}$, there always exist 9 numbers $a_{i_{1}}, a_{i... | [
"To solve this problem, we need to find the smallest positive integer \\( n \\) that is not less than 9, such that for any given \\( n \\) integers \\( a_1, a_2, \\ldots, a_n \\), there always exist 9 numbers \\( a_{i_1}, a_{i_2}, \\ldots, a_{i_9} \\) and \\( b_i \\in \\{4,7\\} \\) (for \\( i = 1, 2, \\ldots, 9 \\)... | {
"ground_truth": "13",
"style": "rule"
} |
4,375 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 3\ma... | [
"To find the area of the parallelogram whose diagonals correspond to $\\mathbf{p} + 3\\mathbf{q}$ and $3\\mathbf{p} - 2\\mathbf{q}$, we can use the property that the diagonals of a parallelogram bisect each other. This means the sides of the parallelogram can be found as follows:\n\nIf the diagonals are $\\mathbf{d... | {
"ground_truth": "\\frac{25\\sqrt{2}}{4}",
"style": "rule"
} |
2,776 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given points $O(0,0)$, $B(1,2)$, and point $A$ lies on the coordinate axes, with $S_{\triangle OAB}=2$, find the coordinates of point $A$. Let's think step by step and output the final answer within \box... | [
"To solve this problem, we need to find the coordinates of point $A$ such that the area of triangle $OAB$ is 2, where $O(0,0)$, $B(1,2)$, and $A$ lies on one of the coordinate axes.\n\n1. **Case 1: Point $A$ lies on the x-axis**. Let's assume $A = (x, 0)$. The formula for the area of a triangle with vertices at $(x... | {
"ground_truth": "(2,0), (-2,0), (0,4), (0,-4)",
"style": "rule"
} |
8 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A new proposal suggests that years ending in double zeroes will be considered leap years only if they leave a remainder of 300 or 700 when divided by 1000. Determine how many double-zero ending leap year... | [
"To solve this problem, we need to identify the leap years ending in double zeroes between 1996 and 4096 under the new rule. The new rule states that a year ending in double zeroes will be a leap year only if it leaves a remainder of 300 or 700 when divided by 1000.\n\nHere is the step-by-step reasoning:\n\n1. Iden... | {
"ground_truth": "4",
"style": "rule"
} |
2,510 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=ax^{2}-(a+2)x+\ln x$
$(1)$ When $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;
$(2)$ When $a > 0$, if the minimum value of $f(x)$... | [
"### Part (1): Equation of the Tangent Line when \\(a = 1\\)\n\n1. **Find the function \\(f(x)\\) and its derivative:**\n \\[\n f(x) = x^2 - 3x + \\ln x\n \\]\n The derivative of \\(f(x)\\) is:\n \\[\n f'(x) = 2x - 3 + \\frac{1}{x}\n \\]\n \n2. **Find the slope of the tangent line at \\(x = 1\\):**\... | {
"ground_truth": "0\\leqslant a\\leqslant 8",
"style": "rule"
} |
883 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a right triangle $\triangle XYZ$ with a right angle at $Y$. Medians from point $X$ and point $Y$ are drawn to halves of segments $\overline{YZ}$ and $\overline{XZ}$, respectively. If the lengths... | [
"To solve the problem, we can use the properties of medians in a right triangle and the Pythagorean theorem. Here's a step-by-step approach:\n\n1. Let \\( \\triangle XYZ \\) be a right triangle with \\( \\angle Y = 90^\\circ \\). Let \\( XY = a \\), \\( YZ = b \\), and \\( XZ = c \\). Since \\( \\triangle XYZ \\) i... | {
"ground_truth": "3\\sqrt{174}",
"style": "rule"
} |
636 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $x-y=20$ and $xy=9$, what is the value of $x^2+y^2$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the value of \\(x^2 + y^2\\) given the equations \\(x - y = 20\\) and \\(xy = 9\\), we can use the identity:\n\n\\[x^2 + y^2 = (x - y)^2 + 2xy\\]\n\nGiven \\(x - y = 20\\) and \\(xy = 9\\), we can substitute these values into the identity:\n\n\\[x^2 + y^2 = 20^2 + 2 \\cdot 9\\]\n\\[x^2 + y^2 = 400 + 18\\]\... | {
"ground_truth": "418",
"style": "rule"
} |
1,071 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the parabola $y^2=4x$ and the line $y=x+b$, determine the relationship between the condition "$b=-1$" and the line passing through the focus of the parabola. Let's think step by step and output the... | [
"1. **Understand the Parabola and its Focus:**\n The given parabola is \\(y^2 = 4x\\). For a parabola of the form \\(y^2 = 4ax\\), the focus is at \\((a, 0)\\). Here, \\(4a = 4\\), so \\(a = 1\\). Therefore, the focus of the parabola \\(y^2 = 4x\\) is \\((1, 0)\\).\n\n2. **Condition for the Line to Pass Through t... | {
"ground_truth": "-1",
"style": "rule"
} |
2,485 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function $f(x) = ax^3 + 2x^2 - a^2x$ has a local minimum at $x = 1$. Find the real number $a$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the problem, we need to use the first and second derivative tests for local extrema. The steps are as follows:\n\n1. Compute the first derivative of $f(x)$ and set it to zero at $x = 1$ to find the possible values of $a$.\n2. Compute the second derivative of $f(x)$ and evaluate it at $x = 1$ to determine ... | {
"ground_truth": "4",
"style": "rule"
} |
2,844 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the complex number $z$ that satisfies $\frac{\overline{z}}{1-i}=i$, where $i$ is the imaginary unit and $\overline{z}$ is the complex conjugate of $z$, find the value of $z$. Let's think step by st... | [
"To solve the given equation $\\frac{\\overline{z}}{1-i}=i$, we need to find the complex number $z$. Here's a step-by-step approach:\n\n1. Start by isolating $\\overline{z}$:\n \\[\n \\overline{z} = i(1-i)\n \\]\n2. Simplify the right-hand side:\n \\[\n \\overline{z} = i - i^2 = i + 1\n \\]\n Since $i... | {
"ground_truth": "1-i",
"style": "rule"
} |
4,684 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC$, we have $a=3$, $b=4$, and $\sin A = \frac{3}{5}$. Find the value of $\sin B$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find \\(\\sin B\\) in triangle \\(ABC\\) given \\(a = 3\\), \\(b = 4\\), and \\(\\sin A = \\frac{3}{5}\\), we can use the Law of Sines. The Law of Sines states that:\n\n\\[\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B}\n\\]\n\nRearranging this, we can solve for \\(\\sin B\\):\n\n\\[\n\\sin B = \\frac{b \\cdot \\sin ... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} |
2,885 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a < 0$, and $|a| \cdot x \leq a$. Simplify: $|x+1| - |x-3| = \ ?$ Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The sum of all integers with an absolute value less than 4 is \\boxed{}.⥂\nThe sum of all integers with an absolute value less than 4 is \\boxed{}.",
"Let $x$ be a real number. Simplify the expression $|x-1| - |x+2| + 3|x+3| = \\ ?$ Let's think step by step and output the final answer within \\boxed{}.",
"Give... | {
"ground_truth": "-4",
"style": "rule"
} |
1,746 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $l$, $m$, $n$ be three different lines, and $a$ be a plane. Consider the following statements:
① If $l \perp a$, then $l$ intersects $a$;
② If $m \subset a$, $n \subset a$, $l \perp m$, $l \perp ... | [
"To determine the correctness of the given statements, we need to analyze each one based on the properties of lines and planes in 3D space.\n\n1. **Statement ①:** If $l \\perp a$, then $l$ intersects $a$.\n - This statement is true. If a line is perpendicular to a plane, it must intersect the plane at a single po... | {
"ground_truth": "\\text{①③④}",
"style": "rule"
} |
651 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When traveling, obey traffic rules: "stop at red lights, go at green lights." At a crossroads, the traffic signal light shows red for $30$ seconds, green for $25$ seconds, and yellow for $5$ seconds ever... | [
"To find the probability of looking up at the traffic light and seeing a green light, we need to calculate the proportion of time the light is green relative to the total cycle time.\n\nThe total cycle time of the traffic light is the sum of the time for red, green, and yellow lights. In this case, it is $30$ secon... | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} |
3,674 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Laura is training for a marathon and decides to include additional exercises in her training schedule. She walks for $15$ minutes at a speed of $3$ miles per hour, bikes $30$ miles at $3x + 2$ miles per ... | [
"To solve this problem, we need to set up an equation based on the total time Laura spends on her workout. We'll convert all times to hours (since the speeds are given in miles per hour), then sum up the times for each activity and set the total equal to \\( \\frac{205}{60} \\) hours (since 205 minutes is \\(\\frac... | {
"ground_truth": "6.35 \\text{ mph}",
"style": "rule"
} |
4,670 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=kx^{3}+3(k-1)x^{2}-k^{2}+1$ that attains extreme values at $x=0,x=4$.
(I) Find the value of the constant $k$;
(II) Let $g(x)=f(x)+c$, and $\forall x\in [-1,2]$, $g(x)\geqslant 2... | [
"### Part I: Finding the value of the constant $k$\nTo find the value of $k$, we need to use the fact that the function $f(x)=kx^3+3(k-1)x^2-k^2+1$ attains extreme values at $x=0$ and $x=4$. This means that the first derivative of $f(x)$ with respect to $x$ should be zero at these points.\n\n1. Compute the first de... | {
"ground_truth": "c\\leqslant -\\frac{49}{9}",
"style": "rule"
} |
3,493 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a_n\}$ where each term is positive, and the sum of the first $n$ terms is $S_n$, $a_1 > 1$, and $6S_n = a_n^2 + 3a_n + 2, n \in \mathbb{N}^*$.
(1) Find the general formula for the s... | [
"### Part (1): Finding the General Formula for $\\{a_n\\}$\n\nGiven the recursive relation:\n\\[ 6S_n = a_n^2 + 3a_n + 2 \\]\n\nFirst, we need to find the first term $a_1$. When $n = 1$, we have:\n\\[ 6S_1 = a_1^2 + 3a_1 + 2 \\]\nSince $S_1 = a_1$, we get:\n\\[ 6a_1 = a_1^2 + 3a_1 + 2 \\]\n\\[ a_1^2 - 3a_1 + 2 = 0 ... | {
"ground_truth": "T_n = 4 - \\frac{3n + 4}{2^n}",
"style": "rule"
} |
3,126 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given vectors $\overset{}{m}=(2\sin B, -\sqrt{3})$ and $\overset{}{n}=(\cos 2B, 2\cos^2\frac{B}{2}-1)... | [
"Let's solve the problem step by step.\n\n### Part 1: Find the measure of acute angle \\( B \\)\n\nGiven the vectors \\(\\vec{m} = (2\\sin B, -\\sqrt{3})\\) and \\(\\vec{n} = (\\cos 2B, 2\\cos^2 \\frac{B}{2} - 1)\\), and \\(\\vec{m} \\parallel \\vec{n}\\), we know that \\(\\vec{m} = k \\vec{n}\\) for some scalar \\... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} |
1,254 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the value of the fraction $\dfrac{x+1}{x-1}$ is $0$, calculate the value of $x$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the problem, we need to find the value of \\(x\\) that makes the fraction \\(\\dfrac{x+1}{x-1} = 0\\).\n\n1. A fraction is equal to zero if and only if its numerator is zero (and the denominator is non-zero). So, we set the numerator equal to zero and solve for \\(x\\):\n \\[\n x + 1 = 0\n \\]\n2. S... | {
"ground_truth": "-1",
"style": "rule"
} |
4,945 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number of ways to select a two-person welcoming committee is $15$, and students can serve on both committees. Calculate the number of different ways to select a four-person planning committee. Let's ... | [
" The number of ways to select a two-person welcoming committee from 6 students is given by the combination formula \\(\\binom{6}{2} = 15\\). This is already provided to us, so we don't need to calculate it again.\n\n Now, we need to calculate the number of ways to select a four-person planning committee from the... | {
"ground_truth": "15",
"style": "rule"
} |
4,663 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Trees are planted every 5 meters along both sides of a road, with trees planted at both the starting and ending points. A total of 72 trees were planted. How long is this road? Let's think step by step a... | [
"To solve this problem, we need to determine the length of the road given that trees are planted every 5 meters along both sides of the road, with trees planted at both the starting and ending points, and a total of 72 trees were planted.\n\nHere's the step-by-step reasoning:\n\n1. **Determine the number of trees o... | {
"ground_truth": "355 \\text{ meters}",
"style": "rule"
} |
1,622 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=4\sin x\cos \left( x-\dfrac{\pi}{3} \right)-\sqrt{3}$.
(I) Find the minimum positive period and the zero points of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ in... | [
"To solve the problem, we will follow these steps:\n\n1. **Find the minimum positive period of \\( f(x) \\):**\n The function \\( f(x) = 4 \\sin x \\cos \\left( x - \\frac{\\pi}{3} \\right) - \\sqrt{3} \\) involves trigonometric functions. We need to simplify this expression and find its period.\n\n2. **Find the ... | {
"ground_truth": "-\\sqrt{2}",
"style": "rule"
} |
2,600 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\arctan ( \tan 75^\circ - 2 \tan 30^\circ )$. (Express your answer in degrees as an angle between $0^\circ$ and $180^\circ$.) Let's think step by step and output the final answer within \boxed{}... | [
"To solve the problem, we need to compute \\(\\arctan ( \\tan 75^\\circ - 2 \\tan 30^\\circ )\\).\n\nFirst, let's calculate the values of \\(\\tan 75^\\circ\\) and \\(\\tan 30^\\circ\\).\n\n\\[\n\\tan 75^\\circ = \\tan (45^\\circ + 30^\\circ) = \\frac{\\tan 45^\\circ + \\tan 30^\\circ}{1 - \\tan 45^\\circ \\tan 30^... | {
"ground_truth": "75^\\circ",
"style": "rule"
} |
1,860 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, sides $a = 2$, $b = 2$, and $c = \sqrt{7}$. Using the Law of Cosines, calculate the angle, in degrees, opposite to side $c$. Let's think step by step and output the final answer withi... | [
"To find the angle opposite to side $c$ in the triangle with sides $a = 2$, $b = 2$, and $c = \\sqrt{7}$, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides $a$, $b$, and $c$ and angle $\\gamma$ opposite to side $c$, the following equation holds:\n\\[\nc^2 = a^2 + b^2 - 2ab \\... | {
"ground_truth": "82.82",
"style": "rule"
} |
4,759 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the result of $(5.92 + 2.4) - 3.32$ as a decimal. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the problem $(5.92 + 2.4) - 3.32$, we can follow these steps:\n\n1. First, add $5.92$ and $2.4$.\n2. Then, subtract $3.32$ from the result of the addition.\n\nLet's perform these steps using Python to ensure accuracy.\n```python\n# Step 1: Add 5.92 and 2.4\naddition_result = 5.92 + 2.4\n\n# Step 2: Subtra... | {
"ground_truth": "5.00",
"style": "rule"
} |
1,087 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the area of the shape enclosed by the curve $y=x^2$ and the lines $y=x$, $y=2x$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the area of the shape enclosed by the curve \\( y = x^2 \\) and the lines \\( y = x \\) and \\( y = 2x \\), we need to follow these steps:\n\n1. **Find the points of intersection**:\n - The intersection of \\( y = x^2 \\) and \\( y = x \\) gives us \\( x^2 = x \\), which simplifies to \\( x^2 - x = 0 \\)... | {
"ground_truth": "\\frac{7}{6}",
"style": "rule"
} |
4,361 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which negative integer values of \( n \) is the function \( f \), defined by the equation
\[
f(x) = \cos 7x \cdot \sin \frac{25x}{n^2}
\]
a periodic function with period \( T = 7\pi \)? Let's thi... | [
"To determine for which negative integer values of \\( n \\) the function \\( f(x) = \\cos(7x) \\cdot \\sin\\left(\\frac{25x}{n^2}\\right) \\) is periodic with period \\( T = 7\\pi \\), we need to ensure that both trigonometric components of \\( f(x) \\) satisfy the periodic condition.\n\n1. The function \\( \\cos(... | {
"ground_truth": "-1, -5",
"style": "rule"
} |
497 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence ${a_n}$ whose first $n$ terms sum is $S_n = nb_n$, where ${b_n}$ is an arithmetic sequence with the first term $1$ and a common difference $2$.
(I) Find the general formula for the sequ... | [
"To solve the problem, let's break it into two parts as described:\n\n### Part I: Finding the general formula for the sequence ${a_n}$\n\n1. **Identify the terms of the sequence ${b_n}$**:\n Since ${b_n}$ is an arithmetic sequence with the first term $1$ and a common difference $2$, we can write:\n \\[\n b_n ... | {
"ground_truth": "\\frac{n}{4n + 1}",
"style": "rule"
} |
2,495 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 120 ... | [
"Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 120 miles more than Ian. Calculate the difference in the distances driven by Jan and Ian. Let's think step by step a... | {
"ground_truth": "195",
"style": "rule"
} |
4,465 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The measures of the interior angles of a convex hexagon are six consecutive integers. What is the number of degrees in the largest angle? Let's think step by step and output the final answer within \boxe... | [
"The sum of the interior angles of a convex hexagon is given by:\n\\[ (6-2) \\times 180^\\circ = 4 \\times 180^\\circ = 720^\\circ \\]\n\nGiven that the measures of the six interior angles are consecutive integers, let's denote these angles as:\n\\[ a, a+1, a+2, a+3, a+4, a+5 \\]\n\nThe sum of these angles must equ... | {
"ground_truth": "122.5^\\circ",
"style": "rule"
} |
860 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are three teachers who teach six classes, with each teacher instructing two classes. Calculate the total number of possible teaching arrangements. Let's think step by step and output the final answ... | [
"First, let's analyze the problem:\n\nWe have three teachers, let's call them Teacher A, Teacher B, and Teacher C. We have six classes, and each teacher needs to teach two classes. The problem is to find the total number of ways to assign the classes to the teachers.\n\nHere's how we can approach the problem:\n\n1.... | {
"ground_truth": "90",
"style": "rule"
} |
2,354 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An equilateral triangle has sides 12 units long. Another equilateral triangle with sides 6 units long is cut off at the top, creating an isosceles trapezoid with the remaining portion. What is the ratio ... | [
"The problem involves two equilateral triangles: a larger one with side length 12 units and a smaller one with side length 6 units. The smaller triangle is cut off from the top of the larger triangle, leaving an isosceles trapezoid.\n\nFirst, we will find the area of the larger triangle and the smaller triangle. Th... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} |
2,398 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rachel has two identical basil plants and two aloe plants. She also has two identical white lamps and two identical red lamps she can put each plant under (she can put more than one plant under a lamp, b... | [
"Rachel has two identical basil plants and two aloe plants. She also has two identical white lamps and two identical red lamps. Each plant must be under exactly one lamp. \n\nTo solve this problem, we need to consider the following:\n\n1. Rachel has four plants (two basil and two aloe), and four lamps (two white an... | {
"ground_truth": "8",
"style": "rule"
} |
3,133 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\overrightarrow{a}=\left(-1,2\right)$, $\overrightarrow{b}=\left(3,m\right)$, if $\overrightarrow{a}\bot (\overrightarrow{a}+\overrightarrow{b})$, then $m=$____. Let's think step by step and outpu... | [
"Given:\n\\[\n\\overrightarrow{a} = (-1, 2), \\quad \\overrightarrow{b} = (3, m)\n\\]\nWe need to find \\(m\\) such that \\(\\overrightarrow{a} \\perp (\\overrightarrow{a} + \\overrightarrow{b})\\).\n\n1. First, calculate \\(\\overrightarrow{a} + \\overrightarrow{b}\\):\n\\[\n\\overrightarrow{a} + \\overrightarrow{... | {
"ground_truth": "-1",
"style": "rule"
} |
3,122 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \... | [
"To solve this problem, we need to find the smallest \\( n \\) such that the conditions given are satisfied. Let's break down the conditions and solve the problem step by step.\n\n1. The set \\( A = \\{1, 2, \\cdots, n\\} \\).\n2. There exists a positive integer \\( m \\) and sets \\( A_1, A_2, \\cdots, A_m \\) suc... | {
"ground_truth": "13",
"style": "rule"
} |
4,794 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive $3$-digit numbers are multiples of $25$, but not of $45$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Determine the sum of $1^3 - 2^3 + 3^3 - 4^3 + \\dots + 99^3 - 100^3$. Let's think step by step and output the final answer within \\boxed{} 100 \\).\n\nAssistant: Let's break down the problem into smaller parts and solve each one step-by-step using Python and SymPy.\n\n### Problem 1: Sum the series \\(1 - 4 + 9 - ... | {
"ground_truth": "32",
"style": "rule"
} |
4,078 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function f(x) is an even function defined on the set of real numbers R, and is decreasing on the interval [0, +∞), if f(a) ≥ f(3), determine the range of the real number a. Let's think step by ... | [
"To solve this problem, we need to use the properties of even functions and the given conditions.\n\n1. Since \\( f(x) \\) is an even function, it satisfies \\( f(x) = f(-x) \\) for all \\( x \\in \\mathbb{R} \\). This means that \\( f(a) = f(|a|) \\) and \\( f(3) = f(|3|) = f(3) \\).\n2. Since \\( f(x) \\) is decr... | {
"ground_truth": "[-3, 3]",
"style": "rule"
} |
1,567 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point $C$ has coordinates $(5, y)$. When Point $C$ is reflected over the x-axis, it lands at Point $D$. What is the sum of the four coordinate values of points $C$ and $D$, if the original y-coordinate i... | [
"To solve this problem, we need to determine the coordinates of Point $D$, which is the reflection of Point $C$ over the x-axis. If Point $C$ has coordinates $(5, -3)$, then Point $D$ will have coordinates $(5, 3)$. \n\nThe coordinates of Point $C$ are $(5, -3)$, and the coordinates of Point $D$ are $(5, 3)$. The s... | {
"ground_truth": "10",
"style": "rule"
} |
2,036 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a geometric sequence $\{a_n\}$ with a common ratio $q>0$ and the sum of the first $n$ terms denoted as $S_n$. If $2a_3$, $a_5$, and $3a_4$ form an arithmetic sequence, and $a_2a_4a_6=64$, then $a_n... | [
"Given a geometric sequence $\\{a_n\\}$ with a common ratio $q > 0$ and the sum of the first $n$ terms denoted as $S_n$. If $2a_3$, $a_5$, and $3a_4$ form an arithmetic sequence, and $a_2a_4a_6=64$, we need to find $a_n$ and $S_n$.\n\n1. **Express the terms of the geometric sequence:**\n \\[\n a_n = a_1 q^{n-1}... | {
"ground_truth": "\\frac {2^{n}-1}{2}",
"style": "rule"
} |
3,740 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In rhombus \(ABCD\) and in triangle \(ABC\), both of which contain its larger diagonal, circles are inscribed. Find the ratio of the radii of these circles if the acute angle of the rhombus is \(\alpha\)... | [
"## Step-by-step Solution\n\nTo solve this problem, we need to find the ratio of the radii of the inscribed circles of a rhombus and a triangle. Let's break down the problem:\n\n1. **Understanding the Problem:**\n - Rhombus \\(ABCD\\) has an acute angle \\(\\alpha\\).\n - Triangle \\(ABC\\) also contains the la... | {
"ground_truth": "2 \\cos^2 \\frac{\\alpha}{4}",
"style": "rule"
} |
507 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a triangle with side lengths 17 cm, 144 cm, and 145 cm, determine the area of the triangle in square centimeters. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im... | [
"To determine the area of the triangle with sides 17 cm, 144 cm, and 145 cm, we can use Heron's formula. Heron's formula states that the area of a triangle whose sides have lengths \\(a\\), \\(b\\), and \\(c\\) is given by:\n\n\\[\n\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}\n\\]\n\nwhere \\(s\\) is the semi-perimeter ... | {
"ground_truth": "1224",
"style": "rule"
} |
1,668 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given sets $A=\{x|x^2-2x<0, x\in \mathbb{R}\}$ and $B=\{x||x|>1, x\in \mathbb{R}\}$, find $A\cap B$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the intersection \\(A \\cap B\\), we need to first determine the elements in sets \\(A\\) and \\(B\\).\n\n1. Determine the set \\(A\\):\n\\[ A = \\{x \\mid x^2 - 2x < 0, x \\in \\mathbb{R}\\} \\]\nThis inequality can be factored as:\n\\[ x(x - 2) < 0 \\]\nThe roots of the equation \\(x(x-2) = 0\\) are \\(x... | {
"ground_truth": "(1,2)",
"style": "rule"
} |
4,592 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A non-zero natural number that can be expressed as the difference of squares of two non-zero natural numbers is called a "wise number". For example, $16 = 5^2 - 3^2$, so 16 is a "wise number". In the seq... | [
"A wise number is a natural number that can be expressed as the difference of squares of two non-zero natural numbers. That is, for a wise number \\( n \\), there exist natural numbers \\( a \\) and \\( b \\) such that \\( n = a^2 - b^2 = (a+b)(a-b) \\). This means \\( n \\) can be written as the product of two int... | {
"ground_truth": "2656",
"style": "rule"
} |
4,003 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In box A, there are 3 white balls and 2 red balls. In box B, there are 4 white balls and 1 red ball. Now, 2 balls are randomly taken from box A and put into box B, then 2 balls are randomly taken from bo... | [
"To solve the problem, we need to use combinatorial probability and expectation.\n\n### Part (1): Expected Value of \\(X\\)\nThe random variable \\(X\\) represents the number of red balls taken from box A. There are two possible values for \\(X\\): 0 or 1 or 2. To find the expected value \\(E[X]\\), we need to calc... | {
"ground_truth": "\\frac{3}{70}",
"style": "rule"
} |
4,196 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The triangle $\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \, \text{cm}$, $PB = 4 \, \text{cm}$, and $PC = 5 \, \text{cm}$. Calculate the length of the sides of the triangle $\tr... | [
"To solve this problem, we need to use some geometric properties and algebra. Let's consider an equilateral triangle with side length $s$. We will use the coordinates for the vertices of the equilateral triangle and the distance formula to set up the equations.\n\nWe can place the equilateral triangle in the coordi... | {
"ground_truth": "\\sqrt{25 + 12 \\sqrt{3}}\\, \\text{cm}",
"style": "rule"
} |
3,729 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the inequality
\[\frac{2}{x - 1} - \frac{3}{x - 3} + \frac{5}{x - 5} - \frac{2}{x - 7} < \frac{1}{15}.\] Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>a... | [
"To solve the inequality, we will first combine the fractions on the left-hand side and then compare the resulting expression to the right-hand side. We will then find the intervals where the inequality holds true.\n\nThe inequality we need to solve is:\n\\[\n\\frac{2}{x - 1} - \\frac{3}{x - 3} + \\frac{5}{x - 5} -... | {
"ground_truth": "(-\\infty,-8) \\cup (-7,-1) \\cup (1,3) \\cup (5,7) \\cup (8,\\infty)",
"style": "rule"
} |
84 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The dilation, centered at $-1 + 2i,$ with scale factor 4, takes $3 + 4i$ to which complex number? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The dilation of a complex number $z$ around a center $c$ with a scale factor $k$ can be computed using the formula:\n\n\\[ \\text{dilated\\_value} = k(z - c) + c \\]\n\nIn this problem, the center $c$ is $-1 + 2i$, the scale factor $k$ is 4, and the original complex number $z$ is $3 + 4i$. We will use the formula ... | {
"ground_truth": "15 + 10i",
"style": "rule"
} |
3,610 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\gcd(101^{11} + 1, 101^{11} + 101^3 + 1)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Let's start by defining the two numbers for which we need to find the greatest common divisor (GCD). The numbers are $a = 101^{11} + 1$ and $b = 101^{11} + 101^3 + 1$.\n\nWe can use the Euclidean algorithm to find the GCD of these two numbers. The Euclidean algorithm states that $\\gcd(a, b) = \\gcd(b, a \\mod b)$... | {
"ground_truth": "1",
"style": "rule"
} |
4,942 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are two lathes processing parts of the same model. The yield rate of the first lathe is $15\%$, and the yield rate of the second lathe is $10\%$. Assuming that the yield rates of the two lathes do ... | [
"### Part 1\nWe need to find the probability of both lathes producing excellent parts simultaneously. Given that the yield rates of the first and second lathes are $15\\%$ and $10\\%$ respectively, the probability of both lathes producing excellent parts simultaneously is the product of their individual probabiliti... | {
"ground_truth": "13\\%",
"style": "rule"
} |
4,841 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eight boys stood equally spaced on a circle of radius 50 feet. Each boy walked to all of the other non-adjacent boys on the circle, shook their hands, and then returned to his original spot on the circle... | [
"To solve this problem, we need to consider the geometry of the situation. The boys are standing on a circle of radius 50 feet, and they are equally spaced. This means that the distance between any two adjacent boys is the same, and it can be calculated as the circumference of the circle divided by the number of bo... | {
"ground_truth": "800 + 1200\\sqrt{2}\\text{ feet}",
"style": "rule"
} |
4,199 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What are the values of $a$ and $c$ that make the matrix
\[\begin{pmatrix} a & \frac{7}{17} \\ c & \frac{10}{17} \end{pmatrix}\]
a projection matrix? Let's think step by step and output the final answer w... | [
"What are the values of $a$ and $c$ that make the matrix\n\\[\\begin{pmatrix} a & \\frac{7}{17} \\\\ c & \\frac{10}{17} \\end{pmatrix}\\]\na projection matrix? Let's think step by step and output the final answer within \\boxed{}",
"To determine the values of \\(a\\) and \\(c\\) that make the given matrix a proje... | {
"ground_truth": "\\left( \\frac{9}{17}, \\frac{10}{17} \\right)",
"style": "rule"
} |
1,516 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a circle C that is tangent to two circles $x^2+(y+4)^2=1$ and $x^2+(y-2)^2=1$, let the equation of the locus of the center of circle C be L. Suppose the minimum distance from a point on L to a poin... | [
"### Part (Ⅰ)\nTo find the equation of the locus L of the center of circle C, let's consider the given conditions. The circle C is tangent to both circles \\(x^2 + (y + 4)^2 = 1\\) and \\(x^2 + (y - 2)^2 = 1\\). Let the center of the circle C be \\((h, k)\\) and its radius be \\(r\\).\n\nSince circle C is tangent t... | {
"ground_truth": "(2, 1) \\text{ or } (-2, 1)",
"style": "rule"
} |
2,591 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the greatest common divisor $\gcd(47^5 + 1, 47^5 + 47^3 + 1)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"First, we need to compute the greatest common divisor (GCD) of \\(47^5 + 1\\) and \\(47^5 + 47^3 + 1\\). We can use the Euclidean algorithm for this. The Euclidean algorithm tells us that:\n\n\\[\n\\gcd(a, b) = \\gcd(b, a \\mod b)\n\\]\n\nSo we need to find \\( (47^5 + 47^3 + 1) \\mod (47^5 + 1) \\).\n\nLet's deno... | {
"ground_truth": "1",
"style": "rule"
} |
3,300 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the data set 2, 4, 5, 6, 8, what is the variance of this data set? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the variance of the data set, we need to follow these steps:\n\n1. Calculate the mean of the data set.\n2. Calculate the squared difference between each data point and the mean.\n3. Find the average of these squared differences. This is the variance.\n\nLet's do this step by step using Python code.\n\n1. C... | {
"ground_truth": "4",
"style": "rule"
} |
957 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the year $500$ BC is expressed as $-500$ years, then calculate the year $2023$ AD in the same notation. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assis... | [
"If the year 500 AD is expressed as +500 years, then calculate the year 2023 BC in the same notation. Please reason step by step, and put your final answer within \\boxed{}.",
"If the year $500$ BC is expressed as $-500$ years, then calculate the year $2023$ AD in the same notation. Let's think step by step and o... | {
"ground_truth": "2023",
"style": "rule"
} |
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