idx int64 0 61 | prompt stringlengths 255 1.88k | answers listlengths 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... |
23 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a modified sequence rule that Zan has created:
1) If a number is 30 or less, triple the number.
2) If a number is more than 30, subtract 15 from it until it falls 30 or less.
Let $G$ be the fir... | [
"First, let's understand the sequence rule and the condition for a number to be \"sweet\". The sequence rule is:\n1) If a number is 30 or less, triple the number.\n2) If a number is more than 30, subtract 15 from it until it falls 30 or less.\n\nA number $G$ is a \"sweet number\" if none of the terms in the sequenc... |
46 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the cartesian coordinate system $(xOy)$, the equation of curve $(C_{1})$ is $\frac{x^2}{3} + y^2 = 1$. Establish a polar coordinate system with coordinate origin $O$ as the pole and the positive half ... | [
"### Part (I): Polar equation of curve \\( C_1 \\) and Cartesian equation of curve \\( C_2 \\)\n#### Polar equation of curve \\( C_1 \\)\nThe equation of curve \\( C_1 \\) in Cartesian coordinates is \\(\\frac{x^2}{3} + y^2 = 1\\). To convert it to polar coordinates, we use the relations \\( x = \\rho \\cos \\theta... |
41 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many terms of the arithmetic sequence 95, 90, 85, $\dots$ appear before the number 5 appears? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To determine how many terms of the arithmetic sequence \\(95, 90, 85, \\ldots\\) appear before the number 5 appears, we need to analyze the sequence. The first term \\(a_1\\) of the sequence is 95, and the common difference \\(d\\) is -5. The general formula for the \\(n\\)-th term of an arithmetic sequence is giv... |
7 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=\begin{cases} x+2 & (x\leqslant -1) \\ x^{2} & (-1< x < 2) \\ 2x & (x\geqslant 2) \end{cases}$
$(1)$ Find $f(2)$, $f\left(\dfrac{1}{2}\right)$, $f[f(-1)]$;
$(2)$ If $f(a)=3$,... | [
"### Part 1: Find \\(f(2)\\), \\(f\\left(\\frac{1}{2}\\right)\\), and \\(f[f(-1)]\\)\n1. To find \\(f(2)\\), we use the third piece of the function because \\(2 \\geq 2\\):\n\\[ f(2) = 2 \\cdot 2 = 4 \\]\n\n2. To find \\(f\\left(\\frac{1}{2}\\right)\\), we use the second piece of the function because \\(-1 < \\frac... |
49 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an equilateral triangle \( \triangle ABC \) with side length 1, \( B \) and \( C \) have \( n \) equally spaced points along the segment \( BC \), named \( P_1, P_2, \ldots, P_{n-1} \) in the direc... | [
"To solve this problem, we need to consider the geometry of the equilateral triangle and the dot products of the vectors involved.\n\n1. **Vectors in the Equilateral Triangle:**\n - Let \\( \\overrightarrow{AB} = \\mathbf{u} \\) and \\( \\overrightarrow{AC} = \\mathbf{v} \\).\n - Since \\( \\triangle ABC \\) is... |
1 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$, respectively. Line through $B$ parallel to $\overline{AC}$ intersects $\overline{AD}$ at $T$ so that $AT/DT = 2$ and $BT... | [
"To solve this problem, we will use coordinate geometry and properties of similar triangles. Let's assume the following coordinates for the points in the triangle:\n\n- \\(A = (0, 0)\\)\n- \\(B = (b, 0)\\)\n- \\(C = (0, c)\\)\n- \\(D = (d, \\frac{cd}{b})\\) (since \\(D\\) is on \\(BC\\), and \\(BD\\) is parallel to... |
14 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For $\mathbf{v} = \begin{pmatrix} 12 \\ -8 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -9 \\ 6 \end{pmatrix}$, compute $\text{proj}_{\mathbf{w}} \mathbf{v}$. Let's think step by step and output the ... | [
"The projection of vector $\\mathbf{v}$ onto vector $\\mathbf{w}$ is given by the formula:\n\\[\n\\text{proj}_{\\mathbf{w}} \\mathbf{v} = \\frac{\\mathbf{v} \\cdot \\mathbf{w}}{\\mathbf{w} \\cdot \\mathbf{w}} \\mathbf{w}\n\\]\n\nFirst, we need to calculate the dot product $\\mathbf{v} \\cdot \\mathbf{w}$:\n\\[\n\\m... |
59 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest area that triangle \( OAB \) can have if its sides \( OA \) and \( OB \) lie on the graph of the function \( y=2|x|-x+1 \), and the line \( AB \) passes through the point \( M(0, 2) ... | [
"To solve this problem, we need to find the smallest area of triangle \\(OAB\\) where \\(O\\) is the origin \\((0,0)\\), \\(A\\) and \\(B\\) are points on the graph of the function \\(y = 2|x| - x + 1\\), and the line \\(AB\\) passes through the point \\(M(0, 2)\\).\n\nFirst, let's understand the function \\(y = 2|... |
20 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Sunday, taxi driver Xiao Zhang volunteered to shuttle tourists for free on an east-west highway. It is stipulated that east is positive and west is negative. The taxi's itinerary is as follows (in kil... | [
"On Sunday, taxi driver Xiao Zhang volunteered to shuttle tourists for free on an east-west highway. It is stipulated that east is positive and west is negative. The taxi's itinerary is as follows (in kilometers): +10, -3, +4, -2, +13, -8, -7, -5, -2.\n\n$(1)$ When Xiao Zhang drops off the last tourist at their des... |
32 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate ∅(2) + ∅(3) + ∅(5) where ∅(x) is defined to be the average of x and x^3. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"First, we need to define the function ∅(x) which is the average of x and x^3. The function can be written as:\n\\[\n∅(x) = \\frac{x + x^3}{2}\n\\]\nWe need to calculate ∅(2), ∅(3), and ∅(5) and then sum them up. Let's start by calculating each term individually.\n\n1. Calculate ∅(2):\n\\[\n∅(2) = \\frac{2 + 2^3}{2... |
4 | <|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 defined on $\mathbb{R}$ and $f(x+4)=f(x-2)$. If $f(x)={6}^{-x}$ when $x\in [-3,0]$, then $f(919)$ $=$ Let's think step by step and output the final answer within \bo... | [
"To solve this problem, we need to carefully analyze the properties of the function $f(x)$.\n\n1. **Even Function**: Since $f(x)$ is an even function, we know that $f(x) = f(-x)$ for all $x \\in \\mathbb{R}$.\n2. **Periodicity**: The function $f(x)$ is periodic with a period of $6$, because $f(x+4) = f(x-2)$ implie... |
55 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Describe a construction of quadrilateral $ABCD$ given:
(i) the lengths of all four sides;
(ii) that $AB$ and $CD$ are parallel;
(iii) that $BC$ and $DA$ do not intersect. Let's think s... | [
"To construct the quadrilateral \\(ABCD\\) given the lengths of all four sides, the fact that \\(AB\\) and \\(CD\\) are parallel, and that \\(BC\\) and \\(DA\\) do not intersect, we can follow these steps:\n\n1. Let's denote the lengths of the sides as follows: \\(AB = a\\), \\(BC = b\\), \\(CD = c\\), and \\(DA = ... |
21 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a_{n}\}$ that satisfies $a_n-a_{n+1}=\frac{a_na_{n+1}}{2^{n-1}}$, and $a_{2}=-1$, if $a_{k}=16a_{8}$, then calculate the positive integer $k$. Let's think step by step and output the ... | [
"Given the sequence $\\{a_{n}\\}$ that satisfies $a_n - a_{n+1} = \\frac{a_n a_{n+1}}{2^{n-1}}$ and $a_{2} = -1$, and $a_{k} = 16a_{8}$, we need to find the positive integer $k$.\n\nFirst, let's rewrite the recurrence relation:\n\\[ a_n - a_{n+1} = \\frac{a_n a_{n+1}}{2^{n-1}}. \\]\n\nDividing both sides by $a_n a_... |
4 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the probability distribution list of the random variable $X$ as $P(X=k)=m\left( \frac{2}{3} \right)^{k}$, where $k=1$, $2$, $3$, determine the value of $m$. Let's think step by step and output the ... | [
"To find the value of \\( m \\), we need to use the property of probability distributions which states that the sum of all probabilities must equal 1. Therefore, we need to sum the probabilities for \\( k = 1, 2, 3 \\) and set the sum equal to 1. This gives us the equation:\n\\[ P(X=1) + P(X=2) + P(X=3) = 1 \\]\n\\... |
5 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given in the Cartesian coordinate system $xOy$, a line $l$ passing through a fixed point $P$ with an inclination angle of $\alpha$ has the parametric equation: $\begin{cases} x=t\cos \alpha \\ y=-2+t\sin... | [
"The polar coordinates of the center of the circle are $(3, \\frac{\\pi}{2})$. Converting this to Cartesian coordinates, the center of the circle is $(0, 3)$. The radius of the circle is $3$. Therefore, the equation of the circle in Cartesian coordinates is:\n\\[x^2 + (y-3)^2 = 9.\\]\nThe parametric equation of the... |
47 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $6$ cards labeled $1$, $2$, $3$, $4$, $5$, and $6$ are drawn without replacement, calculate the probability that the product of the numbers of the $2$ cards is a multiple of $4$. Let's think step b... | [
"Step 1: Determine the total number of ways to choose 2 cards out of the 6 cards.\nThe total number of ways to choose 2 cards out of 6 is given by the combination formula \\( \\binom{6}{2} \\):\n\\[\n\\binom{6}{2} = \\frac{6!}{2!(6-2)!} = \\frac{6 \\times 5}{2 \\times 1} = 15\n\\]\nSo, there are 15 possible ways to... |
44 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate plane $(xOy)$, points $A(2,2)$ and $B(0,4)$ are given. Circle $C$ has line segment $AB$ as its diameter.
(1) Find the equation of circle $C$.
(2) Let point $P$ be a point on c... | [
"### Part 1: Find the equation of circle \\(C\\)\n\n1. **Find the center of the circle**:\n The center of the circle \\(C\\) is the midpoint of the line segment \\(AB\\). The coordinates of the midpoint are calculated as follows:\n \\[\n \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) = \\left( \\... |
51 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The function \( f \) is such that \( f(x+2y) - f(3x-2y) = 2y - x \) for all \( x, y \). Find all possible values of the expression \(\frac{f(5t) - f(t)}{f(4t) - f(3t)}\). Let's think step by step and out... | [
"To solve the problem, we need to find the functional form of \\( f \\). Let's start by substituting specific values for \\( x \\) and \\( y \\) to derive a general form for \\( f \\).\n\nGiven the functional equation:\n\\[ f(x+2y) - f(3x-2y) = 2y - x \\]\n\n### Step 1: Substitute \\( x = 0 \\) and \\( y = 0 \\)\n\... |
51 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all functions \( f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+} \) that satisfy \( f(x) < 2x - \frac{x}{1 + x^{\frac{3}{2}}} \) for \( x > 0 \) and \( f(f(x)) = \frac{5}{2} f(x) - x \). Let's think st... | [
"To solve this problem, we need to find a function \\( f: \\mathbb{R}_{+} \\rightarrow \\mathbb{R}_{+} \\) that satisfies the given conditions. Let's break it down into manageable parts and use reasoning to derive the function.\n\n1. **Understand the Inequality Condition:**\n The function \\( f(x) \\) must satisf... |
15 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with an eccentricity of $$\frac { \sqrt {6}}{3}$$, and a circle centered at the origin O with the major axis of ellips... | [
"### Part 1: Find the standard equation of the ellipse C\n\n#### Step 1: Determine the eccentricity\nThe eccentricity \\( e \\) of the ellipse is given by:\n\\[ e = \\frac{\\sqrt{a^2 - b^2}}{a} = \\frac{\\sqrt{6}}{3} \\]\nSo,\n\\[ \\frac{a^2 - b^2}{a^2} = \\frac{6}{9} = \\frac{2}{3} \\]\n\\[ a^2 - b^2 = \\frac{2}{3... |
17 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$. Let's think step by step and output the final an... | [
"To determine the value of \\( m \\) such that the function \\( f(x) = (m^2 - m - 1)x^{m^2 + m - 3} \\) is a decreasing function on the interval \\((0, +\\infty)\\), we need to ensure that the derivative of \\( f(x) \\) with respect to \\( x \\) is negative for all \\( x \\) in this interval. The derivative of \\( ... |
53 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \[n = f_1\cdot f_2\cdots f_k,\] where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in whi... | [
"To determine $D(72)$, we first need to find all the possible ways to write 72 as a product of factors greater than 1, considering the order of factors. We can break this down into the following steps:\n\n1. Factorize 72 into its prime factors.\n2. Generate all possible ordered factorizations of 72.\n\nThe prime fa... |
42 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the point \\((2,3)\\) lies on the hyperbola \\(C\\): \\( \dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)\\), and the focal distance of \\(C\\) is \\(4\\), then its eccentricity is ... | [
"To solve the problem, we need to determine the eccentricity \\(e\\) of the hyperbola \\(C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\\). We are given that the point \\((2,3)\\) lies on the hyperbola and that the focal distance (distance between the foci) is 4.\n\nThe steps are as follows:\n\n1. **Substitute the Poin... |
55 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain measurement, the result $\xi$ follows a normal distribution $N(1, \sigma^2)$ ($\sigma>0$). If the probability of $\xi$ falling within the interval $(0, 1)$ is $0.4$, find the probability of ... | [
"In the normal distribution \\(N(1, \\sigma^2)\\), the mean \\(\\mu\\) is 1. The probability of \\(\\xi\\) falling within the interval \\((0, 1)\\) is given by:\n\\[ P(0 < \\xi < 1) = 0.4. \\]\nBecause the normal distribution is symmetric around the mean, we can infer:\n\\[ P(1 < \\xi < 2) = P(0 < \\xi < 1) = 0.4. ... |
38 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the line $ax+by+c=0$ has an inclination angle $\alpha$, and $\sin\alpha + \cos\alpha = 0$, determine the value of $a-b$. Let's think step by step and output the final answer within \boxed{}<|im_end... | [
"The inclination angle $\\alpha$ of the line $ax + by + c = 0$ is the angle the line makes with the positive x-axis. The slope of the line, which is $-\\frac{a}{b}$, is equal to $\\tan(\\alpha)$.\n\nGiven that $\\sin(\\alpha) + \\cos(\\alpha) = 0$, we can solve for $\\alpha$. We know that $\\sin(\\alpha) = -\\cos(\... |
41 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$ , for all positive integers... | [
"To solve this problem, we need to analyze the sequence of polynomials $\\left( P_{n}(X)\\right)_{n\\in \\mathbb{Z}_{>0}}$ and determine for which values of $n$ the polynomial $P_n(X)$ is divisible by $X^2 + 1$.\n\nThe sequence is defined as follows:\n1. $P_{1}(X) = 2X$\n2. $P_{2}(X) = 2(X^2 + 1)$\n3. $P_{n+2}(X) =... |
4 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the terms in an infinite geometric series is 20, and the sum of their cubes is 80. Find the first term. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_star... | [
"The solution process is as follows:\n\n1. Let the first term of the geometric series be \\(a\\) and the common ratio be \\(r\\).\n2. The sum of the infinite geometric series is given by \\(S = \\frac{a}{1 - r} = 20\\).\n3. The sum of the cubes of the terms in the infinite geometric series can be represented as an ... |
15 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the triangle shown, what is the positive difference between the greatest and least possible integral values of $x$?
[asy]
defaultpen(linewidth(0.7));
pair a,b,c;
b = (1,3); c = (4,0);
draw(a--b--c--a... | [
"In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Given the sides of the triangle as \\(x\\), \\(7\\), and \\(10\\), we can write the following inequalities:\n\n1. \\(x + 7 > 10\\)\n2. \\(x + 10 > 7\\)\n3. \\(7 + 10 > x\\)\n\nFrom the first inequality, \\(x +... |
10 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( A=(0,6) \) and \( B=(0,10) \). Points \( A' \) and \( B' \) are on the line \( y=x \), and \( \overline{AA'} \) and \( \overline{BB'} \) intersect at \( C=(3,7) \). Find the length of \( \overline... | [
"To solve the problem, let's start by determining the coordinates of points \\( A' \\) and \\( B' \\). Since points \\( A' \\) and \\( B' \\) are on the line \\( y = x \\), we can denote \\( A' = (a, a) \\) and \\( B' = (b, b) \\).\n\nGiven that \\( \\overline{AA'} \\) and \\( \\overline{BB'} \\) intersect at \\( C... |
4 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a hyperbola that shares the same foci with the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ and passes through the point $P\left(-\dfrac{\sqrt{5}}{2}, -\sqrt{6}\right)$, find the standard equat... | [
"The standard form of a hyperbola centered at the origin with foci on the x-axis is $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$. The foci of the hyperbola are located at $(\\pm c, 0)$ where $c = \\sqrt{a^2 + b^2}$.\nGiven the hyperbola $\\frac{x^2}{16} - \\frac{y^2}{9} = 1$, the foci are at $(\\pm \\sqrt{16 + 9}, 0) ... |
0 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\sin 2\alpha = \frac{1}{4}$, calculate $\sin^{2}\left(\alpha + \frac{\pi}{4}\right)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Given that $\\sin 2\\alpha = \\frac{1}{4}$, we need to find the value of $\\sin^2\\left(\\alpha + \\frac{\\pi}{4}\\right)$. We can use the identity for the sine of a sum of angles and the double angle formula.\n\nFirst, recall the identity:\n\\[\n\\sin^2\\left(\\alpha + \\frac{\\pi}{4}\\right) = \\left(\\frac{\\sq... |
49 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence $\{a_n\}$, $a_n = (n+1)(\frac{7}{8})^n$, the maximum term in the sequence $\{a_n\}$ is the $\_\_\_\_\_\_$th term. Let's think step by step and output the final answer within \boxed{}<|im_... | [
"In the sequence $\\{a_n\\}$, $a_n = (n+1) \\left(\\frac{7}{8}\\right)^n$, the maximum term in the sequence $\\{a_n\\}$ is the \\_\\_\\_\\_\\_\\_th term.\nTo find the maximum term in the sequence $a_n = (n+1) \\left(\\frac{7}{8}\\right)^n$, we need to determine the value of $n$ for which $a_n$ is maximized. This ca... |
7 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the point \(\left( 7, \frac{\pi}{3}, -3 \right)\) in cylindrical coordinates to rectangular coordinates. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|... | [
"Cylindrical coordinates \\((r, \\theta, z)\\) can be converted to rectangular coordinates \\((x, y, z)\\) using the following transformations:\n\\[\nx = r \\cos(\\theta)\n\\]\n\\[\ny = r \\sin(\\theta)\n\\]\n\\[\nz = z\n\\]\nGiven the cylindrical coordinates \\(\\left( 7, \\frac{\\pi}{3}, -3 \\right)\\):\n\n1. \\(... |
15 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangular coordinate system $(xOy)$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=2$, and the parametric equation of curve $C_{2}$ is $ \begin{cases} x=2-t \\ y=t \end{cases} (t \text{ is the p... | [
"First, let's find the intersection points of curves $C_{1}$ and $C_{2}$. \n\n1. The equation of curve $C_{1}$ is $x^2 + y^2 = 2$.\n2. The parametric equation of curve $C_{2}$ is $x = 2 - t$ and $y = t$.\n\nBy substituting the parametric equations into the equation of $C_{1}$, we get:\n\\[\n(2 - t)^2 + t^2 = 2.\n\\... |
39 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)= \frac{x^2+ax+11}{x+1} (a \in \mathbb{R})$, if for any $x \in \mathbb{N}^*$, $f(x) \geqslant 3$ always holds, then the range of values for $a$ is ______. Let's think step by step... | [
"Given the function $f(x) = \\frac{x^2 + ax + 11}{x + 1}$, we want to find the range of values for $a$ such that $f(x) \\geq 3$ for any $x \\in \\mathbb{N}^*$. This means that for any positive integer $x$, the inequality $\\frac{x^2 + ax + 11}{x + 1} \\geq 3$ must hold.\n\nWe start by rewriting the inequality:\n\\[... |
28 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the solution that satisfies the system of inequalities:
$$
\left\{\begin{array}{c}
19p < 10, \\
p > 0.5
\end{array}\right., \text{ resulting in } \frac{1}{2} < p < \frac{10}{19}
$$ Let's think step... | [
"We need to solve the system of inequalities:\n\n\\[\n\\left\\{\n\\begin{array}{c}\n19p < 10, \\\\\np > 0.5\n\\end{array}\n\\right.\n\\]\n\nFirst, we'll solve each inequality separately:\n\n1. From \\(19p < 10\\), we get:\n\\[\np < \\frac{10}{19}\n\\]\n\n2. From \\(p > 0.5\\), we get:\n\\[\np > \\frac{1}{2}\n\\]\n\... |
56 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A parallelogram has one angle measuring 150 degrees, and its consecutive sides are 10 inches and 18 inches long. A height is dropped from a point on the extension of one side, 2 inches beyond the vertex,... | [
"Let's start by analyzing the problem. We have a parallelogram with one angle of 150 degrees, and the sides are 10 inches and 18 inches long. The height is dropped from a point 2 inches beyond the vertex on the extension of one side to the opposite side.\n\n1. **Calculate the Area of the Parallelogram:**\n The ar... |
0 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A choir has 90 members who need to be arranged in rows with the same number of members per row. The number of members per row must be between 4 and 15. Determine the number of possible row-lengths for ar... | [
"Given that there are 90 members in the choir and we need to arrange them in rows with the same number of members per row, the number of members per row must be a divisor of 90. Additionally, the number of members per row must be between 4 and 15. We need to find all divisors of 90 within this range and count them.... |
2 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's grandmother's age is fifteen times Jessica's age. If Jessica's grandmother was 60 years old when Jessica was born, how many years old is Jessica now? Let's think step by step and output the fin... | [
"To solve this problem, we need to set up an equation based on the information given. Let's denote Jessica's current age by \\( J \\) and her grandmother's current age by \\( G \\).\n\nAccording to the problem:\n1. Jessica's grandmother's age is fifteen times Jessica's age. So, \\( G = 15J \\).\n2. Jessica's grandm... |
9 | <|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 $a$, $b$, and $c$ respectively, and $b^2+c^2=bc+a^2$.
(1) Find the measure of angle $A$.
(2) If $a=\sqrt{3}$, find the maximum value... | [
"### Step 1: Finding the measure of angle \\( A \\)\nGiven the equation \\( b^2 + c^2 = bc + a^2 \\), we can use the Law of Cosines to find the measure of angle \\( A \\). The Law of Cosines states:\n\\[ a^2 = b^2 + c^2 - 2bc \\cos A. \\]\nBy substituting \\( a^2 \\) from the Law of Cosines into the given equation,... |
54 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the area of rectangle ABCD is 100 square meters, points P and Q are located one-fourth the length of sides AD and CD respectively from vertices A and C, and point R is located one-fourth the l... | [
"```\nTo solve the problem, we will use coordinate geometry. Let's assume that the rectangle ABCD is placed in a coordinate system with A at the origin (0,0), B at (a,0), D at (0,b), and C at (a,b). Since the area of the rectangle is 100 square meters, we have \\(a \\cdot b = 100\\).\n\nPoints P, Q, and R are locat... |
60 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The coordinates of the point that is symmetric with point $P(2,-4)$ about the origin are ____. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The coordinates of the point that is symmetric with point \\(P(2, -4)\\) about the origin are \\(\\boxed{(-2, 4)}\\).",
"The coordinates of the point that is symmetric with point \\(P(2, -4)\\) about the origin are \\((-2, 4)\\).\n\n\\[\n\\boxed{(-2, 4)}\n\\]",
"When a point is reflected about the origin, the ... |
40 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The centers of the circumcircle and the incircle of a triangle are symmetrically located with respect to one of the sides of the triangle. Find the angles of the triangle. Let's think step by step and ou... | [
"The centers of the circumcircle and the incircle of a triangle are symmetrically located with respect to one of the sides of the triangle. This means that the triangle is isosceles with the base equal to the height from the vertex opposite the base. The only triangle that satisfies this property is an isosceles ri... |
0 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu* Let's think step by step and output the final answer wit... | [
"To solve this problem, we need to determine the number of trailing zeros in the factorial of a number. The number of trailing zeros in \\(k!\\) is given by the number of times \\(k!\\) can be divided by 10. Since \\(10 = 2 \\times 5\\), and in factorial, there are usually more multiples of 2 than 5, the number of ... |
6 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a circle with a diameter of 10 meters? Express your answer in terms of \(\pi\). Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The area of a circle is given by the formula:\n\\[\nA = \\pi r^2\n\\]\nwhere \\(r\\) is the radius of the circle. The diameter \\(d\\) of the circle is 10 meters, and the radius \\(r\\) is half of the diameter:\n\\[\nr = \\frac{d}{2} = \\frac{10}{2} = 5 \\text{ meters}\n\\]\nNow we can substitute the radius into t... |
15 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \[g(n) = \left\{
\begin{array}{cl}
n^2-5 & \text{ if }n<0, \\
3n-25 & \text{ if }n \geq 0.
\end{array}
\right.\] Determine the positive difference between the two values of $b$ that satisfy the equat... | [
"To solve this problem, we need to follow these steps:\n\n1. Calculate \\( g(-3) \\).\n2. Calculate \\( g(3) \\).\n3. Substitute these values into the equation \\( g(-3) + g(3) + g(b) = 0 \\).\n4. Solve for \\( b \\) considering the piecewise nature of \\( g(n) \\).\n5. Calculate the positive difference between the... |
7 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular tetrahedron, how many unordered pairs of edges determine the same plane?
A) 3
B) 6
C) 12
D) 15
E) 21 Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start... | [
"To solve this problem, let's analyze the structure of a regular tetrahedron. A tetrahedron has 4 triangular faces, 6 edges, and 4 vertices. Each face of the tetrahedron is an equilateral triangle, and each edge is shared by two faces.\n\nWe need to find how many unordered pairs of edges determine the same plane. S... |
55 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)$ whose domain is $\mathbb{R}$, when $x < 0$, $f(x) = x^3 - 1$; when $-1 \leqslant x \leqslant 1$, $f(-x) = -f(x)$; when $x > \frac{1}{2}$, $f\left(x+ \frac{1}{2}\right) = f\left(... | [
"To evaluate $f(6)$, we need to understand the behavior of the function $f(x)$ based on the given properties:\n\n1. For $x < 0$, $f(x) = x^3 - 1$.\n2. For $-1 \\leq x \\leq 1$, $f(-x) = -f(x)$, which means $f(x)$ is an odd function in this interval.\n3. For $x > \\frac{1}{2}$, $f\\left(x + \\frac{1}{2}\\right) = f\... |
47 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the sets $M=\{x\mid x \geq -1\}$ and $N=\{x\mid 2-x^2 \geq 0\}$, find the union of $M$ and $N$, denoted $M \cup N$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|... | [
"First, we need to determine what set $N$ represents. The set $N$ is defined by the inequality $2 - x^2 \\geq 0$. Let's solve this inequality.\n\n\\[\n2 - x^2 \\geq 0 \\implies x^2 \\leq 2 \\implies -\\sqrt{2} \\leq x \\leq \\sqrt{2}\n\\]\n\nSo, the set $N$ is the interval $[-\\sqrt{2}, \\sqrt{2}]$.\n\nNow, we need... |
43 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Five friends—Masha, Nastya, Irina, Olya, and Anya—gather every day in the park after buying ice cream from the shop around the corner. One day, they had a conversation.
Irina: I was the first in line!
... | [
"First, let's understand the problem:\n\n1. Irina was the first in line. So, Irina is at position 1.\n2. Olya was the last in line. So, Olya is at position 5.\n3. Only one person was after Anya. So, Anya is at position 4.\n4. There were five people ahead of Masha. So, Masha is at position 6, which is not possible s... |
47 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set $P = \{1, 2\}$, calculate the number of sets $Q$ that satisfy $Q \subseteq P$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve this problem, we need to understand the concept of subsets. For a given set $P$, the number of subsets of $P$ (including the empty set and $P$ itself) is given by $2^n$, where $n$ is the number of elements in the set $P$. \n\nHere, the set $P = \\{1, 2\\}$ has $2$ elements. Therefore, the number of subset... |
37 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(Full score: 10 points)
Given _p_: ≤2, _q_: $x^2 - 2x + 1 - m^2 \leq 0$ ($m > 0$), if the negation of _p_ is a necessary but not sufficient condition for the negation of _q_, find the range of values fo... | [
"(Full score: 10 points)\n\nTo solve the problem, we need to understand the logical relationship between the negations of \\(p\\) and \\(q\\).\n\n1. **Understand the Statements \\(p\\) and \\(q\\):**\n - \\(p\\): \\(x \\leq 2\\)\n - \\(q\\): \\(x^2 - 2x + 1 - m^2 \\leq 0\\)\n\n2. **Negate the Statements \\(p\\)... |
32 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Square $PQRS$ is inscribed in right triangle $LMN$, as shown below. If $LP = 30$ units and $SN = 70$ units, what is the area of square $PQRS$?
[asy]
draw((9,15.6)--(10.7,14.6)--(11.7,16.3));
draw((0,0)--... | [
"1. Identify and label the given values:\n - $LP = 30$ units\n - $SN = 70$ units\n - Let the side length of the square $PQRS$ be $s$.\n\n2. Analyze the problem:\n - Since $PQRS$ is a square, all its sides are equal, and each side is $s$ units.\n - The triangles $\\triangle LQP$ and $\\triangle SRN$ are si... |
23 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $A=60^{\circ}$, $b=1$, and the area of the triangle $S_{\triangle} = \frac{\sqrt{3}}{2}$. Find the value of $\frac{a+b+c}{\sin A + \sin B + \sin C}$ . Let's think step by step and out... | [
"### Step 1: Understand the given information\n- $\\angle A = 60^\\circ$\n- Side $b = 1$\n- Area of $\\triangle ABC = \\frac{\\sqrt{3}}{2}$\n\n### Step 2: Use the area formula to find side $c$\nThe area of a triangle can be given by the formula:\n\\[ \\text{Area} = \\frac{1}{2} \\cdot b \\cdot c \\cdot \\sin(A) \\]... |
16 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$\cos 37^{\circ}\cos 23^{\circ}-\sin 37^{\circ}\sin 23^{\circ}=$ ______. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"$\\cos 37^{\\circ}\\cos 23^{\\circ}-\\sin 37^{\\circ}\\sin 23^{\\circ}=$\n\n\\[\n\\boxed{0.2249510540316801}\n\\]\n\nBased on the given information, the final answer is boxed as follows: \\(\\boxed{0.2249510540316801}\\).",
"\\[\n\\cos 37^{\\circ}\\cos 23^{\\circ}-\\sin 37^{\\circ}\\sin 23^{\\circ}=\\cos (37^{\\... |
14 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend... | [
"To solve this problem, we need to calculate the number of ways to ascend from one side of the mountain and descend from any other side. We have the following paths leading to the top of the mountain from each side:\n\n- East: 2 paths\n- West: 1 path\n- South: 3 paths\n- North: 4 paths\n\nTo find the maximum number... |
53 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let digits \( P, Q, R, S \) be selected from \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) with \( P < Q < R < S \). If the sum \( \frac{P}{R} + \frac{Q}{S} \) is to be as small as possible, what is this minimum val... | [
"To solve the problem, we need to find the minimum value of the expression \\(\\frac{P}{R} + \\frac{Q}{S}\\) where \\(P, Q, R, S\\) are distinct digits selected from \\(\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}\\) with \\(P < Q < R < S\\).\n\nTo achieve the smallest possible sum, we should select the smallest possible values... |
5 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In his book "On the Sphere and Cylinder," Archimedes derived an equivalent trigonometric identity $\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{(2 n-1) \pi}{2 n}=\frac{1}{\tan \frac{\pi}... | [
"In his book \"On the Sphere and Cylinder,\" Archimedes derived an equivalent trigonometric identity \\(\\sin \\frac{\\pi}{2n} + \\sin \\frac{2\\pi}{2n} + \\ldots + \\sin \\frac{(2n-1)\\pi}{2n} = \\frac{1}{\\tan \\frac{\\pi}{4n}}\\) while proving the formula for the volume of a sphere. If both sides are multiplied ... |
31 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, given that $a_1=3$, $a_{n+1}=2S_n+3$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $b_n=(2n-1)a_n$, find... | [
"### Part 1: Find the general formula for the sequence $\\{a_n\\}$\n\nGiven:\n1. $a_1 = 3$\n2. $a_{n+1} = 2S_n + 3$\n\nWe know that $S_n = a_1 + a_2 + \\cdots + a_n$. Hence, $S_n = S_{n-1} + a_n$. Using this, we can express $a_{n+1}$ in terms of $a_n$.\n\nFrom the recurrence relation:\n\\[a_{n+1} = 2S_n + 3\\]\n\\[... |
58 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $1 \in \{x, x^2\}$, then $x = \_\_\_\_\_\_\_\_\_\_\_\_$. 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 consider the two possible cases: \\( x = 1 \\) or \\( x^2 = 1 \\).\n\n1. If \\( x = 1 \\), then \\( x \\) is already 1, so it satisfies the condition.\n2. If \\( x^2 = 1 \\), then \\( x \\) can be either 1 or -1. Since we already considered \\( x = 1 \\), the other possibility is \... |
11 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $\{a_n\}$, the common difference $d > 0$, $a_{2009}$ and $a_{2010}$ are the two roots of the equation $x^2 - 3x - 5 = 0$, and $S_n$ is the sum of the first $n$ terms of the seq... | [
"To solve this problem, we need to find the smallest natural number \\( n \\) for which the sum of the first \\( n \\) terms of the arithmetic sequence \\( \\{a_n\\} \\) is greater than 0. Let's break this down into manageable parts:\n\n1. **Find the roots of the quadratic equation \\( x^2 - 3x - 5 = 0 \\)**:\n T... |
32 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The necessary and sufficient condition for the equation x<sup>2</sup>+4xy+3y<sup>2</sup>+λy-1=0 to represent two intersecting lines is _____. Let's think step by step and output the final answer within \... | [
"To determine the necessary and sufficient condition for the equation \\(x^2 + 4xy + 3y^2 + \\lambda y - 1 = 0\\) to represent two intersecting lines, we need to analyze the conic section represented by the equation.\n\nA general conic section equation is \\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\\). The discriminant ... |
57 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)$ defined as follows:
$$f(x) = \begin{cases} x^2, & x \leqslant 1 \\ x + \frac{6}{x} - 6, & x > 1 \end{cases}$$
Find $f(f(-2))$ and the minimum value of $f(x)$. Let's think step b... | [
"First, we need to find \\( f(f(-2)) \\).\n\n1. Calculate \\( f(-2) \\):\nSince \\(-2 \\leq 1\\), we use the first part of the piecewise function:\n\\[ f(-2) = (-2)^2 = 4. \\]\n\n2. Calculate \\( f(4) \\):\nSince \\(4 > 1\\), we use the second part of the piecewise function:\n\\[ f(4) = 4 + \\frac{6}{4} - 6 = 4 + 1... |
15 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all real solutions of the equation
$$
x^{2}+2 x \sin (x y)+1=0
$$ Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve the equation \\(x^2 + 2x \\sin(xy) + 1 = 0\\), we can analyze it step by step.\n\n1. **Rewrite the Equation:**\n The equation \\(x^2 + 2x \\sin(xy) + 1 = 0\\) can be treated as a quadratic equation in terms of \\(x\\). For real solutions, the discriminant of the quadratic equation must be non-negative. ... |
43 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An open box is created by cutting a square of side $x$ inches from each corner of a 12 in. by 18 in. rectangular metal sheet and folding up the flaps, determine the volume expression of the box formed. L... | [
"Consider a function $f(x) = \\frac{1}{2}x^3 - 3x^2 + 4x - 1$. Find the critical points of $f(x)$ and determine whether each critical point corresponds to a local maximum, local minimum, or neither, using the second derivative test. Please box your final answer.\n Cusert\nConsider a line segment $\\overline{AB}$ of... |
8 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given propositions $p$: The domain of the function $y=\log(ax^2-ax+1)$ is $\mathbb{R}$, and proposition $q$: The function $y=x^{a^2-2a-3}$ is decreasing on the interval $(0, +\infty)$, if "p OR q" is a t... | [
"To solve this problem, we need to analyze the truth values of propositions $p$ and $q$:\n\n1. **Proposition $p$: The domain of the function $y=\\log(ax^2-ax+1)$ is $\\mathbb{R}$.**\n\n For the domain of $y=\\log(ax^2-ax+1)$ to be all real numbers, the expression inside the logarithm, $ax^2 - ax + 1$, must be pos... |
16 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A bowl contains 15 jellybeans (five red, three blue, two green, and five white). If you pick four jellybeans from the bowl at random and without replacement, what is the probability that exactly two will... | [
"A bowl contains 15 jellybeans (five red, three blue, two green, and five white). If you pick four jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red and one is green? Express your answer as a common fraction. Let's think step by step and output the fina... |
50 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the product of the matrices:
\[\begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & -2 \\ -2 & 3 & 1 \end{pmatrix} \begin{pmatrix} 3 & -3 & 1 \\ 2 & 1 & -4 \\ 5 & 0 & 0 \end{pmatrix}.\] Let's think step by step... | [
"To find the product of the given matrices:\n\\[\n\\begin{pmatrix} 2 & 0 & -1 \\\\ 1 & 3 & -2 \\\\ -2 & 3 & 1 \\end{pmatrix} \\begin{pmatrix} 3 & -3 & 1 \\\\ 2 & 1 & -4 \\\\ 5 & 0 & 0 \\end{pmatrix},\n\\]\nwe will perform matrix multiplication.\n\nMatrix multiplication of \\( A \\) and \\( B \\) is given by:\n\\[\n... |
19 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the hyperbola $\dfrac{x^2}{a^2} - y^2 = 1\ (a > 0)$ with its right focus $F$, draw a line perpendicular to an asymptote of the hyperbola passing through point $F$. Let $P$ be the foot of the perpen... | [
"Given:\n- Hyperbola equation: $\\dfrac{x^2}{a^2} - y^2 = 1$\n- The right focus $F$ is at $(c, 0)$ where $c = \\sqrt{a^2 + 1}$.\n- The asymptotes of the hyperbola are the lines $y = \\pm \\dfrac{x}{a}$.\n- The $y$-coordinate of point $P$ is $\\dfrac{2\\sqrt{5}}{5}$.\n\nWe will consider the asymptote $y = \\dfrac{x}... |
27 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest integer solution of the equation
$$
\left\lfloor\frac{x}{8}\right\rfloor-\left\lfloor\frac{x}{40}\right\rfloor+\left\lfloor\frac{x}{240}\right\rfloor=210
$$
(If $x$ is a real number, ... | [
"To solve the given equation \\(\\left\\lfloor\\frac{x}{8}\\right\\rfloor - \\left\\lfloor\\frac{x}{40}\\right\\rfloor + \\left\\lfloor\\frac{x}{240}\\right\\rfloor = 210\\) for the smallest integer \\(x\\), we can iterate over possible values of \\(x\\) and check the equation. We need to find the smallest \\(x\\) ... |
23 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express $\sqrt{a} \div\sqrt{b}$ as a common fraction, given:
$$\frac{ {\left( \frac{1}{3} \right)}^2 + {\left( \frac{1}{4} \right)}^2 }{ {\left( \frac{1}{5} \right)}^2 + {\left( \frac{1}{6} \right)}^2} ... | [
"First, we calculate the left-hand side of the given equation:\n\n$$\\frac{ \\left( \\frac{1}{3} \\right)^2 + \\left( \\frac{1}{4} \\right)^2 }{ \\left( \\frac{1}{5} \\right)^2 + \\left( \\frac{1}{6} \\right)^2 }$$\n\nThis can be simplified as follows:\n\n\\[\n\\frac{ \\frac{1}{9} + \\frac{1}{16} }{ \\frac{1}{25} +... |
17 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When the value of $y$ is tripled and then this increased value is divided by 7, the result is 12. What is the value of $y$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
... | [
"To find the value of \\( y \\), we need to solve the equation that represents the given problem. The problem states that when the value of \\( y \\) is tripled and then this increased value is divided by 7, the result is 12. This can be written as the equation:\n\\[\n\\frac{3y}{7} = 12\n\\]\nTo solve for \\( y \\)... |
44 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x) = a\ln x + \frac{1}{2}x^2$ ($a > 0$), if for any two different positive real numbers $x_1$, $x_2$, it always holds that $\frac{f(x_1) - f(x_2)}{x_1 - x_2} \geq 2$, determine the range of the ... | [
"Given the function $f(x) = a\\ln x + \\frac{1}{2}x^2$ ($a > 0$), we want to determine the range of the parameter $a$ such that for any two different positive real numbers $x_1$ and $x_2$, the inequality $\\frac{f(x_1) - f(x_2)}{x_1 - x_2} \\geq 2$ always holds.\n\nThis inequality can be interpreted as the slope of... |
27 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify first, then evaluate: $\left(x+3\right)\left(x-2\right)+x\left(4-x\right)$, where $x=2$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"We need to simplify the expression \\((x+3)(x-2) + x(4-x)\\) first, and then evaluate it at \\(x=2\\).\n\n1. Simplify the expression \\((x+3)(x-2) + x(4-x)\\):\n\n\\[\n(x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6\n\\]\n\\[\nx(4-x) = 4x - x^2\n\\]\n\\[\n(x+3)(x-2) + x(4-x) = (x^2 + x - 6) + (4x - x^2) = x^2 + x - 6... |
3 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify the denominator of the following fraction and bring the resulting expression to its simplest form:
$$
\frac{\sqrt{3+\sqrt{5}}}{\sqrt[3]{\left(4 \sqrt{2}-2 \sqrt{10}\right)}^{2}}
$$ Let's think ... | [
"Simplify the denominator of the following fraction and bring the resulting expression to its simplest form:\n\n$$\n\\frac{\\sqrt{3+\\sqrt{5}}}{\\sqrt[3]{\\left(4 \\sqrt{2}-2 \\sqrt{10}\\right)}^{2}}\n$$\n\nTo solve this, we need to simplify the expression step by step. Let's start with the denominator:\n\n1. First... |
55 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $x > 0$, $y > 0$, and $2x - xy + 6y = 0$, determine the minimum value of $x + y$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Given the equation $2x - xy + 6y = 0$, we can solve for $x$ in terms of $y$ or $y$ in terms of $x$. We'll solve for $y$ in terms of $x$:\n\n\\[2x - xy + 6y = 0\\]\n\\[2x = xy - 6y\\]\n\\[2x = y(x - 6)\\]\n\\[y = \\frac{2x}{x - 6}\\]\n\nWe want to minimize the expression $x + y$. Substitute $y = \\frac{2x}{x - 6}$ ... |
31 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse $C\_1$: $\frac{y^{2}}{16} + \frac{x^{2}}{4} = 1$, and another ellipse $C\_2$ with $C\_1$'s minor axis as its major axis, and having the same eccentricity as $C\_1$.
(I) Find the equatio... | [
"### Part (I)\nThe given ellipse $C_1$ is $\\frac{y^2}{16} + \\frac{x^2}{4} = 1$. For this ellipse, the semi-major axis $a = 4$ and the semi-minor axis $b = 2$. The eccentricity $e$ is given by:\n\\[ e = \\sqrt{1 - \\frac{b^2}{a^2}} = \\sqrt{1 - \\frac{4}{16}} = \\sqrt{\\frac{3}{4}} = \\frac{\\sqrt{3}}{2} \\]\nNow,... |
40 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Fully factor the following expression: $3x^2 + 12x + 12$ Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The code is not necessary for this problem, so let's proceed with reasoning:\n1. Identify common factors and factorize the expression step by step.\n2. Ensure the expression is fully factored.\n\nFirst, identify the common factor for the coefficients and the variable terms. We notice that the expression can be div... |
42 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Real numbers $x$ and $y$ have an arithmetic mean of 20 and a geometric mean of $\sqrt{125}$. Find $x^2+y^2$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>ass... | [
"Real numbers $x$ and $y$ have an arithmetic mean of 20 and a geometric mean of $\\sqrt{125}$. Find $x^2+y^2$. Let's reason step by step, and put the final answer within \\boxed{}.",
"Real numbers $x$ and $y$ have an arithmetic mean of 20 and a geometric mean of $\\sqrt{125}$. Find $x^2+y^2$. Let's reason step by... |
61 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A five-digit natural number \\( \overline{a_1 a_2 a_3 a_4 a_5} \\), where \\(a_i \in \{0,1,2,3,4,5\}\\) for \\(i=1,2,3,4,5\\), is called a "concave number" if and only if \\(a_1 > a_2 > a_3\\) and \\(a_3... | [
"A five-digit natural number \\\\( \\overline{a_1 a_2 a_3 a_4 a_5} \\\\), where \\\\(a_i \\in \\{0,1,2,3,4,5\\}\\\\) for \\\\(i=1,2,3,4,5\\\\), is called a \"concave number\" if and only if \\\\(a_1 > a_2 > a_3\\\\) and \\\\(a_3 < a_4 < a_5\\\\) (e.g., 32014, 53134). Find the number of such concave five-digit natur... |
10 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A two-digit number, when added to the number formed by reversing its digits, gives the square of a natural number. Find all such two-digit numbers. Let's think step by step and output the final answer wi... | [
"Let's denote the two-digit number as \\(10a + b\\), where \\(a\\) and \\(b\\) are digits, and \\(a\\) ranges from 1 to 9 (since it's a two-digit number) and \\(b\\) ranges from 0 to 9. The number formed by reversing its digits is \\(10b + a\\). Adding these two numbers gives:\n\\[\n(10a + b) + (10b + a) = 11a + 11... |
41 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the maximum and minimum values of the exponential function $f(x) = a^x$ on the interval $[1, 2]$ differ by $\frac{a}{2}$, then find the value of $a$. Let's think step by step and output the final answ... | [
"Let's reason step by step and put the final answer within \\boxed{}.",
"Given that the exponential function $f(x) = a^x$ has a maximum and minimum values on the interval $[1, 2]$ that differ by $\\frac{a}{2}$, we need to find the value of $a$. Since $f(x) = a^x$ is a monotonic function, the maximum and minimum v... |
53 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that a school has 120 students and 4 teachers, and the class sizes are 60, 30, 20, 10, calculate t - s where t is the average number of students in a class when a teacher is randomly selected, and ... | [
"To solve this problem, we need to calculate two different averages:\n\n1. The average number of students in a class when a teacher is randomly selected (t).\n2. The average number of students in the class of a randomly selected student (s).\n\n### Step-by-step Calculation:\n\n1. **Calculate t:**\n - When a teach... |
26 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine how many $5$-digit palindromes exist. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To determine how many 5-digit palindromes exist, we need to understand the structure of a 5-digit palindrome. A 5-digit palindrome is a number that reads the same forward and backward. It can be represented in the form $\\overline{abcba}$, where $a$, $b$, and $c$ are digits, and $a \\neq 0$ because it is a 5-digit... |
2 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $\sqrt{\frac{1}{25} + \frac{1}{36} - \frac{1}{144}}$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To calculate the expression \\(\\sqrt{\\frac{1}{25} + \\frac{1}{36} - \\frac{1}{144}}\\), we need to first find the common denominator for the fractions inside the square root. The common denominator for 25, 36, and 144 is 3600. We can rewrite the expression as follows:\n\n\\[\n\\sqrt{\\frac{1}{25} + \\frac{1}{36}... |
36 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Definition: The length of an interval $[c,d] (c < d)$ is $d-c$. Given that the domain of the function $y=|\log_{2}x|$ is $[a,b]$, and the range is $[0,2]$, the difference between the maximum and minimum ... | [
"To find the difference between the maximum and minimum length of the interval $[a, b]$ for the domain of the function $y = |\\log_2 x|$ given that the range is $[0, 2]$, we need to determine the possible values of $a$ and $b$.\n\n1. **Understanding the Function:**\n The function $y = |\\log_2 x|$ has the followi... |
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