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[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor what value of $n$ is $i + 2i^2 + 3i^3 + \\cdots + ni^n = 48 + 49i$? Find the value of $n$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
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] |
97
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[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\\frac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make? Express your answer in the form of an improper fraction \\frac{k}{m}, and find the value of k + m.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
39
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[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. Find the sum of all possible values of $|a-d|$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
18
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[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPoints \\(A\\) and \\(C\\) lie on a circle centered at \\(O\\), each of \\(\\overline{BA}\\) and \\(\\overline{BC}\\) are tangent to the circle, and \\(\\triangle ABC\\) is equilateral. The circle intersects \\(\\overline{BO}\\) at \\(D\\). The original answer is in the form \\(\\frac{k}{m}\\). Please find the value of \\(k + m\\).\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
3
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[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTrapezoid \\(ABCD\\) has \\(AD \\parallel BC\\), \\(BD = 1\\), \\(\\angle DBA = 23^\\circ\\), and \\(\\angle BDC = 46^\\circ\\). The ratio \\(BC: AD\\) is \\(9: 5\\). What is the value of \\(CD\\)? The original answer is in the form \\(\\frac{k}{m}\\), where \\(\\frac{k}{m}\\) is a fully simplified fraction. Please provide the value of \\(k + m\\).\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] |
9
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