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[
{
"content": "Compute the sum of the cubes of the roots of the equation $x^{2022} + 55x^{2019} + 4x^4 + 505 = 0$.",
"role": "user"
},
{
"content": "Let $r_1$, $r_2$, $\\ldots$, $r_{2022}$ be the roots of the equation $x^{2022} + 55x^{2019} + 4x^4 + 505 = 0$.\n\nThe sum of the cubes of the roots is:\... | Compute the sum of the cubes of the roots of the equation $x^{2022} + 55x^{2019} + 4x^4 + 505 = 0$. | Compute the sum of the cubes of the roots of the equation $x^{2022} + 55x^{2019} + 4x^4 + 505 = 0$. | ai2-adapt-dev/openmath-2-math | 0.997921 | Alright, let's see. I need to compute the sum of the cubes of the roots of the equation \( x^{2022} + 55x^{2019} + 4x^4 + 505 = 0 \). Hmm, okay. So, this is a polynomial equation of degree 2022, right? Which means there are 2022 roots (including complex ones). The question is asking for the sum of each root cubed, so i... | To compute the sum of the cubes of the roots of the equation \( x^{2022} + 55x^{2019} + 4x^4 + 505 = 0 \), we use Vieta's formulas and Newton's identities.
1. **Vieta's Formulas**:
- The polynomial is \( x^{2022} + 55x^{2019} + 4x^4 + 505 \).
- The coefficients of \( x^{2021} \), \( x^{2020} \), and \( x^{2018} ... | 18,357 | <think>
Alright, let's see. I need to compute the sum of the cubes of the roots of the equation \( x^{2022} + 55x^{2019} + 4x^4 + 505 = 0 \). Hmm, okay. So, this is a polynomial equation of degree 2022, right? Which means there are 2022 roots (including complex ones). The question is asking for the sum of each root cub... | [
{
"from": "human",
"value": "Compute the sum of the cubes of the roots of the equation $x^{2022} + 55x^{2019} + 4x^4 + 505 = 0$."
},
{
"from": "gpt",
"value": "<think>\nAlright, let's see. I need to compute the sum of the cubes of the roots of the equation \\( x^{2022} + 55x^{2019} + 4x^4 + 505 ... |
[
{
"content": "Let $\\alpha$ be a real number such that the sequence $\\left(\\frac{2n+\\alpha}{n^2+4n+6}\\right)$ converges to a finite limit as $n$ tends to infinity. Determine the sum of the values of $\\alpha$ that satisfy this condition.",
"role": "user"
},
{
"content": "Let $\\left(\\frac{2n+\\... | Let $\alpha$ be a real number such that the sequence $\left(\frac{2n+\alpha}{n^2+4n+6}\right)$ converges to a finite limit as $n$ tends to infinity. Determine the sum of the values of $\alpha$ that satisfy this condition. | Let $\alpha$ be a real number such that the sequence $\left(\frac{2n+\alpha}{n^2+4n+6}\right)$ converges to a finite limit as $n$ tends to infinity. Determine the sum of the values of $\alpha$ that satisfy this condition. | ai2-adapt-dev/openmath-2-math | 0.998367 | Okay, so I need to find the sum of all real numbers α such that the sequence (2n + α)/(n² + 4n + 6) converges to a finite limit as n approaches infinity. Hmm, let's start by recalling some concepts about limits of sequences.
First, I remember that when dealing with the limit of a rational function (a polynomial divide... | To determine the sum of the values of \(\alpha\) such that the sequence \(\left(\frac{2n + \alpha}{n^2 + 4n + 6}\right)\) converges to a finite limit as \(n\) tends to infinity, we start by analyzing the behavior of the sequence.
The denominator \(n^2 + 4n + 6\) is a quadratic polynomial, while the numerator \(2n + \a... | 16,667 | <think>
Okay, so I need to find the sum of all real numbers α such that the sequence (2n + α)/(n² + 4n + 6) converges to a finite limit as n approaches infinity. Hmm, let's start by recalling some concepts about limits of sequences.
First, I remember that when dealing with the limit of a rational function (a polynomia... | [
{
"from": "human",
"value": "Let $\\alpha$ be a real number such that the sequence $\\left(\\frac{2n+\\alpha}{n^2+4n+6}\\right)$ converges to a finite limit as $n$ tends to infinity. Determine the sum of the values of $\\alpha$ that satisfy this condition."
},
{
"from": "gpt",
"value": "<think>\... |
[{"content":"Find the sum of all the coefficients of the expansion of $(x+y)^5$, where $x$ and $y$ a(...TRUNCATED) | "Find the sum of all the coefficients of the expansion of $(x+y)^5$, where $x$ and $y$ are real numb(...TRUNCATED) | "Find the sum of all the coefficients of the expansion of $(x+y)^5$, where $x$ and $y$ are real numb(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.997894 | "Alright, so I have this problem here: Find the sum of all the coefficients of the expansion of (x +(...TRUNCATED) | "To find the sum of all the coefficients of the expansion of \\((x + y)^5\\), we can use the fact th(...TRUNCATED) | 20,079 | "<think>\nAlright, so I have this problem here: Find the sum of all the coefficients of the expansio(...TRUNCATED) | [{"from":"human","value":"Find the sum of all the coefficients of the expansion of $(x+y)^5$, where (...TRUNCATED) |
[{"content":"Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2+2x+3=0$. Find the value of(...TRUNCATED) | "Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2+2x+3=0$. Find the value of $\\frac{(\\(...TRUNCATED) | "Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2+2x+3=0$. Find the value of $\\frac{(\\(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998786 | "Okay, let me try to figure out this problem step by step. The question is: Given the quadratic equa(...TRUNCATED) | "Given the quadratic equation \\(x^2 + 2x + 3 = 0\\) with roots \\(\\alpha\\) and \\(\\beta\\), we n(...TRUNCATED) | 8,378 | "<think>\nOkay, let me try to figure out this problem step by step. The question is: Given the quadr(...TRUNCATED) | [{"from":"human","value":"Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2+2x+3=0$. Find(...TRUNCATED) |
[{"content":"Determine the 25th digit after the decimal point of the product of the decimal equivale(...TRUNCATED) | "Determine the 25th digit after the decimal point of the product of the decimal equivalents for the (...TRUNCATED) | "Determine the 25th digit after the decimal point of the product of the decimal equivalents for the (...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999045 | "Okay, let me try to figure out the 25th digit after the decimal point when I multiply 1/11 by 1/13.(...TRUNCATED) | "To determine the 25th digit after the decimal point of the product of the decimal equivalents for t(...TRUNCATED) | 30,515 | "<think>\nOkay, let me try to figure out the 25th digit after the decimal point when I multiply 1/11(...TRUNCATED) | [{"from":"human","value":"Determine the 25th digit after the decimal point of the product of the dec(...TRUNCATED) |
[{"content":"Let's consider a set of $n$ objects with distinct weights, where the weights of the obj(...TRUNCATED) | "Let's consider a set of $n$ objects with distinct weights, where the weights of the objects are con(...TRUNCATED) | "Let's consider a set of $n$ objects with distinct weights, where the weights of the objects are con(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998134 | "Okay, so I need to figure out how many subsets of n objects with weights from 1 to n have a total e(...TRUNCATED) | "To determine the number of subsets of \\( n \\) objects with distinct weights from 1 to \\( n \\) s(...TRUNCATED) | 7,478 | "<think>\nOkay, so I need to figure out how many subsets of n objects with weights from 1 to n have (...TRUNCATED) | [{"from":"human","value":"Let's consider a set of $n$ objects with distinct weights, where the weigh(...TRUNCATED) |
[{"content":"Subtract from the reciprocal of $\\frac{5}{13}$ the sum of the reciprocals of $7$ and $(...TRUNCATED) | Subtract from the reciprocal of $\frac{5}{13}$ the sum of the reciprocals of $7$ and $19$. | Subtract from the reciprocal of $\frac{5}{13}$ the sum of the reciprocals of $7$ and $19$. | ai2-adapt-dev/openmath-2-math | 0.999735 | "Alright, so I need to solve this problem: Subtract from the reciprocal of 5/13 the sum of the recip(...TRUNCATED) | "To solve the problem of subtracting the sum of the reciprocals of 7 and 19 from the reciprocal of \(...TRUNCATED) | 9,031 | "<think>\nAlright, so I need to solve this problem: Subtract from the reciprocal of 5/13 the sum of (...TRUNCATED) | [{"from":"human","value":"Subtract from the reciprocal of $\\frac{5}{13}$ the sum of the reciprocals(...TRUNCATED) |
[{"content":"Let $P(x)$ be a quadratic polynomial with real coefficients such that\n\n\\[P(x)P(-x)=3(...TRUNCATED) | "Let $P(x)$ be a quadratic polynomial with real coefficients such that\n\n\\[P(x)P(-x)=36x^4-28x^2+1(...TRUNCATED) | "Let $P(x)$ be a quadratic polynomial with real coefficients such that\n\n\\[P(x)P(-x)=36x^4-28x^2+1(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998804 | "Okay, let's see. I need to find the sum of all possible values of the coefficient of x in P(x), whe(...TRUNCATED) | "Given a quadratic polynomial \\( P(x) \\) with real coefficients such that \\( P(x)P(-x) = 36x^4 - (...TRUNCATED) | 2,815 | "<think>\nOkay, let's see. I need to find the sum of all possible values of the coefficient of x in (...TRUNCATED) | [{"from":"human","value":"Let $P(x)$ be a quadratic polynomial with real coefficients such that\n\n\(...TRUNCATED) |
[{"content":"A regular decagon can be split into ten congruent sectors of equal area. A certain line(...TRUNCATED) | "A regular decagon can be split into ten congruent sectors of equal area. A certain linear function (...TRUNCATED) | "A regular decagon can be split into ten congruent sectors of equal area. A certain linear function (...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998602 | "Alright, let's tackle this problem step by step. So, we have a regular decagon, which is a 10-sided(...TRUNCATED) | "A regular decagon centered at the origin can be represented by vertices at the 10th roots of unity (...TRUNCATED) | 31,028 | "<think>\nAlright, let's tackle this problem step by step. So, we have a regular decagon, which is a(...TRUNCATED) | [{"from":"human","value":"A regular decagon can be split into ten congruent sectors of equal area. A(...TRUNCATED) |
[{"content":"In the equation $x^2 - 2px + q = 0$, the sum of the roots is 3 times the product of the(...TRUNCATED) | "In the equation $x^2 - 2px + q = 0$, the sum of the roots is 3 times the product of the roots. If t(...TRUNCATED) | "In the equation $x^2 - 2px + q = 0$, the sum of the roots is 3 times the product of the roots. If t(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.997611 | "Okay, let's see. I need to find the value of p in the equation x² - 2px + q = 0. The problem says (...TRUNCATED) | "Given the quadratic equation \\( x^2 - 2px + q = 0 \\), we know the sum of the roots is 3 times the(...TRUNCATED) | 26,317 | "<think>\nOkay, let's see. I need to find the value of p in the equation x² - 2px + q = 0. The prob(...TRUNCATED) | [{"from":"human","value":"In the equation $x^2 - 2px + q = 0$, the sum of the roots is 3 times the p(...TRUNCATED) |
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