conversations listlengths 2 2 | question stringlengths 11 28.8k |
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[
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"from": "user",
"value": "Return your final response within \\boxed{}. A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\nsize(150... | A cone has a volume of $12288\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]
import markers;
size(150);
import geometry;
draw(scale(1,.2)*arc((0,0),1,0,180),dashed);
draw(scale(1,.2)*arc((0,... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the projection of the vector $\\bar{a}=3 \\bar{i}+4 \\bar{j}-\\bar{k}$ in the direction of the vector $\\bar{b}=\\bar{i}+\\bar{j}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the projection o... | Find the projection of the vector $\bar{a}=3 \bar{i}+4 \bar{j}-\bar{k}$ in the direction of the vector $\bar{b}=\bar{i}+\bar{j}$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( P \\in \\mathbb{R}[X] \\) be a monic polynomial of degree 2020 such that \\( P(n) = n \\) for all \\( n \\in \\{0, 1, \\ldots, 2019\\} \\). Calculate \\( P(2020) \\)."
},
{
"from": "assistant",
"value": "<think>\nOk... | Let \( P \in \mathbb{R}[X] \) be a monic polynomial of degree 2020 such that \( P(n) = n \) for all \( n \in \{0, 1, \ldots, 2019\} \). Calculate \( P(2020) \). |
[
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"from": "user",
"value": "Return your final response within \\boxed{}. Find all subsequences \\( \\{a_{1}, a_{2}, \\cdots, a_{n}\\} \\) of the sequence \\(\\{1, 2, \\cdots, n\\}\\) that satisfy the property \\( i + 1 \\mid 2(a_{1} + a_{2} + \\cdots + a_{j}) \\) for \\(1 \\leq i \\leq n\\)."
},
{
"... | Find all subsequences \( \{a_{1}, a_{2}, \cdots, a_{n}\} \) of the sequence \(\{1, 2, \cdots, n\}\) that satisfy the property \( i + 1 \mid 2(a_{1} + a_{2} + \cdots + a_{j}) \) for \(1 \leq i \leq n\). |
[
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"from": "user",
"value": "Return your final response within \\boxed{}. During a trip, person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the... | During a trip, person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the coastal road, how much time did the entire trip take in minutes?
(A) 50
(B) 70
(C) 8... |
[
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"from": "user",
"value": "Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.You are given two images img1 and img2 both of size n x n, represented as binary, square matrices of the same size. (A binary matrix has only... | Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.You are given two images img1 and img2 both of size n x n, represented as binary, square matrices of the same size. (A binary matrix has only 0s and 1s as values.)
We translate one ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given an acute triangle \\(ABC\\). The altitude \\(AA_1\\) is extended through vertex \\(A\\) by a segment \\(AA_2 = BC\\). The altitude \\(CC_1\\) is extended through vertex \\(C\\) by a segment \\(CC_2 = AB\\). Find the angles of tri... | Given an acute triangle \(ABC\). The altitude \(AA_1\) is extended through vertex \(A\) by a segment \(AA_2 = BC\). The altitude \(CC_1\) is extended through vertex \(C\) by a segment \(CC_2 = AB\). Find the angles of triangle \(A_2BC_2\). |
[
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"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in [Russian] and [Bengali].\n\nThere's a zebra crossing appearing in the mi... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in [Russian] and [Bengali].
There's a zebra crossing appearing in the middle of nowhere with N blocks in it. The c... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given the function\n$$\ny(x) = (\\sqrt{1+x} + \\sqrt{1-x} + 2) \\left(\\sqrt{1-x^2} + 1 \\right),\n$$\n\nwhere \\(x \\in [0,1]\\). Determine the minimum value of \\(y(x)\\)."
},
{
"from": "assistant",
"value": "<think>\nOka... | Given the function
$$
y(x) = (\sqrt{1+x} + \sqrt{1-x} + 2) \left(\sqrt{1-x^2} + 1 \right),
$$
where \(x \in [0,1]\). Determine the minimum value of \(y(x)\). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at... | Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways ca... |
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