idx int64 0 14.6k | gt stringlengths 1 101 | prompt stringlengths 663 9.69k | res bool 2
classes | predic bool 1
class | my_solu sequencelengths 1 1 | pred sequencelengths 1 1 |
|---|---|---|---|---|---|---|
0 | 4 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\pi$ and the volume of the cylinder is $3\pi r^3$. Find the value of $r$.
Please reason step by step, and put ... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\\pi$ and the volume of the cylinder is $3\\pi r^3$. Find the value of $r$.\nPlease reason step by step... | [
""
] |
1 | 4 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\pi$ and the volume of the cylinder is $3\pi r^3$. Find the value of $r$.
Please reason step by step, and put ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\\pi$ and the volume of the cylinder is $3\\pi r^3$. Find the value of $r$.\nPlease reason step by step... | [
""
] |
2 | 4 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\pi$ and the volume of the cylinder is $3\pi r^3$. Find the value of $r$.
Please reason step by step, and put ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA cone and a cylinder share the same base with radius $r$. The height of the cone is equal to the height of the cylinder. The volume of the cone is 32$\\pi$ and the volume of the cylinder is $3\\pi r^3$. Find the value of $r$.\nPlease reason step by step... | [
""
] |
3 | (1, 0) | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $P$ be a point on the line segment $\overline{AB}$ of $\triangle ABC$, with $A(0,0)$, $B(3,0)$, and $C(1,4)$. Determine the coordinates of $P$ such that the circumradius of $\triangle PBC$ is minimum.
Please reason step by step, and put your final answer wi... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nLet $P$ be a point on the line segment $\\overline{AB}$ of $\\triangle ABC$, with $A(0,0)$, $B(3,0)$, and $C(1,4)$. Determine the coordinates of $P$ such that the circumradius of $\\triangle PBC$ is minimum.\nPlease reason step by step, and put your fina... | [
""
] |
4 | (1, 0) | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $P$ be a point on the line segment $\overline{AB}$ of $\triangle ABC$, with $A(0,0)$, $B(3,0)$, and $C(1,4)$. Determine the coordinates of $P$ such that the circumradius of $\triangle PBC$ is minimum.
Please reason step by step, and put your final answer wi... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nLet $P$ be a point on the line segment $\\overline{AB}$ of $\\triangle ABC$, with $A(0,0)$, $B(3,0)$, and $C(1,4)$. Determine the coordinates of $P$ such that the circumradius of $\\triangle PBC$ is minimum.\nPlease reason step by step, and put your fina... | [
""
] |
5 | -\frac{1}{2}, -\frac{1}{9}, 1 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|st... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot... | [
""
] |
6 | -\frac{1}{2}, -\frac{1}{9}, 1 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|st... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot... | [
""
] |
7 | -\frac{1}{2}, -\frac{1}{9}, 1 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|st... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat are the possible values of $a$ for which the equation $ax^3+2ax^2+ax+2=0$ has at least one rational root? Express your answers separated by commas and in increasing order.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot... | [
""
] |
8 | 255 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
If... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header... | [
""
] |
9 | 255 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
Le... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header... | [
""
] |
10 | 255 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
A ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the smallest positive integer that has exactly three distinct prime factors, and whose sum of prime factors is equal to 25?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header... | [
""
] |
11 | 384 - 72\pi | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The square pyramid has a base side length of 12 cm and a height of 8 cm. The top square has a side length of 6 cm and is perpendicular to the base. What is the volume of the remaining solid, in $\text{cm}^3$, after a cylinder with a height of 8 cm and a radius ... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe square pyramid has a base side length of 12 cm and a height of 8 cm. The top square has a side length of 6 cm and is perpendicular to the base. What is the volume of the remaining solid, in $\\text{cm}^3$, after a cylinder with a height of 8 cm and a... | [
""
] |
12 | 384 - 72\pi | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The square pyramid has a base side length of 12 cm and a height of 8 cm. The top square has a side length of 6 cm and is perpendicular to the base. What is the volume of the remaining solid, in $\text{cm}^3$, after a cylinder with a height of 8 cm and a radius ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe square pyramid has a base side length of 12 cm and a height of 8 cm. The top square has a side length of 6 cm and is perpendicular to the base. What is the volume of the remaining solid, in $\\text{cm}^3$, after a cylinder with a height of 8 cm and a... | [
""
] |
13 | 71 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs will win th... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs wil... | [
""
] |
14 | 71 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs will win th... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs wil... | [
""
] |
15 | 71 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs will win th... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs wil... | [
""
] |
16 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in base $8$, s... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nConsider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in bas... | [
""
] |
17 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in base $8$, s... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nConsider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in bas... | [
""
] |
18 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in base $8$, s... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nConsider a number $N$ written in base $b$. Define the function $f(N)$ to be the number of digits in the representation of $N$ in base $b$ that are greater than $\\frac{b}{2}$. For example, if $N = 17$ and $b = 8$, then $N$ is represented as $21_8$ in bas... | [
""
] |
19 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
On the library wall, there is a sign that reads "BOOKS" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?
Please reason step ... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nOn the library wall, there is a sign that reads \"BOOKS\" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?\nPlease re... | [
""
] |
20 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
On the library wall, there is a sign that reads "BOOKS" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?
Please reason step ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nOn the library wall, there is a sign that reads \"BOOKS\" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?\nPlease re... | [
""
] |
21 | 3 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
On the library wall, there is a sign that reads "BOOKS" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?
Please reason step ... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nOn the library wall, there is a sign that reads \"BOOKS\" with 5 letters, each on a separate plaque. If two vowels and one consonant are removed and placed in a box, how many distinct possible collections of letters could be placed in the box?\nPlease re... | [
""
] |
22 | 20.0 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A semicircle of diameter 7 cm is inscribed in a rectangle with two opposite vertices at the ends of the diameter. The rectangle is itself inscribed in a semicircle of diameter 10 cm. What is the area of the region between the two semicircles, to the nearest ten... | true | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA semicircle of diameter 7 cm is inscribed in a rectangle with two opposite vertices at the ends of the diameter. The rectangle is itself inscribed in a semicircle of diameter 10 cm. What is the area of the region between the two semicircles, to the near... | [
""
] |
23 | 20.0 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A semicircle of diameter 7 cm is inscribed in a rectangle with two opposite vertices at the ends of the diameter. The rectangle is itself inscribed in a semicircle of diameter 10 cm. What is the area of the region between the two semicircles, to the nearest ten... | false | false | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA semicircle of diameter 7 cm is inscribed in a rectangle with two opposite vertices at the ends of the diameter. The rectangle is itself inscribed in a semicircle of diameter 10 cm. What is the area of the region between the two semicircles, to the near... | [
""
] |
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