id
stringlengths 22
32
| setting_key
stringclasses 7
values | messages
listlengths 1
1
| ground_truth
stringlengths 1
79
| dataset
stringclasses 1
value | omega_config
stringclasses 7
values | omega_split
stringclasses 1
value |
|---|---|---|---|---|---|---|
trans_prob_letter_test_022
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 8 can be formed from {\"o\": 7,\"p\": 6,\"q\": 5,\"r\": 4}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
63444
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_023
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 9 can be formed from {\"s\": 8,\"t\": 7,\"u\": 6,\"v\": 5}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
259143
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_024
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 12 can be formed from {\"a\": 11,\"b\": 10,\"c\": 9,\"d\": 8,\"e\": 7}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
243982380
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_025
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 10 can be formed from {\"f\": 9,\"g\": 8,\"h\": 7,\"i\": 6,\"j\": 5}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9694180
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_026
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 11 can be formed from {\"k\": 10,\"l\": 9,\"m\": 8,\"n\": 7,\"o\": 6}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
48719704
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_027
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 8 can be formed from {\"p\": 7,\"q\": 6,\"r\": 5,\"s\": 4,\"t\": 3}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
364060
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_028
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 9 can be formed from {\"u\": 8,\"v\": 7,\"w\": 6,\"x\": 5,\"y\": 4}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1908240
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_029
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 7 can be formed from {\"z\": 6,\"a\": 5,\"b\": 4,\"c\": 3,\"d\": 2}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
63595
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_030
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 6 can be formed from {\"e\": 5,\"f\": 4,\"g\": 3,\"h\": 2,\"i\": 1}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8634
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_031
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 5 can be formed from {\"j\": 4,\"k\": 3,\"l\": 2,\"m\": 1,\"n\": 1}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1410
|
math
|
trans_prob_letter
|
test
|
trans_integrations_test_000
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{x + \\sqrt{x}}{1 + \\sqrt{x}} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{2}{3}x^{3/2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_001
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{e^{x+1}}{e^x + 1} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
e \log(e^x + 1)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_002
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\sqrt[3]{3 \\sin(x) - \\sin(3x)} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
-\frac{\sqrt{3}}{4} \cos(x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_003
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{x \\log(x) \\cos(x) - \\sin(x)}{x \\log^2(x)} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{\sin(x)}{\log(x)}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_004
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\sqrt[3]{\\frac{x}{4/2}} \\sqrt{\\frac{x}{5/3}} \\sqrt[4]{x^{6/\\sqrt{4}}} \\cdots dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{x^2}{2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_005
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{e^{2x}}{(x^2 + x)(xe^x)^4 + 1} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{1}{2} \arctan(x^2 e^{2x})
|
math
|
trans_integrations
|
test
|
trans_integrations_test_006
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (\\sec^4(x) - \\tan^4(x)) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2 \tan(x) - x
|
math
|
trans_integrations
|
test
|
trans_integrations_test_007
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{\\sin(4x) \\cos(x)}{\\cos(2x) \\sin(x)} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2x + \sin(2x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_008
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\sin(x) \\sinh(x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{1}{2}(\sin(x) \cosh(x) - \cos(x) \sinh(x))
|
math
|
trans_integrations
|
test
|
trans_integrations_test_009
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{\\cos(x) + \\cos(x + \\frac{2\\pi}{3}) + \\cos(x - \\frac{2\\pi}{3})}{2} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
trans_integrations
|
test
|
trans_integrations_test_010
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{(x - 1)\\log(x+1)}{(x + 1)\\log(x-1)} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
x
|
math
|
trans_integrations
|
test
|
trans_integrations_test_011
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (x \\log x + 2x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{1}{2}x^2 \log x + \frac{3}{4}x^2
|
math
|
trans_integrations
|
test
|
trans_integrations_test_012
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{dx}{x \\log x + 2x}. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\log(\log x + 2)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_013
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{\\cos x + \\cot x + \\csc x + 1}{\\sin x + \\tan x + \\sec x + 1} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\log(\sin x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_014
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{x^{2024} - 1}{x^{506} - 1} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
x + \frac{x^{507}}{507} + \frac{x^{1013}}{1013} + \frac{x^{1519}}{1519}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_015
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\csc^2(x) \\tan^{2024}(x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{\tan^{2023}(x)}{2023}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_016
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\cos^x(x)(\\log(\\cos x) - x \\tan x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\cos^x(x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_017
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (x + 1 - e^{-x})e^{xe^x} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
e^{(e^x-1)x}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_018
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\left(\\frac{\\arctan(x)}{1 - x^2} + \\frac{\\text{arctanh}(x)}{1 + x^2}\\right) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\arctan(x)\text{arctanh}(x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_019
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\sum_{k=0}^{\\infty} \\sin\\left(\\frac{k\\pi}{2}\\right) \\frac{x^k}{k!} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{\log(x^2 + 1)}{2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_020
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{x^4}{3 - 6x + 6x^2 - 4x^3 + 2x^4} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{x^2}{2} + \frac{1}{4}\log(3 - 6x + 6x^2 - 4x^3 + 2x^4)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_021
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int x^{\\frac{1}{\\log x}} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
ex
|
math
|
trans_integrations
|
test
|
trans_integrations_test_022
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\text{sech}(x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2 \arctan(e^x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_023
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{e^x}{(1 + e^x) \\log(1 + e^x)} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\log(\log(1 + e^x))
|
math
|
trans_integrations
|
test
|
trans_integrations_test_024
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (1 + x + x^2 + x^3 + x^4)(1 - x + x^2 - x^3 + x^4) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
x + \frac{x^3}{3} + \frac{x^5}{5} + \frac{x^7}{7} + \frac{x^9}{9}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_025
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (x + \\sin(x) + x \\cos(x) + \\sin(x) \\cos(x)) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{(x + \sin(x))^2}{2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_026
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (\\sin^2(x) + \\cos^2(x) + \\tan^2(x) + \\cot^2(x) + \\sec^2(x) + \\csc^2(x)) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2 \tan(x) - 2 \cot(x) - x
|
math
|
trans_integrations
|
test
|
trans_integrations_test_027
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (2 \\log x + 1)e^{(\\log x)^2} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
xe^{(\log x)^2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_028
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int ((1 - x)^3 + (x - x^2)^3 + (x^2 - 1)^3 - 3(1 - x)(x - x^2)(x^2 - 1)) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
trans_integrations
|
test
|
trans_integrations_test_029
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (\\sin^6 x + \\cos^6 x + 3 \\sin^2 x \\cos^2 x) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
x
|
math
|
trans_integrations
|
test
|
trans_integrations_test_030
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (x + e + 1)^x e^{ex} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
x^{e+1}e^x
|
math
|
trans_integrations
|
test
|
trans_integrations_test_031
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{1 + 2x^{2022}}{x + x^{2023}} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{1}{2022} \log(x^{2022} + x^{4044})
|
math
|
trans_integrations
|
test
|
trans_integrations_test_032
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int (3 \\sin(20x) \\cos(23x) + 20 \\sin(43x)) dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\sin(20x) \sin(23x)
|
math
|
trans_integrations
|
test
|
trans_integrations_test_033
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{\\sin x}{2e^x + \\cos x + \\sin x} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{x - \log(2e^x + \cos x + \sin x)}{2}
|
math
|
trans_integrations
|
test
|
trans_integrations_test_034
|
trans_integrations
|
[
{
"content": "Evaluate the indefinite integral \\int \\frac{\\log(x/\\pi)}{(\\log x)^{\\log(\\pi e)}} dx. Do not include the constant +C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
\frac{x}{(\log x)^{\log \pi}}
|
math
|
trans_integrations
|
test
|
trans_gridworld_test_000
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of an 8x8 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_001
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 10x10 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_002
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 12x12 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_003
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 14x14 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_004
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 16x16 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_005
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of an 18x18 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_006
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 20x20 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_007
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 22x22 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
10
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_008
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 24x24 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
11
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_009
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 26x26 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_010
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 28x28 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
13
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_011
|
trans_gridworld
|
[
{
"content": "Suppose you have an agent in the bottom left (i=0, j=0) cell of a 30x30 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
14
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_012
|
trans_gridworld
|
[
{
"content": "In a 10x10 grid, how many different paths are there from the bottom left (0, 0) to the top right (9, 9), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
10
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_013
|
trans_gridworld
|
[
{
"content": "In a 20x20 grid, how many different paths are there from the bottom left (0, 0) to the top right (19, 19), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (2, 9), (2, 10), (2, 11), (2, 12), (2, 13), (2, 14), (2, 15), (2, 16), (2, 17), (2, 18)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
20
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_014
|
trans_gridworld
|
[
{
"content": "In a 10x10 grid, how many different paths are there from the bottom left (0, 0) to the top right (9, 9), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2860
|
math
|
trans_gridworld
|
test
|
trans_gridworld_test_015
|
trans_gridworld
|
[
{
"content": "In a 20x20 grid, how many different paths are there from the bottom left (0, 0) to the top right (19, 19), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), (10, 10), (11, 11), (12, 12), (13, 13), (14, 14), (15, 15), (16, 16), (17, 17), (18, 18)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
955277400
|
math
|
trans_gridworld
|
test
|
trans_circles_test_000
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. What is the distance between D_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2.41
|
math
|
trans_circles
|
test
|
trans_circles_test_001
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. What is the distance between A_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1.73
|
math
|
trans_circles
|
test
|
trans_circles_test_002
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously. What is the distance between D_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4.41
|
math
|
trans_circles
|
test
|
trans_circles_test_003
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously. What is the distance between A_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3.14
|
math
|
trans_circles
|
test
|
trans_circles_test_004
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct D_5. What is the distance between D_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
11.24
|
math
|
trans_circles
|
test
|
trans_circles_test_005
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct A_5. What is the distance between A_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8.2
|
math
|
trans_circles
|
test
|
trans_circles_test_006
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct D_10. What is the distance between D_10 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
74.84
|
math
|
trans_circles
|
test
|
trans_circles_test_007
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct D_10. What is the distance between A_10 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
56.91
|
math
|
trans_circles
|
test
|
trans_circles_test_008
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct D_20. What is the distance between D_20 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2469.74
|
math
|
trans_circles
|
test
|
trans_circles_test_009
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, drawing its horizontal and vertical diameters and constructing points A_3, B_3, and D_3 analogously, and so on until you construct D_20. What is the distance between A_20 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1889.88
|
math
|
trans_circles
|
test
|
trans_circles_test_010
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. What is the distance between D_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
trans_circles
|
test
|
trans_circles_test_011
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_2, and construct D_3 analogously. What is the distance between D_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
trans_circles
|
test
|
trans_circles_test_012
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_2, and construct D_3 analogously, and so on until you construct D_5. What is the distance between D_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
math
|
trans_circles
|
test
|
trans_circles_test_013
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_2, and construct D_3 analogously, and so on until you construct D_10. What is the distance between D_10 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
10
|
math
|
trans_circles
|
test
|
trans_circles_test_014
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_2, and construct D_3 analogously, and so on until you construct D_20. What is the distance between D_20 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
20
|
math
|
trans_circles
|
test
|
trans_circles_test_015
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. What is the distance between D_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
trans_circles
|
test
|
trans_circles_test_016
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_1, and construct D_3 analogously. What is the distance between D_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
trans_circles
|
test
|
trans_circles_test_017
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_1, and construct D_3 analogously, and so on until you construct D_5. What is the distance between D_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
16
|
math
|
trans_circles
|
test
|
trans_circles_test_018
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_1, and construct D_3 analogously, and so on until you construct D_10. What is the distance between D_10 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
512
|
math
|
trans_circles
|
test
|
trans_circles_test_019
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through the center of C_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through the center of C_1, and construct D_3 analogously, and so on until you construct D_20. What is the distance between D_20 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
524288
|
math
|
trans_circles
|
test
|
trans_circles_test_020
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. What is the distance between D_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2.61
|
math
|
trans_circles
|
test
|
trans_circles_test_021
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, and construct D_3 analogously. What is the distance between D_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5.12
|
math
|
trans_circles
|
test
|
trans_circles_test_022
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, and construct D_3 analogously, and so on until you construct D_5. What is the distance between D_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
20.38
|
math
|
trans_circles
|
test
|
trans_circles_test_023
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, and construct D_3 analogously, and so on until you construct D_10. What is the distance between D_10 and the center of C_1? Round your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
652
|
math
|
trans_circles
|
test
|
trans_circles_test_024
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, and construct D_3 analogously, and so on until you construct D_20. What is the distance between D_20 and the center of C_1? Round your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
667544
|
math
|
trans_circles
|
test
|
trans_circles_test_025
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter, but this time, call the lower endpoint D_2. What is the distance between D_2 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
trans_circles_test_026
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter, but this time, call the lower endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, again letting D_3 be the higher endpoint on C_2's vertical diameter. What is the distance between D_3 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1.58
|
math
|
trans_circles
|
test
|
trans_circles_test_027
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_5. What is the distance between D_5 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2.75
|
math
|
trans_circles
|
test
|
trans_circles_test_028
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_10. What is the distance between D_10 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12.84
|
math
|
trans_circles
|
test
|
trans_circles_test_029
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_20. What is the distance between D_20 and the center of C_1? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
423.74
|
math
|
trans_circles
|
test
|
trans_circles_test_030
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter, but this time, call the lower endpoint D_2. Among all the points constructed on the y-axis, what is the distance between the closest two points? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
trans_circles_test_031
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter, but this time, call the lower endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, again letting D_3 be the higher endpoint on C_2's vertical diameter. Among all the points constructed on the y-axis, what is the distance between the closest two points? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
trans_circles_test_032
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_5. Among all the points constructed on the y-axis, what is the distance between the closest two points? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
trans_circles_test_033
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_10. Among all the points constructed on the y-axis, what is the distance between the closest two points? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
trans_circles_test_034
|
trans_circles
|
[
{
"content": "Let circle C_1 be positioned in the coordinate plane with a radius of 1 and its center on the y-axis. Draw its horizontal diameter and call its endpoints A_1 and B_1. Draw its vertical diameter and call the higher endpoint D_1. Then, let circle C_2 be the circle centered at D_1 that passes through A_1 and B_1. Likewise, draw its horizontal diameter and call its endpoints A_1 and B_1, and draw its vertical diameter and call its higher endpoint D_2. Then, repeat this process, constructing a circle C_3 centered at D_2 that passes through A_2 and B_2, letting D_3 be the higher endpoint on C_3's vertical diameter. Repeat this pattern, letting D_n be the higher endpoint on C_n's vertical diameter when n is odd, and the lower endpoint when n is even, until you construct D_20. Among all the points constructed on the y-axis, what is the distance between the closest two points? Round your answer to the nearest two decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.41
|
math
|
trans_circles
|
test
|
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