idx int64 0 61 | prompt stringlengths 255 1.88k | answers sequencelengths 8 8 |
|---|---|---|
19 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the constant term of the expansion of $\left(3x + \dfrac{2}{5x}\right)^8$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the constant term in the expansion of $\\left(3x + \\dfrac{2}{5x}\\right)^8$, we can use the binomial theorem. The binomial theorem states that:\n\n\\[\n(a + b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^k\n\\]\n\nFor the expression $\\left(3x + \\dfrac{2}{5x}\\right)^8$, let $a = 3x$ and $b = \\dfrac{2}{... |
14 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mary believes that the amount of sleep she gets the night before an exam and her score on that exam are inversely related. On her first exam, she got six hours of sleep and scored 60 on the exam. To the ... | [
"To solve this problem, we need to first understand the relationship between the amount of sleep and the exam score. According to the problem, the amount of sleep \\( S \\) and the exam score \\( E \\) are inversely related, which means \\( S \\times E = k \\) for some constant \\( k \\).\n\nFrom the first exam, we... |
59 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider sets $\{6, 7, 8, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$? Let's think step by step a... | [
"Consider sets $\\{6, 7, 8, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$? Let's think step by step and output the final answer within \\boxed{}",
"The problem involves determining the largest possible value ... |
7 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the distance from moving point $M(x, y)$ to point $F(2, 0)$ is equal to the distance from point $M$ to the line $x=-2$, then the equation of the trajectory of point $M$ is ______. Let's think step by ... | [
"To find the equation of the trajectory of point \\( M(x, y) \\), we use the given condition that the distance from \\( M \\) to the point \\( F(2, 0) \\) is equal to the distance from \\( M \\) to the line \\( x = -2 \\).\n\nThe distance from \\( M(x, y) \\) to \\( F(2, 0) \\) is given by the formula for the dista... |
6 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The numbers $x, y, z$ are proportional to $2, 3, 5$. The sum of $x$, $y$, and $z$ is $100$. The number $y$ is given by the equation $y \equal{} ax \minus{} 10$. Find the value of $a$. Let's th... | [
"The numbers $x, y, z$ are proportional to $2, 3, 5$. This means that there exists a constant $k$ such that:\n\\[ x = 2k, \\quad y = 3k, \\quad z = 5k \\]\n\nWe also know that the sum of $x$, $y$, and $z$ is 100:\n\\[ x + y + z = 100 \\]\nSubstituting the proportional values, we get:\n\\[ 2k + 3k + 5k = 100 \\]\n\\... |
29 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When $a=-1$, the value of the square root $\sqrt{7-a}$ is ____. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"When $a=-1$, the value of the square root $\\sqrt{7-a}$ is:\n\\[\n\\sqrt{7 - (-1)} = \\sqrt{7 + 1} = \\sqrt{8} = \\sqrt{4 \\cdot 2} = 2\\sqrt{2}\n\\]\nSo the final answer is \\(\\boxed{2\\sqrt{2}}\\).",
"When $a=-1$, the value of the square root $\\sqrt{7-a}$ is $\\sqrt{7 - (-1)} = \\sqrt{7 + 1} = \\sqrt{8} = 2\... |
61 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all positive integer triples $(x, y, z) $ that satisfy the equation $$ x^4+y^4+z^4=2x^2y^2+2y^2z^2+2z^2x^2-63. $$ Let's think step by step and output the final answer within \boxed{}<|im_end|>... | [
"First, we can rewrite the given equation \\(x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2z^2 + 2z^2x^2 - 63\\). Rearrange it to:\n\n\\[x^4 + y^4 + z^4 - 2x^2y^2 - 2y^2z^2 - 2z^2x^2 = -63.\\]\n\nThe left-hand side can be expressed in terms of squares of differences:\n\n\\[(x-y)^2 + (y-z)^2 + (z-x)^2 = 2(x^2 + y^2 + z^2 - xy - y... |
38 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $\sqrt{48} \div \sqrt{3} - \sqrt{\frac{1}{2}} \times \sqrt{12} + \sqrt{24}$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"First, we need to simplify each term:\n\n1. $\\sqrt{48} \\div \\sqrt{3} = \\sqrt{48 \\div 3} = \\sqrt{16} = 4$\n2. $\\sqrt{\\frac{1}{2}} \\times \\sqrt{12} = \\sqrt{\\frac{1}{2} \\times 12} = \\sqrt{6}$\n3. $\\sqrt{24} = 2\\sqrt{6}$\n\nNow, we can substitute these simplified terms into the original expression:\n\n... |
33 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $cos(π-α)=-\frac{3}{5}$, calculate the value of $tan(\frac{{3π}}{2}-α)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Given \\( \\cos(\\pi - \\alpha) = -\\frac{3}{5} \\), we first use the identity \\( \\cos(\\pi - \\alpha) = -\\cos(\\alpha) \\). This gives us:\n\\[ -\\cos(\\alpha) = -\\frac{3}{5} \\implies \\cos(\\alpha) = \\frac{3}{5} \\]\n\nNow, we need to find the value of \\( \\tan\\left(\\frac{3\\pi}{2} - \\alpha\\right) \\)... |
12 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four trainee teachers, A, B, C, and D, are to be assigned to three different classes with each class receiving at least one teacher. However, Teachers A and B cannot be assigned to the same class. Find t... | [
"The problem can be broken down as follows:\n\n1. First, determine the total number of ways to assign 4 teachers to 3 classes such that each class gets at least one teacher, without considering the constraint that A and B cannot be in the same class.\n2. Then, subtract the number of assignments where A and B are in... |
8 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arrange nine $1$s and four $0$s in a row, where no two $0$s are adjacent. There are ____ different ways to arrange them. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|i... | [
"First, arrange the nine $1$s in a row. There is only one way to do this:\n\n\\[ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\]\n\nThis creates ten slots (positions) where a $0$ could be placed, as shown below:\n\n\\[ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_ \\ 1 \\ \\_... |
12 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\text{A}*\text{B}$ means $\frac{\text{A}+\text{B}}{2}$, calculate the result of $(5*7)*9$.
A) 5
B) 6
C) 7.5
D) 8
E) 9 Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|... | [
"$\\frac{5+7}{2}=6$\nSo, $5*7=6$\nThe expression now becomes $6*9$\n$\\frac{6+9}{2}=7.5$\nThe answer is $\\boxed{7.5}$",
"Let's calculate this step by step. According to the definition, \\( \\text{A} * \\text{B} = \\frac{\\text{A} + \\text{B}}{2} \\).\n\nFirst, we need to calculate \\( 5 * 7 \\):\n\\[ 5 * 7 = \\f... |
58 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all values of parameters \(a\) and \(b\) for which the polynomial
\[ f(x) = x^5 - 3x^4 + ax^3 + bx^2 - 5x - 5 \]
is divisible by \(x^2 - 1\) without remainder. Let's think step by step and output ... | [
"To find the values of \\(a\\) and \\(b\\) for which the polynomial \\(f(x) = x^5 - 3x^4 + ax^3 + bx^2 - 5x - 5\\) is divisible by \\(x^2 - 1\\) without remainder, we need to ensure that \\(f(x)\\) has both \\(1\\) and \\(-1\\) as roots. This means that \\(f(1) = 0\\) and \\(f(-1) = 0\\).\n\n1. Substitute \\(x = 1\... |
14 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the line $2ax-by+2=0$ $(a > 0,b > 0)$ always bisects the circumference of the circle $x^{2}+y^{2}+2x-4y+1=0$, then the minimum value of $\dfrac{1}{a}+ \dfrac{1}{b}$ is ______. Let's think step by step... | [
"First, we need to understand that if the line $2ax - by + 2 = 0$ always bisects the circumference of the circle $x^2 + y^2 + 2x - 4y + 1 = 0$, then the line must pass through the center of the circle.\n\nWe start by finding the center of the circle. The equation of the circle can be rewritten in standard form by c... |
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