conversations listlengths 2 2 |
|---|
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the degree of the polynomial \\( (4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's break down the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 ... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sq... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To solve the expression \\(\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil\\), we will brea... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's break down the expression step by step:\n\n1. **Evaluate the expression inside the parenthe... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's evaluate the expression step by step:\n\nThe expression is \\(\\left\\lceil 3\\left(6 - \\f... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To evaluate \\(\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil\\), let's break it down step... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To evaluate the expression \\(\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil\\), let's bre... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's evaluate the expression step by step:\n\nThe expression given is $\\left\\lceil3\\left(6-\\... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To evaluate the expression \\(\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil\\), let's bre... |
[
{
"from": "human",
"value": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To evaluate the expression \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil\\), we'll... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and output the f... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we n... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's substitute the given values \\( x = 2 \\) and \\( y = 5 \\) into the expression \\(... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To solve the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's solve the expression \\(\\frac{x^4 + 2y^2}{6}\\) step by step given \\(x = 2\\) and... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's substitute the given values \\( x = 2 \\) and \\( y = 5 \\) into the expression \\(... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we wi... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's solve the expression \\(\\frac{x^4 + 2y^2}{6}\\) step by step with the given values... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we wi... |
[
{
"from": "human",
"value": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "Let's substitute the given values of \\(x\\) and \\(y\\) into the expression \\(\\frac{x^... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and output the final answer w... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To determine the c... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"value": "To find the center... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? Let's think step by step and output the final answer within \\boxed{{}}."
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros do... |
[
{
"from": "human",
"value": "Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros do... |
[
{
"from": "human",
"value": "Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros do... |
[
{
"from": "human",
"value": "Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros do... |
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