<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_66><Z_407><Z_477><Z_339><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_252><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?<|im_end|>
<|im_start|>assistant
<Z_323><Z_321><Z_192><Z_119><Z_347><ANSWER>

Question: The number  $r$  can be expressed as a four-place decimal  $0.abcd,$  where  $a, b, c,$  and  $d$  represent digits, any of which could be zero.  It is desired to approximate  $r$  by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to  $r$  is  $\frac 27.$   What is the number of possible values for  $r$ ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_358><Z_191><Z_391><Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_217><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____<|im_end|>
<|im_start|>assistant
<Z_83><Z_64><Z_194><Z_196><ANSWER>

Question: $a$ and $b$ are opposites of each other, then $2006a+2+2006b=$ _____


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_5><Z_477><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_373><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?<|im_end|>
<|im_start|>assistant
<Z_87><Z_349><Z_219><ANSWER>

Question: A person has workers and money. If he gives each worker 5 coins, he will have 30 coins left, and if he gives each worker 7 coins, he will be short 30 coins. How many workers does he have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_176><Z_256><Z_157><Z_61><Z_311><Z_322><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_149><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_498><Z_79><Z_242><Z_68><Z_371><Z_373><ANSWER>

Question: Suppose $d \neq 0$. Calculate the simplified form of $\left(15d + 16 + 17d^2\right) + \left(3d + 2\right)$, and express it as $ad + b + cd^2$ where $a$, $b$, and $c$ are integers. Then find $a+b+c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_176><Z_237><Z_76><Z_489><Z_202><Z_110><Z_425><Z_354><Z_244><Z_52><Z_216><Z_427><Z_51><Z_244><Z_135><Z_350><Z_280><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.<|im_end|>
<|im_start|>assistant
<Z_207><Z_323><Z_392><Z_455><Z_153><Z_77><Z_91><Z_467><Z_38><Z_181><Z_263><Z_144><Z_35><Z_437><Z_70><Z_382><Z_167><ANSWER>

Question:
Given a sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. For all $n \in \mathbb{N}^*$, the point $\left(n, \frac{S\_n}{n}\right)$ lies on the graph of the function $f(x) = x + \frac{a\_n}{2x}$.

1. Find the values of $a\_1$, $a\_2$, and $a\_3$. Conjecture an expression for $a\_n$ and prove it using mathematical induction.
2. Divide the sequence $\{a\_n\}$ consecutively into groups of 1, 2, 3, and 4 terms, starting from the first term. Compute the sum of the terms within each group and form a new sequence $\{b\_n\}$ using these sums, maintaining the original order of the groups. Find the value of $b\_{2018} - b\_{1314}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_176><Z_187><Z_375><Z_385><Z_432><Z_206><Z_232><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.<|im_end|>
<|im_start|>assistant
<Z_157><Z_510><Z_101><Z_496><Z_247><Z_100><Z_124><ANSWER>

Question: Suppose sets $A = \{1, a, b\}$ and $B = \{a, a^2, ab\}$ are equal, that is $A = B$. Determine the real numbers $a$ and $b$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_323><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?<|im_end|>
<|im_start|>assistant
<Z_361><ANSWER>

Question: tom drives from town w to town b , driving at a constant speed of 60 miles per hour . from town b tom immediately continues to town c . the distance between w and b is twice the distance between b and c . if the average speed of the whole journey was 36 mph , then what is tom ' s speed driving from b to c in miles per hour ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_66><Z_427><Z_254><Z_199><Z_405><Z_510><Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?<|im_end|>
<|im_start|>assistant
<Z_366><Z_61><Z_277><Z_100><Z_336><Z_31><Z_71><ANSWER>

Question:
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Mean Streak & 138 feet \\ \hline
The Tower of Power & 300 feet \\ \hline
The Maximum Ride & 198 feet \\ \hline
\end{tabular} What is the positive difference between the mean and the median of these values?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|im_end|>
<|im_start|>assistant
<Z_176><Z_499><Z_429><Z_256><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|im_end|>
<|im_start|>assistant
<Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.<|im_end|>
<|im_start|>assistant
<Z_137><Z_509><Z_438><Z_140><ANSWER>

Question: Calculate $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_72><Z_338><Z_318><Z_394><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_135><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 2010 for which the sum of the digits is 10?<|im_end|>
<|im_start|>assistant
<Z_162><Z_185><Z_324><Z_85><ANSWER>

Question: What is the first year after 2010 for which the sum of the digits is 10?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

Question: A person can row at 10 kmph in still water. If the velocity of the current is 2 kmph and it takes him a certain amount of time to row to a place and come back, which is 96 km away. How long does it take him to row to the place and come back?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_358><Z_152><Z_181><Z_406><Z_223><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_187><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?<|im_end|>
<|im_start|>assistant
<Z_41><Z_348><Z_300><Z_406><Z_381><ANSWER>

Question: If \( f(x) = x^2 \) and \( g(x) = 3x^2 + 4 \), what is \( f(g(2)) \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_195><Z_293><Z_162><Z_150><Z_73><Z_390><Z_415><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_370><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?<|im_end|>
<|im_start|>assistant
<Z_200><Z_510><Z_321><Z_275><Z_431><Z_298><Z_258><ANSWER>

Question: Point \(A\) is at \((0, 0)\) and point \(B\) is on the line \(y = 6\). The slope of segment \(AB\) is \(\frac{3}{4}\). What is the sum of the \(x\)- and \(y\)-coordinates of point \(B\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_176><Z_152><Z_167><Z_270><Z_511><Z_156><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?<|im_end|>
<|im_start|>assistant
<Z_273><Z_341><Z_412><Z_174><Z_184><Z_287><Z_401><ANSWER>

Question: A boat can travel with a certain speed in still water. If the rate of the stream is 5 km/hr, then the boat takes 8 hours to cover a distance of 168 km downstream. What is the speed of the boat in still water?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_302><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?<|im_end|>
<|im_start|>assistant
<Z_202><ANSWER>

Question: What is the area of a triangle whose hypotenuse is 15 centimeters (cm) and the base is 9 centimeters (cm)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_172><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_130><ANSWER>

Question: Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_195><Z_287><Z_409><Z_485><Z_320><Z_335><Z_73><Z_278><Z_453><Z_128><Z_260><Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_213><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?<|im_end|>
<|im_start|>assistant
<Z_384><Z_390><Z_284><Z_323><Z_432><Z_138><Z_101><Z_374><Z_98><Z_439><Z_9><Z_274><ANSWER>

Question:
Joãozinho wrote the numbers from 1 to 100000 on the board, then replaced each number by the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234, we get \(7+2+3+4=16\) and \(1+6=7\).

(a) What number replaced the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_5><Z_461><Z_116><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_345><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?<|im_end|>
<|im_start|>assistant
<Z_488><Z_362><Z_358><ANSWER>

Question: Acme T-Shirt Company now charges a $\$70$ set-up fee plus $\$11$ for each shirt printed. Beta T-Shirt Company has introduced a set-up fee of $\$10$, but charges $\$15$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_176><Z_293><Z_162><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_508><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?<|im_end|>
<|im_start|>assistant
<Z_41><Z_369><Z_259><Z_140><ANSWER>

Question: Cynthia harvested 67 potatoes from her garden. After washing them, she cut 13 of them into wedges. She then halved the remaining potatoes and made french fries with one half, and potato chips with the other half. If one potato can be cut into 8 wedges or make 20 potato chips, how many more potato chips than wedges did Cynthia make?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_334><Z_310><Z_507><Z_49><Z_167><Z_488><Z_244><Z_234><Z_404><Z_256><Z_142><Z_379><Z_287><Z_180><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the smallest positive integer whose cube ends in $392$.<|im_end|>
<|im_start|>assistant
<Z_38><Z_273><Z_376><Z_415><Z_342><Z_508><Z_49><Z_358><Z_347><Z_422><Z_483><Z_123><Z_436><Z_115><Z_441><Z_236><ANSWER>

Question: Find the smallest positive integer whose cube ends in $392$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_5><Z_235><Z_162><Z_280><Z_255><Z_382><Z_478><Z_392><Z_126><Z_431><Z_61><Z_229><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_381><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest two-digit whole number, the product of whose digits is 12?<|im_end|>
<|im_start|>assistant
<Z_306><Z_202><Z_364><Z_460><Z_463><Z_10><Z_156><Z_13><Z_99><Z_48><Z_487><Z_48><ANSWER>

Question: What is the greatest two-digit whole number, the product of whose digits is 12?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_66><Z_452><Z_301><Z_122><Z_160><Z_425><Z_232><Z_386><Z_498><Z_232><Z_300><Z_293><Z_476><Z_300><Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_326><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .<|im_end|>
<|im_start|>assistant
<Z_369><Z_420><Z_74><Z_448><Z_169><Z_95><Z_232><Z_229><Z_263><Z_250><Z_455><Z_362><Z_428><Z_260><Z_56><ANSWER>

Question: the cost of the paint is rs . 36.50 per kg . if 1 kg of paint covers 16 square feet , how much will it cost to paint outside of a cube having 8 feet each side .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_176><Z_125><Z_498><Z_189><Z_459><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_511><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?<|im_end|>
<|im_start|>assistant
<Z_220><Z_319><Z_508><Z_504><Z_14><ANSWER>

Question: A man can row his boat with the stream at some speed and against the stream at 10 km/h. The man's rate is 6 km/h. What is his speed with the stream?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_5><Z_477><Z_413><Z_510><Z_265><Z_423><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?<|im_end|>
<|im_start|>assistant
<Z_503><Z_284><Z_129><Z_239><Z_304><Z_287><ANSWER>

Question: Three people need three weeks and three days to complete a task. How many days does it take for four people to complete the same task?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_176><Z_34><Z_125><Z_287><Z_376><Z_132><Z_379><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?<|im_end|>
<|im_start|>assistant
<Z_309><Z_213><Z_26><Z_203><Z_163><Z_3><Z_72><ANSWER>

Question:
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_176><Z_307><Z_124><Z_311><Z_127><Z_346><Z_463><Z_497><Z_181><Z_353><Z_391><Z_503><Z_55><Z_152><Z_450><Z_177><Z_433><Z_354><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_491><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?<|im_end|>
<|im_start|>assistant
<Z_200><Z_168><Z_360><Z_145><Z_431><Z_155><Z_460><Z_118><Z_92><Z_477><Z_179><Z_11><Z_466><Z_365><Z_294><Z_193><Z_200><Z_26><Z_219><ANSWER>

Question: The average score of a cricketer in 10 matches is 60, and in other 15 matches is 70. What is the average score in all the 25 matches?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|im_end|>
<|im_start|>assistant
<Z_176><Z_251><Z_354><Z_415><Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|im_end|>
<|im_start|>assistant
<Z_3><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.<|im_end|>
<|im_start|>assistant
<Z_509><Z_389><Z_403><Z_290><Z_285><ANSWER>

Question: Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_195><Z_484><Z_416><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_504><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many three-digit whole numbers have no 5's and no 8's as digits?<|im_end|>
<|im_start|>assistant
<Z_461><Z_304><Z_297><ANSWER>

Question: How many three-digit whole numbers have no 5's and no 8's as digits?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_176><Z_405><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?<|im_end|>
<|im_start|>assistant
<Z_345><Z_438><ANSWER>

Question: A wheel A rotates 20 times every minute and moves 35 cm during each rotation, while another wheel B rotates 30 times every minute and moves 50 cm during each rotation. How many meters does the combined distance of wheel A and wheel B move in one hour?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_176><Z_315><Z_231><Z_300><Z_188><Z_446><Z_484><Z_160><Z_200><Z_484><Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_363><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?<|im_end|>
<|im_start|>assistant
<Z_336><Z_225><Z_81><Z_257><Z_301><Z_264><Z_216><Z_197><Z_271><Z_159><Z_320><ANSWER>

Question: Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, how many fish did they catch altogether if Chris went fishing 10 times?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_5><Z_353><Z_61><Z_434><Z_100><Z_267><Z_311><Z_91><Z_499><Z_107><Z_429><Z_103><Z_244><Z_354><Z_277><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_139><Z_260><Z_96><Z_295><Z_204><Z_12><Z_285><Z_74><Z_476><Z_64><Z_285><Z_20><Z_285><Z_461><Z_68><ANSWER>

Question: Dividing a three-digit number with 9 in the hundreds place is divisible by 7. How many of these three digit numbers are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_358><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?<|im_end|>
<|im_start|>assistant
<Z_27><Z_443><ANSWER>

Question: A jar on the family's counter contains change they've been saving for a trip to the ice cream shop. There are some pennies, 85 nickels, 35 dimes, and 26 quarters. All five family members get a double scoop, which costs $3 each. After the trip, they have 48 cents left over. How many pennies were in the jar?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|im_end|>
<|im_start|>assistant
<Z_313><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?<|im_end|>
<|im_start|>assistant
<Z_16><ANSWER>

Question: Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_66><Z_316><Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?<|im_end|>
<|im_start|>assistant
<Z_29><Z_372><Z_295><ANSWER>

Question:
In the picture, several circles are connected by line segments. Kostya selects a natural number \( n \) and places different natural numbers in the circles such that the following property holds:

If the numbers \( a \) and \( b \) are not connected by a line segment, then their sum \( a+b \) must be coprime with \( n \). However, if they are connected by a line segment, the sum \( a+b \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_66><Z_96><Z_106><Z_500><Z_256><Z_317><Z_276><Z_158><Z_334><Z_307><Z_430><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_486><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)<|im_end|>
<|im_start|>assistant
<Z_427><Z_475><Z_109><Z_139><Z_125><Z_276><Z_352><Z_354><Z_14><Z_305><Z_43><ANSWER>

Question: How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_364><Z_483><Z_447><Z_273><Z_490><Z_320><Z_270><Z_329><Z_165><Z_435><Z_342><Z_354><Z_323><Z_456><Z_77><Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.<|im_end|>
<|im_start|>assistant
<Z_174><Z_235><Z_177><Z_404><Z_466><Z_49><Z_226><Z_82><Z_122><Z_438><Z_323><Z_176><Z_493><Z_54><Z_87><Z_217><Z_27><ANSWER>

Question: Imagine cutting a right triangle $BFC$ with legs of length 12 inches from square $ABCD$. Place the triangle on the left side of the square to create a new shape $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. Find the perimeter, in inches, of the newly formed figure $ABFCDE$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_358><Z_271><Z_293><Z_396><Z_353><Z_156><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?<|im_end|>
<|im_start|>assistant
<Z_179><Z_409><Z_128><Z_297><Z_200><Z_305><ANSWER>

Question: Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_195><Z_83><Z_80><Z_473><Z_180><Z_353><Z_299><Z_489><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_7><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.<|im_end|>
<|im_start|>assistant
<Z_120><Z_17><Z_347><Z_405><Z_390><Z_167><Z_275><Z_32><ANSWER>

Question: Determine the largest value of $m$ such that $3x^2 + mx - 60$ can be factored as the product of two linear factors with integer coefficients.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_195><Z_181><Z_395><Z_236><Z_503><Z_68><Z_494><Z_202><Z_83><Z_399><Z_201><Z_326><Z_394><Z_389><Z_273><Z_441><Z_333><Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_241><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?<|im_end|>
<|im_start|>assistant
<Z_258><Z_370><Z_405><Z_245><Z_84><Z_376><Z_7><Z_185><Z_174><Z_111><Z_241><Z_152><Z_380><Z_38><Z_431><Z_251><Z_101><Z_296><ANSWER>

Question:
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 10 \le 0.\]
How many integer values of $b$ are possible?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_417><Z_102><Z_408><Z_84><Z_383><Z_162><Z_193><Z_121><Z_435><Z_379><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.<|im_end|>
<|im_start|>assistant
<Z_494><Z_371><Z_167><Z_401><Z_445><Z_395><Z_349><Z_19><Z_360><Z_334><Z_358><ANSWER>

Question: Find the sum of the rational roots of $f(x) = x^3 - 8x^2 + 13x - 6$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_5><Z_457><Z_477><Z_411><Z_389><Z_265><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_360><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.<|im_end|>
<|im_start|>assistant
<Z_410><Z_244><Z_325><Z_76><Z_370><Z_11><ANSWER>

Question: Find the units digit of \(n\) given that \(ab = 34^8\) and \(a\) has a units digit of 9.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_358><Z_473><Z_342><Z_361><Z_133><Z_180><Z_313><Z_498><Z_57><Z_351><Z_429><Z_267><Z_461><Z_275><Z_186><Z_42><Z_268><Z_342><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?<|im_end|>
<|im_start|>assistant
<Z_423><Z_178><Z_314><Z_383><Z_302><Z_453><Z_231><Z_101><Z_264><Z_229><Z_295><Z_357><Z_380><Z_30><Z_196><Z_428><Z_462><Z_215><ANSWER>

Question: A can do a work in 12 days. When he had worked for 3 days, B joined him. They completed the work in some more days. B alone can finish the work in 6 days. How many more days did they work together to complete the work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_5><Z_219><Z_480><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$<|im_end|>
<|im_start|>assistant
<Z_94><Z_362><Z_510><ANSWER>

Question:
For which natural numbers \( n \) is it true that

$$
\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \ldots \cdot \log _{n}(n+1)=10?
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_66><Z_234><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_348><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.<|im_end|>
<|im_start|>assistant
<Z_314><Z_251><ANSWER>

Question: Find the constant term in the expansion of $(2x+ \frac {1}{x})^{6}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_66><Z_490><Z_118><Z_254><Z_467><Z_471><Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?<|im_end|>
<|im_start|>assistant
<Z_75><Z_363><Z_177><Z_301><Z_173><Z_292><Z_158><ANSWER>

Question: If the product of 6 integers has a certain sign, at most how many of the integers can have the opposite sign?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]<|im_end|>
<|im_start|>assistant
<Z_176><Z_345><Z_404><Z_316><Z_312><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]<|im_end|>
<|im_start|>assistant
<Z_461><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]<|im_end|>
<|im_start|>assistant
<Z_173><Z_327><Z_27><Z_110><Z_378><ANSWER>

Question:
Find the sum of the infinite series

\[
\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+\ldots
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Fully simplify the following expression: $[(3+4+5+6)^2\div4] + [(3\cdot6+9)^2\div3]$.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Fully simplify the following expression: $[(3+4+5+6)^2\div4] + [(3\cdot6+9)^2\div3]$.<|im_end|>
<|im_start|>assistant
<Z_358><Z_60><Z_462><Z_319><Z_259><Z_184><Z_469><Z_373><Z_70><Z_232><Z_98><Z_465><Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Fully simplify the following expression: $[(3+4+5+6)^2\div4] + [(3\cdot6+9)^2\div3]$.<|im_end|>
<|im_start|>assistant
<Z_182><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Fully simplify the following expression: $[(3+4+5+6)^2\div4] + [(3\cdot6+9)^2\div3]$.<|im_end|>
<|im_start|>assistant
<Z_405><Z_457><Z_9><Z_354><Z_28><Z_467><Z_450><Z_511><Z_491><Z_100><Z_127><Z_18><Z_298><ANSWER>

Question: Fully simplify the following expression: $[(3+4+5+6)^2\div4] + [(3\cdot6+9)^2\div3]$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a Zoo, there are different animals. There are 5 giraffes and twice as many penguins. Penguins make up 20% of all the animals in the Zoo. How many elephants are there in the Zoo if they make up 4% of all the animals?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a Zoo, there are different animals. There are 5 giraffes and twice as many penguins. Penguins make up 20% of all the animals in the Zoo. How many elephants are there in the Zoo if they make up 4% of all the animals?<|im_end|>
<|im_start|>assistant
<Z_176><Z_489><Z_81><Z_280><Z_313><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a Zoo, there are different animals. There are 5 giraffes and twice as many penguins. Penguins make up 20% of all the animals in the Zoo. How many elephants are there in the Zoo if they make up 4% of all the animals?<|im_end|>
<|im_start|>assistant
<Z_305><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a Zoo, there are different animals. There are 5 giraffes and twice as many penguins. Penguins make up 20% of all the animals in the Zoo. How many elephants are there in the Zoo if they make up 4% of all the animals?<|im_end|>
<|im_start|>assistant
<Z_235><Z_122><Z_201><Z_81><Z_501><ANSWER>

Question: In a Zoo, there are different animals. There are 5 giraffes and twice as many penguins. Penguins make up 20% of all the animals in the Zoo. How many elephants are there in the Zoo if they make up 4% of all the animals?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A two-digit integer $AB$ equals $\frac{1}{7}$ of the four-digit integer $AAAB$, where $A$, $B$, and $C$ represent distinct digits from 0 to 9. What is the smallest possible value of the four-digit integer $AAAB$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A two-digit integer $AB$ equals $\frac{1}{7}$ of the four-digit integer $AAAB$, where $A$, $B$, and $C$ represent distinct digits from 0 to 9. What is the smallest possible value of the four-digit integer $AAAB$?<|im_end|>
<|im_start|>assistant
<Z_195><Z_206><Z_473><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A two-digit integer $AB$ equals $\frac{1}{7}$ of the four-digit integer $AAAB$, where $A$, $B$, and $C$ represent distinct digits from 0 to 9. What is the smallest possible value of the four-digit integer $AAAB$?<|im_end|>
<|im_start|>assistant
<Z_230><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A two-digit integer $AB$ equals $\frac{1}{7}$ of the four-digit integer $AAAB$, where $A$, $B$, and $C$ represent distinct digits from 0 to 9. What is the smallest possible value of the four-digit integer $AAAB$?<|im_end|>
<|im_start|>assistant
<Z_98><Z_81><Z_233><ANSWER>

Question: A two-digit integer $AB$ equals $\frac{1}{7}$ of the four-digit integer $AAAB$, where $A$, $B$, and $C$ represent distinct digits from 0 to 9. What is the smallest possible value of the four-digit integer $AAAB$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Robin was making baggies of cookies with 6 cookies in each bag. She had 23 chocolate chip cookies and some oatmeal cookies. She could make 8 baggies. How many oatmeal cookies did she have?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Robin was making baggies of cookies with 6 cookies in each bag. She had 23 chocolate chip cookies and some oatmeal cookies. She could make 8 baggies. How many oatmeal cookies did she have?<|im_end|>
<|im_start|>assistant
<Z_66><Z_281><Z_408><Z_287><Z_278><Z_473><Z_263><Z_183><Z_328><Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Robin was making baggies of cookies with 6 cookies in each bag. She had 23 chocolate chip cookies and some oatmeal cookies. She could make 8 baggies. How many oatmeal cookies did she have?<|im_end|>
<|im_start|>assistant
<Z_505><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Robin was making baggies of cookies with 6 cookies in each bag. She had 23 chocolate chip cookies and some oatmeal cookies. She could make 8 baggies. How many oatmeal cookies did she have?<|im_end|>
<|im_start|>assistant
<Z_172><Z_64><Z_70><Z_108><Z_506><Z_51><Z_314><Z_279><Z_312><Z_361><ANSWER>

Question: Robin was making baggies of cookies with 6 cookies in each bag. She had 23 chocolate chip cookies and some oatmeal cookies. She could make 8 baggies. How many oatmeal cookies did she have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $100$ and $200$ have three different digits in decreasing order?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $100$ and $200$ have three different digits in decreasing order?<|im_end|>
<|im_start|>assistant
<Z_195><Z_243><Z_81><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $100$ and $200$ have three different digits in decreasing order?<|im_end|>
<|im_start|>assistant
<Z_466><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $100$ and $200$ have three different digits in decreasing order?<|im_end|>
<|im_start|>assistant
<Z_42><Z_220><Z_424><Z_131><ANSWER>

Question: How many integers between $100$ and $200$ have three different digits in decreasing order?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the town carnival, Billy rode the ferris wheel 7 times and the bumper cars 3 times. Each ride cost 5 tickets. How many tickets did he use?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the town carnival, Billy rode the ferris wheel 7 times and the bumper cars 3 times. Each ride cost 5 tickets. How many tickets did he use?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the town carnival, Billy rode the ferris wheel 7 times and the bumper cars 3 times. Each ride cost 5 tickets. How many tickets did he use?<|im_end|>
<|im_start|>assistant
<Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At the town carnival, Billy rode the ferris wheel 7 times and the bumper cars 3 times. Each ride cost 5 tickets. How many tickets did he use?<|im_end|>
<|im_start|>assistant
<Z_70><ANSWER>

Question: At the town carnival, Billy rode the ferris wheel 7 times and the bumper cars 3 times. Each ride cost 5 tickets. How many tickets did he use?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(x) = 5x - 1$ and $g(x) = (x+2)^2 + 3$. What is $f(g(3))$?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(x) = 5x - 1$ and $g(x) = (x+2)^2 + 3$. What is $f(g(3))$?<|im_end|>
<|im_start|>assistant
<Z_5><Z_429><Z_199><Z_453><Z_268><Z_291><Z_305><Z_162><Z_233><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(x) = 5x - 1$ and $g(x) = (x+2)^2 + 3$. What is $f(g(3))$?<|im_end|>
<|im_start|>assistant
<Z_488><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $f(x) = 5x - 1$ and $g(x) = (x+2)^2 + 3$. What is $f(g(3))$?<|im_end|>
<|im_start|>assistant
<Z_3><Z_496><Z_321><Z_286><Z_16><Z_177><Z_463><Z_274><Z_455><ANSWER>

Question: Let $f(x) = 5x - 1$ and $g(x) = (x+2)^2 + 3$. What is $f(g(3))$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.<|im_end|>
<|im_start|>assistant
<Z_176><Z_497><Z_312><Z_441><Z_247><Z_173><Z_503><Z_131><Z_260><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.<|im_end|>
<|im_start|>assistant
<Z_510><Z_131><Z_254><Z_84><Z_480><Z_85><Z_290><Z_144><Z_206><ANSWER>

Question: There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_296><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$.<|im_end|>
<|im_start|>assistant
<Z_192><ANSWER>

Question: $\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is twice the number at the second intersection. At the fourth intersection, the number of road signs is 20 fewer than those at the third intersection. Calculate the total number of road signs at the four intersections.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is twice the number at the second intersection. At the fourth intersection, the number of road signs is 20 fewer than those at the third intersection. Calculate the total number of road signs at the four intersections.<|im_end|>
<|im_start|>assistant
<Z_358><Z_333><Z_333><Z_413><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is twice the number at the second intersection. At the fourth intersection, the number of road signs is 20 fewer than those at the third intersection. Calculate the total number of road signs at the four intersections.<|im_end|>
<|im_start|>assistant
<Z_153><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is twice the number at the second intersection. At the fourth intersection, the number of road signs is 20 fewer than those at the third intersection. Calculate the total number of road signs at the four intersections.<|im_end|>
<|im_start|>assistant
<Z_283><Z_496><Z_97><Z_397><ANSWER>

Question: A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is twice the number at the second intersection. At the fourth intersection, the number of road signs is 20 fewer than those at the third intersection. Calculate the total number of road signs at the four intersections.