<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school is organizing a trip to the museum. 4 buses were hired to take the children and teachers to their destination. The second bus has twice the number of people on it as the first bus. The third bus has 6 fewer people than the second bus. The fourth bus has 9 more people than the first bus. There are 75 people going to the museum in total. How many people are on the first bus?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school is organizing a trip to the museum. 4 buses were hired to take the children and teachers to their destination. The second bus has twice the number of people on it as the first bus. The third bus has 6 fewer people than the second bus. The fourth bus has 9 more people than the first bus. There are 75 people going to the museum in total. How many people are on the first bus?<|im_end|>
<|im_start|>assistant
<Z_369><Z_414><Z_446><Z_123><Z_178><Z_248><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school is organizing a trip to the museum. 4 buses were hired to take the children and teachers to their destination. The second bus has twice the number of people on it as the first bus. The third bus has 6 fewer people than the second bus. The fourth bus has 9 more people than the first bus. There are 75 people going to the museum in total. How many people are on the first bus?<|im_end|>
<|im_start|>assistant
<Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school is organizing a trip to the museum. 4 buses were hired to take the children and teachers to their destination. The second bus has twice the number of people on it as the first bus. The third bus has 6 fewer people than the second bus. The fourth bus has 9 more people than the first bus. There are 75 people going to the museum in total. How many people are on the first bus?<|im_end|>
<|im_start|>assistant
<Z_150><Z_345><Z_284><Z_24><Z_167><Z_429><ANSWER>

Question: The school is organizing a trip to the museum. 4 buses were hired to take the children and teachers to their destination. The second bus has twice the number of people on it as the first bus. The third bus has 6 fewer people than the second bus. The fourth bus has 9 more people than the first bus. There are 75 people going to the museum in total. How many people are on the first bus?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = x^2 - x + 2025$. What is the greatest common divisor of $g(103)$ and $g(104)$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = x^2 - x + 2025$. What is the greatest common divisor of $g(103)$ and $g(104)$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_218><Z_247><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = x^2 - x + 2025$. What is the greatest common divisor of $g(103)$ and $g(104)$?<|im_end|>
<|im_start|>assistant
<Z_180><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = x^2 - x + 2025$. What is the greatest common divisor of $g(103)$ and $g(104)$?<|im_end|>
<|im_start|>assistant
<Z_47><Z_149><Z_228><Z_282><ANSWER>

Question: Let $g(x) = x^2 - x + 2025$. What is the greatest common divisor of $g(103)$ and $g(104)$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x$ and $y$ are both positive numbers, and $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{9}{y}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x$ and $y$ are both positive numbers, and $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{9}{y}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x$ and $y$ are both positive numbers, and $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{9}{y}$.<|im_end|>
<|im_start|>assistant
<Z_417><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x$ and $y$ are both positive numbers, and $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{9}{y}$.<|im_end|>
<|im_start|>assistant
<Z_92><Z_503><Z_39><ANSWER>

Question: Given that $x$ and $y$ are both positive numbers, and $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{9}{y}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).<|im_end|>
<|im_start|>assistant
<Z_24><Z_400><Z_250><Z_322><Z_450><Z_450><Z_147><Z_151><Z_202><Z_140><Z_413><Z_332><Z_256><Z_59><Z_488><Z_254><Z_433><Z_296><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).<|im_end|>
<|im_start|>assistant
<Z_311><Z_443><Z_86><Z_309><Z_153><Z_84><Z_177><Z_354><Z_211><Z_135><Z_325><Z_321><Z_315><Z_166><Z_498><Z_61><Z_21><Z_324><ANSWER>

Question: For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A soccer team has 24 players. They have scored some goals on the season. There have been 15 games played. 1/3 of the players have averaged 1 goal each per game. The other players have scored 30 goals in total. How many goals have they scored on the season?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A soccer team has 24 players. They have scored some goals on the season. There have been 15 games played. 1/3 of the players have averaged 1 goal each per game. The other players have scored 30 goals in total. How many goals have they scored on the season?<|im_end|>
<|im_start|>assistant
<Z_477><Z_63><Z_48><Z_113><Z_488><Z_132><Z_366><Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A soccer team has 24 players. They have scored some goals on the season. There have been 15 games played. 1/3 of the players have averaged 1 goal each per game. The other players have scored 30 goals in total. How many goals have they scored on the season?<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A soccer team has 24 players. They have scored some goals on the season. There have been 15 games played. 1/3 of the players have averaged 1 goal each per game. The other players have scored 30 goals in total. How many goals have they scored on the season?<|im_end|>
<|im_start|>assistant
<Z_161><Z_155><Z_370><Z_109><Z_373><Z_94><Z_422><Z_158><ANSWER>

Question: A soccer team has 24 players. They have scored some goals on the season. There have been 15 games played. 1/3 of the players have averaged 1 goal each per game. The other players have scored 30 goals in total. How many goals have they scored on the season?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set?<|im_end|>
<|im_start|>assistant
<Z_369><Z_41><Z_443><Z_368><Z_205><Z_336><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set?<|im_end|>
<|im_start|>assistant
<Z_489><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set?<|im_end|>
<|im_start|>assistant
<Z_132><Z_404><Z_62><Z_482><Z_124><Z_53><ANSWER>

Question: LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A dog has discovered \( 5214_{6} \) ways to navigate through a complex maze. How many ways are there in base 10?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A dog has discovered \( 5214_{6} \) ways to navigate through a complex maze. How many ways are there in base 10?<|im_end|>
<|im_start|>assistant
<Z_369><Z_425><Z_74><Z_449><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A dog has discovered \( 5214_{6} \) ways to navigate through a complex maze. How many ways are there in base 10?<|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
A dog has discovered \( 5214_{6} \) ways to navigate through a complex maze. How many ways are there in base 10?<|im_end|>
<|im_start|>assistant
<Z_340><Z_464><Z_196><Z_339><Z_406><ANSWER>

Question: A dog has discovered \( 5214_{6} \) ways to navigate through a complex maze. How many ways are there in base 10?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_227><Z_162><Z_434><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?<|im_end|>
<|im_start|>assistant
<Z_338><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?<|im_end|>
<|im_start|>assistant
<Z_18><Z_317><Z_317><Z_372><ANSWER>

Question: If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]<|im_end|>
<|im_start|>assistant
<Z_369><Z_181><Z_73><Z_201><Z_378><Z_141><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]<|im_end|>
<|im_start|>assistant
<Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]<|im_end|>
<|im_start|>assistant
<Z_489><Z_273><Z_454><Z_448><Z_114><Z_484><ANSWER>

Question:
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?<|im_end|>
<|im_start|>assistant
<Z_477><Z_31><Z_320><Z_367><Z_228><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?<|im_end|>
<|im_start|>assistant
<Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?<|im_end|>
<|im_start|>assistant
<Z_472><Z_50><Z_18><Z_321><Z_440><ANSWER>

Question: What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?

Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other hand, the sum of the interior angles of an \( n \)-sided convex polygon is \( 180(n-2) \). Thus, we have the equation:

\[ 172a + 173b = 180(a + b - 2) \]

Or equivalently:

\[ 8a + 7b = 360 \]<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?

Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other hand, the sum of the interior angles of an \( n \)-sided convex polygon is \( 180(n-2) \). Thus, we have the equation:

\[ 172a + 173b = 180(a + b - 2) \]

Or equivalently:

\[ 8a + 7b = 360 \]<|im_end|>
<|im_start|>assistant
<Z_24><Z_102><Z_149><Z_510><Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?

Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other hand, the sum of the interior angles of an \( n \)-sided convex polygon is \( 180(n-2) \). Thus, we have the equation:

\[ 172a + 173b = 180(a + b - 2) \]

Or equivalently:

\[ 8a + 7b = 360 \]<|im_end|>
<|im_start|>assistant
<Z_346><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?

Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other hand, the sum of the interior angles of an \( n \)-sided convex polygon is \( 180(n-2) \). Thus, we have the equation:

\[ 172a + 173b = 180(a + b - 2) \]

Or equivalently:

\[ 8a + 7b = 360 \]<|im_end|>
<|im_start|>assistant
<Z_51><Z_312><Z_212><Z_464><Z_185><ANSWER>

Question:
What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?

Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other hand, the sum of the interior angles of an \( n \)-sided convex polygon is \( 180(n-2) \). Thus, we have the equation:

\[ 172a + 173b = 180(a + b - 2) \]

Or equivalently:

\[ 8a + 7b = 360 \]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes some time to complete the journey. The distance between city A and city B is 900 kms and city A and city C is 300 kms. The ratio of their average speed of travel is 4 (Eddy : Freddy). How long does it take for Freddy to complete his journey?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes some time to complete the journey. The distance between city A and city B is 900 kms and city A and city C is 300 kms. The ratio of their average speed of travel is 4 (Eddy : Freddy). How long does it take for Freddy to complete his journey?<|im_end|>
<|im_start|>assistant
<Z_477><Z_286><Z_232><Z_283><Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes some time to complete the journey. The distance between city A and city B is 900 kms and city A and city C is 300 kms. The ratio of their average speed of travel is 4 (Eddy : Freddy). How long does it take for Freddy to complete his journey?<|im_end|>
<|im_start|>assistant
<Z_405><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Eddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes some time to complete the journey. The distance between city A and city B is 900 kms and city A and city C is 300 kms. The ratio of their average speed of travel is 4 (Eddy : Freddy). How long does it take for Freddy to complete his journey?<|im_end|>
<|im_start|>assistant
<Z_25><Z_495><Z_155><Z_393><Z_67><ANSWER>

Question: Eddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes some time to complete the journey. The distance between city A and city B is 900 kms and city A and city C is 300 kms. The ratio of their average speed of travel is 4 (Eddy : Freddy). How long does it take for Freddy to complete his journey?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a celebration, Tim receives a cuboidal container that, when filled to capacity, holds 216 candies. Shortly afterward, Elly receives a larger cuboidal container. Elly's container is three times as long, twice as high, and four times as wide as Tim's. If both containers are filled with candies, how many candies does Elly get assuming the candies are small enough to make the containers approximate figures for volume calculations?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a celebration, Tim receives a cuboidal container that, when filled to capacity, holds 216 candies. Shortly afterward, Elly receives a larger cuboidal container. Elly's container is three times as long, twice as high, and four times as wide as Tim's. If both containers are filled with candies, how many candies does Elly get assuming the candies are small enough to make the containers approximate figures for volume calculations?<|im_end|>
<|im_start|>assistant
<Z_369><Z_268><Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a celebration, Tim receives a cuboidal container that, when filled to capacity, holds 216 candies. Shortly afterward, Elly receives a larger cuboidal container. Elly's container is three times as long, twice as high, and four times as wide as Tim's. If both containers are filled with candies, how many candies does Elly get assuming the candies are small enough to make the containers approximate figures for volume calculations?<|im_end|>
<|im_start|>assistant
<Z_397><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For a celebration, Tim receives a cuboidal container that, when filled to capacity, holds 216 candies. Shortly afterward, Elly receives a larger cuboidal container. Elly's container is three times as long, twice as high, and four times as wide as Tim's. If both containers are filled with candies, how many candies does Elly get assuming the candies are small enough to make the containers approximate figures for volume calculations?<|im_end|>
<|im_start|>assistant
<Z_463><Z_291><Z_196><ANSWER>

Question: For a celebration, Tim receives a cuboidal container that, when filled to capacity, holds 216 candies. Shortly afterward, Elly receives a larger cuboidal container. Elly's container is three times as long, twice as high, and four times as wide as Tim's. If both containers are filled with candies, how many candies does Elly get assuming the candies are small enough to make the containers approximate figures for volume calculations?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 14x + 60$ can be written as a combination of a square of a binomial and an integer. Find the integer.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 14x + 60$ can be written as a combination of a square of a binomial and an integer. Find the integer.<|im_end|>
<|im_start|>assistant
<Z_24><Z_494><Z_151><Z_104><Z_140><Z_157><Z_52><Z_181><Z_498><Z_366><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 14x + 60$ can be written as a combination of a square of a binomial and an integer. Find the integer.<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 14x + 60$ can be written as a combination of a square of a binomial and an integer. Find the integer.<|im_end|>
<|im_start|>assistant
<Z_489><Z_38><Z_55><Z_258><Z_381><Z_114><Z_166><Z_507><Z_395><Z_204><Z_464><ANSWER>

Question: The expression $x^2 + 14x + 60$ can be written as a combination of a square of a binomial and an integer. Find the integer.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate three-fifths of the product of 5 and 9.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate three-fifths of the product of 5 and 9.<|im_end|>
<|im_start|>assistant
<Z_477><Z_497><Z_361><Z_86><Z_148><Z_248><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate three-fifths of the product of 5 and 9.<|im_end|>
<|im_start|>assistant
<Z_475><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate three-fifths of the product of 5 and 9.<|im_end|>
<|im_start|>assistant
<Z_1><Z_86><Z_178><Z_32><Z_282><Z_125><Z_356><ANSWER>

Question: Calculate three-fifths of the product of 5 and 9.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the area of triangle $ABC$, where $\angle A = 90^\circ$, $\angle B = 45^\circ$, and the hypothenuse $\overline{AC} = 20$ units.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the area of triangle $ABC$, where $\angle A = 90^\circ$, $\angle B = 45^\circ$, and the hypothenuse $\overline{AC} = 20$ units.<|im_end|>
<|im_start|>assistant
<Z_369><Z_379><Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the area of triangle $ABC$, where $\angle A = 90^\circ$, $\angle B = 45^\circ$, and the hypothenuse $\overline{AC} = 20$ units.<|im_end|>
<|im_start|>assistant
<Z_137><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the area of triangle $ABC$, where $\angle A = 90^\circ$, $\angle B = 45^\circ$, and the hypothenuse $\overline{AC} = 20$ units.<|im_end|>
<|im_start|>assistant
<Z_222><Z_11><Z_355><ANSWER>

Question: Calculate the area of triangle $ABC$, where $\angle A = 90^\circ$, $\angle B = 45^\circ$, and the hypothenuse $\overline{AC} = 20$ units.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an election with two candidates, the winner received 62% of votes and won by a certain number of votes. The winning candidate received 837 votes. How many votes did the winner win by?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an election with two candidates, the winner received 62% of votes and won by a certain number of votes. The winning candidate received 837 votes. How many votes did the winner win by?<|im_end|>
<|im_start|>assistant
<Z_477><Z_102><Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an election with two candidates, the winner received 62% of votes and won by a certain number of votes. The winning candidate received 837 votes. How many votes did the winner win by?<|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
In an election with two candidates, the winner received 62% of votes and won by a certain number of votes. The winning candidate received 837 votes. How many votes did the winner win by?<|im_end|>
<|im_start|>assistant
<Z_155><Z_190><Z_368><ANSWER>

Question: In an election with two candidates, the winner received 62% of votes and won by a certain number of votes. The winning candidate received 837 votes. How many votes did the winner win by?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country?<|im_end|>
<|im_start|>assistant
<Z_261><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country?<|im_end|>
<|im_start|>assistant
<Z_294><ANSWER>

Question: In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine has $\$15$ to spend and wishes to buy some trading cards. Each card now costs $\$1.25$, tax included, and there is a fixed transaction fee of $\$2$ for the entire purchase. Jasmine also wants to ensure she has at least $\$3$ remaining after her purchases. What is the maximum number of trading cards she can buy?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine has $\$15$ to spend and wishes to buy some trading cards. Each card now costs $\$1.25$, tax included, and there is a fixed transaction fee of $\$2$ for the entire purchase. Jasmine also wants to ensure she has at least $\$3$ remaining after her purchases. What is the maximum number of trading cards she can buy?<|im_end|>
<|im_start|>assistant
<Z_369><Z_454><Z_463><Z_244><Z_345><Z_316><Z_316><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine has $\$15$ to spend and wishes to buy some trading cards. Each card now costs $\$1.25$, tax included, and there is a fixed transaction fee of $\$2$ for the entire purchase. Jasmine also wants to ensure she has at least $\$3$ remaining after her purchases. What is the maximum number of trading cards she can buy?<|im_end|>
<|im_start|>assistant
<Z_315><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine has $\$15$ to spend and wishes to buy some trading cards. Each card now costs $\$1.25$, tax included, and there is a fixed transaction fee of $\$2$ for the entire purchase. Jasmine also wants to ensure she has at least $\$3$ remaining after her purchases. What is the maximum number of trading cards she can buy?<|im_end|>
<|im_start|>assistant
<Z_185><Z_228><Z_250><Z_387><Z_473><Z_154><Z_154><Z_244><ANSWER>

Question: Jasmine has $\$15$ to spend and wishes to buy some trading cards. Each card now costs $\$1.25$, tax included, and there is a fixed transaction fee of $\$2$ for the entire purchase. Jasmine also wants to ensure she has at least $\$3$ remaining after her purchases. What is the maximum number of trading cards she can buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $3^x + 3^x + 3^x + 3^x = 1458$, what is the value of $(x+2)(x-2)$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $3^x + 3^x + 3^x + 3^x = 1458$, what is the value of $(x+2)(x-2)$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_159><Z_199><Z_188><Z_114><Z_240><Z_456><Z_421><Z_253><Z_139><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $3^x + 3^x + 3^x + 3^x = 1458$, what is the value of $(x+2)(x-2)$?<|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
Given that $3^x + 3^x + 3^x + 3^x = 1458$, what is the value of $(x+2)(x-2)$?<|im_end|>
<|im_start|>assistant
<Z_504><Z_443><Z_9><Z_167><Z_249><Z_1><Z_271><Z_312><Z_343><Z_297><ANSWER>

Question: Given that $3^x + 3^x + 3^x + 3^x = 1458$, what is the value of $(x+2)(x-2)$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$.  What is the $x$-coordinate of the vertex of the parabola?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$.  What is the $x$-coordinate of the vertex of the parabola?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$.  What is the $x$-coordinate of the vertex of the parabola?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$.  What is the $x$-coordinate of the vertex of the parabola?<|im_end|>
<|im_start|>assistant
<Z_413><ANSWER>

Question: The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$.  What is the $x$-coordinate of the vertex of the parabola?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?<|im_end|>
<|im_start|>assistant
<Z_477><Z_85><Z_470><Z_181><Z_184><Z_467><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?<|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
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?<|im_end|>
<|im_start|>assistant
<Z_61><Z_511><Z_443><Z_351><Z_54><Z_26><ANSWER>

Question: In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When 242 is divided by a certain divisor, the remainder obtained is 8. When 698 is divided by the same divisor, the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is a specific number. The value of the divisor is 13. What is the remainder obtained when dividing the sum of 242 and 698 by the divisor?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When 242 is divided by a certain divisor, the remainder obtained is 8. When 698 is divided by the same divisor, the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is a specific number. The value of the divisor is 13. What is the remainder obtained when dividing the sum of 242 and 698 by the divisor?<|im_end|>
<|im_start|>assistant
<Z_477><Z_210><Z_334><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When 242 is divided by a certain divisor, the remainder obtained is 8. When 698 is divided by the same divisor, the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is a specific number. The value of the divisor is 13. What is the remainder obtained when dividing the sum of 242 and 698 by the divisor?<|im_end|>
<|im_start|>assistant
<Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When 242 is divided by a certain divisor, the remainder obtained is 8. When 698 is divided by the same divisor, the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is a specific number. The value of the divisor is 13. What is the remainder obtained when dividing the sum of 242 and 698 by the divisor?<|im_end|>
<|im_start|>assistant
<Z_265><Z_83><Z_375><ANSWER>

Question: When 242 is divided by a certain divisor, the remainder obtained is 8. When 698 is divided by the same divisor, the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is a specific number. The value of the divisor is 13. What is the remainder obtained when dividing the sum of 242 and 698 by the divisor?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I paid 1,000 won and bought several pencils for 210 won and a glue for 270 won. How many pencils did you buy in total when you received 100 won in change?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I paid 1,000 won and bought several pencils for 210 won and a glue for 270 won. How many pencils did you buy in total when you received 100 won in change?<|im_end|>
<|im_start|>assistant
<Z_258><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I paid 1,000 won and bought several pencils for 210 won and a glue for 270 won. How many pencils did you buy in total when you received 100 won in change?<|im_end|>
<|im_start|>assistant
<Z_474><ANSWER>

Question: I paid 1,000 won and bought several pencils for 210 won and a glue for 270 won. How many pencils did you buy in total when you received 100 won in change?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chef served 3 different foods for a banquet: 25 plates of lobster rolls, 14 plates of spicy hot noodles, and some plates of seafood noodles. The chef made 55 plates of food altogether. How many plates of seafood noodles did the chef make?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chef served 3 different foods for a banquet: 25 plates of lobster rolls, 14 plates of spicy hot noodles, and some plates of seafood noodles. The chef made 55 plates of food altogether. How many plates of seafood noodles did the chef make?<|im_end|>
<|im_start|>assistant
<Z_477><Z_91><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chef served 3 different foods for a banquet: 25 plates of lobster rolls, 14 plates of spicy hot noodles, and some plates of seafood noodles. The chef made 55 plates of food altogether. How many plates of seafood noodles did the chef make?<|im_end|>
<|im_start|>assistant
<Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A chef served 3 different foods for a banquet: 25 plates of lobster rolls, 14 plates of spicy hot noodles, and some plates of seafood noodles. The chef made 55 plates of food altogether. How many plates of seafood noodles did the chef make?<|im_end|>
<|im_start|>assistant
<Z_129><Z_101><Z_71><ANSWER>

Question: A chef served 3 different foods for a banquet: 25 plates of lobster rolls, 14 plates of spicy hot noodles, and some plates of seafood noodles. The chef made 55 plates of food altogether. How many plates of seafood noodles did the chef make?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a larger class of 50 students, 12 students received an A in history and 25 received an A in math, including six that received an A in both courses. How many students did not receive an A in either of these courses?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a larger class of 50 students, 12 students received an A in history and 25 received an A in math, including six that received an A in both courses. How many students did not receive an A in either of these courses?<|im_end|>
<|im_start|>assistant
<Z_477><Z_293><Z_456><Z_68><Z_102><Z_219><Z_217><Z_61><Z_185><Z_490><Z_274><Z_321><Z_75><Z_294><Z_463><Z_375><Z_472><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a larger class of 50 students, 12 students received an A in history and 25 received an A in math, including six that received an A in both courses. How many students did not receive an A in either of these courses?<|im_end|>
<|im_start|>assistant
<Z_164><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a larger class of 50 students, 12 students received an A in history and 25 received an A in math, including six that received an A in both courses. How many students did not receive an A in either of these courses?<|im_end|>
<|im_start|>assistant
<Z_291><Z_338><Z_135><Z_232><Z_242><Z_261><Z_294><Z_325><Z_418><Z_172><Z_115><Z_71><Z_492><Z_33><Z_158><Z_310><Z_267><ANSWER>

Question: In a larger class of 50 students, 12 students received an A in history and 25 received an A in math, including six that received an A in both courses. How many students did not receive an A in either of these courses?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the line $l: x+y=5$ and the circle $C: x^2+y^2=16$, which intersect at points $A(x_1,y_1)$ and $B(x_2,y_2)$, find the value of $x_1y_2+x_2y_1$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the line $l: x+y=5$ and the circle $C: x^2+y^2=16$, which intersect at points $A(x_1,y_1)$ and $B(x_2,y_2)$, find the value of $x_1y_2+x_2y_1$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_273><Z_450><Z_272><Z_71><Z_229><Z_46><Z_246><Z_83><Z_283><Z_246><Z_351><Z_463><Z_413><Z_498><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the line $l: x+y=5$ and the circle $C: x^2+y^2=16$, which intersect at points $A(x_1,y_1)$ and $B(x_2,y_2)$, find the value of $x_1y_2+x_2y_1$.<|im_end|>
<|im_start|>assistant
<Z_93><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the line $l: x+y=5$ and the circle $C: x^2+y^2=16$, which intersect at points $A(x_1,y_1)$ and $B(x_2,y_2)$, find the value of $x_1y_2+x_2y_1$.<|im_end|>
<|im_start|>assistant
<Z_382><Z_186><Z_198><Z_202><Z_303><Z_252><Z_363><Z_156><Z_435><Z_322><Z_82><Z_359><Z_45><Z_276><Z_399><Z_468><ANSWER>

Question: Given the line $l: x+y=5$ and the circle $C: x^2+y^2=16$, which intersect at points $A(x_1,y_1)$ and $B(x_2,y_2)$, find the value of $x_1y_2+x_2y_1$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$.  Determine $|c|$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$.  Determine $|c|$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_248><Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$.  Determine $|c|$.<|im_end|>
<|im_start|>assistant
<Z_354><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$.  Determine $|c|$.<|im_end|>
<|im_start|>assistant
<Z_230><Z_162><Z_307><ANSWER>

Question:
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$.  Determine $|c|$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When a number is divided by 7, the remainder is 3. What is the remainder when five times the number minus 11 is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When a number is divided by 7, the remainder is 3. What is the remainder when five times the number minus 11 is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_477><Z_172><Z_87><Z_276><Z_194><Z_188><Z_273><Z_169><Z_449><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When a number is divided by 7, the remainder is 3. What is the remainder when five times the number minus 11 is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
When a number is divided by 7, the remainder is 3. What is the remainder when five times the number minus 11 is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_81><Z_445><Z_280><Z_260><Z_109><Z_314><Z_484><Z_330><Z_41><Z_303><ANSWER>

Question: When a number is divided by 7, the remainder is 3. What is the remainder when five times the number minus 11 is divided by 7?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At school today, Charlize was 20 minutes late. Four of her classmates were each a certain amount of time later than she was. The total time for which the five students were late is 140 minutes. How many minutes later were each of her classmates compared to Charlize?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At school today, Charlize was 20 minutes late. Four of her classmates were each a certain amount of time later than she was. The total time for which the five students were late is 140 minutes. How many minutes later were each of her classmates compared to Charlize?<|im_end|>
<|im_start|>assistant
<Z_24><Z_227><Z_377><Z_187><Z_214><Z_420><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At school today, Charlize was 20 minutes late. Four of her classmates were each a certain amount of time later than she was. The total time for which the five students were late is 140 minutes. How many minutes later were each of her classmates compared to Charlize?<|im_end|>
<|im_start|>assistant
<Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
At school today, Charlize was 20 minutes late. Four of her classmates were each a certain amount of time later than she was. The total time for which the five students were late is 140 minutes. How many minutes later were each of her classmates compared to Charlize?<|im_end|>
<|im_start|>assistant
<Z_483><Z_364><Z_491><Z_174><Z_337><Z_26><ANSWER>

Question: At school today, Charlize was 20 minutes late. Four of her classmates were each a certain amount of time later than she was. The total time for which the five students were late is 140 minutes. How many minutes later were each of her classmates compared to Charlize?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive four-digit integers are divisible by both 11 and 5?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive four-digit integers are divisible by both 11 and 5?<|im_end|>
<|im_start|>assistant
<Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many positive four-digit integers are divisible by both 11 and 5?<|im_end|>
<|im_start|>assistant
<Z_182><ANSWER>

Question: How many positive four-digit integers are divisible by both 11 and 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?<|im_end|>
<|im_start|>assistant
<Z_24><Z_106><Z_143><Z_98><Z_185><Z_267><Z_68><Z_214><Z_279><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?<|im_end|>
<|im_start|>assistant
<Z_271><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?<|im_end|>
<|im_start|>assistant
<Z_127><Z_396><Z_299><Z_223><Z_194><Z_496><Z_392><Z_308><Z_504><Z_391><ANSWER>

Question: A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).<|im_end|>
<|im_start|>assistant
<Z_477><Z_229><Z_473><Z_502><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).<|im_end|>
<|im_start|>assistant
<Z_271><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).<|im_end|>
<|im_start|>assistant
<Z_54><Z_245><Z_484><Z_66><ANSWER>

Question:
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bertha plays tennis. Every ten games, one of her tennis balls wears out. Every five games, she loses a ball. Every four games, she buys a canister of three balls. She started with a certain number of balls and gave one to her partner. After twenty games, she has 10 tennis balls. How many balls did she have before giving one to her partner?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bertha plays tennis. Every ten games, one of her tennis balls wears out. Every five games, she loses a ball. Every four games, she buys a canister of three balls. She started with a certain number of balls and gave one to her partner. After twenty games, she has 10 tennis balls. How many balls did she have before giving one to her partner?<|im_end|>
<|im_start|>assistant
<Z_24><Z_375><Z_97><Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bertha plays tennis. Every ten games, one of her tennis balls wears out. Every five games, she loses a ball. Every four games, she buys a canister of three balls. She started with a certain number of balls and gave one to her partner. After twenty games, she has 10 tennis balls. How many balls did she have before giving one to her partner?<|im_end|>
<|im_start|>assistant
<Z_269><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bertha plays tennis. Every ten games, one of her tennis balls wears out. Every five games, she loses a ball. Every four games, she buys a canister of three balls. She started with a certain number of balls and gave one to her partner. After twenty games, she has 10 tennis balls. How many balls did she have before giving one to her partner?<|im_end|>
<|im_start|>assistant
<Z_57><Z_133><Z_96><Z_75><ANSWER>

Question: Bertha plays tennis. Every ten games, one of her tennis balls wears out. Every five games, she loses a ball. Every four games, she buys a canister of three balls. She started with a certain number of balls and gave one to her partner. After twenty games, she has 10 tennis balls. How many balls did she have before giving one to her partner?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

Question:
Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle ABC$, with opposite sides $a$, $b$, and $c$ corresponding to angles $A$, $B$, and $C$ respectively, it is given that $b=2$, $c=\sqrt{3}$, and $\angle A=\frac{\pi}{6}$.
(1) Find the area of $\triangle ABC$;
(2) Find the length of side $BC$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle ABC$, with opposite sides $a$, $b$, and $c$ corresponding to angles $A$, $B$, and $C$ respectively, it is given that $b=2$, $c=\sqrt{3}$, and $\angle A=\frac{\pi}{6}$.
(1) Find the area of $\triangle ABC$;
(2) Find the length of side $BC$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_382><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle ABC$, with opposite sides $a$, $b$, and $c$ corresponding to angles $A$, $B$, and $C$ respectively, it is given that $b=2$, $c=\sqrt{3}$, and $\angle A=\frac{\pi}{6}$.
(1) Find the area of $\triangle ABC$;
(2) Find the length of side $BC$.<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $\triangle ABC$, with opposite sides $a$, $b$, and $c$ corresponding to angles $A$, $B$, and $C$ respectively, it is given that $b=2$, $c=\sqrt{3}$, and $\angle A=\frac{\pi}{6}$.
(1) Find the area of $\triangle ABC$;
(2) Find the length of side $BC$.<|im_end|>
<|im_start|>assistant
<Z_49><Z_339><Z_363><ANSWER>

Question:
In triangle $\triangle ABC$, with opposite sides $a$, $b$, and $c$ corresponding to angles $A$, $B$, and $C$ respectively, it is given that $b=2$, $c=\sqrt{3}$, and $\angle A=\frac{\pi}{6}$.
(1) Find the area of $\triangle ABC$;
(2) Find the length of side $BC$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble.<|im_end|>
<|im_start|>assistant
<Z_477><Z_269><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble.<|im_end|>
<|im_start|>assistant
<Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble.<|im_end|>
<|im_start|>assistant
<Z_385><Z_263><ANSWER>

Question: A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?<|im_end|>
<|im_start|>assistant
<Z_477><Z_217><Z_237><Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?<|im_end|>
<|im_start|>assistant
<Z_286><Z_282><Z_73><Z_77><ANSWER>

Question: Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $n = \int_{1}^{e} \frac{6}{x}dx$, find the constant term in the expansion of $(x^2 - \frac{1}{x})^n$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $n = \int_{1}^{e} \frac{6}{x}dx$, find the constant term in the expansion of $(x^2 - \frac{1}{x})^n$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_250><Z_202><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $n = \int_{1}^{e} \frac{6}{x}dx$, find the constant term in the expansion of $(x^2 - \frac{1}{x})^n$.<|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
Given that $n = \int_{1}^{e} \frac{6}{x}dx$, find the constant term in the expansion of $(x^2 - \frac{1}{x})^n$.<|im_end|>
<|im_start|>assistant
<Z_230><Z_442><Z_391><ANSWER>

Question: Given that $n = \int_{1}^{e} \frac{6}{x}dx$, find the constant term in the expansion of $(x^2 - \frac{1}{x})^n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, a student took 87 as divisor instead of the correct divisor. His answer was 24. The correct answer is 58. What divisor should the student have taken?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, a student took 87 as divisor instead of the correct divisor. His answer was 24. The correct answer is 58. What divisor should the student have taken?<|im_end|>
<|im_start|>assistant
<Z_369><Z_214><Z_410><Z_394><Z_134><Z_468><Z_373><Z_335><Z_265><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, a student took 87 as divisor instead of the correct divisor. His answer was 24. The correct answer is 58. What divisor should the student have taken?<|im_end|>
<|im_start|>assistant
<Z_507><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a division, a student took 87 as divisor instead of the correct divisor. His answer was 24. The correct answer is 58. What divisor should the student have taken?<|im_end|>
<|im_start|>assistant
<Z_191><Z_92><Z_443><Z_489><Z_341><Z_235><Z_368><Z_297><Z_210><ANSWER>

Question: In a division, a student took 87 as divisor instead of the correct divisor. His answer was 24. The correct answer is 58. What divisor should the student have taken?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $N$ be the product of the divisors of $60$. How many distinct prime factors does $N$ have?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $N$ be the product of the divisors of $60$. How many distinct prime factors does $N$ have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_206><Z_433><Z_503><Z_58><Z_276><Z_177><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $N$ be the product of the divisors of $60$. How many distinct prime factors does $N$ have?<|im_end|>
<|im_start|>assistant
<Z_163><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $N$ be the product of the divisors of $60$. How many distinct prime factors does $N$ have?<|im_end|>
<|im_start|>assistant
<Z_331><Z_200><Z_9><Z_269><Z_163><Z_215><Z_380><Z_54><ANSWER>

Question: Let $N$ be the product of the divisors of $60$. How many distinct prime factors does $N$ have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In 5 football games thus far this season, Barry has run for 98, 107, 85, 89, and some yards. To keep his season average above 100 yards, he needs to gain a minimum of 130 yards this Sunday. How many yards did Barry run in the fifth game?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In 5 football games thus far this season, Barry has run for 98, 107, 85, 89, and some yards. To keep his season average above 100 yards, he needs to gain a minimum of 130 yards this Sunday. How many yards did Barry run in the fifth game?<|im_end|>
<|im_start|>assistant
<Z_24><Z_48><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In 5 football games thus far this season, Barry has run for 98, 107, 85, 89, and some yards. To keep his season average above 100 yards, he needs to gain a minimum of 130 yards this Sunday. How many yards did Barry run in the fifth game?<|im_end|>
<|im_start|>assistant
<Z_14><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In 5 football games thus far this season, Barry has run for 98, 107, 85, 89, and some yards. To keep his season average above 100 yards, he needs to gain a minimum of 130 yards this Sunday. How many yards did Barry run in the fifth game?<|im_end|>
<|im_start|>assistant
<Z_27><Z_209><Z_249><ANSWER>

Question: In 5 football games thus far this season, Barry has run for 98, 107, 85, 89, and some yards. To keep his season average above 100 yards, he needs to gain a minimum of 130 yards this Sunday. How many yards did Barry run in the fifth game?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jemma saw 7 grasshoppers on her African daisy plant.  Then, hopping on the grass under the plant, she found 2 dozen baby grasshoppers.  How many grasshoppers did  Jemma find altogether?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jemma saw 7 grasshoppers on her African daisy plant.  Then, hopping on the grass under the plant, she found 2 dozen baby grasshoppers.  How many grasshoppers did  Jemma find altogether?<|im_end|>
<|im_start|>assistant
<Z_24><Z_260><Z_264><Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jemma saw 7 grasshoppers on her African daisy plant.  Then, hopping on the grass under the plant, she found 2 dozen baby grasshoppers.  How many grasshoppers did  Jemma find altogether?<|im_end|>
<|im_start|>assistant
<Z_154><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jemma saw 7 grasshoppers on her African daisy plant.  Then, hopping on the grass under the plant, she found 2 dozen baby grasshoppers.  How many grasshoppers did  Jemma find altogether?<|im_end|>
<|im_start|>assistant
<Z_388><Z_270><Z_249><Z_358><ANSWER>

Question: Jemma saw 7 grasshoppers on her African daisy plant.  Then, hopping on the grass under the plant, she found 2 dozen baby grasshoppers.  How many grasshoppers did  Jemma find altogether?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the quadratic equation in $x$: $x^{2}-kx+k-1=0$.
$(1)$ Prove that: Regardless of the value of $k$, the equation always has real roots.
$(2)$ Given that $x=2$ is a root of the equation, find the value of $k$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the quadratic equation in $x$: $x^{2}-kx+k-1=0$.
$(1)$ Prove that: Regardless of the value of $k$, the equation always has real roots.
$(2)$ Given that $x=2$ is a root of the equation, find the value of $k$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_155><Z_90><Z_425><Z_429><Z_64><Z_250><Z_371><Z_292><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the quadratic equation in $x$: $x^{2}-kx+k-1=0$.
$(1)$ Prove that: Regardless of the value of $k$, the equation always has real roots.
$(2)$ Given that $x=2$ is a root of the equation, find the value of $k$.<|im_end|>
<|im_start|>assistant
<Z_193><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the quadratic equation in $x$: $x^{2}-kx+k-1=0$.
$(1)$ Prove that: Regardless of the value of $k$, the equation always has real roots.
$(2)$ Given that $x=2$ is a root of the equation, find the value of $k$.<|im_end|>
<|im_start|>assistant
<Z_50><Z_400><Z_420><Z_425><Z_394><Z_446><Z_81><Z_287><Z_422><ANSWER>

Question:
Given the quadratic equation in $x$: $x^{2}-kx+k-1=0$.
$(1)$ Prove that: Regardless of the value of $k$, the equation always has real roots.
$(2)$ Given that $x=2$ is a root of the equation, find the value of $k$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_59><Z_111><Z_137><Z_123><Z_322><Z_215><Z_502><Z_33><Z_238><Z_448><Z_433><Z_214><Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?<|im_end|>
<|im_start|>assistant
<Z_208><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?<|im_end|>
<|im_start|>assistant
<Z_196><Z_412><Z_230><Z_276><Z_360><Z_169><Z_316><Z_365><Z_52><Z_118><Z_511><Z_407><Z_443><Z_32><ANSWER>

Question: Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nico borrows 5 books from the library on Monday. On Monday, he reads the first book with a total of 20 pages and starts the second book with 45 pages in it, but he reads only half of it. On Tuesday, he finishes the second book, reads the third book with a total of 32 pages and starts the fourth book of 60 pages, but manages to read only two-fifths of it. On Wednesday, he finishes the fourth book and reads the fifth book with unknown number of pages. If he has read a total of 200 pages from Monday to Wednesday, how many pages did the fifth book have?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nico borrows 5 books from the library on Monday. On Monday, he reads the first book with a total of 20 pages and starts the second book with 45 pages in it, but he reads only half of it. On Tuesday, he finishes the second book, reads the third book with a total of 32 pages and starts the fourth book of 60 pages, but manages to read only two-fifths of it. On Wednesday, he finishes the fourth book and reads the fifth book with unknown number of pages. If he has read a total of 200 pages from Monday to Wednesday, how many pages did the fifth book have?<|im_end|>
<|im_start|>assistant
<Z_369><Z_149><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nico borrows 5 books from the library on Monday. On Monday, he reads the first book with a total of 20 pages and starts the second book with 45 pages in it, but he reads only half of it. On Tuesday, he finishes the second book, reads the third book with a total of 32 pages and starts the fourth book of 60 pages, but manages to read only two-fifths of it. On Wednesday, he finishes the fourth book and reads the fifth book with unknown number of pages. If he has read a total of 200 pages from Monday to Wednesday, how many pages did the fifth book have?<|im_end|>
<|im_start|>assistant
<Z_436><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nico borrows 5 books from the library on Monday. On Monday, he reads the first book with a total of 20 pages and starts the second book with 45 pages in it, but he reads only half of it. On Tuesday, he finishes the second book, reads the third book with a total of 32 pages and starts the fourth book of 60 pages, but manages to read only two-fifths of it. On Wednesday, he finishes the fourth book and reads the fifth book with unknown number of pages. If he has read a total of 200 pages from Monday to Wednesday, how many pages did the fifth book have?<|im_end|>
<|im_start|>assistant
<Z_182><Z_45><Z_465><ANSWER>

Question: Nico borrows 5 books from the library on Monday. On Monday, he reads the first book with a total of 20 pages and starts the second book with 45 pages in it, but he reads only half of it. On Tuesday, he finishes the second book, reads the third book with a total of 32 pages and starts the fourth book of 60 pages, but manages to read only two-fifths of it. On Wednesday, he finishes the fourth book and reads the fifth book with unknown number of pages. If he has read a total of 200 pages from Monday to Wednesday, how many pages did the fifth book have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiaomeng is reading a storybook. On the first day, she reads 12 pages. Starting from the second day, she reads 1 more page each day than the previous day. On the last day, she reads 26 pages and finishes the book. How many pages are there in the entire storybook?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiaomeng is reading a storybook. On the first day, she reads 12 pages. Starting from the second day, she reads 1 more page each day than the previous day. On the last day, she reads 26 pages and finishes the book. How many pages are there in the entire storybook?<|im_end|>
<|im_start|>assistant
<Z_477><Z_61><Z_56><Z_469><Z_469><Z_81><Z_462><Z_208><Z_99><Z_165><Z_173><Z_487><Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Xiaomeng is reading a storybook. On the first day, she reads 12 pages. Starting from the second day, she reads 1 more page each day than the previous day. On the last day, she reads 26 pages and finishes the book. How many pages are there in the entire storybook?<|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
Xiaomeng is reading a storybook. On the first day, she reads 12 pages. Starting from the second day, she reads 1 more page each day than the previous day. On the last day, she reads 26 pages and finishes the book. How many pages are there in the entire storybook?<|im_end|>
<|im_start|>assistant
<Z_350><Z_474><Z_77><Z_77><Z_278><Z_158><Z_200><Z_396><Z_509><Z_177><Z_306><Z_340><Z_427><ANSWER>

Question: Xiaomeng is reading a storybook. On the first day, she reads 12 pages. Starting from the second day, she reads 1 more page each day than the previous day. On the last day, she reads 26 pages and finishes the book. How many pages are there in the entire storybook?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers such that \( x_1 + x_2 + \ldots + x_n = 1 \) and
\[ x_1^2 + x_2^2 + \ldots + x_n^2 \le \frac{1}{25}. \]
Find the smallest possible value of \( n \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers such that \( x_1 + x_2 + \ldots + x_n = 1 \) and
\[ x_1^2 + x_2^2 + \ldots + x_n^2 \le \frac{1}{25}. \]
Find the smallest possible value of \( n \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_420><Z_446><Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers such that \( x_1 + x_2 + \ldots + x_n = 1 \) and
\[ x_1^2 + x_2^2 + \ldots + x_n^2 \le \frac{1}{25}. \]
Find the smallest possible value of \( n \).<|im_end|>
<|im_start|>assistant
<Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers such that \( x_1 + x_2 + \ldots + x_n = 1 \) and
\[ x_1^2 + x_2^2 + \ldots + x_n^2 \le \frac{1}{25}. \]
Find the smallest possible value of \( n \).<|im_end|>
<|im_start|>assistant
<Z_186><Z_247><Z_46><Z_65><ANSWER>

Question:
Let \( x_1, x_2, \ldots, x_n \) be nonnegative real numbers such that \( x_1 + x_2 + \ldots + x_n = 1 \) and
\[ x_1^2 + x_2^2 + \ldots + x_n^2 \le \frac{1}{25}. \]
Find the smallest possible value of \( n \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can the number 1,000,000 be expressed as the product of three natural numbers? Products that differ only in the order of the factors are considered identical.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can the number 1,000,000 be expressed as the product of three natural numbers? Products that differ only in the order of the factors are considered identical.<|im_end|>
<|im_start|>assistant
<Z_477><Z_347><Z_225><Z_267><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can the number 1,000,000 be expressed as the product of three natural numbers? Products that differ only in the order of the factors are considered identical.<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can the number 1,000,000 be expressed as the product of three natural numbers? Products that differ only in the order of the factors are considered identical.<|im_end|>
<|im_start|>assistant
<Z_189><Z_65><Z_428><Z_488><ANSWER>

Question: In how many different ways can the number 1,000,000 be expressed as the product of three natural numbers? Products that differ only in the order of the factors are considered identical.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_158><Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).<|im_end|>
<|im_start|>assistant
<Z_2><Z_367><Z_511><ANSWER>

Question: On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?<|im_end|>
<|im_start|>assistant
<Z_477><Z_478><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?<|im_end|>
<|im_start|>assistant
<Z_446><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?<|im_end|>
<|im_start|>assistant
<Z_267><Z_398><Z_386><ANSWER>

Question: How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suzanne read the first 15 pages of her book on Monday. She read 16 more pages than that on Tuesday. Then there were 18 pages left. How many pages are in Suzanne's book altogether?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suzanne read the first 15 pages of her book on Monday. She read 16 more pages than that on Tuesday. Then there were 18 pages left. How many pages are in Suzanne's book altogether?<|im_end|>
<|im_start|>assistant
<Z_369><Z_308><Z_98><Z_439><Z_265><Z_391><Z_294><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suzanne read the first 15 pages of her book on Monday. She read 16 more pages than that on Tuesday. Then there were 18 pages left. How many pages are in Suzanne's book altogether?<|im_end|>
<|im_start|>assistant
<Z_436><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suzanne read the first 15 pages of her book on Monday. She read 16 more pages than that on Tuesday. Then there were 18 pages left. How many pages are in Suzanne's book altogether?<|im_end|>
<|im_start|>assistant
<Z_193><Z_37><Z_207><Z_108><Z_361><Z_117><Z_84><Z_500><ANSWER>

Question: Suzanne read the first 15 pages of her book on Monday. She read 16 more pages than that on Tuesday. Then there were 18 pages left. How many pages are in Suzanne's book altogether?