<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some students in the classroom, and their average age is 8 years old. If the age of Teacher Li is included, the average age becomes 11 years old. Given that Teacher Li's age is 32 years old, how many people are there in the classroom in total?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some students in the classroom, and their average age is 8 years old. If the age of Teacher Li is included, the average age becomes 11 years old. Given that Teacher Li's age is 32 years old, how many people are there in the classroom in total?<|im_end|>
<|im_start|>assistant
<Z_5><Z_174><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some students in the classroom, and their average age is 8 years old. If the age of Teacher Li is included, the average age becomes 11 years old. Given that Teacher Li's age is 32 years old, how many people are there in the classroom in total?<|im_end|>
<|im_start|>assistant
<Z_287><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are some students in the classroom, and their average age is 8 years old. If the age of Teacher Li is included, the average age becomes 11 years old. Given that Teacher Li's age is 32 years old, how many people are there in the classroom in total?<|im_end|>
<|im_start|>assistant
<Z_12><Z_105><Z_0><ANSWER>

Question: There are some students in the classroom, and their average age is 8 years old. If the age of Teacher Li is included, the average age becomes 11 years old. Given that Teacher Li's age is 32 years old, how many people are there in the classroom in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest integer $x$ such that $|8x^2-53x+21|$ is prime?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest integer $x$ such that $|8x^2-53x+21|$ is prime?<|im_end|>
<|im_start|>assistant
<Z_66><Z_337><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest integer $x$ such that $|8x^2-53x+21|$ is prime?<|im_end|>
<|im_start|>assistant
<Z_235><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest integer $x$ such that $|8x^2-53x+21|$ is prime?<|im_end|>
<|im_start|>assistant
<Z_364><Z_75><ANSWER>

Question: What is the smallest integer $x$ such that $|8x^2-53x+21|$ is prime?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?<|im_end|>
<|im_start|>assistant
<Z_176><Z_453><Z_424><Z_400><Z_408><Z_389><Z_425><Z_98><Z_330><Z_500><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?<|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
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?<|im_end|>
<|im_start|>assistant
<Z_415><Z_405><Z_140><Z_309><Z_125><Z_32><Z_476><Z_364><Z_23><Z_52><ANSWER>

Question: Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?<|im_end|>
<|im_start|>assistant
<Z_176><Z_446><Z_265><Z_339><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?<|im_end|>
<|im_start|>assistant
<Z_454><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?<|im_end|>
<|im_start|>assistant
<Z_48><Z_466><Z_193><Z_486><ANSWER>

Question: Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby can deadlift 300 pounds at 13. When he is 18, he can deadlift 100 pounds more than a certain percentage of his previous deadlift. He added 110 pounds per year. What is the percentage of his previous deadlift that Bobby can deadlift 100 pounds more than when he is 18?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby can deadlift 300 pounds at 13. When he is 18, he can deadlift 100 pounds more than a certain percentage of his previous deadlift. He added 110 pounds per year. What is the percentage of his previous deadlift that Bobby can deadlift 100 pounds more than when he is 18?<|im_end|>
<|im_start|>assistant
<Z_176><Z_418><Z_154><Z_509><Z_488><Z_141><Z_503><Z_208><Z_399><Z_256><Z_483><Z_363><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby can deadlift 300 pounds at 13. When he is 18, he can deadlift 100 pounds more than a certain percentage of his previous deadlift. He added 110 pounds per year. What is the percentage of his previous deadlift that Bobby can deadlift 100 pounds more than when he is 18?<|im_end|>
<|im_start|>assistant
<Z_245><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bobby can deadlift 300 pounds at 13. When he is 18, he can deadlift 100 pounds more than a certain percentage of his previous deadlift. He added 110 pounds per year. What is the percentage of his previous deadlift that Bobby can deadlift 100 pounds more than when he is 18?<|im_end|>
<|im_start|>assistant
<Z_11><Z_511><Z_177><Z_461><Z_413><Z_279><Z_510><Z_285><Z_348><Z_213><Z_194><Z_360><ANSWER>

Question: Bobby can deadlift 300 pounds at 13. When he is 18, he can deadlift 100 pounds more than a certain percentage of his previous deadlift. He added 110 pounds per year. What is the percentage of his previous deadlift that Bobby can deadlift 100 pounds more than when he is 18?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school conducted a health survey among all 1600 male and female students, using stratified sampling to select a sample of 200. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ___.<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school conducted a health survey among all 1600 male and female students, using stratified sampling to select a sample of 200. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ___.<|im_end|>
<|im_start|>assistant
<Z_358><Z_380><Z_65><Z_284><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school conducted a health survey among all 1600 male and female students, using stratified sampling to select a sample of 200. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ___.<|im_end|>
<|im_start|>assistant
<Z_143><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school conducted a health survey among all 1600 male and female students, using stratified sampling to select a sample of 200. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ___.<|im_end|>
<|im_start|>assistant
<Z_223><Z_220><Z_70><Z_148><ANSWER>

Question: A school conducted a health survey among all 1600 male and female students, using stratified sampling to select a sample of 200. It is known that 10 fewer girls were sampled than boys. The number of female students in the school should be ___.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If we express $4x^2 - 8x + 5$ in the form $a(x - h)^2 + k$, what is $a + h + k$?<|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
If we express $4x^2 - 8x + 5$ in the form $a(x - h)^2 + k$, what is $a + h + k$?<|im_end|>
<|im_start|>assistant
<Z_195><Z_107><Z_287><Z_256><Z_399><Z_268><Z_333><Z_192><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If we express $4x^2 - 8x + 5$ in the form $a(x - h)^2 + k$, what is $a + h + k$?<|im_end|>
<|im_start|>assistant
<Z_128><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If we express $4x^2 - 8x + 5$ in the form $a(x - h)^2 + k$, what is $a + h + k$?<|im_end|>
<|im_start|>assistant
<Z_421><Z_8><Z_431><Z_147><Z_469><Z_1><Z_338><Z_306><ANSWER>

Question: If we express $4x^2 - 8x + 5$ in the form $a(x - h)^2 + k$, what is $a + h + k$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $g(x),$ for positive integer values of $x,$ by:
\[ g(x) = \left\{
\begin{aligned}
\log_{16} x & \quad \text{if } \log_{16} x \text{ is an integer} \\
2 + g(x - 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $g(256).$<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $g(x),$ for positive integer values of $x,$ by:
\[ g(x) = \left\{
\begin{aligned}
\log_{16} x & \quad \text{if } \log_{16} x \text{ is an integer} \\
2 + g(x - 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $g(256).$<|im_end|>
<|im_start|>assistant
<Z_5><Z_55><Z_353><Z_467><Z_156><Z_471><Z_473><Z_174><Z_126><Z_260><Z_488><Z_183><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $g(x),$ for positive integer values of $x,$ by:
\[ g(x) = \left\{
\begin{aligned}
\log_{16} x & \quad \text{if } \log_{16} x \text{ is an integer} \\
2 + g(x - 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $g(256).$<|im_end|>
<|im_start|>assistant
<Z_438><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a function $g(x),$ for positive integer values of $x,$ by:
\[ g(x) = \left\{
\begin{aligned}
\log_{16} x & \quad \text{if } \log_{16} x \text{ is an integer} \\
2 + g(x - 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $g(256).$<|im_end|>
<|im_start|>assistant
<Z_123><Z_147><Z_479><Z_205><Z_92><Z_311><Z_395><Z_307><Z_189><Z_18><Z_106><Z_77><ANSWER>

Question:
Define a function $g(x),$ for positive integer values of $x,$ by:
\[ g(x) = \left\{
\begin{aligned}
\log_{16} x & \quad \text{if } \log_{16} x \text{ is an integer} \\
2 + g(x - 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $g(256).$


<|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
\[3x + \frac{1}{x^2}\]
for \(x > 0\).<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of
\[3x + \frac{1}{x^2}\]
for \(x > 0\).<|im_end|>
<|im_start|>assistant
<Z_5><Z_156><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
\[3x + \frac{1}{x^2}\]
for \(x > 0\).<|im_end|>
<|im_start|>assistant
<Z_441><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
\[3x + \frac{1}{x^2}\]
for \(x > 0\).<|im_end|>
<|im_start|>assistant
<Z_228><Z_284><ANSWER>

Question:
Find the minimum value of
\[3x + \frac{1}{x^2}\]
for \(x > 0\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are some boys and 44 girls. There are 6 happy boys and 4 sad girls. There are 4 boys who are neither happy nor sad. How many boys are there?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are some boys and 44 girls. There are 6 happy boys and 4 sad girls. There are 4 boys who are neither happy nor sad. How many boys are there?<|im_end|>
<|im_start|>assistant
<Z_176><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are some boys and 44 girls. There are 6 happy boys and 4 sad girls. There are 4 boys who are neither happy nor sad. How many boys are there?<|im_end|>
<|im_start|>assistant
<Z_204><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are some boys and 44 girls. There are 6 happy boys and 4 sad girls. There are 4 boys who are neither happy nor sad. How many boys are there?<|im_end|>
<|im_start|>assistant
<Z_472><Z_205><ANSWER>

Question: Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are some boys and 44 girls. There are 6 happy boys and 4 sad girls. There are 4 boys who are neither happy nor sad. How many boys are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the owner of a furniture shop charges his customer 20 % more than the cost price . if a customer paid rs . 3600 for a computer table , then what was the cost price of the computer table ?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the owner of a furniture shop charges his customer 20 % more than the cost price . if a customer paid rs . 3600 for a computer table , then what was the cost price of the computer table ?<|im_end|>
<|im_start|>assistant
<Z_5><Z_312><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the owner of a furniture shop charges his customer 20 % more than the cost price . if a customer paid rs . 3600 for a computer table , then what was the cost price of the computer table ?<|im_end|>
<|im_start|>assistant
<Z_506><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the owner of a furniture shop charges his customer 20 % more than the cost price . if a customer paid rs . 3600 for a computer table , then what was the cost price of the computer table ?<|im_end|>
<|im_start|>assistant
<Z_246><Z_8><ANSWER>

Question: the owner of a furniture shop charges his customer 20 % more than the cost price . if a customer paid rs . 3600 for a computer table , then what was the cost price of the computer table ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(x, y)$ listed in the following table lie on a straight line. Determine the value of $t$ if the point $(45, t)$ lies on the same line.
\begin{tabular}{c|c}
$x$ & $y$ \\ \hline
2 & 10 \\
6 & 26 \\
10 & 42 \\
\end{tabular}<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(x, y)$ listed in the following table lie on a straight line. Determine the value of $t$ if the point $(45, t)$ lies on the same line.
\begin{tabular}{c|c}
$x$ & $y$ \\ \hline
2 & 10 \\
6 & 26 \\
10 & 42 \\
\end{tabular}<|im_end|>
<|im_start|>assistant
<Z_5><Z_510><Z_301><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(x, y)$ listed in the following table lie on a straight line. Determine the value of $t$ if the point $(45, t)$ lies on the same line.
\begin{tabular}{c|c}
$x$ & $y$ \\ \hline
2 & 10 \\
6 & 26 \\
10 & 42 \\
\end{tabular}<|im_end|>
<|im_start|>assistant
<Z_160><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The points $(x, y)$ listed in the following table lie on a straight line. Determine the value of $t$ if the point $(45, t)$ lies on the same line.
\begin{tabular}{c|c}
$x$ & $y$ \\ \hline
2 & 10 \\
6 & 26 \\
10 & 42 \\
\end{tabular}<|im_end|>
<|im_start|>assistant
<Z_315><Z_114><Z_475><Z_480><ANSWER>

Question:
The points $(x, y)$ listed in the following table lie on a straight line. Determine the value of $t$ if the point $(45, t)$ lies on the same line.
\begin{tabular}{c|c}
$x$ & $y$ \\ \hline
2 & 10 \\
6 & 26 \\
10 & 42 \\
\end{tabular}


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(The full score for this question is 15 points) When the purchase price of a product is 80 yuan and it is sold for 90 yuan each, 400 units can be sold. It is known that for every 1 yuan increase in price, the sales volume decreases by 10 units. To achieve the maximum profit, what should be the selling price per unit?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(The full score for this question is 15 points) When the purchase price of a product is 80 yuan and it is sold for 90 yuan each, 400 units can be sold. It is known that for every 1 yuan increase in price, the sales volume decreases by 10 units. To achieve the maximum profit, what should be the selling price per unit?<|im_end|>
<|im_start|>assistant
<Z_72><Z_414><Z_316><Z_354><Z_112><Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(The full score for this question is 15 points) When the purchase price of a product is 80 yuan and it is sold for 90 yuan each, 400 units can be sold. It is known that for every 1 yuan increase in price, the sales volume decreases by 10 units. To achieve the maximum profit, what should be the selling price per unit?<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
(The full score for this question is 15 points) When the purchase price of a product is 80 yuan and it is sold for 90 yuan each, 400 units can be sold. It is known that for every 1 yuan increase in price, the sales volume decreases by 10 units. To achieve the maximum profit, what should be the selling price per unit?<|im_end|>
<|im_start|>assistant
<Z_470><Z_20><Z_349><Z_494><Z_424><Z_112><ANSWER>

Question: (The full score for this question is 15 points) When the purchase price of a product is 80 yuan and it is sold for 90 yuan each, 400 units can be sold. It is known that for every 1 yuan increase in price, the sales volume decreases by 10 units. To achieve the maximum profit, what should be the selling price per unit?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \).<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \).<|im_end|>
<|im_start|>assistant
<Z_195><Z_181><Z_350><Z_183><Z_452><Z_57><Z_312><Z_389><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \).<|im_end|>
<|im_start|>assistant
<Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \).<|im_end|>
<|im_start|>assistant
<Z_119><Z_504><Z_420><Z_279><Z_94><Z_86><Z_507><Z_113><ANSWER>

Question: Find \( n \) if \( n\cdot (n+1)! + (n+1)! = 5040 \), where \( n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1 \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bryan took a look at his books as well. If he has 56 books in each of his 9 bookshelves, how many books does he have in total?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bryan took a look at his books as well. If he has 56 books in each of his 9 bookshelves, how many books does he have in total?<|im_end|>
<|im_start|>assistant
<Z_5><Z_125><Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bryan took a look at his books as well. If he has 56 books in each of his 9 bookshelves, how many books does he have in total?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Bryan took a look at his books as well. If he has 56 books in each of his 9 bookshelves, how many books does he have in total?<|im_end|>
<|im_start|>assistant
<Z_120><Z_40><Z_212><ANSWER>

Question: Bryan took a look at his books as well. If he has 56 books in each of his 9 bookshelves, how many books does he have in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2743 is lent into two parts so that the interest on the first part for 8 years at 3% per annum may be equal to the interest on the second part for a certain number of years at 5% per annum. The second sum is Rs. 1688. For how many years is the second part lent?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2743 is lent into two parts so that the interest on the first part for 8 years at 3% per annum may be equal to the interest on the second part for a certain number of years at 5% per annum. The second sum is Rs. 1688. For how many years is the second part lent?<|im_end|>
<|im_start|>assistant
<Z_5><Z_254><Z_323><Z_500><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2743 is lent into two parts so that the interest on the first part for 8 years at 3% per annum may be equal to the interest on the second part for a certain number of years at 5% per annum. The second sum is Rs. 1688. For how many years is the second part lent?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2743 is lent into two parts so that the interest on the first part for 8 years at 3% per annum may be equal to the interest on the second part for a certain number of years at 5% per annum. The second sum is Rs. 1688. For how many years is the second part lent?<|im_end|>
<|im_start|>assistant
<Z_217><Z_342><Z_311><Z_235><ANSWER>

Question: A sum of Rs. 2743 is lent into two parts so that the interest on the first part for 8 years at 3% per annum may be equal to the interest on the second part for a certain number of years at 5% per annum. The second sum is Rs. 1688. For how many years is the second part lent?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_346><Z_500><Z_431><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_147><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.<|im_end|>
<|im_start|>assistant
<Z_259><Z_433><Z_447><Z_283><Z_172><ANSWER>

Question: A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the number $900$. What is the sum of all the positive even factors of this number?<|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
Consider the number $900$. What is the sum of all the positive even factors of this number?<|im_end|>
<|im_start|>assistant
<Z_5><Z_318><Z_270><Z_416><Z_497><Z_215><Z_509><Z_284><Z_141><Z_81><Z_159><Z_478><Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the number $900$. What is the sum of all the positive even factors of this number?<|im_end|>
<|im_start|>assistant
<Z_478><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the number $900$. What is the sum of all the positive even factors of this number?<|im_end|>
<|im_start|>assistant
<Z_196><Z_198><Z_105><Z_26><Z_435><Z_361><Z_422><Z_225><Z_230><Z_335><Z_229><Z_60><Z_355><ANSWER>

Question: Consider the number $900$. What is the sum of all the positive even factors of this number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of \(12^2 + 2(12)(5) + 5^2\)?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of \(12^2 + 2(12)(5) + 5^2\)?<|im_end|>
<|im_start|>assistant
<Z_66><Z_502><Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of \(12^2 + 2(12)(5) + 5^2\)?<|im_end|>
<|im_start|>assistant
<Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the value of \(12^2 + 2(12)(5) + 5^2\)?<|im_end|>
<|im_start|>assistant
<Z_433><Z_275><Z_73><ANSWER>

Question: What is the value of \(12^2 + 2(12)(5) + 5^2\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a set of 21 numbers whose average is 44. The average of the first 10 numbers is 48, and the average of the last 10 numbers is 41. Additionally, the average of the middle 7 numbers (consisting of the last number of the first 10 and the first number of the last 10, and 5 unknown numbers in between) is 45. Find the 5 missing numbers in the middle set.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a set of 21 numbers whose average is 44. The average of the first 10 numbers is 48, and the average of the last 10 numbers is 41. Additionally, the average of the middle 7 numbers (consisting of the last number of the first 10 and the first number of the last 10, and 5 unknown numbers in between) is 45. Find the 5 missing numbers in the middle set.<|im_end|>
<|im_start|>assistant
<Z_195><Z_488><Z_489><Z_92><Z_463><Z_488><Z_108><Z_180><Z_446><Z_478><Z_265><Z_103><Z_293><Z_453><Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a set of 21 numbers whose average is 44. The average of the first 10 numbers is 48, and the average of the last 10 numbers is 41. Additionally, the average of the middle 7 numbers (consisting of the last number of the first 10 and the first number of the last 10, and 5 unknown numbers in between) is 45. Find the 5 missing numbers in the middle set.<|im_end|>
<|im_start|>assistant
<Z_81><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a set of 21 numbers whose average is 44. The average of the first 10 numbers is 48, and the average of the last 10 numbers is 41. Additionally, the average of the middle 7 numbers (consisting of the last number of the first 10 and the first number of the last 10, and 5 unknown numbers in between) is 45. Find the 5 missing numbers in the middle set.<|im_end|>
<|im_start|>assistant
<Z_371><Z_501><Z_60><Z_1><Z_62><Z_104><Z_356><Z_135><Z_327><Z_368><Z_10><Z_252><Z_460><Z_306><Z_277><ANSWER>

Question: Consider a set of 21 numbers whose average is 44. The average of the first 10 numbers is 48, and the average of the last 10 numbers is 41. Additionally, the average of the middle 7 numbers (consisting of the last number of the first 10 and the first number of the last 10, and 5 unknown numbers in between) is 45. Find the 5 missing numbers in the middle set.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ table, positive numbers are placed such that the product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any $2 \times 2$ square is equal to 2. What number is in the center cell? Find all possible values and prove that there are no others.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ table, positive numbers are placed such that the product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any $2 \times 2$ square is equal to 2. What number is in the center cell? Find all possible values and prove that there are no others.<|im_end|>
<|im_start|>assistant
<Z_195><Z_289><Z_486><Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ table, positive numbers are placed such that the product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any $2 \times 2$ square is equal to 2. What number is in the center cell? Find all possible values and prove that there are no others.<|im_end|>
<|im_start|>assistant
<Z_151><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $3 \times 3$ table, positive numbers are placed such that the product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any $2 \times 2$ square is equal to 2. What number is in the center cell? Find all possible values and prove that there are no others.<|im_end|>
<|im_start|>assistant
<Z_256><Z_463><Z_93><Z_421><ANSWER>

Question: In a $3 \times 3$ table, positive numbers are placed such that the product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any $2 \times 2$ square is equal to 2. What number is in the center cell? Find all possible values and prove that there are no others.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed had 30 more marbles than Doug. Ed lost 21 of his marbles at the playground. If Ed had 91 marbles. How many more marbles did Ed have than Doug then?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed had 30 more marbles than Doug. Ed lost 21 of his marbles at the playground. If Ed had 91 marbles. How many more marbles did Ed have than Doug then?<|im_end|>
<|im_start|>assistant
<Z_72><Z_420><Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed had 30 more marbles than Doug. Ed lost 21 of his marbles at the playground. If Ed had 91 marbles. How many more marbles did Ed have than Doug then?<|im_end|>
<|im_start|>assistant
<Z_235><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ed had 30 more marbles than Doug. Ed lost 21 of his marbles at the playground. If Ed had 91 marbles. How many more marbles did Ed have than Doug then?<|im_end|>
<|im_start|>assistant
<Z_458><Z_35><Z_436><ANSWER>

Question: Ed had 30 more marbles than Doug. Ed lost 21 of his marbles at the playground. If Ed had 91 marbles. How many more marbles did Ed have than Doug then?


<|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 integers from $-50$ to $70$, inclusive?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the integers from $-50$ to $70$, inclusive?<|im_end|>
<|im_start|>assistant
<Z_5><Z_345><Z_48><Z_233><Z_70><Z_388><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 integers from $-50$ to $70$, inclusive?<|im_end|>
<|im_start|>assistant
<Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of the integers from $-50$ to $70$, inclusive?<|im_end|>
<|im_start|>assistant
<Z_129><Z_485><Z_138><Z_403><Z_209><Z_177><ANSWER>

Question: What is the sum of the integers from $-50$ to $70$, inclusive?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, at the end of 7 days, the total number of bees in the hive (including the queen) is 27201. How many bees did the queen have at the beginning?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, at the end of 7 days, the total number of bees in the hive (including the queen) is 27201. How many bees did the queen have at the beginning?<|im_end|>
<|im_start|>assistant
<Z_66><Z_118><Z_478><Z_349><Z_475><Z_339><Z_234><Z_300><Z_246><Z_256><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, at the end of 7 days, the total number of bees in the hive (including the queen) is 27201. How many bees did the queen have at the beginning?<|im_end|>
<|im_start|>assistant
<Z_424><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, at the end of 7 days, the total number of bees in the hive (including the queen) is 27201. How many bees did the queen have at the beginning?<|im_end|>
<|im_start|>assistant
<Z_204><Z_201><Z_59><Z_220><Z_197><Z_230><Z_343><Z_395><Z_330><Z_29><ANSWER>

Question: 3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, at the end of 7 days, the total number of bees in the hive (including the queen) is 27201. How many bees did the queen have at the beginning?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden.
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden.
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_66><Z_376><Z_118><Z_394><Z_65><Z_259><Z_405><Z_472><Z_417><Z_340><Z_478><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden.
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden.
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_326><Z_507><Z_498><Z_66><Z_467><Z_501><Z_415><Z_45><Z_269><Z_315><Z_511><ANSWER>

Question:
Determine the number of possible paths moving only south and east along the line segments from $A$ to $B$ on a 10x4 grid where certain paths (specific line segments) are forbidden.
[asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 10; ++i)
if (i != 4 && i != 5 && i != 7)
    draw((2*i,0)--(2*i,4));
for(int j = 0; j <= 4; ++j)
    draw((0,j)--(20,j));
draw((2*4,0)--(2*4,1), red);
draw((2*5,0)--(2*5,1), red);
draw((2*4,3)--(2*4,4), red);
draw((2*5,3)--(2*5,4), red);
draw((2*7,0)--(2*7,2), red);
draw((2*7,3)--(2*7,4), red);
label("$A$",(0,4),NW);
label("$B$",(20,0),E);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Peter has four horses. Each one eats 4 pounds of oats, twice a day, and 3 pounds of grain once a day. How many pounds of food does he need to feed his horses for 3 days?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Peter has four horses. Each one eats 4 pounds of oats, twice a day, and 3 pounds of grain once a day. How many pounds of food does he need to feed his horses for 3 days?<|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
Peter has four horses. Each one eats 4 pounds of oats, twice a day, and 3 pounds of grain once a day. How many pounds of food does he need to feed his horses for 3 days?<|im_end|>
<|im_start|>assistant
<Z_384><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Peter has four horses. Each one eats 4 pounds of oats, twice a day, and 3 pounds of grain once a day. How many pounds of food does he need to feed his horses for 3 days?<|im_end|>
<|im_start|>assistant
<Z_444><ANSWER>

Question: Peter has four horses. Each one eats 4 pounds of oats, twice a day, and 3 pounds of grain once a day. How many pounds of food does he need to feed his horses for 3 days?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse?<|im_end|>
<|im_start|>assistant
<Z_5><Z_237><Z_184><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse?<|im_end|>
<|im_start|>assistant
<Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse?<|im_end|>
<|im_start|>assistant
<Z_456><Z_80><Z_419><ANSWER>

Question: What is the area, in square inches, of a right triangle with a 15-inch leg and a 17-inch hypotenuse?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter minus 8. What is the number of units in the perimeter of this rectangle?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter minus 8. What is the number of units in the perimeter of this rectangle?<|im_end|>
<|im_start|>assistant
<Z_176><Z_385><Z_409><Z_232><Z_453><Z_96><Z_328><Z_394><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter minus 8. What is the number of units in the perimeter of this rectangle?<|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
A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter minus 8. What is the number of units in the perimeter of this rectangle?<|im_end|>
<|im_start|>assistant
<Z_282><Z_275><Z_300><Z_268><Z_408><Z_497><Z_326><Z_375><ANSWER>

Question: A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter minus 8. What is the number of units in the perimeter of this rectangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Betty has 3.0 boxes containing 24 oranges each. How many oranges does Betty have?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Betty has 3.0 boxes containing 24 oranges each. How many oranges does Betty have?<|im_end|>
<|im_start|>assistant
<Z_5><Z_94><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Betty has 3.0 boxes containing 24 oranges each. How many oranges does Betty have?<|im_end|>
<|im_start|>assistant
<Z_294><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Betty has 3.0 boxes containing 24 oranges each. How many oranges does Betty have?<|im_end|>
<|im_start|>assistant
<Z_128><Z_233><Z_203><ANSWER>

Question: Betty has 3.0 boxes containing 24 oranges each. How many oranges does Betty have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one and a half women drink one and a half tea in one and a half minutes, then 9 women can drink 18 tea in some minutes. How many minutes does it take for the 9 women to drink 18 tea?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one and a half women drink one and a half tea in one and a half minutes, then 9 women can drink 18 tea in some minutes. How many minutes does it take for the 9 women to drink 18 tea?<|im_end|>
<|im_start|>assistant
<Z_5><Z_238><Z_358><Z_502><Z_127><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one and a half women drink one and a half tea in one and a half minutes, then 9 women can drink 18 tea in some minutes. How many minutes does it take for the 9 women to drink 18 tea?<|im_end|>
<|im_start|>assistant
<Z_308><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If one and a half women drink one and a half tea in one and a half minutes, then 9 women can drink 18 tea in some minutes. How many minutes does it take for the 9 women to drink 18 tea?<|im_end|>
<|im_start|>assistant
<Z_29><Z_453><Z_448><Z_334><Z_491><Z_40><ANSWER>

Question: If one and a half women drink one and a half tea in one and a half minutes, then 9 women can drink 18 tea in some minutes. How many minutes does it take for the 9 women to drink 18 tea?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?<|im_end|>
<|im_start|>assistant
<Z_66><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?<|im_end|>
<|im_start|>assistant
<Z_102><Z_354><ANSWER>

Question: A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes Nissa 10 seconds to clip each of her cats' nails, 90 seconds to clean each of her ears, and some minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total if it takes 640 seconds?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes Nissa 10 seconds to clip each of her cats' nails, 90 seconds to clean each of her ears, and some minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total if it takes 640 seconds?<|im_end|>
<|im_start|>assistant
<Z_5><Z_300><Z_75><Z_213><Z_174><Z_485><Z_244><Z_379><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes Nissa 10 seconds to clip each of her cats' nails, 90 seconds to clean each of her ears, and some minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total if it takes 640 seconds?<|im_end|>
<|im_start|>assistant
<Z_207><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes Nissa 10 seconds to clip each of her cats' nails, 90 seconds to clean each of her ears, and some minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total if it takes 640 seconds?<|im_end|>
<|im_start|>assistant
<Z_146><Z_13><Z_442><Z_62><Z_259><Z_33><Z_420><Z_194><ANSWER>

Question: It takes Nissa 10 seconds to clip each of her cats' nails, 90 seconds to clean each of her ears, and some minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total if it takes 640 seconds?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with some tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Ralph did not hit 110 tennis balls. How many tennis balls did he load up the machine with?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with some tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Ralph did not hit 110 tennis balls. How many tennis balls did he load up the machine with?<|im_end|>
<|im_start|>assistant
<Z_195><Z_256><Z_200><Z_319><Z_135><Z_281><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with some tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Ralph did not hit 110 tennis balls. How many tennis balls did he load up the machine with?<|im_end|>
<|im_start|>assistant
<Z_429><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with some tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Ralph did not hit 110 tennis balls. How many tennis balls did he load up the machine with?<|im_end|>
<|im_start|>assistant
<Z_145><Z_386><Z_305><Z_182><Z_435><Z_365><ANSWER>

Question: Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with some tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Ralph did not hit 110 tennis balls. How many tennis balls did he load up the machine with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?<|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
On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_195><Z_325><Z_422><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?<|im_end|>
<|im_start|>assistant
<Z_275><Z_83><Z_282><ANSWER>

Question: On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandy gets 3 marks for each correct sum and loses 2 marks for each incorrect sum. Sandy attempts 30 sums and gets 22 sums correct. How many marks did Sandy obtain in total?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandy gets 3 marks for each correct sum and loses 2 marks for each incorrect sum. Sandy attempts 30 sums and gets 22 sums correct. How many marks did Sandy obtain in total?<|im_end|>
<|im_start|>assistant
<Z_176><Z_442><Z_320><Z_198><Z_354><Z_121><Z_484><Z_338><Z_270><Z_322><Z_34><Z_235><Z_452><Z_319><Z_331><Z_237><Z_507><Z_364><Z_312><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandy gets 3 marks for each correct sum and loses 2 marks for each incorrect sum. Sandy attempts 30 sums and gets 22 sums correct. How many marks did Sandy obtain in total?<|im_end|>
<|im_start|>assistant
<Z_409><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sandy gets 3 marks for each correct sum and loses 2 marks for each incorrect sum. Sandy attempts 30 sums and gets 22 sums correct. How many marks did Sandy obtain in total?<|im_end|>
<|im_start|>assistant
<Z_16><Z_483><Z_183><Z_65><Z_443><Z_511><Z_307><Z_374><Z_361><Z_358><Z_483><Z_194><Z_84><Z_76><Z_234><Z_469><Z_34><Z_222><Z_337><ANSWER>

Question: Sandy gets 3 marks for each correct sum and loses 2 marks for each incorrect sum. Sandy attempts 30 sums and gets 22 sums correct. How many marks did Sandy obtain in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain lab experiment with white and brown mice only, 2/3 of the mice are white. There are 14 white mice in the experiment. How many brown mice are in the experiment?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain lab experiment with white and brown mice only, 2/3 of the mice are white. There are 14 white mice in the experiment. How many brown mice are in the experiment?<|im_end|>
<|im_start|>assistant
<Z_66><Z_412><Z_233><Z_235><Z_442><Z_353><Z_157><Z_418><Z_295><Z_502><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain lab experiment with white and brown mice only, 2/3 of the mice are white. There are 14 white mice in the experiment. How many brown mice are in the experiment?<|im_end|>
<|im_start|>assistant
<Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a certain lab experiment with white and brown mice only, 2/3 of the mice are white. There are 14 white mice in the experiment. How many brown mice are in the experiment?<|im_end|>
<|im_start|>assistant
<Z_192><Z_428><Z_232><Z_31><Z_310><Z_322><Z_488><Z_272><Z_120><Z_495><ANSWER>

Question: In a certain lab experiment with white and brown mice only, 2/3 of the mice are white. There are 14 white mice in the experiment. How many brown mice are in the experiment?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miyoung is ranked 12th from the front and 12th from the back in math scores of her class. Miyoung's school has a total of 3 grades, each grade has 12 classes, and each class has the same number of students. How many students are there in this school?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miyoung is ranked 12th from the front and 12th from the back in math scores of her class. Miyoung's school has a total of 3 grades, each grade has 12 classes, and each class has the same number of students. How many students are there in this school?<|im_end|>
<|im_start|>assistant
<Z_358><Z_65><Z_452><Z_341><Z_450><Z_392><Z_287><Z_500><Z_276><Z_329><Z_81><Z_277><Z_299><Z_320><Z_367><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miyoung is ranked 12th from the front and 12th from the back in math scores of her class. Miyoung's school has a total of 3 grades, each grade has 12 classes, and each class has the same number of students. How many students are there in this school?<|im_end|>
<|im_start|>assistant
<Z_339><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miyoung is ranked 12th from the front and 12th from the back in math scores of her class. Miyoung's school has a total of 3 grades, each grade has 12 classes, and each class has the same number of students. How many students are there in this school?<|im_end|>
<|im_start|>assistant
<Z_195><Z_420><Z_472><Z_263><Z_344><Z_153><Z_342><Z_51><Z_441><Z_368><Z_372><Z_32><Z_30><Z_478><Z_311><ANSWER>

Question: Miyoung is ranked 12th from the front and 12th from the back in math scores of her class. Miyoung's school has a total of 3 grades, each grade has 12 classes, and each class has the same number of students. How many students are there in this school?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Amy, Jeremy, and Chris have a combined age of 132.  Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy.  How old is Jeremy?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Amy, Jeremy, and Chris have a combined age of 132.  Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy.  How old is Jeremy?<|im_end|>
<|im_start|>assistant
<Z_176><Z_293><Z_244><Z_458><Z_293><Z_333><Z_445><Z_151><Z_214><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Amy, Jeremy, and Chris have a combined age of 132.  Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy.  How old is Jeremy?<|im_end|>
<|im_start|>assistant
<Z_298><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Amy, Jeremy, and Chris have a combined age of 132.  Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy.  How old is Jeremy?<|im_end|>
<|im_start|>assistant
<Z_321><Z_354><Z_241><Z_390><Z_448><Z_168><Z_421><Z_94><Z_164><ANSWER>

Question: Amy, Jeremy, and Chris have a combined age of 132.  Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy.  How old is Jeremy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?<|im_end|>
<|im_start|>assistant
<Z_176><Z_159><Z_268><Z_391><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?<|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
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?<|im_end|>
<|im_start|>assistant
<Z_201><Z_381><Z_321><Z_503><Z_367><ANSWER>

Question: An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all complex numbers $z$ such that $z \overline{z}^3 + \overline{z} z^3 = 450$, where both the real and imaginary parts of $z$ are integers. Plot these complex numbers in the complex plane and calculate the area of the quadrilateral formed by these complex numbers as vertices.<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all complex numbers $z$ such that $z \overline{z}^3 + \overline{z} z^3 = 450$, where both the real and imaginary parts of $z$ are integers. Plot these complex numbers in the complex plane and calculate the area of the quadrilateral formed by these complex numbers as vertices.<|im_end|>
<|im_start|>assistant
<Z_5><Z_493><Z_48><Z_254><Z_157><Z_380><Z_259><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find all complex numbers $z$ such that $z \overline{z}^3 + \overline{z} z^3 = 450$, where both the real and imaginary parts of $z$ are integers. Plot these complex numbers in the complex plane and calculate the area of the quadrilateral formed by these complex numbers as vertices.<|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
Find all complex numbers $z$ such that $z \overline{z}^3 + \overline{z} z^3 = 450$, where both the real and imaginary parts of $z$ are integers. Plot these complex numbers in the complex plane and calculate the area of the quadrilateral formed by these complex numbers as vertices.<|im_end|>
<|im_start|>assistant
<Z_244><Z_141><Z_181><Z_506><Z_233><Z_467><Z_161><ANSWER>

Question: Find all complex numbers $z$ such that $z \overline{z}^3 + \overline{z} z^3 = 450$, where both the real and imaginary parts of $z$ are integers. Plot these complex numbers in the complex plane and calculate the area of the quadrilateral formed by these complex numbers as vertices.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Teas worth Rs. 126 per kg and some amount per kg are mixed with a third variety in the ratio 1 : 1 : 2. The mixture is worth Rs. 152 per kg, and the price of the third variety per kg is Rs. 173.5. What is the price of the second variety per kg?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Teas worth Rs. 126 per kg and some amount per kg are mixed with a third variety in the ratio 1 : 1 : 2. The mixture is worth Rs. 152 per kg, and the price of the third variety per kg is Rs. 173.5. What is the price of the second variety per kg?<|im_end|>
<|im_start|>assistant
<Z_176><Z_413><Z_342><Z_366><Z_233><Z_465><Z_79><Z_497><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Teas worth Rs. 126 per kg and some amount per kg are mixed with a third variety in the ratio 1 : 1 : 2. The mixture is worth Rs. 152 per kg, and the price of the third variety per kg is Rs. 173.5. What is the price of the second variety per kg?<|im_end|>
<|im_start|>assistant
<Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Teas worth Rs. 126 per kg and some amount per kg are mixed with a third variety in the ratio 1 : 1 : 2. The mixture is worth Rs. 152 per kg, and the price of the third variety per kg is Rs. 173.5. What is the price of the second variety per kg?<|im_end|>
<|im_start|>assistant
<Z_209><Z_438><Z_238><Z_229><Z_218><Z_40><Z_477><Z_47><ANSWER>

Question: Teas worth Rs. 126 per kg and some amount per kg are mixed with a third variety in the ratio 1 : 1 : 2. The mixture is worth Rs. 152 per kg, and the price of the third variety per kg is Rs. 173.5. What is the price of the second variety per kg?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of 81 consecutive integers is $9^5$. What is their median?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of 81 consecutive integers is $9^5$. What is their median?<|im_end|>
<|im_start|>assistant
<Z_195><Z_380><Z_138><Z_131><Z_389><Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of 81 consecutive integers is $9^5$. What is their median?<|im_end|>
<|im_start|>assistant
<Z_484><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of 81 consecutive integers is $9^5$. What is their median?<|im_end|>
<|im_start|>assistant
<Z_407><Z_489><Z_379><Z_396><Z_252><Z_470><ANSWER>

Question: The sum of 81 consecutive integers is $9^5$. What is their median?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the length of a rectangular solid whose width is 8 meters, depth is 5 meters, and total surface area is 314 square meters?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the length of a rectangular solid whose width is 8 meters, depth is 5 meters, and total surface area is 314 square meters?<|im_end|>
<|im_start|>assistant
<Z_176><Z_389><Z_319><Z_265><Z_385><Z_183><Z_199><Z_298><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the length of a rectangular solid whose width is 8 meters, depth is 5 meters, and total surface area is 314 square meters?<|im_end|>
<|im_start|>assistant
<Z_500><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the length of a rectangular solid whose width is 8 meters, depth is 5 meters, and total surface area is 314 square meters?<|im_end|>
<|im_start|>assistant
<Z_333><Z_230><Z_498><Z_18><Z_265><Z_360><Z_505><Z_10><ANSWER>

Question: What is the length of a rectangular solid whose width is 8 meters, depth is 5 meters, and total surface area is 314 square meters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 1200 m long train crosses a tree in some time. It takes 220 sec to pass a platform 1000 m long. How much time does it take to cross the tree?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 1200 m long train crosses a tree in some time. It takes 220 sec to pass a platform 1000 m long. How much time does it take to cross the tree?<|im_end|>
<|im_start|>assistant
<Z_66><Z_139><Z_303><Z_233><Z_478><Z_133><Z_234><Z_292><Z_383><Z_349><Z_358><Z_388><Z_193><Z_441><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 1200 m long train crosses a tree in some time. It takes 220 sec to pass a platform 1000 m long. How much time does it take to cross the tree?<|im_end|>
<|im_start|>assistant
<Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 1200 m long train crosses a tree in some time. It takes 220 sec to pass a platform 1000 m long. How much time does it take to cross the tree?<|im_end|>
<|im_start|>assistant
<Z_431><Z_351><Z_424><Z_17><Z_96><Z_143><Z_161><Z_325><Z_270><Z_114><Z_49><Z_233><Z_148><Z_498><Z_163><ANSWER>

Question: A 1200 m long train crosses a tree in some time. It takes 220 sec to pass a platform 1000 m long. How much time does it take to cross the tree?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Workers at a campaign office have some fliers to send out. They send out 1/5 of them in the morning and 1/4 of the remaining ones out during the afternoon. They have 1500 fliers left for the next day. How many fliers did they have initially?<|im_end|>
<|im_start|>assistant
<Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Workers at a campaign office have some fliers to send out. They send out 1/5 of them in the morning and 1/4 of the remaining ones out during the afternoon. They have 1500 fliers left for the next day. How many fliers did they have initially?<|im_end|>
<|im_start|>assistant
<Z_358><Z_179><Z_80><Z_430><Z_92><Z_42><Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Workers at a campaign office have some fliers to send out. They send out 1/5 of them in the morning and 1/4 of the remaining ones out during the afternoon. They have 1500 fliers left for the next day. How many fliers did they have initially?<|im_end|>
<|im_start|>assistant
<Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Workers at a campaign office have some fliers to send out. They send out 1/5 of them in the morning and 1/4 of the remaining ones out during the afternoon. They have 1500 fliers left for the next day. How many fliers did they have initially?<|im_end|>
<|im_start|>assistant
<Z_483><Z_42><Z_158><Z_504><Z_421><Z_266><Z_381><ANSWER>

Question: Workers at a campaign office have some fliers to send out. They send out 1/5 of them in the morning and 1/4 of the remaining ones out during the afternoon. They have 1500 fliers left for the next day. How many fliers did they have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 11 days was finished 3 days earlier after 10 more men joined. How many men were employed in total to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 11 days was finished 3 days earlier after 10 more men joined. How many men were employed in total to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_195><Z_440><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 11 days was finished 3 days earlier after 10 more men joined. How many men were employed in total to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_494><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A work which could be finished in 11 days was finished 3 days earlier after 10 more men joined. How many men were employed in total to finish the work earlier?<|im_end|>
<|im_start|>assistant
<Z_350><Z_208><Z_504><ANSWER>

Question: A work which could be finished in 11 days was finished 3 days earlier after 10 more men joined. How many men were employed in total to finish the work earlier?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let GCF(a, b) represent the greatest common factor of a and b, and LCM(c, d) represent the least common multiple of c and d. Calculate GCF(LCM(9, 15), LCM(10, 21)).<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let GCF(a, b) represent the greatest common factor of a and b, and LCM(c, d) represent the least common multiple of c and d. Calculate GCF(LCM(9, 15), LCM(10, 21)).<|im_end|>
<|im_start|>assistant
<Z_5><Z_381><Z_293><Z_289><Z_233><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let GCF(a, b) represent the greatest common factor of a and b, and LCM(c, d) represent the least common multiple of c and d. Calculate GCF(LCM(9, 15), LCM(10, 21)).<|im_end|>
<|im_start|>assistant
<Z_416><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let GCF(a, b) represent the greatest common factor of a and b, and LCM(c, d) represent the least common multiple of c and d. Calculate GCF(LCM(9, 15), LCM(10, 21)).<|im_end|>
<|im_start|>assistant
<Z_14><Z_484><Z_240><Z_194><Z_483><ANSWER>

Question: Let GCF(a, b) represent the greatest common factor of a and b, and LCM(c, d) represent the least common multiple of c and d. Calculate GCF(LCM(9, 15), LCM(10, 21)).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).<|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
For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

Question:
For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many real numbers $b$ does the quadratic equation $x^2 + bx + 10b = 0$ have only integer roots for $x$?<|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
For how many real numbers $b$ does the quadratic equation $x^2 + bx + 10b = 0$ have only integer roots for $x$?<|im_end|>
<|im_start|>assistant
<Z_5><Z_152><Z_346><Z_256><Z_100><Z_206><Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many real numbers $b$ does the quadratic equation $x^2 + bx + 10b = 0$ have only integer roots for $x$?<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many real numbers $b$ does the quadratic equation $x^2 + bx + 10b = 0$ have only integer roots for $x$?<|im_end|>
<|im_start|>assistant
<Z_205><Z_332><Z_202><Z_61><Z_444><Z_199><Z_178><ANSWER>

Question: For how many real numbers $b$ does the quadratic equation $x^2 + bx + 10b = 0$ have only integer roots for $x$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Square each integer $n$ in the range $1 \leq n \leq 15$ and find the remainders when the squares are divided by $16$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $16$?<|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
Square each integer $n$ in the range $1 \leq n \leq 15$ and find the remainders when the squares are divided by $16$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $16$?<|im_end|>
<|im_start|>assistant
<Z_176><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Square each integer $n$ in the range $1 \leq n \leq 15$ and find the remainders when the squares are divided by $16$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $16$?<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Square each integer $n$ in the range $1 \leq n \leq 15$ and find the remainders when the squares are divided by $16$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $16$?<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

Question: Square each integer $n$ in the range $1 \leq n \leq 15$ and find the remainders when the squares are divided by $16$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $16$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2665 is lent into two parts so that the interest on the first part for some years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. The second sum is Rs. 1640. For how many years was the first part lent?<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2665 is lent into two parts so that the interest on the first part for some years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. The second sum is Rs. 1640. For how many years was the first part lent?<|im_end|>
<|im_start|>assistant
<Z_5><Z_463><Z_182><Z_278><Z_400><Z_89><Z_375><Z_201><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2665 is lent into two parts so that the interest on the first part for some years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. The second sum is Rs. 1640. For how many years was the first part lent?<|im_end|>
<|im_start|>assistant
<Z_408><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A sum of Rs. 2665 is lent into two parts so that the interest on the first part for some years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. The second sum is Rs. 1640. For how many years was the first part lent?<|im_end|>
<|im_start|>assistant
<Z_490><Z_192><Z_378><Z_448><Z_250><Z_405><Z_302><Z_226><ANSWER>

Question: A sum of Rs. 2665 is lent into two parts so that the interest on the first part for some years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. The second sum is Rs. 1640. For how many years was the first part lent?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?<|im_end|>
<|im_start|>assistant
<Z_358><Z_295><Z_318><Z_251><Z_277><Z_485><Z_235><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?<|im_end|>
<|im_start|>assistant
<Z_454><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?<|im_end|>
<|im_start|>assistant
<Z_208><Z_171><Z_333><Z_227><Z_134><Z_63><Z_378><ANSWER>

Question: The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_1=1$ and $a_{n+1}=2S_n+3$. Find $S_5$.<|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
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_1=1$ and $a_{n+1}=2S_n+3$. Find $S_5$.<|im_end|>
<|im_start|>assistant
<Z_5><Z_293><Z_180><Z_478><Z_61><Z_391><Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_1=1$ and $a_{n+1}=2S_n+3$. Find $S_5$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_1=1$ and $a_{n+1}=2S_n+3$. Find $S_5$.<|im_end|>
<|im_start|>assistant
<Z_408><Z_134><Z_285><Z_205><Z_351><Z_133><Z_293><ANSWER>

Question: Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_1=1$ and $a_{n+1}=2S_n+3$. Find $S_5$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?<|im_end|>
<|im_start|>assistant
<Z_195><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?<|im_end|>
<|im_start|>assistant
<Z_176><Z_365><Z_386><Z_33><Z_323><Z_244><Z_153><Z_135><Z_248><Z_270><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?<|im_end|>
<|im_start|>assistant
<Z_500><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?<|im_end|>
<|im_start|>assistant
<Z_77><Z_388><Z_364><Z_326><Z_460><Z_52><Z_446><Z_322><Z_402><Z_358><ANSWER>

Question: How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?