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

There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

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

There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)<|im_end|>
<|im_start|>assistant
<Z_369><Z_419><Z_321><Z_391><Z_475><ANSWER>

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

There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)<|im_end|>
<|im_start|>assistant
<Z_91><ANSWER>

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

There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)<|im_end|>
<|im_start|>assistant
<Z_34><Z_431><Z_228><Z_235><Z_120><ANSWER>

Question:
There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A firm is comprised of partners, associates, and managers with the ratio of 2 : 63 : 20. If 35 more associates and 10 more managers were hired, the ratio of partners to associates to managers would become 1 : 34 : 15. How many partners are currently in the firm?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A firm is comprised of partners, associates, and managers with the ratio of 2 : 63 : 20. If 35 more associates and 10 more managers were hired, the ratio of partners to associates to managers would become 1 : 34 : 15. How many partners are currently in the firm?<|im_end|>
<|im_start|>assistant
<Z_24><Z_381><Z_144><Z_152><Z_264><Z_467><Z_128><Z_322><Z_244><Z_328><Z_211><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A firm is comprised of partners, associates, and managers with the ratio of 2 : 63 : 20. If 35 more associates and 10 more managers were hired, the ratio of partners to associates to managers would become 1 : 34 : 15. How many partners are currently in the firm?<|im_end|>
<|im_start|>assistant
<Z_180><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A firm is comprised of partners, associates, and managers with the ratio of 2 : 63 : 20. If 35 more associates and 10 more managers were hired, the ratio of partners to associates to managers would become 1 : 34 : 15. How many partners are currently in the firm?<|im_end|>
<|im_start|>assistant
<Z_306><Z_441><Z_481><Z_219><Z_427><Z_372><Z_343><Z_120><Z_141><Z_325><Z_127><ANSWER>

Question: A firm is comprised of partners, associates, and managers with the ratio of 2 : 63 : 20. If 35 more associates and 10 more managers were hired, the ratio of partners to associates to managers would become 1 : 34 : 15. How many partners are currently in the firm?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of permutations of 8 letters \( A, B, C, D, E, F, G, H \) where exactly 4 elements are not in their original positions.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of permutations of 8 letters \( A, B, C, D, E, F, G, H \) where exactly 4 elements are not in their original positions.<|im_end|>
<|im_start|>assistant
<Z_369><Z_171><Z_87><Z_502><Z_433><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of permutations of 8 letters \( A, B, C, D, E, F, G, H \) where exactly 4 elements are not in their original positions.<|im_end|>
<|im_start|>assistant
<Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of permutations of 8 letters \( A, B, C, D, E, F, G, H \) where exactly 4 elements are not in their original positions.<|im_end|>
<|im_start|>assistant
<Z_152><Z_255><Z_264><Z_387><Z_503><Z_307><ANSWER>

Question: Find the number of permutations of 8 letters \( A, B, C, D, E, F, G, H \) where exactly 4 elements are not in their original positions.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circular artificial lake has a circumference of 1200 meters. It is planned to plant a willow tree every 10 meters along the lake's edge, with a poplar tree planted between every two willow trees. How many willow and poplar trees are needed in total?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circular artificial lake has a circumference of 1200 meters. It is planned to plant a willow tree every 10 meters along the lake's edge, with a poplar tree planted between every two willow trees. How many willow and poplar trees are needed in total?<|im_end|>
<|im_start|>assistant
<Z_369><Z_87><Z_453><Z_273><Z_321><Z_307><Z_255><Z_67><Z_144><Z_193><Z_346><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circular artificial lake has a circumference of 1200 meters. It is planned to plant a willow tree every 10 meters along the lake's edge, with a poplar tree planted between every two willow trees. How many willow and poplar trees are needed in total?<|im_end|>
<|im_start|>assistant
<Z_17><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A circular artificial lake has a circumference of 1200 meters. It is planned to plant a willow tree every 10 meters along the lake's edge, with a poplar tree planted between every two willow trees. How many willow and poplar trees are needed in total?<|im_end|>
<|im_start|>assistant
<Z_421><Z_297><Z_193><Z_340><Z_386><Z_481><Z_181><Z_203><Z_110><Z_137><Z_448><ANSWER>

Question: A circular artificial lake has a circumference of 1200 meters. It is planned to plant a willow tree every 10 meters along the lake's edge, with a poplar tree planted between every two willow trees. How many willow and poplar trees are needed 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 $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(-3, 0)$ and $B(-2, 5)$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(-3, 0)$ and $B(-2, 5)$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_391><Z_176><Z_99><Z_175><Z_176><Z_344><Z_54><Z_424><Z_247><Z_140><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(-3, 0)$ and $B(-2, 5)$?<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(-3, 0)$ and $B(-2, 5)$?<|im_end|>
<|im_start|>assistant
<Z_112><Z_7><Z_487><Z_119><Z_80><Z_264><Z_173><Z_147><Z_167><Z_497><Z_133><ANSWER>

Question: What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(-3, 0)$ and $B(-2, 5)$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest possible number of whole 3-by-4 rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest possible number of whole 3-by-4 rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest possible number of whole 3-by-4 rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?<|im_end|>
<|im_start|>assistant
<Z_362><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 possible number of whole 3-by-4 rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?<|im_end|>
<|im_start|>assistant
<Z_312><ANSWER>

Question: What is the smallest possible number of whole 3-by-4 rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien invades Earth and abducts 1000 people. It returns 52.8% of the people abducted. After that, it takes 135 people to Planet Zog, 88 people to Planet Xelbor, and 45 people to Planet Qyruis. The remaining abductees are taken to the alien's home planet. How many people did the alien take to its home planet?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien invades Earth and abducts 1000 people. It returns 52.8% of the people abducted. After that, it takes 135 people to Planet Zog, 88 people to Planet Xelbor, and 45 people to Planet Qyruis. The remaining abductees are taken to the alien's home planet. How many people did the alien take to its home planet?<|im_end|>
<|im_start|>assistant
<Z_369><Z_425><Z_352><Z_296><Z_250><Z_63><Z_89><Z_503><Z_318><Z_232><Z_77><Z_289><Z_50><Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien invades Earth and abducts 1000 people. It returns 52.8% of the people abducted. After that, it takes 135 people to Planet Zog, 88 people to Planet Xelbor, and 45 people to Planet Qyruis. The remaining abductees are taken to the alien's home planet. How many people did the alien take to its home planet?<|im_end|>
<|im_start|>assistant
<Z_489><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien invades Earth and abducts 1000 people. It returns 52.8% of the people abducted. After that, it takes 135 people to Planet Zog, 88 people to Planet Xelbor, and 45 people to Planet Qyruis. The remaining abductees are taken to the alien's home planet. How many people did the alien take to its home planet?<|im_end|>
<|im_start|>assistant
<Z_282><Z_345><Z_201><Z_484><Z_135><Z_362><Z_352><Z_495><Z_319><Z_477><Z_489><Z_498><Z_137><Z_205><ANSWER>

Question: An alien invades Earth and abducts 1000 people. It returns 52.8% of the people abducted. After that, it takes 135 people to Planet Zog, 88 people to Planet Xelbor, and 45 people to Planet Qyruis. The remaining abductees are taken to the alien's home planet. How many people did the alien take to its home planet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 1950 for which the sum of the digits is 18?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 1950 for which the sum of the digits is 18?<|im_end|>
<|im_start|>assistant
<Z_24><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 1950 for which the sum of the digits is 18?<|im_end|>
<|im_start|>assistant
<Z_364><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the first year after 1950 for which the sum of the digits is 18?<|im_end|>
<|im_start|>assistant
<Z_237><Z_405><ANSWER>

Question: What is the first year after 1950 for which the sum of the digits is 18?


<|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 all positive integers less than 200 that are squares of perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers less than 200 that are squares of perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_477><Z_84><Z_217><Z_434><Z_83><Z_341><Z_321><Z_313><Z_144><Z_59><Z_456><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 all positive integers less than 200 that are squares of perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_89><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 all positive integers less than 200 that are squares of perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_174><Z_208><Z_348><Z_53><Z_253><Z_350><Z_263><Z_487><Z_453><Z_68><Z_185><ANSWER>

Question: What is the sum of all positive integers less than 200 that are squares of perfect cubes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]<|im_end|>
<|im_start|>assistant
<Z_477><Z_195><Z_84><Z_319><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]<|im_end|>
<|im_start|>assistant
<Z_405><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]<|im_end|>
<|im_start|>assistant
<Z_486><Z_292><Z_290><Z_362><ANSWER>

Question:
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_477><Z_28><Z_213><Z_246><Z_320><Z_87><Z_130><Z_473><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_304><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_227><Z_62><Z_278><Z_434><Z_79><Z_419><Z_161><Z_252><ANSWER>

Question: The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food. 144 students suggested adding mashed potatoes, 467 suggested adding bacon to the menu, and some students suggested adding tomatoes. There were 65 more students who suggested mashed potatoes than those that suggested tomatoes. How many students suggested adding tomatoes?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food. 144 students suggested adding mashed potatoes, 467 suggested adding bacon to the menu, and some students suggested adding tomatoes. There were 65 more students who suggested mashed potatoes than those that suggested tomatoes. How many students suggested adding tomatoes?<|im_end|>
<|im_start|>assistant
<Z_369><Z_264><Z_269><Z_504><Z_274><Z_266><Z_105><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food. 144 students suggested adding mashed potatoes, 467 suggested adding bacon to the menu, and some students suggested adding tomatoes. There were 65 more students who suggested mashed potatoes than those that suggested tomatoes. How many students suggested adding tomatoes?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sofia asked the students to suggest specific types of food. 144 students suggested adding mashed potatoes, 467 suggested adding bacon to the menu, and some students suggested adding tomatoes. There were 65 more students who suggested mashed potatoes than those that suggested tomatoes. How many students suggested adding tomatoes?<|im_end|>
<|im_start|>assistant
<Z_265><Z_202><Z_397><Z_415><Z_475><Z_16><Z_221><ANSWER>

Question: Sofia asked the students to suggest specific types of food. 144 students suggested adding mashed potatoes, 467 suggested adding bacon to the menu, and some students suggested adding tomatoes. There were 65 more students who suggested mashed potatoes than those that suggested tomatoes. How many students suggested adding tomatoes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper, assuming the third line intersects only one of the circles and not the other line or circle?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper, assuming the third line intersects only one of the circles and not the other line or circle?<|im_end|>
<|im_start|>assistant
<Z_477><Z_134><Z_344><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper, assuming the third line intersects only one of the circles and not the other line or circle?<|im_end|>
<|im_start|>assistant
<Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper, assuming the third line intersects only one of the circles and not the other line or circle?<|im_end|>
<|im_start|>assistant
<Z_78><Z_13><Z_149><ANSWER>

Question: What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper, assuming the third line intersects only one of the circles and not the other line or circle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of 3 ropes are in the ratio 4 : 5 : 6. If the sum of the weights of the longest and the shortest rope is 100 meters more than the length of the third rope, what is the length of the shortest rope?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of 3 ropes are in the ratio 4 : 5 : 6. If the sum of the weights of the longest and the shortest rope is 100 meters more than the length of the third rope, what is the length of the shortest rope?<|im_end|>
<|im_start|>assistant
<Z_477><Z_503><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of 3 ropes are in the ratio 4 : 5 : 6. If the sum of the weights of the longest and the shortest rope is 100 meters more than the length of the third rope, what is the length of the shortest rope?<|im_end|>
<|im_start|>assistant
<Z_155><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of 3 ropes are in the ratio 4 : 5 : 6. If the sum of the weights of the longest and the shortest rope is 100 meters more than the length of the third rope, what is the length of the shortest rope?<|im_end|>
<|im_start|>assistant
<Z_112><Z_197><ANSWER>

Question: The length of 3 ropes are in the ratio 4 : 5 : 6. If the sum of the weights of the longest and the shortest rope is 100 meters more than the length of the third rope, what is the length of the shortest rope?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.<|im_end|>
<|im_start|>assistant
<Z_24><Z_59><Z_501><Z_403><Z_494><Z_147><Z_116><Z_164><Z_260><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.<|im_end|>
<|im_start|>assistant
<Z_334><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.<|im_end|>
<|im_start|>assistant
<Z_200><Z_292><Z_98><Z_26><Z_57><Z_81><Z_245><Z_53><Z_495><Z_464><ANSWER>

Question: Using the digits 1, 2, 3, 4, and 5, form distinct five-digit numbers without repeating any digit and ensuring that the digit 4 doesn't appear in either the first or last position. Additionally, among the digits 1, 3, and 5, exactly two must be adjacent to each other. Determine the total number of different five-digit numbers that meet these conditions.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = \frac{x^2}{e^x}$.
(1) If a tangent line to the curve of function $f(x)$ passes through the point M(0,0), find the equation of that tangent line.
(2) Determine the maximum and minimum values of function $f(x)$ on the interval $[-3, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = \frac{x^2}{e^x}$.
(1) If a tangent line to the curve of function $f(x)$ passes through the point M(0,0), find the equation of that tangent line.
(2) Determine the maximum and minimum values of function $f(x)$ on the interval $[-3, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_242><Z_266><Z_425><Z_391><Z_263><Z_178><Z_487><Z_150><Z_503><Z_181><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = \frac{x^2}{e^x}$.
(1) If a tangent line to the curve of function $f(x)$ passes through the point M(0,0), find the equation of that tangent line.
(2) Determine the maximum and minimum values of function $f(x)$ on the interval $[-3, +\infty)$.<|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
Given the function $f(x) = \frac{x^2}{e^x}$.
(1) If a tangent line to the curve of function $f(x)$ passes through the point M(0,0), find the equation of that tangent line.
(2) Determine the maximum and minimum values of function $f(x)$ on the interval $[-3, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_229><Z_364><Z_424><Z_454><Z_14><Z_325><Z_457><Z_381><Z_20><Z_294><Z_294><ANSWER>

Question:
Given the function $f(x) = \frac{x^2}{e^x}$.
(1) If a tangent line to the curve of function $f(x)$ passes through the point M(0,0), find the equation of that tangent line.
(2) Determine the maximum and minimum values of function $f(x)$ on the interval $[-3, +\infty)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie had 7 dimes in her bank. She gave her dad some dimes, and her mother gave her 4 dimes. Melanie has 3 dimes now. How many dimes did Melanie give to her dad?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie had 7 dimes in her bank. She gave her dad some dimes, and her mother gave her 4 dimes. Melanie has 3 dimes now. How many dimes did Melanie give to her dad?<|im_end|>
<|im_start|>assistant
<Z_369><Z_461><Z_263><Z_378><Z_351><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie had 7 dimes in her bank. She gave her dad some dimes, and her mother gave her 4 dimes. Melanie has 3 dimes now. How many dimes did Melanie give to her dad?<|im_end|>
<|im_start|>assistant
<Z_182><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie had 7 dimes in her bank. She gave her dad some dimes, and her mother gave her 4 dimes. Melanie has 3 dimes now. How many dimes did Melanie give to her dad?<|im_end|>
<|im_start|>assistant
<Z_272><Z_96><Z_500><Z_160><Z_258><ANSWER>

Question: Melanie had 7 dimes in her bank. She gave her dad some dimes, and her mother gave her 4 dimes. Melanie has 3 dimes now. How many dimes did Melanie give to her dad?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sara had 24 peaches and 37 pears at her fruit dish. She went to the orchard and picked some peaches. She picked 37 peaches. How many peaches are there now?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sara had 24 peaches and 37 pears at her fruit dish. She went to the orchard and picked some peaches. She picked 37 peaches. How many peaches are there now?<|im_end|>
<|im_start|>assistant
<Z_24><Z_504><Z_87><Z_443><Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sara had 24 peaches and 37 pears at her fruit dish. She went to the orchard and picked some peaches. She picked 37 peaches. How many peaches are there now?<|im_end|>
<|im_start|>assistant
<Z_496><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sara had 24 peaches and 37 pears at her fruit dish. She went to the orchard and picked some peaches. She picked 37 peaches. How many peaches are there now?<|im_end|>
<|im_start|>assistant
<Z_448><Z_206><Z_174><Z_424><Z_326><ANSWER>

Question: Sara had 24 peaches and 37 pears at her fruit dish. She went to the orchard and picked some peaches. She picked 37 peaches. How many peaches are there now?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Set X consists of all two-digit primes and set Y consists of all positive multiples of 4 less than 100. If the two sets are merged into one, what will be the range of the new set?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Set X consists of all two-digit primes and set Y consists of all positive multiples of 4 less than 100. If the two sets are merged into one, what will be the range of the new set?<|im_end|>
<|im_start|>assistant
<Z_369><Z_353><Z_349><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Set X consists of all two-digit primes and set Y consists of all positive multiples of 4 less than 100. If the two sets are merged into one, what will be the range of the new set?<|im_end|>
<|im_start|>assistant
<Z_46><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Set X consists of all two-digit primes and set Y consists of all positive multiples of 4 less than 100. If the two sets are merged into one, what will be the range of the new set?<|im_end|>
<|im_start|>assistant
<Z_425><Z_265><Z_369><ANSWER>

Question: Set X consists of all two-digit primes and set Y consists of all positive multiples of 4 less than 100. If the two sets are merged into one, what will be the range of the new set?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?<|im_end|>
<|im_start|>assistant
<Z_477><Z_210><Z_348><Z_200><Z_414><Z_146><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?<|im_end|>
<|im_start|>assistant
<Z_196><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?<|im_end|>
<|im_start|>assistant
<Z_122><Z_412><Z_173><Z_201><Z_359><Z_425><Z_388><ANSWER>

Question: In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A drink vendor has 60 liters of Maaza, 220 liters of Pepsi, 500 liters of Sprite, 315 liters of Fanta, and 125 liters of 7UP. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A drink vendor has 60 liters of Maaza, 220 liters of Pepsi, 500 liters of Sprite, 315 liters of Fanta, and 125 liters of 7UP. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?<|im_end|>
<|im_start|>assistant
<Z_477><Z_507><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A drink vendor has 60 liters of Maaza, 220 liters of Pepsi, 500 liters of Sprite, 315 liters of Fanta, and 125 liters of 7UP. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?<|im_end|>
<|im_start|>assistant
<Z_265><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A drink vendor has 60 liters of Maaza, 220 liters of Pepsi, 500 liters of Sprite, 315 liters of Fanta, and 125 liters of 7UP. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?<|im_end|>
<|im_start|>assistant
<Z_335><Z_107><Z_398><ANSWER>

Question: A drink vendor has 60 liters of Maaza, 220 liters of Pepsi, 500 liters of Sprite, 315 liters of Fanta, and 125 liters of 7UP. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. What is the least number of cans required?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family of 12 monkeys collected some piles of bananas. 6 piles had 9 hands, with each hand having 14 bananas, while the remaining piles had 12 hands, with each hand having 9 bananas. If they divide the bananas equally amongst themselves, each monkey would get 99 bananas. How many piles of bananas did they collect?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family of 12 monkeys collected some piles of bananas. 6 piles had 9 hands, with each hand having 14 bananas, while the remaining piles had 12 hands, with each hand having 9 bananas. If they divide the bananas equally amongst themselves, each monkey would get 99 bananas. How many piles of bananas did they collect?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family of 12 monkeys collected some piles of bananas. 6 piles had 9 hands, with each hand having 14 bananas, while the remaining piles had 12 hands, with each hand having 9 bananas. If they divide the bananas equally amongst themselves, each monkey would get 99 bananas. How many piles of bananas did they collect?<|im_end|>
<|im_start|>assistant
<Z_238><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A family of 12 monkeys collected some piles of bananas. 6 piles had 9 hands, with each hand having 14 bananas, while the remaining piles had 12 hands, with each hand having 9 bananas. If they divide the bananas equally amongst themselves, each monkey would get 99 bananas. How many piles of bananas did they collect?<|im_end|>
<|im_start|>assistant
<Z_419><ANSWER>

Question: A family of 12 monkeys collected some piles of bananas. 6 piles had 9 hands, with each hand having 14 bananas, while the remaining piles had 12 hands, with each hand having 9 bananas. If they divide the bananas equally amongst themselves, each monkey would get 99 bananas. How many piles of bananas did they collect?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the product of $1101101_2$ and $111_2$. Express your product in binary and convert it to a decimal.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the product of $1101101_2$ and $111_2$. Express your product in binary and convert it to a decimal.<|im_end|>
<|im_start|>assistant
<Z_24><Z_509><Z_298><Z_413><Z_235><Z_357><Z_326><Z_321><Z_178><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the product of $1101101_2$ and $111_2$. Express your product in binary and convert it to a decimal.<|im_end|>
<|im_start|>assistant
<Z_137><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the product of $1101101_2$ and $111_2$. Express your product in binary and convert it to a decimal.<|im_end|>
<|im_start|>assistant
<Z_165><Z_430><Z_247><Z_317><Z_22><Z_50><Z_398><Z_250><Z_86><ANSWER>

Question: Find the product of $1101101_2$ and $111_2$. Express your product in binary and convert it to a decimal.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If k is a non-negative integer and 15^k is a divisor of 759,325, then 3^k minus some expression equals 1. What is the expression that needs to be subtracted from 3^k to equal 1?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If k is a non-negative integer and 15^k is a divisor of 759,325, then 3^k minus some expression equals 1. What is the expression that needs to be subtracted from 3^k to equal 1?<|im_end|>
<|im_start|>assistant
<Z_24><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If k is a non-negative integer and 15^k is a divisor of 759,325, then 3^k minus some expression equals 1. What is the expression that needs to be subtracted from 3^k to equal 1?<|im_end|>
<|im_start|>assistant
<Z_149><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If k is a non-negative integer and 15^k is a divisor of 759,325, then 3^k minus some expression equals 1. What is the expression that needs to be subtracted from 3^k to equal 1?<|im_end|>
<|im_start|>assistant
<Z_262><Z_457><ANSWER>

Question: If k is a non-negative integer and 15^k is a divisor of 759,325, then 3^k minus some expression equals 1. What is the expression that needs to be subtracted from 3^k to equal 1?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $x^2 - kx + 18 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $x^2 - kx + 18 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_502><Z_201><Z_71><Z_134><Z_318><Z_330><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $x^2 - kx + 18 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?<|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
The equation $x^2 - kx + 18 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?<|im_end|>
<|im_start|>assistant
<Z_43><Z_496><Z_241><Z_480><Z_466><Z_443><Z_286><Z_43><ANSWER>

Question: The equation $x^2 - kx + 18 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some men do a work in 80 days, 20 men will do it in 32 days. How many men were in the first group?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some men do a work in 80 days, 20 men will do it in 32 days. How many men were in the first group?<|im_end|>
<|im_start|>assistant
<Z_477><Z_29><Z_158><Z_398><Z_199><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some men do a work in 80 days, 20 men will do it in 32 days. How many men were in the first group?<|im_end|>
<|im_start|>assistant
<Z_154><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If some men do a work in 80 days, 20 men will do it in 32 days. How many men were in the first group?<|im_end|>
<|im_start|>assistant
<Z_311><Z_359><Z_97><Z_209><Z_197><ANSWER>

Question: If some men do a work in 80 days, 20 men will do it in 32 days. How many men were in the first group?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\).<|im_end|>
<|im_start|>assistant
<Z_477><Z_228><Z_483><Z_129><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\).<|im_end|>
<|im_start|>assistant
<Z_189><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\).<|im_end|>
<|im_start|>assistant
<Z_66><Z_396><Z_36><Z_159><ANSWER>

Question:
Given the function
$$
f(x) = 10x^2 + mx + n \quad (m, n \in \mathbb{Z})
$$
which has two distinct real roots in the interval \((1, 3)\). Find the maximum possible value of \(f(1) f(3)\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the circumference of a circle is equal to 72 pi . find the radius of this circle .<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the circumference of a circle is equal to 72 pi . find the radius of this circle .<|im_end|>
<|im_start|>assistant
<Z_477><Z_43><Z_320><Z_244><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the circumference of a circle is equal to 72 pi . find the radius of this circle .<|im_end|>
<|im_start|>assistant
<Z_150><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the circumference of a circle is equal to 72 pi . find the radius of this circle .<|im_end|>
<|im_start|>assistant
<Z_397><Z_410><Z_429><Z_291><Z_53><ANSWER>

Question: the circumference of a circle is equal to 72 pi . find the radius of this circle .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $60^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $60^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$<|im_end|>
<|im_start|>assistant
<Z_24><Z_114><Z_434><Z_492><Z_425><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $60^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$<|im_end|>
<|im_start|>assistant
<Z_311><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $60^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$<|im_end|>
<|im_start|>assistant
<Z_495><Z_363><Z_416><Z_12><Z_67><ANSWER>

Question: In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $60^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can Lisa select three out of eight flowers to make a bouquet?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can Lisa select three out of eight flowers to make a bouquet?<|im_end|>
<|im_start|>assistant
<Z_477><Z_75><Z_400><Z_56><Z_494><Z_382><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can Lisa select three out of eight flowers to make a bouquet?<|im_end|>
<|im_start|>assistant
<Z_480><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can Lisa select three out of eight flowers to make a bouquet?<|im_end|>
<|im_start|>assistant
<Z_406><Z_309><Z_241><Z_127><Z_231><Z_282><ANSWER>

Question: In how many ways can Lisa select three out of eight flowers to make a bouquet?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A store sells a set of 150 pencils for $24.75. An additional shipping cost of $8.25 is applied to the order. What is the average cost, in cents, for each pencil including shipping charges? Round your answer to the nearest whole number.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A store sells a set of 150 pencils for $24.75. An additional shipping cost of $8.25 is applied to the order. What is the average cost, in cents, for each pencil including shipping charges? Round your answer to the nearest whole number.<|im_end|>
<|im_start|>assistant
<Z_24><Z_217><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A store sells a set of 150 pencils for $24.75. An additional shipping cost of $8.25 is applied to the order. What is the average cost, in cents, for each pencil including shipping charges? Round your answer to the nearest whole number.<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A store sells a set of 150 pencils for $24.75. An additional shipping cost of $8.25 is applied to the order. What is the average cost, in cents, for each pencil including shipping charges? Round your answer to the nearest whole number.<|im_end|>
<|im_start|>assistant
<Z_310><Z_202><ANSWER>

Question: A store sells a set of 150 pencils for $24.75. An additional shipping cost of $8.25 is applied to the order. What is the average cost, in cents, for each pencil including shipping charges? Round your answer to the nearest whole number.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The cost to park a car in a certain parking garage is $10.00 for up to a certain number of hours of parking and $1.75 for each hour in excess of that number of hours. The average (arithmetic mean) cost per hour to park a car in the parking garage for 9 hours is $2.4722222222222223. What is the number of hours for the initial parking cost?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The cost to park a car in a certain parking garage is $10.00 for up to a certain number of hours of parking and $1.75 for each hour in excess of that number of hours. The average (arithmetic mean) cost per hour to park a car in the parking garage for 9 hours is $2.4722222222222223. What is the number of hours for the initial parking cost?<|im_end|>
<|im_start|>assistant
<Z_477><Z_377><Z_170><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The cost to park a car in a certain parking garage is $10.00 for up to a certain number of hours of parking and $1.75 for each hour in excess of that number of hours. The average (arithmetic mean) cost per hour to park a car in the parking garage for 9 hours is $2.4722222222222223. What is the number of hours for the initial parking cost?<|im_end|>
<|im_start|>assistant
<Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The cost to park a car in a certain parking garage is $10.00 for up to a certain number of hours of parking and $1.75 for each hour in excess of that number of hours. The average (arithmetic mean) cost per hour to park a car in the parking garage for 9 hours is $2.4722222222222223. What is the number of hours for the initial parking cost?<|im_end|>
<|im_start|>assistant
<Z_4><Z_210><Z_12><ANSWER>

Question: The cost to park a car in a certain parking garage is $10.00 for up to a certain number of hours of parking and $1.75 for each hour in excess of that number of hours. The average (arithmetic mean) cost per hour to park a car in the parking garage for 9 hours is $2.4722222222222223. What is the number of hours for the initial parking cost?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What multiple of 15 is closest to 2021?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What multiple of 15 is closest to 2021?<|im_end|>
<|im_start|>assistant
<Z_369><Z_244><Z_80><Z_362><Z_214><Z_144><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What multiple of 15 is closest to 2021?<|im_end|>
<|im_start|>assistant
<Z_15><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What multiple of 15 is closest to 2021?<|im_end|>
<|im_start|>assistant
<Z_461><Z_232><Z_250><Z_199><Z_368><Z_355><ANSWER>

Question: What multiple of 15 is closest to 2021?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_219><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).<|im_end|>
<|im_start|>assistant
<Z_30><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).<|im_end|>
<|im_start|>assistant
<Z_409><Z_354><Z_26><ANSWER>

Question: Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express $343_{8} - 265_{7}$ as a base 10 integer.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express $343_{8} - 265_{7}$ as a base 10 integer.<|im_end|>
<|im_start|>assistant
<Z_24><Z_210><Z_162><Z_263><Z_294><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express $343_{8} - 265_{7}$ as a base 10 integer.<|im_end|>
<|im_start|>assistant
<Z_368><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Express $343_{8} - 265_{7}$ as a base 10 integer.<|im_end|>
<|im_start|>assistant
<Z_84><Z_477><Z_155><Z_506><Z_50><ANSWER>

Question: Express $343_{8} - 265_{7}$ as a base 10 integer.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_242><Z_267><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).<|im_end|>
<|im_start|>assistant
<Z_311><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).<|im_end|>
<|im_start|>assistant
<Z_279><Z_326><Z_329><ANSWER>

Question: Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the average ( arithmetic mean ) of 7 consecutive integers is 20 , then the product of the greatest and least integer is<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the average ( arithmetic mean ) of 7 consecutive integers is 20 , then the product of the greatest and least integer is<|im_end|>
<|im_start|>assistant
<Z_369><Z_218><Z_503><Z_147><Z_71><Z_150><Z_345><Z_418><Z_425><Z_50><Z_53><Z_153><Z_106><Z_100><Z_357><Z_273><Z_304><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the average ( arithmetic mean ) of 7 consecutive integers is 20 , then the product of the greatest and least integer is<|im_end|>
<|im_start|>assistant
<Z_22><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the average ( arithmetic mean ) of 7 consecutive integers is 20 , then the product of the greatest and least integer is<|im_end|>
<|im_start|>assistant
<Z_451><Z_228><Z_413><Z_386><Z_376><Z_443><Z_196><Z_17><Z_274><Z_507><Z_139><Z_362><Z_123><Z_316><Z_340><Z_7><Z_370><ANSWER>

Question: if the average ( arithmetic mean ) of 7 consecutive integers is 20 , then the product of the greatest and least integer is


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile?<|im_end|>
<|im_start|>assistant
<Z_369><Z_162><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile?<|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
If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile?<|im_end|>
<|im_start|>assistant
<Z_162><Z_238><Z_413><ANSWER>

Question: If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with some halls with 40 rooms each. The hotel has a total of 4248 rooms. How many halls are on each floor of the second wing?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with some halls with 40 rooms each. The hotel has a total of 4248 rooms. How many halls are on each floor of the second wing?<|im_end|>
<|im_start|>assistant
<Z_477><Z_465><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with some halls with 40 rooms each. The hotel has a total of 4248 rooms. How many halls are on each floor of the second wing?<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with some halls with 40 rooms each. The hotel has a total of 4248 rooms. How many halls are on each floor of the second wing?<|im_end|>
<|im_start|>assistant
<Z_102><Z_309><Z_15><ANSWER>

Question: Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with some halls with 40 rooms each. The hotel has a total of 4248 rooms. How many halls are on each floor of the second wing?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $21, x, y, z, 53$, what is the value of $y$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $21, x, y, z, 53$, what is the value of $y$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_308><Z_263><Z_227><Z_272><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $21, x, y, z, 53$, what is the value of $y$?<|im_end|>
<|im_start|>assistant
<Z_258><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $21, x, y, z, 53$, what is the value of $y$?<|im_end|>
<|im_start|>assistant
<Z_115><Z_37><Z_29><Z_95><Z_19><ANSWER>

Question: In the arithmetic sequence $21, x, y, z, 53$, what is the value of $y$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.<|im_end|>
<|im_start|>assistant
<Z_24><Z_309><Z_221><Z_419><Z_128><Z_321><Z_217><Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.<|im_end|>
<|im_start|>assistant
<Z_230><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.<|im_end|>
<|im_start|>assistant
<Z_475><Z_110><Z_49><Z_308><Z_32><Z_381><Z_474><Z_19><ANSWER>

Question: A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The weights of 10 students are 41, 48, 50, 53, 49, 50, 53, 53, 51, 67 (in kg). The range of this set of data is ____ kg.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The weights of 10 students are 41, 48, 50, 53, 49, 50, 53, 53, 51, 67 (in kg). The range of this set of data is ____ kg.<|im_end|>
<|im_start|>assistant
<Z_477><Z_340><Z_322><Z_100><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The weights of 10 students are 41, 48, 50, 53, 49, 50, 53, 53, 51, 67 (in kg). The range of this set of data is ____ kg.<|im_end|>
<|im_start|>assistant
<Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The weights of 10 students are 41, 48, 50, 53, 49, 50, 53, 53, 51, 67 (in kg). The range of this set of data is ____ kg.<|im_end|>
<|im_start|>assistant
<Z_465><Z_172><Z_505><Z_475><ANSWER>

Question: The weights of 10 students are 41, 48, 50, 53, 49, 50, 53, 53, 51, 67 (in kg). The range of this set of data is ____ kg.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_330><Z_87><Z_274><Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.<|im_end|>
<|im_start|>assistant
<Z_486><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.<|im_end|>
<|im_start|>assistant
<Z_313><Z_508><Z_23><Z_133><Z_415><ANSWER>

Question:
The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 17 fewer children than adults on the train at the first station. At the next station, 57 adults and 18 children got on and 44 got off. If there are 502 people on the train right now, how many adults got on the train at the first station?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 17 fewer children than adults on the train at the first station. At the next station, 57 adults and 18 children got on and 44 got off. If there are 502 people on the train right now, how many adults got on the train at the first station?<|im_end|>
<|im_start|>assistant
<Z_477><Z_140><Z_382><Z_200><Z_37><Z_382><Z_219><Z_184><Z_460><Z_401><Z_481><Z_430><Z_199><Z_305><Z_150><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 17 fewer children than adults on the train at the first station. At the next station, 57 adults and 18 children got on and 44 got off. If there are 502 people on the train right now, how many adults got on the train at the first station?<|im_end|>
<|im_start|>assistant
<Z_460><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were 17 fewer children than adults on the train at the first station. At the next station, 57 adults and 18 children got on and 44 got off. If there are 502 people on the train right now, how many adults got on the train at the first station?<|im_end|>
<|im_start|>assistant
<Z_9><Z_77><Z_332><Z_450><Z_511><Z_297><Z_210><Z_231><Z_217><Z_209><Z_179><Z_65><Z_24><Z_474><Z_369><ANSWER>

Question: There were 17 fewer children than adults on the train at the first station. At the next station, 57 adults and 18 children got on and 44 got off. If there are 502 people on the train right now, how many adults got on the train at the first station?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of the distinct prime factors of 420?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum and product of the distinct prime factors of 420?<|im_end|>
<|im_start|>assistant
<Z_477><Z_214><Z_330><Z_434><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 and product of the distinct prime factors of 420?<|im_end|>
<|im_start|>assistant
<Z_37><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 and product of the distinct prime factors of 420?<|im_end|>
<|im_start|>assistant
<Z_338><Z_233><Z_51><Z_442><ANSWER>

Question: What is the sum and product of the distinct prime factors of 420?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?<|im_end|>
<|im_start|>assistant
<Z_24><Z_447><Z_352><Z_199><Z_162><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?<|im_end|>
<|im_start|>assistant
<Z_36><Z_97><Z_410><Z_133><Z_227><ANSWER>

Question: In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\boxed{0.027\;^{- \frac {1}{3}}-(- \frac {1}{7})^{-2}+256\;^{ \frac {3}{4}}-3^{-1}+( \sqrt {2}-1)^{0}} \_\_\_\_\_\_ \;\;.$<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\boxed{0.027\;^{- \frac {1}{3}}-(- \frac {1}{7})^{-2}+256\;^{ \frac {3}{4}}-3^{-1}+( \sqrt {2}-1)^{0}} \_\_\_\_\_\_ \;\;.$<|im_end|>
<|im_start|>assistant
<Z_185><ANSWER>

Question: Compute: $\boxed{0.027\;^{- \frac {1}{3}}-(- \frac {1}{7})^{-2}+256\;^{ \frac {3}{4}}-3^{-1}+( \sqrt {2}-1)^{0}} \_\_\_\_\_\_ \;\;.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the longest possible sequence of distinct positive integers with the first member 1 and the last member 31 such that each member is divisible by its predecessor? How many such sequences have this length?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the longest possible sequence of distinct positive integers with the first member 1 and the last member 31 such that each member is divisible by its predecessor? How many such sequences have this length?<|im_end|>
<|im_start|>assistant
<Z_369><Z_307><Z_238><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the longest possible sequence of distinct positive integers with the first member 1 and the last member 31 such that each member is divisible by its predecessor? How many such sequences have this length?<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the longest possible sequence of distinct positive integers with the first member 1 and the last member 31 such that each member is divisible by its predecessor? How many such sequences have this length?<|im_end|>
<|im_start|>assistant
<Z_107><Z_156><Z_263><ANSWER>

Question: What is the longest possible sequence of distinct positive integers with the first member 1 and the last member 31 such that each member is divisible by its predecessor? How many such sequences have this length?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mia is at a picnic with six other friends. They decide to play a team game and need to form teams of four. However, only four people can be on a team. How many different teams of four can the seven of them form?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mia is at a picnic with six other friends. They decide to play a team game and need to form teams of four. However, only four people can be on a team. How many different teams of four can the seven of them form?<|im_end|>
<|im_start|>assistant
<Z_477><Z_308><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mia is at a picnic with six other friends. They decide to play a team game and need to form teams of four. However, only four people can be on a team. How many different teams of four can the seven of them form?<|im_end|>
<|im_start|>assistant
<Z_243><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mia is at a picnic with six other friends. They decide to play a team game and need to form teams of four. However, only four people can be on a team. How many different teams of four can the seven of them form?<|im_end|>
<|im_start|>assistant
<Z_6><Z_455><Z_495><ANSWER>

Question: Mia is at a picnic with six other friends. They decide to play a team game and need to form teams of four. However, only four people can be on a team. How many different teams of four can the seven of them form?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If x > 0, x / some number + x / 25 is 6% of x. What is the number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If x > 0, x / some number + x / 25 is 6% of x. What is the number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_382><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If x > 0, x / some number + x / 25 is 6% of x. What is the number?<|im_end|>
<|im_start|>assistant
<Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If x > 0, x / some number + x / 25 is 6% of x. What is the number?<|im_end|>
<|im_start|>assistant
<Z_20><Z_205><ANSWER>

Question: If x > 0, x / some number + x / 25 is 6% of x. What is the number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the positive integers x, x + 2, x + 4, x + 7, and x + 22, the mean is how much greater than the median?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the positive integers x, x + 2, x + 4, x + 7, and x + 22, the mean is how much greater than the median?<|im_end|>
<|im_start|>assistant
<Z_369><Z_487><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the positive integers x, x + 2, x + 4, x + 7, and x + 22, the mean is how much greater than the median?<|im_end|>
<|im_start|>assistant
<Z_220><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the positive integers x, x + 2, x + 4, x + 7, and x + 22, the mean is how much greater than the median?<|im_end|>
<|im_start|>assistant
<Z_384><Z_211><ANSWER>

Question: For the positive integers x, x + 2, x + 4, x + 7, and x + 22, the mean is how much greater than the median?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jorge bought some tickets for $7 each. For purchasing so many, he is given a discount of 50%. He spent $84 on tickets. How many tickets did he buy?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jorge bought some tickets for $7 each. For purchasing so many, he is given a discount of 50%. He spent $84 on tickets. How many tickets did he buy?<|im_end|>
<|im_start|>assistant
<Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jorge bought some tickets for $7 each. For purchasing so many, he is given a discount of 50%. He spent $84 on tickets. How many tickets did he buy?<|im_end|>
<|im_start|>assistant
<Z_136><ANSWER>

Question: Jorge bought some tickets for $7 each. For purchasing so many, he is given a discount of 50%. He spent $84 on tickets. How many tickets did he buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.<|im_end|>
<|im_start|>assistant
<Z_24><Z_346><Z_276><Z_184><Z_205><Z_94><Z_288><Z_388><Z_419><Z_217><Z_124><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.<|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
If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.<|im_end|>
<|im_start|>assistant
<Z_44><Z_180><Z_380><Z_151><Z_462><Z_232><Z_179><Z_454><Z_240><Z_376><Z_176><Z_436><ANSWER>

Question: If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all real numbers \(x\), if the value of every quadratic function \( f(x) = ax^2 + bx + c \) (where \(a < b\)) is always a non-negative real number, find the minimum value of \( M = \frac{a+b+c}{b-a} \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all real numbers \(x\), if the value of every quadratic function \( f(x) = ax^2 + bx + c \) (where \(a < b\)) is always a non-negative real number, find the minimum value of \( M = \frac{a+b+c}{b-a} \).<|im_end|>
<|im_start|>assistant
<Z_477><Z_465><Z_413><Z_288><Z_227><Z_446><Z_418><Z_503><Z_245><Z_139><Z_253><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all real numbers \(x\), if the value of every quadratic function \( f(x) = ax^2 + bx + c \) (where \(a < b\)) is always a non-negative real number, find the minimum value of \( M = \frac{a+b+c}{b-a} \).<|im_end|>
<|im_start|>assistant
<Z_473><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For all real numbers \(x\), if the value of every quadratic function \( f(x) = ax^2 + bx + c \) (where \(a < b\)) is always a non-negative real number, find the minimum value of \( M = \frac{a+b+c}{b-a} \).<|im_end|>
<|im_start|>assistant
<Z_181><Z_39><Z_228><Z_112><Z_122><Z_267><Z_50><Z_29><Z_113><Z_205><Z_267><Z_35><ANSWER>

Question: For all real numbers \(x\), if the value of every quadratic function \( f(x) = ax^2 + bx + c \) (where \(a < b\)) is always a non-negative real number, find the minimum value of \( M = \frac{a+b+c}{b-a} \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ashok secured an average of 77 marks in some subjects. If the average of marks in 5 subjects is 74, and he secured 92 marks in the last subject, how many subjects are there in total?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ashok secured an average of 77 marks in some subjects. If the average of marks in 5 subjects is 74, and he secured 92 marks in the last subject, how many subjects are there in total?<|im_end|>
<|im_start|>assistant
<Z_477><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ashok secured an average of 77 marks in some subjects. If the average of marks in 5 subjects is 74, and he secured 92 marks in the last subject, how many subjects are there in total?<|im_end|>
<|im_start|>assistant
<Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ashok secured an average of 77 marks in some subjects. If the average of marks in 5 subjects is 74, and he secured 92 marks in the last subject, how many subjects are there in total?<|im_end|>
<|im_start|>assistant
<Z_134><Z_408><ANSWER>

Question: Ashok secured an average of 77 marks in some subjects. If the average of marks in 5 subjects is 74, and he secured 92 marks in the last subject, how many subjects are there in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(Q\) lies on the line \(x = 4\) and is 8 units from the point \((-1, 3)\). Find the sum of all possible \(y\)-coordinates that satisfy the given conditions.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(Q\) lies on the line \(x = 4\) and is 8 units from the point \((-1, 3)\). Find the sum of all possible \(y\)-coordinates that satisfy the given conditions.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(Q\) lies on the line \(x = 4\) and is 8 units from the point \((-1, 3)\). Find the sum of all possible \(y\)-coordinates that satisfy the given conditions.<|im_end|>
<|im_start|>assistant
<Z_496><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \(Q\) lies on the line \(x = 4\) and is 8 units from the point \((-1, 3)\). Find the sum of all possible \(y\)-coordinates that satisfy the given conditions.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

Question: Point \(Q\) lies on the line \(x = 4\) and is 8 units from the point \((-1, 3)\). Find the sum of all possible \(y\)-coordinates that satisfy the given conditions.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?<|im_end|>
<|im_start|>assistant
<Z_192><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?<|im_end|>
<|im_start|>assistant
<Z_426><ANSWER>

Question: Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?<|im_end|>
<|im_start|>assistant
<Z_24><Z_422><Z_57><Z_366><Z_150><Z_510><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?<|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
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?<|im_end|>
<|im_start|>assistant
<Z_202><Z_243><Z_318><Z_175><Z_188><Z_217><Z_132><ANSWER>

Question: Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes some tractors 12 days to plough a field. It would take 15 tractors 6.4 days to plough the same field. How many tractors were in the first scenario?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes some tractors 12 days to plough a field. It would take 15 tractors 6.4 days to plough the same field. How many tractors were in the first scenario?<|im_end|>
<|im_start|>assistant
<Z_24><Z_79><Z_386><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes some tractors 12 days to plough a field. It would take 15 tractors 6.4 days to plough the same field. How many tractors were in the first scenario?<|im_end|>
<|im_start|>assistant
<Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
It takes some tractors 12 days to plough a field. It would take 15 tractors 6.4 days to plough the same field. How many tractors were in the first scenario?<|im_end|>
<|im_start|>assistant
<Z_346><Z_320><Z_182><Z_203><ANSWER>

Question: It takes some tractors 12 days to plough a field. It would take 15 tractors 6.4 days to plough the same field. How many tractors were in the first scenario?