Search is not available for this dataset
idx int64 0 20.8k | my_solu sequencelengths 1 1 | new_prompt stringlengths 4 5.74k | old_prompt stringlengths 104 4.5k | type stringclasses 9
values | solution stringlengths 15 11.7k | old_solution sequencelengths 1 1 |
|---|---|---|---|---|---|---|
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,827 | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: Adding all three equations together, we get $2(... | <bos><start_of_turn>user
Suppose $a, b, c$ are real numbers satisfying the system \begin{align*}
b + c & = 13, \\
c + a & = 14, \\
a + b & = 15.
\end{align*} Calculate the value of $\sqrt{abc(a+b+c)}$.
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | We can solve this system of equations by adding all three equations together. Doing so, we get:
\begin{align*}
(a+b) + (b+c) + (c+a) &= 15 + 13 + 14 \\
2(a+b+c) &= 42 \\
a+b+c &= 21.
\end{align*}
Now we can substitute this value of $a+b+c$ back into any of the original equations. Let's use $b+c=13$:
\begin{align*}
(... | [
"<bos><start_of_turn>user\nSuppose $a, b, c$ are real numbers satisfying the system \\begin{align*}\nb + c & = 13, \\\\\nc + a & = 14, \\\\\na + b & = 15.\n\\end{align*} Calculate the value of $\\sqrt{abc(a+b+c)}$.\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: Adding all three equations ... |
10,828 | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: The prime numbe... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve this problem, we need to check if switching the positions of the digits of a prime number between 12 and 99 results in another prime number.
Let's list all the prime numbers between 12 and 99:
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
For each prime, we switch the posit... | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... |
10,828 | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: The prime numbe... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve this problem, we need to check if switching the positions of the digits of a prime number between 12 and 99 results in another prime number.
Let's list all the prime numbers between 12 and 99:
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
For each prime, we switch the posit... | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... |
10,828 | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: The prime numbe... | <bos><start_of_turn>user
Consider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| gpt-3.5-turbo | To solve this problem, we need to check if switching the positions of the digits of a prime number between 12 and 99 results in another prime number.
Let's list all the prime numbers between 12 and 99:
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
For each prime, we switch the posit... | [
"<bos><start_of_turn>user\nConsider all the prime numbers with two digits that lie between 12 and 99. Which of these primes, if you switch the positions of the digits, still result in a prime number, and what is the sum of these primes?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The pr... |
10,829 | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | gpt-3.5-turbo | To find the coordinates of the midpoint of segment $P'R'$, we first need to find the coordinates of $P'$, $Q'$, and $R'$.
Let's translate each point of triangle $PQR$ to obtain triangle $P'Q'R'$.
The coordinates of $P'$ can be obtained by adding 4 to the x-coordinate and subtracting 3 from the y-coordinate of $P$:
$P... | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... |
10,829 | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | gpt-3.5-turbo | To find the coordinates of the midpoint of segment $P'R'$, we first need to find the coordinates of $P'$, $Q'$, and $R'$.
Let's translate each point of triangle $PQR$ to obtain triangle $P'Q'R'$.
The coordinates of $P'$ can be obtained by adding 4 to the x-coordinate and subtracting 3 from the y-coordinate of $P$:
$P... | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... |
10,829 | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | gpt-3.5-turbo | To find the coordinates of the midpoint of segment $P'R'$, we first need to find the coordinates of $P'$, $Q'$, and $R'$.
Let's translate each point of triangle $PQR$ to obtain triangle $P'Q'R'$.
The coordinates of $P'$ can be obtained by adding 4 to the x-coordinate and subtracting 3 from the y-coordinate of $P$:
$P... | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... |
10,829 | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | <bos><start_of_turn>user
In the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are the coo... | gpt-3.5-turbo | To find the coordinates of the midpoint of segment $P'R'$, we first need to find the coordinates of $P'$, $Q'$, and $R'$.
Let's translate each point of triangle $PQR$ to obtain triangle $P'Q'R'$.
The coordinates of $P'$ can be obtained by adding 4 to the x-coordinate and subtracting 3 from the y-coordinate of $P$:
$P... | [
"<bos><start_of_turn>user\nIn the coordinate plane, the points $P(3, 2)$, $Q(7, 5)$ and $R(3, 8)$ form triangle $PQR$. If triangle $PQR$ is translated 4 units to the right and 3 units down to create triangle $P'Q'R'$, where $P'$ is the image of $P$, $Q'$ is the image of $Q$, and $R'$ is the image of $R$, what are t... |
10,830 | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite $12! + 14!$ as $12!(1 + 13\cdot 14... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | Factor out $12!$ from both terms: $12!+14!=12!(1+13\cdot 14)=12!\cdot 183$. Factor $183=3\cdot 61$. Since $12!$ has no prime factors greater than 11, $\boxed{61}$ is the greatest prime factor of $12!+14!$. | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... |
10,830 | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite $12! + 14!$ as $12!(1 + 13\cdot 14... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | Factor out $12!$ from both terms: $12!+14!=12!(1+13\cdot 14)=12!\cdot 183$. Factor $183=3\cdot 61$. Since $12!$ has no prime factors greater than 11, $\boxed{61}$ is the greatest prime factor of $12!+14!$. | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... |
10,830 | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite $12! + 14!$ as $12!(1 + 13\cdot 14... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | Factor out $12!$ from both terms: $12!+14!=12!(1+13\cdot 14)=12!\cdot 183$. Factor $183=3\cdot 61$. Since $12!$ has no prime factors greater than 11, $\boxed{61}$ is the greatest prime factor of $12!+14!$. | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... |
10,830 | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite $12! + 14!$ as $12!(1 + 13\cdot 14... | <bos><start_of_turn>user
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | Factor out $12!$ from both terms: $12!+14!=12!(1+13\cdot 14)=12!\cdot 183$. Factor $183=3\cdot 61$. Since $12!$ has no prime factors greater than 11, $\boxed{61}$ is the greatest prime factor of $12!+14!$. | [
"<bos><start_of_turn>user\nWhat is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite $12! + 14!$ as $12!... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,831 | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: To solve this problem, we need to determi... | <bos><start_of_turn>user
If x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| GSM_SV | To solve this problem, we need to determine the value of x, which represents the number of carpenters needed to make 50 chairs in 10 days.
We know that 12 carpenters are needed to make 75 chairs in 10 days.
Let's set up the equation using the concept of proportionality:
Number of carpenters / Number of chairs = Number ... | [
"<bos><start_of_turn>user\nIf x carpenters can make 50 chairs in 10 days, 12 carpenters are needed to make 75 chairs in 10 days. What is the value of unknown variable x? What is the value of unknown variable x?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: To solve this problem, we need t... |
10,832 | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square, we have $-5(r^2-8r+16)+80-12= -5(r-4)^2+68$. ки\nStep... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки
| <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | We complete the square: \begin{align*}
-5r^2 + 40r - 12 & = (-5r^2 + 40r) - 12\\
&= -5(r^2 - 8r + 16) -12 + 5 \cdot 16\\
&= -5(r - 4)^2 + 68
\end{align*} The maximum value of $-5(r-4)^2$ is $0$, since the square of a real number is never negative. Thus, the maximum value of the expression is $\boxed{68}$. | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... |
10,832 | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки
Step 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ки
| <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | We complete the square: \begin{align*}
-5r^2 + 40r - 12 & = (-5r^2 + 40r) - 12\\
&= -5(r^2 - 8r + 16) -12 + 5 \cdot 16\\
&= -5(r - 4)^2 + 68
\end{align*} The maximum value of $-5(r-4)^2$ is $0$, since the square of a real number is never negative. Thus, the maximum value of the expression is $\boxed{68}$. | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... |
10,832 | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки
Step 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ки
Step 3:... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | We complete the square: \begin{align*}
-5r^2 + 40r - 12 & = (-5r^2 + 40r) - 12\\
&= -5(r^2 - 8r + 16) -12 + 5 \cdot 16\\
&= -5(r - 4)^2 + 68
\end{align*} The maximum value of $-5(r-4)^2$ is $0$, since the square of a real number is never negative. Thus, the maximum value of the expression is $\boxed{68}$. | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... |
10,832 | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки
Step 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ки
Step 3:... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | We complete the square: \begin{align*}
-5r^2 + 40r - 12 & = (-5r^2 + 40r) - 12\\
&= -5(r^2 - 8r + 16) -12 + 5 \cdot 16\\
&= -5(r - 4)^2 + 68
\end{align*} The maximum value of $-5(r-4)^2$ is $0$, since the square of a real number is never negative. Thus, the maximum value of the expression is $\boxed{68}$. | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... |
10,832 | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки
Step 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ки
Step 3:... | <bos><start_of_turn>user
What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | We complete the square: \begin{align*}
-5r^2 + 40r - 12 & = (-5r^2 + 40r) - 12\\
&= -5(r^2 - 8r + 16) -12 + 5 \cdot 16\\
&= -5(r - 4)^2 + 68
\end{align*} The maximum value of $-5(r-4)^2$ is $0$, since the square of a real number is never negative. Thus, the maximum value of the expression is $\boxed{68}$. | [
"<bos><start_of_turn>user\nWhat is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: We can rewrite the expression as $-5(r^2-8r+2.4)$. ки\nStep 2: Completing the square inside the parentheses, we have $-5((r-4)^2 - 12.6)$. ... |
10,833 | [
"<bos><start_of_turn>user\nGiselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The ratio of green paint to whi... | <bos><start_of_turn>user
Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: The ratio of green paint to white paint ... | <bos><start_of_turn>user
Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | The ratio of white paint to green paint is $4:2$ which simplifies to $2:1$, so Giselle should use twice as much white paint as green paint. Since she uses $12$ quarts of white paint, she should use $12 \div 2 = \boxed{6}$ quarts of green paint. | [
"<bos><start_of_turn>user\nGiselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The ratio of green paint to whi... |
10,833 | [
"<bos><start_of_turn>user\nGiselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The ratio of green paint to whi... | <bos><start_of_turn>user
Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?
Please reason step by step.<end_of_turn>
<start_of_turn>model
Step 1: The ratio of green paint to white paint ... | <bos><start_of_turn>user
Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?
Please reason step by step.<end_of_turn>
<start_of_turn>model
| math | The ratio of white paint to green paint is $4:2$ which simplifies to $2:1$, so Giselle should use twice as much white paint as green paint. Since she uses $12$ quarts of white paint, she should use $12 \div 2 = \boxed{6}$ quarts of green paint. | [
"<bos><start_of_turn>user\nGiselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?\nPlease reason step by step.<end_of_turn>\n<start_of_turn>model\nStep 1: The ratio of green paint to whi... |
10,834 | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | gpt-3.5-turbo | To find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive integer $n,$ we need to determine the fixed points of the functions $h, h \circ h, h \circ h \circ h, \ldots.$
Let's examine the first few functions in the sequence, starting with $h(x):$
\[h(x) = \frac{3x + ... | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... |
10,834 | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | gpt-3.5-turbo | To find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive integer $n,$ we need to determine the fixed points of the functions $h, h \circ h, h \circ h \circ h, \ldots.$
Let's examine the first few functions in the sequence, starting with $h(x):$
\[h(x) = \frac{3x + ... | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... |
10,834 | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | gpt-3.5-turbo | To find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive integer $n,$ we need to determine the fixed points of the functions $h, h \circ h, h \circ h \circ h, \ldots.$
Let's examine the first few functions in the sequence, starting with $h(x):$
\[h(x) = \frac{3x + ... | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... |
10,834 | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | <bos><start_of_turn>user
Let $h(x) = \frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \circ h_{n - 1}$ for all $n \ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive inte... | gpt-3.5-turbo | To find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some positive integer $n,$ we need to determine the fixed points of the functions $h, h \circ h, h \circ h \circ h, \ldots.$
Let's examine the first few functions in the sequence, starting with $h(x):$
\[h(x) = \frac{3x + ... | [
"<bos><start_of_turn>user\nLet $h(x) = \\frac{3x + 9}{x}.$ Consider the sequence $(h_n)$ of functions where $h_1 = h$ and $h_n = h \\circ h_{n - 1}$ for all $n \\ge 2.$ For instance, $h_2(x) = h(h(x))$ and so on. Find the number of elements in the set $S$ of all real numbers $x$ such that $h_n(x) = x$ for some posi... |
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