<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$ is a root of the equation $x^{2}-3x-1011=0$, find the value of the algebraic expression $2a^{2}-6a+1$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$ is a root of the equation $x^{2}-3x-1011=0$, find the value of the algebraic expression $2a^{2}-6a+1$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_231><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$ is a root of the equation $x^{2}-3x-1011=0$, find the value of the algebraic expression $2a^{2}-6a+1$.<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a$ is a root of the equation $x^{2}-3x-1011=0$, find the value of the algebraic expression $2a^{2}-6a+1$.<|im_end|>
<|im_start|>assistant
<Z_62><Z_252><ANSWER>

Question: Given that $a$ is a root of the equation $x^{2}-3x-1011=0$, find the value of the algebraic expression $2a^{2}-6a+1$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)<|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
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)<|im_end|>
<|im_start|>assistant
<Z_369><Z_45><Z_245><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)<|im_end|>
<|im_start|>assistant
<Z_225><Z_323><Z_373><ANSWER>

Question: Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho^{2}=4 \sqrt {2}\rho\sin (\theta+ \frac {\pi}{4})-4$.
(Ⅰ) Find the Cartesian equation of curve $C_{2}$ and identify what type of curve it represents;
(Ⅱ) If curve $C_{1}$ intersects with curve $C_{2}$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho^{2}=4 \sqrt {2}\rho\sin (\theta+ \frac {\pi}{4})-4$.
(Ⅰ) Find the Cartesian equation of curve $C_{2}$ and identify what type of curve it represents;
(Ⅱ) If curve $C_{1}$ intersects with curve $C_{2}$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_224><Z_279><Z_215><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho^{2}=4 \sqrt {2}\rho\sin (\theta+ \frac {\pi}{4})-4$.
(Ⅰ) Find the Cartesian equation of curve $C_{2}$ and identify what type of curve it represents;
(Ⅱ) If curve $C_{1}$ intersects with curve $C_{2}$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.<|im_end|>
<|im_start|>assistant
<Z_408><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho^{2}=4 \sqrt {2}\rho\sin (\theta+ \frac {\pi}{4})-4$.
(Ⅰ) Find the Cartesian equation of curve $C_{2}$ and identify what type of curve it represents;
(Ⅱ) If curve $C_{1}$ intersects with curve $C_{2}$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.<|im_end|>
<|im_start|>assistant
<Z_182><Z_494><Z_354><Z_83><ANSWER>

Question:
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho^{2}=4 \sqrt {2}\rho\sin (\theta+ \frac {\pi}{4})-4$.
(Ⅰ) Find the Cartesian equation of curve $C_{2}$ and identify what type of curve it represents;
(Ⅱ) If curve $C_{1}$ intersects with curve $C_{2}$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is :<|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 a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is :<|im_end|>
<|im_start|>assistant
<Z_369><Z_151><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is :<|im_end|>
<|im_start|>assistant
<Z_429><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is :<|im_end|>
<|im_start|>assistant
<Z_32><Z_474><Z_406><ANSWER>

Question: if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is :


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $m$ terms of an arithmetic sequence $\{a_n\}$ is $30$, and the sum of the first $2m$ terms is $100$. Find the sum of the first $3m$ terms.<|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 sum of the first $m$ terms of an arithmetic sequence $\{a_n\}$ is $30$, and the sum of the first $2m$ terms is $100$. Find the sum of the first $3m$ terms.<|im_end|>
<|im_start|>assistant
<Z_369><Z_259><Z_394><Z_268><Z_502><Z_255><Z_316><Z_447><Z_425><Z_91><Z_423><Z_467><Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $m$ terms of an arithmetic sequence $\{a_n\}$ is $30$, and the sum of the first $2m$ terms is $100$. Find the sum of the first $3m$ terms.<|im_end|>
<|im_start|>assistant
<Z_358><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first $m$ terms of an arithmetic sequence $\{a_n\}$ is $30$, and the sum of the first $2m$ terms is $100$. Find the sum of the first $3m$ terms.<|im_end|>
<|im_start|>assistant
<Z_322><Z_143><Z_500><Z_188><Z_219><Z_511><Z_264><Z_369><Z_398><Z_72><Z_383><Z_292><Z_101><ANSWER>

Question: The sum of the first $m$ terms of an arithmetic sequence $\{a_n\}$ is $30$, and the sum of the first $2m$ terms is $100$. Find the sum of the first $3m$ terms.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?<|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
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?<|im_end|>
<|im_start|>assistant
<Z_24><Z_240><Z_343><Z_464><Z_485><Z_464><Z_487><Z_252><Z_478><Z_101><Z_330><Z_469><Z_472><Z_245><Z_64><Z_441><Z_470><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?<|im_end|>
<|im_start|>assistant
<Z_284><Z_129><Z_124><Z_106><Z_453><Z_153><Z_55><Z_348><Z_357><Z_378><Z_200><Z_436><Z_221><Z_270><Z_499><Z_248><Z_268><Z_240><ANSWER>

Question: Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pet store had six kittens. They got some more kittens and now they have a total of 9 kittens. How many kittens did the pet store get?<|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 pet store had six kittens. They got some more kittens and now they have a total of 9 kittens. How many kittens did the pet store get?<|im_end|>
<|im_start|>assistant
<Z_369><Z_343><Z_363><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pet store had six kittens. They got some more kittens and now they have a total of 9 kittens. How many kittens did the pet store get?<|im_end|>
<|im_start|>assistant
<Z_138><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pet store had six kittens. They got some more kittens and now they have a total of 9 kittens. How many kittens did the pet store get?<|im_end|>
<|im_start|>assistant
<Z_481><Z_109><Z_96><Z_134><ANSWER>

Question: A pet store had six kittens. They got some more kittens and now they have a total of 9 kittens. How many kittens did the pet store get?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A class has 32 students and they all need to present their projects. Every period is 40 minutes long. Students are required to present their project for 5 minutes, followed by 3 minutes of questions and answers. Additionally, 4 students will be giving group presentations that last for 12 minutes each, including questions and answers. How many periods will it take for every student to present their project, taking into account the additional requirements and varying presentation durations?<|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 class has 32 students and they all need to present their projects. Every period is 40 minutes long. Students are required to present their project for 5 minutes, followed by 3 minutes of questions and answers. Additionally, 4 students will be giving group presentations that last for 12 minutes each, including questions and answers. How many periods will it take for every student to present their project, taking into account the additional requirements and varying presentation durations?<|im_end|>
<|im_start|>assistant
<Z_369><Z_76><Z_432><Z_209><Z_322><Z_214><Z_84><Z_314><Z_454><Z_151><Z_497><Z_428><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A class has 32 students and they all need to present their projects. Every period is 40 minutes long. Students are required to present their project for 5 minutes, followed by 3 minutes of questions and answers. Additionally, 4 students will be giving group presentations that last for 12 minutes each, including questions and answers. How many periods will it take for every student to present their project, taking into account the additional requirements and varying presentation durations?<|im_end|>
<|im_start|>assistant
<Z_216><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A class has 32 students and they all need to present their projects. Every period is 40 minutes long. Students are required to present their project for 5 minutes, followed by 3 minutes of questions and answers. Additionally, 4 students will be giving group presentations that last for 12 minutes each, including questions and answers. How many periods will it take for every student to present their project, taking into account the additional requirements and varying presentation durations?<|im_end|>
<|im_start|>assistant
<Z_50><Z_280><Z_170><Z_139><Z_129><Z_388><Z_173><Z_27><Z_394><Z_506><Z_181><Z_403><ANSWER>

Question: A class has 32 students and they all need to present their projects. Every period is 40 minutes long. Students are required to present their project for 5 minutes, followed by 3 minutes of questions and answers. Additionally, 4 students will be giving group presentations that last for 12 minutes each, including questions and answers. How many periods will it take for every student to present their project, taking into account the additional requirements and varying presentation durations?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 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
Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 5?<|im_end|>
<|im_start|>assistant
<Z_24><Z_150><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 5?<|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
Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 5?<|im_end|>
<|im_start|>assistant
<Z_49><Z_66><ANSWER>

Question: Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 5?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different lines pass through at least two points in a 4-by-4 grid of lattice points?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different lines pass through at least two points in a 4-by-4 grid of lattice points?<|im_end|>
<|im_start|>assistant
<Z_477><Z_331><Z_183><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different lines pass through at least two points in a 4-by-4 grid of lattice points?<|im_end|>
<|im_start|>assistant
<Z_142><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different lines pass through at least two points in a 4-by-4 grid of lattice points?<|im_end|>
<|im_start|>assistant
<Z_171><Z_478><Z_218><ANSWER>

Question: How many different lines pass through at least two points in a 4-by-4 grid of lattice points?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.<|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
Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.<|im_end|>
<|im_start|>assistant
<Z_369><Z_226><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.<|im_end|>
<|im_start|>assistant
<Z_261><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.<|im_end|>
<|im_start|>assistant
<Z_42><Z_473><Z_8><ANSWER>

Question: Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A salesman's income consists of a commission and a base salary of $400 per week. Over the past 5 weeks, his weekly income totals have been different each week. His average commission per week over the next two weeks must be $315 so that his average weekly income is $500 over the 7-week period. What were his weekly income totals for the past 5 weeks?<|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 salesman's income consists of a commission and a base salary of $400 per week. Over the past 5 weeks, his weekly income totals have been different each week. His average commission per week over the next two weeks must be $315 so that his average weekly income is $500 over the 7-week period. What were his weekly income totals for the past 5 weeks?<|im_end|>
<|im_start|>assistant
<Z_369><Z_343><Z_307><Z_236><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A salesman's income consists of a commission and a base salary of $400 per week. Over the past 5 weeks, his weekly income totals have been different each week. His average commission per week over the next two weeks must be $315 so that his average weekly income is $500 over the 7-week period. What were his weekly income totals for the past 5 weeks?<|im_end|>
<|im_start|>assistant
<Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A salesman's income consists of a commission and a base salary of $400 per week. Over the past 5 weeks, his weekly income totals have been different each week. His average commission per week over the next two weeks must be $315 so that his average weekly income is $500 over the 7-week period. What were his weekly income totals for the past 5 weeks?<|im_end|>
<|im_start|>assistant
<Z_410><Z_220><Z_337><Z_228><ANSWER>

Question: A salesman's income consists of a commission and a base salary of $400 per week. Over the past 5 weeks, his weekly income totals have been different each week. His average commission per week over the next two weeks must be $315 so that his average weekly income is $500 over the 7-week period. What were his weekly income totals for the past 5 weeks?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.<|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 every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.<|im_end|>
<|im_start|>assistant
<Z_24><Z_211><Z_92><Z_321><Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.<|im_end|>
<|im_start|>assistant
<Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.<|im_end|>
<|im_start|>assistant
<Z_155><Z_497><Z_256><Z_407><Z_270><ANSWER>

Question: For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve for \(y\) in the equation \(4y - 8 = 2y + 18\). Then compute the value of \(3(y^2 + 6y + 12)\).<|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
Solve for \(y\) in the equation \(4y - 8 = 2y + 18\). Then compute the value of \(3(y^2 + 6y + 12)\).<|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
Solve for \(y\) in the equation \(4y - 8 = 2y + 18\). Then compute the value of \(3(y^2 + 6y + 12)\).<|im_end|>
<|im_start|>assistant
<Z_400><ANSWER>

Question: Solve for \(y\) in the equation \(4y - 8 = 2y + 18\). Then compute the value of \(3(y^2 + 6y + 12)\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, some of the water balloons pop on the ground. Max and Zach have 170 filled water balloons in total. How many water balloons popped on the ground?<|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
Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, some of the water balloons pop on the ground. Max and Zach have 170 filled water balloons in total. How many water balloons popped on the ground?<|im_end|>
<|im_start|>assistant
<Z_369><Z_245><Z_105><Z_243><Z_114><Z_406><Z_357><Z_279><Z_346><Z_147><Z_173><Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, some of the water balloons pop on the ground. Max and Zach have 170 filled water balloons in total. How many water balloons popped on the ground?<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, some of the water balloons pop on the ground. Max and Zach have 170 filled water balloons in total. How many water balloons popped on the ground?<|im_end|>
<|im_start|>assistant
<Z_27><Z_380><Z_158><Z_64><Z_274><Z_234><Z_35><Z_22><Z_325><Z_391><Z_55><Z_69><ANSWER>

Question: Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, some of the water balloons pop on the ground. Max and Zach have 170 filled water balloons in total. How many water balloons popped on the ground?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 5. What is the area of the triangle using Heron’s formula?<|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 triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 5. What is the area of the triangle using Heron’s formula?<|im_end|>
<|im_start|>assistant
<Z_477><Z_217><Z_494><Z_173><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 5. What is the area of the triangle using Heron’s formula?<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 5. What is the area of the triangle using Heron’s formula?<|im_end|>
<|im_start|>assistant
<Z_240><Z_320><Z_279><Z_190><ANSWER>

Question: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 5. What is the area of the triangle using Heron’s formula?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A straight line in the xy-plane has a slope of 2 and a certain y-intercept. On this line, the x-coordinate of the point whose y-coordinate is 540 is 269. What is the y-intercept of the line?<|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 straight line in the xy-plane has a slope of 2 and a certain y-intercept. On this line, the x-coordinate of the point whose y-coordinate is 540 is 269. What is the y-intercept of the line?<|im_end|>
<|im_start|>assistant
<Z_369><Z_295><Z_92><Z_219><Z_328><Z_322><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A straight line in the xy-plane has a slope of 2 and a certain y-intercept. On this line, the x-coordinate of the point whose y-coordinate is 540 is 269. What is the y-intercept of the line?<|im_end|>
<|im_start|>assistant
<Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A straight line in the xy-plane has a slope of 2 and a certain y-intercept. On this line, the x-coordinate of the point whose y-coordinate is 540 is 269. What is the y-intercept of the line?<|im_end|>
<|im_start|>assistant
<Z_223><Z_440><Z_444><Z_165><Z_511><Z_125><ANSWER>

Question: A straight line in the xy-plane has a slope of 2 and a certain y-intercept. On this line, the x-coordinate of the point whose y-coordinate is 540 is 269. What is the y-intercept of the line?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The smallest natural number $n$ that has exactly 10 positive divisors.<|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 smallest natural number $n$ that has exactly 10 positive divisors.<|im_end|>
<|im_start|>assistant
<Z_369><Z_488><Z_243><Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The smallest natural number $n$ that has exactly 10 positive divisors.<|im_end|>
<|im_start|>assistant
<Z_299><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The smallest natural number $n$ that has exactly 10 positive divisors.<|im_end|>
<|im_start|>assistant
<Z_299><Z_44><Z_238><Z_321><ANSWER>

Question: The smallest natural number $n$ that has exactly 10 positive divisors.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two distinct primes, each greater than 50, are multiplied. What is the least possible product of these two primes?<|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
Two distinct primes, each greater than 50, are multiplied. What is the least possible product of these two primes?<|im_end|>
<|im_start|>assistant
<Z_24><Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two distinct primes, each greater than 50, are multiplied. What is the least possible product of these two primes?<|im_end|>
<|im_start|>assistant
<Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two distinct primes, each greater than 50, are multiplied. What is the least possible product of these two primes?<|im_end|>
<|im_start|>assistant
<Z_277><Z_55><ANSWER>

Question: Two distinct primes, each greater than 50, are multiplied. What is the least possible product of these two primes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.<|im_end|>
<|im_start|>assistant
<Z_477><Z_330><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.<|im_end|>
<|im_start|>assistant
<Z_454><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.<|im_end|>
<|im_start|>assistant
<Z_89><Z_124><ANSWER>

Question: Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when $17^{63}$ is divided by 7?<|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 remainder when $17^{63}$ is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_24><Z_98><Z_453><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when $17^{63}$ is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_429><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the remainder when $17^{63}$ is divided by 7?<|im_end|>
<|im_start|>assistant
<Z_483><Z_28><Z_95><ANSWER>

Question: What is the remainder when $17^{63}$ is divided by 7?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .<|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  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .<|im_end|>
<|im_start|>assistant
<Z_24><Z_329><Z_186><Z_81><Z_415><Z_157><Z_256><Z_247><Z_378><Z_296><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .<|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
Let  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .<|im_end|>
<|im_start|>assistant
<Z_176><Z_101><Z_155><Z_179><Z_291><Z_164><Z_290><Z_397><Z_56><Z_203><ANSWER>

Question: Let  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting at $(0,0),$ an object moves in the coordinate plane using steps each of length one in either left, right, up, or down directions, with all directions equally probable. Let \( q \) be the probability that the object reaches $(3,3)$ in either six or eight steps. Given that \( q \) can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are co-prime positive integers, determine \( a+b \).<|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
Starting at $(0,0),$ an object moves in the coordinate plane using steps each of length one in either left, right, up, or down directions, with all directions equally probable. Let \( q \) be the probability that the object reaches $(3,3)$ in either six or eight steps. Given that \( q \) can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are co-prime positive integers, determine \( a+b \).<|im_end|>
<|im_start|>assistant
<Z_24><Z_266><Z_155><Z_441><Z_428><Z_363><Z_193><Z_68><Z_377><Z_245><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Starting at $(0,0),$ an object moves in the coordinate plane using steps each of length one in either left, right, up, or down directions, with all directions equally probable. Let \( q \) be the probability that the object reaches $(3,3)$ in either six or eight steps. Given that \( q \) can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are co-prime positive integers, determine \( a+b \).<|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
Starting at $(0,0),$ an object moves in the coordinate plane using steps each of length one in either left, right, up, or down directions, with all directions equally probable. Let \( q \) be the probability that the object reaches $(3,3)$ in either six or eight steps. Given that \( q \) can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are co-prime positive integers, determine \( a+b \).<|im_end|>
<|im_start|>assistant
<Z_449><Z_403><Z_169><Z_495><Z_45><Z_335><Z_174><Z_133><Z_319><Z_0><ANSWER>

Question: Starting at $(0,0),$ an object moves in the coordinate plane using steps each of length one in either left, right, up, or down directions, with all directions equally probable. Let \( q \) be the probability that the object reaches $(3,3)$ in either six or eight steps. Given that \( q \) can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are co-prime positive integers, determine \( a+b \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Cliff's impressive rock collection, there are half as many igneous rocks as sedimentary rocks. Of the igneous rocks, 1/3 are shiny and the rest are matte, while 1/5 of the sedimentary rocks are shiny. Cliff has a total of 270 rocks. How many shiny igneous rocks does Cliff have?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Cliff's impressive rock collection, there are half as many igneous rocks as sedimentary rocks. Of the igneous rocks, 1/3 are shiny and the rest are matte, while 1/5 of the sedimentary rocks are shiny. Cliff has a total of 270 rocks. How many shiny igneous rocks does Cliff have?<|im_end|>
<|im_start|>assistant
<Z_24><Z_187><Z_381><Z_296><Z_149><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Cliff's impressive rock collection, there are half as many igneous rocks as sedimentary rocks. Of the igneous rocks, 1/3 are shiny and the rest are matte, while 1/5 of the sedimentary rocks are shiny. Cliff has a total of 270 rocks. How many shiny igneous rocks does Cliff have?<|im_end|>
<|im_start|>assistant
<Z_293><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Cliff's impressive rock collection, there are half as many igneous rocks as sedimentary rocks. Of the igneous rocks, 1/3 are shiny and the rest are matte, while 1/5 of the sedimentary rocks are shiny. Cliff has a total of 270 rocks. How many shiny igneous rocks does Cliff have?<|im_end|>
<|im_start|>assistant
<Z_446><Z_203><Z_150><Z_167><Z_494><Z_176><ANSWER>

Question: In Cliff's impressive rock collection, there are half as many igneous rocks as sedimentary rocks. Of the igneous rocks, 1/3 are shiny and the rest are matte, while 1/5 of the sedimentary rocks are shiny. Cliff has a total of 270 rocks. How many shiny igneous rocks does Cliff have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an examination, 300 students appeared. Out of these students, 27% got first division, 54% got second division, and the remaining just passed. Assuming that no student failed, how many students just passed?<|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 an examination, 300 students appeared. Out of these students, 27% got first division, 54% got second division, and the remaining just passed. Assuming that no student failed, how many students just passed?<|im_end|>
<|im_start|>assistant
<Z_369><Z_117><Z_106><Z_268><Z_84><Z_468><Z_46><Z_493><Z_165><Z_274><Z_168><Z_422><Z_110><Z_487><Z_264><Z_494><Z_194><Z_313><Z_110><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an examination, 300 students appeared. Out of these students, 27% got first division, 54% got second division, and the remaining just passed. Assuming that no student failed, how many students just passed?<|im_end|>
<|im_start|>assistant
<Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In an examination, 300 students appeared. Out of these students, 27% got first division, 54% got second division, and the remaining just passed. Assuming that no student failed, how many students just passed?<|im_end|>
<|im_start|>assistant
<Z_214><Z_84><Z_434><Z_111><Z_154><Z_412><Z_38><Z_203><Z_400><Z_328><Z_65><Z_438><Z_39><Z_102><Z_7><Z_341><Z_457><Z_191><Z_229><ANSWER>

Question: In an examination, 300 students appeared. Out of these students, 27% got first division, 54% got second division, and the remaining just passed. Assuming that no student failed, how many students just passed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that requires using at least 3 weights to measure it?<|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 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that requires using at least 3 weights to measure it?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that requires using at least 3 weights to measure it?<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that requires using at least 3 weights to measure it?<|im_end|>
<|im_start|>assistant
<Z_456><ANSWER>

Question: Given 1 gram, 2 grams, 3 grams, and 5 grams weights, each available in one piece, you can measure weights from 1 gram to 11 grams. Some weights can be measured in more than one way; for example, 3 grams can be measured using the 3 grams weight or using both 1 gram and 2 grams weights. What is the minimum weight that requires using at least 3 weights to measure it?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).<|im_end|>
<|im_start|>assistant
<Z_369><Z_322><Z_154><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).<|im_end|>
<|im_start|>assistant
<Z_340><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).<|im_end|>
<|im_start|>assistant
<Z_378><Z_269><Z_97><ANSWER>

Question: Let \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the simple interest on a certain amount in at 4 % rate 5 years amounted to rs . 2000 less than the principal . what was the principal ?<|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 simple interest on a certain amount in at 4 % rate 5 years amounted to rs . 2000 less than the principal . what was the principal ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_100><Z_256><Z_310><Z_176><Z_345><Z_94><Z_227><Z_267><Z_264><Z_440><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the simple interest on a certain amount in at 4 % rate 5 years amounted to rs . 2000 less than the principal . what was the principal ?<|im_end|>
<|im_start|>assistant
<Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if the simple interest on a certain amount in at 4 % rate 5 years amounted to rs . 2000 less than the principal . what was the principal ?<|im_end|>
<|im_start|>assistant
<Z_100><Z_231><Z_424><Z_25><Z_488><Z_260><Z_450><Z_442><Z_2><Z_459><Z_308><ANSWER>

Question: if the simple interest on a certain amount in at 4 % rate 5 years amounted to rs . 2000 less than the principal . what was the principal ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine bakes a total of $30$ pies, which are either raspberry, peach, or plum. The ratio of raspberry to peach to plum pies is $2:5:3$. How many plum pies did Jasmine bake?<|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
Jasmine bakes a total of $30$ pies, which are either raspberry, peach, or plum. The ratio of raspberry to peach to plum pies is $2:5:3$. How many plum pies did Jasmine bake?<|im_end|>
<|im_start|>assistant
<Z_477><Z_308><Z_243><Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jasmine bakes a total of $30$ pies, which are either raspberry, peach, or plum. The ratio of raspberry to peach to plum pies is $2:5:3$. How many plum pies did Jasmine bake?<|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
Jasmine bakes a total of $30$ pies, which are either raspberry, peach, or plum. The ratio of raspberry to peach to plum pies is $2:5:3$. How many plum pies did Jasmine bake?<|im_end|>
<|im_start|>assistant
<Z_220><Z_140><Z_283><Z_214><ANSWER>

Question: Jasmine bakes a total of $30$ pies, which are either raspberry, peach, or plum. The ratio of raspberry to peach to plum pies is $2:5:3$. How many plum pies did Jasmine bake?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 boys and 2 girls lining up for a photo, with the requirement that the girls must be adjacent. There are ______ ways to arrange them.<|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
There are 3 boys and 2 girls lining up for a photo, with the requirement that the girls must be adjacent. There are ______ ways to arrange them.<|im_end|>
<|im_start|>assistant
<Z_24><Z_84><Z_443><Z_200><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 boys and 2 girls lining up for a photo, with the requirement that the girls must be adjacent. There are ______ ways to arrange them.<|im_end|>
<|im_start|>assistant
<Z_35><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 3 boys and 2 girls lining up for a photo, with the requirement that the girls must be adjacent. There are ______ ways to arrange them.<|im_end|>
<|im_start|>assistant
<Z_275><Z_364><Z_175><Z_178><ANSWER>

Question: There are 3 boys and 2 girls lining up for a photo, with the requirement that the girls must be adjacent. There are ______ ways to arrange them.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A squirrel runs up a cylindrical post, in a perfect spiral path making one circuit for each rise of some feet. The post is 16 feet tall and 3 feet in circumference. The squirrel travels 12 feet. What is the rise in feet for each circuit the squirrel makes?<|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 squirrel runs up a cylindrical post, in a perfect spiral path making one circuit for each rise of some feet. The post is 16 feet tall and 3 feet in circumference. The squirrel travels 12 feet. What is the rise in feet for each circuit the squirrel makes?<|im_end|>
<|im_start|>assistant
<Z_477><Z_496><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A squirrel runs up a cylindrical post, in a perfect spiral path making one circuit for each rise of some feet. The post is 16 feet tall and 3 feet in circumference. The squirrel travels 12 feet. What is the rise in feet for each circuit the squirrel makes?<|im_end|>
<|im_start|>assistant
<Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A squirrel runs up a cylindrical post, in a perfect spiral path making one circuit for each rise of some feet. The post is 16 feet tall and 3 feet in circumference. The squirrel travels 12 feet. What is the rise in feet for each circuit the squirrel makes?<|im_end|>
<|im_start|>assistant
<Z_215><Z_405><Z_59><ANSWER>

Question: A squirrel runs up a cylindrical post, in a perfect spiral path making one circuit for each rise of some feet. The post is 16 feet tall and 3 feet in circumference. The squirrel travels 12 feet. What is the rise in feet for each circuit the squirrel makes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Red cards are worth 3 credits, while blue cards are worth 5 credits. You need any combination of 20 cards to play a game. With a certain amount of credits to buy cards, you will have 8 red cards when you play. How many credits do you have to buy cards?<|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
Red cards are worth 3 credits, while blue cards are worth 5 credits. You need any combination of 20 cards to play a game. With a certain amount of credits to buy cards, you will have 8 red cards when you play. How many credits do you have to buy cards?<|im_end|>
<|im_start|>assistant
<Z_477><Z_443><Z_316><Z_84><Z_42><Z_219><Z_68><Z_86><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Red cards are worth 3 credits, while blue cards are worth 5 credits. You need any combination of 20 cards to play a game. With a certain amount of credits to buy cards, you will have 8 red cards when you play. How many credits do you have to buy cards?<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Red cards are worth 3 credits, while blue cards are worth 5 credits. You need any combination of 20 cards to play a game. With a certain amount of credits to buy cards, you will have 8 red cards when you play. How many credits do you have to buy cards?<|im_end|>
<|im_start|>assistant
<Z_58><Z_167><Z_355><Z_171><Z_225><Z_115><Z_414><Z_254><Z_191><ANSWER>

Question: Red cards are worth 3 credits, while blue cards are worth 5 credits. You need any combination of 20 cards to play a game. With a certain amount of credits to buy cards, you will have 8 red cards when you play. How many credits do you have to buy cards?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonnegative solutions are there to the equation $x^2 = -6x$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonnegative solutions are there to the equation $x^2 = -6x$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonnegative solutions are there to the equation $x^2 = -6x$?<|im_end|>
<|im_start|>assistant
<Z_177><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many nonnegative solutions are there to the equation $x^2 = -6x$?<|im_end|>
<|im_start|>assistant
<Z_306><Z_1><ANSWER>

Question: How many nonnegative solutions are there to the equation $x^2 = -6x$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different isosceles triangles have integer side lengths and a perimeter of 25?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different isosceles triangles have integer side lengths and a perimeter of 25?<|im_end|>
<|im_start|>assistant
<Z_369><Z_307><Z_80><Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different isosceles triangles have integer side lengths and a perimeter of 25?<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many different isosceles triangles have integer side lengths and a perimeter of 25?<|im_end|>
<|im_start|>assistant
<Z_30><Z_463><Z_463><Z_55><ANSWER>

Question: How many different isosceles triangles have integer side lengths and a perimeter of 25?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x^2 + y^2 = 16x + 8y + 20$, find the largest possible value of $2x + 3y$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x^2 + y^2 = 16x + 8y + 20$, find the largest possible value of $2x + 3y$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_101><Z_390><Z_384><Z_63><Z_366><Z_377><Z_470><Z_472><Z_247><Z_276><Z_157><Z_394><Z_465><Z_58><Z_58><Z_469><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x^2 + y^2 = 16x + 8y + 20$, find the largest possible value of $2x + 3y$.<|im_end|>
<|im_start|>assistant
<Z_212><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $x^2 + y^2 = 16x + 8y + 20$, find the largest possible value of $2x + 3y$.<|im_end|>
<|im_start|>assistant
<Z_434><Z_57><Z_371><Z_465><Z_128><Z_145><Z_408><Z_121><Z_366><Z_87><Z_468><Z_57><Z_440><Z_430><Z_288><Z_304><Z_275><ANSWER>

Question: Given that $x^2 + y^2 = 16x + 8y + 20$, find the largest possible value of $2x + 3y$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$, $b$, and $c$ be the roots of the equation $$
6x^3 - 803x + 1606 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$<|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
Let $a$, $b$, and $c$ be the roots of the equation $$
6x^3 - 803x + 1606 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$<|im_end|>
<|im_start|>assistant
<Z_477><Z_53><Z_185><Z_92><Z_220><Z_419><Z_232><Z_433><Z_210><Z_341><Z_63><Z_246><Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$, $b$, and $c$ be the roots of the equation $$
6x^3 - 803x + 1606 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$<|im_end|>
<|im_start|>assistant
<Z_373><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a$, $b$, and $c$ be the roots of the equation $$
6x^3 - 803x + 1606 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$<|im_end|>
<|im_start|>assistant
<Z_61><Z_78><Z_254><Z_321><Z_308><Z_66><Z_38><Z_96><Z_502><Z_22><Z_406><Z_98><Z_432><ANSWER>

Question:
Let $a$, $b$, and $c$ be the roots of the equation $$
6x^3 - 803x + 1606 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|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
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_369><Z_497><Z_171><Z_395><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Convert the binary number $101101_{(2)}$ to decimal.<|im_end|>
<|im_start|>assistant
<Z_489><Z_459><Z_422><Z_22><ANSWER>

Question: Convert the binary number $101101_{(2)}$ to decimal.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the expansion of $(1-2x)^5(1+ax)^4$, the coefficient of $x$ is 2. Find the value of the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the expansion of $(1-2x)^5(1+ax)^4$, the coefficient of $x$ is 2. Find the value of the real number $a$.<|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
Given the expansion of $(1-2x)^5(1+ax)^4$, the coefficient of $x$ is 2. Find the value of the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_217><ANSWER>

Question: Given the expansion of $(1-2x)^5(1+ax)^4$, the coefficient of $x$ is 2. Find the value of the real number $a$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and some small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. He eats 48 pieces that day. How many small pizzas did he buy?<|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
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and some small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. He eats 48 pieces that day. How many small pizzas did he buy?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and some small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. He eats 48 pieces that day. How many small pizzas did he buy?<|im_end|>
<|im_start|>assistant
<Z_342><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and some small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. He eats 48 pieces that day. How many small pizzas did he buy?<|im_end|>
<|im_start|>assistant
<Z_436><ANSWER>

Question: Albert is wondering how much pizza he can eat in one day. He buys 2 large pizzas and some small pizzas. A large pizza has 16 slices and a small pizza has 8 slices. He eats 48 pieces that day. How many small pizzas did he buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A typical tournament of tennis has 4 rounds. There are 8 games in the first round, 4 in the second round, 2 in the third round, and 1 during the finals. If each game requires new tennis balls, and on average each game uses a certain number of cans of tennis balls, a total of 225 tennis balls are used at the end of the tournament, with each can having 3 tennis balls. How many cans of tennis balls are used per game on average?<|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 typical tournament of tennis has 4 rounds. There are 8 games in the first round, 4 in the second round, 2 in the third round, and 1 during the finals. If each game requires new tennis balls, and on average each game uses a certain number of cans of tennis balls, a total of 225 tennis balls are used at the end of the tournament, with each can having 3 tennis balls. How many cans of tennis balls are used per game on average?<|im_end|>
<|im_start|>assistant
<Z_477><Z_446><Z_127><Z_430><Z_449><Z_253><Z_501><Z_106><Z_378><Z_169><Z_387><Z_106><Z_393><Z_451><Z_60><Z_146><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A typical tournament of tennis has 4 rounds. There are 8 games in the first round, 4 in the second round, 2 in the third round, and 1 during the finals. If each game requires new tennis balls, and on average each game uses a certain number of cans of tennis balls, a total of 225 tennis balls are used at the end of the tournament, with each can having 3 tennis balls. How many cans of tennis balls are used per game on average?<|im_end|>
<|im_start|>assistant
<Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A typical tournament of tennis has 4 rounds. There are 8 games in the first round, 4 in the second round, 2 in the third round, and 1 during the finals. If each game requires new tennis balls, and on average each game uses a certain number of cans of tennis balls, a total of 225 tennis balls are used at the end of the tournament, with each can having 3 tennis balls. How many cans of tennis balls are used per game on average?<|im_end|>
<|im_start|>assistant
<Z_115><Z_98><Z_341><Z_286><Z_17><Z_275><Z_332><Z_242><Z_331><Z_472><Z_151><Z_235><Z_64><Z_429><Z_38><Z_371><ANSWER>

Question: A typical tournament of tennis has 4 rounds. There are 8 games in the first round, 4 in the second round, 2 in the third round, and 1 during the finals. If each game requires new tennis balls, and on average each game uses a certain number of cans of tennis balls, a total of 225 tennis balls are used at the end of the tournament, with each can having 3 tennis balls. How many cans of tennis balls are used per game on average?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of two numbers is 84942 and their HCF is 33, find their LCM.<|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 the product of two numbers is 84942 and their HCF is 33, find their LCM.<|im_end|>
<|im_start|>assistant
<Z_24><Z_41><Z_413><Z_276><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of two numbers is 84942 and their HCF is 33, find their LCM.<|im_end|>
<|im_start|>assistant
<Z_335><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the product of two numbers is 84942 and their HCF is 33, find their LCM.<|im_end|>
<|im_start|>assistant
<Z_487><Z_259><Z_266><Z_65><ANSWER>

Question: If the product of two numbers is 84942 and their HCF is 33, find their LCM.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]
for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$<|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 equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]
for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$<|im_end|>
<|im_start|>assistant
<Z_24><Z_35><Z_246><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]
for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]
for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$<|im_end|>
<|im_start|>assistant
<Z_122><Z_212><Z_452><ANSWER>

Question:
The equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]
for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paul eats a lot when he studies. He loves sandwiches and eats them at the same rate every three days. He eats 2 sandwiches the first day, then doubles that number of sandwiches for the second day. On the third day, he doubles the number of sandwiches he ate on the second day. How many sandwiches would Paul eat if he studied 6 days in a row?<|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
Paul eats a lot when he studies. He loves sandwiches and eats them at the same rate every three days. He eats 2 sandwiches the first day, then doubles that number of sandwiches for the second day. On the third day, he doubles the number of sandwiches he ate on the second day. How many sandwiches would Paul eat if he studied 6 days in a row?<|im_end|>
<|im_start|>assistant
<Z_477><Z_353><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paul eats a lot when he studies. He loves sandwiches and eats them at the same rate every three days. He eats 2 sandwiches the first day, then doubles that number of sandwiches for the second day. On the third day, he doubles the number of sandwiches he ate on the second day. How many sandwiches would Paul eat if he studied 6 days in a row?<|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
Paul eats a lot when he studies. He loves sandwiches and eats them at the same rate every three days. He eats 2 sandwiches the first day, then doubles that number of sandwiches for the second day. On the third day, he doubles the number of sandwiches he ate on the second day. How many sandwiches would Paul eat if he studied 6 days in a row?<|im_end|>
<|im_start|>assistant
<Z_90><Z_59><ANSWER>

Question: Paul eats a lot when he studies. He loves sandwiches and eats them at the same rate every three days. He eats 2 sandwiches the first day, then doubles that number of sandwiches for the second day. On the third day, he doubles the number of sandwiches he ate on the second day. How many sandwiches would Paul eat if he studied 6 days in a row?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?<|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 student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?<|im_end|>
<|im_start|>assistant
<Z_369><Z_156><Z_366><Z_71><Z_173><Z_453><Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?<|im_end|>
<|im_start|>assistant
<Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?<|im_end|>
<|im_start|>assistant
<Z_347><Z_159><Z_278><Z_327><Z_141><Z_41><Z_190><ANSWER>

Question: A student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the bracelet shown, the beads are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming plays a bead counting game with the rule that starting from bead number 1, he counts each bead in a clockwise manner with natural numbers, but skips any number that contains the digit 7 or is a multiple of 7. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. On which bead will he land when he reaches the count of 100?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the bracelet shown, the beads are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming plays a bead counting game with the rule that starting from bead number 1, he counts each bead in a clockwise manner with natural numbers, but skips any number that contains the digit 7 or is a multiple of 7. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. On which bead will he land when he reaches the count of 100?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the bracelet shown, the beads are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming plays a bead counting game with the rule that starting from bead number 1, he counts each bead in a clockwise manner with natural numbers, but skips any number that contains the digit 7 or is a multiple of 7. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. On which bead will he land when he reaches the count of 100?<|im_end|>
<|im_start|>assistant
<Z_458><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the bracelet shown, the beads are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming plays a bead counting game with the rule that starting from bead number 1, he counts each bead in a clockwise manner with natural numbers, but skips any number that contains the digit 7 or is a multiple of 7. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. On which bead will he land when he reaches the count of 100?<|im_end|>
<|im_start|>assistant
<Z_378><ANSWER>

Question: In the bracelet shown, the beads are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming plays a bead counting game with the rule that starting from bead number 1, he counts each bead in a clockwise manner with natural numbers, but skips any number that contains the digit 7 or is a multiple of 7. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. On which bead will he land when he reaches the count of 100?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer $n$ that satisfies these conditions:
(i) $n^2 = (m + 1)^3 - m^3$ for some integer $m$.
(ii) $2n + 103$ is a perfect square.<|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 largest integer $n$ that satisfies these conditions:
(i) $n^2 = (m + 1)^3 - m^3$ for some integer $m$.
(ii) $2n + 103$ is a perfect square.<|im_end|>
<|im_start|>assistant
<Z_24><Z_68><Z_494><Z_29><Z_227><Z_295><Z_252><Z_102><Z_144><Z_115><Z_296><Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer $n$ that satisfies these conditions:
(i) $n^2 = (m + 1)^3 - m^3$ for some integer $m$.
(ii) $2n + 103$ is a perfect square.<|im_end|>
<|im_start|>assistant
<Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the largest integer $n$ that satisfies these conditions:
(i) $n^2 = (m + 1)^3 - m^3$ for some integer $m$.
(ii) $2n + 103$ is a perfect square.<|im_end|>
<|im_start|>assistant
<Z_178><Z_173><Z_96><Z_182><Z_46><Z_476><Z_488><Z_169><Z_68><Z_457><Z_90><Z_16><ANSWER>

Question:
Find the largest integer $n$ that satisfies these conditions:
(i) $n^2 = (m + 1)^3 - m^3$ for some integer $m$.
(ii) $2n + 103$ is a perfect square.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
70% of a certain number is greater than 60% of 80 by 22. What is that 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
70% of a certain number is greater than 60% of 80 by 22. What is that number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_181><Z_463><Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
70% of a certain number is greater than 60% of 80 by 22. What is that number?<|im_end|>
<|im_start|>assistant
<Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
70% of a certain number is greater than 60% of 80 by 22. What is that number?<|im_end|>
<|im_start|>assistant
<Z_426><Z_488><Z_9><Z_16><ANSWER>

Question: 70% of a certain number is greater than 60% of 80 by 22. What is that number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the positive integer $t$ satisfies $10^{t-1} < 2^{64} < 10^t$, then $t=$ _____

(Given that $\lg 2 \approx 0.3010$).<|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 the positive integer $t$ satisfies $10^{t-1} < 2^{64} < 10^t$, then $t=$ _____

(Given that $\lg 2 \approx 0.3010$).<|im_end|>
<|im_start|>assistant
<Z_369><Z_312><Z_227><Z_465><Z_432><Z_423><Z_53><Z_264><Z_335><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the positive integer $t$ satisfies $10^{t-1} < 2^{64} < 10^t$, then $t=$ _____

(Given that $\lg 2 \approx 0.3010$).<|im_end|>
<|im_start|>assistant
<Z_156><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the positive integer $t$ satisfies $10^{t-1} < 2^{64} < 10^t$, then $t=$ _____

(Given that $\lg 2 \approx 0.3010$).<|im_end|>
<|im_start|>assistant
<Z_454><Z_153><Z_481><Z_290><Z_485><Z_352><Z_413><Z_233><Z_222><ANSWER>

Question:
If the positive integer $t$ satisfies $10^{t-1} < 2^{64} < 10^t$, then $t=$ _____

(Given that $\lg 2 \approx 0.3010$).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$<|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
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$<|im_end|>
<|im_start|>assistant
<Z_369><Z_239><Z_231><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$<|im_end|>
<|im_start|>assistant
<Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$<|im_end|>
<|im_start|>assistant
<Z_29><Z_1><Z_379><ANSWER>

Question: For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arun thinks his weight is greater than 61 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than a certain number. His mother's view is that his weight cannot be greater than 64 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 63 kg. What is the upper limit of Arun's brother's opinion about his weight?<|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
Arun thinks his weight is greater than 61 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than a certain number. His mother's view is that his weight cannot be greater than 64 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 63 kg. What is the upper limit of Arun's brother's opinion about his weight?<|im_end|>
<|im_start|>assistant
<Z_24><Z_101><Z_458><Z_247><Z_101><Z_492><Z_437><Z_350><Z_470><Z_390><Z_403><Z_439><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arun thinks his weight is greater than 61 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than a certain number. His mother's view is that his weight cannot be greater than 64 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 63 kg. What is the upper limit of Arun's brother's opinion about his weight?<|im_end|>
<|im_start|>assistant
<Z_202><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Arun thinks his weight is greater than 61 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than a certain number. His mother's view is that his weight cannot be greater than 64 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 63 kg. What is the upper limit of Arun's brother's opinion about his weight?<|im_end|>
<|im_start|>assistant
<Z_0><Z_140><Z_312><Z_91><Z_199><Z_89><Z_191><Z_317><Z_330><Z_18><Z_233><Z_161><ANSWER>

Question: Arun thinks his weight is greater than 61 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than a certain number. His mother's view is that his weight cannot be greater than 64 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 63 kg. What is the upper limit of Arun's brother's opinion about his weight?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_24><Z_483><Z_188><Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|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
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_163><Z_71><Z_498><Z_382><ANSWER>

Question: In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?