idx int64 0 10k | prompt stringlengths 227 4.55k | answers listlengths 8 8 | gt dict |
|---|---|---|---|
2,582 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $-\frac{\pi}{2} < \alpha < 0$ and $\sin{\alpha} + \cos{\alpha} = \frac{1}{5}$, find the value of $\frac{1}{\cos^2{\alpha} - \sin^2{\alpha}}$. Let's think step by step and output the final answ... | [
"To find the value of $\\frac{1}{\\cos^2{\\alpha} - \\sin^2{\\alpha}}$ given that $\\sin \\alpha + \\cos \\alpha = \\frac{1}{5}$ and $-\\frac{\\pi}{2} < \\alpha < 0$, we can follow these steps:\n\n1. Square the given equation to find $\\sin \\alpha \\cos \\alpha$.\n2. Use the identity $\\cos^2 \\alpha - \\sin^2 \\a... | {
"ground_truth": "\\frac{25}{7}",
"style": "rule"
} |
2,592 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Anna and Berta are playing a game where they take turns removing marbles from the table. Anna makes the first move. If at the beginning of a turn there are \( n \geq 1 \) marbles on the table, then the p... | [
"To determine the smallest number \\( N \\geq 100000 \\) such that Berta can force a win, we need to analyze the game strategy.\n\n1. **Game Analysis**:\n - If there is only 1 marble, the player whose turn it is takes it and wins.\n - If there are 2 marbles, the player whose turn it is takes 1 marble, leaving... | {
"ground_truth": "131070",
"style": "rule"
} |
908 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the complex number $z=(1+i)(2a-i)$ ($i$ is the imaginary unit) corresponds to a point on the imaginary axis in the complex plane, express $z$ in terms of $a$. Let's think step by step and output th... | [
"To determine the value of $z$ in terms of $a$, we need to multiply the complex numbers $(1+i)$ and $(2a-i)$ and then simplify the result. We also need to ensure that the resulting complex number lies on the imaginary axis, meaning its real part must be zero.\n\n1. Let's start by multiplying the two complex numbers... | {
"ground_truth": "-2i",
"style": "rule"
} |
8,006 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that Amy bought and then sold \(15n\) avocados for some positive integer \(n\), she made a profit of \$100. She paid \$2 for every 3 avocados, and she sold every 5 avocados for \$4. Calculate the v... | [
"Let's break down the problem into smaller parts and solve it step by step.\n\n1. **Calculate the cost of buying the avocados**:\n - Amy bought \\(15n\\) avocados. Since she paid \\$2 for every 3 avocados, the cost per avocado is \\(\\frac{2}{3}\\) dollars.\n - Therefore, the total cost of buying \\(15n\\) avoc... | {
"ground_truth": "50",
"style": "rule"
} |
50 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a set of data $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, $x_{5}$ with a variance of $2$, what is the standard deviation of the data $2x_{1}$, $2x_{2}$, $2x_{3}$, $2x_{4}$, $2x_{5}$? Let's think step by st... | [
"The standard deviation of a data set measures the amount of variation or dispersion of a set of values. The variance is the square of the standard deviation. If the variance of the original data set $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ is 2, then the standard deviation of the original data set is $\\sqrt{2}$. \n\nW... | {
"ground_truth": "2 \\sqrt{2}",
"style": "rule"
} |
8,510 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate:<br/>
(1) $x(x+6)=8(x+3)$;<br/>
(2) $\frac{3{a}^{2}-3{b}^{2}}{{a}^{2}b+a{b}^{2}}\div \left(1-\frac{{a}^{2}+{b}^{2}}{2ab}\right)$. Let's think step by step and output the final answer within \bo... | [
"### Part 1: Solve the equation \\(x(x + 6) = 8(x + 3)\\)\nFirst, let's expand and simplify the equation:\n\\[x(x + 6) = 8(x + 3)\\]\n\\[x^2 + 6x = 8x + 24\\]\nRearrange the equation to set it to zero:\n\\[x^2 + 6x - 8x - 24 = 0\\]\n\\[x^2 - 2x - 24 = 0\\]\nNow, we can solve this quadratic equation by factoring:\n\... | {
"ground_truth": "-\\frac{6}{a-b}",
"style": "rule"
} |
4,423 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the solution set of the inequality $kx^2+2kx+2<0$ is empty, determine the range of the real number $k$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assis... | [
"To determine the range of the real number \\(k\\) for which the inequality \\(kx^2 + 2kx + 2 < 0\\) has an empty solution set, we need to analyze the quadratic expression. The inequality \\(kx^2 + 2kx + 2 < 0\\) will have an empty solution set if the quadratic equation \\(kx^2 + 2kx + 2 = 0\\) either has no real r... | {
"ground_truth": "[0,2]",
"style": "rule"
} |
7,176 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the sequence $\{a_n\}$, $a_1=2$, $a_{n+1}=a_n+c\cdot2^n$ (where $c$ is a constant, $n=1,2,3,\ldots$), and $a_1, a_2, a_3$ form a geometric sequence with a common ratio not equal to 1.
(Ⅰ) Find the v... | [
"### Part (Ⅰ): Find the value of $c$\n\nGiven the sequence $\\{a_n\\}$ defined by $a_1 = 2$ and $a_{n+1} = a_n + c \\cdot 2^n$, we need to find the value of $c$ such that $a_1, a_2, a_3$ form a geometric sequence.\n\nFirst, let's find the expressions for $a_2$ and $a_3$:\n\\[\na_2 = a_1 + c \\cdot 2^1 = 2 + 2c\n\\]... | {
"ground_truth": "a_n=2^n",
"style": "rule"
} |
3,594 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Evaluate the integral $$∫_{ -1 }^{ 8 } 3x \, dx$$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Evaluate the integral $$∫_{ -1 }^{ 8 } 3x \\, dx$$. Let's think step by step and output the final answer within \\boxed{} and \\boxed{} IGHL\nLet's break down the problem step-by-step and evaluate the integral using Python and SymPy.\n\n1. **Understand the Integral:**\n The integral we need to evaluate is \\( \... | {
"ground_truth": "\\frac {45}{4}",
"style": "rule"
} |
5,173 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circle $C_1$ has its center $O$ on a line $\ell$. Two points $P$ and $Q$ on $\ell$ are tangents to $C_1$, with $O$ between $P$ and $Q$. The distances are $OP = 9$, and $OQ = 15$. If another point $R$ e... | [
"A circle $C_1$ has its center $O$ on a line $\\ell$. Two points $P$ and $Q$ on $\\ell$ are tangents to $C_1$, with $O$ between $P$ and $Q$. The distances are $OP = 9$, and $OQ = 15$. If another point $R$ exists on $C_1$ such that $\\angle PRQ = 90^\\circ$, what is the radius of $C_1$?\nA) 10\nB) 12\nC) $\\sqrt{153... | {
"ground_truth": "\\sqrt{153}",
"style": "rule"
} |
6,650 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = \log_{3}x + x - 5$, its root $x_{0}$ is in the interval $[a, b]$, and $b - a = 1$, where $a, b \in \mathbb{N}^{*}$. Find the value of $a + b$. Let's think step by step and outp... | [
"To find the interval $[a, b]$ where the function $f(x) = \\log_{3}x + x - 5$ has a root and $b - a = 1$, we need to evaluate the function at integer points until we find a sign change, indicating that a root is between those points.\n\n1. Evaluate the function at some integer points to find the interval where the ... | {
"ground_truth": "7",
"style": "rule"
} |
3,485 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Salma tosses five coins. What is the probability that she will get at least two heads? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Salma tosses five coins. What is the probability that she will get at least two heads? Let's think step by step and output the final answer within \\boxed{}.",
"To solve the problem of finding the probability that Salma will get at least two heads when tossing five coins, we can follow these steps:\n\n1. **Deter... | {
"ground_truth": "\\frac{13}{16}",
"style": "rule"
} |
3,152 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To make the fraction $\frac{1}{{x-4}}$ meaningful, determine the range of values of $x$ that satisfy the condition for the denominator. Let's think step by step and output the final answer within \boxed{... | [
"For the fraction $\\frac{1}{x-4}$ to be meaningful, the denominator must be non-zero. Therefore, we need to find the values of $x$ such that $x - 4 \\neq 0$. This means $x \\neq 4$. So, the range of values for $x$ is all real numbers except 4. We can express this as the set of all real numbers excluding 4, which w... | {
"ground_truth": "x \\neq 4",
"style": "rule"
} |
7,492 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the inequality
$$
\sqrt{10x-21} - \sqrt{5x^2 - 21x + 21} \geqslant 5x^2 - 31x + 42
$$
In the answer, provide the sum of all integer values of \( x \) that satisfy the inequality. Let's think step... | [
"To solve the inequality \\(\\sqrt{10x-21} - \\sqrt{5x^2 - 21x + 21} \\geqslant 5x^2 - 31x + 42\\), we need to determine the values of \\(x\\) that satisfy this condition. This involves several steps, including determining the domain of the inequality, and checking the inequality for values within that domain.\n\n1... | {
"ground_truth": "7",
"style": "rule"
} |
484 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Why can't general geometric statements be justified using instrumental measurements? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To address the question of whether general geometric statements can be justified using instrumental measurements, let's break down the problem and explore it from a logical and mathematical perspective.\n\n1. **Understanding Geometric Statements:**\n Geometric statements are based on axioms and theorems derived ... | {
"ground_truth": "\\text{229. Instrumental measurements can only be approximately performed and only for a few, but not for all possible figures of the given type.}",
"style": "rule"
} |
6,159 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two ants crawled along their own closed routes on a $7 \times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What... | [
"To solve this problem, let's first understand the constraints and the objective. We have a $7 \\times 7$ board, and each ant needs to visit each of the 64 vertices exactly once and return to the starting point, which means each ant will crawl along 64 sides.\n\nThe objective is to find the minimum number of sides ... | {
"ground_truth": "16",
"style": "rule"
} |
5,457 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If receiving $100$ yuan is denoted as $+100$ yuan, what value represents $-80$ yuan? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"When we denote receiving $100$ yuan as $+100$ yuan, it means we are using a positive sign to represent a gain or a positive quantity of money. So, $-80$ yuan would represent a loss or a negative quantity of money. Specifically, $-80$ yuan represents spending or losing $80$ yuan. Therefore, the value that represent... | {
"ground_truth": "\\text{A: spending }80\\text{ yuan}",
"style": "rule"
} |
4,577 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $\sin x = 4 \cos x$, what is $\sin x \cos x$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To solve for $\\sin x \\cos x$ given that $\\sin x = 4 \\cos x$, we can follow these steps:\n\n1. Use the given equation $\\sin x = 4 \\cos x$ to express $\\sin x$ in terms of $\\cos x$.\n2. Use the Pythagorean identity $\\sin^2 x + \\cos^2 x = 1$ to find the value of $\\cos x$.\n3. Substitute the value of $\\cos ... | {
"ground_truth": "\\frac{4}{17}",
"style": "rule"
} |
23 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One day, Xiaoming was chatting with his older brother. Xiaoming said to his brother: "I am particularly looking forward to the year 2013, because 2, 0, 1, and 3 are four different digits, and this is the... | [
"Let's solve this step-by-step using Python code. We need to find the next 'lucky year' after 2013 and then backtrack to find the birth year of Xiaoming's elder brother. A 'lucky year' is defined as a year where the digits are all different. Let's start by identifying the next 'lucky year' after 2013.\n\nFirst, we ... | {
"ground_truth": "1987",
"style": "rule"
} |
6,874 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some integers that are not palindromes, like 196, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For integers between 100 and 1000,... | [
"To solve this problem, we need to follow these steps:\n\n1. Identify all integers between 100 and 1000 that are not palindromes.\n2. For each non-palindrome, apply the reverse-and-add process for up to three steps.\n3. Check if the number becomes a palindrome after exactly three steps.\n4. Sum all such numbers tha... | {
"ground_truth": "13590",
"style": "rule"
} |
3,911 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a quadratic function $y=ax^{2}-2ax+3$ (where $x$ is the independent variable), the corresponding function values of $y$ are all positive when $0 \lt x \lt 3$. Determine the range of values for $a$.... | [
"To determine the range of values for \\(a\\) such that the quadratic function \\(y = ax^2 - 2ax + 3\\) is positive for \\(0 < x < 3\\), we can follow these steps:\n\n1. **Find the vertex of the quadratic function**: The quadratic function \\(y = ax^2 - 2ax + 3\\) can be written in the form \\(y = a(x^2 - 2x) + 3\\... | {
"ground_truth": "-1\\leqslant a < 0 \\text{ or } 0 < a < 3",
"style": "rule"
} |
5,865 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A point moving in the positive direction of the $OX$ axis has its horizontal coordinate given by $x(t) = 5(t + 1)^2 + \frac{a}{(t + 1)^5}$, where $a$ is a positive constant. Find the minimum value of $a$... | [
"To find the minimum value of \\(a\\) such that \\(x(t) \\geqslant 24\\) for all \\(t \\geqslant 0\\), we need to analyze the function \\(x(t) = 5(t + 1)^2 + \\frac{a}{(t + 1)^5}\\) and find the conditions under which it is always greater than or equal to 24.\n\n1. First, we need to find the critical points of the ... | {
"ground_truth": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^{7}}",
"style": "rule"
} |
2,816 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the digits 1, 2, 3, 4, 5, 6, how many even three-digit numbers less than 700 can be formed if each digit can be used more than once? Let's think step by step and output the final answer within \box... | [
"To solve this problem, we need to determine how many even three-digit numbers less than 700 can be formed using the digits 1, 2, 3, 4, 5, and 6, with repetition allowed.\n\nHere's the step-by-step reasoning:\n\n1. **Determine the range of the first digit**: The first digit of the three-digit number must be less th... | {
"ground_truth": "108",
"style": "rule"
} |
1,528 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $O$ is the origin, the focus of the parabola $C:y^{2}=2px\left(p \gt 0\right)$ is $F$, $P$ is a point on $C$, $PF$ is perpendicular to the $x$-axis, $Q$ is a point on the $x$-axis, and $PQ\bot... | [
"Given the parabola \\(C: y^2 = 2px\\) with focus \\(F\\left(\\frac{p}{2}, 0\\right)\\) and directrix \\(x = -\\frac{p}{2}\\), and the condition that \\(PF\\) is perpendicular to the \\(x\\)-axis, we know that point \\(P\\) has coordinates \\(\\left(\\frac{p}{2}, y_0\\right)\\) where \\(y_0^2 = 2p \\cdot \\frac{p}{... | {
"ground_truth": "x = -\\frac{3}{2}",
"style": "rule"
} |
68 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the hyperbola $C_{1}$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \ (a > 0, b > 0)$ passes through the focus of the parabola $C_{2}$: $y^{2} = 2px \ (p > 0)$, and the asymptotes of the hyperb... | [
"1. **Understanding the Problem:**\n - The hyperbola \\(C_1: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\\) passes through the focus of the parabola \\(C_2: y^2 = 2px\\).\n - The asymptotes of the hyperbola form an equilateral triangle with the directrix of the parabola.\n\n2. **Focus of the Parabola \\(C_2\\):**\n... | {
"ground_truth": "\\dfrac{2\\sqrt{3}}{3}",
"style": "rule"
} |
9,983 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Esquecinaldo has a terrible memory for numbers but a great one for remembering sequences of operations. To remember his 5-digit bank code, he recalls that:
- None of the digits are zero.
- The first two... | [
"To solve this problem, we need to find a 5-digit number that satisfies all the given conditions. We'll break down the problem and use Python to iterate through possible values to find the correct code.\n\n1. The first two digits form a power of 5.\n2. The last two digits form a power of 2.\n3. The middle digit is ... | {
"ground_truth": "25916",
"style": "rule"
} |
7,026 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $\log_{10}{m}= b-\log_{10}{n}$, calculate the value of $m$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the value of \\( m \\) given the equation \\(\\log_{10}{m} = b - \\log_{10}{n}\\), we can follow these steps:\n\n1. **Combine the logarithms on the right-hand side**: Use the property of logarithms that \\(\\log_{10}{a} + \\log_{10}{b} = \\log_{10}{(ab)}\\).\n2. **Exponentiate both sides** to remove the lo... | {
"ground_truth": "\\frac{10^b}{n}",
"style": "rule"
} |
204 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the complex number $(1-i)(a-i)$ corresponds to a point in the second quadrant on the complex plane, determine the range of the real number $a$. Let's think step by step and output the final answer wit... | [
"To determine the range of the real number \\(a\\) such that the complex number \\((1-i)(a-i)\\) corresponds to a point in the second quadrant on the complex plane, we need to follow these steps:\n\n1. Compute the product \\((1-i)(a-i)\\).\n2. Determine the real and imaginary parts of the resulting complex number.\... | {
"ground_truth": "(-\\infty,-1)",
"style": "rule"
} |
9,246 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Line \( l \) passes through point \( A(2,0) \) and intersects the parabolic curve \( y = x^{2} + 2 \) at points \( B \) and \( C \). Points \( B_{1} \) and \( C_{1} \) are the feet of the perpendiculars ... | [
"To solve this problem, we need to follow these steps:\n\n1. Find the equation of the line \\( l \\) passing through the point \\( A(2,0) \\) and intersecting the parabola \\( y = x^2 + 2 \\) at points \\( B \\) and \\( C \\).\n2. Find the coordinates of points \\( B \\) and \\( C \\).\n3. Determine the coordinates... | {
"ground_truth": "12x - 3y - 4 = 0",
"style": "rule"
} |
9,743 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $xy = c^2$ and $\frac{1}{x^2} + \frac{1}{y^2} = bc$, find $(x - y)^2$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Given the equations:\n\\[ xy = c^2 \\]\n\\[ \\frac{1}{x^2} + \\frac{1}{y^2} = bc \\]\n\nWe need to find the value of \\((x - y)^2\\).\n\nFirst, let's rewrite the second equation in a more convenient form:\n\\[ \\frac{1}{x^2} + \\frac{1}{y^2} = \\frac{x^2 + y^2}{x^2 y^2} = bc. \\]\nSince \\(xy = c^2\\), it follows ... | {
"ground_truth": "bc^4 - 2c^2",
"style": "rule"
} |
7,123 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of $\frac{15!}{6!9!}$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Great! Let's solve the problem step by step. The expression we need to evaluate is $\\frac{15!}{6!9!}$. This is a straightforward calculation that can be simplified as follows:\n\n\\[\n\\frac{15!}{6!9!} = \\frac{15 \\times 14 \\times 13 \\times 12 \\times 11 \\times 10 \\times 9!}{6! \\times 9!} = \\frac{15 \\time... | {
"ground_truth": "770",
"style": "rule"
} |
185 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The parabola $y^2 = 12x$ and the circle $x^2 + y^2 - 4x - 6y = 0$ intersect at two points $C$ and $D$. Find the distance $CD$. Let's think step by step and output the final answer within \boxed{}<|im_end... | [
"The parabola $y^2 = 12x$ and the circle $x^2 + y^2 - 4x - 6y = 0$ intersect at two points $C$ and $D$. Find the distance $CD$. Let's think step by step and output the final answer within \\boxed{}囷",
"To find the distance between the points of intersection of the parabola \\(y^2 = 12x\\) and the circle \\(x^2 + ... | {
"ground_truth": "CD = 6\\sqrt{3}",
"style": "rule"
} |
7,221 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is $(-1)^1+(-1)^2+\cdots+(-1)^{2021}$ ? 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 evaluate the sum of the series $(-1)^1 + (-1)^2 + \\cdots + (-1)^{2021}$. Let's break it down:\n\n1. **Identify the Pattern:**\n - For odd $n$, $(-1)^n = -1$.\n - For even $n$, $(-1)^n = 1$.\n \n2. **Count the Number of Terms:**\n - The series starts at $n=1$ and ends at $n... | {
"ground_truth": "-1",
"style": "rule"
} |
4,694 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\{a\_n\}$ be an arithmetic sequence with a non-zero common difference. The sum of its first $n$ terms is denoted by $S\_n$. Given that $S\_9=90$, and $a\_1$, $a\_2$, $a\_4$ form a geometric sequence... | [
"Let's solve the problem step-by-step.\n\n### Step 1: Find the general term formula for the sequence $\\{a_n\\}$\n\nThe general term of an arithmetic sequence is given by:\n\\[a_n = a_1 + (n-1)d\\]\nwhere $a_1$ is the first term and $d$ is the common difference.\n\nThe sum of the first $n$ terms of an arithmetic se... | {
"ground_truth": "T\\_n=\\frac{n}{4n+4}",
"style": "rule"
} |
3,533 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A right circular cone is inscribed in a right rectangular prism. The base of the prism measures $3a$ by $2a$ and the height of the prism (and cone) is $h$. What is the ratio of the volume of the cone to ... | [
"The volume of the cone is given by the formula $\\frac{1}{3}\\pi r^2 h$, where $r$ is the radius of the base of the cone and $h$ is its height. The volume of the right rectangular prism is given by the formula $lwh$, where $l, w,$ and $h$ are the length, width, and height of the prism, respectively.\n\nSince the c... | {
"ground_truth": "\\frac{\\pi}{18}",
"style": "rule"
} |
4,112 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that a coat costs $100 and a pair of trousers costs $50, calculate the total amount of money saved when the coat is bought at a 30% discount and the trousers at a 20% discount, and express this sav... | [
"To calculate the total amount of money saved when the coat is bought at a 30% discount and the trousers at a 20% discount, and to express this saving as a percentage of the total original prices, we can follow these steps:\n\n1. Calculate the original total price of the coat and trousers.\n2. Calculate the discoun... | {
"ground_truth": "26.67\\%, \\ 40",
"style": "rule"
} |
7,970 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each integer $n\geq 5$, let $b_n$ denote the base-$n$ number $0.\overline{144}_n$. The product $b_5b_6 \dotsm b_{100}$ can be expressed as $\frac{p}{q!}$, where p and q are positive integers and q is... | [
"To solve this problem, we need to understand the repeating decimal $0.\\overline{144}_n$ in base $n$. This repeating decimal represents the fraction $\\frac{144_n}{n^3-1}$, where $144_n$ is the number $1n^2 + 4n + 4$ in base $n$.\n\nSo, the repeating decimal $0.\\overline{144}_n$ can be expressed as:\n\\[\nb_n = \... | {
"ground_truth": "1943994",
"style": "rule"
} |
7,581 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the focus of the parabola $y = -\frac{1}{8} x^2$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The standard form of a parabola that opens up or down is $y = \\frac{1}{4p} x^2$, where $p$ is the distance from the vertex to the focus (and also from the vertex to the directrix).\n\nGiven the equation $y = -\\frac{1}{8} x^2$, we can compare it with the standard form and find the value of $p$. The standard form ... | {
"ground_truth": "(0,2)",
"style": "rule"
} |
3,082 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The track and field events of the Hangzhou Asian Games concluded on October 5th. In the final two events, the men's and women's marathon races, Chinese athlete He Jie won the men's group championship wit... | [
"### Part 1: Writing down the function \\(Q(t)\\)\n\nThe athlete runs for a total of 4 hours. Let's break down the function \\(Q(t)\\) into two parts: the first hour when the athlete is in the stable stage and the remaining time when the athlete is in the fatigue stage.\n\n#### Stable Stage (0 \\leq t \\leq 1)\nThe... | {
"ground_truth": "t=2h",
"style": "rule"
} |
4,130 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jamie has a jar of coins containing the same number of nickels, dimes, and quarters. The total value of the coins in the jar is $\$$14.40. How many dimes does Jamie have? Let's think step by step and out... | [
"Jamie has a jar of coins containing the same number of nickels, dimes, and quarters. The total value of the coins in the jar is \\(\\$$14.40. Let's denote the number of nickels, dimes, and quarters by \\( n \\). Each nickel is worth \\$0.05, each dime is worth \\$0.10, and each quarter is worth \\$0.25. Therefore,... | {
"ground_truth": "36",
"style": "rule"
} |
4,244 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that (-1), a_{1}, a_{2}, (-9) form an arithmetic sequence, and (-9), b_{1}, b_{2}, b_{3}, (-1) form a geometric sequence, find the value of b_{2}(a_{2}-a_{1}). Let's think step by step and output t... | [
"To solve this problem, we need to determine the values of \\(a_1\\), \\(a_2\\), \\(b_1\\), \\(b_2\\), and \\(b_3\\). Let's break it down into manageable steps.\n\n1. **Arithmetic Sequence:**\n The terms of the arithmetic sequence are \\(-1\\), \\(a_1\\), \\(a_2\\), \\(-9\\). Let the common difference be \\(d\\).... | {
"ground_truth": "8",
"style": "rule"
} |
416 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC$, where $AB = 13$, $BC = 12$, and $CA = 5$. A circle $P$ with the smallest radius passes through $C$ and is tangent to $CA$ at its midpoint. Let points $Q$ and $R$ represent the intersec... | [
"To solve this problem, we need to find the length of the segment \\(QR\\) where the circle \\(P\\) intersects sides \\(AB\\) and \\(BC\\) of the triangle \\(ABC\\). Here are the steps to solve it:\n\n1. **Identify the coordinates of the points:**\n - Place point \\(C\\) at the origin \\((0,0)\\).\n - Point \\(... | {
"ground_truth": "5",
"style": "rule"
} |
5,255 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Charles has two six-sided dice. One of the dice is fair, and the other die is biased such that it shows a six with a probability of $\frac{3}{4}$, and each of the other five sides has a probability of $\... | [
"To solve this problem, we can use Bayes' theorem and the properties of conditional probabilities. Here's the step-by-step reasoning:\n\n1. **Define the events:**\n - Let $F$ be the event that Charles chose the fair die.\n - Let $B$ be the event that Charles chose the biased die.\n - Let $S_1, S_2, S_3, S_4$ ... | {
"ground_truth": "\\frac{774}{1292}",
"style": "rule"
} |
2,624 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the similarity ratio of two similar polygons is $1:5$, calculate their area ratio. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The area ratio of two similar polygons is the square of the similarity ratio. So if the similarity ratio is $1:5$, the area ratio is $1^2:5^2 = 1:25$.\n\nTo express this in a clear format, we can use Python to verify the calculation.\n```python\n# Define the similarity ratio\nsimilarity_ratio = 1 / 5\n\n# Calculat... | {
"ground_truth": "1:25",
"style": "rule"
} |
2,812 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The value of $\frac{2\sin80°-\sin20°}{\cos20°}$ is ____. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"We need to evaluate the expression $\\frac{2\\sin80^\\circ - \\sin20^\\circ}{\\cos20^\\circ}$. We can use trigonometric identities and properties to simplify this expression.\n\nFirst, let's use the identity $\\sin80^\\circ = \\cos10^\\circ$. So we can rewrite the expression as:\n\\[\n\\frac{2\\cos10^\\circ - \\si... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} |
9,109 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $AB = 15, BC = 13$, and $CA = 14$. Point $N$ is the midpoint of side $AC$, and $K$ is the foot of the altitude from $B$ to $AC$. Determine the length of $KN$.
A) 7
B) 8
C) 4.5
D) 5
E)... | [
"In $\\triangle ABC$, $AB = 15, BC = 13$, and $CA = 14$. Point $N$ is the midpoint of side $AC$, and $K$ is the foot of the altitude from $B$ to $AC$. Determine the length of $KN$.\nA) 7\nB) 8\nC) 4.5\nD) 5\nE) 5.5\nThe correct answer is \\boxed{7.5}",
"In the quadrilateral $ABCD$, $AB = 14, BC = 13$, and $AC = 1... | {
"ground_truth": "D) 5",
"style": "rule"
} |
6,984 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is half of the absolute value of the difference of the squares of 21 and 19? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"We'll calculate\n\\[\n\\frac{1}{2} \\times \\left| 21^2 - 19^2 \\right|\n\\]\n\n1. Compute \\(21^2\\).\n2. Compute \\(19^2\\).\n3. Compute the difference \\(21^2 - 19^2\\).\n4. Compute the absolute value of the difference.\n5. Compute half of the absolute value.\n\nLet's execute these steps in Python to ensure acc... | {
"ground_truth": "40",
"style": "rule"
} |
4,844 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(A\) and \(B\) be independent events. In terms of \(\mathrm{P}(A)\) and \(\mathrm{P}(B)\), express the probabilities of the events consisting of exactly \(k\), at least \(k\), and at most \(k\) even... | [
"Let's solve the problem step-by-step. We have two independent events \\(A\\) and \\(B\\). We need to find the probabilities of the following events:\n\n1. Exactly \\(k\\) events occurring from \\(A\\) and \\(B\\) for \\(k=0,1,2\\).\n2. At least \\(k\\) events occurring from \\(A\\) and \\(B\\) for \\(k=0,1,2\\).\n... | {
"ground_truth": "\\begin{aligned}\n& \\mathrm{P}(\\xi \\leqslant 2) = 1, \\\\\n& \\mathrm{P}(\\xi = 0) = 1 - a - b + ab, \\\\\n& \\mathrm{P}(\\xi = 1) = a + b - 2ab, \\\\\n& \\mathrm{P}(\\xi = 2) = ab, \\\\\n& \\mathrm{P}(\\xi \\geqslant 1) = a + b - ab, \\\\\n& \\mathrm{P}(\\xi \\leqslant 1) = 1 - ab.\n\\end{align... |
7,320 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $f(x) = m \cdot 2^x + x^2 + nx$, if $\{x|f(x)=0\} = \{x|f(f(x))=0\} \neq \emptyset$, then the range of values for $m+n$ is \_\_\_\_\_\_. Let's think step by step and output the final answer within ... | [
"To solve the problem, we need to understand the condition given in the problem statement: $\\{x|f(x)=0\\} = \\{x|f(f(x))=0\\} \\neq \\emptyset$. This implies that if $x_0$ is a root of $f(x)$, then $f(x_0) = 0$, and hence $f(f(x_0)) = f(0) = 0$. This means that the constant term of $f(x)$ must be zero. Therefore, ... | {
"ground_truth": "[0, 4)",
"style": "rule"
} |
7,069 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The price of the commodity at the end of May was the same as it had been at the beginning of January, with the price undergoing a series of percentage changes: a 30% increase, a 15% decrease, a 10% incre... | [
"Let's break down the problem step by step and use Python to ensure accuracy.\n\n1. Let the initial price of the commodity be \\( P \\).\n2. After a 30% increase, the new price is \\( P \\times (1 + 0.30) = 1.30P \\).\n3. After a 15% decrease, the new price is \\( 1.30P \\times (1 - 0.15) = 1.30P \\times 0.85 = 1.1... | {
"ground_truth": "31\\%",
"style": "rule"
} |
3,981 | <|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 circle that passes through points $M(5,2)$ and $N(3,2)$ and has its center on the line $y=2x-3$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|... | [
"The equation of a circle with center $(h,k)$ and radius $r$ is given by:\n\\[\n(x-h)^2 + (y-k)^2 = r^2\n\\]\n\nWe know the circle passes through points $M(5,2)$ and $N(3,2)$. So, substituting these points into the circle's equation gives us:\n\\[\n(5-h)^2 + (2-k)^2 = r^2 \\quad \\text{(1)}\n\\]\n\\[\n(3-h)^2 + (2-... | {
"ground_truth": "{\\left(x-4\\right)}^{2}+{\\left(y-5\\right)}^{2}=10",
"style": "rule"
} |
2,197 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given vectors $\overrightarrow{a} = (2, 3)$ and $\overrightarrow{b} = (1, m)$, and knowing that $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of the real number $m$. Let's... | [
"Given that the vectors $\\overrightarrow{a} = (2, 3)$ and $\\overrightarrow{b} = (1, m)$ are perpendicular, their dot product must be zero. The dot product of two vectors $\\overrightarrow{a} = (a_1, a_2)$ and $\\overrightarrow{b} = (b_1, b_2)$ is given by:\n\n\\[\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = ... | {
"ground_truth": "-\\frac{2}{3}",
"style": "rule"
} |
1,740 | <|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 directrix of the parabola \( y = -4x^2 + 4 \). Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the equation of the directrix of the given parabola \\( y = -4x^2 + 4 \\), we need to follow these steps:\n\n1. **Convert the equation to the standard form of a parabola:**\n The standard form of a parabola that opens up or down is \\( (x - h)^2 = 4p(y - k) \\), where \\((h, k)\\) is the vertex of the pa... | {
"ground_truth": "y = \\frac{65}{16}",
"style": "rule"
} |
2,377 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Extend line $PD$ to intersect line $BC$ at point $F$. Construct lines $DG$ and $PT$ parallel to $AQ$.
Introduce the following notations: $AP = x$, $PB = \lambda x$, $BQ = y$, $QC = \mu y$, $PE = u$, $E... | [
"```python\n# Given the ratio PE:ED = 3:20, we can directly box the final answer based on the reasoning provided.\n\n# The ratio PE:ED is given as 3:20\nratio_PE_to_ED = 3 / 20\n\n# The final answer is the ratio PE:ED\nboxed_answer = ratio_PE_to_ED\nboxed_answer = boxed_answer * 23 # This is just for verification ... | {
"ground_truth": "P E : E D = 3 : 20",
"style": "rule"
} |
4,290 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine how many positive integers $n$ satisfy $300 < n^2 < 1200$. 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 determine the number of positive integers \\( n \\) such that \\( 300 < n^2 < 1200 \\). This means we need to find the integers \\( n \\) that satisfy \\( \\sqrt{300} < n < \\sqrt{1200} \\).\n\nFirst, let's calculate the approximate values of \\( \\sqrt{300} \\) and \\( \\sqrt{1200... | {
"ground_truth": "17",
"style": "rule"
} |
4,610 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose \( a \) and \( b \) are the roots of \( x^{2}+x \sin \alpha+1=0 \) while \( c \) and \( d \) are the roots of the equation \( x^{2}+x \cos \alpha-1=0 \). Find the value of \( \frac{1}{a^{2}}+\fra... | [
"To solve the problem, we need to find the value of \\(\\frac{1}{a^2} + \\frac{1}{b^2} + \\frac{1}{c^2} + \\frac{1}{d^2}\\), where \\(a\\) and \\(b\\) are the roots of the equation \\(x^2 + x \\sin \\alpha + 1 = 0\\) and \\(c\\) and \\(d\\) are the roots of the equation \\(x^2 + x \\cos \\alpha - 1 = 0\\).\n\nFirst... | {
"ground_truth": "1",
"style": "rule"
} |
388 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that Zara has 5 marbles (Aggie, Bumblebee, Steelie, Tiger, and Cat's Eye), determine the number of arrangements of these marbles in a row where the Steelie is not next to the Tiger or the Cat's Eye... | [
"To solve this problem, we need to follow these steps:\n\n1. Calculate the total number of possible arrangements of the 5 marbles.\n2. Calculate the number of arrangements where the Steelie is next to the Tiger or the Cat's Eye.\n3. Subtract the number of invalid arrangements from the total number of arrangements t... | {
"ground_truth": "48",
"style": "rule"
} |
3,352 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
One of the asymptotes of a hyperbola has equation \( y = 2x \). The foci of the hyperbola have the same \( y \)-coordinate, which is \( -3 \). Find the equation of the other asymptote of the hyperbola, g... | [
"The equation of a hyperbola in standard form is given by:\n\\[\n\\frac{(y - k)^2}{a^2} - \\frac{(x - h)^2}{b^2} = 1\n\\]\nor\n\\[\n\\frac{(x - h)^2}{b^2} - \\frac{(y - k)^2}{a^2} = 1,\n\\]\nwhere \\((h, k)\\) is the center of the hyperbola, \\(a\\) and \\(b\\) are constants, and the asymptotes are given by \\(y = ... | {
"ground_truth": "y = -2x - 6",
"style": "rule"
} |
5,547 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The product $1! \cdot 2! \cdot 3! \cdots \cdots \cdot 99! \cdot 100!$ has ___ consecutive zeros at the end. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assi... | [
"To determine the number of trailing zeros in the product \\(1! \\cdot 2! \\cdot 3! \\cdots 99! \\cdot 100!\\), we need to count the number of times 10 is a factor in this product. Since \\(10 = 2 \\times 5\\) and there are usually more factors of 2 than factors of 5, we need to count the number of times 5 is a fac... | {
"ground_truth": "1124",
"style": "rule"
} |
8,401 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assi... | [
"To solve this problem, we need to find the greatest common divisor (GCD) of the expressions \\(8b^2 + 85b + 200\\) and \\(2b + 10\\) when \\(b\\) is an even multiple of 7786. Let's denote \\(b = 7786k\\) where \\(k\\) is an even integer.\n\nFirst, let's substitute \\(b = 7786k\\) into the expressions:\n\\[8b^2 + 8... | {
"ground_truth": "10",
"style": "rule"
} |
6,922 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a line in the Cartesian plane where an increase of 4 $x$-units results in an increase of 9 $y$-units, by how many units does the $y$-value increase when the $x$-value increases by 12 units? Let's t... | [
"```python\n# Given that an increase of 4 units in x results in an increase of 9 units in y,\n# we can find the rate of change of y with respect to x as follows:\n# (increase in y) / (increase in x) = 9 / 4\n\n# We want to find the increase in y when x increases by 12 units.\n# Let's denote the increase in y as del... | {
"ground_truth": "27",
"style": "rule"
} |
7,223 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the triangle \(ABC\) with the usual notations \(\gamma = 3 \alpha\), \(a = 27\), \(c = 48\), determine the length of side \(b\). Let's think step by step and output the final answer within \boxed{}<|i... | [
"We will use the Law of Sines and the Law of Cosines to find the length of side \\(b\\).\n\nGiven:\n- \\( \\gamma = 3 \\alpha \\)\n- \\( a = 27 \\)\n- \\( c = 48 \\)\n\nFirst, we can use the Law of Sines to express the angles in terms of the sides. The Law of Sines states that:\n\\[ \\frac{a}{\\sin \\alpha} = \\fra... | {
"ground_truth": "35",
"style": "rule"
} |
9,503 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The interval where the zero point of the function $f\left(x\right)=x^{3}+2x-5$ lies. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"The function $f(x) = x^3 + 2x - 5$ is a continuous function. We need to find an interval $[a, b]$ such that $f(a)$ and $f(b)$ have opposite signs. This indicates that there is a zero point of the function in the interval $(a, b)$.\n\nLet's evaluate the function at some points to determine where the function change... | {
"ground_truth": "\\left(1,2\\right)",
"style": "rule"
} |
947 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, PQRS is a square and M is the midpoint of PS. Find the ratio of the area of triangle QMS to the area of square PQRS. Let's think step by step and output the final answer within \boxed{}<|... | [
"To solve this problem, we need to find the ratio of the area of triangle \\(QMS\\) to the area of square \\(PQRS\\).\n\n1. Let's assume the side length of the square \\(PQRS\\) is \\(a\\).\n2. The coordinates of the points can be set as follows:\n - \\(P = (0, 0)\\)\n - \\(Q = (a, 0)\\)\n - \\(R = (a, a)\\)\... | {
"ground_truth": "1:4",
"style": "rule"
} |
6,329 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Simplify the expressions, then evaluate.
(1) $-2a^2 + 3 - (3a^2 - 6a + 1) + 3$
(2) $\frac{1}{2}x - 2\left(x - \frac{1}{3}y^2\right) + \left(-\frac{3}{2}x + \frac{1}{3}y^2\right)$, where $x=-2$, $y=-3$. L... | [
"Simplify the expressions, then evaluate.\n(1) $-2a^2 + 3 - (3a^2 - 6a + 1) + 3$\n(2) $\\frac{1}{2}x - 2\\left(x - \\frac{1}{3}y^2\\right) + \\left(-\\frac{3}{2}x + \\frac{1}{3}y^2\\right)$, where $x=-2$, $y=-3$.\n\nFirst, let's simplify each expression:\n\n1. $-2a^2 + 3 - (3a^2 - 6a + 1) + 3$\n \\[\n = -2a^2 +... | {
"ground_truth": "15",
"style": "rule"
} |
3,509 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = px^8 + qx^6 - rx^4 + 5$. If $g(12) = 3$, find $g(12) + g(-12)$. 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 properties of the function $g(x)$. The function is given as $g(x) = px^8 + qx^6 - rx^4 + 5$. Notice that all the powers of $x$ in the function are even. This means that $g(x)$ is an even function, i.e., $g(x) = g(-x)$. This property simplifies our task significantly... | {
"ground_truth": "6",
"style": "rule"
} |
7,714 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ with foci $F_1$ and $F_2$, if point $P$ is a point on the ellipse such that $\Delta PF_1F_2$ is a right triangle, find the area of $\Delta PF_1F_2$.... | [
"```python\n# Given ellipse: x^2/16 + y^2/9 = 1\n# Here a^2 = 16 and b^2 = 9, so a = 4 and b = 3.\n# The distance from the center to each focus (c) is given by c = sqrt(a^2 - b^2) = sqrt(16 - 9) = sqrt(7).\n\n# For the triangle to be a right triangle, one of the following conditions must be satisfied:\n# 1. PF1 is ... | {
"ground_truth": "\\frac{9\\sqrt{7}}{4}",
"style": "rule"
} |
895 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the 55th digit after the decimal point in the decimal representation of $\frac{1}{17}$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the 55th digit after the decimal point in the decimal representation of \\(\\frac{1}{17}\\), we need to first find the decimal representation of \\(\\frac{1}{17}\\) and identify its repeating sequence. We can use Python to help us with this task.\n\nFirst, let's find the decimal representation of \\(\\frac... | {
"ground_truth": "2",
"style": "rule"
} |
2,935 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{10}$ be the 10 zeros of the polynomial $z^{10} - 2^{30}$. For each $j$, let $w_j$ be one of $z_j$, $iz_j$, or $-iz_j$. Find the maximum possible value of the real pa... | [
"### Question\nGiven the polynomial $z^{10} - 2^{30}$, find the roots and then determine the maximum possible value of the real part of the sum of the modified roots where each root $z_j$ can be transformed into $z_j$, $iz_j$, or $-iz_j$.\n\n### Analysis\n1. **Finding the roots of the polynomial**: The polynomial $... | {
"ground_truth": "8 + 8\\sqrt{5}",
"style": "rule"
} |
1,310 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all solutions to the inequality \[\sqrt[3]{x}+ \frac{1}{\sqrt[3]{x}+ 2} \le 0,\] expressing your answer in interval notation. Let's think step by step and output the final answer within \boxed{}<|im... | [
"To solve the inequality \\(\\sqrt[3]{x} + \\frac{1}{\\sqrt[3]{x} + 2} \\leq 0\\), we'll follow these steps:\n\n1. **Introduce a substitution**: Let \\(y = \\sqrt[3]{x}\\). The inequality becomes \\(y + \\frac{1}{y + 2} \\leq 0\\).\n2. **Find the critical points**: Determine where the expression \\(y + \\frac{1}{y ... | {
"ground_truth": "(-8, -1]",
"style": "rule"
} |
6,111 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \(\frac{a(b-c)}{b(c-a)}=\frac{b(c-a)}{c(b-a)}=r\) and \(r>0\), find \(r\). Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"Given that:\n\\[\n\\frac{a(b-c)}{b(c-a)} = \\frac{b(c-a)}{c(b-a)} = r\n\\]\nwe can set up the following equations:\n1. \\(\\frac{a(b-c)}{b(c-a)} = r\\)\n2. \\(\\frac{b(c-a)}{c(b-a)} = r\\)\n\nFrom equation 1, we get:\n\\[\na(b-c) = rb(c-a)\n\\]\nFrom equation 2, we get:\n\\[\nb(c-a) = rc(b-a)\n\\]\n\nLet's solve t... | {
"ground_truth": "\\frac{\\sqrt{5}-1}{2}",
"style": "rule"
} |
9,778 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A folded rectangle. Two opposite vertices of a rectangle with sides \( x \) and \( y \) are brought together. Find the length of the crease. Let's think step by step and output the final answer within \b... | [
"To find the length of the crease when two opposite vertices of a rectangle with sides \\( x \\) and \\( y \\) are brought together, we need to analyze the geometry of the problem. When the vertices are brought together, the crease will be the perpendicular bisector of the line segment connecting the two vertices. ... | {
"ground_truth": "\\frac{y \\sqrt{x^2 + y^2}}{x}",
"style": "rule"
} |
9,012 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the graph of the quadratic function $y=x^{2}-6x+c$ passes through three points $A(0$,$y_{1})$ and $B(4$,$y_{2})$, determine the relationship between $y_{1}$ and $y_{2}$. Let's think step by step and o... | [
"To determine the relationship between \\(y_1\\) and \\(y_2\\), we need to evaluate the quadratic function \\(y = x^2 - 6x + c\\) at the given points \\(A(0, y_1)\\) and \\(B(4, y_2)\\).\n\n1. For point \\(A(0, y_1)\\):\n\\[y_1 = 0^2 - 6 \\cdot 0 + c = c\\]\n\n2. For point \\(B(4, y_2)\\):\n\\[y_2 = 4^2 - 6 \\cdot ... | {
"ground_truth": "y_1 > y_2",
"style": "rule"
} |
8,000 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each letter represents a non-zero digit. What is the value of $t?$
\begin{align*}
a + b &= v \\
v + d &= t \\
t + a &= r \\
b + d + r &= 18
\end{align*} Let's think step by step and output the final ans... | [
"Let's define the variables and solve the equations step by step using Python. We know that each letter represents a non-zero digit, i.e., a digit from 1 to 9.\n\nWe'll start by defining the equations and then use a brute-force approach to find the values of the variables that satisfy all the equations.\n\nThe equa... | {
"ground_truth": "9",
"style": "rule"
} |
8,391 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a straight line passes through points A(2, -3) and B(1, 4), then the slope-intercept form of the line's equation is \_\_\_\_\_\_ . Let's think step by step and output the final answer within \boxed{}<... | [
"To find the equation of the line in slope-intercept form \\(y = mx + b\\) that passes through the points \\(A(2, -3)\\) and \\(B(1, 4)\\), we need to follow these steps:\n\n1. Calculate the slope \\(m\\) of the line.\n2. Use the point-slope form of the equation of a line to find the y-intercept \\(b\\).\n3. Conver... | {
"ground_truth": "y = -7x + 11",
"style": "rule"
} |
2,652 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the value of the product:
\[
\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\... | [
"Calculate the value of the product step by step:\n\n\\[\n\\left(1+\\frac{1}{1}\\right)\\cdot\\left(1+\\frac{1}{2}\\right)\\cdot\\left(1+\\frac{1}{3}\\right)\\cdot\\left(1+\\frac{1}{4}\\right)\\cdot\\left(1+\\frac{1}{5}\\right)\\cdot\\left(1+\\frac{1}{6}\\right)\\cdot\\left(1+\\frac{1}{7}\\right)\\cdot\\left(1+\\fr... | {
"ground_truth": "\\textbf{(B) }9",
"style": "rule"
} |
6,476 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Glenburgh parade squad consists of $20m$ members, and when they line up in rows of 28, there are 6 members left over. If $m$ is an integer and there are fewer than 1200 members, what is the maximum n... | [
"To find the maximum number of members that could be in the Glenburgh parade squad, we need to solve for the largest integer $20m$ that is less than 1200 and leaves a remainder of 6 when divided by 28.\n\nFirst, we need to find the general form of the number of members. We know that:\n\n\\[ 20m \\equiv 6 \\pmod{28}... | {
"ground_truth": "1160",
"style": "rule"
} |
9,845 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=2\cos(\omega x+\varphi)$ ($\omega \gt 0$, $0 \lt \varphi \lt \frac{π}{2}$), the graph passes through the point $(0, \sqrt{2})$, and is monotonically decreasing on the interval $[... | [
"Given the function \\( f(x) = 2\\cos(\\omega x + \\varphi) \\), the graph passes through the point \\((0, \\sqrt{2})\\), and is monotonically decreasing on the interval \\([0, \\frac{\\pi}{2}]\\). We need to find the maximum value of \\(\\omega\\).\n\n1. **Substitute the point \\((0, \\sqrt{2})\\) into the functio... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} |
6,718 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+... | [
"To solve this problem, we need to understand how to calculate $f(n)$, the number of ways to write $n$ as a sum of powers of 2, keeping track of the order of the summation. This is essentially the problem of finding the number of compositions of $n$ where each part is a power of 2.\n\nThe key observation here is th... | {
"ground_truth": "2047",
"style": "rule"
} |
9,258 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the function $y=\log_{a}x+1$ ($a>0$ and $a\neq1$) must pass through the point (fill in the coordinates of the point). Let's think step by step and output the final answer within \boxed{}<|im... | [
"The function is $y = \\log_a(x) + 1$. We need to find a point $(x, y)$ that this function must pass through regardless of the value of $a$.\n\n1. Let's consider the point where $x = 1$. We know that $\\log_a(1) = 0$ for any base $a$. Therefore, substituting $x = 1$ into the function, we get:\n\\[ y = \\log_a(1) + ... | {
"ground_truth": "(1,1)",
"style": "rule"
} |
6,072 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There is a row of 15 elephants, each weighing an integer number of kilograms. If we take any elephant, except the one on the far right, and add twice the weight of its right neighbor to its weight, we ge... | [
"To solve the problem, let's denote the weights of the 15 elephants as \\(w_1, w_2, \\ldots, w_{15}\\), where \\(w_i\\) is the weight of the \\(i\\)-th elephant from the left. According to the problem, for each elephant except the one on the far right, the following equation holds:\n\n\\[w_i + 2w_{i+1} = 15000 \\qu... | {
"ground_truth": "5 \\text{ tons}",
"style": "rule"
} |
8,889 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When plotted in the standard rectangular coordinate system, a trapezoid \( PQRS \) has vertices at \( P (1, 0) \), \( Q (1, 3) \), \( R (5, 9) \), \( S (5, 3) \). What is the area of trapezoid \( PQRS \)... | [
"To find the area of trapezoid \\( PQRS \\), we can use the formula for the area of a trapezoid:\n\n\\[\n\\text{Area} = \\frac{1}{2} (b_1 + b_2) h\n\\]\n\nwhere \\( b_1 \\) and \\( b_2 \\) are the lengths of the parallel sides, and \\( h \\) is the height.\n\nFirst, let's identify the parallel sides and their lengt... | {
"ground_truth": "18",
"style": "rule"
} |
1,656 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the following propositions:
① If $|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|$, then $\vec{a}$ and $\vec{b}$ are collinear and have the same direction;
② The function $f(x) = \cos(\sin x)$ has a... | [
"Let's analyze each proposition one by one using reasoning and Python code where necessary.\n\n### Proposition ①\nThe proposition states that if \\( |\\vec{a} + \\vec{b}| = |\\vec{a}| + |\\vec{b}| \\), then \\(\\vec{a}\\) and \\(\\vec{b}\\) are collinear and have the same direction.\n\nThis is true because the equa... | {
"ground_truth": "①②④",
"style": "rule"
} |
8,679 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel, maintaining a fixed center, has an outside diameter of 8 feet. Determine the number of revolutions required for a point on the rim to travel two miles.
A) $\frac{2640}{\pi}$
B) $\frac{1320}{\pi}... | [
"The circumference of the wheel is the distance traveled by a point on the rim for each revolution. The circumference \\(C\\) of a circle is given by \\(C = 2\\pi r\\), where \\(r\\) is the radius of the circle. Given that the diameter of the wheel is 8 feet, the radius \\(r\\) is 4 feet. Therefore, the circumferen... | {
"ground_truth": "\\frac{1320}{\\pi}",
"style": "rule"
} |
7,591 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the sequence $\{t_{n}\}$ satisfies ${t_{n+1}}={t_n}-\frac{{f({{t_n}})}}{{{f'}({{t_n}})}}$, then the sequence is called the "tangent line - zero sequence". Given that the function $f\left(x\right)=x^{2... | [
"Given that the function $f(x) = x^2 + px + q$ has two zeros at $1$ and $2$, we can determine the values of $p$ and $q$. The function can be written as:\n\\[ f(x) = (x - 1)(x - 2) = x^2 - 3x + 2 \\]\nSo, $p = -3$ and $q = 2$. The function is $f(x) = x^2 - 3x + 2$ and its derivative is $f'(x) = 2x - 3$.\n\nThe tange... | {
"ground_truth": "2^{2024} - 2",
"style": "rule"
} |
9,704 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given $y=f(x)$ be a differentiable function defined on $\mathbb{R}$, and its derivative $f'(x)$ satisfies $f(x) > f'(x)$, and $f(0)=2$, determine the solution set of the inequality $f(x) < 2e^{x}$. Let's... | [
"To solve the problem, let's start by analyzing the given conditions and the inequality.\n\n1. The function \\( y = f(x) \\) is differentiable and satisfies \\( f(x) > f'(x) \\).\n2. We are also given \\( f(0) = 2 \\).\n3. We need to determine the solution set of the inequality \\( f(x) < 2e^x \\).\n\nFirst, let's ... | {
"ground_truth": "(0,+\\infty)",
"style": "rule"
} |
7,448 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An object moves along a straight line with velocity $v$, and $v(t) = 2t - 3$ (where $t$ is in seconds, and $v$ is in meters per second). Calculate the distance traveled by the object from the moment $t =... | [
"To find the distance traveled by the object, we need to integrate the absolute value of the velocity function over the given time interval. The velocity function is given by $v(t) = 2t - 3$. First, we need to determine where the velocity function changes sign within the interval $[0, \\frac{3}{2}]$. The velocity f... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} |
7,851 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sequence \(\left(a_{n}\right)\) is defined by the following relations: \(a_{1}=1\), \(a_{2}=2\), \(a_{n}=a_{n-1}-a_{n-2}+n\) for \(n \geq 3\). Find \(a_{2019}\). Let's think step by step and output t... | [
"To find the value of \\(a_{2019}\\) in the sequence defined by \\(a_1 = 1\\), \\(a_2 = 2\\), and \\(a_n = a_{n-1} - a_{n-2} + n\\) for \\(n \\geq 3\\), we can generate the sequence iteratively until we reach \\(a_{2019}\\). We'll use Python to ensure accuracy.\n\nFirst, we'll initialize the first two terms of the ... | {
"ground_truth": "2020",
"style": "rule"
} |
6,339 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the function $f(x) = x^2$, then $f'(1) = \_\_\_\_\_\_$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the derivative of the function $f(x) = x^2$ at the point $x = 1$, we can use the definition of the derivative or we can use the power rule of differentiation. Here, I'll use the power rule, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.\n\nFor the function $f(x) = x^2$, the derivative $f'(x)$ ... | {
"ground_truth": "2",
"style": "rule"
} |
7,494 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the equations:<br/>
$(1) 4x = 2x + 6$;<br/>
$(2) 3x + 5 = 6x - 1$;<br/>
$(3) 3x - 2(x-1) = 2 + 3(4-x)$;<br/>
$(4) \frac{x-3}{5} - \frac{x+4}{2} = -2$. Let's think step by step and output the final ... | [
"Solve the equations:<br/>\n$(1) 4x = 2x + 6$;<br/>\n$(2) 3x + 5 = 6x - 1$;<br/>\n$(3) 3x - 2(x-1) = 2 + 3(4-x)$;<br/>\n$(4) \\frac{x-3}{5} - \\frac{x+4}{2} = -2$. Let's solve each equation step by step using Python and SymPy:\n\n1. Solve the equation \\(4x = 2x + 6\\).\n2. Solve the equation \\(3x + 5 = 6x - 1\\).... | {
"ground_truth": "x = -2",
"style": "rule"
} |
226 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A piece of cheese is located at \( (16, 8) \). A mouse starts at \( (2, -4) \) running up along the line \( y = -3x + 10 \). At what point \( (a, b) \) along its path does the mouse begin to get farther ... | [
"To solve this problem, we need to find the point on the line \\( y = -3x + 10 \\) that is closest to the cheese located at \\( (16, 8) \\). The closest point on a line to a given point is the point where the perpendicular from the given point to the line intersects the line. Once we find this point, we can conclud... | {
"ground_truth": "8.8",
"style": "rule"
} |
6,587 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x)=(x^3+2x^2+ax-a)e^x$, find the value of $f'(0)$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find $f'(0)$, we need to first find the derivative of $f(x)$. The function $f(x)$ is given by:\n\\[ f(x) = (x^3 + 2x^2 + ax - a)e^x. \\]\nUsing the product rule, the derivative of $f(x)$ is:\n\\[ f'(x) = \\frac{d}{dx}[(x^3 + 2x^2 + ax - a)e^x]. \\]\nThe product rule states that:\n\\[ \\frac{d}{dx}[u(x)v(x)] = u... | {
"ground_truth": "0",
"style": "rule"
} |
9,187 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A shipping box has interior dimensions 8-inches by 5-inches by 14-inches. Inside this shipping box, as many solid 4-inch cubes as possible are packed, with all of the cubes entirely inside the box. How m... | [
"The dimensions of the box are 8 inches by 5 inches by 14 inches, and each cube has a side length of 4 inches.\n\n1. First, we need to determine how many cubes fit into the dimensions of the box.\n\n2. We will then calculate the volume of the box and the volume taken up by the cubes.\n\n3. Finally, we will find the... | {
"ground_truth": "68.57\\%",
"style": "rule"
} |
6,227 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the set $P=\{0, 2, 4, 6\}$, and the set $Q=\{x \in \mathbb{N} | x \leq 3\}$, find the intersection of sets P and Q. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|... | [
"First, let's define the sets P and Q in Python. \n\nSet P is given as P = {0, 2, 4, 6}. \n\nSet Q is defined as Q = {x ∈ ℕ | x ≤ 3}, which means Q = {0, 1, 2, 3} because the natural numbers include 0 and all positive integers.\n\nThe intersection of sets P and Q (denoted as P ∩ Q) is the set of elements that are c... | {
"ground_truth": "\\{0, 2\\}",
"style": "rule"
} |
2,935 | <|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 the odd integers from 15 through 45, inclusive? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the sum of the odd integers from 15 through 45, inclusive, we can follow these steps:\n\n1. Identify the range of odd integers from 15 to 45.\n2. Use a loop to iterate through the range and sum the odd numbers.\n\nHere's the Python code to do this:\n```python\n# Initialize the sum\nsum_of_odds = 0\n\n# Ite... | {
"ground_truth": "480",
"style": "rule"
} |
6,142 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( g(x) = x^3 - 3x \). Find the value of \( g(g(g(g(g(g(g(-2))))))) \). 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 evaluate the function \\( g(x) = x^3 - 3x \\) iteratively starting from \\( x = -2 \\). We will compute \\( g(-2) \\), then \\( g(g(-2)) \\), and so on, up to \\( g(g(g(g(g(g(g(-2))))))) \\).\n\nFirst, let's define the function \\( g(x) \\) and then compute the value step by step u... | {
"ground_truth": "-2",
"style": "rule"
} |
7,644 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The coefficient of $x^3y^3$ in the expansion of $(x-y)(2x-y)^5$ is $\_\_\_\_\_\_$. (Answer with a number) Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assist... | [
"To find the coefficient of $x^3y^3$ in the expansion of $(x-y)(2x-y)^5$, we can break the problem down into smaller steps.\n\n1. **Expand the Binomial $(2x-y)^5$:**\n Use the binomial theorem to expand $(2x-y)^5$. The binomial theorem states that $(a + b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^k$. So, $(2x-... | {
"ground_truth": "-120",
"style": "rule"
} |
5,575 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $i$ be the imaginary unit. Find the value of the complex number $\dfrac {2}{1-i}=$ \_\_\_\_\_\_. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| [
"To find the value of the complex number \\(\\frac{2}{1-i}\\), we need to rationalize the denominator. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \\(1-i\\) is \\(1+i\\). Here’s the step-by-step solution:\n\n1. Multiply the numerator and t... | {
"ground_truth": "1+i",
"style": "rule"
} |
6,797 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = a\sin(\pi x + \alpha) + b\cos(\pi x + \beta)$, where $a$, $b$, $\alpha$, and $\beta$ are non-zero real numbers, and $f(2017) = -1$, find the value of $f(2018)$. Let's think ste... | [
"To find the value of \\( f(2018) \\), let's use the properties of trigonometric functions and the given information. The function is given by:\n\\[ f(x) = a\\sin(\\pi x + \\alpha) + b\\cos(\\pi x + \\beta) \\]\n\nFirst, we know that \\( f(2017) = -1 \\). So, let's substitute \\( x = 2017 \\) into the function:\n\\... | {
"ground_truth": "1",
"style": "rule"
} |
9,629 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dots are spaced two units apart, horizontally and vertically. Create a polygon by connecting these points in the order: (0,0), (6,0), (6,6), (0,6) and back to (0,0). Calculate the area of the polygon enc... | [
"Dots are spaced two units apart, horizontally and vertically. Create a polygon by connecting these points in the order: (0,0), (6,0), (6,6), (0,6) and back to (0,0). Calculate the area of the polygon enclosed. Let's think step by step and output the final answer within \\boxed{}.",
"We can calculate the area of ... | {
"ground_truth": "36",
"style": "rule"
} |
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